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LabVIEW System Identification Toolkit User Manual

1. Feedback in a Closed Loop System Feedback is common in control systems With feedback the system output corresponds to a reference input Feedback also reduces the effect of input disturbances One example of a closed loop system is a system that regulates room temperature as shown in Figure 6 1 In this example the reference input is the temperature 7 at which you want the room to stay The thermostat senses the actual temperature Ty tqi of the room Based on the difference between T 4 and Ter the thermostat activates the heater or the air conditioner The thermostat returns T 4 i aS the feedback to compare again with 7 Then the thermostat uses the difference between Tactuat Nd T to regulate the temperature at the next moment National Instruments Corporation 6 1 System Identification Toolkit User Manual Chapter 6 System Identification Toolkit User Manual 6 2 Model Estimation Methods in Closed Loop Systems Set Temperature Actual Temperature Heater Air Conditioner Controller gt Tactual Figure 6 1 A Closed Loop System Regulating Room Temperature You must verify if feedback exists before choosing a model estimation method because not all open loop model estimation methods work correctly with closed loop data Note You need to have knowledge about whether the data you collect is from an open loop system or a closed loop system according to the real
2. By examining the frequency response data you see that filtering is necessary to remove this resonance peak at 42 Hz The following section describes how to use the System Identification Toolkit to apply a filter to the flexible arm data System Identification Toolkit User Manual 9 4 ni com Chapter 9 System Identification Case Study Applying a Filter to the Raw Data To eliminate the resonance peak at 42 Hz you can apply a filter to the raw data By first applying a lowpass filter with a cutoff frequency of 25 Hz you eliminate the high frequency noise from the raw data set Figure 9 5 shows how to use SI Lowpass Filter to apply a lowpass filter to the raw data set Figure 9 5 Non Parametric FRF with Prefiltering VI You can see the effects of the lowpass filter by comparing the frequency response of the filtered data set in Figure 9 6 to the frequency response of the non filtered data set in Figure 9 4 By using a lowpass filter you can see that the resonance at approximately 42 Hz is no longer part of the data set you will use to estimate the model magnitude response Amplitude 0 7 0 795775 phase response 400 Phase 200 Si 0 795775 Frequency Hz Figure 9 6 Frequency Response of the Filtered Data Set National Instruments Corporation 9 5 System Identification Toolkit User Manual Chapter 9 System Identification Case Study Downsampling the Raw Data
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4. Developing Models with the System Identification Toolkit The System Identification Toolkit provides a library of VIs and an assistant You can apply these VIs or the assistant to develop and validate the plant model of a system that uses a large set of raw data System Identification Vis The System Identification Toolkit provides VIs that you can use to preprocess raw data from a dynamic system and develop a model that reflects the behavior of that system The Data Preprocessing VIs enable you to analyze the response of a plant or dynamic system to a certain stimulus After analyzing the data you can use the Parametric Model Estimation Nonparametric Model Estimation Partially Known Model Estimation or Recursive Model Estimation VIs to estimate a model for the plant or dynamic system Finally you can use the Model Validation or Model Analysis VIs to determine whether the model accurately describes the dynamics of the identified system The System Identification VIs enable you to customize a LabVIEW block diagram to achieve specific goals You also can use other LabVIEW VIs and functions to enhance the functionality of the application Creating a LabVIEW application using the System Identification VIs requires basic knowledge about programming in LabVIEW Refer to the LabVIEW Fundamentals and Getting Started with LabVIEW manuals for more information about the LabVIEW programming environment System Identification Assistant If you d
5. gt P k 1 P k K k 97k P 10 Stop if the error is small enough else set k k 1 and repeat steps 4 10 Kalman Filter The Kalman filter is a linear optimum filter that minimizes the mean of the squared error recursively The convergence rate of the Kalman filter is relatively fast but the implementation is more complex than that of LMS based algorithms Recall that the equation J k E e k defines the cost function The following procedure lists the steps of the Kalman filter algorithm 1 Initialize the parametric vector k using a small positive number National Instruments Corporation 7 7 System Identification Toolkit User Manual Chapter 7 Recursive Model Estimation Methods 2 10 gt Initialize the data vector k gt 0 0 0 0 Initialize the k x k matrix P 0 P 0 ooon oom so D co a 0 gt gt For k 1 update the data vector k based on k 1 and the current input data u k and output data y k Compute the predicted response y k by solving the following equation A gt gt y k P7k W k Compute the error e k by solving the following equation e k y k y k Update the Kalman gain vector k defined by the following equation gt P k 9 k Kb p gt Ou 97 K PC O Qy is the measurement noise and P k is a k x k matrix whose initial value is defined by P O in step 3 Update the parametric vector
6. Sampling theory in conjunction with the Nyquist criterion enables you to reduce the sampling rate from 500 Hz to 50 Hz Applying a filter and downsampling the data set reduces the number of samples in and the computational complexity of the data set The goal is to use as few samples as possible to evaluate the behavior of the system Sampling theory enables you to downsample or decimate the data set Downsampling reduces the sampling rate 500 Hz by a factor of 10 Thus downsampling enables you to acquire the data at a sampling rate of 50 Hz The Nyquist criterion states that you need to sample the signal at a minimum of twice the highest frequency in the system Recall that the input data is equally spaced over the frequency band 0 122 24 4 Hz Therefore according to the Nyquist criterion you need to sample at a minimum of 50 Hz to avoid any antialiasing The benefit of sampling at 50 Hz is that you still acquire all the data in the frequency band yet you eliminate the resonance peak at 42 Hz Therefore in Figure 9 7 the SI Lowpass Filter VI sets the cutoff frequency to 25 In addition to applying a lowpass filter to the data you must downsample the reduced data set The SI Down Sample VI in Figure 9 7 uses a decimation factor of 10 Figure 9 7 Non Parametric FRF with Prefiltering and Down Sampling VI The SI Lowpass Filter VI applies a lowpass filter before downsampling the data set to avoid aliasing at the 42
7. and reasons for choosing one representation over another Parametric Model Representations The LabVIEW System Identification Toolkit provides four categories of parametric models general linear polynomial transfer function zero pole gain and state space General linear polynomial transfer function and zero pole gain models are all polynomial models General Linear Polynomial Model General linear polynomial models apply only for discrete systems Generally you can describe a discrete system using the following equation which is known as the general linear polynomial model p k 2 Gz O u k H z BeCK where u k and y k are the input and output of the system respectively e k is the disturbance of the system which usually is zero mean white noise G z7 8 is the transfer function of the deterministic part of the system H z 0 is the transfer function of the stochastic part of the system National Instruments Corporation 4 1 System Identification Toolkit User Manual Chapter 4 Parametric Model Estimation Methods The deterministic transfer function specifies the relationship between the output and the input signal The stochastic transfer function specifies how the random disturbance affects the output signal Often the deterministic and stochastic parts of a system are referred to as system dynamics and stochastic dynamics respectively The term z is the backward shift operator which is defined
8. k k 1 GOFFE System Identification Toolkit User Manual 6 6 ni com Chapter 6 Model Estimation Methods in Closed Loop Systems If you define Gq as the closed loop transfer function between the reference signal and the response signal and let Gy satisfy the following equation Gee Go 2 1 GOP 2 you can estimate G with r k as the input and y k as the output using an open loop method because r k and e k are uncorrelated You then can calculate Gg after you calculate G as the following equation shows G G 6 3 1 G F For feedforward path closed loop systems as shown in Figure 6 4 you use the following two equations to describe the systems 6 4 y k Go z u k e k r k yW FY 2 6 5 u k By combining Equations 6 4 and 6 5 you can represent the feedforward path closed loop relationship with the following equation F 2 G 2 1 A a ee GF se Gar If you define Gq as the feedforward path closed loop transfer function and let Gy satisfy the following equation _ F 2Z Gy z 1 1 Go z F 2 you can estimate G with r k as the input and y k as the output using an open loop method because r k and e k are uncorrelated You then can calculate Gg after you calculate G as the following equation shows G Gy gt 6 6 1 G F With indirect identification you calculate G by performing polynomial operations on G and F Because of the limitations of polynomial operations
9. 5 u k n h n e k n 0 You now must design the prewhitening filter so that u k is white noise The SI Estimate Impulse Response VI uses an AR model for this purpose Refer to Chapter 4 Parametric Model Estimation Methods for more information about AR model estimation National Instruments Corporation 3 5 System Identification Toolkit User Manual Chapter 3 Nonparametric Model Estimation Methods Accuracy of the Impulse Response The accuracy of the impulse response estimation using the correlation analysis method depends on the performance of the prewhitening filter specifically whether the filter produces a white noise result u k for u k The performance of the filter depends on the signal and the AR order of the filter The rule of thumb for selecting the AR order is trial and error If u k is not white enough the result from the correlation method is not reliable You can increase the AR order to improve the accuracy of the impulse response The SI Estimate Impulse Response VI provides the outputs whiteness test and rejected to indicate whether you have properly set the AR order and consequently whether the impulse response estimation is reliable The following example shows how the whiteness property of the input signal affects the correlation analysis method and how to use the outputs whiteness test and rejected to justify the impulse response estimation Figure 3 3 shows the front panel of a VI that simulates a s
10. 7 k k 1 K e k Rk Update the P k matrix P k 1 P k R k 6 k P k Qp Qp is the correlation matrix of the process noise Stop if the error is small enough else set k k 1 and repeat steps 4 10 System Identification Toolkit User Manual 7 8 ni com Analyzing Validating and Converting Models After estimating a model for the plant of a system you can observe model characteristics by analyzing the model You also can verify that the model simulates the real world plant by validating the model The LabVIEW System Identification Toolkit provides tools that enable you to analyze and validate models According to linear system theory you can represent a linear system with different models Each model representation has benefits and drawbacks for characterizing a dynamic system Certain model representations are more suitable for certain analysis techniques With the System Identification Toolkit you have the flexibility to convert models from one representation to another to identify the best fit model for the system This chapter describes model analysis and validation methods This chapter also describes how to convert models with the Model Conversion VIs Analyzing Models Model analysis allows you to observe some characteristics like frequency response stability and order of the model you obtain The System Identification Toolkit enables you to investigate model estimation results and present t
11. Chapter 3 Nonparametric Model Estimation Methods Figure 3 6 shows the block diagram of this VI Simulate the system Generate the input signal 0 2Kk F 08u k A 0 3K Estimate the impulse response estimated impulse response hin IIR Filter vi Gaussian White Noise vi z Le Coase Reverse Coefficients Ho Figure 3 6 Block Diagram of Time Delay Example VI In Figure 3 5 the two initial values of the estimated impulse response are smaller than the confidence level You can have 99 0 confidence that values less than the confidence level are insignificant and you can consider those values to be equal to 0 Therefore you can conclude that the time delay of the system is 2 because the beginning of the first two values of the impulse response are zero Another common application of the impulse response is to detect feedback in systems using the least squares method If feedback exists in a system the impulse response of the system becomes significantly large at negative lags and the correlation between the input signal and disturbance e k is nonzero The correlation analysis method assumes the input signal and the disturbance e k are independent from each other Thus this method cannot estimate accurately the impulse response of the system that contains feedback Only the least squares method can provide reliable results Refer to Chapter 6 Model Estimation Methods in Closed Loop Systems fo
12. Hz resonance Together the lowpass filter and downsampling remove the high frequency disturbance and make the process faster and more efficient Notice that the window length parameter of the SI Estimate Frequency Response VI in Figure 9 7 is around 400 instead of 4096 as shown in Figure 9 3 You can reduce the window length by a factor of 10 because the number of samples in the reduced data set is one tenth of the number of samples in the raw data set System Identification Toolkit User Manual 9 6 ni com Chapter 9 System Identification Case Study Figure 9 8 shows the frequency response after applying a filter to and downsampling the raw data set magnitude response Frequency Hz Figure 9 8 Frequency Response of the Filtered Data Set after Downsampling Filtering and downsampling are beneficial because they eliminate the nonrealistic parts of the frequency response and reduce the amount of work required in the model estimation process Refer to Chapter 2 Acquiring and Preprocessing Data for more information about filtering and downsampling data Estimating the Model One of the biggest challenges in model estimation is selecting the correct model and the order of the model The System Identification Toolkit supports three different criteria to aid in the estimation of the order of a model e FPE Final Prediction Error Criterion e AlIC Akaike s Information Criterion e MDL Minimum Description Le
13. INSTRUMENTS TESTING PLATFORMS AND BECAUSE A USER OR APPLICATION DESIGNER MAY USE NATIONAL INSTRUMENTS PRODUCTS IN COMBINATION WITH OTHER PRODUCTS IN A MANNER NOT EVALUATED OR CONTEMPLATED BY NATIONAL INSTRUMENTS THE USER OR APPLICATION DESIGNER IS ULTIMATELY RESPONSIBLE FOR VERIFYING AND VALIDATING THE SUITABILITY OF NATIONAL INSTRUMENTS PRODUCTS WHENEVER NATIONAL INSTRUMENTS PRODUCTS ARE INCORPORATED IN A SYSTEM OR APPLICATION INCLUDING WITHOUT LIMITATION THE APPROPRIATE DESIGN PROCESS AND SAFETY LEVEL OF SUCH SYSTEM OR APPLICATION Contents About This Manual CONVENTIONS 3555 sscscetiestend exci theaves NNE EEA NE cubes Lavedee be E E wees E EE acs 1X Related DocCUuMentatiOn ccccecccecesccssessssssessscscecsscecescecececessesesesescscncnesseessseseseesess x Chapter 1 Introduction to System Identification Model Based Control Design Process 0 ec eseeseeseseeseesseeseceseeaececeeaeceeesseesaeeseesseeaees 1 2 Analyzing Data and Creating a Dynamic System Model eee 1 2 Designing a Controller s 22 sissciceseh sheds pecescesass hacen EEA AAE 1 4 Simulating the Dynamic System cesses ceceeecneceseseeeesesaeesseeseeseeeas 1 4 Deploying the Controller ee iesise h arana Aaea aii 1 4 Model Types and Model Representations 0 0 0 eee eecceeecseceseceeeeseeseeeseeseenseeaseeeeaes 1 5 Model Ty pes isi sists sats sesthitheeteseeet ins Soadeed ER E E EE Sees belevs coven tazens 1 5 Linear versus Nonlinear Models 0
14. Minimum Description Length Criterion MDL are criteria you can use to estimate the model order The SI Estimate Orders of System Model VI implements the AIC FPE and MDL methods to search for the optimal model order in the range of interest You also can plot the prediction error as a function of the model dimension and then visually find the minimum in the curve or apply an F test to obtain an appropriate estimation of the model order National Instruments Corporation 4 13 System Identification Toolkit User Manual Chapter 4 Parametric Model Estimation Methods Akaike s Information Criterion The Akaike s Information Criterion AIC is a weighted estimation error based on the unexplained variation of a given time series with a penalty term when exceeding the optimal number of parameters to represent the system For the AIC an optimal model is the one that minimizes the following equation AIC A 2p N Nis the number of data points V is an index related to the prediction error or the residual sum of squares and p defines the number of parameters in the model Final Prediction Error Criterion The Final Prediction Error Criterion FPE estimates the model fitting error when you use the model to predict new outputs For the FPE an optimal model is the one that minimizes the following equation FPE v 1 5a N p You want to choose a model that minimizes the FPE which represents a balance between the number of param
15. T 2 3 1 1 1 1 1 ma 400 600 800 1000 1200 1449 Time HEA N Figure 5 2 Estimating a Local Optimum and a Global Optimum from Different Initial Guesses As Figure 5 2 shows if you set C to an initial guess of 0 1 you obtain an optimized value of 0 02 You can see the Estimated response global plot and the Measured response plot match in the Response graph This response from the estimated model is close to the real world model response However if you set the initial guess of C to 1 5 you get an optimized value of 1 41 The Estimated response local plot does not match the Measured response plot in the Response graph Thus with this initial guess the estimated model response does not represent the real world model response accurately Partially Known Model Estimation Case Study This section contains an example that uses the prior knowledge you have about a system to define and estimate state space models You use the same procedure when estimating continuous transfer function models However you apply different methods to define continuous transfer function models Refer to the Defining and Estimating Partially Known Models section of this chapter for more information about defining transfer function models System Identification Toolkit User Manual 5 6 ni com Chapter 5 Partially Known Model Estimation Methods Figure 5 3 shows an RLC circuit where u is the input voltage y is the output voltage i is the current
16. aliasing filters A common rule of thumb is to sample signals at 10 times the bandwidth of the system or the bandwidth of interest for the model If uncertainty exists in the system bandwidth and a fast data acquisition environment is available you can sample as fast as possible then use a digital filter and decimation to reduce the sampling rate to the desired value Decimation is a form of downsampling the data set Refer to the Filtering and Downsampling section for more information about filtering and downsampling a data set Applying an Anti Aliasing Filter According to the Nyquist sampling theorem the sampling rate must be greater than twice the maximum frequency component of the signal of interest In other words the maximum frequency of the input signal must be greater than half the sampling rate This criterion in practice is often difficult to ensure Even if you are sure that the measured signal has an upper limit on its frequency external factors such as signals from the powerline interference or radio stations can contain frequencies higher than the Nyquist frequency These frequencies might then alias into the frequency range of interest and give you inaccurate results To ensure that you limit the frequency content of the input signal add a lowpass filter before the sampler and the analog to discrete converter A lowpass filter passes low frequencies and attenuates high frequencies This filter is an anti aliasing filt
17. and Preprocessing Data Figure 2 3 shows an example of a pseudo random binary sequence Psuedo Random Binary Sequence Figure 2 3 Pseudo random Binary Sequence Chirp Waveform The chirp waveform also known as a swept sine wave is a sinusoid waveform with a frequency that varies continuously over a certain range of values lt lt for a specific period of time 0 lt t lt T The resulting signal has a crest factor Cp of J2 You can modify the signal to excite specific signal spectra In comparison to other signals like the white noise stimulus a chirp waveform is easier to generate and control Figure 2 4 shows an example of a chirp waveform i 1 1 i i i 0 2500 5000 7500 10000 12500 15000 Sample Figure 2 4 Chirp or Swept Sine Wave System Identification Toolkit User Manual 2 6 ni com Chapter 2 Acquiring and Preprocessing Data Selecting a Sampling Rate The time constants of a system influence the selection of a sampling rate Sampling at rates substantially greater than the system bandwidth leads to data redundancy numerical issues and modeling of high frequency artifacts likely due to noise Sampling at rates slower than system dynamics leads to difficulties determining an accurate system model and problems introduced by aliasing You can use an anti aliasing filter to counter the effects of aliasing Refer to the Applying an Anti Aliasing Filter section for more information about anti
18. and operational stability of the engine Biologists and physiologists use system identification techniques in areas such as eye pupil response and heart rate control Meteorologists and economists build mathematical models based on historical data for use in forecasting This manual focuses on how to use system identification in the model based control design process which involves identifying a model of a plant analyzing and synthesizing a controller for the plant simulating the plant and controller and deploying the controller A plant is the real world physical system that you want to control System identification is the initial step identifying a model of a plant in the model based control design process System identification is an iterative process You first acquire raw data from a real world system then format and process the data as necessary and finally select a mathematical algorithm that you can use to identify a mathematical model of the system You then can use the resulting mathematical model to analyze the dynamic characteristics and simulate the time response of the system You also can use the mathematical model to design a model based controller The LabVIEW System Identification Toolkit assists you in identifying large multivariable models of high order systems from large amounts of data The System Identification Toolkit provides two tools an assistant and a library of VIs for identifying these linear systems Bo
19. any other analyses you might conclude that this model is an accurate representation of the system However the pole zero analysis in the Minimum Description Length Criterion section showed poles outside of the unit circle So you already determined that this model is unstable Thus despite acceptable autocorrelation and cross correlation values concluding that this model is accurate is incorrect Thus if you only performed a residual analysis you might not discover that this model is actually unstable When validating a model perform multiple analyses to ensure the accuracy of the model National Instruments Corporation 9 17 System Identification Toolkit User Manual Chapter 9 System Identification Case Study Estimating a State Space Model For a state space model order estimation is equivalent to estimating the number of significant singular values which correspond to the number of states in the model After identifying a state space model that represents the system you can use the same validation and verification technique used in the Simulation and Prediction section and the Residual Analysis section Refer to the State Space Model section of Chapter 4 Parametric Model Estimation Methods for more information about estimating state space models The examples in this section use the same flexible robotic arm data and the same preprocessing techniques Finding the Singular Values The block diagram in Figure 9 20 shows
20. as measurement noise on the output The identification method of the Box Jenkins model is the prediction error method which is the same as that of the ARMAX model You can use the SI Estimate BJ Model VI to estimate Box Jenkins models AR Model When C z D z and F z equal 1 and B z equals 0 the general linear polynomial model reduces to the AR model The following equation describes an AR model A Z y k e k Figure 4 6 depicts the signal flow of an AR model e k 1 y k A z Figure 4 6 Signal Flow of an AR Model The AR model does not include the dynamics between the input and output Therefore the AR model is more suitable for representing signals rather than a system because a system generally has an input and an output Time series analysis methods such as power spectrum envelope estimation prewhitening and linear prediction coding commonly use the AR model Refer to the Time Series Analysis Tools User Manual at ni com manuals for more information about time series analysis methods If you consider A z to be a filter A z y k is the filtering of A z on the signal y k The result of the filtering is white noise e k as shown in the AR model equation Hence the filter A z also is known as the prewhitening filter From the frequency domain standpoint the prewhitening filter A z suppresses the spectrum at frequencies where the magnitude of the spectrum is large Suppressing the high m
21. by the following equations z x k x k 1 z x k x k 2 z x k x k n z defines the number of delay samples between the input and the output G z7 0 and A z7 0 are rational polynomials as defined by the following equations 21 E B z 9 C28 Fe yr 0 ey Ce A z 8 D z 8 The vector 9 is the set of model parameters Equations in the following sections of this manual do not display O to make the equations easier to read The following equations define A z B z C z D z and F z 1 2 ky A z l a z 4 z a 2 E 1 2 kp 1 B z byt biz b b 1z 1 2 k C z l eyz C 40 2 D z 1 dz d d z System Identification Toolkit User Manual 4 2 ni com Chapter 4 Parametric Model Estimation Methods k F z 14 f z Hifz TF thZ where ka kp Ke ka and ky are the model orders The following equation describes a general linear polynomial model A z y k 222 hy B Atk n eww Figure 4 1 depicts the signal flow of a general linear polynomial model ia C z D z u k a gt a wh Figure 4 1 Signal Flow of a General Linear Polynomial Model A general linear polynomial model provides flexibility for both system dynamics and stochastic dynamics You can use the SI Estimate General Linear Model VI to estimate general linear polynomial models Setting one or more of A z C
22. how to use the SI Estimate Orders of System Model VI to find the optimal order and the number of significant singular values Figure 9 20 Estimate Orders of State Space Model VI The Singular Values graph in Figure 9 21 shows a singular value plot with four leading singular values System Identification Toolkit User Manual 9 18 ni com Chapter 9 System Identification Case Study Singular Yalues eo Singular Yalue Sako rd cee en Roe Gees Ree nen bane Pane Oo 2 2 3 4 amp amp Ff 8 SF 10 ih is 4 Number order optimal order Figure 9 21 Singular Value Plot for State Space Model By looking both at the Singular Values graph and the optimal order you can see that there are four states in this state space model Validating the Estimated State Space Model You can validate the state space model in the same way that you validated the ARX model You use the SI Model Simulation VI and the SI Model Prediction VI to determine the accuracy of the state space model Figure 9 22 shows the complete process from estimating the state space model to simulating and predicting the response of the model SI Model Simulation vi inulation ISI Estimate State Space Mi ISI Lowpass Filter vi r a gj aaeeea ae a a oe 2 SISO WOT i i T7 cutof
23. is with the flexible arm as shown in Figure 3 11 The input of this system is the reaction torque of the structure on the ground This input is a multi sine wave with 200 frequency points equally spaced over the frequency band from 0 122 Hz to 24 4 Hz The output of this system is the acceleration of the flexible arm The frequency response of this system is not significant outside of the range of interest which is the frequency band of the input signal or 0 122 Hz to 24 4 Hz However notice that the magnitude response has a peak around 42 Hz The peak around 42 Hz may be the result of noise or nonlinearity or another input source You can use lowpass filtering to remove the 42 Hz peak before applying parametric estimation magnitude response 1 1 100 248 6 2000 1 1 1 1 0 795775 10 248 6 Frequency Hz Figure 3 11 Frequency Response of a Flexible Arm System Identification Toolkit User Manual 3 18 ni com Parametric Model Estimation Methods Parametric models describe systems in terms of difference or differential equations depending on whether a system is represented by a discrete or continuous model Compared to nonparametric models parametric models might provide more accurate estimation if users have prior knowledge about the system dynamics to determine model orders time delays and so on This chapter describes parametric model representations the assumptions about each estimation method
24. k k 1 and repeat steps 4 8 System Identification Toolkit User Manual 7 4 ni com Chapter 7 Recursive Model Estimation Methods The LMS algorithm is one of the most widely used and understood adaptive algorithms Selecting the step size u is important with the LMS algorithm because the selection of the step size u directly affects the rate of convergence and the stability of the algorithm The convergence rate of the LMS algorithm is usually proportional to the step size u The larger the step size LL the faster the convergence rate However a large step size u can cause the LMS algorithm to become unstable The following equation describes the range of the step size u 0 lt U lt Umax Umax 1S the maximum step size that maintains stability in the LMS algorithm Unax 1S related to the statistical property of the stimulus signal A uniformly optimized step size u that achieves a fast convergence speed while maintaining the stability in the system does not exist regardless of the statistical property of the stimulus signal For better performance use a self adjustable step size u and the normalized least mean squares NLMS algorithm Normalized Least Mean Squares The following equation defines a popular self adjustable step size u k that you use in the normalized least mean squares algorithm uk E e loco gt Again k represents the data vector is a very small positive number gt 2 that prevents the de
25. models provide a more complete representation of the system especially for multiple input multiple output MIMO systems than polynomial models because state space models are similar to first principle models that can provide more degree of freedom in describing MIMO systems The identification procedure for state space models does not involve nonlinear optimization so the estimation reaches a solution regardless of the initial guess Moreover the parameter settings for the state space model are simpler You need to select only the order or the number of states of the model The order can come from prior knowledge of the system You also can determine the order by analyzing the singular values of the information matrix However the states that the state space identification procedure identifies might not reflect the physical characteristics of a system accurately Using a similarity transformation you can identify equivalent models with states that better represent the system Similarity transformations enable you to transform the states without misrepresenting the input output behavior of the system Refer to the book Linear systems for more information about similarity transforms Refer to Chapter 9 System Identification Case Study for an example of how to estimate the order of the state space model of a system When model order is high state space models are preferable to polynomial models Polynomial models with high order might encoun
26. of samples in the data set After preprocessing the data you acquired from a dynamic system the result is a data set that you can use to estimate a model that reflects the system dynamics Refer to Chapter 3 Nonparametric Model Estimation Methods for more information about the nonparametric model estimation System Identification Toolkit User Manual 2 10 ni com Chapter 2 Acquiring and Preprocessing Data methods that use the impulse response and frequency response Refer to Chapter 4 Parametric Model Estimation Methods for more information about the parametric model structures and the parametric model estimation methods Refer to Chapter 7 Recursive Model Estimation Methods for more information about recursive model estimation methods Data Scaling In multiple input multiple output MIMO systems to have inputs and outputs of different amplitude ranges is common Such a diversity in amplitudes can make the model estimation calculation ill conditioned which deteriorates the precision of the dynamic response For example consider the values A and B in Figure 2 7 Valves A and B operate between 0 100 and 50 60 opening range respectively Level Figure 2 7 Tanks The pressure in their respective stream lines are P and Pg Assume that Pg can be much larger than P you might need to normalize the range of operation of valve B for numerical robustness You can use the following relationship to normalize the range of
27. operation ALevel E ALevel A A A B 50 10 The SI Normalize VI ensures that all stimulus and response signals have a zero mean and unit variance over the sample data range used for model estimation This process standardizes the range of the equation for all signals considered for model estimation This data preprocessing step considers all inputs and outputs equally important from the numerical calculation viewpoint National Instruments Corporation 2 11 System Identification Toolkit User Manual Nonparametric Model Estimation Methods After acquiring and preprocessing the data from a linear time invariant system the next step in the system identification process is to estimate the model The two most common techniques to estimate models that represent linear time invariant systems are nonparametric estimation and parametric estimation This chapter describes the nonparametric estimation methods You can describe linear time invariant models with transfer functions or by using the impulse responses or frequency response of the system The impulse response and frequency response are two ways of estimating a nonparametric model The impulse response reveals the time domain properties of the system such as time delay and damping whereas the frequency response reveals the frequency domain properties such as the natural frequency of a dynamic system Nonparametric model estimation is simple and more efficient but often less accu
28. optimum depends on the limit range you set and the random initial value the SI Estimate Partially Known State Space Model VI selects To increase the possibility of finding the global optimum complete the following steps 1 Use prior knowledge to set the range as narrow as possible 2 Perform multiple estimates with the range you set You might get different optimization results because the SI Estimate Partially Known State Space Model VI randomly selects an initial value within the range each time you run the VI If you repeatedly obtain the same result this result might be the optimum you want to find If you obtain inconsistent results either choose the result that best meets the system requirements or continue with step 3 to adjust the limit range 3 Select one of the previous results you got in step 2 according to the prior knowledge you have of the system Narrow the range in which the result falls Run the SI Estimate Partially Known State Space Model VI multiple times A consistent result you get might be the optimum you want to find Otherwise repeat this step until you find a consistent result You set limits in the SI Estimate Partially Known Continuous Transfer Function Model VI the same way you do in the SI Estimate Partially Known State Space Model VI Setting Parameter Constraints with an Initial Guess If you have information about a certain parameter and can estimate a value for that parameter you can refine est
29. prediction error correlates to the stimulus signal Prediction errors are usually uncorrelated with all stimulus signals in an open loop system The block diagram in Figure 9 17 shows how you can use the SI Model Residual Analysis VI with the ARX model identified in the User Defined Criterion section to analyze the residuals National Instruments Corporation 9 15 System Identification Toolkit User Manual Chapter 9 System Identification Case Study 8 waveform Duration vi Autocorrelation of Prediction Error ISI Down Sampling vi SI Estimate ARX Model vi py Flexible Arm YF ISI Data Samples vi SISO WDT 7 ISI Model Residual Analysis vi p j 5150 WOT 7 cutoff frequenc decimation Factor list portion of Figure 9 17 Residual Analysis VI Figure 9 18 shows an example of ideal results where both autocorrelation and cross correlation are inside the confidence region except those in the vicinity of T 0 This result indicates that the estimated model accurately describes the system Autocorrelation of Prediction Error 1 auto std dev 2 in 1 Amplitude Oo 1 0 6 4 0 HAN Cross
30. realization methods The deterministic stochastic subspace method uses principal component analysis to estimate parameters This method uses both stimulus and System Identification Toolkit User Manual 4 10 ni com Chapter 4 Parametric Model Estimation Methods response signals to estimate state space models The realization method uses the impulse response to estimate only the deterministic state space model This method does not include stochastic parts of the system in the model structure Therefore the difference between these two methods is that the deterministic stochastic subspace method includes noise in the model structure whereas the realization method does not Refer to the LabVIEW System Identification Toolkit Algorithm References manual SIreference pdf available in the Labview manuals directory for more information about the deterministic stochastic subspace method and the realization method For continuous state space models the System Identification Toolkit supports partially known model estimation methods You must provide an initial guess for each parameter before conducting estimation Refer to Chapter 5 Partially Known Model Estimation Methods for more information about partially known continuous state space model estimation methods User Defined Model When the model you want to estimate does not fall into the above four categories you can define your own model by revising a template VI You can find template VI
31. representation to another and convert between continuous and discrete models You have the flexibility to obtain best fit models in different applications The following chapters describe data acquisition and preprocessing techniques model estimation methods model analysis and validation techniques and model conversion tools System Identification Toolkit User Manual 1 8 ni com Acquiring and Preprocessing Data The first step in identifying an unknown system is data acquisition You can acquire data from NI data acquisition hardware and software or you can use data from a pre stored file For verification and validation reasons you need to acquire two sets of input output data samples or split the data into two sets You use one set of samples to estimate the mathematical model of the system You use the second set of samples to validate the resulting model If the resulting model does not meet the predefined specifications such as the mean square error MSE modify the settings and re verify the resulting model with the data sets After acquiring the data you need to preprocess the raw data samples Preprocessing involves steps such as removing trends filtering noise and so on The LabVIEW System Identification Toolkit provides Data Preprocessing VIs that enable you to analyze the raw data and determine whether the data accurately reflects the response of the system you want to identify This chapter briefly describes the data
32. such as an abnormal pulse a temporary sensor failure or transmitter failure can corrupt the raw data samples These disturbances or outliers can severely distort the resulting model estimation However you often can recognize outliers by visually inspecting the data as shown in Figure 2 6 Amplitude 1 1 1 i 1 1 100 120 140 160 180 200 Time Figure 2 6 Data with Outliers Visually inspecting the graph tells you that the data acquired between 85 100 seconds is abnormal When preprocessing data you want to remove all outliers in the data set You must remove the outliers manually Filtering and Downsampling You might be interested in only a specific frequency range of the frequency response for a model You can filter and enhance the data in the frequency range to improve the fit in the regions of interest If the sampling frequency is much higher than the bandwidth of the system the sampling frequency might substantially increase the computation burden for complicated identification algorithms You can decrease the sampling frequency by taking every n sample to construct a new downsampled data set Applying an anti alias filter on the data before downsampling prevents corruption of the downsampled data set You can use the SI Lowpass Filter VI or the SI Bandpass Filter VI to apply a lowpass or bandpass filter respectively to the data from the system You then can use the SI Downsampling VI to reduce the number
33. system at those frequencies The frequency response is a complex valued sequence The magnitude of G w is the magnitude response of the system and the phase of G is the phase response of the system This method of obtaining the frequency response is straightforward but takes a long time to complete and is sensitive to noise For these reasons the System Identification Toolkit uses the spectral analysis method to estimate the frequency response function Spectral Analysis Method You can use the spectral analysis method with any input signal However the frequency bandwidth of the input signal must cover the range of interest Because the frequency response is the Fourier transform of the impulse response applying the Fourier transform to both sides of the cross correlation function yields the following equation 1O 0 jo D e e G e uu System Identification Toolkit User Manual 3 12 ni com Chapter 3 Nonparametric Model Estimation Methods G e is the frequency response of the system e is the auto spectral density of the stimulus signal u k e is the cross spectral density between the stimulus signal u k and the response signal y k You then can use the following equation to compute the frequency response G e P e G e Pule You can compute e and e by applying a fast Fourier transform FFT to the autocorrelation R and the cross correlation Ry respect
34. the coupling between the deterministic and stochastic dynamics can bias the estimation of the ARX model You can set the model order higher than the actual model order to minimize the estimation error especially when the signal to noise ratio is low However increasing the model order can change some dynamic characteristics of the model such as the stability of the model The identification method for the ARX model is the least squares method which is a special case of the prediction error method The least squares method is the most efficient polynomial estimation method because this method solves linear regression equations in analytic form Moreover the solution is unique Refer to the LabVIEW System Identification Toolkit Algorithm References manual SIreference pdf available in the labview manuals directory for more information about the least squares and prediction error methods You can use the SI Estimate ARX Model VI to estimate ARX models ARMAX Model When D z and F z equal 1 the general linear polynomial model reduces to the ARMAX model The following equation describes an ARMAX model A z y k z B z u k C z e k B z u k n C z e k System Identification Toolkit User Manual 4 4 ni com Chapter 4 Parametric Model Estimation Methods Figure 4 3 depicts the signal flow of an ARMAX model gas C z u k 1 y k B z gt gt AG gt Figure 4 3 Sign
35. the imaginary part of the complex frequency response against the real part You can view the Nyquist plot as a combination of the Bode magnitude plot and the Bode phase plot In polar coordinates the Nyquist plot shows the phase as the angle and the magnitude as the distance from the origin as shown in Figure 8 2 imaginary Nyquist plot Nyquist Upper Limit Lower Limit confidence level 22 Figure 8 2 A Nyquist Plot with Sample Data The SI Nyquist Plot VI also calculates the upper and lower limits according to the confidence level you set You can use the SI Nyquist Plot VI to generate the Nyquist plot and display this plot using the SI Nyquist Plot Indicator The Nyquist plot is commonly used to predict the stability of a system National Instruments Corporation 8 3 System Identification Toolkit User Manual Chapter 8 Analyzing Validating and Converting Models Pole Zero Plot The pole zero plot displays the poles and zeros of a system By observing the locations of the poles and zeros you can conclude if the system is stable In a stable system all the poles fall within the unit circle Figure 8 3 shows an example of a stable model Pole Zero Plot imaginary Pole erry Zero Le S Pole Confidence ara Zero Confidence perre Unit Circe Axis EEA Confidence Level J 99 Figure 8 3 A Pole Zero Plot with Sample Data You can use the SI Pole Zero Plot VI to generate the p
36. the system Because system identification involves many variables such as sampling frequency type of mathematical model model order and so on you usually have a number of models you can use Each model describes the behavior of the system to some extent or in a particular mode of operation Furthermore multiple applicable algorithms might be available for the same model The algorithms you select depend on the model structure stochastic assumptions and numerical properties of the algorithm The System Identification Toolkit includes different adaptive techniques for recursive system identification and different algorithms for model estimation Refer to Chapter 3 Nonparametric Model Estimation Methods Chapter 4 Parametric Model Estimation Methods Chapter 5 Partially Known Model Estimation Methods Chapter 6 Model Estimation Methods in Closed Loop Systems and Chapter 7 Recursive Model Estimation Methods for more information about the various estimation methods that the System Identification Toolkit supports National Instruments Corporation 1 3 System Identification Toolkit User Manual Chapter 1 Introduction to System Identification Designing a Controller In the second phase of the design process you synthesize and analyze a controller The LabVIEW Control Design Toolkit provides a set of VIs for classical and modern linear control analysis and design techniques With these VIs you can create and analyze linear time inv
37. world system configuration If you do not have such information you can determine if feedback exists by obtaining the impulse response of a plant You can use the Least Squares instances of the SI Estimate Impulse Response VI to estimate the impulse response of a plant Figure 6 2 shows a comparison of the impulse responses of the plant in a closed loop system and an open loop system The values outside the upper limit and lower limit range at the negative lag which appears between 10 and 0 on the x axis are considered significant values Significant values in the impulse response at negative lags imply feedback in data As shown in Figure 6 2 significant values exist in the Closed loop data plot Therefore feedback exists in the closed loop system No significant impulse response values exist in the Open loop data plot Thus feedback does not exist in the open loop system Amplitude Impulse Response Closed loop data ae Open loop data i Upper limit as Lower limit as confidence level Figure 6 2 Impulse Responses of Open Loop and Closed Loop Data ni com Chapter 6 Model Estimation Methods in Closed Loop Systems You can use the SI Detect Feedback VI to detect feedback in a system This VI calculates the impulse response of a plant using the stimulus and response signals of the plant and then calculates upper and lower limits according to the confidence level you set The SI Detect Feedback VI displays t
38. z D z and F z equal to 1 can create simpler models such as autoregressive with exogenous terms ARX autoregressive moving average with exogenous terms ARMAX output error and Box Jenkins models which you commonly use in real world applications ARX Model When C z D z and F z equal 1 the general linear polynomial model reduces to an ARX model The following equation describes an ARX model A z y k 2 B z u k e k B z u k n e k 3 Note The backward shift operator makes z u k u k n National Instruments Corporation 4 3 System Identification Toolkit User Manual Chapter 4 Parametric Model Estimation Methods Figure 4 2 depicts the signal flow of an ARX model e k me gt B z gt gt Ae gt Y K Figure 4 2 Signal Flow of an ARX Model The ARX model is the simplest model incorporating the stimulus signal However the ARX model captures some of the stochastic dynamics as part of the system dynamics In this model the transfer function of the deterministic part G z7 0 of the system and the transfer function of the stochastic part H z 0 of the system have the same set of poles This coupling can be unrealistic The system dynamics and stochastic dynamics of a system do not share the same set of poles all the time You can reduce this disadvantage if the signal to noise ratio is high When the disturbance e k of a system is not white noise
39. 2 pole zero plot Pole zero eea bo o Pole Confidence Eee ona AAN Zero Confidence Unit Circle Axis gt fa r an Q amp optimal orders 4 B j 6 6 delay Figure 9 12 Pole Zero Plot for a MDL Model Compare Figure 9 12 which uses the MDL criterion and Figure 9 11 which uses the AIC criterion Because there are no overlapping pole zero pairs in Figure 9 12 you can conclude that the MDL criterion fits better than the AIC criterion in this particular example In addition to examining redundancy you also can use the pole zero plot for other purposes For example both Figure 9 11 and Figure 9 12 show poles outside the unit circle Having poles outside the unit circle implies that this model is not optimal because the ARX system based on the AIC or MDL criteria is unstable One way to stabilize the system is to change the order In addition to the FPE AIC and MDL criteria you can set user defined orders in the SI Estimate Orders of System Model VI National Instruments Corporation 9 11 System Identification Toolkit User Manual Chapter 9 System Identification Case Study User Defined Criterion If you know nothing about the system you might have to rely on trial and error to determine the optimal orders of the model However if you have some knowledge about a system you can customize the estimation to find a model that fits a certain model dimension For this model assume you know th
40. 6 9 System Identification Toolkit User Manual Chapter 6 Model Estimation Methods in Closed Loop Systems Therefore the deterministic stochastic subspace method works with data from a closed loop system e Use the SI Estimate Transfer Function VI or the SI Transfer Function Estimation Express VI to estimate a transfer function model of the plant in a closed loop system You can apply direct indirect and joint input output identification to compute transfer function models e To identify zero pole gain models for a plant you first must identify the plant using other model representations You then can convert other model representations to zero pole gain models using the Model Conversion VIs Refer to Chapter 8 Analyzing Validating and Converting Models for more information about converting models Refer to Chapter 4 Parametric Model Estimation Methods for more information about the models and model algorithms described in this section Refer to the LabVIEW System Identification Toolkit Algorithm References manual SIreference pdf available in the labview manuals directory for more information about the prediction error method the deterministic stochastic subspace method and the realization method Refer to the LabVIEW Help for more information about the VIs described in this section System Identification Toolkit User Manual 6 10 ni com Recursive Model Estimation Methods The model estimation methods in Chapter 3 No
41. Based Control Design Process Analyzing Data and Creating a Dynamic System Model In the initial phase of the design process you must obtain a mathematical model of the plant you want to control One way to obtain a model is by using a numerical process known as system identification This process involves acquiring data from a plant and then numerically analyzing stimulus and response data to estimate the parameters of the plant National Instruments provides data acquisition DAQ and modular instrumentation software and hardware that you can use to stimulate and measure the response of the plant You then can use the System Identification Toolkit to estimate and create accurate mathematical models of the plant System Identification Toolkit User Manual 1 2 ni com Chapter 1 Introduction to System Identification System identification is a process that includes acquiring preprocessing and identifying mathematical models based on raw data from a real world system You then validate that the resulting model fits the observed system behavior If the results are unsatisfactory you revise the parameters and iterate through the process Figure 1 2 shows a typical system identification flowchart Modeling Data gt Select Model EE gt Applications Characterization Figure 1 2 System Identification Application Flowchart A real world system seldom has one model that perfectly describes all the observed behaviors of
42. Block Diagram of Lag Window Example VI In this example VI the input signal u x is a swept sine wave whose normalized frequency is from 0 to 0 5 The number of data points in the input signal is 4096 The length of the lag window therefore must be less than or equal to 4096 Figure 3 9 and Figure 3 10 show the resulting frequency responses when the window length is 4096 and 64 respectively National Instruments Corporation 3 15 System Identification Toolkit User Manual Chapter 3 Nonparametric Model Estimation Methods Estimated FRF Ideal FRF Magnitude response 25 50 75 100 0 000488281 0 2 Frequency Magnitude window length H4096 Figure 3 9 Frequency Response with Large Window Length Estimated FRF Ideal FRF Magnitude response Magnitude 1o0 l 0 000488281 0 2 Frequency window length J 64 Figure 3 10 Frequency Response with Small Window Length The frequency response curve is smoother and the variance is smaller when the length of the lag window is small However when the length of the lag window is too small you cannot distinguish between the two close peaks in the frequency response as shown in Figure 3 10 When the length of the lag window is large the SI Estimate Frequency Response VI accurately estimates the peaks as shown in Figure 3 9 The bias is small with a large lag window but the variance of the estimated frequency response is lar
43. Correlation of Prediction Error and Input 0 6 cross ki std dev 0 2 Time ps 1 0 Amplitude Figure 9 18 Residual Analysis for A Order 4 B Order 5 and Delay 0 When you verify and validate the identified model you must use multiple analysis techniques to determine if the estimated model accurately represents the system Some analysis techniques can be misleading For example if you performed a residual analysis on the model identified in the Minimum Description Length Criterion section you might conclude that this model is an accurate representation of the system Figure 9 19 shows the autocorrelation and cross correlation residual analysis for the model in System Identification Toolkit User Manual 9 16 ni com Chapter 9 System Identification Case Study the Minimum Description Length Criterion section Recall that this model has the following orders e A order 6 e Border 6 e delay 0 Autocorrelation of Prediction Error 1 auto os std dev 2 o0 Ed 0 6 1 1 I I I 1 1 0 1 Z z 4 51557 Time H Cross Correlation of Prediction Error and Input 0 6 cross 0 4 std dev w l 0 2 l 0 Amplitude 0 3 1 1 J 1 1 1 0 1 2 z 4 SERS Time Figure 9 19 Residual Analysis of ARX Model with A Order 6 B Order 6 and Delay 0 Figure 9 19 shows that both the autocorrelation and cross correlation are inside the confidence region Therefore without performing
44. Estimate the impulse response pulse response hin i ejected Figure 3 4 Block Diagram of Prewhitening Example VI In this example VI the is white noise checkbox determines whether the SI Estimate Impulse Response VI generates zero mean white noise as an input to the system When you place a checkmark in the is white noise checkbox and run the VI the generated input signal is zero mean white noise and the estimated impulse response closely approximates the true impulse response When you do not place a checkmark in the is white noise checkbox the generated input signal is not zero mean white noise As a result the estimated impulse response is different from the true impulse response These results indicate that the correlation analysis method is accurate and reliable when the input signal is zero mean white noise The AR order box determines the level of prewhitening When AR order equals 0 the SI Estimate Impulse Response VI does not apply prewhitening to the system When AR order is small and you do not place a checkmark in the is white noise checkbox the variance of the impulse response is large because the input signal is not always white noise The greater the value of AR order the better the VI whitens the signal but the more computation time and memory the VI requires The whiteness test indicator of this VI shows whether the input is zero mean white noise This indicator displays th
45. LabVIEW System Identification Toolkit User Manual August 2006 c7 NATIONAL 371001C 01 INSTRUMENTS Worldwide Technical Support and Product Information ni com National Instruments Corporate Headquarters 11500 North Mopac Expressway Austin Texas 78759 3504 USA Tel 512 683 0100 Worldwide Offices Australia 1800 300 800 Austria 43 0 662 45 79 90 0 Belgium 32 0 2 757 00 20 Brazil 55 11 3262 3599 Canada 800 433 3488 China 86 21 6555 7838 Czech Republic 420 224 235 774 Denmark 45 45 76 26 00 Finland 385 0 9 725 725 11 France 33 0 1 48 14 24 24 Germany 49 0 89 741 31 30 India 91 80 41190000 Israel 972 0 3 6393737 Italy 39 02 413091 Japan 81 3 5472 2970 Korea 82 02 3451 3400 Lebanon 961 0 1 33 28 28 Malaysia 1800 887710 Mexico 01 800 010 0793 Netherlands 31 0 348 433 466 New Zealand 0800 553 322 Norway 47 0 66 90 76 60 Poland 48 22 3390150 Portugal 351 210 311 210 Russia 7 095 783 68 51 Singapore 1800 226 5886 Slovenia 386 3 425 4200 South Africa 27 0 11 805 8197 Spain 34 91 640 0085 Sweden 46 0 8 587 895 00 Switzerland 41 56 200 51 51 Taiwan 886 02 2377 2222 Thailand 662 278 6777 United Kingdom 44 0 1635 523545 For further support information refer to the Technical Support and Professional Services appendix To comment on National Instruments documentation refer to the National Instruments Web site at ni com info and enter the info code feedback 2004 2006 National Instruments Corporation All right
46. NG FOR A LEVEL OF RELIABILITY SUITABLE FOR USE IN OR IN CONNECTION WITH SURGICAL IMPLANTS OR AS CRITICAL COMPONENTS IN ANY LIFE SUPPORT SYSTEMS WHOSE FAILURE TO PERFORM CAN REASONABLY BE EXPECTED TO CAUSE SIGNIFICANT INJURY TO A HUMAN 2 IN ANY APPLICATION INCLUDING THE ABOVE RELIABILITY OF OPERATION OF THE SOFTWARE PRODUCTS CAN BE IMPAIRED BY ADVERSE FACTORS INCLUDING BUT NOT LIMITED TO FLUCTUATIONS IN ELECTRICAL POWER SUPPLY COMPUTER HARDWARE MALFUNCTIONS COMPUTER OPERATING SYSTEM SOFTWARE FITNESS FITNESS OF COMPILERS AND DEVELOPMENT SOFTWARE USED TO DEVELOP AN APPLICATION INSTALLATION ERRORS SOFTWARE AND HARDWARE COMPATIBILITY PROBLEMS MALFUNCTIONS OR FAILURES OF ELECTRONIC MONITORING OR CONTROL DEVICES TRANSIENT FAILURES OF ELECTRONIC SYSTEMS HARDWARE AND OR SOFTWARE UNANTICIPATED USES OR MISUSES OR ERRORS ON THE PART OF THE USER OR APPLICATIONS DESIGNER ADVERSE FACTORS SUCH AS THESE ARE HEREAFTER COLLECTIVELY TERMED SYSTEM FAILURES ANY APPLICATION WHERE A SYSTEM FAILURE WOULD CREATE A RISK OF HARM TO PROPERTY OR PERSONS INCLUDING THE RISK OF BODILY INJURY AND DEATH SHOULD NOT BE RELIANT SOLELY UPON ONE FORM OF ELECTRONIC SYSTEM DUE TO THE RISK OF SYSTEM FAILURE TO AVOID DAMAGE INJURY OR DEATH THE USER OR APPLICATION DESIGNER MUST TAKE REASONABLY PRUDENT STEPS TO PROTECT AGAINST SYSTEM FAILURES INCLUDING BUT NOT LIMITED TO BACK UP OR SHUT DOWN MECHANISMS BECAUSE EACH END USER SYSTEM IS CUSTOMIZED AND DIFFERS FROM NATIONAL
47. W Help available by selecting Help Search the LabVIEW Help for more information about specific System Identification VIs in this case study System Identification Toolkit User Manual 9 20 ni com References This manual contains information about using the LabVIEW System Identification Toolkit VIs in LabVIEW applications In most cases the underlying theory is omitted This list of references contains more information about the theory and algorithms implemented in the System Identification Toolkit Astrom K J and B Wittenmark 1974 Adaptive control 2nd ed Addison Wesley De Moor B 1988 Mathematical concepts and techniques for modelling of static and dynamic systems Ph D thesis Katholieke Universiteit Leuven UDC 519 17 Gill P E W Murray and M H Wright 1981 Practical optimization Academic Press Goodwin G C and R L Payne 1977 Dynamic system identification Experiment design and data analysis Academic Press Kailath T 1980 Linear systems Prentice Hall Ljung L 1999 System Identification Theory for the User 2nd ed Prentice Hall Ljung L and T Glad 1994 Modeling of dynamic systems Prentice Hall Ljung L and T S derstr m 1983 Theory and practice of recursive identification MIT Press Oppenheim A V and R W Schafer 1989 Discrete time signal processing 2nd ed Prentice Hall Oppenheim A V A S Willsky and I T Young 1997 Signals and systems Prenti
48. aE E erates EAS EAEE E 8 1 B d Plotis nne e E ee A E R EEA 8 2 Nyquist Plots S ET T EE E E 8 3 Pole Zero Ploti araa teins Ra ee a al eS 8 4 Vadan Models sccsech io essere sessesdensoctesscvab oes e a ENRERE E G EREE RE 8 5 Validation Methods a a A EE A aAa A aa TARS 8 5 Model Simulation ie eree e ean ae eena ee aAa i E EEEE EEEE 8 6 Model Predict n konii a E O a 8 6 Residual Analysis eena E RRE 8 6 Convertine Mode S sities a dagen hed Gaeta i 8 8 National Instruments Corporation vij System Identification Toolkit User Manual Contents Chapter 9 System Identification Case Study Data Preprocessing isis ssivezes cassis s e a ia sasubeoseaaseaves ces TEE E a ove ia 9 1 Examining the Time Response Data sessseeseeessesesrsresreresresrrresresrrrrsrrsrssese 9 2 Examining the Frequency Response Data eessesseeesesessserereserrrrrsrsresreresresrse 9 3 Applying a Filter to the Raw Data sseeesessseesseesssrsresrsresresesresreresresrsresereeseeet 9 5 Downsampling the Raw Data 0 eee ceceseeeeeeseeeeeeseeeeesecaeensesseeeaeenees 9 6 Estimating the M del iiinis ieoi e E E NER KEE aaRS 9 7 Akaike s Information Criterion seesseessseeseeesrereereerereersrestssrererrsrestesrsresrerest 9 8 Verifying the Results ennnen epads reed ibe EE E T 9 9 Minimum Description Length Criterion eee eeeeseeseeeeeeeeeeseeeeeesetsees 9 10 User Defined Criterion vic oerna iaiia naen aa 9 12 ARX Model Validation 0 cscccdec
49. acquisition process and the assumptions the System Identification Toolkit makes This chapter also describes how to preprocess raw data using the Data Preprocessing VIs Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for more information about the Data Preprocessing VIs Acquiring Data from a System One of the biggest advantages of using the System Identification Toolkit is the integration with LabVIEW NI data acquisition hardware and NI DAQ Refer to the LabVIEW Help available in LabVIEW by selecting Help Search the LabVIEW Help for more information about setting up and configuring a data acquisition system and how to use LabVIEW to acquire data samples National Instruments Corporation 2 1 System Identification Toolkit User Manual Chapter 2 Acquiring and Preprocessing Data Identifying a system involves a number of choices with regard to the system output signals you want to measure and the input signals you want to manipulate The choices you make about how to manipulate system inputs types of signal conditioning signal ranges and sampling behavior affect the validity of the model you obtain You can use different modeling techniques on the same experimental data set but if the data set does not reflect the behavior of interest then you need to acquire a more descriptive data set Because the system identification process is often an experimental process it is often time consuming and poss
50. agnitude frequencies results in a flat spectrum Refer to Chapter 3 Nonparametric National Instruments Corporation 4 7 System Identification Toolkit User Manual Chapter 4 Parametric Model Estimation Methods Model Estimation Methods for examples of using the AR model for prewhitening As shown in the AR model equation if you know the AR coefficients A z and the noise e k you can reconstruct the signal y k A z and e k completely characterize a signal A z normally has a small number of coefficients e k has a small dynamic range and requires a smaller number of bits for encoding Therefore you can use the AR model for compression purposes in a process known as linear prediction coding LPC Speech and vibration signal processing methods such as compression and pattern recognition commonly use LPC You also can use A z and e k to estimate the power spectrum of the signal y k Use the SI Estimate AR Model VI to estimate AR models Transfer Function Model You can use a transfer function to define either a continuous system or a discrete system using the following equations respectively y t G s u t e t y k G z u k e k Let the parameters of the model be the numerator and denominator coefficients of the transfer function The transfer function provides a mathematical representation of the relationship between one input and one output The following equations define the continuous and discrete transfer func
51. ailable by selecting Help Search the LabVIEW Help for more information about specific System Identification VIs in this case study Data Preprocessing After you gather data the next step in the system identification process is to preprocess the data The input to the system in this case study is the reaction torque of the structure on the ground This input is a swept sine wave with 200 frequency points equally spaced over the frequency band from 0 122 Hz to 24 4 Hz The output of this system is the acceleration of the flexible arm The acceleration contains information about the flexible resonances and anti resonances 1 The flexible robotic arm data was adopted from a case study in the MATRIXx Interactive System Identification Module Part 2 manual Hendrik Van Brussel and Jan Swevers of the laboratory of Production Manufacturing and Automation of the Katholieke Universiteit Leuven provided this data which they obtained in the framework of the Belgian Programme on Interuniversity Attracion Poles National Instruments Corporation 9 1 System Identification Toolkit User Manual Chapter 9 System Identification Case Study The data set contains 4096 samples at a sampling rate of 500 Hz or sampling time of 0 002 seconds Thus the total time of the response is 8 192 seconds The following sections show you how to preprocess the raw data by examining the time and frequency responses of the system Based on those analyses you can fil
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53. al Flow of an ARMAX Model Unlike the ARX model the system structure of an ARMAX model includes the stochastic dynamics ARMAX models are useful when you have dominating disturbances that enter early in the process such as at the input For example a wind gust affecting an aircraft is a dominating disturbance early in the process The ARMAX model has more flexibility than the ARX model in handling models that contain disturbances The identification method of the ARMAX model is the prediction error method The SI Estimate ARMAX Model VI uses the Gauss Newton method to optimize the mean square value of the prediction error when searching for the optimal ARMAX model This searching process is iterative and might converge to a local minimum Therefore you must validate the estimated model If the estimated model passes the validation test you can use this model even if the SI Estimate ARMAX Model VI might locate only a local minimum Refer to Chapter 5 Partially Known Model Estimation Methods for more information about optimization and local minima Output Error Model When A z C z and D z equal 1 the general linear polynomial model reduces to the output error model The following equation describes an output error model z B z F z u k e k Fit e k y k National Instruments Corporation 4 5 System Identification Toolkit User Manual Chapter 4 Parametric Model Estimation Methods Figure 4 4 depicts the sign
54. al flow of an output error model e k ae uk Figure 4 4 Signal Flow of an Output Error Model The output error model describes the system dynamics separately from the stochastic dynamics The output error model does not use any parameters for simulating the disturbance characteristics The identification method of the output error model is the prediction error method which is the same as that of the ARMAX model If the disturbance e k is white noise all minima are global However a local minimum can exist if the disturbance is not white noise You can use the SI Estimate OE Model VI to estimate output error models Box Jenkins Model When A z equals 1 the general linear polynomial model reduces to the Box Jenkins model The following equation describes a Box Jenkins model z z B z F z C z D z B z F z C z e k Di y k hk u k n e k Figure 4 5 depicts the signal flow of a Box Jenkins model e k t C z D z u k R 7 gt Vth Z Figure 4 5 Signal Flow of a Box Jenkins Model System Identification Toolkit User Manual 4 6 ni com Chapter 4 Parametric Model Estimation Methods The Box Jenkins model provides a complete model of a system The Box Jenkins model represents disturbance properties separately from system dynamics This model is useful when you have disturbances that enter late in the process such
55. al operation range you can combine the positive and negative step changes of different magnitudes in the system inputs e You want the input signal to deliver as much input power to the system as possible However in the real world you must ensure that this input power stays within the limits of the physical system The crest factor Cp defined by the following equation describes this property a maxu t f N 1 2 limy 5 7 u t t 1 The smaller the crest factor the better the signal excitation resulting in larger total energy delivery and enhanced signal to noise ratio The theoretical lower bound for crest factor is 1 Common Stimulus Signals The system response data is dependent on the physics of the system you want to study Some systems tend to respond faster than others and never reach steady state Other systems have large time constants and delays For these reasons defining a stimulus signal that provides enough excitation to the system is important so that the response captures the important features of the system dynamics The following sections describe common stimulus signals you can use in different process applications National Instruments Corporation 2 3 System Identification Toolkit User Manual Chapter 2 Acquiring and Preprocessing Data Filtered Gaussian White Noise Filtered Gaussian white noise is a simple signal that can generate virtually any signal spectra in conjunction with the proper lin
56. alyzing Validating and Converting Models Autocorrelation The following equation defines the autocorrelation of the residuals N R a gt e be k 2 k 1 Ideally the residual is white noise and therefore the autocorrelation R T is zero when T is nonzero A large autocorrelation when T is nonzero indicates that the residual is not zero mean white noise and also implies that the model structure is not relevant to the system or that you might need to increase the model order In real world applications the autocorrelation R T cannot be zero when T is nonzero because of the limited length of data points However the SI Model Residual Analysis VI assesses if the autocorrelation value is sufficiently small to be ignored If the value of autocorrelation falls within the confidence range the autocorrelation value is insignificant and you can consider this value to be equal to zero Cross Correlation The following equation defines the cross correlation between residuals and past inputs N Erare iy e k u k t k 1 If the residual is independent of the input the cross correlation is zero for all t If the residual correlates with the input the cross correlation is nonzero suggesting that the model did not capture all deterministic variations from the data Therefore you need to revise the model variation The SI Model Residual Analysis VI assesses if the value of cross correlation is sufficiently small If the value of cr
57. amples and syntax examples This font is also used for the proper names of disk drives paths directories programs subprograms subroutines device names functions operations variables filenames and extensions Italic text in this font denotes text that is a placeholder for a word or value that you must supply Related Documentation The following documents contain information that you might find helpful as you read this manual e LabVIEW Help e LabVIEW Fundamentals e Getting Started with LabVIEW e NI Express Workbench Help e LabVIEW System Identification Toolkit Algorithm References e LabVIEW Control Design Toolkit User Manual e Signal Processing Toolset User Manual e Time Series Analysis Tools User Manual Refer to Appendix A References for a list of textbooks and technical papers that National Instruments used to develop the System Identification Toolkit System Identification Toolkit User Manual X ni com Introduction to System Identification System identification involves building mathematical models of a dynamic system based on a set of measured stimulus and response data samples You can use system identification in a wide range of applications including mechanical engineering biology physiology meteorology economics and model based control design For example engineers use a system model of the relationship between the fuel flow and the shaft speed of a turbojet engine to optimize the efficiency
58. and u is the capacitor voltage In this example y equals the capacitor voltage uc R Cc Up NNA L Figure 5 3 An RLC Circuit Representing a State Space Model y Suppose R is 1 5 Q and L and C are unknown You can complete the following steps to identify the values of L and C 1 Apply a wide band voltage to u and measure the output y simultaneously The Continuous State Space Model of an RLC Circuit example VI uses a chirp signal from 0 5 Hz to 6 Hz as the stimulus signal The response to the chirp signal is the response signal This example VI then preprocesses the stimulus and response signals to remove the offset level in these signals 2 Define a model for this circuit Because you have information about the approximate values of L and C you can build a partially known state space model or a partially known transfer function model 3 Estimate the model you defined in step 2 and then estimate L and C The Continuous State Space Model of an RLC Circuit example VI guides you through defining and estimating a state space model for the RLC circuit You can access this example VI by selecting Help Find Examples to display the NI Example Finder and then navigating to the Toolkits and Modules System Identification folder Refer to the NI Developer Zone at ni com zone for an example of using a partially known transfer function model to estimate the RLC circuit National Instruments Corpor
59. ariant LTT system models and design automatic control systems You can use the Control Design Toolkit to analyze the plant model you identified with the System Identification Toolkit The Control Design VIs help you determine an appropriate controller structure You then can synthesize a controller to achieve the desired performance criteria of the system based on the dynamic behavior of the plant and or control system Finally you can analyze the overall system by combining the controller with the identified plant model Simulating the Dynamic System In the third phase of the design process you simulate the dynamic system The LabVIEW Simulation Module allows you to simulate dynamic systems in LabVIEW You can investigate the time response of the dynamic system to complex time varying inputs before deploying a controller For this process you can use a simple LTI model a higher order model or a nonlinear model of the plant Deploying the Controller The last stage of the design process is to deploy the controller to a real time target LabVIEW and the LabVIEW Real Time Module provide a common platform that you can use to implement or prototype the embedded control system You also can use the Simulation Module and the Real Time Module as the platform for implementing the control system National Instruments also provides products for I O and signal conditioning that you can use to gather and process data Using these tools which ar
60. at the system is stable therefore no poles exist outside the unit circle Because both the AIC and MDL criterion did not produce stable models the model orders do not describe the system accurately On the block diagram as shown in Figure 9 13 you can customize the method parameter Instead of AIC or MDL you can select lt Other gt and enter the desired model dimension in the textbox Assume you know that the model dimension is nine ISI Data Samples vi SI Lowpass Filter vi CE for SI Estimate ARX Model vi ae eR EERREREESS Teng oor Flexible Arm v SISO WDT cutoff Frequenc Figure 9 13 Pole Zero Cancellation VI Figure 9 14 shows the corresponding pole zero plot graph with a model dimension of nine which corresponds to the following optimal orders e Aorder 4 e Border 5 e delay 0 System Identification Toolkit User Manual 9 12 ni com Chapter 9 System Identification Case Study pole zero plot Pole be 3c Zero econ Pole Confidence pees Zero Confidence Ee Unit Circle Axis BAAS optimal orders imaginary Figure 9 14 Pole Zero Plot with User Defined Orders Compare the pole zero plot in Figure 9 14 with Figure 9 11 and Figure 9 12 Figure 9 14 has no overlapping pole zero pairs and all the poles are within the unit circle By visually inspecting the pole zero plot you can see that thi
61. ation 5 7 System Identification Toolkit User Manual Chapter 5 Partially Known Model Estimation Methods You can use the following first order differential equation to represent the relationship between the capacitor voltage and the current of this RLC circuit u ait 5 1 You can use the following first order differential equation to represent the voltage relationship in this RLC circuit Ri uct Li u 5 2 By manipulating Equations 5 1 and 5 2 you can deduce the continuous state space model for this RLC circuit using the following two equations ud 0 1 C c 0l i H2 R L li U L L y u hal ly The System Identification Toolkit provides the SI Create Partially Known State Space Model VI with which you can build the symbolic state space model for this circuit as shown in Figure 5 4 SI Create Partially Known State Space Model vi reeek psc Continuous SISO Figure 5 4 Defining a Partially Known State Space Model You specify the symbolic state space model using formula strings such as 1 C 1 L and 1 5 L with L and C as variables Then you define L and System Identification Toolkit User Manual 5 8 ni com Chapter 5 Partially Known Model Estimation Methods C with the variables input as shown in Figure 5 5 Using prior knowledge you know that L is a positive value around the initial value of 0 1 H and C is a value between 0 F and 0 3 F You already know the value o
62. ce Hall National Instruments Corporation A 1 System Identification Toolkit User Manual Appendix A References Overschee P V and B De Moor 1993 N4SID Subspace algorithms for the stochastic identification problem Automatica 29 no 3 649 660 1994 Subspace algorithms for the identification of combined deterministic stochastic systems Automatica 30 no 1 75 93 Powell M J D 1964 An efficient method for finding the minimum of a function of several variables without calculating derivatives Computer Journal 7 155 162 Priestley M 1981 Spectral analysis and time series Academic Press System Identification Toolkit User Manual A 2 ni com Technical Support and Professional Services Visit the following sections of the National Instruments Web site at ni com for technical support and professional services Support Online technical support resources at ni com support include the following Self Help Resources For answers and solutions visit the award winning National Instruments Web site for software drivers and updates a searchable KnowledgeBase product manuals step by step troubleshooting wizards thousands of example programs tutorials application notes instrument drivers and so on Free Technical Support All registered users receive free Basic Service which includes access to hundreds of Application Engineers worldwide in the NI Developer Exchange at ni com exchange National Instr
63. ct the response of the system using past input and output data You also can use model residual analysis to test using statistical techniques the whiteness of the prediction error and the independency between the prediction error and the input signal The methods you select to validate the model depend on the purpose for which you created the model You can use one or all of these methods to validate the model National Instruments Corporation 8 5 System Identification Toolkit User Manual Chapter 8 Analyzing Validating and Converting Models Model Simulation The SI Model Simulation VI determines the outputs of a system for given inputs After you build a model for the system using the input and output data you measured you can use the model to simulate the response of the system by using the model equations You then can evaluate the behavior of the system You also can use simulation to validate the model by comparing the simulated response with the measured response Model Prediction The SI Model Prediction VI determines the response of a system at time t based on the output information available at time k and all the inputs applied from time t k to time t k represents the size of the prediction window Therefore model prediction can determine how useful a model is in estimating future responses of the system given all information at time t and an expected input profile in the future Some control techniques take advan
64. d a tutorial about how to use the assistant The System Identification Assistant uses the same system identification concepts described in this manual However this manual does not include information about how to use the assistant Refer to the NI Express Workbench Help available in the NI Express Workbench environment by selecting Help Express Workbench Help for more information about the steps in the assistant and a tutorial about how to use the assistant 3 bold italic The following conventions are used in this manual The symbol leads you through nested menu items and dialog box options to a final action The sequence File Page Setup Options directs you to pull down the File menu select the Page Setup item and select Options from the last dialog box This icon denotes a note which alerts you to important information Bold text denotes items that you must select or click in the software such as menu items and dialog box options Bold text also denotes parameter names Italic text denotes variables emphasis a cross reference or an introduction to a key concept Italic text also denotes text that is a placeholder for a word or value that you must supply National Instruments Corporation ix System Identification Toolkit User Manual About This Manual monospace monospace italic Text in this font denotes text or characters that you should enter from the keyboard sections of code programming ex
65. deal Dirac delta function is unrealistic If you apply an approximate impulse with a small duration to the input of a system the output of the system is the approximation of the impulse response of the system The smaller the duration of the impulse the closer the output of the system is to the true impulse response However an impulse carries little energy and might not excite the system and noise might corrupt the output of the system An impulse with a large amplitude and duration can improve the signal to noise ratio of the output signal However the large amplitude impulse can damage the hardware of the system and a long duration impulse leads to inaccuracy For these reasons the System Identification Toolkit uses the least squares method and the correlation analysis method to estimate the impulse response Least Squares If both the input signal u k and output signal y k of a system are available you can obtain the value of h k as shown in Equation 3 1 This method does not require a Dirac delta function as the input signal of the system Instead you can use common stimulus signal and the corresponding response signal from the system to compute the impulse response mathematically You can obtain the impulse response for both positive and negative lags The Least Squares instances of the SI Estimate Impulse Response VI implements the least squares method to solve the equation Refer to the LabVIEW System Identification Toolkit A
66. dentification approach the System Identification Toolkit enables you to estimate the plant in a closed loop system with general linear polynomial state space transfer function and zero pole gain models If you want to apply the indirect or joint input output approach to identify the plant this toolkit provides you with transfer function models Select the System Identification VIs using the following guidelines e Use the Polynomial Model Estimation VIs or the SI Model Estimation Express VI to estimate ARX ARMAX output error Box Jenkins and general linear models For ARX models the System Identification Toolkit uses the least squares method which is a special case of the prediction error method For all other models this toolkit uses the prediction error method This method can accurately identify a plant model in a closed loop system Hence you can use the Polynomial Model Estimation VIs to estimate the model of a plant in a closed loop system e Use the SI Estimate State Space Model VI or the SI Model Estimation Express VI to estimate a state space model of the plant in a closed loop system You can choose the deterministic stochastic subspace method that the SI Estimate State Space Model VI implements The deterministic stochastic subspace method uses principal component analysis to estimate parameters This method does not assume a zero correlation between the input signal and the output noise National Instruments Corporation
67. e Impulse Response VI has inputs to specify how many points of the impulse response to observe With the least squares method you must ensure the sum of num of points t lt 0 and num of points t gt 0 is no larger than the signal length With the correlation analysis method you can set num of points to be as large as the signal length Applications of the Impulse Response The impulse response not only indicates the stability and causality of the system if feedback exists in the system but also provides information on properties such as the damping dominating time constant and time delay Some of this information such as the time delay is useful for parametric model estimation Therefore you can use nonparametric impulse response estimation before parametric model estimation to help estimate the parameters The following example demonstrates how to use the SI Estimate Impulse Response VI to estimate the impulse response and determine the time delay of a system by using the correlation analysis method National Instruments Corporation 3 9 System Identification Toolkit User Manual Chapter 3 Nonparametric Model Estimation Methods Figure 3 5 shows the front panel of a VI that simulates a system defined by the following equation y k 0 2u k 2 0 8u k 3 0 3u k 4 system K 0 2 k 22 0 8k 0 3 k 4 Figure 3 5 Front Panel of Time Delay Example VI System Identification Toolkit User Manual 3 10 ni com
68. e autocorrelation of the stimulus signal after whitening If most of the autocorrelation is within the confidence region the input signal is well prewhitened and the estimation of the impulse response is reliable If the autocorrelation is outside of the confidence region the estimation is unreliable When the estimation is unreliable rejected is TRUE and indicates a 5 risk of rejecting an impulse response estimation that might be reliable System Identification Toolkit User Manual 3 8 ni com Chapter 3 Nonparametric Model Estimation Methods If you apply proper prewhitening the correlation analysis method is accurate and reliable for any input signal To obtain the best prewhitening settings start with a small AR order value like 2 and observe the whiteness test and rejected outputs of the SI Estimate Impulse Response VI If necessary increase the value of AR order Generally the smaller the bandwidth of the input signal the larger the AR order you need However avoid setting the value of AR order greater than 500 Selecting Impulse Response Length Theoretically the length of the impulse response might be infinite For some systems the impulse response quickly reaches zero and the number of nonzero points is finite For other systems the impulse response never reaches zero Realistically you only can obtain the first N points of the impulse response due to limited signal length and limited memory size Therefore the SI Estimat
69. e built on the LabVIEW platform you can experiment with different approaches at each stage in the design process and quickly identify the optimal design solution for an embedded control system Refer to the National Instruments Web site at ni com for more information about these National Instruments products System Identification Toolkit User Manual 1 4 ni com Chapter 1 Introduction to System Identification Model Types and Model Representations Model Types You can represent a dynamic system using several types of dynamic system models The following sections provide information about the different types and representations of dynamic system models that you can use with the System Identification Toolkit You base the type of a dynamic system model on the properties of the dynamic system that the model represents The following sections provide information about the different types of models you can create with the System Identification Toolkit Linear versus Nonlinear Models Dynamic system models are either linear or nonlinear A linear model obeys the principles of superposition and homogeneity as shown by Equations 1 1 and 1 2 respectively yi f ur y2 f u2 f u Uz f ui f U2 y y2 1 1 f am aif U1 ayy 1 2 where u and uz are the system inputs and y and yz are the system outputs Conversely nonlinear models do not obey the principles of superposition or homogeneity Nonlinear effects in
70. e standard deviation of the stimulus white noise and 8 t is the Dirac function Substituting R t into the cross correlation function System Identification Toolkit User Manual 3 4 ni com Chapter 3 Nonparametric Model Estimation Methods between the stimulus signal u k and the response signal y k yields the following equation R T 0 Y k TAK A T k 0 You can rearrange the terms of this equation to obtain the following equation defining the impulse response h k ro Bal u Prewhitening The correlation analysis method that estimates the impulse response is useful only when the input signal u k is a zero mean white noise signal However the input signal is not white noise in most real world applications The input u k and output y k signals therefore must be preconditioned before you apply the correlation analysis method Prewhitening is a preconditioning technique for the correlation analysis method Prewhitening involves applying a filter to the input signal u k and the output signal y k to obtain a prewhitened input signal u k and a prewhitened output signal y k If the filter is well designed such that u k is white noise you can perform a correlation analysis on u k and y k to estimate the impulse response The impulse response that you estimate with u k and y k is equivalent to the impulse response that you estimate with u k and y k because the following equation remains true co y k
71. ear filtering The theoretical crest factor Cy for a Gaussian is infinite but clipping the Gaussian amplitude to the input signal limits results in a corresponding reduction in crest factor while minimally affecting the generated spectrum Figure 2 1 shows an example of a Filtered Gaussian white noise Filtered Gaussian White Noise pu g 2 4 07 a 1 1 1 1 0 10 20 30 40 50 60 70 80 90 100 Figure 2 1 Filtered Gaussian White Noise Random Binary Signal A random binary signal is a random process that can assume one of two possible values at any time A simple method of generating a random binary signal is to take Gaussian white noise filter it for the desired spectra and then convert it to a binary signal by taking the sign of the filtered signal The desired spectra is a function of the system time constraints The appropriate scaling must provide a meaningful response to the system well above the noise level You can scale the signal to any desired amplitude The resulting signal has a minimum crest factor Cy of 1 Some differences in the resulting spectra are expected so you must perform off line analysis of the signal Binary signals are useful for identifying linear systems However the dual level signal does not allow for validation against nonlinearities If a system is nonlinear you can use an input interval corresponding to the desired operating point You might need to work with more than two input
72. eate Partially Known State Space Model VI for example to define a state space model that represents an RLC circuit consisting of a resistor R an inductor L and a capacitor C Using prior knowledge you describe the relationship of R L and C with the following equations Los EKOR B 0 1 1 xC c i o You also can use prior knowledge to define the initial guesses and upper and lower limits of R L and C The SI Create Partially Known State Space Model VI uses variables rather than numerical values to construct a symbolic model As Figure 5 1 shows you use variable names such as R L and C in the symbolic A symbolic B symbolic C and symbolic D inputs to define the RLC circuit Then you specify values for R L and C in the variables input Figure 5 1 Using Variables to Create a Symbolic State Space Model National Instruments Corporation 5 3 System Identification Toolkit User Manual Chapter 5 Partially Known Model Estimation Methods The SI Create Partially Known Continuous Transfer Function Model VI enables you to define partially known continuous transfer function models The following equation represents a continuous transfer function model 2 K 1 a s ays sT 4 e 29 Jo 2 s w s wy i J where XK is the transfer function gain T is the delay T is the first order time constant w is the natural frequency r is the damping ratio s represents the time You can apply the prior know
73. ecord graph shows that the water level changes in the vicinity of the operating point of six meters If you use the water level record for system identification you must remove the six meter operating point value Water tank Water level record ays 10 D Water level 4 N 1 1 1 1 1 1 1 1 1 40 60 80 100 120 140 160 180 200 Time o Figure 2 5 Operating Point of Water Tank The SI Remove Trend VI enables you to remove the offset from the raw data set You must set the trend type to mean to use this preprocessing technique Removing Trends External influences might add some low frequency or periodic components which are not relevant to the specific modeling problem to the data Examples of external influences include variations due to the 24 hour day cycle in power plants seasonal influences in biological and economical systems thermal expansion in rolling mills 50 Hz and 60 Hz powerline interferences and so on The amplitude of these trends can be large and can corrupt the results of signal analysis and parametric identification algorithms The SI Remove Trend VI provides a way for you to remove these external influences or trends from the raw data set You must set the trend type to linear to use this preprocessing technique National Instruments Corporation 2 9 System Identification Toolkit User Manual Chapter 2 Outliers Acquiring and Preprocessing Data Various unexpected events
74. ed saturation or quantization effects that occur during acquisition or preprocessing You also can plot the data waveform and the spectral density function of the data to discover periodic disturbances Traditionally you examine data samples either in the time domain or the frequency domain An effective approach is to display the data in the joint time frequency domain which provides a better understanding about the measured signals Refer to the Signal Processing Toolset User Manual available at ni com manuals for more information about joint time frequency domain techniques for data processing Removing Offsets and Trends The SI Remove Trend VI enables you to remove offsets and trends from the raw data set You can specify which you want to remove using the input trend type The following sections describe the difference between removing offsets and removing trends System Identification Toolkit User Manual 2 8 ni com Chapter 2 Acquiring and Preprocessing Data Removing Offsets The estimated system model is a linearized version of the true system around the operating point You must subtract the operating points from the raw data samples because linearization is done with respect to the signal values relative to the operating point which is the offset level of the signal Figure 2 5 shows an example of removing the offset level of a signal The goal of the water tank is to keep the water level at six meters The Water level r
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76. em Identification Procedure You first must acquire data before estimating a model for the plant in a system Then you can preprocess raw data by using a number of data preprocessing techniques the System Identification Toolkit provides to get high quality data When data is collected you can use nonparametric model estimation methods to estimate the impulse response and the frequency response of a system Nonparametric model estimation is often less accurate However you can use a nonparametric model estimation method to obtain useful information about a system before applying parametric model estimation which provides more insight into a system Also if you have some information about the system you can set constraints on parameters by using partially known model estimation methods the System Identification Toolkit provides When you have a plant in a closed loop system you can use the VIs this toolkit provides to identify the plant You also can use recursive model estimation methods with the System Identification Toolkit to update a system model while the system is running After you estimate a model you can analyze the model to obtain some information such as frequency response about the model and investigate model estimation results You also can validate an estimated model to determine how accurately the model describes the real world plant of a system With the System Identification Toolkit you can convert system models from one
77. er because by attenuating the frequencies greater than the Nyquist frequency the filter prevents the sampling of aliased components When you use a filter before the sampler and analog to discrete converter the anti aliasing filter is an analog filter Using an analog filter satisfies the Nyquist sampling theorem National Instruments Corporation 2 7 System Identification Toolkit User Manual Chapter 2 Acquiring and Preprocessing Data Similarly you can use a digital filter to remove frequency content above the system bandwidth and then decimate or downsample the data to the desired sampling rate Preprocessing Data from a System A number of preprocessing techniques ensure that the incoming data samples are free from external noise scaling problems outliers and other corruptions These preprocessing techniques include the following methods e Visually inspecting data e Removing offsets and trends e Removing outliers e Filtering and downsampling Validating the quality of the data at each step in the preprocessing procedure is important in ensuring that you accurately identify a model in later steps of the system identification process The following sections describe these preprocessing techniques and how you can use the System Identification Toolkit to apply these techniques Visually Inspecting the Data Visual inspection of the data is the best way to detect major signal corruptions or errors such as outliers clipp
78. eseneeeaeerees 2 7 Preprocessing Data from a System 0 0 ec eeeecceseceseeeecesecseeeeeeseceeessesseeseesseeaeeeeeateaes 2 8 Visually Inspecting the Data 00 eee cece ceeeeseceeeeeeeseeeseeaecneeeaesneenaeseeeeaees 2 8 National Instruments Corporation v System Identification Toolkit User Manual Contents Removing Offsets and Trends ee eeceeeeceeeseeeseeseeeeeeaeeeeeaseecesesseseaeeaees 2 8 Removing Offsets seiniin naa aaa 2 9 Removing Trends nsssiercaneniinaeiio i diel seve te A a s 2 9 Outlets naoa Shes a e a a a T 2 10 Filtering and Downsampling ssseessseesseessesrseerssrsresrerertsrestesrsresesesresrsresrsee 2 10 IDEVE ESL it a PARET T 2 11 Chapter 3 Nonparametric Model Estimation Methods Impuls R s poise neninn cad en a E cava EE exes NEE waa E 3 2 Least Sqare Soora a a O A ER eae 3 3 Correlation Analysis ssrin iana ei a a e EE 3 4 Prewhitenih g nin i a R 3 5 Accuracy of the Impulse Response seesseseeeseeereeeeersrsererrsrrsrsrese 3 6 Selecting Impulse Response Length 0 eee ee eee eeeeeceeeeeesseeeaeeseeeseeaeees 3 9 Applications of the Impulse Response cece sceeseeseescsseseesesseeetsessenees 3 9 Frequency Response sii c c cetascrssdssssaseedsattaseesssadessacadescheveescandeasessssbscousessensaapseacanscesed s 3 12 Spectral Analysis Method sisses 3 12 Accuracy of the Lag Window 00 ce eccscesesseceseeeceseeeeeseceeeaecnesesesneeeaeesees 3 14 Applications of the Frequency R
79. esponse ee eeeeeeseeseeeeeseeeeeesesseeeseesees 3 17 Chapter 4 Parametric Model Estimation Methods Parametric Model Representations ec eeeeseesececceseseeeeseeeceseeseeeseeaecneeeaeeneeeaeesees 4 1 General Linear Polynomial Model 0 eee ceeeeceseeseeeeeeeeceeeeseseeeeseeeeeaeeseees 4 1 ARX Modele E E ed ee 4 3 ARMAX Modela peoeores nimien EK AE REEE 4 4 Output Error Model sssr a a a Ra 4 5 Box Jenkins Modelek oee e aea reie E R EASE E 4 6 AR Mod l sneen nerea E R a cies nee R ES 4 7 FTranster Function Mod l rren e eenen or ana EA E A ERS 4 8 Zero Pole Gain Model e aae raae aea R A R 4 9 State Space M del eare e e A R A e eras a Aa EE oE EEEE TEN 4 10 User Defined M de lnscn a a E E 4 11 Polynomial Models versus State Space Models sessseeseeeeeeeseesreersersrrersee 4 12 Determining Parameters for the Prediction Error Method eccesseeseeeeneeeeeeeeeenees 4 12 Akaike s Information Criterion eccceccescesecesceesneceseeeececeaeeeeeceneesaeeesneesase 4 14 Final Prediction Error Criterion cccceescesceesecenceseeeceeecsaeeeneeeaeersaeeeseeesaes 4 14 Minimum Description Length Criterion eee eeeeceeseeeseeeeceeeeseeeseenees 4 14 System Identification Toolkit User Manual vi ni com Contents Chapter 5 Partially Known Model Estimation Methods Comparing Unknown and Partially Known Model Estimation Methods 5 1 Defining and Estimating Partially Known Models eee eceeseeeeeseeeeeeseeee
80. eters and the explained variation Minimum Description Length Criterion The Minimal Description Length Criterion MDL is based on V plus a penalty for the number of terms used For the MDL an optimal model is the one that minimizes the following equation MDL v i pelan N You want to choose a model that minimizes the MDL which allows the shortest description of data you measure System Identification Toolkit User Manual 4 14 ni com Partially Known Model Estimation Methods The model estimation methods introduced in previous chapters assume that all model parameters are adjustable However in many real world applications you cannot adjust all the parameters arbitrarily because the parameters might have constraints For example in some chemical processes water must flow only in one direction When estimating the flow rate of water you know that the flow rate cannot be negative Thus the constraint is that the flow rate must be a positive value You must consider this constraint and any other constraints when you estimate the flow rate of water in this process Models for which you can set constraints on the parameters are called partially known models This chapter compares unknown and partially known model estimation methods describes the methods you can use to define partially known models and explains how user defined initial values affect estimation results This chapter also provides a case study to estimate a partia
81. f Frequenc decimation Factor 1st portion number of states 10 H i p ahead prediction ee SISO WDT 7 Figure 9 22 Simulation amp Prediction with State Space Model VI The simulation and 1 step ahead prediction graphs in Figure 9 22 show simulation and prediction plots for a state space model National Instruments Corporation 9 19 System Identification Toolkit User Manual Chapter 9 System Identification Case Study Simulated simulation Measured Amplitude Predicted 1 step ahead prediction Measured Amplitude Figure 9 23 Simulation and Prediction Graphs for a State Space Model Refer to the Validating Models section of Chapter 8 Analyzing Validating and Converting Models for more information about validating a model Additional Examples In this chapter you learned how to start from raw data and find an accurate model to represent the system from which you acquired the data The examples in this chapter are located in the labview examples system identification SICaseStudy1 11b The labview examples system identification directory also contains other examples you can use to become familiar with the System Identification VIs You also can access the System Identification example VIs by selecting Help Find Examples to display the NI Example Finder and then navigating to the Toolkits and Modules System Identification folder Refer to the LabVIE
82. f the resistance R 1 5 LE oe Ol Ce cr Jo Wan Past Figure 5 5 Setting Model Parameters of the RLC circuit Next you can estimate the state space model with the SI Estimate Partially Known State Space Model VI as shown in Figure 5 6 Eymbolic 4 fave p jEr create Partially Known State Space Model vi stimated Model Emboic8 2 a vr mi Ea Lave stimated Values of Unknown Variables Bymbolic C f Continuous SISO Y tes Response iq Figure 5 6 Estimating a Partially Known State Space Model National Instruments Corporation 5 9 System Identification Toolkit User Manual Chapter 5 Partially Known Model Estimation Methods The SI Estimate Partially Known State Space Model VI estimates each parameter of the model You obtain the estimated model and optimized variables of the model after this VI performs an optimization In this example you obtain the values 0 20 H for L and 0 02 F for C as shown in Figure 5 7 Estimated Values of Unknown Variables Figure 5 7 Optimized Variables of the RLC Circuit The Continuous State Space Model of an RLC Circuit example VI uses the SI Draw Model VI and the values of L and C you obtain to display the estimated model in a picture indicator as shown in Figure 5 8 estimated model g o 49 8046 o RO soo zs37 MOF soz YO vt 1 0 x Figure 5 8 The Estimated Model of the RLC Circuit You then can determine how accurately this model s
83. ge with a large lag window System Identification Toolkit User Manual 3 16 ni com Chapter 3 Nonparametric Model Estimation Methods Setting the length of the lag window to 5 10 of the number of data points when estimating the frequency response often results in a good trade off between the bias and variance However the selection of the length of the lag window is not trivial The length also depends on the signals the properties of the system and the purpose of application For example if you want to know the passband of a system use a smaller lag window If you want to identify the dynamic properties of a system such as its natural frequency use a larger lag window Applications of the Frequency Response The frequency response gives the characteristics of the system in the frequency domain You can use the frequency response to obtain useful information before applying parametric estimation For example you can use the frequency response to determine whether you need to pre filter the signals or what the model order of the system is You also can use nonparametric frequency response to verify parametric model estimation results by comparing the frequency response of the parametric model with the nonparametric frequency response National Instruments Corporation 3 17 System Identification Toolkit User Manual Chapter 3 Nonparametric Model Estimation Methods One example of a real world application of the frequency response
84. he RLS algorithm 1 Initialize the parametric vector k using a small positive number 0 e gt 2 Initialize the data vector 4 gt o 0 0 0 0 3 Initialize the k x k matrix P 0 P 0 ooom oom eo Oo noon 0 gt gt 4 Fork 1 update the data vector k based on k 1 and the current input data u k and output data y k 5 Compute the predicted response y k by using the following equation p k OM BUD System Identification Toolkit User Manual 7 6 ni com Chapter 7 Recursive Model Estimation Methods 6 Compute the error e k by solving the following equation elk y k y k 7 Update the gain vector k defined by the following equation 5 P k 94 Ra P00 A Tk PCK OCK The properties of a system might vary with time so you need to ensure that the algorithm tracks the variations You can use the forgetting factor which is an adjustable parameter to track these variations The smaller the forgetting factor A the less previous information this algorithm uses When you use small forgetting factors the adaptive filter is able to track time varying systems that vary rapidly The range of the forgetting factor is between zero and one typically 0 98 lt A lt 1 P k is ak x k matrix whose initial value is defined by P 0 in step 3 8 Update the parametric vector W k 1 k 1 k elk Rk 9 Update the P k matrix
85. he result of feedback detection in an impulse response graph If significant values exist in the impulse response at negative lags the system has feedback This VI also uses a Boolean indicator to show the result of detection Refer to the Impulse Response section of Chapter 3 Nonparametric Model Estimation Methods for more information about impulse responses Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for more information about the SI Estimate Impulse Response VI and the SI Detect Feedback VI Understanding Closed Loop Model Estimation Methods Closed loop model estimation methods use data from a closed loop system to build a model for a plant that a controller regulates Figure 6 3 shows a system that consists of a plant and a controller In this system Go is the plant F is the controller H is the stochastic part of the plant u is the stimulus signal y is the response signal r is the reference signal that is an external signal and e is the output noise In control engineering this system is known as a feedback path closed loop system which is a typical closed loop system Bas e e e ee bbe tee ete H Stimulus Signal i r u l iY Reference Signal Go r Response Signal Plant Controller F Figure 6 3 A Feedback Path Closed Loop System National Instruments Corporation 6 3 System Identification Toolkit User Manual Chapter 6 Mode
86. hese results in graphs with the following three tools the Bode plot the Nyquist plot and the pole zero plot National Instruments Corporation 8 1 System Identification Toolkit User Manual Chapter 8 Analyzing Validating and Converting Models Bode Plot The Bode plot contains a Bode magnitude plot and a Bode phase plot The Bode magnitude plot describes magnitude against frequency and the Bode phase plot describes phase against frequency These two plots together describe the frequency response of the plant model you estimate as shown in Figure 8 1 Bode magnitude plot Bode Magnitude Upper Limit Lower Limit Magnitude 0 01 frequency Hz Bode phase plot 100 8 200 300 i o 3 confidence level a oo i 0 01 i 0 5 Frequency Hz Figure 8 1 A Bode Plot with Sample Data The SI Bode Plot VI also calculates the upper and lower limits according to the confidence level you set You can obtain information such as the gain of the system and the cutoff frequency by evaluating the Bode plot You can use the SI Bode Plot VI to produce the Bode magnitude and Bode phase plots You then can display the Bode magnitude and phase using the SI Bode Plot Indicator System Identification Toolkit User Manual 8 2 ni com Nyquist Plot Chapter 8 Analyzing Validating and Converting Models The Nyquist plot describes the gain and phase of a frequency response in polar coordinates by plotting
87. ibly costly Therefore you must think about the design of process prior to experimenting with various identification techniques The following sections describe the various data acquisition and system stimulation assumptions you must consider before identifying a system model These sections also provide information about the trade offs associated with each choice Accounting for Factors that Influence a System The key to the system identification process is having some knowledge of the system for which you want to identify a model This knowledge provides the basis for determining which signals are outputs which in turn determines sensor placement and which signals are inputs that you can use to excite the system Simple tests might be necessary to determine influences coupling time delays and time constants to aid in the modeling effort Also you need to consider signals that are not directly capable of being manipulated but still affect the system You need to include those signals as inputs to the system model For example consider the effect of wind gusts on the pitch dynamics of an airplane The airplane responds in pitch to the elevator angle as a direct input A wind gust affects the pitch of an airplane which in turn influences the dynamics of the airplane but the wind gust is not directly adjustable To create an accurate model of the airplane you might want to include wind gusts as an input variable Choosing a Stimulus Signa
88. ies using the same set of poles The ARMAX output error and Box Jenkins models use the resulting orders of the poles and zeros as the B and F model parameters and the first or second order models for the noise characteristics Refer to Chapter 8 Analyzing Validating and Converting Models for more information about pole zero plot and confidence intervals 5 Determine if additional signals influence the output if you cannot obtain a suitable model at this point You can incorporate measurements of these signals as extra input signals If you still cannot obtain a suitable model additional physical insight into the problem might be necessary Compensating for nonlinear sensors or actuators and handling important physical nonlinearities often are necessary in addition to using a ready made model From the prediction error standpoint the higher the order of the model is the better the model fits the data because the model has more degrees of freedom However you need more computation time and memory for higher orders The parsimony principle says to choose the model with the smallest degree of freedom or number of parameters if all the models fit the data well and pass the verification test The criteria to assess the model order therefore not only must rely on the prediction error but also must incorporate a penalty when the order increases Akaike s Information Criterion AIC Akaike s Final Prediction Error Criterion FPE and the
89. imation Methods u k Unknown System a Adaptive Model y k e k eye Figure 7 1 Recursive System Identification Diagram The stimulus signal u k is the input to both the unknown system and the recursive model The response of the system y k and the predicted response of the adaptive model y k are combined to determine the error of the system The error of the system is defined by the following equation e k y k y k 7 1 The adaptive model generates the predicted response 4 1 based on u k 1 after adjusting the parametric vector k based on the error e k Refer to the Adaptive Algorithms section of this chapter for more information about the definition of x in different algorithms Figure 7 1 shows how the error information e k is sent back to the adaptive model which adjusts the parametric vector k to account for the error You iterate on this process until you minimize the magnitude of the least mean square error e k Before you apply the recursive model estimation you must first select the parametric model structure that determines the parametric vector k Then you must select the method that automatically adjusts the parametric vector such that the error e k goes to the minimum The LabVIEW System Identification Toolkit provides Recursive Model Estimation VIs that support the following model structures e ARX e ARMAX e Output Error e Box Jenkin
90. imation by using that value as an initial guess The SI Estimate Partially Known State Space Model VI and the SI Estimate Partially Known Continuous Transfer Function Model VI perform optimization using the initial guess you provide These two VIs then use the upper and lower limit settings you specify as boundary constraints during the optimization process The initial guess you provide greatly affects the performance of any optimization technique Whether an optimization process reaches a global optimum depends on the initial guess With some initial guesses optimization processes might locate only a local optimum which is the smallest difference between the estimated output and the real output within a certain smaller range rather than in the whole range of interest Therefore National Instruments Corporation 5 5 System Identification Toolkit User Manual Chapter 5 Partially Known Model Estimation Methods to decrease the risk of locating a local optimum instead of the global optimum try different initial guesses Figure 5 2 shows an example of different estimations resulting from different initial guesses and illustrates the importance of setting different initial guesses to find the global optimum variable 1 a 0 02007 variable 2 Measured response Ww Estimated response global ca initial guess Response graph Estimated response local as A 1 5 upper limit A i Inf V lower limit E i Inf amp
91. imulates the real world plant by validating the model Refer to Chapter 4 Parametric Model Estimation Methods for more information about validating models Refer to Chapter 9 System Identification Case Study for an example of validating a model You can select Help Find Examples to display the NI Example Finder and then navigate to the Toolkits and Modules System Identification book to view more examples of using the System Identification Toolkit to estimate the partially known model of a plant System Identification Toolkit User Manual 5 10 ni com Model Estimation Methods in Closed Loop Systems Unlike the systems discussed in the previous chapters systems in many real world applications contain feedback Feedback is a process in which the output signal of a plant is passed or fed back to the input to regulate the next output Systems without feedback are open loop systems Systems with feedback are closed loop systems In an open loop system the stimulus signal and the output noise do not correlate with each other In a closed loop system the stimulus signal correlates to the output noise Though you can apply many open loop model estimation methods to closed loop data not all open loop model estimation methods handle the correlation between the stimulus signal and output noise well This chapter describes closed loop model estimation methods the corresponding assumptions and the advantages and disadvantages of these methods
92. ively As shown in the autocorrelation function R and the cross correlation function R described in the Correlation Analysis section the number of data points you need to compute R and R decreases as the lag T increases Therefore R and R become inaccurate for a large lag T When computing e and e you can apply a lag window w t to Ruu and R before conducting the FFT operation to improve the accuracy of the frequency response estimation as shown in the following equations N P e R pul T Wp Te t N N P e gt Ry TW T T N The lag window approaches zero when the lag T is large The window weighs out the points of R and R with large lag T thereby improving the accuracy of the frequency response estimation The SI Estimate Frequency Response VI uses a Hanning window as the lag window Refer to the book System Identification Theory for the User for more information about using a Hanning window 1 Ljung L 1999 System Identification Theory for the User 2nd ed Prentice Hall National Instruments Corporation 3 13 System Identification Toolkit User Manual Chapter 3 Nonparametric Model Estimation Methods Accuracy of the Lag Window The frequency response with the lag window G e is equivalent to the moving average version of the frequency response without the lag window G e The average smooths the frequency response but the smooth frequency respon
93. kit The state space model describes a system using difference or differential equations with an auxiliary state vector The following equations describe a discrete state space model x k 1 Ax k Bu k Ke k y k Cx k Du k e k The following equations describe a continuous state space model xX Ax t Bu t Ke t y Cx t Du t e t where xis the state vector k is the model sampling time multiplied by the discrete time step where the discrete time step equals 0 1 2 t is the time for the continuous model A is the system matrix that describes the dynamics of the states of the system B is the input matrix that relates the inputs to the states C is the output matrix that relates the outputs to the states D is the transmission matrix that relates the inputs to the outputs K is the Kalman gain The dimension of the state vector x is the only setting you need to provide for the state space model The state space transfer matrices A B C and D often reflect physical characteristics of a system The state space model is the most convenient model in describing multivariable systems State space models often are preferable to polynomial models especially in modern control applications that focus on multivariable systems The System Identification Toolkit provides the SI Estimate State Space Model VI to estimate discrete state space models using two methods the deterministic stochastic subspace and the
94. l The choice of stimulus signals has an important role in the observed system behavior and the accuracy of the estimated model These signals determine the operating points of the system While the system under test often limits the choice of signals you want an input signal to exhibit certain characteristics to produce a response that provides the information needed System Identification Toolkit User Manual 2 2 ni com Chapter 2 Acquiring and Preprocessing Data for developing an accurate model The following sections summarize these characteristics e To obtain meaningful dynamic behavior you must test the system under conditions similar to the actual operating conditions When you complete experiments in these conditions you identify the system in the same conditions under which you will implement the resulting model This criterion is extremely important for nonlinear systems e You want the inputs to the system under test to excite the system Exciting the system is dependent on the spectrum of the input signal Specifically you must excite the system with an input frequency similar to the frequency at which such inputs change during normal operations e You want the amplitude of the step input to cover a wide range of variations Therefore in the data you use for model estimation you need to cover the normal operation range of system inputs especially when you use the calculated model for model based control To cover the norm
95. l Estimation Methods in Closed Loop Systems In some cases the controller comes before the plant in a closed loop system This system is known as a feedforward path closed loop system as shown in Figure 6 4 r u Reference Signal LEO contoter i i Hl Signal Go sania Response Signal Fy Stimulus i Figure 6 4 A Feedforward Path Closed Loop System Depending on the amount of prior knowledge you have about the feedback the controller and the reference signal of a system you can categorize closed loop model estimation approaches into the following three groups Direct identification Uses the stimulus signal and the response signal to identify the plant model as if the plant is in an open loop system You can apply the direct identification approach to compute many models such as the general linear polynomial state space transfer function and zero pole gain models Indirect identification Identifies a closed loop system using the reference signal and the response signal and then determines the plant model based on the known controller of the closed loop system You can apply the indirect identification approach to compute transfer function models Joint input output identification Considers the stimulus signal and the response signal as outputs of a cascaded system The reference signal and the noise jointly perturb the system and the plant model is identified from this join
96. ledge you have about the parameters K T4 T w and r to the static gain delay s T s natural freq rad s and damping ratio inputs respectively of the SI Create Partially Known Continuous Transfer Function Model VI by defining the initial guesses and upper and lower limits Refer to the Transfer Function Model section of Chapter 4 Parametric Model Estimation Methods for more information about continuous transfer function models With the System Identification Toolkit and a partially known model you can set constraints on each parameter of a state space or continuous transfer function model in two ways with an upper and lower limit or with an initial guess Setting Parameter Constraints with a Range If you have prior knowledge of a parameter you can set constraints by providing upper and lower limits for the parameter With the limit range the SI Estimate Partially Known State Space Model VI randomly selects a value within the range as an initial guess of the parameter From this initial value the VI then performs optimization to minimize the difference between the estimated output and the measured real output The goal of constraint optimization is to find a global optimum or the smallest System Identification Toolkit User Manual 5 4 ni com Chapter 5 Partially Known Model Estimation Methods difference between the estimated output and the real output with parameters of physical meaning Successfully finding the global
97. levels in these cases You can combine multiple binary signals of different levels to form the stimulus signal System Identification Toolkit User Manual 2 4 ni com Chapter 2 Acquiring and Preprocessing Data Figure 2 2 shows an example of a random binary signal Random Binary Sequence 2 1 55 1 0 5 0 0 5 1 1 5 2 1 1 1 1 1 1 1 1 0 10 20 30 40 50 60 70 80 90 100 Sample Amplitude Figure 2 2 Random Binary Sequence Pseudo Random Binary Sequence A Pseudo Random Binary Sequence also known as Maximal Length Sequence MLS is a periodic deterministic signal with properties similar to white noise You often generate a pseudo random binary sequence using an n bit shift register with feedback through an exclusive OR function While appearing random the sequence actually repeats every 2n 1 values When using a whole period the pseudo random binary sequence has special mathematical advantages that make it attractive as a stimulus signal In particular you can attribute variations in response signals between two periods of the stimulus to noise due to the periodic nature of the signal Also like the white random binary noise the pseudo random binary sequence has a low crest factor Cy You can use the SI Generate Pseudo Random Binary Sequence VI to generate a Pseudo Random Binary Sequence National Instruments Corporation 2 5 System Identification Toolkit User Manual Chapter 2 Acquiring
98. lgorithm References manual SIreference pdf available in the Labview manuals directory for more information about the least squares method National Instruments Corporation 3 3 System Identification Toolkit User Manual Chapter 3 Nonparametric Model Estimation Methods Correlation Analysis The correlation analysis method uses the cross correlation between the input and output signals as an estimation of the impulse response as shown by the following equation co y k u k n h n e k k 0 The input signal must be zero mean white noise with a spectral density that is equally distributed across the whole frequency range The SI Estimate Impulse Response VI can prewhiten input signals that are not white noise Assuming the input u k of the system is a stationary stochastic process and statistically independent of the disturbance e k the following equation is true R T Y Rulk DAK k 0 R represents the cross correlation function between the stimulus signal u k and the response signal y k as defined by the following equation l N max t 0 1 Ryl SL e tut k min t 0 R u represents the autocorrelation of the stimulus signal u k as defined by the following equation N t 1 Rud Yue talk k 0 N is the number of data points If the stimulus signal is a zero mean white noise signal the autocorrelation function reduces to the following equation R t 6 8 t where is th
99. lly known model Comparing Unknown and Partially Known Model Estimation Methods When you use the model estimation methods described in previous chapters you assume systems are unknown However in practice many systems are partially known because you have information about the underlying dynamics or some of the physical parameters Unknown and partially known models also are known as black box and grey box models respectively Conventional black box model estimation methods do not use the prior knowledge you have about a model When estimating a black box model you cannot set constraints on parameters You can set only the model order that specifies the number of parameters to calculate With black box estimation methods you use either an algorithm or a trial and error method to vary model parameters until the behavior of the model matches the National Instruments Corporation 5 1 System Identification Toolkit User Manual Chapter 5 Partially Known Model Estimation Methods measured input output data You can use the estimated parameters to reproduce the response of the system accurately However the parameters might not have any physical meanings With grey box model estimation methods you can incorporate prior knowledge of a system by setting constraints on certain parameters These constraints reflect the knowledge you have of the physical system You therefore can obtain a more realistic parameter estimation Such constraint
100. luding photocopying recording storing in an information retrieval system or translating in whole or in part without the prior written consent of National Instruments Corporation National Instruments respects the intellectual property of others and we ask our users to do the same NI software is protected by copyright and other intellectual property laws Where NI software may be used to reproduce software or other materials belonging to others you may use NI software only to reproduce materials that you may reproduce in accordance with the terms of any applicable license or other legal restriction Trademarks National Instruments NI ni com and LabVIEW are trademarks of National Instruments Corporation Refer to the Terms of Use section on ni com legal for more information about National Instruments trademarks Other product and company names mentioned herein are trademarks or trade names of their respective companies Members of the National Instruments Alliance Partner Program are business entities independent from National Instruments and have no agency partnership or joint venture relationship with National Instruments Patents For patents covering National Instruments products refer to the appropriate location Help Patents in your software the patents txt file on your CD or ni com patents WARNING REGARDING USE OF NATIONAL INSTRUMENTS PRODUCTS 1 NATIONAL INSTRUMENTS PRODUCTS ARE NOT DESIGNED WITH COMPONENTS AND TESTI
101. m between every two sampling points Continuous system models are analog You derive continuous models of a physical system from differential equations of the system The coefficients of continuous models have clear physical meanings For example you can derive the continuous transfer function of an RC circuit if you know the details of the circuit The coefficients of the continuous transfer function are the functions of R and C in the circuit You use continuous models if you need to match the coefficients of a model to some physical components in the system Discrete system models are digital You derive discrete models of a physical system from difference equations or by converting continuous models to discrete models In computer based applications signals and operations are digital Thus you can use discrete models to implement a digital controller or to simulate the behavior of a physical system at discrete instants You also can use discrete models in the accurate model based design of a discrete controller for a plant Model Representations You can use the System Identification Toolkit to represent dynamic system models in the following four categories general linear polynomial transfer function zero pole gain and state space Refer to Chapter 4 Parametric Model Estimation Methods for more information about these models System Identification Toolkit User Manual 1 6 ni com Chapter 1 Introduction to System Identification
102. model estimation method then can estimate T z because r k and e k are uncorrelated signals After you obtain the value of T z you can compute k 7T z r k You then can represent u k as follows u k u k F z T z e k 6 7 Using Equation 6 7 you obtain an input signal i k which is constructed from r k and is uncorrelated with the measurement noise 2 By manipulating Equation 6 1 you can rewrite y k as follows yk Go z u k To z e k Because i k is uncorrelated with e k the original closed loop model estimation problem between u k and y k becomes an open loop problem between i k and y k System Identification Toolkit User Manual 6 8 ni com Chapter 6 Model Estimation Methods in Closed Loop Systems You use the same methodology to compute y k for a feedforward path closed loop system where F Z Plz 1 G F You rewrite y k as follows y k u k Go z 1 To 2 Go 2 le k The two stage method does not require you to know anything about the feedback or the controller structure and controller parameters Also you treat the closed loop model estimation as an open loop model estimation within each of the two steps Therefore you can use any method that works with open loop models Whether the real world output noise is white noise or color noise the two stage method provides reliable estimations Using System Identification Vis for Model Estimation If you want to use the direct i
103. ndancy in the data If a pole and a zero overlap the pole and zero cancel out each other which indicates the estimated optimal order is too high The pole zero plot graph in Figure 9 11 shows a pole zero plot with three overlapping pole zero pairs Due to numerical error it is unlikely that a zero and a pole perfectly overlap You can use the confidence region to justify whether the pole and the zero cancel out each other National Instruments Corporation 9 9 System Identification Toolkit User Manual Chapter 9 System Identification Case Study pole zero plot Pole EPI Zero can Pole Confidence piser Zero Confidence area Unit Circle Axis QS imaginary optimal orders Figure 9 11 Pole Zero Plot for an ARX Model Because there are three pole zero pairs you can conclude that the AIC criterion does not produce the most optimal orders Minimum Description Length Criterion Because the AIC criterion produced a model with non optimal orders you can try estimating the model orders with the MDL criterion By using the MDL criterion the lowest prediction error corresponds to a model dimension of 12 as shown in Figure 9 9 The SI Estimate Orders of System Model VI returns the following optimal orders e A order 6 e Border 6 e delay 0 System Identification Toolkit User Manual 9 10 ni com Chapter 9 System Identification Case Study Figure 9 12 shows a pole zero plot of a model with a model dimension of 1
104. ngth criterion Sometimes the results you obtain with these three criteria might be inconsistent You can use a pole zero plot for further investigation and to verify the results of the order estimation Refer to the Akaike s Information Criterion section the Final Prediction Error Criterion section and the National Instruments Corporation 9 7 System Identification Toolkit User Manual Chapter 9 System Identification Case Study Minimum Description Length Criterion section of Chapter 4 Parametric Model Estimation Methods for more information about these criteria Figure 9 9 shows a prediction error plot generated by the SI Estimate Orders of System Model VI for an ARX model The y axis is the prediction error and the x axis is the model dimension The three different color bars on the chart represent the FPE AIC and MDL criteria prediction error 100 FPE AIC 10 MDL 1 Pred Error 0 1 0 01 1 1 1 1 7 1 1 1 1 2 A 6 B I0 i2 i i6 1820 Model Dimension Prediction Error Figure 9 9 Prediction Error Plot for an ARX Model The following sections show you how to use the AIC MDL and a user defined criterion to determine the A and B orders of the ARX model Akaike s Information Criterion System Identification Toolkit User Manual 9 8 The block diagram in Figure 9 10 uses the SI Estimate Orders of System Model VI for order estimation To estimate the orders of a model the SI Estimate Orders
105. nominator from equaling zero when lec Wl approaches zero The step size u k is time varying because the step size changes with the time index k Substituting u k into the parametric vector k equation yields the following equation gt gt 2 Wk 1 W k u k je k o k Compared to the LMS algorithm the NLMS algorithm is always stable if the step size u k is between zero and two regardless of the statistical property of the stimulus signal u k National Instruments Corporation 7 5 System Identification Toolkit User Manual Chapter 7 Recursive Model Estimation Methods The procedure of the NLMS algorithm is the same as the LMS algorithm except for the estimation of the time varying step size U k Recursive Least Squares The recursive least squares RLS algorithm and Kalman filter algorithm use the following equations to modify the cost function J k defined in the Adaptive Algorithms section Refer to the Kalman Filter section for information about the Kalman filter algorithm N 1 Jk Ele k iy e k i i 0 Compare this modified cost function which uses the previous N error terms to the cost function J k E e k which uses only the current error information e k The modified cost function J k is more robust The corresponding convergence rate in the RLS algorithm is faster but the implementation is more complex than that of LMS based algorithms The following procedure describes how to implement t
106. nparametric Model Estimation Methods Chapter 4 Parametric Model Estimation Methods Chapter 5 Partially Known Model Estimation Methods and Chapter 6 Model Estimation Methods in Closed Loop Systems use nonrecursive methods to estimate a model of the plant in a system Nonrecursive model estimation identifies a model for a plant based on input output data gathered at a time prior to the current time However many real world applications such as adaptive control and adaptive prediction having a model of the system update while the system is running is necessary or helpful In this type of application you obtain the mathematical model of the system in real time Recursive model estimation is a common system identification technique that enables you to develop a model that adjusts based on real time data coming from the system Recursive model estimation processes the measured input output data recursively as the data becomes available This chapter discusses recursive model estimation techniques and various adaptive algorithms associated with each method Defining Recursive Model Estimation Figure 7 1 represents a general recursive system identification application A system identification application consists of an unknown system that has an input signal or stimulus signal u k and an output signal or response signal y k National Instruments Corporation 7 1 System Identification Toolkit User Manual Chapter 7 Recursive Model Est
107. o not have prior knowledge about programming in LabVIEW you can use the System Identification Assistant to develop a model that reflects the behavior of a certain dynamic system You access the System Identification Assistant through the NI Express Workbench The Express Workbench is a framework that can host multiple interactive National Instruments tools and assistants Using the System Identification Assistant you can create a project that encompasses the whole system identification process In a single project you can load or acquire raw data into the System Identification Assistant preprocess the data estimate a model that describes the system and then validate the accuracy of the model The Express Workbench provides windows in which you can see the raw data the response data the National Instruments Corporation 1 7 System Identification Toolkit User Manual Chapter 1 Introduction to System Identification estimated model the validation results and the mathematical equations that describe the model After creating a project in the Express Workbench you can convert the project to a LabVIEW block diagram and customize the block diagram in LabVIEW This conversion enables you to enhance the capabilities of the application Refer to the MI Express Workbench Help available in the NI Express Workbench environment by selecting Help Express Workbench Help for more information about using the assistant to develop models Syst
108. o verify the ARX model created in the User Defined Criterion section SI Lowpass Filter vi SI Down Sampling vi run SISO WDT Y cutoff frequenc decimation factor ist portion 3 10 Figure 9 15 Simulation amp Prediction with ARX Model VI Estimate ARX Model vi ISI Model Simulation vi SO WDT 7 Model P rediction vi The simulation and 1 step ahead prediction graphs enable you to visually determine how accurate the model is Figure 9 16 shows the results of the simulation and prediction as well as the actual response of the system System Identification Toolkit User Manual 9 14 ni com Chapter 9 System Identification Case Study simulated simulation measured simulated 1 step ahead prediction measured Figure 9 16 Simulation and Prediction Graphs for an ARX Model Notice how the actual response or the measured response is different from the simulated response in the simulation graph The SI Model Simulation VI simulates the response of the system without considering the actual response of and the noise dynamics in the system Residual Analysis In addition to simulation and prediction you can perform a residual analysis to validate the system model Residual analysis tests whether the
109. of System Model VI requires two data sets one for estimation and one for validation You do not need to acquire two data sets from a system rather you can partition one data set into two using the SI Split Signals VI The SI Split Signals VI divides the preprocessed data samples into a portion for model estimation and a portion for model validation ni com Chapter 9 System Identification Case Study In Figure 9 10 the 1st portion is 66 which means the SI Estimate Orders of System Model VI will use 66 of the data samples for estimation and the remainder of the data samples for validation SI Data Samples vi Figure 9 10 Estimate Orders of ARX Model VI The SI Estimate Orders of System Model VI generates the prediction error plot for the ARX model and the optimal A order and B order based on the AIC criterion By using the AIC criterion the lowest prediction error corresponds to a model dimension of 19 as shown in Figure 9 9 For an ARX model the model dimension is equal to the sum of the A order B order and delay values The SI Estimate Orders of System Model VI returns the following optimal orders e lt Aorder 9 Border 10 e delay 0 Verifying the Results After determining the orders of the model you want to verify the results to ensure the model accurately describes the system One method is to plot a pole zero map and visually inspect the plot to determine whether there is any redu
110. ole zero plot and display this plot using the SI Pole Zero Plot Indicator You also can use the pole zero plot to determine if you can reduce model orders By observing the pole zero placements you can determine if any pole zero pairs have overlapping confidence intervals A confidence interval is a region the SI Pole Zero Plot VI calculates from the confidence level you set The existence of overlapping confidence intervals implies that pole zero cancellations exist and that the model order might be unnecessarily high The pole zero plot shown in Figure 8 3 is an optimal model with the appropriate order because the pole zero pairs do not have overlapping confidence intervals If the model order is too high you can try reducing the model order You then can use the F test criterion to assess if the reduction in model order leads to a significant increase in the prediction error If the reduction in System Identification Toolkit User Manual 8 4 ni com Chapter 8 Analyzing Validating and Converting Models model order leads to a significant increase in the prediction error do not reduce the model order Validating Models Model estimation determines the best model of the system within the chosen model structure Model estimation does not determine if the model provides the most accurate description of the system After you obtain a model you must validate the model to determine how well the behavior of the model corresponds to the data yo
111. oss correlation falls within the confidence range the value is insignificant and you can consider this value to be equal to zero National Instruments Corporation 8 7 System Identification Toolkit User Manual Chapter 8 Analyzing Validating and Converting Models Converting Models With the System Identification Toolkit you can apply the Model Conversion VIs to convert system models from one representation to another from continuous to discrete models and from discrete to continuous models You can convert models you created in this toolkit to models you can use in another toolkit You also can convert a model after you estimate the model or after you analyze or validate the model You can use the Model Conversion VIs to switch between different model representations or types For example when estimating a digital system you can convert an existing continuous model to a discrete model to approximate the real time behavior of the system You do not need to create a new discrete model for the digital system Using the Model Conversion VIs you also can convert models you create in the System Identification Toolkit into transfer function zero pole gain or state space models that you then can use with the LabVIEW Control Design Toolkit This model conversion process enables you to identify a model for an unknown system with the System Identification Toolkit and then design a controller for this system using the Control Design Toolki
112. r more information about feedback closed loop systems and feedback detection National Instruments Corporation 3 11 System Identification Toolkit User Manual Chapter 3 Nonparametric Model Estimation Methods Frequency Response In theory the results from impulse response estimation and the results from frequency response estimation are equivalent For example the Fourier transform of the impulse response h n which you can compute using impulse response estimation equals the frequency response G e However this equivalence does not hold in most real world applications because of different preprocessing schemes in impulse response estimation and frequency response estimation The frequency response provides the complete frequency domain characteristics of the system including the passband and the natural frequency of the system A sinusoidal input signal has the following general form u t sin Mot For a linear time invariant system the response of a linear time invariant system to a sinusoidal input also is a sinusoidal signal but potentially with a different magnitude and phase as shown in the following equation y t bsin ot 8 where b and are the magnitude and phase respectively of the frequency response of the system to an input sinusoidal frequency If you apply input signals with a number of sinusoids at different frequencies then you can obtain an estimate of the frequency response G of the
113. rate than parametric estimation However you can use a nonparametric model estimation method to obtain useful information about a system before applying parametric model estimation For example you can use nonparametric model estimation to determine whether the system requires preconditioning what the time delay of the system is what model order to select and so on You also can use nonparametric model estimation to verify parametric models For example you can compare the Bode plot of a parametric model with the frequency response of the nonparametric model Refer to Chapter 4 Parametric Model Estimation Methods for more information about parametric model estimation methods The LabVIEW System Identification Toolkit uses least squares method and correlation analysis method to estimate the impulse response and spectral analysis method to estimate the frequency response The following sections describe the impulse response and frequency response methods National Instruments Corporation 3 1 System Identification Toolkit User Manual Chapter 3 Nonparametric Model Estimation Methods Impulse Response An impulse input as shown in Figure 3 1 to a dynamic system is defined differently depending on whether the system is discrete or continuous For acontinuous dynamic system an impulse input is a unit area signal with an infinite amplitude and infinitely small duration occurring at a specified time At all other times the input signal
114. real world systems include saturation dead zone friction backlash and quantization effects relays switches and rate limiters Many real world systems are nonlinear but you can simulate most real world systems with linear models to simplify a design or analysis procedure Time Variant versus Time Invariant Models Dynamic system models are either time variant or time invariant The parameters of a time variant model change with time For example you can use a time variant model to describe an automobile As fuel burns the mass of the vehicle changes with time National Instruments Corporation 1 5 System Identification Toolkit User Manual Chapter 1 Introduction to System Identification Conversely the parameters of a time invariant model do not change with time For an example of a time invariant model consider a simple robot Generally the dynamic characteristics of robots do not change over short periods of time Continuous versus Discrete Models Dynamic system models are either continuous or discrete Both continuous and discrete system models can be linear or nonlinear and time invariant or time variant Continuous models describe how the behavior of a system varies continuously with time which means you can obtain the properties of a system at any certain moment from the continuous model Discrete models describe the behavior of a system at separate time instants which means you cannot obtain the behavior of the syste
115. s e General Linear System Identification Toolkit User Manual 7 2 ni com Chapter 7 Recursive Model Estimation Methods The Recursive Model Estimation VIs have a recursive method parameter that enables you to specify which recursive estimation method to use Refer to the Parametric Model Representations section of Chapter 4 Parametric Model Estimation Methods for information about each of these models Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for more information about the Recursive Model Estimation VIs and mathematical definitions of each of these models You can compute each model recursively using the following four types of adaptive algorithms e Least mean square LMS e Normalized least mean squares NLMS e Recursive least squares RLS e Kalman filter KF The following sections provide more information about each of these adaptive algorithms Adaptive Algorithms Adaptive algorithms are fundamental in recursive system identification The adaptive method you use affects the performance of recursive system identification application The goal of all recursive algorithms is to adjust the parametric vector k until you minimize the cost function J k The following equation defines the cost function J k Ik Ele k where EF is the expectation Again e k represents the difference between the predicted response y k and the response y k of the unknown system as
116. s usually follow one of the following guidelines e A parameter must be as close to a value as possible e A parameter must be between two values Two or more parameters must correlate to each other With these constraints you can specify the system model more accurately Parameter constraints increase the possibility of the System Identification VIs locating the optimal parameters that describe the real world model Parameter constraints also improve the accuracy of locating these optimal parameters Defining and Estimating Partially Known Models Before estimating partially known models you first must define those models Using prior knowledge you choose a model for the plant in a system and set parameter constraints for the model You then can estimate the model to represent the real world plant The LabVIEW System Identification Toolkit provides two VIs with which you can define partially known models the SI Create Partially Known State Space Model VI and the SI Create Partially Known Continuous Transfer Function Model VI The SI Create Partially Known State Space Model VI enables you to define partially known continuous or discrete state space models Refer to the State Space Model section of Chapter 4 Parametric Model Estimation Methods for more information about state space model definitions System Identification Toolkit User Manual 5 2 ni com Chapter 5 Partially Known Model Estimation Methods You can use the SI Cr
117. s in the labview vi 1lib addons System Identification User Defined Model Templates 11b directory Then you can estimate the model you define using the SI Estimate User Defined Model VI The SI Estimate User Defined Model VI enables you to estimate some other model representations in addition to the general linear polynomial transfer function zero pole gain and state space models that the System Identification Toolkit directly supports For example you can use this VI to estimate nonlinear models With this VI you also can estimate your own linear models that you define early Refer to the Estimate Hammerstein Model example VI the Estimate Hammerstein Wiener Model example VI the Estimate Wiener Model example VI and the Parameterize Nonlinear Differential Equation example VI for more information about using the SI Estimate User Defined Model VI to estimate nonlinear models You can access these example VIs by selecting Help Find Examples to display the NI Example Finder and then navigating to the Toolkits and Modules System Identification folder Refer to the LabVIEW Help for more information about the SI Estimate User Defined Model VI National Instruments Corporation 4 11 System Identification Toolkit User Manual Chapter 4 Parametric Model Estimation Methods Polynomial Models versus State Space Models Selecting the correct model type and model order is crucial for successfully estimating a parametric model In general state space
118. s model is stable and not redundant Using these model orders you now can estimate and verify the system model ARX Model Validation The goal of model validation is to determine whether or not the estimated model accurately reflects the actual system Using the model orders found in the User Defined Criterion section you can simulate and predict the response of the system You can compare these responses to the actual response and determine the accuracy of the estimated model You also can analyze the residuals to determine the accuracy of the estimated model Refer to the Validating Models section of Chapter 8 Analyzing Validating and Converting Models for more information about validating a model The following sections describe how to apply these techniques to model validation National Instruments Corporation 9 13 System Identification Toolkit User Manual Chapter 9 System Identification Case Study Simulation and Prediction You can use the SI Model Simulation VI and SI Model Prediction VI to determine the accuracy of the estimated model The SI Model Simulation VI simulates the system model and the SI Model Prediction VI performs a prediction of the system model The results of the SI Model Prediction VI might differ from the SI Model Simulation VI because the SI Model Prediction VI periodically makes corrections to the estimated response based on the actual response of the system Figure 9 15 shows how you use these VIs t
119. s reserved Important Information Warranty The media on which you receive National Instruments software are warranted not to fail to execute programming instructions due to defects in materials and workmanship for a period of 90 days from date of shipment as evidenced by receipts or other documentation National Instruments will at its option repair or replace software media that do not execute programming instructions if National Instruments receives notice of such defects during the warranty period National Instruments does not warrant that the operation of the software shall be uninterrupted or error free A Return Material Authorization RMA number must be obtained from the factory and clearly marked on the outside of the package before any equipment will be accepted for warranty work National Instruments will pay the shipping costs of returning to the owner parts which are covered by warranty National Instruments believes that the information in this document is accurate The document has been carefully reviewed for technical accuracy In the event that technical or typographical errors exist National Instruments reserves the right to make changes to subsequent editions of this document without prior notice to holders of this edition The reader should consult National Instruments if errors are suspected In no event shall National Instruments be liable for any damages arising out of or related to this document or the information cont
120. se also can deviate more from the true frequency response Adjusting the length of the lag window can balance the trade off between variance and bias of the frequency response estimation The larger the length of the lag window the fewer points of G e the SI Estimate Frequency Response VI averages to compute G e and hence the larger the variance and the smaller the bias of the frequency estimation The following example demonstrates how the length of the lag window affects the frequency response estimation Figure 3 7 shows the front panel of a VI that simulates a system defined by the following equation y k 1 46y k 1 2 5y k 2 1 46y k 3 t yk 4 uk 0 45u k 1 u k 2 Estimated FRF e Magnitude response Ideal FRF T 50 2 0 Magnitude 50 75 100 i 0 000488281 0 2 0 4 a5 Frequency window length J 2048 Figure 3 7 Front Panel of Lag Window Example VI System Identification Toolkit User Manual 3 14 ni com Chapter 3 Nonparametric Model Estimation Methods Figure 3 8 shows the block diagram of this VI Simulate the system 1 4 01452 1 42 2 IH 2 1 1 462 14 2 52 2 1 462 JH2 4 Gaussian White Noise vi Estimate the frequency response Generate a sweep signal Freq in log fee j SI Estimate Frequency Response vi IIR Filter vi Figure 3 8
121. shown in Figure 7 1 When the cost function J k is sufficiently small the parametric vector w x is considered optimal for the estimation of the actual system National Instruments Corporation 7 3 System Identification Toolkit User Manual Chapter 7 Recursive Model Estimation Methods Least Mean Squares The least mean squares LMS method uses the following equations to define the cost function J k E e k The parametric vector k updates according to the following equation B k 1 Bb pelk Olk kis the number of iterations u is step size which is a positive constant and k is the data vector from the past input data u k and output data y k k is defined by the following equation PW y t 1 yt k u t iout The following procedure describes how to implement the LMS algorithm 1 Initialize the step size u 2 Initialize the parametric vector W k using a small positive number gt T w 0 3 Initialize the data vector O k lu k y i 2 T 0 0 0 0 gt gt 4 Fork 1 update the data vector k based on k 1 and the current input data u k and output data y k 5 Compute the predicted response y k using the following equation HU OTC 6 Compute the error e k by solving the following equation elk y k y k 7 Update the parameter vector k 1 K we HO H 8 Stop if the error is small enough else set
122. ssctadesisssntascdsasees sbeasessscaesesapcnsedagsavstscosessucescdescasezeabecsais 9 13 Simulation and Prediction 0 0 eee cee eseeseeeseeseeseeeseeseeesecseeesesseseaeeseesaeeaeees 9 14 Residtial Analysis ccs sssse csseschoi stant eseseaeveni es tiseaceseeeenedagiaietieeaes qedetestastaoeeaiets 9 15 Estimating a State Space Model eee eeceesescceseceeceseeseeesessecesecseesseceeesesaesneeeseenaes 9 18 Finding the Singular Values cece eeceesecseceseeseeeseeeceseeaeeseeeaecnsenaeeseeeaee 9 18 Validating the Estimated State Space Model eee eeeeseeeecereeseeeeesees 9 19 Additional Examples inss eninin ni a aaea caveeced eseastg tte ast 9 20 Appendix A References Appendix B Technical Support and Professional Services System Identification Toolkit User Manual viii ni com About This Manual Conventions The LabVIEW System Identification Toolkit provides a library of VIs and an assistant for developing models of a system based on a large set of raw data Both tools enable you to complete the entire system identification process from analyzing the raw data to validating the identified model This manual discusses the main steps in the system identification process and how to use the System Identification VIs to create applications that can accomplish the various tasks in the process Refer to the LabVIEW Help available in LabVIEW by selecting Help Search the Lab VIEW Help for more information about the steps in the assistant an
123. stem is more important than the stochastic part Therefore you can take advantage of the relationship between input and output signals of the transfer function model to describe the deterministic part of the system Zero Pole Gain Model If you rewrite the equations for the transfer function model to show the locations of the zeroes and poles of the dynamic system you obtain the zero pole gain model The following equations represent the continuous and discrete zero pole gain models respectively Gia K s 2 s Z 8 Z s p S P S Py K z 2Z Z Zy Z Zp G z Z p Z p2 8 p where K is the gain z are the zeroes and p are the poles The System Identification Toolkit does not provide a VI to estimate zero pole gain models directly because you can use the SI Model Conversion VI to convert another model representation to a zero pole gain model Refer to Chapter 8 Analyzing Validating and Converting Models for more information about converting other model representations to zero pole gain models Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for more information about the SI Model Conversion VI National Instruments Corporation 4 9 System Identification Toolkit User Manual Chapter 4 Parametric Model Estimation Methods State Space Model In addition to polynomial models you also can estimate state space models with the System Identification Tool
124. t Refer to the Model Based Control Design Process section of Chapter 1 Introduction to System Identification for more information about the integration of the System Identification Toolkit and the Control Design Toolkit in the model based control design process Refer to the National Instruments Web site at ni com for more information about the Control Design Toolkit Refer to the LabVIEW Help available by selecting Help Search the Lab VIEW Help for more information about the VIs described in this chapter System Identification Toolkit User Manual 8 8 ni com System Identification Case Study This chapter contains a case study that guides you through the system identification process The case study uses sample data that the LabVIEW System Identification Toolkit provides in the SI Data Samples VI The SI Data Samples VI includes data sets for a DC motor a flexible robot arm a ball and beam apparatus an RC circuit and so on The case study in this chapter uses the flexible arm data to demonstrate the system identification process and to compare different estimation methods The flexible arm is a nonlinear dynamic system The System Identification Toolkit enables you to build models for systems linearly This chapter guides you through obtaining a linear representation of a nonlinear system The VIs for this case study are located in the labview examples system identification SICaseStudy1 11b Refer to the LabVIEW Help av
125. t input output system You can apply the joint input output identification approach to compute transfer function models You can choose a suitable model identification approach according to the information you have about the closed loop system Table 6 1 summarizes the information you must have to use each identification approach System Identification Toolkit User Manual 6 4 ni com Table 6 1 Required Information for Closed Loop Identification Approaches Chapter 6 Model Estimation Methods in Closed Loop Systems Input Output v Vv J Stimulus Signal Response Signal pire ae sabres Direct J J Indirect J J J Joint With the LabVIEW System Identification Toolkit you can choose to use the direct indirect or joint input output identification approaches for different types of closed loop systems The direct identification approach supports single input single output SISO multiple input single output MISO and multiple input multiple output MIMO systems The indirect and joint input output identification approaches support SISO systems only The following sections discuss the algorithms and assumptions of each closed loop identification approach in detail Direct Identification National Instruments Corporation 6 5 If the stimulus and response signals of a closed loop system are available but you do not have any other information about the system you can use only the
126. tage of model prediction to improve control performance For example model predictive control uses some of the prediction properties of a model to determine if a particular limitation or constraint is active in the future This method allows the controller to take preventive actions before such constraints become active If you have the measured input and output of a system you also can validate the model of the system by comparing the predicted output and the measured output If the prediction error is small the model is acceptable Residual Analysis Residual analysis is the third validation method that the System Identification Toolkit provides The response that an estimated model predicts and the actual response from the system are different This difference is called the prediction error or residual The following equation defines the residual e k e k y k y k y k is the measured output and y k is the output from the one step ahead prediction If the model is capable of describing the true system the residual is zero mean white noise and independent of the input signal You can use autocorrelation analysis to test if the residual is zero mean white noise You can use cross correlation analysis to test if the residual is independent of the input signal The SI Model Residual Analysis VI calculates both the autocorrelation and the cross correlation values System Identification Toolkit User Manual 8 6 ni com Chapter 8 An
127. techniques developed for open loop models to estimate the closed loop system However you cannot apply all open loop identification methods to estimate the model of a plant in a closed loop system Some open loop model identification methods assume zero correlation between the stimulus signal and output noise In closed loop systems this correlation is nonzero Thus if you use certain open loop model estimation methods such as the instrument variable IV method and the correlation analysis methods with closed loop data you might estimate a model incorrectly You can use the prediction error method to identify the plant in a closed loop system The direct identification approach is used commonly in real world applications This approach is convenient because you do not need to have additional information about a closed loop system such as the reference signal or the controller However the estimation might not be accurate if the model type you select for a plant does not describe the output noise of the system accurately For example if the output noise of a plant is color noise and you select an OE model which assumes the output noise is white System Identification Toolkit User Manual Chapter 6 Model Estimation Methods in Closed Loop Systems noise the estimation for the OE model might be biased when you use direct identification The bias might be small though if the signal to noise ratio SNR of the system is high Indirect Iden
128. ter and downsample the data set to reduce the amount of data in the raw data set for simpler identification Examining the Time Response Data Using the data in the SI Data Sample VI for the flexible robotic arm you can view the input and output data as shown in Figure 9 1 SI Data Samples vi stimulus signal torque Flexible Arm Y 5150 woT Y ma Figure 9 1 Flexible Arm Data Set VI B Note The names of the block diagram figures in this chapter reflect the names of the example VIs located in the labview examples system identification SICaseStudy1 11b The stimulus signal torque output corresponds to the input data or the torque and the response signal acceleration output corresponds to the output data or the acceleration Figure 9 2 shows the input and output data on graphs during the length of the response By looking at the graphs you can inspect the data for outliers clipped saturation or quantization effects that you can remove because they are not representative of the system behavior System Identification Toolkit User Manual 9 2 ni com Chapter 9 System Identification Case Study response signal acceleration sional Li 6 8 192 Figure 9 2 Flexible Arm Data Set Plotted in the Time Domain Figure 9 2 shows no obvious nominal trend or outlier values in the input or output time waveforms Examining the Frequency Response Data In addition
129. ter numerical problems in computation Determining Parameters for the Prediction Error Method The identification method for most of the polynomial models is the prediction error method Determining the delay and model order for the prediction error method is typically a trial and error process The following steps can help you obtain a suitable model These steps are not the only methods you can use nor are these steps a comprehensive procedure 1 Obtain useful information about the model order by observing the number of resonance peaks in the nonparametric frequency response function Normally the number of peaks in the magnitude response equals half the order of A z F z 1 Kailath T 1980 Linear systems Prentice Hall System Identification Toolkit User Manual 4 12 ni com Chapter 4 Parametric Model Estimation Methods 2 Obtain a reasonable estimate of the delay by observing the impulse response or by testing reasonable values in a medium sized ARX model Choose the delay that provides the best model fit based on prediction errors or another criterion 3 Test various ARX model orders with this delay choosing those orders that provide the best fit 4 Reduce the model order by plotting the poles and zeros with confidence intervals and looking for potential cancellations of pole zero pairs The resulting model might be unnecessarily high in order because the ARX model describes both the system dynamics and noise propert
130. th tools enable you to complete the entire system identification process from analyzing raw data to validating the identified model National Instruments Corporation 1 1 System Identification Toolkit User Manual Chapter 1 Introduction to System Identification This chapter provides an overview of the model based control design process and the steps in the process where you can use National Instruments software and hardware This chapter also describes various representations of physical models that you can use to identify the plant model of a system Finally this chapter provides information about the two tools in the System Identification Toolkit that enable you to identify system models Model Based Control Design Process The model based control design process involves modeling a plant analyzing and synthesizing a controller for the plant simulating the plant and controller and deploying the controller While the System Identification Toolkit provides solutions for analyzing raw data and creating plant models National Instruments also provides solutions for the other three components in the process as shown in Figure 1 1 Plant Modeling Control Design Simulation Deployment and Analysis LabVIEW System e gt LabVIEW lt gt LabVIEW LabVIEW Identification Control Design Simulation Real Time Toolkit Toolkit Module Module LabVIEW Figure 1 1 National Instruments Tools for the Model
131. the orders of the numerator and denominator might change after National Instruments Corporation 6 7 System Identification Toolkit User Manual Chapter 6 Model Estimation Methods in Closed Loop Systems manipulation Thus the SI Estimate Transfer Function Model VI or the SI Transfer Function Estimation Express VI which you can use with the indirect identification approach might return an error regarding the mismatch between the order you set and the order of the estimated model In this case you must adjust the tolerance setting of these two VIs so that the numerator and denominator orders match the orders you set A larger tolerance facilitates zero pole cancellations which reduce the numerator and denominator polynomial orders Joint Input Output Identification If you do not have any knowledge about the controller structure but the stimulus response and reference signals are all available you can use the joint input output identification approach to estimate the transfer function model of a plant in a closed loop system This approach uses the transfer functions from different input output signal pairs to estimate a closed loop system The System Identification Toolkit implements the following two stage method for the joint input output approach 1 Let T z satisfy the following equation 1 TE GOF E By manipulating Equations 6 1 and 6 2 you can rewrite u k as follows u k T z r k F z To z e k Any open loop
132. tification The indirect identification approach which estimates the transfer function model of a plant in a closed loop system first identifies the transfer function model of the closed loop system based on the reference signal and the response signal This approach then retrieves the transfer function model of the plant from the identified closed loop system The indirect identification approach can identify the transfer function of the plant accurately even when the SNR of the system is low and no matter whether the output noise is white noise or color noise However this approach requires prior knowledge about the controller of the system and the reference signal also must be available In addition any inaccuracy or nonlinearity of the controller in the system might affect estimating the model of the plant With indirect identification you can use the following two equations to describe the feedback path closed loop system shown in Figure 6 3 V k Go z u k e k 6 1 u k r k F z y k 6 2 where Go z is the open loop transfer function of the plant F z is the transfer function of a linear time invariant LTT controller u k is the stimulus signal of the system y k is the response signal of the system r k is the reference signal of the system e k is the output noise of the system By combining Equations 6 1 and 6 2 you can represent the closed loop relationship with the following equation Go z WSTIR
133. tions where the numerator and denominator are polynomials Continuous Transfer Function Model m 1 bot bst b _ 8 b G s n 1 n aj taSt a _ 18 a 8 Discrete Transfer Function Model m 1 _ bgt Diet oi FO n 12 b 2 G z n 1 n Apt aZ 4 4 12 4 2 You can use the SI Estimate Transfer Function Model VI to estimate both continuous and discrete models For discrete models this VI implements the prediction error method For continuous models this VI internally System Identification Toolkit User Manual 4 8 ni com Chapter 4 Parametric Model Estimation Methods performs the following three consecutive steps to estimate the model The SI Estimate Transfer Function Model VI first calculates a discrete model with the prediction error method This VI then applies the Zero Order Hold method to convert the discrete model to a continuous model Finally this VI uses the Gauss Newton method to optimize the continuous model this VI converted in the second step Refer to the LabVIEW Control Design Toolkit User Manual available at ni com manuals for more information about the Zero Order Hold method Transfer function models describe only the deterministic part of the system For stochastic control general linear polynomial models commonly are used because these models separately describe the deterministic and stochastic parts of a system However in classical control engineering the deterministic part of the sy
134. to examining the time response data you also want to examine the frequency response data You can use the SI Estimate Frequency Response VI to view the frequency response of the measured output signal as shown in Figure 9 3 Flexible Arm at a SISO WDT Y window length Figure 9 3 Non Parametric FRF VI National Instruments Corporation 9 3 System Identification Toolkit User Manual Chapter 9 System Identification Case Study The input data is periodic over 4096 samples which is the signal length Notice that in Figure 9 3 the window length 4096 is the same as the signal length so as to obtain a smaller bias in the frequency response estimation Figure 9 4 shows the magnitude and phase responses of the measured output signal The magnitude response graph shows three resonances and two anti resonances in the frequency domain Resonances are vibrations of large amplitude in a system caused by exciting the system at its natural frequency magnitude response 2000 1 1 1 1 0 795775 10 248 6 Frequency Hz Figure 9 4 Frequency Response of the Flexible Arm Data Set Notice the resonance at approximately 42 Hz You can deduce that this resonance is caused by noise or nonlinear system behavior because the 42 Hz falls outside the frequency range of the input data 0 122 24 4 Hz At 42 Hz there is no input energy thus implying that the response at 42 Hz is not a result of the input
135. u measured to any prior knowledge of the system and to the purpose for which you use the model Model validation also determines if the model is flexible enough to describe the system If the model is inadequate you must revise the system identification process or consider using another method The best way to validate a model is to experiment with the model under real world conditions If the model works as you expect the model estimation is successful However experimenting with the model under real world conditions might be dangerous For example introducing arbitrary perturbations to the input of a chemical plant might lead to a harmful explosion Therefore before you incorporate the model into real world applications validate the model by using plots and common sense or by using statistical tests on the prediction error The System Identification Toolkit provides three of the most common validation methods model simulation model prediction and model residual analysis 3 Note When validating the model you obtain you must use a set of data that is different from the data you used to estimate the model Validation Methods After you build a model you can use at least three different methods to validate the model and evaluate its flexibility You can use model simulation to understand the underlying dynamic relationship between the model inputs and outputs You can use model prediction to test the ability of the model to predi
136. uments Application Engineers make sure every question receives an answer For information about other technical support options in your area go to ni com services or contact your local branch at ni com contact Training and Certification Visit ni com training for self paced training eLearning virtual classrooms interactive CDs and Certification program information You also can register for instructor led hands on courses at locations around the world System Integration If you have time constraints limited in house technical resources or other project challenges National Instruments Alliance Partner members can help To learn more call your local NI office or visit ni com alliance If you searched ni com and could not find the answers you need contact your local office or NI corporate headquarters Phone numbers for our worldwide offices are listed at the front of this manual You also can visit the Worldwide Offices section of ni com niglobal to access the branch office Web sites which provide up to date contact information support phone numbers email addresses and current events National Instruments Corporation B 1 System Identification Toolkit User Manual
137. value is zero For a discrete system an impulse is a physical pulse that has unit amplitude at the first sample period and zero amplitude for all other times u t Area 1 0 gt 0 t At 0 w Figure 3 1 Impulse Response Because the impulse signal excites all frequencies and the duration of this signal is infinitely small you can see the natural response of the system Figure 3 2 shows that the impulse response of a linear time invariant system is equal to the output y k of the system when you apply an impulse signal to the input u k of the system The impulse response provides the complete characteristic information of a system System h n u k yk Figure 3 2 Impulse Response Definition System Identification Toolkit User Manual 3 2 ni com Chapter 3 Nonparametric Model Estimation Methods If you know the impulse response A n and the input signal u k of a system then you can compute the output y k of the system by using the following equation co y k F u k n h n e k 3 1 AEE where e k is the disturbance of the system According to impulse response theory when you apply a Dirac delta function to the input of a system the output of the system is the impulse response You can think of the Dirac delta function 5 x as a function that has the value of infinity for x 0 the value zero elsewhere and a total integral of one However generating an i
138. ystem defined by the following equation y k 0 2u k 0 8u k 1 0 3u k 2 System Identification Toolkit User Manual 3 6 ni com Chapter 3 Nonparametric Model Estimation Methods output signal y k pe 10 20 4 1 0 500 System Simulation Prewhitening input signal u k whiteness test 1 aza AR order i H J 100 1 1 1 3 o 500 1000 1279 c 0 25 _lis white noise oa 0 25 7 1 i 1 0 500 1000 1279 Lag auto correlation after whiten Rg system 95 confidence region iW AX 0 2 A 0 8k 4 O 30K7 h n 0 2 0 8 0 3 Estimation Results estimated impulse response hin X rejected Jo fos foso ozs o 00 zA Braa Aa Wet 1 1 1000 1279 Figure 3 3 Front Panel of Prewhitening Example VI Figure 3 4 shows the block diagram of this VI This example VI demonstrates the accuracy of the impulse response estimation in the following circumstances National Instruments Corporation Zero mean pseudo white noise input signal without prewhitening Zero mean pseudo white noise input signal with prewhitening Non zero mean white noise input signal without prewhitening Non zero mean white noise input signal with prewhitening 3 7 System Identification Toolkit User Manual Chapter 3 Nonparametric Model Estimation Methods Generate the input signal Forward Coefficients Ho Simulate the system 0 208 0 8k 0 3Uk 2

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