Vibration Control Active Mass Damper - One Floor (AMD-1)
Contents
1. q_amd1_e_map Full Order Observer 3 o x File Edit View Simulation Format Tools Help WinCon Full Order Observer Xo_dot A Xo B U G Y Yo Yo C Xo D U Figure 8 Actual AMD 1 Full Order State Observer Step2 Before being able to run the actual control loops the different controller gains must be initialized in the Matlab workspace since they are to be used by the Simulink controller diagram Start by running the Matlab script called setup_lab_amdl m However ensure beforehand that the CONTROLLER_TYPE flag is set to MANUAL This file initializes all the AMD 1 model parameters user defined parameters and AMD 1 s state space matrices as defined in pre lab Assignment 3 First the two PV controller gains K and K calculated in pre lab Assignment 1 and satisfying the de sired time requirements must be entered in the Matlab workspace To assign K and K type their values in the Matlab command window by following the Matlab nota tions used for the controller gains as presented in Table A 2 of Appendix A Second the state feedback gain vector K must be calculated and entered in the Matlab work space Use Matlab to carry out the pole placement calculations satisfying the design requirement expressed by Equation 3 Third and last the observer gain matrix G must also be calculated and entered in the Matlab workspace Use Matlab to carry out the pole placement calculations satisfying the design requirement expres2. Revision 02 Page 4 Vibration Control Laboratory Student Handout The particular pole placements presented in the following by Equations 3 and 4 were chosen for the AMD closed loop system to meet the specifications stated above 5 2 1 State Feedback Design Pole Locations Determine the AMD 1 full state feedback gain vector K such that the closed loop poles i e eigenvalues due to the state feedback law are placed at the following locations 6 15 j 6 15 j 8 16 3 5 2 2 Full Order State Observer Design Pole Locations However in order to be based on a full state feedback i e all structural displacements and velocities law the AMD 1 s active structural control strategy should be based on a state observer This is explained by the fact that the AMD 1 s floor deflection and velocity are not measured directly only its acceleration is This particular design for the AMD 1 plant is due to full scale real life applications where floor deflections and velocities are difficult to measure directly However they can be estimated from acceleration measurements since accelerometers provide an affordable and reliable way of sensing a building dynamic behaviour Determine the AMD 1 full order observer gain matrix G such that the closed loop poles i e eigenvalues due to the observer error dynamics are placed at the following locations 20 25 30 35 4 Comparing Equations 3 and 4 it can be noted that
3. B AMD 1 Equations Of Motion EOM 25 Document Number 561 Revision 02 Page i Vibration Control Laboratory Student Handout 1 Objectives As illustrated in Figure 1 below the purpose of the AMD 1 experiment is to design a control system that dampens the vibration of a bench scale building like tall structure using an Active Mass Damper AMD mounted at the top The Active Mass Damper One Floor AMD 1 experiment can be used in earthquake mitigation studies and to investigate Control Structure Interaction CSI It is conceptually similar to active mass dampers used to suppress vibrations in tall structures e g high rise buildings and to protect not only against earthquakes but also for example strong winds e g hurricanes The Active Mass Damper One Floor AMD 1 plant is illustrated in Figure 1 below and is fully described in Reference 1 This laboratory takes advantage that the dynamics of the active mass i e cart are tightly coupled to these of the building like structure to which it is attached Therefore the active mass can either be used to excite or to dampen the flexible structure vibration The purpose of the AMD 1 laboratory is to design a switching mode control system that first excites the vibration mode of the one story structure and then dampens the structure oscillation Figure 1 AMD 1 Experiment During the course of this experiment you will become fa
4. Figure 9 In order to observe the structure natural be haviour versus its actively damped or controlled behaviour the AMD mode is only switched on once every over 10 second period while the excitation disturbance re peats itself once every 10 second period Step5 In order to observe the system s real time responses from the actual system open the following WinCon Scopes xf_ddot m s 2 xf mm and PV Position Controller PV Control xc mm You should now be able to monitor on the fly as the system goes through the synchronized excitation AMD control sequences the measured top floor acceleration the controller mode the estimated floor deflection and the linear cart setpoint and actual position respectively Furthermore you can also open the sink Vm V in a WinCon Scope This allows you to monitor on line the actual commanded motor voltage which is proportional to the control effort spent sent to the power amplifier Hint 1 To open a WinCon Scope click on the Scope button of the WinCon Server window and choose the display that you want to open e g xf_ddot m s 2 from the selection list Hint 2 For a good visualization of the actual system responses you should set the WinCon scope buffer to 20 seconds To do so use the Update Buffer menu item from the desired WinCon scope Step6 Over an example run of 20 seconds your actual AMD 1 s floor acceleration and control mode sequences should look similar to the ones
5. the dynamics of the observer error i e rate at which the estimation error goes to zero are at least three times faster i e farther to the left in the s plane than the plant itself This should ensure that the estimator does not interfere with the plant s dynamics Document Number 561 Revision 02 Page 5 Vibration Control Laboratory Student Handout 6 Pre Lab Assignments 6 1 Assignment 1 Proportional Velocity PV Controller Design In this design procedure for the PV controller the one floor flexible structure on top of which the linear cart is mounted is ignored The system is considered to consist of the IP01 or IP02 based linear servo plant alone A schematic of the AMD 1 linear cart system s input and output is represented in Figure 7 below Motor Voltage Linear Cart Cart Position V Servo Plant Xi m Figure 2 The AMD 1 Linear Cart Input and Output 1 Based on your previous work from Reference 7 write down the open loop transfer function of your IP01 or IP02 based system G s as defined below BTs 5 2 The Proportional Velocity PV position controller implemented in this lab for your lin ear servo plant introduces two corrective terms one is proportional by K to the cart po sition error while the other is proportional by K to the cart velocity Equation 6 be low expresses the resulting PV control law V N K x x K a D 6 Quickly re iterate your pre
6. 15 Vibration Control Laboratory Student Handout Fla_amdi_e_mqp PY Position Controller 4 3 0 x File Edit view Simulation Format Tools Help WinCon Structure Excitation Mode PY Controller PY Control xe mm wer 2 s s 2 zetafwetstwct 2 Derivative Filter Figure 7 Diagram used for the Real Time Implementation of the AMD 1 Cart PV Position Controller CAUTION The velocity signal used in the control inner loop of the actual linear cart is ob tained by first differentiating the position signal i e encoder counts and then by low pass filtering the obtained signal in order to eliminate its high frequency content As a matter of fact high frequency noise which is moreover amplified during differentiation causes long term damage to the motor To protect your DC motor the recommended cut off frequency is 50 Hz When the Mode Switching Sequence block outputs 2 instead of 1 that turns on the Active Mass Damping AMD capabilities of the linear cart located on top of the structure The AMD behaviour is achieved by implementing a full order observer based state feedback control loop as illustrated in Figure 4 above by the orange blocks The full order state observer defined by Equations 20 and 21 is imple mented in the subsystem called Full Order Observer and depicted in Figure 8 below Document Number 561 Revision 02 Page 16 Vibration Control Laboratory Student Handout
7. J K M J KM f m g c m g f c mp and Document Number 561 Revision 02 Page 27 Vibration Control Laboratory Student Handout 2 2 ofa MBr z0 gP n M vp J E x 1 K dea n Gi aoe 2 2 2 2 2 2 t Mr M J K M J K M Mr M J KM JKM c mp J m g c m g f c mp f m g c m g f B 14 2 M Fr c c mp Mr M J K M J KM p f m g c m g f c m Equations B 13 and B 14 represent the Equations Of Motion EOM of the system It can be noticed in the case of the AMD 1 system that the EOM are linear As a remark if both BL and Jm are neglected Equations B 13 and B 14 become d K x t M M F a x 1 M MM B 15 and K xf t F x x t M M B 16 Document Number 561 Revision 02 Page 28
8. Laboratory Student Handout 7 In Lab Procedure 7 1 Experimental Setup And Wiring Even if you do not configure the experimental setup entirely yourself you should be at least completely familiar with it and understand it If in doubt refer to References 1 4 5 and or 6 The first task upon entering the lab is to ensure that the complete system is wired as fully described in Reference 1 You should be familiar with the complete wiring and connec tions of your Active Mass Damper One Floor AMD 1 system If you are still unsure of the wiring please ask for assistance from the Teaching Assistant assigned to the lab When you are confident with your connections you can power up the UPM You are now ready to begin the lab 7 2 Real Time Implementation Of The AMD 1 Switching Mode Controller 7 2 1 Objectives m To implement in real time with WinCon the previously designed PV position controller in order to command your actual AMD 1 linear servo plant m To design through pole placement a full order observer for the actual AMD 1 system m To design through pole placement a state feedback law for the actual AMD 1 system by using the obtained estimated state vector m To implement in real time with WinCon the previously determined observer based state feedback to dampen the vibration of the AMD 1 structure by appropriately driving the active mass on top of it 7 2 2 Experimental Procedure Please follow the steps d
9. Linear Motion Servo Plant AMD 1 Linear Experiment 9 Vibration Control GIU ANS ER INNOVATE EDUCATE Active Mass Damper One Floor AMD 1 Student Handout Vibration Control Laboratory Student Handout Table of Contents LOD 1 Aia 2 OLI RA 2 ALn 3 4 1 Main COMpon nts icessiscssnrissopie iernii tii AER EEEE ETEA EEE EiS 3 Pe os WIDE RE ROOT 3 5 Controllers Design SPECIHCANOi Silio 4 5 1 Excitation Mode PV Controller Design Specifications 4 5 2 Active Mass Damping AMD Mode 4 5 2 1 State Feedback Design Pole LOCaNONS sscciaversccsdacnsunanvederoessbndseurevcarsenenessloesines 5 5 2 2 Full Order State Observer Design Pole Locations 5 6 Pre Lab Assisi iii air 6 6 1 Assignment 1 Proportional Velocity PV Controller Design 6 6 2 AMD 1 System Representation and Notations i 8 6 3 Assignment 2 Determination of the AMD 1 System s Linear Equations Of Motion POM aaa 9 6 4 Assignment 3 AMD 1 State Space Representation 9 6 5 Assignment 4 Full Order Observer conii 11 TM EaD Prodi rca 13 7 1 Experimental Setup And Wifig arnie 13 7 2 Real Time Implementation Of The AMD 1 Switching Mode Controller 13 2350 120065 CR FOR RO RE E IA 13 722 Experimental Proc dUrTE rr anita ria 13 Appendix A Nomenelatii iaia 22 Appendix
10. T T T Excitation 1 Excitation 2 40 4 AMD ON 20 4 AMD OFF Mana er Natural Vibration Active Damping 20 4 40 Lo mm x mm Mode 10x 60 0 2 4 6 8 10 12 14 16 18 20 Time s Figure 9 AMD 1 Actual Experiment As previously mentioned the AMD 1 s linear cart is commanded by a switching mode controller In excitation mode i e Mode Switching Sequence equals to 1 or AMD OFF the cart is subject to the PV position controller In Active Mass Damping Document Number 561 Revision 02 Page 18 Vibration Control Laboratory Student Handout AMD mode i e Mode Switching Sequence equals to 2 or AMD ON the cart is subject to an observer based state feedback controller The AMD 1 experiment is setup such that the cart excites the structural dynamics once every 10 seconds by fol lowing a full period sinusoidal excitation at the structure s natural frequency This is illustrated in Figure 9 above by Excitation 1 and Excitation 2 This initial and re peatable mass movement causes a vibration in the AMD 1 flexible structure which persists at its natural rate if the AMD mode controller is not switched on as depicted by the first 10 seconds in Figure 9 The AMD mode is only turned on after the second structural excitation This causes the top floor vibrations to dampen out quickly as shown by the last 10 seconds in
11. To successfully carry out this laboratory the prerequisites are i To be familiar with your Active Mass Damper One Floor AMD 1 main components e g actuator sensors your data acquisition card e g Q8 MultiQ and your power amplifier e g UPM as described in References 1 3 4 and 5 ii To have successfully completed the pre laboratory described in Reference 2 Students are therefore expected to be familiar in using WinCon to control and monitor the plant in real time and in designing their controller through Simulink as detailed in Reference 6 iii To have successfully completed the laboratory described in Reference 7 iv To be familiar with the complete wiring of your Active Mass Damper One Floor AMD 1 plant as per dictated in Reference 1 v To be familiar with the design theory of full order state observers as described for example in Reference 8 3 References 1 Active Mass Damper One Floor AMD 1 User Manual 2 IPO1 and IP02 Linear Experiment 0 Integration with WinCon Student Handout 3 IPO1 and IP02 User Manual 4 Data Acquisition Card User Manual 5 Universal Power Module User Manual 6 WinCon User Manual 7 IPOI and IP02 Linear Experiment 1 PV Position Control Student Handout Document Number 561 Revision 02 Page 2 Vibration Control Laboratory Student Handout 8 P R B langer Control Engineering A Modern Approach 1995 Saun
12. bserver has for inputs the sys tem s input s and output s and calculates as outputs the states estimates The observer is basically a replica of the actual plant with a corrective term G Y Y multiplied by the ob server gain matrix Document Number 561 Revision 02 Page 11 Vibration Control Laboratory Student Handout The obtained estimated state vector can then be used for state feedback law as expressed below U V KX 22 Let us define the estimation error vector as follows X X X 23 Using Equations 13 20 21 and 23 the estimation error dynamically behaves accord ingly to the following relationship d gg A EC X 24 Therefore from Equation 24 the estimation error will asymptotically go to zero if and only if A GC is stable that is to say iff G is determined such as A GC has all its eigenvalues in the left hand plane However a theorem shows that if A C is observable then A GC can always be made stable by a proper choice of G Therefore it is possible to estimate the state s of a system if and only if that system is observable Please refer to your in class notes as required Determine whether A C is observable Hint By definition a system is observable iff its observability matrix has full rank i e number of states The observability matrix is defined as follows T W C CA CA CA 25 Document Number 561 Revision 02 Page 12 Vibration Control
13. der time derivatives In our case X is defined such that its transpose is as follows d d T X SAN x t Ahh 14 Furthermore it is reminded that the system s measured output vector is y ua ale 15 Also in Equation 13 the input U is set in a first time to be F the driving force of the linear motorized cart Thus we have U F 16 As a remark it can be seen from Equations 15 and 16 that the AMD 1 system consists of two outputs for one input 2 From the system s state space representation previously found transform the state space matrices for the case where the system s input U is equal to the linear cart s DC motor voltage Vm instead of the linear force F The system s input U can now be expressed by U V n 17 Hint In order to convert the previously found force equation state space representation to voltage input it is reminded that the driving force F generated by the DC motor and acting on the cart through the motor pinion has already been determined in previous laboratories As shown for example in Equation B 9 of Reference 7 F can be expressed by 2 d ae N Ky 1 E Km 0 A N K Np K Vp c 2 R r Rp Tp m mp 18 Document Number 561 Revision 02 Page 10 Vibration Control Laboratory Student Handout 3 Evaluate the matrices A B C and D found in question 2 that is to say in case the system s input U is equal to the cart s DC motor voltage as expres
14. ders College Publishing 4 Experimental Setup 4 1 Main Components To setup this experiment the following hardware and software are required Power Module Quanser UPM 1503 2405 or equivalent Data Acquisition Board Quanser Q8 MultiQ PCI MQ3 or equivalent GB Active Mass Damper Plant Quanser Active Mass Damper One Floor AMD 1 as represented in Figure 1 above B Real Time Control Software The WinCon Simulink RTX configuration as detailed in Reference 6 or equivalent For a complete and detailed description of the main components comprising this setup please refer to the manuals corresponding to your configuration 4 2 Wiring To wire up the system please follow the default wiring procedure for your Active Mass Damper One Floor AMD 1 system as fully described in Reference 1 When you are confident with your connections you can power up the UPM Document Number 561 Revision 02 Page 3 Vibration Control Laboratory Student Handout 5 Controllers Design Specifications The Active Mass Damper One Floor AMD 1 experiment essentially consists of a switching mode controller using a pre defined logic sequence to alternate between a mode of repeatable structural excitation and the Active Mass Damping AMD mode to stiffen the structure Each mode is achieved through its own closed loop control scheme However the power amplifier e g UPM should not go into saturation in any case A
15. dinates x and x have the following formal formulations 0 0 L 5L 0 ar at x t as e B 7 and 2_1 2 1 0 d dx B 8 ot On x t f K In Equations B 7 and B 8 above L is called the Lagrangian and is defined to be equal to Document Number 561 Revision 02 Page 26 Vibration Control Laboratory Student Handout L T V B 9 For our system the generalized forces can be defined as follows d 0 D FAD By g2 and 2 0 B 10 It should be noted that the nonlinear Coulomb friction applied to the linear cart has been neglected Furthermore the viscous damping force applied to the structure floor has also been neglected Calculating Equation B 7 results in a more explicit expression for the first Lagrange s equation such that 2 2 da HJ K x t M Fa r 7 M O F 8 130 B 11 p r mp Likewise calculating Equation B 8 also results in a more explicit form for the second Lagrange s equation as shown below 2 d d M 0 a is B 12 Finally solving the set of the two Lagrange s equations as previously expressed in Equations B 11 and B 12 for the second order time derivative of the two Lagrangian coordinates results in the following two equations gt 2MB a F OM x t K hap MB MB G89 a 2 2 z t 2 2 2 dt Mr M J KM JKM Mr M J K M J K M c mp f m g c m g f c mp f m g c m g f B 13 2 A M F MF Mr M
16. displayed in Figure 10 below Corresponding to the same experimental run the time recors of the AMD 1 s cart measured position and excitation setpoint are depicted in Figure 11 below Document Number 561 Revision 02 Page 19 Vibration Control Laboratory Student Handout Figure 10 AMD 1 Floor Actual Acceleration And Controller Mode Figure 11 AMD 1 Cart Actual Position Response And Excitation Setpoint Document Number 561 Revision 02 Page 20 Vibration Control Laboratory Student Handout Step7 Compare the structure s actively damped and natural behaviours in terms of settling time and amplitude of the floor deflection xf and or acceleration xf_ddot Assess the actual performance of your Active Mass Damping controller Measure your floor estimated deflection response settling time Are the design specifications satisfied Explain Hint In order to accurately measure a signal amplitude and time values from your WinCon Scope plot you can first select Freeze Plot from the WinCon Scope Update menu and then reduce the window s time interval by opening the Set Time Interval input box through the Scope s Axis Time menu item You should now be able to scroll through your plotted data Alternatively you can also save your Scope trace s to a Matlab file for further data pr
17. escribed below Step1 If you have not done so yet you can start up Matlab now Depending on your sys tem configuration open the Simulink model file of name type g_amdI_ZZ mdl or q_amd1_e_ZZ mdl where ZZ stands for either for mq3 mqp q8 or nie Ask the TA assigned to this lab if you are unsure which Simulink model is to be used in the lab You should obtain a diagram similar to the one shown in Figure 4 below Document Number 561 Revision 02 Page 13 Vibration Control Laboratory Student Handout la amdi_e_mqp Ele Edit View Simulation Format Tools Help WinCon Active Mass Damper One Floor AMD 1 Actual Experiment 1 Structure Excitation Mode 2 Active Mass Damping Mode gt gt Mode Switching xf_ddot Sequence ma Excitation Disturbance Position Setpoint m UPM Voltage Saturation PV Position Controller AMD 1_E MOPCI Mode Switch Disturbance Setpoint Xo xc xf xc_dot xf_dot Full Order Observer 0 100000 u Figure 4 Real Time Implementation of the AMD 1 Controller In order to use your actual AMD 1 system the controller diagram directly interfaces with your system hardware as shown in Figure 5 below qg_amdi_e_mqp AMD 1_E MQPCI File Edit View Simulation Format Tools Help WinCon Interface to the IP02 Based AMD 1 Plant g
18. int 2 You can use the method of your choice to model the system s dynamics However the modelling developed in Appendix B uses the energy based Lagrangian approach In this case since the system has two Degrees Of Freedom DOF there should be two Lagrangian coordinates a k a generalized coordinates The chosen two coordinates are namely x and xs Also the input to the system is defined to be F the linear force applied by the motorized cart 6 4 Assignment 3 AMD 1 State Space Representation In order to design and implement a state feedback controller for our system a state space representation of that system needs to be derived Moreover it is reminded that state space matrices by definition represent a set of linear differential equations that describe the sys tem s dynamics Since the two EOM of the AMD 1 as found in Assignment 2 should al Document Number 561 Revision 02 Page 9 Vibration Control Laboratory Student Handout ready be linear they can be written under the state space matrices representation Answer the following questions 1 Determine from the system s two equations of motion the state space representation of our AMD 1 plant That is to say determine the state space matrices A B C and D verifying the following relationships ox AX BU and Y CX DU 13 where X is the system s state vector In practice X is often chosen to include the generalized coordinates as well as their first or
19. lled as a standard linear spring mass system as represented in Figure 3 above Using the system s specifications given in Reference 1 the floor assembly mass Mrs can be calculated from M M M 9 The AMD 1 s top floor linear stiffness constant Ks for small angular structure oscillations is given in Reference 1 Ks models the lateral stiffness of the structure However it is consistent with the following relationship obtained from the standard Ordinary Differential Equation ODE describing the free oscillatory motion of a mass Document Number 561 Revision 02 Page 8 Vibration Control Laboratory Student Handout K 4 M M nf 10 In the presented modelling approach the structure viscous damping coefficient By is neglected 6 3 Assignment 2 Determination of the AMD 1 System s Linear Equations Of Motion EOM The determination of the AMD 1 s structure plus cart equations of motion is derived in Appendix B If Appendix B has not been supplied with this handout derive the system s equations of motion following the system s schematic and notations previously defined and illustrated in Figure 3 Also put the resulting EOM under the following format AVI Pak or lt or x Xe di ot to ore 11 Can CA a ae arr lars te arte a 12 Hint 1 By neglecting the Coulomb a k a static friction of the cart system the two EOM should be linear They represent a pure spring mass damper system and H
20. lso both controllers control effort which is proportional to the motor input voltage Vm should stay within the system s physical limitations 5 1 Excitation Mode PV Controller Design Specifications The self excitation mode uses the active mass to generate a repeatable disturbance to the AMD 1 s building like structure In this case the linear cart i e active mass control loop consists of a Proportional Velocity PV position controller In the present laboratory i e the pre lab and in lab sessions you will design and implement a control strategy based on the Proportional Velocity PV control scheme in order for your linear cart system to satisfy the following performance closed loop requirements 1 The cart position Percent Overshoot PO should be less than 10 i e PO lt 10 1 2 The time to first peak should be less than 150 ms i e t S 0 15 s 2 5 2 Active Mass Damping AMD Mode In the Active Mass Damping AMD mode the linear cart is controlled by using a state feedback law based on a full order observer This time around the purpose of the control strategy is to dampen the AMD 1 top floor oscilations created in the excitation mode by stiffening the structure The general AMD design requirements are expressed below in terms of settling time and decayed amplitude of the floor deflection xf for an initial deflection of around 30 mm t lt 1 0 s for x lt 1 5 mm Document Number 561
21. miliar with the design of Proportional plus Velocity PV position controller to drive the linear cart i e active mass such that it excites the flexible structure natural vibrations Then a vibration reduction control strategy will be designed in order to dampen the structure oscillations Such a AMD control strategy will be implemented using a full state feedback law based on a full order observer Pole placement will be used to tune the control scheme Document Number 561 Revision 02 Page 1 Vibration Control Laboratory Student Handout At the end of the session you should know the following How to mathematically model the IP01 or IP02 based linear servo plant from first principles in order to obtain the open loop transfer function in the Laplace domain How to design and tune a Proportional Velocity PV position controller to meet the required design specifications How to mathematically model the Active Mass Damper One Floor AMD 1 plant using the Lagrange s method and to obtain its state space representation How to design and tune a full order state observer based on the structure s acceleration feedback signal How to design and tune a state feedback controller satisfying the closed loop system s desired design specifications How to implement in real time the total AMD 1 mode switching between vibration excitation and active mass damping control scheme and evaluate its actual performance 2 Prerequisites
22. nd the state feed back law is designed to dampen the structure oscillation by driving the active mass i e cart Both control loops are the actual implementations of the pre laboratory as signments previously carried out Opening the Disturbance Setpoint sub system should show a diagram similar to Fig ure 6 This diagram generates the cart position setpoint to follow in order to best ex cite the AMD 1 s flexible structure The setpoint basically consists of one full period of a sine wave of frequency 2 5 Hz i e for a duration of 0 4 seconds and amplitude 6 cm The sinusoidal excitation s frequency has been chosen accordingly to the struc ture s natural frequency as given in Reference 1 This excitation repeats itself every 10 seconds 3q_amdi_e_map Disturbance Setpoint A 3 z oj x File Edit Yiew Simulation Format Tools Help WinCon Excitation Disturbance Cart Position Setpoint m Be 2 Sine Wave 2 5 Hz Amplitude Tstart 8 Period Tend Figure 6 Excitation Mode Cart Position Setpoint Generation Figure 4 above also includes a subsystem named PV Position Controller which im plements your linear cart PV controller s two feedback loops This PV controller is based on the actual measurement of the cart position coming from the encoder The obtained position signal is then differentiated and low pass filtered to attenuate any high frequency noise Document Number 561 Revision 02 Page
23. ocessing Do so by using the File Save selection list from the WinCon Scope menu bar Step8 If your AMD 1 responses do not meet the desired design specifications review your PV and or observer and or state feedback gain calculations and or alter the closed loop pole locations until they do If you are still unable to achieve the required performance level ask your T A for advice Step9 Include in your lab report your final values for K K K and G as well as the resulting plots of the actual system responses e g floor acceleration estimated deflection controller mode cart position command voltage Ensure to properly document all your results and observations Step10 You can move on and begin your report for this lab Document Number 561 Revision 02 Page 21 Vibration Control Laboratory Student Handout Appendix A Nomenclature Table A 1 below provides a complete listing of the symbols and notations used in the Active Mass Damper One Floor AMD 1 mathematical modelling as presented in this laboratory The numerical values of the system parameters can be found in Reference 1 Symbol Description Matlab Simulink Notation Mi Structure Top Floor Mass Mtf M Rack Mass Mr fa Floor a k a Roof Natural Frequency fn Ke Floor a k a Roof Linear Stiffness Constant Kf Mrs Floor Roof i e floor 1 rack Total Mass Mf Xf Floor Horizontal Deflection Relative To The Ground xf oy Floor Horizontal Veloci
24. of the translational and rotational kinetic energies arising from the motorized linear cart since the cart s direction of translation is orthogonal to that of the rotor s rotation and the translational kinetic energy of the flexible structure s floor In other words the total kinetic energy of the AMD 1 system can be formulated as below ee ae in Tt B 2 First the translational kinetic energy of the motorized cart can be expressed as a function of its centre of gravity s linear velocity as shown by the following equation 2 n im 2x0 0 A Document Number 561 Revision 02 Page 25 Vibration Control Laboratory Student Handout Second the rotational kinetic energy due to the cart s DC motor can be characterized by 2 2 d JK 10 Tr 1 g dt B 4 c 2 2 r mp Third and last the structure floor s translational kinetic energy can be characterized as follows 2 T zM g0 B 5 Thus by replacing Equations B 3 B 4 and B 5 into Equation B 2 the system s total kinetic energy results to be such as 2 1 1 nE d d d Dista 0 BR a 370 i B 6 2 1 1 d M 3 M 8 It can be seen from Equation B 6 that the total kinetic energy can be expressed in terms of the generalized coordinates first time derivatives Let us now consider the Lagrange s equations for our system By definition the two Lagrange s equations resulting from the previously defined two generalized coor
25. ption Matlab Simulink Notation PO Percent Overshoot PO tp Peak Time tp One Cart Undamped Natural Frequency wn c Ge Cart Damping Ratio zeta _c K Cart Proportional Gain Kp K Cart Velocity Gain Kv Kea Cart Desired Position i e Reference Signal xc d Document Number 561 Revision 02 Page 23 Vibration Control Laboratory Student Handout Symbol Description Matlab Simulink Notation Continuous Time Laplace Operator Table A 2 PV Cart Controller Nomenclature Table A 3 below provides a complete listing of the symbols and notations used in the full order observer and state feedback controller design as presented in this laboratory Description Matlab Simulink Notation A B C D State Space Matrices of the AMD 1 System Actual State Vector Actual Output Vector System Input State Feedback Gain Vector Estimated State Vector Estimated Output Vector Full Order State Observer Gain Matrix Observer s Observability Matrix State Vector Estimation Error Table A 3 Control Loop Nomenclature Document Number 561 Revision 02 Page 24 Vibration Control Laboratory Student Handout Appendix B AMD 1 Equations Of Motion EOM This Appendix derives the general dynamic equations of the Active Mass Damper One Floor AMD 1 system The Lagrange s method is used to obtain the dynamic model of the system In this approach the single input to the system is considered to be F Furthermore it is reminded tha
26. sed by Equa tion 4 Refer to your in class notes regarding the full order observer design theory as needed Hint Pole placement calculations can be achieved in Matlab by using the function place or acker CAUTION Document Number 561 Revision 02 Page 17 A A Vibration Control Laboratory Student Handout Have your lab assistant check your controller gain values Do not proceed to the next step without his or her approval Step3 You are now ready to go ahead with compiling and running your actual switch ing mode controller for the AMD 1 system First compile the real time code corre sponding to your diagram by using the WinCon Build option from the Simulink menu bar After successful compilation and download to the WinCon Client you should see the green START button available on the WinCon Server window You are now in a position to use WinCon Server to run in real time your actual closed loop system CAUTION Before starting your actual controller manually move the linear cart located on the top floor to the middle of the track i e mid stroke position Also make sure that the AMD 1 ground floor is properly clamped mounted to a table Step4 You can now run your Active Mass Damper experiment on the actual AMD 1 plant by clicking on the START STOP button of the WinCon Server window This should start the AMD 1 control sequence as illustrated by real data in Figure 9 below 60 T T T T T T
27. sed in Equation 16 Hint Evaluate state space matrices by using the model parameter values given in Reference 1 Ask your laboratory instructor what system configuration you are going to use in your in lab session In case no additional information is provided assume that the two additional masses are mounted on top of the linear cart 4 Calculate the open loop poles from the system s state space representation as previously evaluated in question 3 Is it stable What is the type of the system What can you infer regarding the system s dynamic behaviour Do you see the need for a closed loop controller Explain Hint The characteristic equation of the open loop system can be expressed as shown below det s A 0 19 where det is the determinant function s is the Laplace operator and the identity matrix Therefore the system s open loop poles can be seen as the eigenvalues of the state space matrix A 6 5 Assignment 4 Full Order Observer Since all the states contained in X as defined in Equation 14 cannot be directly measured e g Xr a state observer needs to be built to estimate them The system s variables directly measured are expressed in the output vector defined in Equation 15 This laboratory focuses on the design of a full order observer The full order observer structure is defined as follows d Fo AX tBU G Y Y 20 and Y CX DU 21 It can be seen from Equations 20 and 21 that the state o
28. t Vm Command W Quanser Consulting Pr TRICRAGLE ai MultiQ PCI DAC ma DAC Limit Cable Gain Pre Compensation Analog Output 0 To UPM Driving the Motor E xt_ddot_meas m s 2 Floor 1 Acceleration Quanser Consulting MultiQ PCI ADC K_ACC a4 _ddot m s 2 Conversion to mist2 Analog Input 2 Floor 1 Accelerometer Input XC_LIM_ENABLE O xe_meas mm Cart Position Watchdog Cart Position Quanser Consulting Multi0 PCI ENC P xe m Conversion to m Encoder Input 0 Cart Position Figure 5 Interface Subsystem to the Actual AMD 1 Plant Using the MultiQ PCI Card Document Number 561 Revision 02 Page 14 Vibration Control Laboratory Student Handout To familiarize yourself with the diagram it is suggested that you open the model sub systems to get a better idea of their composing blocks as well as take note of the I O connections The real time model shown in Figure 4 above implements a switching mode control ler A pre defined Mode Switching Sequence alternatively controls the linear cart us ing either a Proportional Velocity PV position controller or an observer based state feedback law In short the PV controller is mostly used in the excitation mode to self excite the vibration in the building like structure On the other ha
29. t the reference frame used is defined in Figure 3 on page 8 To carry out the Lagrange s approach the Lagrangian of the system needs to be determined This is done through the calculation of the system s total potential and kinetic energies Let us first calculate the system s total potential energy Vr The potential energy in a system is the amount of energy that that system or system element has due to some kind of work being or having been done to it It is usually caused by its vertical displacement from normality gravitational potential energy or by a spring related sort of displacement elastic potential energy Here there is no gravitational potential energy since both AMD 1 cart and structure are assumed to stay at a constant elevation i e no vertical displacement from normality for small angular structure oscillations However as represented in Figure 3 the AMD 1 top floor is modelled as a linear spring mass system Therefore the AMD 1 s total potential energy is only due to its elastic potential energy It results that the total potential energy of the AMD 1 plant can be fully expressed as 2 V 5 K x t B 1 It can be seen from Equation B 1 that the total potential energy can be expressed in terms of the system s generalized coordinates alone Let us now determine the system s total kinetic energy 77 The kinetic energy measures the amount of energy in a system due to its motion Here the total kinetic energy is the sum
30. ty Relative To The Ground xf dot 32 ae X Floor Horizontal Acceleration Relative To The Ground xf ddot M Total Mass of the Cart System Mc Vm Cart Motor Armature Voltage Vm Tei Cart Motor Armature Current Im Rm Cart Motor Armature Resistance Rm Ln Cart Motor Armature Inductance Lm K Cart Motor Torque Constant Kt Nm Cart Motor Efficiency Eff m Km Cart Back ElectroMotive Force EMF Constant Km Jm Cart Rotor Moment of Inertia Jm Be Equivalent Viscous Damping Coefficient as seen at the Beq Motor Pinion K Cart Planetary Gearbox Gear Ratio Kg Ne Cart Planetary Gearbox Efficiency Eff g Document Number 561 Revision 02 Page 22 Vibration Control Laboratory Student Handout Description Matlab Simulink Notation Cart Motor Pinion Radius Cart Driving Force Produced by the Motor i e Control Force Cart Linear Position Relative To The Floor Cart Linear Velocity Relative To The Floor Total Potential Energy of the AMD 1 System Cart s Translational Kinetic Energy Cart Rotor s Rotational Kinetic Energy Structure Top Floor s Translational Kinetic Energy Total Kinetic Energy of the AMD 1 System Generalized Force Applied on the Generalized Coordinate Xe Generalized Force Applied on the Generalized Coordinate Xf Table A 1 AMD 1 Model Nomenclature Table A 2 below provides a complete listing of the symbols and notations used in the PV position controller design as used in this laboratory Symbol Descri
31. vious work from Pre Lab Assignment 3 i e the PV Document Number 561 Revision 02 Page 6 Vibration Control Laboratory Student Handout Controller Design Section of Reference 7 in order to determine the two PV controller s gains K and K as functions of the second order system s characteristic parameters Wnc and C 4 Using the two Hint formulae provided below express n and amp as functions of the two PV design specifications previously defined PO and t Hint formula 1 T di a N 7 PO 100e Hint formula 2 n 2 8 5 Determine the values of x and corresponding to the desired PV design specifications as defined previously Determine then from your results the numerical values of K and K satisfying the desired time requirements of your closed loop PV plus cart system Document Number 561 Revision 02 Page 7 Vibration Control Laboratory Student Handout 6 2 AMD 1 System Representation and Notations A schematic of the Active Mass Damper One Floor AMD 1 plant is represented in Figure 3 below The AMD 1 scaled building is a Single Degree Of Freedom SDOF structure i e single story The AMD 1 system s nomenclature is provided in Appendix A As illustrated in Figure 3 the positive direction of horizontal displacement is towards the right when facing the system Figure 3 Schematic of the AMD 1 Plant For small floor deflection angles the AMD 1 roof is mode
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