1 getting started with fda-cad
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1. iii Compare the stability mode FDP Example 6 1 mat characteristics given above Step input Step Input 40 1 se 9 with those discussed in zz n5 2 4 E E 9 30 X ES 4 jJ N N Example 6 1 ba x alle SE 2 AZ 5 7 5 5 KB use FDA CAD to Y TE o oa the response 0 20 40 60 80 100 0 20 40 60 80 100 plots shown in Fig 6 1 first MES Step Input Step Input check only those output rant N As v o 0 05 B E 0H 4659 EA variables required in the 25 ML B E 5 TE ria o5 x Output Responses 5 00 BE x Be 5 5 0 05 window in FDA CAD Select u MR 0 20 40 60 80 100 0 20 40 60 80 100 the Long Period output Tire sec Time sec response time 100s and T Step Input 0 i remember to set the Input 2 o ES f 1 NG 5 f B Signa to a step of 525 ool LZ 8i j 3 55 5 2 magnitude 0 075rad 1 deg 3 Ei ol il B ga Push the Plot Responses 0 40 60 80 100 0 2 40 6 80 100 Time sec Time sec 34 FDA CAD v 3 User Manual button to open the figure window with the annotated response time histories as selected A copy of the figure window is shown above v If wider plots are preferred to match those shown in Fig 6 1 then FDA CAD should be set to plot no more than four var2. 180 3 10 1 0 Frequency rad sec iv Note the difference in phase scale with that shown in Fig 7 8 This is due to the way in which different computer programs deal with non minimum phase transfer functions Fig 7 8 was obtained with the aid of Program CC This problem of Bode frequency interpretation is discussed in FDP It is left as an exercise for the user to obtain the frequency response plots shown in Fig 7 9 and in Fig 7 10 Adverse response to rudder is an established feature of aircraft dynamics and the consequent non effects minimum phase Example 7 3 Sideslip ang le frequency response to aileron B m Magnitude dB BR D 180 90 Phase deg o 90 180 10 42 0 1 Frequency rad sec FDA CAD v 3 User Manual appear in the Bode plots It is sufficient that the user is aware of these properties when interpreting the output of analysis using FDA CAD Example 11 4 This example is intended to show the effect of single output variable feedback to elevator on the longitudinal dynamics of an aircraft The root locus plot is used for this purpose and FDA CAD enables the analysis to be made quickly and easily It is first necessary to load the a
3. Acceleration State matrix Wing span Input matrix Mean aerodynamic chord mac Output matrix Direct matrix force Acceleration due to gravity Height Moment of inertia in roll Moment of inertia in pitch Moment of inertia in yaw Inertia product about ox and oz axes Feedback gain Feedback gain matrix Rolling moment Mass Pitching moment Mass matrix Yawing moment Roll rate Pitch rate Yaw rate Wing reference area Time Axial velocity Axial component of steady equilibrium velocity Lateral velocity Steady equilibrium velocity Normal velocity Normal component of steady equilibrium velocity Axial displacement State vector Axial force Lateral displacement Output vector Lateral force Normal displacement Normal force 54 FDA CAD v 3 Greek symbols a p Y Subscripts C340 N xz co 0o0o52j3 ocxo User Manual Angle of attack Body incidence Sideslip angle Flight path angle Control angle Roll control stick angle Pitch control stick angle Yaw control stick angle Rudder angle Dutch roll damping ratio Phugoid damping ratio Short period pitching oscillation damping ratio Elevator angle Pitch angle Aileron angle Air density Roll angle Yaw angle Undamped natural frequency Dutch roll Equilibrium Rolling moment Pitching moment Yawing moment Phugoid Roll rate Pitch rate Roll mode Yaw rate Short period pitching oscillation Spiral Mode Axial ve
4. Aileron input derivatives Example 5 2 This example is intended to show the math steps in solving the equations of motion to obtain the response transfer functions This process is of course hidden in FDA CAD However the computational solution can be found and compared with that obtained in the example Load the model data file FDP Example 5 2 mat to FDA CAD to see the flight condition and derivative data which should be compared with that given in the example It is important also to note the units applying and these reflect the source of the information i Press the RUN MODEL button This solves the equations of motion and in the process the longitudinal state equation and a full set of response transfer functions are calculated 30 FDA CAD v 3 User Manual ii Press the Print Report button Open the Matlab Command Window to see the solution report As before the state equation appears first and is followed by the full list of response transfer functions iii The transfer function of interest here is that describing pitch attitude response to elevator this is the fourth transfer function to appear in the list and appears as follows TRANSFER FUNCTIONS FACTORISED FORM Zero pole gain from input to output u Axial velocity 2 3668 845 519 s 4 215 s72 0 033265 0 02201 s42 0 8917s 4 883 Zero pole gain from input to output w Normal velocity 22 1206 s 64 67 5 2 0
5. Sideslip angle the response to aileron 0 2 4 6 8 1 1 4 16 18 2 Time sec variables in the outputs window Step Input Enter a response time of 20s in Rudder Command To p Roll rate o o the Select Response Time 0 2 4 6 8 10 12 14 16 18 20 Time sec window and set the Input Step Input Signal to a step of magnitude 0 075rad 1 deg Push the Plot Responses button to Rudder Command To r Yaw rate o a 0 2 4 6 8 10 12 14 16 18 20 open the figure window with the Time sec Step Input annotated response time Roll attitude histories as selected A copy of Rudder Command To o 0 4 the figure window is shown 0 2 4 6 8 10 12 14 16 18 20 Time sec here Again the y axis scales have been adjusted to exactly match those shown in Fig 7 2 vi The plots can be re scaled and additional information can be added using the standard Matlab tools in the figure window prior to saving or printing Choice of scales for the plots is important if the adverse response properties are not to be missed Such properties are the result of non minimum phase transfer functions and care should be exercised when conducting analysis with FDA CAD to correlate the visible response shapes with their describing t
6. 3 38957 0 102298 3 38957 0 102298 2 78231 1 Step Y E Ss e tt Plot Response Histories Stor 10 Long an College of Aeronautics FDA CAD v 3 User Manual 3 2 1 Menu options The top of screen menu bar shows the Model button as well as the IUTUPERS FDA CAD help button Selecting Load from the Model menu MES i opens the standard Windows browse window from which a mat pcd Save file containing previously saved aircraft data may be selected Select the desired file and click Open This action sets longitudinal or lateral directional analysis from saved information as appropriate it loads the aircraft data and places the filename in the title field at the top of the GUI This data is then visible in the flight condition and derivative data fields in the top half of the screen Alternatively data can be entered by hand and the Save button will bring up the Windows save window in which the file name and save destination can be entered The model data can be saved at any time the screen is visible and when a model is saved its name appears in the title field of all analysis screens as reference 3 2 2 Data entry The input fields are situated in the top of the screen and are separated into flight condition and control derivative data Numerical data may be entered into each field from the keyboard Once a field value is filled continue to the next using the mouse
7. 0 00798 Xdw 0 00000 Zdw 0 00000 Mdw 0 00017 Xq 0 00000 Zq 0 00000 Mq 0 34000 Xeta 0 00381 Zeta 24 45684 Meta 4 52000 ii Key in the wind axes referenced derivatives and change the body incidence value to Odeg in the flight data field Change the aircraft axes to wind axes in the pop up menu then push the Run Model button followed by the Print Report button Compare the report with that obtained for Example 6 1 and note any differences 2365 FDA CAD v 3 iii Push the yellow u button in the output window to open the Bode plot figure window for that variable To facilitate Magnitude dB Example 6 3 u eta Bode plot User Manual 80 S 60 40 20 comparison with Fig 6 7 it is 20 90 convenient to use the Matlab tools to change the scales of the Bode gain and phase 90 Phase deg plots A print out from the 180 figure window is shown and 270 Frequency rad sec it will be seen to compare favourably with Fig 6 7 iv Repeat the process to obtain a Bode plot of pitch attitude in the figure window Again 2 to facilitate comparison with Fig 6 8 use the Matlab tools to change the scales of the Bode gain and phase plots A print out from the figure window is shown below and it will be seen to compare favourably with Fig 6 8 w
8. Time sec Elevator Command To w Normal velocity Lp Step Input r r r F r r r r r 0 02 4 NG q Pitch rate 0 04 Elevator Command sia t t 0 1 2 3 4 5 6 7 8 9 0 Time sec Step Input Ors r r r 0 02 0 04 0 06 0 08 Elevator Command To 0 Pitch attitude F t 5 6 7 8 9 Time sec e 99 A pr 11 5 With reference to FDP Example 11 5 is the lateral directional equivalent of longitudinal Example 11 4 Aircraft model data will be found in data file FDP Example 11 5 mat and its analysis follows the procedures described in the previous example However there are some minor differences i The user must toggle between the lateral feedback structure window and the directional feedback structure window by means of the pop up menu alongside the file name window at the top of the screen The RUN MODEL function includes both SATs FDA CAD v 3 User Manual lateral feedback to aileron and directional feedback to rudder in its solution The model also has provision for an aileron actuator a rudder actuator and an aileron rudder interlink gain In all other respects interpretation of the functional tools in the GUI screen is similar to previou
9. An entry must be made in each field If a field is not completed or its input is not a valid number the MachNo M Atitude Pitch neria O Flight Path y Airdensiy PJ Bedylnidence a Wing area S Airspeed V Mean chord c ESSEN Mass m Greviyconstent 7 Wind Aves _ Dimensional UK program will return a warning message Individual data entries can be corrected or changed at any time It is critically important that the user ensures consistent units for all the data entered into the model The RESET ALL button clears the entire model data and sets all fields to a blank A new model can then be entered manually or loaded from a data file The data is also cleared by pressing the Return to opening screen button and this action is advised before starting work with a new aircraft model 9 FDA CAD v 3 User Manual 3 2 3 Derivative format selection When data is entered manually the derivative format and aircraft reference axes must be set using the drop down menus Three alternative derivative formats are available which Full Order Derivative format Dimensional UK KC ee covers most applications Dimensional Dimensionless UK and Normalised USA format When the model data is saved this information is saved with the data Note that data given in normalised North American format as found in Teper and in Heffley and Jewell
10. Outputs window shows the available variables from which the response time history plots may be selected Only the checked v responses will appear in the response plots and these may be toggled on or off By pressing the yellow buttons a Matlab LTI figure window opens to show the Bode plot for the selected variable The Bode plot can then be printed or saved from this window in the usual way Whilst the LTI figure window is open the user may also obtain alternative plots such as Nichols Nyquist Step Singular value etc which can also be saved and printed Important Some aircraft transfer functions are negative and some are non minimum phase To avoid problems with frequency response interpretation transfer function sign is checked and the sign of negative transfer functions is ignored Thus all frequency response plots are made with positive transfer functions It is strongly recommended that the user refers to the correct transfer function properties obtained by pressing the Print Report button when interpreting Bode and other frequency response plots See chapters 6 and 7 of FDP for more information on this topic 3 2 11 Plotting options By selecting the response time below the Outputs window and pressing the Plot Responses button a new Matlab figure window appears containing the checked response time histories The figure title is shown as the file name of the aircraft model A typical example is
11. Step Input 1 1 1 1 44 37 s 0 0732 s 1 63 eee CA Cg gao 0 0551s 10 082657 44 95 4172 To 6 Pitch attitude A 8 Elevator Command 3 2 13 FDA CAD report The Print Report button generates a report in the Matlab command window from where it can be printed and saved as required The report shows the equations of motion of the 14 FDA CAD v 3 User Manual basic aircraft in state space matrix format the factorised transfer functions obtained in the solution of the equations of motion and the stability mode characteristics An example report is shown below Note in the example shown height h should read height rate dh This is corrected in version 3 0 of FDA CAD a am ss ss SS ns eS Si eu em i tn st xdot Bu Cx Du WIlccerccccclcceu eee uT Eta x T u Ww q theta Control Law eta d delta K u K 0 0 0 0 STATE SPACE MATRICES 0 06728 0 02323 0 9 81 0 396 1 729 48 32 0 0078012 0 24314 3 1589 0 0 l B transposed 0 17 01 44 375 C l 0 0 1 0 0 0 0 1 L 0 396 1 729 1 6804 0 0 0 02 0 Continued below ELT FDA CAD v 3 D transposed 0 0 0 0 0 0 0 0 TRANSFER FUNCTIONS FACTORISED FORM Zero pole gain from input to output u Axial velocity perturbation 0 39514 s 974 3 3 1 85 s 2 0 05508s 0 06255 s Z 4 9s 17 22 Zero
12. rad s 3 47e 002 4 16e 002i 6 40e 001 5 42e 002 3 47e 002 4 16e 0021 6 40e 001 5 42e 002 2 59 000 7 57 0001 3 24 001 8 00 000 2 59 000 7 57 0001 3 24 001 8 00 000 ii Compare the State matrices and B with those given in Example 11 6 and compare the stability modes characteristics also Note some numerical differences due to the fact that the example was originally worked by hand and some approximations were made terms A 2 4 and A 3 4 were both approximated by zero and A 3 3 the pitch damping term was approximated by the stability derivative mg The last approximation causes an error in the value of the short period mode damping but this does not significantly change the interpretation of aircraft dynamics The solution obtained with FDA CAD matches closely that given in the source reference Heffley and Jewell iii Switch to the Stability and Control Augmentation screen Observe that the basic aircraft short period mode and phugoid mode characteristics are shown in the Pole Placement window fields for reference The same information is also shown in the Aircraft Modes window below iv Choose the desired new values for short period and phugoid mode frequency and damping as explained in the example Enter these values in the Pole Placement window fields Push the Pole Placement button and the feedback gain values 49 FDA CAD v 3 User Manual required to achieve these mod
13. 03485s 0 02249 s72 0 033265 0 02201 s42 0 8917s 4 883 Zero pole gain from input to output q Pitch rate 4 658 s s 0 2688 s 0 1335 s72 0 033265 0 02201 s42 0 8917s 4 883 iv Comparison of the pitch attitude response transfer function with equation 5 24 in the example indicates a good match Minor numerical differences are evident and these are almost certainly due to the accuracy of the hand calculation used in preparation of the FDP example This is not a problem as the accuracy of either transfer function is more than adequate for flight dynamics analysis and for flight control system design Note that the units of the input output response described by the transfer function is rad rad or equivalently deg deg Example 5 6 This example can not be worked directly with FDA CAD since it does not accept derivatives in the concise format However it does calculate the concise derivatives which appear as the coefficients in the A and B state matrices Normalised aircraft data Bie FDA CAD v 3 User Manual converted to SI units see data file FDP Example 5 6 mat were obtained from Heffley and Jewell and input to FDA CAD in the usual way Push the RUN MODEL button followed by the Print Report button to obtain the solution of the equations of motion It will be seen that the A and B matrices compare favourably with equation 5 58 with the exception of the concise derivative yp
14. Responses button to Step Input open the figure window showing the four time histories A copy of the figure Elevator Command To 0 Pitch attitude a 0 1 2 3 4 5 6 TA 8 9 10 window is shown and the responses Time sec Step Input should be compared with the those shown in FDP Fig 6 5 Unfortunately Elevator Command To Angle of attack O multiple comparative plots on the same Time sec 1 UI CA 1 FDA CAD v 3 User Manual axes cannot be made from the basic aircraft screen in this version of FDA CAD Example 6 3 This example illustrates the use of FDA CAD to obtain longitudinal Bode frequency response plots from the basic aircraft transfer functions The aircraft model is the same as that used in Example 6 1 but the derivative data is first converted to a wind axes reference i With the model data for Example 6 1 loaded into FDA CAD convert the derivative data to a wind axes reference and obtain the following report General Flight Conditions Mach 0 30 Height 15000 ft Body Incidence 13 30 deg Original Axes notation BODY AXES Xu 0 00501 Zu 0 08570 Mu 0 00183 Xw 0 00464 Zw 0 54500 Mw 0 00777 Xdw 0 00000 Zdw 0 00000 Mdw 0 00018 Xq 0 00000 Zq 0 00000 Mq 0 34000 Xeta 5 63000 Zeta 23 80000 Meta 4 52000 Xu 0 04225 Zu 0 20455 Mu 0 00001 Xw 0 11421 Zw 0 49774 Mw
15. having pitch rate feedback to elevator Select a step Input Signal of one degree magnitude 0 0175 and with no feedback gains set in the feedback gain fields push the RUN MODEL button to obtain the basic aircraft solution Check the four variables u w q and 0 in the Output window select Short Period response time 10s and push the Plot Responses button The plots will appear in a Matlab figure window as described before Toggle the Save Plot 1 button the data is saved and the window is closed Now enter Kg 0 3 in the appropriate feedback gain field and push RUN MODEL to obtain the closed loop equations of motion Repeat the plot process for the same input to see the closed loop time histories in the figure window Toggle the Save Plot 2 button to save the plot data and close the window With both save buttons still toggled push the Compare Plots 1 and 2 button The Matlab figure window opens to show both sets of plots on the same axes as shown below Note that the plots legend has been edited to suit 46 FDA CAD v 3 User Manual FDP Example 11 4 mat Step Input F F F r r r r r r Kq 0 3 Basic Aircraft T Ep fh _t E ft E t E SS 2 3 4 5 6 7 8 9 Time sec Elevator Command To u Axial velocity E Step Input 0 r r r N Pat r r r r r r r 0 1 2 3 4 5 6 7 8 9 10
16. s 2 0 068915 0 002364 s72 11 285 75 64 Zero pole gain from input to output theta Pitch attitude 60 9176 s 1 909 s 0 06782 CHARACTERISTIC POLE LOCATIONS Eigenvalue Damping Freq rad s 3 45 002 3 43e 0021 7 09e 001 4 86e 002 3 45e 002 3 43 0021 7 09e 001 4 86e 002 5 64 000 6 62 0001 6 49e 001 8 70 000 5 64 000 6 62 0001 6 49 001 8 70 000 vi It will be seen that the closed loop transfer functions are a good match with those given in Example 11 6 equations 11 65 and 11 66 with the exception of the numerator of the velocity u response to elevator transfer function This is though to be due to the different solution algorithms used in the software to work the example Because aircraft matrix equations are not always well conditioned the solutions do not always converge on the correct numerical coefficient values However in such cases the time response plots are usually correct su FDA CAD v 3 User Manual 6 REFERENCES 1 Cook M V 2012 Flight Dynamics Principles 3rd Edition Elsevier Ltd Oxford UK 2 Teper G L 1969 Aircraft Stability and Control Data Systems Technology Inc STI Technical Report 176 1 3 Heffley R K and Jewell W F 1972 Aircraft Handling Qualities Data NASA Contractor Report NASA CR 2144 53 FDA CAD v 3 User Manual 7 NOTATION Standard symbols aorA N NSS SE 6
17. shown below The Short Period and Long Period buttons select a response time of 10 seconds or 100 seconds respectively However the user may also enter an alternative response time in the field window A maximum of eight plots may be shown in the figure window By selecting a smaller number of response plots and the most suitable Input Signal the size of each plot is increased to fit the figure window thereby enhancing the analytical usefulness Thus for good resolution check only those variables of primary interest The figure window may then be saved and printed as required 13 FDA CAD v 3 User Manual The typical response plot figure window Time Histories File Edit View Insert Tools Window Help 1 PER Cherlong mat Step c 2 zs Bo 0 s 8 2 Pibe sa z 3 ir HIS 150 iB Full Order Normalised Step Input HE Bis 52 525 o 5 o ra BE BYR mg s2 ae 0 5 10 5 Time sec Time sec Step Input MU j Plot Response Histories Ee NGA Shot f 10 Long E X F 8 E 10 1 Bi ENS gt bara 0 5 10 Time sec 3 2 12 Transfer function inclusion By toggling the Show Transfer Functions button before pressing the Plot Responses button the transfer functions of each of the response plots is shown in the plot This is illustrated below
18. the screens The program may be shut down from the opening screen by clicking on the top right hand title bar button in the usual way 3 2 The Basic Aircraft Screens The longitudinal and lateral directional basic aircraft screens are very similar in appearance The user choice simply sets the annotation of the screen as required The longitudinal and lateral directional screens are shown below In the interests of expediency these images are those developed for version 2 0 of FDA CAD and differ in minor detail only from those developed for version 3 0 The most notable difference is the absence of the large College of Aeronautics button which has been replaced by a smaller Return to opening screen button at the bottom of the window Other minor changes are mostly cosmetic to improve overall appearance The functional properties of the screen objects are described in the following paragraphs FDA CAD v 3 User Manual The longitudinal basic aircraft screen presentation DA CAD v 2 Flight Dynamics Analysis Command Augmentation Design By Konstantinos Siliverdis 2004 lodel Cherlong mat FLIGHT CONDITION DATA CONTROL DERIVATIVE DATA Mach M 0 147053 Xu 0 06728 Zu 0 396 Muf 0 Altitude h 1500 Pitch Inertia ly 1700 xw 0 02323 zw 1 729 Mw 0 2772 FlightPath y 0 Airdensity p 106 xw 0 Zaw 0 00197 Body Incidence
19. 35 1 433 Zero pole gain from input to output rA Yaw rate due to Aileron 0 01875 s 1 589 5 2 3 2465 4 982 s 1 329 s 0 006498 s42 0 25435 1 433 Zero pole gain from input to output phi A Roll attitude due to Aileron 1 62 s 2 0 3624s 1 359 s 1 329 s 0 006498 s42 0 2543s 1 433 Transfer functions describing response to rudder Zero pole gain from input to output beta R Sideslip due to Rudder 0 0288 s 30 21 s 1 296 s 0 01477 s 1 329 s 0 006498 s42 0 25435 1 433 Zero pole gain from input to output pR Roll rate due to Rudder 0 392 s s 2 566 s 1 85 s 1 329 s 0 006498 5 2 0 25435 1 433 Zero pole gain from input to output rR Yaw rate due to Rudder 0 864 s 1 335 842 0 029955 0 1092 s 1 329 s 0 006498 s42 0 25435 1 433 Zero pole gain from input to output phi R Roll attitude due to Rudder 0 392 s 2 566 s 1 85 1 329 s 0 006498 s 2 0 25435 1 433 CHARACTERISTIC POLE LOCATIONS Eigenvalue Damping Freq rad s 6 50e 003 1 00e 000 6 50e 003 1 27 001 1 19 0001 1 06e 001 1 20e 000 1 27 001 1 19e 000i 1 06 001 1 20e 000 1 33e 000 1 00 000 1 33e 000 39 FDA CAD v 3 User Manual iii Note that FDA CAD shows the sideslip response transfer functions in terms of sideslip angle p only To obtain the transfer functions in terms of sideslip velocity v it necessary only to multiply thos
20. 4 2 FDA CAD is used to show the process of derivative conversion The flight condition data and dimensionless derivatives should be loaded from the relevant model data example file The derivative format and axes reference will be correctly set and since the body incidence value is non zero this confirms a body axes reference The units applying are easily identified since the gravity constant g is set to 9 81 m s which confirms that SI units apply Pause the cursor over the data fields to check applicable units i Press Derivative Conversion button ii Derivative conversion menu window opens select conversion to Dimensional Body Axes Press Calculate button iii Open Matlab Command Window to see the conversion report which may be printed or saved as required See below iv Compare the dimensional derivatives calculated with those shown in the first set of three equations in Example 4 2 General Flight Conditions Mach 0 60 Height 35000 ft Body Incidence 9 40 deg 26 FDA CAD v 3 User Manual Original Axes notation BODY AXES Xu 0 00760 Zu 0 72730 Mu 0 03400 Xw 0 04830 Zw 3 12450 Mw 0 21690 Xdw 0 00000 Zdw 0 39970 Mdw 0 59100 Xq 0 00000 Zq 21090 Mq 1 27320 Xeta 0 06180 Zeta 0 37410 Meta 0 55810 Xu 12 68597 Zu 1214 01427 Mu 277 46561 Xw 80 62270 Zw 5215 43735 Mw 1770 06736 Xdw 0 00000 Zdw 18 32502 Mdw 132 47007 Xq 0 00000 Zq 9881 85598 Mq 50798 03442
21. FDA gt CAD FLIGHT DYNAMICS ANALYSIS COMMAND AUGMENTATION DESIGN Version 3 01 USER GUIDE A Matlab Graphical User Interface for Flight Dynamics Analysis Originated by Shane Rees Version 1 0 Written by Konstantinos Siliverdis Version 2 0 Edited by Michael V Cook February 2013 Cranfield University 2013 FDA CAD v 3 User Manual CONTENTS 1 GETTING STARTED WITH FDA CAD h 2 3 4 5 System Requirements Installation Execution Screen Size Directory Path 2 FDA CAD OVERVIEW 2 1 2 2 Flight Dynamics Analysis Graphical User Interface GUI 3 THE OPENING AND BASIC AIRCRAFT SCREENS 3 1 3 2 The Opening Screen The Basic Aircraft Screens 3 2 1 Menu options 3 2 2 Data entry 3 2 3 Derivative format selection 3 2 4 Model order selection 3 2 5 Aircraft axes selection 3 2 6 Stability and control derivative conversion 3 2 7 Solution of the equations of motion 3 2 8 Stability modes 3 2 9 Input signal options 3 2 10 Response options 3 2 11 Plotting options 3 2 12 Transfer function inclusion 3 2 13 FDA CAD report 3 2 14 Exporting variables to Matlab workspace 3 2 15 Return to opening screen 3 2 16 Close FDA CAD 3 2 17 Transfer to stability augmentations screen 4 THESTABILITY AUGMENTATION SCREEN 4 9 Basic airframe state space matrices Stability modes Feedback gain structure Actuator model Pole placement Feedback gain design using the root locus plot R
22. Workspace button places a copy of all the system variables in the Matlab workspace for continuing off line analysis The information transferred includes flight condition data control and stability derivatives axes notation derivative format and the transfer function matrix numerators and denominator 3 2 15 Return to opening screen Pressing the Return to Opening Screen button closes down the current aircraft model resets the data fields to zero and opens the opening screen from which a new study can be initiated 3 2 16 Close FDA CAD Pressing the red Close FDA CAD button closes down the entire program A dialogue box opens first to enable the user to confirm the action The program may also be shut down by clicking on the top right hand title bar button in the usual way 3 2 17 Transfer to stability augmentation screen Pressing the Stability and Control Augmentation button opens a new screen set out for closed loop system design Before proceeding the program opens a dialogue window asking whether the user wants to save the current model or cancel the transfer process and continue working with the basic aircraft This transfer is not currently available in version 3 0 of FDA CAD when the user has been working with the longitudinal reduced order aircraft model 17 FDA CAD v 3 User Manual 4 THE STABILITY AUGMENTATION SCREEN In the stability augmentation screen the user may perform simple c
23. Xeta 18361 94495 Zeta 111152 16188 Meta 810703 90627 Example 4 3 FDA CAD is used to show the process of deriving the longitudinal state equation for an aircraft With the same aircraft data model loaded as for example 4 2 i Press the RUN MODEL button This solves the equations of motion and in the process the longitudinal state equation is calculated ii Press the Print Report button Open the Matlab Command Window to see the solution report The state equation comprises the state matrix A and the input matrix B which appear in the first part of the report as shown below iii Compare the A and B matrices with those in the equation given at the end of Example 4 3 LONGITUDINAL REPORT FOR FDP Example 4 3 mat HA Bu State Equation y Cx Du Output Equation Wits ae eee Ses u T Eta Input variable elevator angle xXAT u w q theta State variables Control Law eta_d delta K u Control law for augmented case equations K Feedback gain matrix for augmented case equations 0 0 0 0 27 FDA CAD v 3 User Manual STATE SPACE MATRICES ES State Matrix transposed Input Matrix Example 4 5 FDA CAD is used to show the process of deriving the longitudinal state equation for an aircraft starting from the American normalised derivative format Aircraft data as given was entered from the keyboard and flight data not given in th
24. a 0 Wingarea S 15 xq 0 Zq 1 6804 Mg 2207 Airspeed V 50 Mean chord c i xe 0 ze 4701 4471 _ Mass m 109 Graviyconsten g 38 Model Dynamics Full Order AircraftAxes windAxes Derivative format Normalised _ Derivative Conversion Aircraft Modes o rad sec C 0 250101 0 110113 0 250101 0 110113 414918 0 59049 414918 0 59049 Outputs Basic Aircraft Longitudinal Dynamics Input Signal Step m EM Ed Plot Response Histories Short 10 Long 0 College of A eronautics A CAD v 2 Flight Dynamics Analysis Command Augmentation Design By Konstantinos Siliverdis 2004 adel eek o FLIGHT CONDITION DATA CONTROL DERIVATIVE DATA Mach No M 0 147053 Yaw Inertia 2 1400 Yul 01444 Lv 01166 0174 Altitude h 1500 Inertia Product x 0 Yp O Lp 2283 Np 1 732 FlightPath 0 Air density p 106 w 0 Lr 1 053 Nr 19s Bodylncidence o 0 Wingarea 5 15 Yda 0 Lda 3101 Nda o0 Airspeed V 50 WingSpen b 3911 203 Lar 06133 Ndr 6583 Mass m 109 Greviyconstent g 38 Model Dynamics AircraftAxes Wind Axes Rall Inertia b 3100 Derivative format Normalised _Derivative Conversion ka Aileron nu Basic Aircraft Lateral Directional Dynamics o sec Input Signal wi pi AN
25. acteristics are shown As the legend states blue text is used to denote real poles while a negative sign in either the damping ratio field or in a blue field denotes an unstable mode The fields are editable and the user can enter the desired stability mode characteristics for the Dutch Roll closed loop aircraft Then by pressing the Pole Placement 33555 9 0154 Spiral Mode 01023 GJ 27823 Roll Mode button the feedback gain fields are updated to show the Blue tex Black text real pole 1 Tp complex pole pair Negative unstable values necessary to achieve the desired dynamics Pressing the RUN MODEL button then calculates the solution of the augmented 2 FDA CAD v 3 User Manual equations of motion and places the closed loop modes characteristics in the relevant fields Note The pole placement tool can not be used when actuator dynamics are included in the model 4 6 Feedback gain design using the root locus plot An alternative method of designing a feedback gain value is by means of a SISO root locus plot Pressing the yellow button in any feedback path in the illustration below Ka the relevant root locus plot is opened in the reserved plotting window on the screen The user may then interact with the plot using the standard Matlab root locus plotting tools By clicking and dragging the mouse on a branch of the locus a text box appears showing the corresponding feedback gain pole damping ratio freq
26. ample 4 6 FDA CAD is used to show the process of deriving the lateral directional state equation for an aircraft starting from the American normalised derivative format Data for the same aircraft at the same flight condition as in Example 4 5 was used for this example and the data file should be loaded into FDA CAD i Press the RUN MODEL button This solves the equations of motion and in the process the lateral directiona state equation is calculated ii Press the Print Report button Open the Matlab Command Window to see the solution report The state equation comprises the state matrix A and the input matrix B which appear in the first part of the report as shown below iii Compare the A and B matrices with those in the equation given by equation 4 89 in Example 4 6 20 FDA CAD v 3 User Manual iv Note that FDA CAD has two inputs in the lateral directional model aileron and rudder Observe that the B matrix has two columns to allow for two inputs As in Example 4 5 note that again American Imperial units apply g 32 2 ft s Observe also that FDA CAD is organised such the first of the four equations of motion in the state equation the first row represents sideslip angle p and not sideslip velocity v This is usually clearly stated in the output material LATERAL DIRECTIONAL REPORT FOR FDP Example 4 6 mat K 0000 Feedback gain matrix for augmented case equations 0000 STATE SPACE MATRICES
27. bination to suit the problem at hand 2 2 Graphical User Interface GUI Most of the Graphical User Interface GUI objects have tool tips which are revealed whenever the mouse pointer is held over a field button pop up menu etc For example typical tips include the function performed by a button the current units applicable to the aircraft data definition of a derivative and other advisory information Warning dialogue boxes are also revealed whenever the user invokes an operation which may give rise to an erroneous or irreversible outcome FDA CAD v 3 User Manual 3 THE OPENING AND BASIC AIRCRAFT SCREENS 3 1 The Opening Screen The opening screen shows imagery linking the program to the book FDP Four buttons appear on the screen About FDA CAD opens a short PDF document describing the history and scope of the program Longitudinal Analysis transfers the user immediately to the longitudinal basic aircraft analysis screen and resets the variable data fields to zero Lateral Directional Analysis transfers the user immediately to the lateral basic aircraft analysis screen and resets the variable data fields to zero The user may then toggle between lateral or directional analysis from the lateral screen Open User Guide opens the PDF file containing this document The top of screen menu bar includes the FDA CAD help button which may also be used to open the User Guide This appears on all of
28. btain the closed loop equations of motion enter the chosen feedback gain value 0 3 in the field in the feedback structure window push the RUN MODEL button and observe the change in the stability modes in the Aircraft Modes window Push the Print Report button to see the entire solution in the Matlab Command Window At any time the user can see a reminder of the basic aircraft state equation by clicking on the Basic Aircraft box in the system structure window 44 FDA CAD v 3 User Manual Small scale plot to show phugoid dynamics Fig 11 13 Pitch Rate feedback to Elevator 0 1 L L L L L L L R 0 38 0 28 0 17 0 08 0 09 0 5 0 08 0 07 r System untitled 7 Gain 0 49 0 06 a 0 06 9 64 Pole 0 00467 0 0736i N E Damping 0 0633 amp 0 05 Overshoot 81 9 5 Frequency rad sec 0 0738 s E 004 0 04 0 8 0 03 T 0 02 0 02 Fo g4 4 0 01 F 0 IE 0 t 0 05 0 045 0 04 0 035 0 03 0 025 0 02 0 015 0 01 0 005 Real Axis vi Having designed a typical feedback loop and having calculated the closed loop system model the user can now design an additional feedback loop by the same process This can be repeated for as many loop closures up to the maximum of four For example a statically unstable aircraft typically requires both a and q feedback to achieve a satisfactory solution In such an examp
29. d ii Press the Print Report button Open the Matlab Command Window to see the solution report As before the state equation appears first followed by the full list of response transfer functions and concludes with a summary of the stability modes characteristics The full report is shown below In particular the state equation matrices A and B may be compared with equation 7 10 The transfer functions describing response to aileron compare with those given in equations 7 12 and the transfer functions describing response to rudder compare with those given in equations 7 13 The characteristic pole locations given by FDA CAD compare with equation 7 14 LATERAL DIRECTIONAL REPORT FOR FDP Example 7 1 mat K 0000 0000 STATE SPACE MATRICES A 0 1008 0 468 2 32 2 0 0057881 1 232 0 397 0 0 0027787 0 0346 0 257 0 0 1 0 0 B_transposed 0 1 62 0 01875 0 Aileron input derivatives 13 484 0 392 0 864 0 Rudder input derivatives Cz 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 38 FDA CAD v 3 User Manual D transposed 0 0 0 0 0 0 0 0 TRANSFER FUNCTIONS FACTORISED FORM Transfer functions describing response to aileron Zero pole gain from input to output beta A Sideslip due to Aileron 0 01875 s 7 896 s 0 1969 s 1 329 s 0 006498 s42 0 2543s 1 433 Zero pole gain from input to output pA Roll rate due to Aileron 1 62 s 5 2 0 3624s 1 359 s 1 329 s 0 006498 s42 0 254
30. e example was obtained from Heffley and The model data was saved and this file should be loaded first i Press the RUN MODEL button This solves the equations of motion and in the process the longitudinal state equation is calculated ii Press the Print Report button Open the Matlab Command Window to see the solution report The state equation comprises the state matrix A and the input matrix B which appear in the first part of the report as shown below iii Compare the A and B matrices with those in equation 4 87 in Example 4 5 iv Note that FDA CAD can only cope with a single input in the longitudinal model elevator or stabiliser angle in this example The second input shown in equation 4 87 is thrust The thrust response solution could be obtained by replacing the elevator derivatives with the normalised thrust derivatives as given Load FDP Example 4 5 thrust mat to see this solution v Clearly this process can be repeated as necessary and it is straightforward to convert the derivative data to another format if required However care is required to maintain visibility of the correct units applying note that in this example American Imperial units apply g 32 2 ft s 28 FDA CAD v 3 User Manual LONGITUDINAL REPORT FOR FDP Example 4 5 mat u T Eta u Ww q theta Control Law eta d delta u K Ex
31. e in terms of B with aircraft velocity V It follows that v then has the units of velocity appropriate to the aircraft model data ivy FDA CAD may now be used to reproduce the response plots shown in Fig 7 1 and in Fig 7 2 First to obtain a set of Example 7 1 mat responses to aileron to match Fig 7 1 uncheck the response Tx Pulse ipu to rudder variables in the N WEE cn c outputs window Enter a is response time of 30s in the 5 2 o Select Response Time field Bo E zd D and set the Input Signal to a 2 sal pulse of magnitude 0 075rad 1 T qo n 1 deg and a width of 2 0s Push p Med the Plot Responses button to i I open the figure window with b the annotated response time a 5 10 15 20 25 30 histories as selected A copy of ou Hee the figure window is shown p A here Note that the y axis HE 0 04 a a e 1 scales have been adjusted to uM cu 7 exactly match those shown in Fig 7 1 this is done by opening the properties editor for each plot in turn within the Matlab figure window and editing the scale limits 40 FDA CAD v 3 User Manual v Second to obtain a set of FDP Example 7 1 mat responses to rudder to match Step Input Fig 7 2 check the response to rudder variables and uncheck Rudder Command To
32. e top left of the plot window states whether the feedback gain should be entered into the model with a positive or negative sign To obtain a hard copy record of the root locus plot click on the Root Locus Plot button in the top left screen tool bar and then click on the undock button to open the usual Matlab figure window The plot may be edited as required and annotated with gain test points as desired For this illustration the plot was rescaled twice once to show the short period mode dynamics 243 FDA CAD v 3 User Manual clearly and once to show the phugoid dynamics clearly Copies of both plots were obtained as shown below Large scale plot to show short period dynamics Fig 11 13 Pitch Rate feedback to Elevator 2 5 T L L 0 7 0 56 0 38 0 2 0 81 2r NG Fj System untitled1 Pa Gain 0 3 a 15r g Pole 1 72 1 86i amp Damping 0 68 2 Overshoot 96 5 44 Frequency rad sec 2 53 g E 1f 4 m 0 955 O5 oggg j 2 5 2 1 5 1 0 5 a 0 na D 3 2 5 2 51 9 1 0 5 0 Real Axis iv This procedure can be repeated for all of the variables discussed in FDP Example 11 4 which is left as an exercise for the user In a typical system design analysis the feedback gain requirements can usually be established in the reserved screen plot window It is only generally necessary to undock the figure when a printed copy of the conclusion is required for the record v To o
33. ect access to the basic aircraft A and B matrices The window is configured such that it can be minimised for repeat opening during the course of a design study 4 2 Stability modes The stability modes window is located at the lower left part of the gmi GUI screen and its function is the same as in the basic aircraft screen Every time the RUN MODEL button is pressed these fields rad sec 5 are updated to show the closed loop modes characteristics mE NID 0 250101 0 110113 including the actuator poles when included in the model panahi apo 4 3 Feedback gains structure The feedback structure is configured such that combinations of feedback variables can be selected up to and including full state feedback The feedback gains are set at zero until changed by the user For longitudinal system design the feedback variables are axial velocity u normal velocity w pitch rate q and pitch attitude 0 and these are fed back to elevator through an actuator when selected For lateral directional system design the feedback variables are sideslip angle roll rate yaw rate r and roll attitude and these may be fed back to aileron and or rudder through an actuator when selected Each yellow feedback path button is labelled with the gain that it represents and by pressing it its SISO root locus plot appears in the plotting area Once the required feedback gain variables are determined by the user these are entered ma
34. es characteristics will appear in the feedback gain fields in the system structure window Push the RUN MODEL button to obtain the solution of the closed loop equations of motion Push the Print Report button to obtain the closed loop solution report in the Matlab Command Window as shown below Observe that as full state feedback is used the stability modes characteristics are exactly as specified LONGITUDINAL REPORT FOR FDP Example 11 6 mat Ka a RN u T Eta xT w q theta Control Law eta_d delta K u K 7 8693e 006 0 00050711 0 099319 0 0006542 STATE SPACE MATRICES A Closd loop state matrix 0 067703 0 010496 1 9235 9 8101 0 021926 2 0645 363 55 0 0013106 0 010257 0 12418 9 138 0 039974 0 0 1 0 B_transposed Closed loop input matrix 0 4023 76 257 60 918 0 C 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 021926 2 0645 11 442 0 0013106 0 0 0026667 0 0 0 0 0026667 0 1 0 l 0 375 D transposed 0 0 0 O 0 O 0 O 50 Feedback gain matrix FDA CAD v 3 User Manual CHARACTERISTIC POLE LOCATIONS Closed loop stability modes Eigenvalue Damping Freq rad s 3 51e 002 4 10e 002i 6 50e 001 5 40e 002 3 51e 002 4 10 0021 6 50e 001 5 40e 002 5 60 000 5 71 0001 7 00 001 8 00 000 5 60 000 5 71e 000i1 7 00 001 8 00 000 v The feedback system design can be simplified by observing that the gains Ky Kw and K are sufficiently small that they ca
35. he augmented aircraft system variables in the Matlab workspace for continuing off line analysis The information transferred includes flight condition data control and stability derivatives axes notation derivative format feedback gains and the transfer function matrix numerators and denominator 4 13 Return to basic aircraft screen Pressing the Return to Basic Aircraft button closes down the current augmented aircraft model and returns to the basic aircraft screen and the basic aircraft model 4 14 Close FDA CAD Pressing the red Close FDA CAD button closes down the entire program A dialogue box opens first to enable the user to confirm the action The program may also be shut down by clicking on the top right hand title bar button in the usual way 25 FDA CAD v 3 User Manual 5 EXAMPLES To illustrate the use of FDA CAD for problem solving a number of the worked examples in FDP have been selected to introduce the interactive procedures capabilities and limitations Aircraft model data for each example can be found in the FDP Examples directory of FDA CAD v3 01 In each example the flight condition and derivative data were entered from the keyboard the axes reference and derivative format were selected the entries were checked for accuracy and then saved as a model data file mat Additional data required to complete all the flight data fields were obtained from the reference sources given Example
36. iables at one time i e two figures vi The plots can be re scaled and additional information can be added using the standard Matlab tools in the figure window prior to saving or printing Example 6 2 This example shows the derivation and solution of the longitudinal reduced order equations of motion together with some typical response plots for the unaugmented aircraft The aircraft model is the same as that used in Example 6 1 i With the model data loaded into FDA CAD select Reduced Order in the Model Dynamics pop up menu Push the RUN MODEL button followed by the Print Report button Compare the reduced order report with that obtained for Example 6 1 ii The model is now third order second order is shown in FDP Example 6 1 FDA CAD retains pitch attitude in the model in order to remain compatible with both a wind axes reference and a body axes reference FDP Example 6 2 mat Reduced order pitch attitude response Step Input o is also useful in handling qualities a assessment o Elevator Command To w Normal velocity iii Set up the response v plotting o Ds 4 5 6 7 8 9 10 Time sec requirement for variables w q 9 a Step input select the response time for 10s and Elevator Command To q Pitch rate select the input to be a 1deg step 0 1 2 3 4 5 6 8 9 10 Time sec Push the Plot
37. ircraft model data file FDP Example 11 4 mat i Push the RUN MODEL button followed by the Print Report button to obtain an i overview of the basic aircraft transfer functions and also the stability modes characteristics If response plots for the unaugmented aircraft are required they should also be obtained at this time and the process for so doing has been discussed in earlier examples It is useful to obtain hard copy print output showing the basic aircraft properties as this information is useful for reference during the design of closed loop feedback gains Once the analysis of the basic unaugmented aircraft is complete press the Stability and Control Augmentation button This opens a new screen showing the longitudinal feedback structure model in FDA CAD Note that alternative structures are not possible The user can switch between screens as necessary during the course of an analytical study iii To investigate pitch rate feedback to elevator and re create Fig 11 13 push the yellow Kq button in the feedback structure window This produces the root locus plot in the reserved plot window which may be interrogated by the user using the standard Matlab tools to evaluate the effect of q feedback on the stability properties of the aircraft The feedback gain to achieve a chosen mode frequency and damping ratio can be identified by moving the mouse cursor along the appropriate locus A reminder in th
38. ith the exception of the phase values Low frequency phase should be zero but as the transfer function governing pitch attitude in this example is non minimum phase the computational solution interprets phase as shown Care is needed to be aware that many aircraft transfer functions are non minimum phase which may produce unexpected phase frequency response Further it is common to find that typical software tools differ in the way that non minimum phase transfer Example 6 3 theta eta Bode plot functions are interpreted It is left as an exercise for the user to obtain the Bode plot for Magnitude dB alpha a response to elevator n Adjust the scales of the Bode plot obtained to match the plots shown in Fig 6 9 Phase deg 180 10 0 10 Frequency rad sec 37 FDA CAD v 3 User Manual Example 7 1 This example shows the full solution of the lateral directional equations of motion and some typical response plots for the unaugmented aircraft The aircraft model data should be loaded into FDA CAD from file FDP Example 7 1 mat i Press the RUN MODEL button This solves the equations of motion and in the process the lateral directional state equation a full set of response transfer functions and the longitudinal stability modes are calculate
39. l stability augmentation design and analysis screen A CAD v 2 Stability Augmentation Screen se Edit View Insert Tools Window Help Figure Lateral Lateral Analysis gt X Cherlat mat Basic Aircraft Bu 1 Du Aileron Rudder Input Signal Step Dutch Roll 33896 00194 Spiral Mode 01023 G 27823 Mode alee lack text SEE cn 1 Tp Sees Ses Negative unstable Aircraft Modes o rad sec ig 03 0 0194075 zl 3 38857 0 102298 02 3 38957 0 102298 2 78231 1 The version 2 0 screens shown here have been edited to create version 3 0 of FDA CAD and minor differences will be seen in both presentation and functionality In particular version 3 0 does not include any handling qualities analysis tools and the Command Path Design button and supporting software has been removed 4 1 Basic airframe state space matrices Basic Aircraft Ax Bu y Cx Du Aircraft Dynamics File Edit View Insert Tools Window Help Longitudinal Dynamics of Basic Aircraft du 0 0672 0 02323 9 2 dw 0 396 1 729 E 3196 dq eoan La 0 2431 ES 3 1589 Ko q d8 0 0 mae 17 01 n 19 FDA CAD v 3 User Manual By pressing the Basic Aircraft button in the control structure block diagram the small Aircraft Dynamics window opens showing the basic aircraft state equation giving the user dir
40. le the gain Ka would be designed first to restore the static margin to the desired value With the chosen value of K entered into the appropriate feedback gain field push the RUN MODEL button to obtain the closed loop equations of motion with the a loop closed Using this closed loop model FDA CAD can now be used to design a value for K to restore the pitch damping to an acceptable value as described above Entering the chosen value of feedback gain into the Kq field and pushing the RUN MODEL button produces the equations of motion with both feedback loops closed Push the Print Report button to see the full solution vii At any time during the analysis the user may add an actuator to the system model The actuator transfer function numerator and denominator are entered into the actuator window fields and pushing the RUN MODEL button updates the closed loop system 45 FDA CAD v 3 User Manual model accordingly The actuator mode then appears in the Aircraft Modes window With no feedback gains in the model the solution is then simply the basic aircraft augmented to include the open loop actuator dynamics viii Response time histories can be obtained at any time during the development of a feedback gain structure and FDA CAD enables two sets of time histories to be compared to illustrate the effects of feedback and other system changes In the present example compare the basic aircraft with the augmented aircraft
41. locity Lateral velocity Normal velocity OX axis Oy axis OZ axis Angle of attack or incidence Rudder Elevator Pitch Aileron 255 FDA CAD v 3 Stability Derivatives Notation FDACAD Xu Xw Np Abbreviations FDA CAD GUI FDP Xu Xw Xdw Xq Xde Zu Zw Zdw Zq Zde Mu Mw Mdw Mq Mde Yv Yp Yr Yda Ydr Lv Lp Lr Lda Ldr Nv Np Nr Nda Ndr User Manual Description Axial velocity due to velocity Axial velocity due to incidence Axial velocity due to downwash lag Axial velocity due to pitch rate Axial velocity due to elevator Normal velocity due to velocity Normal velocity due to incidence Normal velocity due to downwash lag Normal velocity due to pitch rate Normal velocity due to elevator Pitching moment due to velocity Pitching moment due to incidence Pitching moment due to downwash lag Pitching moment due to pitch rate Pitching moment due to elevator Side force due to lateral velocity Side force due to roll rate Side force due to yaw rate Side force due to aileron Side force due to rudder Rolling moment due to lateral velocity Rolling moment due to roll rate Rolling moment due to yaw rate Rolling moment due to aileron Rolling moment due to rudder Yawing moment due to lateral velocity Yawing moment due to roll rate Yawing moment due to yaw rate Yawing moment due to aileron Yawing moment due to rudder Flight Dynamics Analysis Command Augmentation De
42. losed loop feedback gain design add an actuator to the model and plot and compare time responses for two different system designs The feedback gains can be designed either by the pole placement technique or by plotting root loci in the reserved plotting area To avoid excessive congestion in the screen window for lateral directional system design the feedback structure is shown separately for the lateral and for the directional applications by selecting the appropriate axis reference from the pop up menu in the top of the GUI screen to the left of the file name field The screen appearance is very similar to the basic aircraft screen and many of the functional items are the same The longitudinal and lateral stability augmentation screens are shown below The longitudinal stability augmentation design and analysis screen v 2 Stability Augmentation Screen se Edit View Insert Tools Window Help Figure Input Signal Step Cherlong mat Basic Aircraft X Ax Bu Du kag Short Period 41492 99 05905 Phugoid 0 2501 0 1101 Ji Negative unstable Blue text Black text tealpole 1 Tp complex pole pair Aircraft Modes o rad sec i 0 250101 0 110113 0 250101 0 110113 414818 0 53049 414818 0 53049 Outputs gt F NE Plot Re 18 FDA CAD v 3 User Manual The latera
43. may be entered directly when Normalised USA format is selected 3 2 4 Model order selection From the Model Dynamics pop up menu the user can choose between Full Order or Reduced Order analysis for the longitudinal aircraft model only The Full Order Only model is used for lateral directional analysis when the pop up menu window is deactivated Model Dynamics Full Order Dimensional UK 3 2 5 Aircraft axes selection The pop up axes selection menu choice does not influence the model analysis directly but it is used in the derivative conversion process For the model analysis process the axes notation is made clear by the presence or absence of a non zero PO incidence value 3 2 6 Stability and control derivative conversion By pressing the Derivative Conversion button a new menu appears to the left of the screen where the user may choose a particular derivative conversion By pressing the Calculate button a report is generated in the Matlab command window an example of which is shown below From Normalised Wind Axes to 0 06728 0 396 0 02323 1728 C Wind Axes i 0 o C Dimensional UK C Normalised I5 um 2 C Body Axes rdf 2 0 E 1701 C Dimensionless UK 1 Full Order irerait eur Cancel Calculate FDA CAD v 3 User Manual General Flight Conditions Mach 0 15 Height 1500 ft Body Incidence 0 00 deg Original Axe
44. me limited further adjustment to the root locus plot can be made to improve the analytical accuracy of its interpretation From a right click on the plot the properties editor can be opened from which some changes can be made Short Period Phugoid 45027 0 2305 07496 G 01195 T System untitled Gain 0 0417 5h Pole 3 38 2 98i ia CC BE 4 Damping 0 75 Cd 3 3 lt Overshoot 36 2 85 Frequency radisec 4 5 4 7 Root locus plot undocking When the user wishes to adjust and annotate the root locus plot further for the purpose of saving or printing for example it is convenient to open the plot in a standard Matlab figure window From the screen menu tool bar click on Root Locus Plot and undock the figure Figure in FDA CAD version 2 0 as illustrated This action opens a new Matlab figure window from which the plot can be saved printed or further edited 4 8 Input signal options Figure Undock Figure Root Locus plot File Edit View Insert Tools Window Help 922 Pitch Rate feedback Root Locus See paragraph 3 2 9 Input signal options for details 23 FDA CAD v 3 4 9 Response options See section 3 2 10 Response options for details 4 10 Plotting options The plotting options are the same as described in paragraph 3 2 11 Plotting options for the basic aircraft with an additional opti
45. n be safely approximated by zero Accordingly enter a value of zero in the feedback gain fields Choose also to round the pitch rate feedback gain to Kg 0 1 and this value should be entered in the appropriate feedback gain field Push the RUN MODEL button to solve the revised closed loop equations of motion Push the Print Report button and obtain the revised closed loop solution report as shown below LONGITUDINAL REPORT FOR FDP Example 11 6 mat 2 E xdot Bu y Cx Du xT u w q theta Control Law d delta K u K 0 0 0 1 0 Feedback gain matrix showing single gain value STATE SPACE MATRICES A 0 067703 0 0107 1 9233 9 8099 0 021926 2 1031 363 5 0 051198 0 010257 0 15507 9 1795 0 00012247 0 0 1 0 B_transposed 0 4023 76 257 60 918 0 C 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 5 FDA CAD v 3 User Manual 0 021926 2 1031 11 493 0 051198 0 0 0026667 0 0 0 0 0026667 0 1 0 1 0 375 D transposed 0 0 0 O O O O O TRANSFER FUNCTIONS FACTORISED FORM Zero pole gain from input to output u Axial velocity 0 4023 s 305 2 s 6 245 s 1 488 s 2 0 068915 0 002364 s72 11 285 75 64 Zero pole gain from input to output w Normal velocity 76 2574 s 299 6 s 0 07117 s 0 00321 s 2 0 068915 0 002364 s72 11 285 75 64 Zero pole gain from input to output q Pitch rate 60 9176 s s 1 909 5 0 06782
46. nually into the gain fields in the feedback paths Pressing the RUN MODEL button calculates the closed loop model which may 20 FDA CAD v 3 User Manual then be analysed for performance By this means feedback loops can be closed and evaluated one by one until the desired closed loop system performance is achieved 4 4 Actuator model For the actuator model two fields are provided In the upper field the numerator gain is entered and in the lower field the denominator polynomial is inserted in Matlab format For example if the actuator transfer function is 450 A a S 52 305 450 then the denominator must be inserted as 1 space 30 space 450 written 1 30 450 Version 3 0 of FDA CAD supports a zero order numerator gain and first or second order actuator denominator only An error message appears on screen if these limits are violated Pressing the yellow Actuator button opens the actuator Bode plot in a LTI figure window 4 5 Pole placement The general mode characteristics that appear in the aircraft Sannoo modes fields also appear in the pole placement fields For the longitudinal case the Short Period and the Phugoid O Ju Short Period Phugoid mode characteristics are given while for the lateral LEO erum 05905 G 01101 Blue text Black text real pole 1 Tp complex pole pair directional case the Dutch Roll Spiral and Roll mode Negative unstable char
47. on for comparing two sets of plots For this purpose three additional buttons are provided Save Plot 1 Save Plot 2 and Compare Plots 1 and 2 By toggling the Save Plot 1 button after making a set of response plots the figure window is closed and the plot data is stored If then a second set of response plots is made for a modified closed loop system model for example these may also be stored by toggling the Save Plot 2 button By pressing the Compare Plots 1 and 2 button a new figure window opens where the checked responses are compared for the two instances as illustrated here A legend appears on the plots and this may be edited prior to printing if required User Manual Plot Response Histories KAAS PLD Elevator Command To q Pitch rate Elevator Command To 6 Pitch attitude Cherlong mat Note that when comparing plots the Show TF s button is deactivated 4 11 FDA CAD report By pressing the Print Report button a report similar to that generated from the basic aircraft screen see paragraph 3 2 13 FDA CAD report is printed in the Matlab command window The same information is output for the augmented aircraft model and its corresponding feedback gain matrix 4 12 Exporting variables to Matlab workspace 24 FDA CAD v 3 User Manual Pressing the Export to Workspace button places a copy of all t
48. oot locus plot undocking Input signal options Response options 4 10 Plotting options 4 11 FDA CAD report FDA CAD v 3 User Manual 4 12 Exporting variables to Matlab workspace 4 13 Return to basic aircraft screen 4 14 Close FDA CAD 5 EXAMPLES 6 REFERENCES 7 NOTATION FDA CAD v 3 User Manual 1 GETTING STARTED WITH FDA CAD 1 1 System Requirements Matlab Version 7 0 4 or higher Matlab Control Toolbox Version 6 2 or higher Adobe Acrobat Reader Version 8 or higher 1 2 Installation Copy all the files to a directory of your choice preferably a new folder named FDA CAD Within Matlab add this folder to the Matlab path This is achieved by selecting the File Menu and choose Set Path click on Add Folder using the file browser select the newly created folder and click OK Alternatively the folder can be added by selecting it to be the temporary current Matlab directory from the option in the Workspace Window toolbar 1 3 Execution Within Matlab set the Current Directory to the newly created folder From the Matlab Command Window type fdacad3 and press enter to start the program 1 4 Screen Size FDA CAD was designed to operate on a screen with size properties e Screen resolution 1024x768 pixels e Screen aspect ratio 4 3 e Screen physical size 12x9 inches 305x229mm 1 3 Directory Path In order that the About and User Guide PDF files can be opened from within FDA CAD it i
49. pole gain from input to output w Normal velocity perturbation 17 01 s 129 2 s 2 0 066675 0 07902 s 2 0 055083 0 06255 s Z 4 95 17 22 Zero pole gain from input to output q Pitch rate 44 3749 s 341 63 s 0 07321 s 2 0 05508s 0 06255 s 2 4 98 17 22 Zero pole gain from input to output theta Pitch attitude 44 3749 s 1 63 3 0 07321 s 2 0 05508s 0 06255 s42 4 9s 17 22 Zero pole gain from input to output Az Normal acceleration 103 9779 s 37 76 s 2 0 02851s 0 004665 s42 0 055085 0 06255 s 2 4 95 17 22 Zero pole gain from input to output alpha Angle of attack 0 3402 s 129 2 s 2 0 06667s 0 07902 s 2 0 05508s 0 06255 s 2 4 9s 17 22 Zero pole gain from input to output gamma Flight path angle 0 3402 s 15 21 s 14 03 2 0 02508 s 2 0 05508s 0 06255 s 2 4 95 17 22 Zero pole gain from input to output h height 17 01 3 15 21 3 14 03 0 02508 2 2 0 055085 0 06255 s 2 4 95 17 22 CHARACTERISTIC POLE LOCATIONS Eigenvalue Damping Freq rad s 2 75e 002 2 49e 001i 1 10 001 2 50 001 2 75e 002 2 49 001 1 10 001 2 50 001 2 45 000 3 35 0001 5 90 001 4 15 000 2 45 000 3 35 0001 5 90 001 4 15 000 16 User Manual FDA CAD v 3 User Manual 3 2 14 Exporting variables to Matlab workspace Pressing the Export to
50. r example a flight data or derivative value is corrected or changed 3 2 8 Stability modes When the equations of motion are solved the stability modes characteristics are shown in the Aircraft Modes rad sec ta aircraft modes window The mode frequencies and 0 250101 0 110113 0 250101 0 110113 damping ratios are shown When the longitudinal 4 14818 0 53043 4 14918 0 59049 reduced order solution is selected then the short period mode characteristics are shown together with the pitch attitude lag since the latter variable is retained in the model to ensure consistency with both wind and body axes referenced solutions 3 2 9 Input signal options This pop up menu provides the user with a choice of input signal types These include Step Input Signal Impulse Pulse Ramp Sine wave and Doublet The input signal is taken to mean aileron elevator or rudder angle depending on context and the default units are radians The input signal properties can be can be selected which includes the magnitude in radians the pulse or doublet width in seconds and the sine wave frequency in radians per second The step input is the default input signal and the Mag button toggles between an input magnitude of 1 radian or 1 degree expressed in radians 0 0175 The user can also enter an alternative value in the magnitude field 212 FDA CAD v 3 User Manual 3 2 10 Response options The
51. ransfer functions Example 7 3 This example illustrates the use of FDA CAD to obtain lateral directional Bode frequency response plots from the basic aircraft transfer functions The aircraft model is the same as that used in Example 7 1 i With the model data loaded to FDA CAD push the RUN MODEL button followed by the Print Report button Retain the report for reference to the transfer functions of interest A1 FDA CAD v 3 User Manual ii Push the yellow button in the aileron column of the output window to open the Bode iii plot figure window for that variable The Bode plot compares directly with Fig 7 7 Use the Matlab tools to change the scales of the Bode gain and phase plots to match those of Fig 7 7 A print out from the figure window is shown below and it will be seen to compare favourably with Fig 7 7 Repeat the process by pushing the yellow P button in the aileron column of the output window to open the Bode plot figure window for that variable As before use the Matlab tools to change the scales of the Bode gain and phase plots to match those of Fig 7 8 A print out from the figure window is shown below and it will be seen to compare favourably with Fig 7 8 Example 7 3 Roll attitude frequency response to ai leron 40 20 Magnitude dB 45 90 Phase deg 135
52. root locus or pole placement computation for designing feedback gains Longitudinal analysis of the basic aircraft may be undertaken using either the full order model or the reduced order model but with this version of FDA CAD longitudinal stability augmentation design may only be undertaken with the full order model All lateral directional analysis can only be undertaken with the full order model The stability augmentation analysis tools also permit the direct comparison of unaugmented and augmented time response plots on the same axes for the selected output variables Longitudinal closed loop system design permits feedback path to elevator gain design for axial velocity Ky normal velocity Ky pitch rate K and pitch attitude K in any combination The design of feedback loop gains to the thrust input is beyond the scope of version 3 0 of FDA CAD v 3 User Manual FDA CAD Longitudinal handling qualities assessment and command path design are also beyond the scope of version 3 0 of FDA CAD Lateral Directional closed loop system design and analysis provides feedback path gains for sideslip B roll rate p yaw rate r and roll attitude to aileron and or rudder Lateral closed loop feedback to aileron and directional closed loop feedback to rudder are dealt with on two different screens The directional screen also incorporates the facility for designing the aileron rudder interlink gain The lateral directional loops may be closed in any com
53. s examples ii It is left as an exercise for the user to investigate the lateral directional feedback design capabilities of FDA CAD Example 11 6 The purpose of this example is to show the pole placement capability of FDA CAD in the design of a simple full state feedback matrix Aircraft model data will be found in the file FDP Example 11 6 mat which should loaded for longitudinal analysis in FDA CAD Note that the data have been transferred to SI units throughout i Push the RUN MODEL button followed by the Print Report button to obtain an overview of the basic aircraft state equation and also the stability modes characteristics Extracts from the report required for the example are shown below LONGITUDINAL REPORT FOR FDP Example 11 6 mat L Bu O u T Eta u w q theta Control Law eta_d delta K u K 0 0 0 0 STATE SPACE MATRICES Basic unaugmented aircraft A 0 0677 0 0107 1 9635 9 8099 0 022527 2 1031 371 13 0 051198 0 010736 0 15507 3 0877 0 00012247 0 0 1 0 B_transposed 0 4023 76 257 60 918 0 48 FDA CAD v 3 User Manual C 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 022527 2 1031 3 8677 0 00016691 0 0 0026667 0 0 0 0 0026667 0 1 0 1 0 375 D transposed 0 0 0 0 76 257 0 0 0 Continues to CHARACTERISTIC POLE LOCATIONS Eigenvalue Damping Freq
54. s necessary to change the path information in the program Open the fdacad3 m file in the Matlab Editor and enter the correct path for your installation in lines 586 590 and 594 FDA CAD v 3 User Manual 2 FDA CAD OVERVIEW 21 Flight Dynamics Analysis Version 3 0 of FDA CAD is designed explicitly to accompany the book Flight Dynamics Principles FDP by the author The program facilitates the rapid solution and analysis of the linear equations of motion of an aircraft and incorporates tools that accommodate various notational styles The notation and symbols correlate with those given in FDP as far as the limitations of the program language permit In general interpretation of the GUI screens and the notation adopted should be obvious to those familiar with the subject FDA CAD facilitates Longitudinal and Lateral Directional stability and control analysis of an aircraft given the flight condition data and stability and control derivatives referenced to either wind or body axes Results obtained from the analysis include the matrix state equation matrix output equation response transfer functions time history plots system pole and zero descriptions and stability mode characteristics A derivative conversion feature enables the user to calculate derivative values in different formats and referred to alternative reference axes Stability augmentation system design tools include provision for the inclusion of an actuator model and
55. s notation WIND AXES 0 06728 Zu 0 39600 Mu 0 00000 0 02323 Zw 1 72900 027720 0 00000 Zdw 0 00000 0 01970 0 00000 Zq 1 68040 Mq 2 20700 0 00000 Zeta 17 00000 Meta 44 71000 0 18449 Zu 1 08589 Mu 0 00000 0 06370 Zw 4 74116 Mw 0 74094 0 00000 Zdw 0 00000 Mdwz 1 64554 0 00000 Zq 2 87993 Mq 3 68701 0 00000 Zeta 0 93288 Meta 2 39016 The conversion does not change the visible model used in the on screen analysis Its purpose is to provide a convenient off line tool for use when derivative values in an alternative format are required by the investigator It is essential to exercise care when using this tool as some conversions will require a non zero value for the body incidence a data field which will of course be set to zero if the original model data is referred to wind axes It is prudent to return the incidence value to its initial model value after the conversion calculation is completed to avoid errors in any further interactive analysis A warning dialogue window opens in these situations to provide a reminder 11 FDA CAD v 3 User Manual 3 2 7 Solution of the equations of motion By pressing the RUN MODEL button the aircraft equations of motion are solved for the given input data If a wind axes notation is selected and the body incidence is non zero a warning message is displayed before proceeding This function can be invoked as many times as necessary when fo
56. sign Graphical User Interface Flight Dynamics Principles 3rd Edition
57. uation 6 6 and the transfer functions with those given in equations 6 8 LONGITUDINAL REPORT FOR FDP Example 6 1 mat Were ok Be SO TF Se Bu y Cx Du OR ie cen m he tte Pom se u T Eta xT u Ww q theta Control Law eta d delta K u K 0 0 0 0 004 FDA CAD v 3 User Manual STATE SPACE MATRICES A 0 00501 0 00464 72 926 31 336 0 0857 0 545 308 5 7 4076 0 0018453 0 007673 0 39491 0 0013186 0 0 1 0 B_transposed 5 63 23 8 4 5158 0 C 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0857 0 545 0 7 4076 0 0 0031546 O 0 0 0 0031546 O 1 0 1 0 317 D transposed 0 0 0 0 23 8 0 0 0 TRANSFER FUNCTIONS FACTORISED FORM Zero pole gain from input to output Az Normal acceleration 23 8 8 845 664 s 5 226 s 0 02774 s 2 0 03325 0 01971 s 2 0 90175 2 662 233 FDA CAD v 3 User Manual Zero pole gain from input to output dh height rate 23 8 s 6 137 s 4 961 s 0 02719 s 2 0 03325 0 01971 s 2 0 90175 2 662 CHARACTERISTIC POLE LOCATIONS Eigenvalue Damping Freq rad s 1 66e 002 1 39e 001i 1 18 001 1 40e 001 Phugoid mode damping and frequency 1 66e 002 1 39e 0011 1 18e 001 1 40e 001 4 51 001 1 57 0001 2 76e 001 1 63e 000 Short period mode damping and frequency 4 51 001 1 57 0001 2 76e 001 1 63 000
58. uency and overshoot corresponding with that specific locus test point Important Some aircraft transfer functions are negative and give rise to an incorrect root locus plot To avoid problems with interpretation and correct gain design when using the root locus plot transfer function sign is checked and the sign of negative transfer functions is ignored Thus all root locus plots are made with positive transfer functions When the natural aircraft transfer function is negative the chosen feedback gain must also be negative to ensure stabilising negative feedback A warning is shown on the root locus screen advising the correct sign of the feedback gain When the pole location corresponding with the desired stability characteristics is chosen the feedback gain value is read from the text box and inserted manually into the appropriate feedback gain field not forgetting the correct sign In the illustration below the aim is to achieve 0 75 damping ratio for the Short Period mode using pitch rate feedback to elevator The required feedback gain is shown in the text box as 0 0417 This value is then inserted in the K feedback gain field with a NEGATIVE sign since the natural aircraft transfer function describing pitch rate response to elevator is negative Pressing the RUN MODEL button the aircraft modes fields as well as the Short Period and Phugoid fields are updated to the new closed loop values 22 FDA CAD v 3 User Manual So
59. yp was approximated by zero and this has disturbed some of the transfer functions as will be seen by comparing those obtained with FDA CAD with those given by equations 5 61 and 5 62 Be aware also that the sideslip transfer functions in the example refer to sideslip velocity whereas those produce by FDA CAD refer to sideslip angle Dividing the gain of the sideslip velocity transfer functions in the example by aircraft velocity converts the sideslip variable to sideslip angle which compares directly with the transfer functions given in the FDA CAD solution Example 6 1 This example shows the full solution of the longitudinal equations of motion and some typical response plots for the unaugmented aircraft The aircraft model data should be loaded into FDA CAD from file FDP Example 6 1 mat i Press the RUN MODEL button This solves the equations of motion and in the process the longitudinal state equation a full set of response transfer functions and the longitudinal stability modes are calculated ii Press the Print Report button Open the Matlab Command Window to see the solution report As before the state equation appears first followed by the full list of response transfer functions and concludes with a summary of the stability modes characteristics The full report is shown below and the transfer functions relevant to the example are highlighted In particular the state equation may be compared with eq
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