Robust Control. 8th class. Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) Tue., 29th May, 2018, 10:45~11:30, S423 Lecture Room

Transcription

1 Robust Control Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) 8th class Tue., 29th May, 2018, 10:45~11:30, S423 Lecture Room 1

2 8. Design Example 8.1 HiMAT: Control (Highly Maneuverable Aircraft Technology) Reference: [SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design, Second Edition, Wiley, [SLH81] M. G. Safonov, A. J. Laub and G. L. Hartmann, Feedback Properties of Multivariable Systems: The Role and Use of the Return Difference Matrix, IEEE Transactions on Automatic Control, Vol. 26, No. 1, pp , [ES84] M. B. Evans and L. J. Schilling, The Role of Simulation in the Development and Flight Test of the HiMAT Vehicle, NASA Technical Memorandum 84912, [CS98] R. Y. Chiang and M. G. Safonov, Robust Control Toolbox User s Guide, The MathWorks,

3 HiMAT: Attitude and flight path control Input 6.86m Output m Objective Control attitude ( ) and flight path ( MIMO System [Ex.] ) separately Control attitude without changing flight path Specification Robustness Spec.: db/dec roll-off and db at 100rad/s (Because of the unmodeled low-damped structual mode) Performance Spec.: Minimize the sens. function as much as possible 3

4 Modeling HiMAT 3 5 Interaction (Coupling) Frequency Response Poles Zeros Unstable 4

5 MIMO Loop Shaping [SP05, p. 343] 6 9 Robustness Performance Unstable Pole: (Roll-off) Time Delay: 0.025s 40 rad/s Actuator dynamics Disturbance (Gust): Performance Weight 1.43 rad/s Uncertainty Weight 30 rad/s 5

6 HiMAT: Uncertainty Weight Uncertain Factors: Actuator Dynamics + Unmodeled dynamics (High freq.) wm = tf([1 0 0], [1000]); WM = eye(2)*wm; Frequency: 2 Oscillation mode (2nd order system) Natural angular frequency Damping factor Target Loop 0.033, 30 rad/s Roll-off: 6 db at 100rad/s db/dec

7 HiMAT: Performance Weight , Ms = 2; A = 0.01; wb = 1.5; wp = tf([1/ms wb], [1 wb*a]); WP = eye(2)*wp; Update Larger (Trade-off) 7

8 HiMAT: Control Problem Formulation Nominal Performance Robust Stability Generalized Plant sysic %Generalized Plant% systemnames = 'Pnom WP WM'; inputvar = '[w(2);u(2)]'; outputvar = '[WP;WM;-w-Pnom]'; input_to_pnom= '[u]'; input_to_wp = '[w+pnom]'; input_to_wm = ' [Pnom]'; G = sysic; Mixed Sensitivity 8

9 HiMAT: -iteration to obtain Find such that Check 1) Appropriately sub-optimal (Default settings) Check 2) No Solution Data Structure Controller [SP05, p. 358] Resetting value of Gamma min based on D_11, D_12, D_21 terms Test bounds: < gamma <= gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f e e e e p e e+06# 1.6e e f e e+06# 1.6e e f e e+07# 1.6e e # f e e e e # f e e e e # f e e e e # f e e e e # f Gamma value achieved: Closed-loop Transfer Function There is No Controller [Khi,CLhi,ghi,hiinfo] = hinfsyn(g,nmeas,ncon, Display, on ); [SV,w]=sigma(CLhi); figure; semilogx(w,sv) 9

10 HiMAT: Controller figure sigma(khi) [Ex.] Poles Zeros Order 8 Model Reduction 10

11 HiMAT: Open-loop Frequency Response corresponding maximum stability margin Loop Transfer Function figure sigma(fhi.lo,wp, inv(wm),wp/ghi,ghi*inv(wm)) NP/RS Test Nominal Performance (NP) Robust Stability (RS) [SV,w]=sigma(WP*Fhi.So); figure; semilogx(w,sv) [SV,w]=sigma(WM*Fhi.To); figure; semilogx(w,sv) 12

12 HiMAT: Closed-loop Performance Nominal Stability (NS) Poles of pole(clhi) zero(clhi) figure; pzmap(clhi) Zeros of Nominal Performance (NP) Robust Stability (RS) 11

13 HiMAT: Step Response (closed loop) Reference Nominal Model Perturbed Model 13

14 8. Design Example 8.1 HiMAT: Control (Highly Maneuverable Aircraft Technology) Reference: [SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design, Second Edition, Wiley, [SLH81] M. G. Safonov, A. J. Laub and G. L. Hartmann, Feedback Properties of Multivariable Systems: The Role and Use of the Return Difference Matrix, IEEE Transactions on Automatic Control, Vol. 26, No. 1, pp , [ES84] M. B. Evans and L. J. Schilling, The Role of Simulation in the Development and Flight Test of the HiMAT Vehicle, NASA Technical Memorandum 84912, [CS98] R. Y. Chiang and M. G. Safonov, Robust Control Toolbox User s Guide, The MathWorks,

15 Class evaluation by students Class Name : Robust Control Code of Class : SCE.C402 Instructor : Prof. Masayuki Fujita How to fill out questionnaire Use Pencil only (Do not use Pen) Mark Carefully 15

16 NASA HiMAT Real Product 1 General characteristics Crew: None Length: 6.86m Wingspan: 4.75m Height: 1.31m Max takeoff weight: 1,542kg Performance Maximum speed: Mach 1.6 Maneuvering: 8G performance 16

17 NASA HiMAT Real Product Testing Product UAV(Unmanned Aerial Vehicle) Teleoperation 2 Testing high technology HiMAT Classic Aircraft 17 20

18 Modeling HiMAT Control Input 4.75m 6.86m Output 3 : angle of attack : attitude angle : elevon actuator : canard actuator states to represent actuator dynamics 18

19 HIMAT: Nominal Plant Model State Space Form (Matrix Representation) 4 % NASA HiMAT model G(s) ag =[ e e e e e e-01; e e e e e e-03; e e e e e e+01; e ; e+01 0; e+01]; bg = [ 0 0; 0 0; 0 0; 0 0; 30 0; 0 30]; cg = [ ; ]; dg = [ 0 0; 0 0]; G=ss(ag,bg,cg,dg); 19

20 HiMAT: Nominal Plant Model Controllability Observability Poles (Stability) Zeros Frequency Response , , -30, -30, i Unstable Step Response for Nominal Plant Model

21 Actuators: Hydraulic serve system 6 Block diagrams of elevon Bode diagram of actuators Saturation Rate limit [deg] 30rad/s Block diagrams of canard Saturation Rate limit [deg] 21

22 Actuators: Hydraulic serve system Block diagrams of elevon 7 Symmetric input [deg] Saturation Rate limit [deg] Asymmetric input [deg] Saturation 0.5 Scaling Saturation Rate limit [deg] Bode diagram of actuators Block diagrams of canard Rate limit [deg] 30rad/s Symmetric canard pulse 24

23 Loop Time: Control Input time 8 Onboard computer Processing Time 25 ms (40Hz) Downlink 110kbits/s Uplink 53.3times/s 23

24 Models: Understand Performance Specifications 9 : Process Noise Process STEP 6 STEP 5 Actuators [Nm] [rad] Sensors Process D/A Decision STEP 7 Controller Information A/D Controller 24

25 HiMAT: Closed-loop Performance (without update ) Nominal Stability (NS) Poles of 〇 10 Zeros of Nothing 〇 Nominal Performance (NP) Pole/Zero Cancellations Robust Stability (RS) 1.5rad/s 25

26 HiMAT: Controller (without update ) figure sigma(khi) 11 Poles Zeros Order 8 Numerical problems or inaccuracies may be caused too high order Difficult to implement 26

27 HiMAT: Open-loop Frequency Response (without update ) corresponding maximum stability margin Loop Transfer Function 12 figure sigma(fhi.lo,wp, inv(wm),wp/ghi,ghi*inv(wm)) NP/RS Test Nominal Performance (NP) Robust Stability (RS) [SV,w]=sigma(WP*Fhi.So); figure; semilogx(w,sv) [SV,w]=sigma(WM*Fhi.To); figure; semilogx(w,sv) 27

28 HiMAT: -iteration to obtain Controller (without update ) Find such that Check 1) Appropriately sub-optimal (Default settings) [SP05, p. 358] Closed-loop Transfer Function 13 Check 2) Test) Tune No Solution and sub-optimality Data Structure There is No Controller interested frequency range [Khi,CLhi,ghi,hiinfo] = hinfsyn(g,nmeas,ncon, Gmax,100, Gmin,100); [SV,w]=sigma(CLhi); figure; semilogx(w,sv) 28

29 HiMAT: Step Response of closed loop (without update )

30 HIMAT: Specifications and Open-loop (Loop Shaping Synthesis) Approximately a bandwidth of 10 rad/s Desired Loop Shape 15 Target loop Gd= tf( 10, [1.001] ) ; [K,CL,GAM,INFO]=loopsyn(G,Gd); sigma(gd, b',g*k,'r',gd/gam, g:',gd*gam, g:',{.1,100}) 30

31 HIMAT: Controller Model Reduction (Loop Shaping Synthesis) 16 Order: 16 9 size(k) hsv = hankelsv(k); semilogy(hsv,'*--'), grid title('hankel singular values of K'), xlabel('order') Kr = reduce(k,9); order(kr) Step responses sigma(k,'b',k-kr,'r-.') legend('k','error K-Kr') 31

32 HIMAT: Sensitivity (Loop Shaping Synthesis) 17-3 T = feedback(g*k,eye(2)); S = eye(2)-t; sigma(inv(s),'m',t,'g',l,'r--',gd,'b',gd/gam,'b:',gd*gam,'b:',{.1,100}) 32

33 HIMAT: Designed Weight (Loop Shaping Synthesis) 18 sigma(info.gs, b',g,'r',info.w, g',{.01,100}); 33

34 HIMAT: Controller (Loop Shaping Synthesis) 19 Order: 16 sigma(k); Numerical problems or inaccuracies may be caused too high order Difficult to implement 34

35 HIMAT: Step Response of closed loop (Loop Shaping Synthesis)