1. AAE 666 – H ∞ Controller Design AAE 666: Nonlinear Dynamics Systems and Control - An H ∞ Controller For The Mixed Sensitivity Problem Moritz Niendorf 05/05/2009

2. AAE 666 – H ∞ Controller Design The Mixed Sensitivity Problem Singular Value Diagram σ i =λ i (G ∗ (jω)G(jω))=λ i (G(jω)G ∗ (jω))

3. AAE 666 – H ∞ Controller Design The Mixed Sensitivity Problem Closed Loop Diagram r→y: T y =(I p + GK) –1 GK Complimentary Sensitivity Function r→e: S y =(I p + GK) –1 Sensitivity Function With S y + T y = I p

4. AAE 666 – H ∞ Controller Design The Mixed Sensitivity Problem Desired shape of T y and S y

5. AAE 666 – H ∞ Controller Design The Mixed Sensitivity Problem Closed Loop Shaping

6. AAE 666 – H ∞ Controller Design The Mixed Sensitivity Problem Closed Loop Shaping

7. AAE 666 – H ∞ Controller Design The Mixed Sensitivity Problem Combining Requirements

8. AAE 666 – H ∞ Controller Design The Mixed Sensitivity Problem Find a controller that: guarantees internal stability of the closed loop satisfies H ∞ norm of G zw = is less than 1

9. AAE 666 – H ∞ Controller Design The H ∞ Control Problem Standard configuration

10. AAE 666 – H ∞ Controller Design The H ∞ Control Problem

11. AAE 666 – H ∞ Controller Design The H ∞ Control Problem

12. AAE 666 – H ∞ Controller Design The H ∞ Control Problem The Suboptimal H ∞ Control Problem: Find a Controller K for the Plant P and γ>0 so that: The Closed Loop System is stable => Algebraic solution The Optimal H ∞ Control Problem: Find a Controller K for the Plant P and a γ<0 so that: The Closed Loop System is stable is minimized => Iterative solution

13. AAE 666 – H ∞ Controller Design The H ∞ Control Problem State Space Solution State Space Representation of The Standard Problem Introduce a Controller K in State Space Form

14. AAE 666 – H ∞ Controller Design The H ∞ Control Problem Application to the Mixed-Sensitivity Problem Define: Recall: =>

15. AAE 666 – H ∞ Controller Design The H ∞ Control Problem Algebraic Riccati Equation Solution to the algebraic Riccati Equation with Hamiltonian Matrix Stabilizing Solution

16. AAE 666 – H ∞ Controller Design Algebraic Riccati Equation Conditions for Solvability of the Standard-Problem (A,B 2 ) is stabilizable, (C 2,A) is detectable rank(D 12 )=dim(u’)=q 2 rank(D 21 )=dim(e’)=p 2 Recall:

17. AAE 666 – H ∞ Controller Design The H ∞ Control Problem Solution Algorithm for the regular problem Conditions for Solvability: does not have imaginary Eigenvalues and exists does not have imaginary Eigenvalues and exists ρ(X ∞ Y ∞ ) < γ 2 with ρ being the largest eigenvalue

18. AAE 666 – H ∞ Controller Design The H ∞ Control Problem Solution Algorithm for the regular problem If the above conditions are satisfied, there exist matrices: A Controller K can then be obtained with: Minimum Entropy Controller

19. AAE 666 – H ∞ Controller Design The H ∞ Control Problem Iteration Algorithm to find the Optimal Controller: Define [γ inf, γ sup ] so that with 1. 2.If γ sup – γ inf < ε, then “stop” 3.If possible compute H ∞ and Y ∞, check ρ(X ∞ Y ∞ ) < γ 2 If a solution exists: γ sup = γ Elseγ inf = γ Go to 1.

20. AAE 666 – H ∞ Controller Design Example Using the Matlab Robust Control Toolbox: [K,CL,GAM,INFO]=MIXSYN(G,W1,W2,W3) or [K,CL,GAM,INFO]=MIXSYN(G,W1,W2,W3,KEY1,VALUE1,KEY2,VALUE2,...) MIXSYN H-infinity mixed-sensitivity synthesis method for robust control design. Controller K stabilizes plant G and minimizes the H-infinity cost function || W1*S || || W2*K*S || || W3*T ||Hinf where S := inv(I+G*K) % sensitivity T := I-S = G*K/(I+G*K) % complementary sensitivity W1, W2 and W3 are stable LTI 'weights'

21. AAE 666 – H ∞ Controller Design Example Using the Matlab Robust Control Toolbox: Inputs: G LTI plant W1,W2,W3 LTI weights (either SISO or compatibly dimensioned MIMO) To omit weight, use empty matrix (e.g., W2=[] omits W2) Optional Inputs: KEY1 MIXSYN supports all of the key & value input pairs VALUE1 available to the function HINFSYN; see the etc. HINFSYN documentation for further information Outputs: K H-infinity Controller CL CL=[W1*S; W2*K*S; W3*T]; weighted closed-loop system GAM GAM=hinfnorm(CL), closed-loop H-infinity norm INFO Information STRUCT, see HINFSYN documentation for details

22. AAE 666 – H ∞ Controller Design Example Using the Matlab Robust Control Toolbox: Pitch Controller for a Jet aircraft. State Space representation with 6 states: α, θ, α_dot, θ_dot, elevon and canard dynamics. 2 Control Inputs: elevon and canard position 2 Outputs: α, θ

23. AAE 666 – H ∞ Controller Design Example Using the Matlab Robust Control Toolbox: % Set up the performance and robustness bounds W1 & W3 s=zpk('s'); % Laplace variable s MS=2;AS=.03;WS=5; W1=(s/MS+WS)/(s+AS*WS); MT=2;AT=.05;WT=20; W3=(s+WT/MT)/(AT*s+WT); % Compute the H-infinity mixed-sensitivity optimal sontroller K1 [K1,CL1,GAM1]=mixsyn(G,W1,[],W3);

24. AAE 666 – H ∞ Controller Design Example % Next compute and plot the closed-loop system. % Compute the loop L1, sensitivity S1, and comp sensitivity T1: L1=G*K1; I=eye(size(L1)); S1=feedback(I,L1); % S=inv(I+L1); T1=I-S1; % Plot the results: % step response plots step(T1,1.5); title('\alpha and \theta command step responses'); % frequency response plots figure; sigma(1/(I+L1),'--',T1,':',L1,'r--',... W1/GAM1,'k--‘,W3/GAM1,'k-.',{.1,100});grid legend(‘\sigma(S) performance',... '\sigma(T) robustness',... '\sigma(L) loopshape',... '\sigma(W1) performance bound',... '\sigma(W3) robustness bound');

25. AAE 666 – H ∞ Controller Design Example

26. AAE 666 – H ∞ Controller Design Example

27. AAE 666 – H ∞ Controller Design Conclusion The Mixed-Sensitivity-Problem has been introduced The H ∞ -Control-Problem has been introduced The Transformation of the Mixed-Sensitivity-Problem has been shown A solution algorithm for the regular H ∞ -Control-Problem has been shown An example using Matlab has been presented

28. AAE 666 – H ∞ Controller Design References Hanel, M., Kämpf, B., Robuste Regelung, Lecture Notes (in German), Universität Stuttgart, 1998 Balas, G. et al, Robust Control Toolbox TM 3 - Getting Started Guide, The Mathworks, 2009 McFarlane, D., Glover, K., A Loop Shaping Design Procedure Using H ∞ Synthesis, IEE Transactions on automatic control, Vol. 37, No.6, June 1992 Stoorvogel, A., The H ∞ control problem: a state space approach, University of Michigan, 2000 Kwakernaak, H., Robust Control and H ∞ -Optimization - Tutorial Paper, Automatica, Vol. 29, No. 2, 1993, pp.255-273 Doyle, et al., State-Space Solutions to Standard H 2 and H ∞ Control Problems, IEEE Transactions on Automatic Control, Vol. 34, No.8, August 1989, pp. 831-847

29. AAE 666 – H ∞ Controller Design