Applied Robust Control, Chap 2, 2012 Spring 1 MEM800-007 Chapter 2 Sensitivity Function Matrices r e K u d y G Loop transfer function matrix: L GK Sensitivity function matrix: S ( I L) Complementary Sensitivity function matrix: 1 T L( I L) 1 1 y L( I L) r( I L) d Tr Sd 1 smaller S smaller worst-case disturbance response smaller T better robust stability u K I L rd R r d 1 ( ) ( ) ( ) smaller R smaller worst-case control input
Applied Robust Control, Chap 2, 2012 Spring 2 Physical meaning of H infinity Norm
Applied Robust Control, Chap 2, 2012 Spring 3 Unstructured Norm-bounded Uncertainties K u G I M G y K u G b M a G y b M T a Small Gain Theorem: Assume the nominal closed-loop system, T, is stable, then 1 the uncertain closed-loop system ( I T ) or ( I ) 1 GK is stable if and only if T( j ) 1 ( j ) for all. smaller T( j ) better robust stability M M
Applied Robust Control, Chap 2, 2012 Spring 4 Singular Values and Singular Value Decomposition Maximum singular value: * max ( X ) max ( X X)
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Applied Robust Control, Chap 2, 2012 Spring 9 Example 1: r e K u d y G 2500 GsKs () () ss ( 5)( s50) a) Find the gain and phase margins., ( ), so b) Find the least upper bound of ( j M ) that T( j) 1 ( ) closed-loop system with j robustly stable. and therefore the uncertain ( ) ( ) is M %File applrbstcntrl_3b_bode_sigma %GK=2500/s(s+5)(s+50) num=2500; den=[1 55 250 0]; L=tf(num,den); figure(1) bode(l);
Applied Robust Control, Chap 2, 2012 Spring 10 20 Bode Diagram 10 0 Magnitude (db) -10-20 -30-40 -50-60 -90-135 Phase (deg) -180-225 -270 10 0 10 1 10 2 Frequency (rad/sec)
Applied Robust Control, Chap 2, 2012 Spring 11 %Nyquist plot figure(2) nyquist(l,{10,100}) 0.2 Nyquist Diagram 0.15 0.1 Imaginary Axis 0.05 0-0.05-0.1-0.15-0.2-1 -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 Real Axis
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Applied Robust Control, Chap 2, 2012 Spring 13 %Complementary function T T = feedback(l,1); % SIGMA frequency response plots figure(3) sigma(t,'g',{.01,100}) 10 Singular Values 0-10 Singular Values (db) -20-30 -40-50 -60 10 0 10 1 10 2 Frequency (rad/sec)
Applied Robust Control, Chap 2, 2012 Spring 14 >> sv=sigma(t,6.25) sv = 1.8323 Find the phase margin based on the singular value plot of T. >> sv=sigma(t,16) sv = 0.2159 Find the gain margin based on the singular value plot of T.
Applied Robust Control, Chap 2, 2012 Spring 15 %Sensitivity function S S=1-T; figure(4) sigma(inv(s),'m',t,'g',l,'r--',{.01,100}) 60 Singular Values 40 20 Singular Values (db) 0-20 -40 Note that -60 10-2 10-1 10 0 10 1 10 2 Frequency (rad/sec) 1 1 ( ) if L 1 S I L L ( ) if L 1 1 T L I L L and therefore we have 1 L( j) S( j) for low frequencies L( j ) T( j ) for high frequencies
Applied Robust Control, Chap 2, 2012 Spring 16 Example 2: r e K u d y G 1 Gs () s 1, K () s 3 a) Find the gain and phase margins. b) Find the least upper bound of M ( j ), ( ), so that T( j) 1 ( ) and therefore the uncertain ( j ) ( ) is closed-loop system with robustly stable. %File 635_3a_bode_sigma %G=1/(s-1), K=3 num=3; den=[1-1]; L=tf(num,den); figure(1) bode(l); %Nyquist plot figure(2) nyquist(l) M
Applied Robust Control, Chap 2, 2012 Spring 17 10 Bode Diagram 0 Magnitude (db) -10-20 -30-90 Phase (deg) -135-180 10-2 10-1 10 0 10 1 10 2 Frequency (rad/sec) Gain crossover frequency = rad/s Phase margin = degree Phase crossover frequency = rad/s Gain margin = db 1.5 Nyquist Diagram 1 0.5 Imaginary Axis 0-0.5-1 -1.5-3 -2.5-2 -1.5-1 -0.5 0 0.5 Real Axis
Applied Robust Control, Chap 2, 2012 Spring 18 %Complementary function T T = feedback(l,1); % SIGMA frequency response plot of T figure(3) sigma(t,'g', {.01,100}) 5 Singular Values 0-5 Singular Values (db) -10-15 -20-25 -30-35 10-2 10-1 10 0 10 1 10 2 Frequency (rad/sec)
Applied Robust Control, Chap 2, 2012 Spring 19 %Sensitivity function S S=1-T; figure(4) sigma(inv(s),'m',t,'g',l,'r--',{.01,100}) 10 Singular Values 5 0 Singular Values (db) -5-10 -15-20 -25-30 -35 10-2 10-1 10 0 10 1 10 2 Frequency (rad/sec)
Applied Robust Control, Chap 2, 2012 Spring 20 Mixed Sensitivity Problem T zw WS 1 WT 3 w W 1 z 1 u G W 3 z 2 K y 1 z WSwW I GK w 1 1 1 ( ) 1 z WTwWGK I GK w 2 3 3 ( )
Applied Robust Control, Chap 2, 2012 Spring 21 Consider the 2-by-2 NASA HiMAT aircraft model: The control variables are elevon and canard actuators ( e and c ). The output variables are angle of attack ( ) and pitch angle ( ). The model has six states, x x x x x x x x x T 1 2 3 4 5 6 e c where x e and x c are the elevator and canard states.
Applied Robust Control, Chap 2, 2012 Spring 22 % filename: applrbstcntrl_4_mixedsensitivity.m % mixsyn H mixed-sensitivity synthesis design on the HiMAT model % Create the NASA Himat model % The state-space matrices for the NASA HiMAT model G(s) ag =[ -2.2567e-02-3.6617e+01-1.8897e+01-3.2090e+01 3.2509e+00-7.6257e-01; 9.2572e-05-1.8997e+00 9.8312e-01-7.2562e-04-1.7080e-01-4.9652e-03; 1.2338e-02 1.1720e+01-2.6316e+00 8.7582e-04-3.1604e+01 2.2396e+01; 0 0 1.0000e+00 0 0 0; 0 0 0 0-3.0000e+01 0; 0 0 0 0 0-3.0000e+01]; bg = [0 0; 0 0; 0 0; 0 0; 30 0; 0 30]; cg = [0 1 0 0 0 0; 0 0 0 1 0 0]; dg = [0 0; 0 0]; G=ss(ag,bg,cg,dg); G.InputName = {'elevon','canard'}; G.OutputName = {'alpha','theta'}; % 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); >> W1 >> W3 % Compute the H-infinity mixed-sensitivity optimal sontroller K1 [K1,CL1,GAM1]=mixsyn(G,W1,[],W3); >> GAM1
Applied Robust Control, Chap 2, 2012 Spring 23 >> size(cl1) >> size(k1) % Compute the loop L1, sensitivity S1, and % complementary sensitivity T1: L1=G*K1; I=eye(size(L1)); S1=feedback(I,L1); % S=inv(I+L1); T1=I-S1; >> size(l1) >> size(t1) >> size(s1) figure(1) step(t1,1.5); title('\alpha and \theta command step responses');
Applied Robust Control, Chap 2, 2012 Spring 24 1.5 and command step responses From: In(1) From: In(2) To: Out(1) 1 0.5 Amplitude 0 1.5 To: Out(2) 1 0.5 0 0 0.5 1 1.5 Time (sec) 0 0.5 1 1.5 figure(2) sigma(i+l1,'--',t1,':',l1,'r--',... W1/GAM1,'k--',GAM1/W3,'k-.',{.1,100});grid legend('1/\sigma(s) performance',... '\sigma(t) robustness',... '\sigma(l) loopshape',... '\sigma(w1) performance bound',... '\sigma(1/w3) robustness bound');
Applied Robust Control, Chap 2, 2012 Spring 25 40 30 20 Singular Values 1/(S) performance (T) robustness (L) loopshape (W1) performance bound (1/W3) robustness bound Singular Values (db) 10 0-10 -20-30 10-1 10 0 10 1 10 2 Frequency (rad/sec)