Applied Robust Control, Chap 2, 2012 Spring 1

MEM800-007 Chapter 2 Sensitivity Function Matrices

Loop transfer function matrix: L GK

Sensitivity function matrix: 1( )S I L

Complementary Sensitivity function matrix:

1( )T L I L

1 1( ) ( )y L I L r I L d Tr Sd

smaller S

smaller worst-case disturbance response

smaller T

better robust stability

1( ) ( ) ( )u K I L r d R r d

smaller R

smaller worst-case control input

GK

u yrd

e

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

Small Gain Theorem:

Assume the nominal closed-loop system, T, is stable, then the uncertain closed-loop system 1( )MI T or

1( )I GK is stable if and only if

( ) 1 ( )MT j j for all .

smaller ( )T j better robust stability

b aM

T

GKy

uG

Mb a

GKy

MI u G

Applied Robust Control, Chap 2, 2012 Spring 4

Singular Values  and  Singular Value Decomposition 

Maximum singular value:    *

max max( ) ( )X X X

Applied Robust Control, Chap 2, 2012 Spring 5

Applied Robust Control, Chap 2, 2012 Spring 6

Applied Robust Control, Chap 2, 2012 Spring 7

Applied Robust Control, Chap 2, 2012 Spring 8

Applied Robust Control, Chap 2, 2012 Spring 9

Example 1:

2500

( ) ( )( 5)( 50)

G s K ss s s

a) Find the gain and phase margins. b) Find the least upper bound of ( )M j , ( ) , so

that ( ) 1 ( )T j and therefore the uncertain

closed-loop system with ( ) ( )M j is

robustly stable.

%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);

GK

u yrd

e

Applied Robust Control, Chap 2, 2012 Spring 10

-60

-50

-40

-30

-20

-10

0

10

20

Mag

nitu

de (

dB)

100

101

102

-270

-225

-180

-135

-90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Applied Robust Control, Chap 2, 2012 Spring 11

%Nyquist plot figure(2) nyquist(L,{10,100})

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2Nyquist Diagram

Real Axis

Imag

inar

y A

xis

Applied Robust Control, Chap 2, 2012 Spring 12

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

010

110

2-60

-50

-40

-30

-20

-10

0

10Singular Values

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

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})

10-2

10-1

100

101

102

-60

-40

-20

0

20

40

60Singular Values

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

Note that 1 1( )S I L L if 1L

1( )T L I L L if 1L

and therefore we have

1

( )( )

L jS j

for low frequencies

( ) ( )L j T j for high frequencies

Applied Robust Control, Chap 2, 2012 Spring 16

Example 2:

1

( )1

G ss

, ( ) 3K s

a) Find the gain and phase margins. b) Find the least upper bound of ( )M j , ( ) , so

that ( ) 1 ( )T j and therefore the uncertain

closed-loop system with ( ) ( )M j is

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)

GK

u yrd

e

Applied Robust Control, Chap 2, 2012 Spring 17

-30

-20

-10

0

10M

agni

tude

(dB

)

10-2

10-1

100

101

102

-180

-135

-90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Gain crossover frequency = rad/s Phase margin = degree Phase crossover frequency = rad/s Gain margin = dB

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5-1.5

-1

-0.5

0

0.5

1

1.5Nyquist Diagram

Real Axis

Imag

inar

y A

xis

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})

10-2

10-1

100

101

102

-35

-30

-25

-20

-15

-10

-5

0

5Singular Values

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

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})

10-2

10-1

100

101

102

-35

-30

-25

-20

-15

-10

-5

0

5

10Singular Values

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

Applied Robust Control, Chap 2, 2012 Spring 20

Mixed Sensitivity Problem

1

3

zw

W ST

W T

11 1 1( )z W Sw W I GK w

1

2 3 3 ( )z W T w W GK I GK w

y

w

K

G

1W

3W

u

1z

2z

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,

1 2 3 4 5 6T

e cx x x x x x x x x

where ex and cx 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

0

0.5

1

1.5From: In(1)

To:

Out

(1)

0 0.5 1 1.50

0.5

1

1.5

To:

Out

(2)

From: In(2)

0 0.5 1 1.5

and command step responses

Time (sec)

Am

plitu

de

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

10-1

100

101

102

-30

-20

-10

0

10

20

30

40

Singular Values

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

1/(S) performance

(T) robustness

(L) loopshape

(W1) performance bound

(1/W3) robustness bound