lecture 11: robust stability and robust...

...Lecture 11: Robust Stability and Robust

Performance

Dr.-Ing. Sudchai Boonto

Department of Control System and Instrumentation EngineeringKing Mongkut�s Unniversity of Technology Thonburi

Thailand

Feedback System with Uncertainty

..r. e. K. u. up.

di

. N. y.−

.

n

.

d

.

∆

P = Fu(N,∆), ∥∆∥∞ ≤ 1

where• N is a nominal plant• ∆ is possibly a diagonal matrix with real and dynamic uncertainties.

..Lecture 11: Robust Stability and Robust Performance ◀ 2/36 ▶ ⊚

Feedback System with UncertaintyTerminologies

• Nominal stability (NS): Feedback system is internally stable when∆ = 0.

• Robust stability (RS): Feedback system is internally stable forany norm-bounded ∆.

• Nominal performance (NP): Feedback system is stable andsatisfies certain performance for ∆ = 0.

• Robust performance (RP): Feedback system is stable andsatisfies certain performance for any norm-bounded ∆.

..Lecture 11: Robust Stability and Robust Performance ◀ 3/36 ▶ ⊚

Norminal Stability

.. K. N.−

• Analysis: Given a controller K, check if the feedback (FB) systemabove is internally stable.

• Synthesis: Design K such that the feedback system is internallystable.

..Lecture 11: Robust Stability and Robust Performance ◀ 4/36 ▶ ⊚

Robust Stability

.. K. N.−

.

∆

• Analysis: Given nominally internally stabilizing controller K, checkif the feedback system above is internally stable for all stablestructured ∆ with ∥∆∥∞ ≤ 1

• Synthesis: Design K such that the feedback system is robustlyinternally stable.

..Lecture 11: Robust Stability and Robust Performance ◀ 5/36 ▶ ⊚

Structure of an Uncertainty

• An uncertainty is called structured if it has a fixed structure, e.g.,• Some components are zero• Some components are real, or dynamic uncertainty• Some components are the same uncertainty

∆ :=

δ1

δ2

δ3

δ3

∆4(s)∆5(s)

• δ1, δ2, δ3 ∈ R,• ∆4(s),∆5(s) ∈ H∞ set of stable functions

..Lecture 11: Robust Stability and Robust Performance ◀ 6/36 ▶ ⊚

LFT RepresentationFor analysis and synthesis purpose, we use an LFT representation byextracting K:

.. K. N.−

.

∆

.. N.

∆

.

K

−K is redefined as K

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Robust Stability Condition

• Assume that fixed M(s) and a structured uncertain ∆(s) are stable• FB system below is internally stable for any structured ∆ with

∥∆∥∞ < 1

.. M.

∆

if and only if

det(I − M(jω)∆(jω)) ̸= 0, ∀ω∀∆ : structured , ∥∆∥∞ ≤ 1

• This condition is impractical to check because it involves uncertain∆.

..Lecture 11: Robust Stability and Robust Performance ◀ 8/36 ▶ ⊚

Robust Stability ConditionSpecial case

• Assume that a fixed M(s) and an unstructured uncertain ∆(s) arestable.

• FB system below is internally stable for any unstructured ∆ with∥∆∥∞ ≤ 1 if and only if

.. M.

∆

∥M∥∞ < 1

• This condition is practical because the condition is without ∆.

..Lecture 11: Robust Stability and Robust Performance ◀ 9/36 ▶ ⊚

Remarks on Robust Stability

• For unstructured uncertainty,• Analysis is a computation of H∞ norm of a system.

• We study the Bounded Real Lemma• In MATLAB, use norm(sys,inf)

• Robust stabilization is by H∞ controller design• In MATLAB, use hinfsyn(sys)

• For structured uncertainty• Analysis is by µ-analysis• Robust stabilization is by µ-synthesis.

..Lecture 11: Robust Stability and Robust Performance ◀ 10/36 ▶ ⊚

Robust Performance

• Analysis: Given robustly internally stabilizing K, check if thefeedback system below satisfies performance for all stablestructured ∆ with ∥∆∥∞ ≤ 1

..r. e. K. u. up.

di

. N. y.−

.

n

.

d

.

∆

• ∥WSS∥∞ < 1

• Synthesis: Design K such that the feedback system satisfies robustperformance.

..Lecture 11: Robust Stability and Robust Performance ◀ 11/36 ▶ ⊚

LFT Representation

• For analysis and synthesis purpose, we use an LFT representationby attracting K:

..r. e.

WS

.

zs

. K. u. up. N. y.−

.

n

.

∆

.

z

.

w.. G.

∆

.

K

.

z

.

w

.

e

.

u

. zs.r

..Lecture 11: Robust Stability and Robust Performance ◀ 12/36 ▶ ⊚

Nominal Performance Condition

• Analysis: Given a nominally stabilizing K, check if

∥Tzsr∥∞ < 1

• Synthesis: Design a nominally stabilizing K such that

∥Tzsr∥∞ < 1

.. G.

∆

.

K

.

z

.

w

.

e

.

u

. zs.r

..Lecture 11: Robust Stability and Robust Performance ◀ 13/36 ▶ ⊚

Robust Performance Condition

• Analysis: Given a robustly stabilizing K, check if

∥Tzsr∥∞ < 1, ∀∆

∆ : Structured , ∥∆∥∞ ≤ 1

• Synthesis: Design a robustly stabilizing K such that the abovecondition is satisfied.

.. G.

∆

.

K

.

z

.

w

.

e

.

u

. zs.r

..Lecture 11: Robust Stability and Robust Performance ◀ 14/36 ▶ ⊚

Robust Performance ConditionReducing robust performance to robust stability

• Robust performance problems are equivalent to robust stabilityproblems with augmented uncertainty

.. G.

∆

.

K

.

z

.

w

.

e

.

u

. zs.r

∥Tzsr∥∞ < 1

.. G.

∆

.

∆p

.

K

.

z

.

w

.

e

.

u

. zs.r .

∆aug

..Lecture 11: Robust Stability and Robust Performance ◀ 15/36 ▶ ⊚

Remarks on Nominal Performance and Robust Performance

• For nomianl performance• Analysis is computation for H∞ norm of a system• Controller design for nominal performance is by H∞ controller design

• For robust performance• Analysis is by µ-analysis• Robust stabilization by µ-synthesis• Same difficulty as the difficulty for robust stability analysis and

robust stabilization for structured uncertainty.

..Lecture 11: Robust Stability and Robust Performance ◀ 16/36 ▶ ⊚

Uncertain systemExample 1

G(s) = 11

bws + 1

(1 + W(s)∆(s)) bw = 5(1 + 0.1δ), δ ∈ [−1, 1]

W(s) = s + 9(0.05)s10

+ 9, ∥∆∥∞ ≤ 1

..

Uncertain system

.

clc; clear all;bw = ureal('bw',5,'Percentage',10);Gnom = tf(1,[1/bw 1]);W = makeweight(0.05,9,10);Delta = ultidyn('Delta',[1 1]);G = Gnom*(1+W*Delta);

..Lecture 11: Robust Stability and Robust Performance ◀ 17/36 ▶ ⊚

Uncertain systemExample 1

−150

−100

−50

0

50

Magnitude (

dB

)

10−2

100

102

104

−180

0

180

360

Phase (

deg)

Bode Diagram

Frequency (rad/s)

..Lecture 11: Robust Stability and Robust Performance ◀ 18/36 ▶ ⊚

Uncertain systemExample 1

• PI controllers

..

xi = 0.707;wn = 3;K1 = tf([(2*xi*wn/5-1) wn*wn/5],[1 0]);wn = 7.5;K2 = tf([(2*xi*wn/5-1) wn*wn/5],[1 0]);

• Complementary sensitivity functions

..

T1 = feedback(G*K1,1);T2 = feedback(G*K2,1);

..Lecture 11: Robust Stability and Robust Performance ◀ 19/36 ▶ ⊚

Uncertain systemExample 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−1

−0.5

0

0.5

1

1.5

2

Step Response

Time (seconds)

Am

plit

ude

T1

T2

..Lecture 11: Robust Stability and Robust Performance ◀ 20/36 ▶ ⊚

Uncertain systemExample 1

• Robust stability analysis

..

[stabmarg1,destabu1,report1] = robuststab(T1)

..

stabmarg1 = LowerBound: 4.0323UpperBound: 4.0323

DestabilizingFrequency: 4.0938report1 = Uncertain system is robustly stable to

modeled uncertainty.-- It can tolerate up to 403% of the modeled

uncertainty.-- A destabilizing combination of 403% of the

modeled uncertainty was found.-- This combination causes an instability at 4.09

rad/seconds.

..Lecture 11: Robust Stability and Robust Performance ◀ 21/36 ▶ ⊚

Uncertain systemExample 1

• Robust stability analysis

..

[stabmarg2,destabu2,report1] = robuststab(T2)

..

stabmarg2 = LowerBound: 1.2616UpperBound: 1.2616

DestabilizingFrequency: 9.8187report1 = Uncertain system is robustly stable to

modeled uncertainty.-- It can tolerate up to 126% of the modeled

uncertainty.-- A destabilizing combination of 126% of the

modeled uncertainty was found.-- This combination causes an instability at 9.82

rad/seconds.

..Lecture 11: Robust Stability and Robust Performance ◀ 22/36 ▶ ⊚

Uncertain systemExample 1

• Sensitivity peak analysis

..

S1 = feedback(1,G*K1);S2 = feedback(1,G*K2);[maxgain1,wcu1] = wcgain(S1);[maxgain2,wcu2] = wcgain(S2);

..Lecture 11: Robust Stability and Robust Performance ◀ 23/36 ▶ ⊚

Uncertain systemExample 1

..

>> maxgain1maxgain1 =

LowerBound: 1.8778UpperBound: 1.8779

CriticalFrequency: 3.0583>> maxgain2maxgain2 =

LowerBound: 4.5400UpperBound: 4.5402

CriticalFrequency: 13.1431

..

bodemag(S1.NominalValue,'b',usubs(S1,wcu1),'b');hold on, grid onbodemag(S2.NominalValue,'r',usubs(S2,wcu2),'r');hold off

..Lecture 11: Robust Stability and Robust Performance ◀ 24/36 ▶ ⊚

Uncertain systemExample 1

10−1

100

101

102

103

−50

−40

−30

−20

−10

0

10

20M

agnitude (

dB

)

Bode Diagram

Frequency (rad/s)

S1

S1

S2

S2

Nominal

Nominal

Worst−caseWorst−case

..Lecture 11: Robust Stability and Robust Performance ◀ 25/36 ▶ ⊚

Uncertain systemExample 2

G(s) =

0 p 1 0−p 0 0 1

1 p 0 0−p 1 0 0

×(1 +

[W1(s)∆1(s) 0

0 W2(s)∆2(s)

])

p = 10(1 + 0.1δ), δ ∈ [−1, 1]

W1(s) =s + 20 · 0.1

s50

+ 20, ∥∆1∥∞ ≤ 1

W2(s) =s + 45 · 0.2

s50

+ 45, ∥∆2∥∞ ≤ 1

..Lecture 11: Robust Stability and Robust Performance ◀ 26/36 ▶ ⊚

Uncertain systemExample 2

..

p = ureal('p',10,'Percentage',10);A = [0 p; -p 0]; B = eye(2);C = [1 p; -p 1];H = ss(A,B,C,[0 0; 0 0]);W1 = makeweight(0.1,20,50);W2 = makeweight(0.2,45,50);Delta1 = ultidyn('Delta1',[1 1]);Delta2 = ultidyn('Delta2',[1 1]);G = H*blkdiag(1+W1*Delta1, 1+W2*Delta2);

..Lecture 11: Robust Stability and Robust Performance ◀ 27/36 ▶ ⊚

Uncertain systemExample 2

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5From: In(1)

To: O

ut(

1)

0 0.5 1 1.5 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

To: O

ut(

2)

From: In(2)

0 0.5 1 1.5 2

Step Response

Time (seconds)

Am

plit

ude

..Lecture 11: Robust Stability and Robust Performance ◀ 28/36 ▶ ⊚

Uncertain systemExample 2

−200

0

200From: In(1)

To

: O

ut(

1)

−720

0

720

To

: O

ut(

1)

−200

0

200

To

: O

ut(

2)

100

105

−720

0

720

To

: O

ut(

2)

From: In(2)

100

105

Bode Diagram

Frequency (rad/s)

Ma

gn

itu

de

(d

B)

; P

ha

se

(d

eg

)

..Lecture 11: Robust Stability and Robust Performance ◀ 29/36 ▶ ⊚

Uncertain systemClosed-loop robust analysis

..r. e. K. u. up.

di

. P. y.−

.

n

.

d

Li = KP, Si = (1 + Li)−1, Ti = I − Si

Lo = KP, So = (1 + Lo)−1, To = I − So

..

>> load mimoKexample>> F = loopsense(G,K)

..Lecture 11: Robust Stability and Robust Performance ◀ 30/36 ▶ ⊚

Uncertain systemClosed-loop robust analysis

The transmission of disturbances at the plant input to the plant output

−150

−100

−50

0

50From: du(1)

To: yP

(1)

100

105

−150

−100

−50

0

50

To: yP

(2)

From: du(2)

100

105

Bode Diagram

Frequency (rad/s)

Magnitude (

dB

)

..Lecture 11: Robust Stability and Robust Performance ◀ 31/36 ▶ ⊚

Uncertain systemWorst-Case Gain Analysis

Bode magnitude of the nominal output sensitivity function.

..bodemag(F.So,'b',F.So.NominalValue,'r',{1e-1 100})

−80

−60

−40

−20

0

From: dy(1)

To: y(1

)

From: dy(2)

10−1

100

101

102

Bode Diagram

Frequency (rad/s)

Magnitude (

dB

)

10−1

100

101

102

−100

−80

−60

−40

−20

0

To: y(2

)

..Lecture 11: Robust Stability and Robust Performance ◀ 32/36 ▶ ⊚

Uncertain systemWorst-Case Gain Analysis

• Nominal peak gain (largest singular value)

..

PeakNom =1.1317

freq =7.0483

• Worst-case gain

..

[maxgain,wcu] = wcgain(F.So)maxgain =

LowerBound: 2.1459UpperBound: 2.1466

CriticalFrequency: 8.4435

..Lecture 11: Robust Stability and Robust Performance ◀ 33/36 ▶ ⊚

Uncertain systemWorst-Case Gain Analysis

• The analysis indicates that the worst-case gain is somewherebetween 2.1 and 2.2. The frequency where the peak is achieved isabout 8.5.

• We can replace the values of Delta1, Delta2 and p that achievethe gain of 2.1, using usubs

..step(F.To.NominalValue,'r',usubs(F.To,wcu),'b',5)

• The perturbed response, which is the worst combination ofuncertain values in terms of output sensitivity amplification, doesnot show significant degradation of the command response.

• The setting time is increased by about 50%, from 2 to 4, and theoff-diagonal coupling is increased by about a factor of about 2, butis still quite small.

..Lecture 11: Robust Stability and Robust Performance ◀ 34/36 ▶ ⊚

Uncertain systemWorst-Case Gain Analysis

0

0.5

1

1.5From: dy(1)

To: yP

(1)

0 1 2 3 4 5

0

0.5

1

1.5

To: yP

(2)

From: dy(2)

0 1 2 3 4 5

Step Response

Time (seconds)

Am

plit

ude

..Lecture 11: Robust Stability and Robust Performance ◀ 35/36 ▶ ⊚

Reference

1. Herbert Werner ”Lecture note on Optimal and Robust Control”,

2012

2. Ryozo Nagamune ”Lecture not on Multivariable Feedback

Control”, 2009

..Lecture 11: Robust Stability and Robust Performance ◀ 36/36 ▶ ⊚