HinfSynLMI

8/13/2019 HinfSynLMI

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Synthesis Discrete-Time LTI

Given an open-loop generalized plant, state equations

xk+1ekyk

=A B

1 B

2C 1 D11 D12C 2 D21 D22

xk

dkuk

Goal: determine if there is a linear controller K ,

k+1uk

=AK BK C K DK

kyk

(5.11)

such that the closed-loop system is well-dened, stable, and has Ged 22 = Ged < 1.

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Fact 1 Characterizing Stability and Gain

Linear system,

xk+1ek =

A BC D

xkdk =: M

xkdk

Then, (A) < 1, max|z | 1

D + C (zI A) 1B < 1 if and only if there exists

0 W = W F n n with

W 1/ 2

I m

A BC D

W 1/ 2

I m < 1

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Fact 2 Constant Matrix Optimization

1. Given R F l l, U F l m and V F p l, where m, p l.

2. Suppose U F l (l m) and V F (l p) l have

U U ,V V

are both invertible

U U = 0m (l m), V V = 0 p (l p)

3. Let Z F l l be a given set of positive denite, Hermitian matrices

Then

inf QF m p

Z Z

Z 1/ 2 (R + UQV ) Z 1/ 2 < 1

if and only if there is a Z Z such that

V (RZR Z ) V 0

and

U

RZ 1R Z 1 U 0.

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Synthesis Discrete-Time LTI (contd)

Write closed-loop system as static feedback around an extended system. Letm denote the (as of yet unknown) state-dimension of the controller. Denean extended plant

xk+1 k+1ek kyk

=

A 0 B1 0 B20 0m 0 I m 0

C 1 0 D11 0 D120 I m 0 0 0

C 2 0 D21 0 D22

xk kdk

k+1uk

Call this 5 5 block matrix M mss . For the controller in (5.11), dene a matrixK ss as

K ss :=AK BK C K DK

R (m+ nu ) (m+ ny )

In this notation, the closed-loop system is governed

xk+1 k+1ek

= F L (M mss , K ss )xk kdk

(5.12)

F L (M mss , K ss ) is

A 0 B10 0m 0

C 1 0 D11

+0 B2

I m 00 D12

K ss I 0 0

0 D22K ss

1 0 I m 0

C 2 0 D21

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Controller achieves stability and Ged < 1 if and only if there exists aW R (n+ m) (n+ m) such that W = W T 0 and

W 12 0

0 I n eF L (M mss , K ss )

W 12 00 I nd

< 1

This is the equation to solve by searching over

1. integers m 0,

2. K ss R (m+ nu ) (m+ ny ) and

3. W R (n+ m) (n+ m)

Without loss of generality D22 = 0, so dening

R :=A 0 B10 0m 0

C 1 0 D11

, U :=0 B2

I m 00 D12

, V :=0 I m 0

C 2 0 D21

gives F L (M mss , K ss ) = R + UK ss V

Clearly, matrices U and V have the form

U =U ,1

0U ,2

, V = [V ,1 0 V ,2]

where the matrices U ,1, U ,2, V ,1 and V ,2 are determined from the data inU and V (could use svd in Matlab).

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For notational purposes, dene

E :=A B 1

C 1 D11

Use Y 1, Y 2 and Y 3 to denote the various entries of the positive denite scalingmatrix W R (n+ m) (n+ m), and X 1, X 2 and X 3 for W 1,

W =Y 1 Y 2Y T 2 Y 3

W 1 =X 1 X 2X T 2 X 3

Theorem: There is a controller achieving closed-loop stability and Ged

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