Robust Control of Time-delay Systems - Illinois …mypages.iit.edu/~qzhong2/RobustControlTDS_beamer.pdfRobust Control of Time-delay Systems Qing-Chang Zhong Distinguished Lecturer,

Robust Control of Time-delaySystems

Qing-Chang Zhong

Distinguished Lecturer, IEEE Power Electronics SocietyMax McGraw Endowed Chair Professor in Energy and Power Engineering

Dept. of Electrical and Computer EngineeringIllinois Institute of Technology

Email: [email protected]

Web: http://mypages.iit.edu/∼qzhong2/

Qing-Chang Zhong (IIT, Chicago, [email protected]) Robust Control of Time-delay Systems

http://mypages.iit.edu/~qzhong2/

Outline

Notations and some preliminaries

Standard H∞ control problem

Delay-type Nehari problem

Controller implementation


Notations

Given a matrix A, AT and A∗ denote its transposeand complex conjugate transpose respectively. A−∗

stands for (A−1)∗ when the inverse A−1 exists.

G (s) = D + C (sI − A)−1B.=

[

A B

C D

]

G∼(s) = [G (−s∗)]∗ =

[

−A∗ −C ∗

B∗ D∗

]

Fl and Fu are the commonly-used lower/upper linearfractional transformation (LFT).


Preliminaries

Two operators

Chain-scattering representation

Homographic transformation


Completion operator

A completion operator πh is defined for h ≥ 0 as:

πh{G}.=

[

A B

Ce−Ah 0

]

−e−sh

[

A B

C D

]

.= G (s)−e−shG (s).

0 0.2 0.4 0.6 0.8 h=1 1.2 1.40

1

2

3

4

Time (sec)

Impu

lse

resp

onse

G

πh(G)

e−shG

shift


Truncation operator

A truncation operator τh is defined for h ≥ 0 as:

τh{G}.=

[

A B

C D

]

− e−sh

[

A eAhB

C 0

]

.= G (s)− e−shG (s)

0 0.2 0.4 0.6 0.8 h=1 1.2 1.40

1

2

3

4

Time (sec)

Impu

lse

resp

onse

G

τh(G)


Representations of systems

Input-output rep.

Cl(M)

(a) the left CSR

Cr(M)

(b) the right CSR

Chain-scattering rep.Qing-Chang Zhong (IIT, Chicago, [email protected]) Robust Control of Time-delay Systems

Chain-scattering representation

For M =

[

M11 M12

M21 M22

]

, the right and left

chain-scattering representations are defined as:

Cr(M).=

[

M12 −M11M−121 M22 M11M

−121

−M−121 M22 M−1

21

]

Cl(M).=

[

M−112 −M−1

12 M11

M22M−112 M12 −M22M

−112 M11

]

provided that M21 and M12, respectively, areinvertible. If both M21 and M12 are invertible, then

Cr (M) · Cl(M) = I .


Homographic transformations (HMT)

The right HMT:

Hr (M ,Q) = (M11Q +M12) (M21Q +M22)−1

The left HMT:

Hl (N ,Q) = −(N11 − QN21)−1(N12 − QN22)

M

❅❅

Q

✲

✛ ✛

✻

N

��

Q

✲

✛ ✛

✻

Hr (M,Q) Hl (N,Q)


Properties of the HMT

Lemma

Hr satisfies the following properties:(i) Hr(Cr(M), S) = Fl(M , S);(ii) Hr (I , S) = S ;(iii) Hr (M1,Hr(M2, S) = Hr(M1M2, S);(iv) If Hr(G , S) = R and G−1 exists, then

S = Hr (G−1,R).

Lemma

Let Λ be any unimodular matrix, then the H∞

control problem ‖Hr(G , K0)‖∞ < γ is solvable iff‖Hr(GΛ, K )‖∞ < γ is solvable. Furthermore,K0 = Hr(Λ, K ) or K = Hr(Λ

−1, K0).Qing-Chang Zhong (IIT, Chicago, [email protected]) Robust Control of Time-delay Systems

Outline






Standard H∞ problem of single-delay systems

Given a γ > 0, find a proper controller K such thatthe closed-loop system is internally stable and

∥

∥Fl(P, Ke−sh)∥

∥

∞< γ.

P

e−shI

K

✛

✛ ✛

✛

y

z

u

w

u1

✲


Key steps to solve the SPh

The SPh is solved via solving two simpler problems:

The one-block problem (OPh): in the form of SPh butwith P(∞) =

[

0 I

I 0

]

.

An extended Nehari problem (ENPh): in the form ofNPh but minimising the H∞-norm of Gβ11 + e−shQβ.

Key steps:

Step 1: reduce SPh to SP0 + OPh

Step 2: reduce OPh to ENPh

Step 3: solve ENPh

Step 4: recover the controllers


Reducing SPh to SP0 + OPh

Cr(P)

❅❅ e−shI

K

✲

✛ ✛ ✛

w

z u

y

u1

✻

Cr (P)

❅❅

Gα

❅❅

Cr(Gβ)

❅❅ e−shI

K

Delay-free problem 1-block delay problem

✲

✛

✲

✛

✲

✛ ✛ ✛

w

z u

y

u1

✻

w1

z1

y

u1

Gα is the controller generator of SP0. Gβ is defined such that Cr (Gβ ).= G−1

α . Gα and Cr (Gβ ) are allbistable.


Reducing OPh to ENPh (I)

Qα(s) = Hr (Cr (Gβ ), e−shK ) = Fl (Gβ, e−shK )

= Gβ11 + Gβ12Ke−sh(I − Gβ22Ke

−sh)−1Gβ21

❤

Gβ11

Gβ12

Gβ22

Gβ21❤

∆2

e−shI

❤

Kh(s)

✛

✲

+

-

K (s)✛

✲ ✲

✛✛

✻

✻

✻

❄

❄

❄

❄

z1

w1

u1

y

u

Introducing a Smith predictor

∆2(s) = −πh{Gβ22}.= −Gβ22 + e−shGβ22


Reducing OPh to ENPh (II)

❤

Gβ11

Gβ12e−shI

Gβ21

Gβ22

❤

Kh(s)

✛

✲

+Qβ(s)✛

✲

✛✛

✻

✻

✻

❄

❄

z1

w1

u2

z2

Qβ.= Hr (

[

Gβ12 0

−G−1β21Gβ22(s) G−1

β21

]

,Kh)

The OPh is now reduced to ENPh:‖Qα(s)‖∞ =

∥

∥Gβ11 + e−shQβ

∥

∥

∞< γ.


Solution to the SPh

Solvability ⇐⇒ :

H0 ∈ dom(Ric) and X = Ric(H0) ≥ 0;

J0 ∈ dom(Ric) and Y = Ric(J0) ≥ 0;

ρ(XY ) < γ2;

γ > γh, where γh = max{γ : detΣ22 = 0}.

Z V−1

❤

Q

❅❅

✲-

u

y✲

✛✛

✻

❄

❄

V−1 =

A+ B2C1 B2 − Σ12Σ−122 C

∗1 Σ−∗

22 B1

C1 I 0−γ−2B∗

1Σ∗21 − C2Σ

∗22 0 I


Outline






The delay-type Nehari problem

Given a minimal state-space realisation Gβ =[

A B

−C 0

]

,

which is not necessarily stable, and h ≥ 0,characterise the optimal value

γopt = inf{∥

∥Gβ(s) + e−shK (s)∥

∥

L∞: K (s) ∈ H∞}

and for a given γ > γopt , parametrise the suboptimalset of proper K ∈ H∞ such that

∥

∥Gβ(s) + e−shK (s)∥

∥

L∞< γ.


The optimal value

It is well known that this problem is solvable iff

γ > γopt.=

∥

∥ΓeshGβ

∥

∥ , (1)

where Γ denotes the Hankel operator. The symboleshGβ is non-causal and, possibly, unstable.

The problem is how to characterise it.


Estimation of γopt

It is easy to see that

γopt ≤ ‖Gβ(s)‖L∞

because at least K can be chosen as 0. It can be seen from(1) that γopt ≥

∥

∥ΓGβ

∥

∥. Hence,

∥

∥ΓGβ

∥

∥ ≤ γopt ≤ ‖Gβ(s)‖L∞ . (2)

When γ ≤ ‖Gβ(s)‖L∞ , the matrix

H =

[

A γ−2BB∗

−C ∗C −A∗

]

has at least one pair of eigenvalues on the jω-axis.


Two AREs

For a minimally-realised Gβ =

[

A B

−C 0

]

having no

jω-axis zero or pole, the following two AREs

[

−Lc I]

Hc

[

I

Lc

]

= 0,[

I −Lo]

Ho

[

LoI

]

= 0

(3)always have unique stabilising solutions Lc ≤ 0 andLo ≤ 0, respectively, where

Hc =

[

A γ−2BB∗

0 −A∗

]

, Ho =

[

A 0−C ∗C −A∗

]

.


The formula for γopt

Theorem For a given minimally-realised transfer

matrix Gβ =

[

A B

−C 0

]

having neither jω-axis zero

nor jω-axis pole, the optimal value γopt of thedelay-type Nehari problem is

γopt = max{γ : det Σ22 = 0},

where

Σ22 =[

−Lc I]

Σ

[

LoI

]

,

Σ =

[

Σ11 Σ12

Σ21 Σ22

]

.= eHh.


Parametrisation of K

For a given γ > γopt , all K (s) ∈ H∞ solving the problem can beparametrised as

K = Hr (

[

I 0Z I

]

W−1, Q),

where ‖Q(s)‖H∞ < γ is a free parameter and

Z = −πh{Fu(

[

Gβ I

I 0

]

, γ−2G∼

β )},

W−1 =

A+ γ−2BB∗Lc Σ−∗

22(Σ∗

12+ LoΣ

∗

11)C∗ −Σ−∗

22B

−C I 0γ−2B∗(Σ∗

21− Σ∗

11Lc) 0 I

.


Representation in a block diagram

The controller K consists of an infinite-dimensional block Z , which is afinite-impulse-response (FIR) block (i.e. a modified Smith predictor), afinite-dimensional block W−1 and a free parameter Q.

❤

Gβ Z

e−shI

W −1

❤

Q

K❅❅✛

✲-

u

y

z

w

✛

✲

✛✛

✻

✻

✻

❄

❄


Example: Gβ(s) = − 1s−a

(a > 0)

Σ22

ah

aγ

The surface Σ22 with respect to ah and aγ

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ah

aγ

aγopt

The contour Σ22 = 0 on the ah-aγ plane

Since I − LcLo = 1 − 4a2γ2, there is∥

∥

∥ΓGβ

∥

∥

∥= 1

2a. As a result, the optimal value γopt satisfies

0.5 ≤ aγopt ≤ 1.


Outline







All the above control laws associated with delay systems include a distributed delay like

v(t) =

ˆ

h

0

eAζ

Bu(t − ζ)dζ,

or in the s-domain, Z(s) = (I − e−(sI−A)h) · (sI − A)−1.

The implementation of Z is not trivial because A maybe unstable. This problem had confused the delaycommunity for several years and was proposed as anopen problem in Automatica. It was reported thatthe quadrature implementation might cause instabilityhowever accurate the implementation is.

My investigation shows that:The quadrature approximation error converges to 0 inthe sense of H∞-norm.

10−2

10−1

100

101

102

103

10−4

10−3

10−2

10−1

100

101

Frequency (rad/sec)

N=1

N=5

N=20

App

roxi

mat

ion

erro

r


Numerical integration

v(t) =

ˆ h

0

eAζBu(t − ζ)dζ

It is well known that

v(t) ≈ vw(t) =h

NΣN−1

i=0 e iAhNBu(t − i

h

N),

but what will happen if transferred into the s-domainusing the Laplace transform?

Zw(s) =h

N· ΣN−1

i=0 e−i hN(sI−A)B

Not strictly proper =⇒ No guarantee of stability !


A trivial result

τ t t−h/N

y(τ)

t 0

y(t) p(t)

t

1

0 h/N

∗=

ˆ hN

0

y(t − τ)dτ =

ˆ t

t− hN

y(τ)dτ = y(t) ∗ p(t),


Approximation of Z

v(t) =

ˆ h

0

eAζBu(t − ζ)dζ

=N−1∑

i=0

ˆ (i+1) hN

i hN

eAζBu(t − ζ)dζ

≈

N−1∑

i=0

e iAhNB

ˆ (i+1) hN

i hN

u(t − ζ)dζ

=N−1∑

i=0

e iAhNB

ˆ hN

0

u(t − ih

N− τi)dτi

=N−1∑

i=0

e iAhNBu(t − i

h

N)∗p(t) = vf (t)


Hidden sampling-hold effect

vw(t) =N−1∑

i=0

e iAhNBu(t − i

h

N)×

h

N

vf (t) =

N−1∑

i=0

e iAhNBu(t − i

h

N)∗p(t)

This is equivalent to the sampling-hold effect.


Implementation in the z-domain

hN ΣN−1

i=0 e ihNABz−i SZOH ✛ ✛✛✛ ✛ uv

(a) Zf (s) =1−e

−s hN

s·∑N−1


(eAhN − I )A−1 ΣN−1

i=0 e ihNABz−iZOH S✛ ✛✛✛✛ uv

(b) Zf 0(s) =1−e

−hN

s

se

hNA−IhN

A−1 ·∑N−1



Implementation in the s-domain

Zf ǫ(s) =1 − e−

hN(s+ǫ)

1 − e−hNǫ

ehNA − I

s/ǫ+ 1A−1·ΣN−1

i=0 e−i hN(sI−A)B .

limN→+∞ ‖Zf ǫ(s)− Z (s)‖∞ = 0, (ǫ ≥ 0).

10−2

10−1

100

101

102

103

10−4

10−3

10−2

10−1

100

101

Frequency (rad/sec)

N=1

N=5

N=20

App

roxi

mat

ion

erro

r


Rational implementation

The δ-operator is defined as δ = (q − 1)/τ, where q

is the shift operator and τ is the sampling period.We have

δ =eτs − 1

τ

because q → eτs when τ → 0. Define Φ = 1τI and

∆ = (eτ(sI−A) − I )Φ,

then

sI − A = limτ→0

∆, e−(sI−A)τ = (Φ−1∆+ I )−1


Z (s) = (I − e−(sI−A)h) · (sI − A)−1

= (I − e−(sI−A)τN)(sI − A)−1B

= (I − (Φ−1∆+ I )−N)(sI − A)−1B .

Since ∆ ≈ sI − A, Z can be approximated as

Zr(s) = (I − (Φ−1(sI − A) + I )−N)(sI − A)−1B

= (I − (Φ−1(sI − A) + I )−1) ·

ΣN−1k=0 (Φ

−1(sI − A) + I )−k(sI − A)−1B

= Φ−1ΣNk=1(Φ

−1(sI − A) + I )−kB

= ΣNk=1Π

kΦ−1B ,

where Π = (sI − A+Φ)−1Φ.


1x2xΠ

Nx 1−Nx

B1−Φbu

u

rv

…

ΦΦ+−=Π −1)( AsI

Π Π

…

Π = (sI − A+ Φ)−1Φ

In order to guaranteezero static error,

Φ = (´ h

N

0 e−Aζdζ)−1.


Fast-converging rational implementation

Define

Φ = (

ˆ hN

0

e−Aζdζ)−1(e−A hN + I )

Γ = (eτ(sI−A) − I )(eτ(sI−A) + I )−1Φ, τ = h/N

Then,sI − A = lim

τ→0Γ,

sI − A|s=0 = Γ|s=0 = −A,

Γ|sI−A=0 = 0.


1x2xΠ

Nx 1−NxB u

rv

…

)()( 1 Φ+−Φ+−=Π − sIAAsI

1)(2 −Φ+−=Ξ AsI

Π Ξ

…

Π = (Φ− sI + A)(sI − A+ Φ)−1,

Ξ = 2(sI − A+ Φ)−1

10−2

10−1

100

101

102

1010

−4

10−3

10−2

10−1

100

101

Impl

emen

tatio

n er

ror

Frequency (rad/sec)

δ−operator

discrete delay

bilinear trans.


Summary






Documents

Robust Control of Time-delay Systems - Illinois …mypages.iit.edu/~qzhong2/RobustControlTDS_beamer.pdfRobust Control of Time-delay Systems Qing-Chang Zhong Distinguished Lecturer,