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COURSE ON LMI

PART II.1

LMIs IN SYSTEMS CONTROL

STATE-SPACE METHODS

STABILITY ANALYSIS

Didier HENRIONwww.laas.fr/henrion

October 2006
http://www.laas.fr/~henrionhttp://www.laas.fr/~henrionhttp://www.laas.fr/~henrionhttp://www.laas.fr/~henrion


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State-space methodsDeveloped by Kalman and colleagues in the 1960s asan alternative to frequency-domain techniques (Bode,Nichols..)

Starting in the 1980s, numerical analysts developedpowerful linear algebra routines for matrix equations:numerical stability, low computational complexity, large-scale problems

Matlab launched by Cleve Moler (1977-1984) heavilyrelies on LINPACK, EISPACK & LAPACK packages

Pseudospectrum of a Toeplitz matrix


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Linear systems stability

The continuous-time linear time invariant (LTI)

system

x(t) =Ax(t) x(0) =x0

where x(t) Rn

is asymptotically stable,meaning

limt

x(t) = 0 x0

if and only if

there exists a quadratic Lyapunov functionV(x) =xPx such that

V(x(t)) > 0V(x(t)) < 0

along system trajectories equivalently, matrix A satisfies

maxi real i(A)


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Lyapunov stabilityNote that V(x) =xPx=x(P+ P)x/2so that Lyapunov matrix P can be chosensymmetric without loss of generality

Since V(x) = xPx + xPx=xAPx + xPAx positivityofV(x) and negativity ofV(x) along system trajectoriescan be expressed as an LMI

AP+ PA 0 P 0

Matrices P satisfying Lyapunovs LMI with A=

0 11 2


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Lyapunov equation

The Lyapunov LMI can be written equivalently

as the Lyapunov equation

AP+ PA + Q= 0

where Q 0

The following statements are equivalent

the system x=Ax is asymptotically stable

for some matrix Q 0 the matrix P solvingthe Lyapunov equation satisfies P 0

for all matrices Q 0 the matrix P solving

the Lyapunov equation satisfies P 0

The Lyapunov LMI can be solved numerically

without IP methods since solving the above

equation amounts to solving alinear systemof

n(n + 1)/2 equations in n(n + 1)/2 unknowns


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Alternative to Lyapunov LMI

Recall the theorem of alternatives for LMI

F(x) =F0+

i

xiFi

Exactly one statement is true there exists x s.t. F(x) 0

there exists a nonzero Z 0 s.t.trace F0Z 0 and trace FiZ= 0 for i >0

Alternative to Lyapunov LMI

F(x) = AP PA 0

0 P 0

is the existence of a nonzero matrix

Z=

Z1 0

0 Z2

0

such that

Z1A + AZ1 Z2= 0


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Alternative to Lyapunov LMI (proof)

Suppose that there exists such a matrix Z= 0and extract Cholesky factor

Z1=U U

Since Z1A + AZ1 0 we must have

AUU =USUwhere S=S1+ S2 and S1= S

1, S2 0

It follows from

AU=US

that U spans an invariant subspace of Aassociated with eigenvalues of S,which all satisfy real i(S) 0

Conversely, suppose i(A) = +j with 0for some i with eigenvector v

Then rank-one matrices

Z1=vv Z2 = 2vv

solve the alternative LMI


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Discrete-time Lyapunov LMI

Similarly, the discrete-time LTI system

xk+1=Axk

is asymptotically stable iff

there exists a quadratic Lyapunov function

V(x) =xPx such that

V(xk)>0V(xk+1) V(xk)


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More general stability regions

Let

D = {s C :

1s

d0 d1d1 d2

1s


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LMI stability regions

We can consider D-stability in LMI regions

D = {s C : D(s) =D0+ D1s + D1s

0}

such as

D dynamicsreal(s)< dominant behaviorreal(s)< , |s| < r oscillations1


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Lyapunov LMI for LMI stability regions

Matrix A has all its eigenvalues in the region

D = {s C : D0+ D1s + D1s

0}

if and only if the following LMI is feasible

D0 P+ D1 AP+ D1 PA

0 P 0

where denotes the Kronecker product

Litterally replace s with A !

Can be extended readily to quadratic matrix

inequality stability regions

D = {s C : D0+ D1s + D1s

+ D2ss 0}

parabolae, hyperbolae, ellipses etc

convex (D2 0) or not


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Uncertain systems and robustness

When modeling systems we face severalsources

of uncertainty, including

non-parametric (unstructured) uncertainty unmodeled dynamics truncated high frequency modes

non-linearities effects of linearization, time-variation..

parametric (structured) uncertainty physical parameters vary within given bounds interval uncertainty (l) ellipsoidal uncertainty (l

2)

diamond uncertainty (l1)

How can we overcome uncertainty ? model predictive control adaptive control robust control

A control law is robust if it is valid over the

whole range of admissible uncertainty (can be

designed off-line, usually cheap)


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Uncertainty modeling

Consider the continuous-time LTI system

x(t) =Ax(t) A A

where matrix A belongs to an uncertainty set

A

For unstructured uncertainties we consider

norm-bounded matrices

A = {A + BC : 2 }

For structured uncertainties we considerpolytopic matrices

A = conv {A1, . . . , AN}

There are other more sophisticated uncertainty

models not covered here

Uncertainty modeling is an important and

difficult step in control system design !


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Robust stability

The continuous-time LTI system

x(t) =Ax(t) A A

is robustly stable when it is asymptoticallystable for all A A

If S denotes the set of stable matrices, thenrobust stability is ensured as soon as A SUnfortunately S is a non-convex cone !

Non-convex setof continuous-time

stable matrices

1 xy z


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Symmetry

If dynamic systems were symmetric, i.e

A=A

continuous-time stability maxi reali(A) < 0 would beequivalent to

A + A 0

and discrete-time stability maxi |i(A)|


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Robust and quadratic stability

Because of non-convexity of the cone of stable

matrices, robust stability is sometimesdifficult

to check numerically, meaning that

computational cost is an exponential functionof the number of system parameters

Remedy:

The continuous-time LTI system x(t) =Ax(t)

isquadratically stableif its robust stability can

be guaranteed with the same quadraticLyapunov function for all A A

Obviously, quadratic stability is more pessimistic,

or more conservative than robust stability:

Quadratic stability = Robust stability

but the converse is not always true


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Quadratic stability for polytopic uncertainty

The system with polytopic uncertainty

x(t) =Ax(t) A conv {A1, . . . , AN}

is quadratically stable iff there exists a matrix

P solving the LMI

ATi P+ PAi 0 P 0

Proof by convexity

i

i(ATi P+ PAi) =A

T()P+ PA() 0

for all i 0 such that

i i = 1

This is a vertex result: stability of a whole

family of matrices is ensured by stability of the

vertices of the family

Usually vertex results ensure

computational tractability


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Quadratic stability for norm-boundeduncertainty

The system with norm-bounded uncertainty

x(t) = (A + BC)x(t) 2

is quadratically stable iff there exists a matrix

P solving the LMI

AP+ PA + CC PB

BP 2I

0 P 0

with 1 =

This is called the bounded-real lemma

proved next with the S-procedure

We can maximize the level of

allowed uncertainty by minimizing scalar


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Norm-bounded uncertainty as feedback

Uncertain system

x= (A + BC)x

can be written as the feedback system

x = Ax + Bw

z = Cxw = z

.x = Ax+Bw

w z

so that for the Lyapunov function V(x) =xPxwe have

V(x) = 2xPx

= 2xP(Ax + Bw)= x(AP+ PA)x + 2xPBw

=

xw

AP+ PA PB

BP 0

xw


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Norm-bounded uncertainty as feedback (2)

Since 2I it follows that

ww =zz 2zz

w

w 2

z

z = x

w CC 0

0 2I x

w 0

Combining with the quadratic inequality

V(x) =

xw

AP+ PA PB

BP 0

xw


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Norm-bounded uncertainty: generalization

Now consider the feedback system

x = Ax + Bwz = Cx+Dw

w = z

with additional feedthrough term Dw

We assume that matrix ID is non-singular

= well-posedness of feedback interconnection

so that we can write

w= z = (Cx + Dw)(ID)w= Cx

w= (ID)1Cx

and derive the linear fractional transformation

(LFT) uncertainty description

x=Ax + Bw = (A + B(ID)1C)x


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Norm-bounded LFT uncertainty

The system with norm-bounded

LFT uncertainty

x= A + B(ID)1

x 2 is quadratically stable iff there exists a matrix

P solving the LMI

AP+ PA + CC PB+ CD

BP+ DC DD 2I

0 P 0

Notice the lower right block DD 2I 0

which ensures non-singularity ofID hence

well-posedness

LFT modeling can be used more generally tocope with rational functions of uncertain pa-

rameters, but this is not covered in this course..


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Sector-bounded uncertainty

Consider the feedback system

x = Ax + Bwz = Cx + Dw

w = f(z)

where vector function f(z) satisfies

zf(z) 0 f(0) = 0

which is a sector condition

f(z)

z

f(z) can also be considered as an uncertainty

but also as a non-linearity


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Quadratic stability for sector-bounded

uncertainty

We want to establish quadratic stability with

the quadratic Lyapunov matrix V(x) = xPx

whose derivative

V(x) = 2xP(Ax + Bf(z))

=

xf(z)

AP+ PA PB

BP 0

xf(z)

must be negative when

2zf(z) = 2(Cx + Df(z))f(z)

=

xf(z)

0 C

C D+ D

xf(z)

is non-positive, so we invoke the S-procedure

to derive the LMI

AP+ PA PB CBP C D D

0 P 0

This is called the positive-real lemma


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Parameter-dependent Lyapunov functions

Quadratic stability: fast variation of parameters computationally tractable conservative, or pessimistic (worst-case)

Robust stability: no variation of parameters computationally difficult (in general) exact (is it really relevant ?)

Is there something in between ?

For example, given an LTI system affected by box,or interval uncertainty

x(t) =A((t))x(t) = (A0+N

i=1 i(t)Ai)x(t)

where

= {i [i, i]}

we may consider parameter-dependentLyapunov matrices, such as

P((t)) =P0+

i i(t)Pi


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Polytopic Lyapunov certificate

Quadratic Lyapunov function V(x) = xP()x must bepositive with negative derivative along systemtrajectory hence

P() 0

and

A()P() + P()A() + P() 0

We have to solve parametrized LMIs

Parametrized LMIs feature non-linear terms in soit isnot enoughto check vertices of , denoted by vert

21 1+ 2 0 on vert but not everywhere on = [0, 1] [0, 1]


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Parametrized LMIs

Central problem in robustness analysis:

find x such that

F(x, ) = F(x) 0,

where is a compact set, typically the unit

simplex or the unit ball

Convex but infinite-dimensional problem

which is difficult in general

Matrix extensions of polynomial positivity

conditions, for which varioushierarchies of LMIs

are available:

Polyas theorem

Schmudgens representation

Putinar representation

See EJC 2006 survey by Carsten Scherer


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H2 norm

The H2 norm is the energy (l2 norm) of the impulseresponse h(t) of a system G

G2 =

0

h(t)h(t)dt

1/2=

1

2

H(j)H(j)d

1/2

For a continuous-time system

x = Ax + Buy = Cx + Du

with transfer function G(s) =C(sIA)1B +D we mustassume D = 0 to have G2 finite


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Computing the H2 norm

Let hi(t) = CeAtBi denote the i-th column of

the impulse response of G, then

G22 =

i

hi22

= i

0

Bi eAtCCeAtBidt

= trace B

0eA

tCCeAtdt

B

Matrix

Po =

0eA

tCCeAtdt

is the observability Grammian solution to theLyapunov equation

APo+ PoA + CC= 0

and hence

G22= trace BPoB

If (A, C) observable then Po 0


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Dual and LMI computation of the H2 norm

Defining the controllability Grammian

Pc =

0eAtBBeA

tdt

solution to the Lyapunov equation

APc+ PcA + BB = 0

we have a dual expression

G22= trace CPcC

Dual lyapunov equations formulated as LMIs

G22 = min trace BPB

s.t. AP+ PA + CC 0P 0

G22 = min trace CQC

s.t. AQ + QA + BB 0Q 0


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H norm

The H norm is the induced energy gain

(l2 to l2)

G = supx2=1

Gx2= supG(j)

It is the worst case gain


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Computing the H norm

In contrast with the H2 norm, computation of

the H norm is iterative

For the continuous-time linear system


with transfer functionG(s) =C(sIA)1B+D

we have G(s) < iff R = 2I DD 0

and the Hamiltonian matrix

A + BR1DC BR1B

C(I+ DR1D)C (A + BR1DC)

has no eigenvalues on the imaginary axis

We can then design a bisection algorithmwith guaranteed quadratic convergence

to find the minimum value of such that

the Hamiltonian has no imaginary eigenvalues


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LMI computation of the H norm

Refer to the part of the course on norm-boundeduncertainty

supz2=1

w = < 1

to prove that for the continuous-time system


with transfer functionG(s) =C(sIA)1B+Dwe have G(s) < iff there exists a matrixP solving the LMI

AP+ PA + CC PB+ CD

BP+ DC DD 2I

0 P 0

Using the Schur complement and a change ofvariables this can be expanded to

A

P+ PA PB C

BP I D

C D I

0 P 0


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Linear systems design

Open-loop continuous-time LTI system

x=Ax + Bu

with state-feedback controller

u=Kx

produces closed-loop system

x= (A + BK)x

Applying Lyapunov LMI stability condition

(A + BK)P+ P(A + BK) 0 P 0

we get bilinear terms..

Bilinear Matrix Inequalities (BMIs) are

non-convex in general !


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Finslers lemma

A very useful trick in robust control..

The following statements are equivalent

1. xAx >0 for all x = 0 s.t. Bx= 02. BAB >0 where BB = 03. A + BB >0 for some scalar 4. A + XB+ BX >0 for some matrix X

Paul Finsler(1894 Heilbronn - 1970 Zurich)


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State-feedback design: null-space projection

We can also use item 2 of Finslers lemma,

projecting onto the (full column rank)

null-space B of B

BB = 0

so that BMI

AP+ PA + KBP+ PBK 0

is equivalent to the projected LMI

B(AQ + QA)B 0 Q 0

Feedback K can be recovered from

Lyapunov matrix Q as

K= BQ1

where is a suitably large scalar

Here we obtained an LMI thanks to a

projection onto a null-space


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Robust state-feedback design

for polytopic uncertainty

Open-loop system x=Ax + Bu with polytopic

uncertainty

(A, B) conv {(A1, B1), . . . , (AN

, BN

)}

androbust state-feedback controller u=Kx

In order to derive synthesis condition,

we start with analysis conditions

(Ai+ BiK)P+ P(Ai+ BiK) 0 i Q 0

and we obtain thequadratic stabilizabilityLMI

AiQ + QAi + BiY + Y

Bi 0 i Q 0

with the linearizing change of variables

Q=P1 Y =KP1


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State-feedback H2 control

Continuous-time LTI open-loop system

x = Ax + Bww+ Buuz = Czx + Dzww+ Dzuu

with state-feedback controller

u=Kx

yields closed-loop system

x = (A + BuK)x + Bwwz = (Cz+ DzuK)x + Dzww

with transfer function

G(s) =Dzw + (Cz + DzuK)(sIABuK)1Bw

between performance signals w and z

H2 performance specification

G(s)2<

We must have Dzw = 0 (finite gain)


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H2 design LMIs

As usual, start with analysis condition:

there exists K such that G(s)2< iff

trace (Cz+ DzuK)Q(Cz+ DzuK) < (A + BuK)Q + Q(A + BuK) + BB 0

The trace inequality can be written as

trace(Cz +DzuK)Q(Cz +DzuK)


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State-feedback H control

Similarly, with closed-loop system

x = (A + BuK)x + Bwwz = (Cz+ DzuK)x + Dzww

and H performance specification

G(s)<

on transfer function betweenw andz we obtain

the design LMI

AQ + QA + BuY + YBu+ BwB

w

CzQ + DzuY + DzwBw DzwDzw

2I

0

Q 0

with resulting H suboptimal state-feedback

K=YQ1

Optimal H control: minimize


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Mixed H2/H control

State-feedback controller systemwith two performance channels

x = (A + BuK)x + Bwwz = (C+ DuK)x + Dwwz2 = (C2+ D2uK)x

and mixed performance specifications

G(s) < G2(s)2< 2

on transfer functions from w to z and z2 respectively

Formulation of H constraint

AQ+ BuKQ+ () + BwBw

CQ+ DuKQ+ DwBw DwDw

2I 0

Q 0BMI formulation of H2 constraint

trace W < 2 W C2Q2+ D2uKQ2

Q2

0

AQ2+ BuKQ2+ () + BwBw 0

Problem:

We cannot linearize simultaneouslyterms KQ and KQ2 !


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Mixed H2/H control design LMI

Remedy:

Enforce Q2 =Q=Q !

Conservative but useful.. Always trade-off

between conservatism and tractability

Resulting mixed H2/H design LMI

AQ + BuY + () + BwBw

CQ + DuY + DwBw DwDw

2I

0

trace W < 2 W C2Q + D2uY

Q

0

AQ + BuY + () + BwBw 0

Guaranteed cost mixed H2/H:

given

minimize 2

Can be used forH2design of uncertain systems

with norm-bounded uncertainty


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Mixed H2/H control: example

Active suspension system (Weiland)

m2q2+ b2(q2 q1) + k2(q2 q1) + F = 0m1q1+ b2(q1 q2) + k2(q1 q2)

+k1(q1 q0) + b1(q1 q0) + F = 0

z = q1 q0

F

q2q2 q1 y = q2

q2 q1 w =q0 u= F

G(s) from q0 to [q1 q0 F]G2(s) from q0 to [q2 q2 q1]

Trade-off between G 1 and G22 2


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Dynamic output-feedback

Continuous-time LTI open-loop system

x = Ax + Bww+ Buuz = Czx + Dzww+ Dzuuy = Cyx + Dyww

with dynamic output-feedback controller

xc = Acxc+ Bcyu = Ccxc+ Dcy

Denote closed-loop system asx = Ax + Bwz = Cx + Dw

with x=

xxc

and

A=

A + BuDcCy BuCc

BcCy Ac

B =

Bw+ BuDcDyw

BcDyw

C=

Cz+ DzuDcCy DzuCc

D =Dzw+ DzuDcDyw

Affine expressions on controller matrices


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H2 output feedback design

H2 design conditions

trace W < W CQ

Q

0

AQ + QA BB I

0

can be linearized with a specificchange of variables

Denote

Q=

Q QQ

P = Q1 =

P PP

so that P and Q can be obtainedfrom P and Q via relation

P Q + PQ=I

Always possible when controller hassame orderthan the open-loop plant


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H output-feedback design

Similarly two-step procedure for full-order Houtput-feedback design:

solve LMI for Lyapunov variables Q, P, W and

controller variables X, Y, U, V

retrieve controller matrices via linear algebra

For reduced-order controller of order nc < n

there exists a solution P,Q to the equation

P Q + PQ=I

iff

rank (P Q I) =nc

rank

Q I

I P

=n + nc

Static output feedback iff P Q=I

Difficult rank constrained LMI !


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H output-feedback design

Alternative LMI formulation viaprojectionontonull-spaces (recall Finslers lemma)

N

AQ + QA QCz Bw

I Dzw

I

N 0

M

A

P+ PA PBw Cz I Dzw I

M 0

Q I

I P

0

where N and M are null-space basis

Bu D

zu 0

N = 0

Cu Dyw 0

M = 0

Controller coefficients retrieved afterwards

with a similar change of variables

Alleviate assumptions of Riccati approach(Dzu and Dyw full rank, no imaginary zeros)


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COURSE ON LMI

PART II.3

LMIs IN SYSTEMS CONTROL

ROBUSTNESS ANALYSIS

POLYNOMIAL METHODS

Didier HENRIONwww.laas.fr/henrion

October 2006
http://www.laas.fr/~henrionhttp://www.laas.fr/~henrionhttp://www.laas.fr/~henrionhttp://www.laas.fr/~henrion


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Polynomial methods

Based on the algebra of polynomials and

polynomial matrices, typically involve

linear Diophantine equations quadratic spectral factorization

Pioneered in central Europe during the 70smainly by Vladimr Kucera from the former

Czechoslovak Academy of Sciences

Network funded by the European commission

www.utia.cas.cz/europoly

Polynomial matrices also occur in Jan Willems

behavorial approach to systems theory

Alternative to state-space methods developed

during the 60s most notably by Rudolf Kalmanin the USA, rather based on

linear Lyapunov equations quadratic Riccati equations
http://www.utia.cas.cz/europolyhttp://www.utia.cas.cz/europoly


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Ratio of polynomials

A scalar transfer function can be viewed as theratio of two polynomials

Example

Consider the mechanical system

k2

1k

m

u

y

y displacement u external force

k1 viscous friction coeff k2 spring constant

m mass

Neglecting static and Coloumb frictions, we

obtain the linear transfer function

G(s) =y(s)

u(s)=

1

ms2 + k1s + k2


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Ratio of polynomial matrices

Similarly, a MIMO transfer function can be

viewed as the ratio ofpolynomial matrices

G(s) =NR(s)D1R (s) =D

1L (s)NL(s)

the so-called matrix fraction description (MFD)

Lightly damped structuressuch as oil derricks,

regional power models, earthquakes models,

mechanical multi-body systems, damped gyro-

scopic systems are most naturally represented

by second order polynomial MFDs

(D0+ D1s + D2s2)y(s) =N0u(s)

Example

The (simplified) oscillations of a wing in an air streamis captured by properties of the quadratic polynomialmatrix[Lancaster 1966]

D(s) = 121 18.9 15.9

0 2.7 0.14511.9 3.64 15.5

+ 7.66 2.45 2.1

0.23 1.04 0.2230.6 0.756 0.658

s+ 17.6 1.28 2.891.28 0.824 0.4132.89 0.413 0.725

s2


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First-order polynomial MFD

Example

RCL network

C

R

L

u y y21

y1 voltage through inductor y2 current through inductor

u voltage

Applying Kirchoffs laws and Laplace transform we get 1 LsCs 1 + RCs

y1(s)y2(s)

=

0Cs

u(s)

and thus the first-order left system MFD

G(s) =

1 LsCs 1 + RCs

1 0Cs

.


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Second-order polynomial MFD

Example

mass-spring system

Vibration of system governed by 2nd-order differential

equation Mx + Cx + Kx= 0 where e.g. n= 250, mi =1, i = 5, i = 10 except 1 =n= 10 and 1 =n = 20

Quadratic matrix polynomial

D(s) =M s2 + Cs + K

with

M=IC= tridiag(10, 30, 10)

K= tridiag(5, 15, 5).


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More examples of polynomial MFDs

Higher degree polynomial matrices can also

be found in aero-acoustics (3rd degree) or in

the study of the spatial stability of the Orr-

Sommerfeld equation for plane Poiseuille flow

in fluid mechanics (4rd degree)

Pseudospectra of Orr-Sommerfeld equation

For more info see Nick Highams homepage at

www.ma.man.ac.uk/higham
http://www.ma.man.ac.uk/~highamhttp://www.ma.man.ac.uk/~highamhttp://www.ma.man.ac.uk/~highamhttp://www.ma.man.ac.uk/~higham


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Stability analysis for polynomials

Well established theory - LMIs are of no use here !

Given a continuous-time polynomial

p(s) =p0+ p1s + +pn1sn1 +pns

n

with pn >0 we define its n n Hurwitz matrix

H(p) =

pn1 pn3 0 0pn pn2... ...

0 pn1 . . . 0 0

0 pn p0 0... ... p1 00 0 p2 p0

Hurwitz stability criterion: Polynomial p(s) is stable iff

all principal minors of H(p) are >0

Adolf Hurwitz(Hanover 1859 - Zurich 1919)


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Robust stability analysis for polynomials

Analyzingstability robustnessof polynomials is

a little bit more interesting..

Here toocomputational complexitydepends on

theuncertainty model

In increasing order of complexity, we will

distinguish between

single parameter uncertainty q[qmin, qmax] interval uncertainty qi [qimin, qimax]

polytopic uncertainty 1q1+ + NqN multilinear uncertainty q0+ q1 q2 q3

LMIs will not show up very soon....justbasic linear algebra


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Single parameter uncertainty

and eigenvalue criterion

Consider the uncertain polynomial

p(s, q) =p0(s) + qp1(s)

where p0(s) nominally stable with positive coefs p1(s) such that degp1(s)


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Higher powers of a single parameterNow consider the continuous-time polynomial

p(s, q) =p0(s) + qp1(s) + q2p2(s) + + qmpm(s)

with p0(s) stable and degp0(s)>degpi(s)

Using the zeros (roots of determinant) of thepolynomialHurwitz matrix

H(p) =H(p0) + qH(p1) + q2H(p2) + + q

mH(pm)

we can show that

qmin = 1/min(M)qmax = 1/

+max(M)

where

M=

0 0 0... . . . ... ...0 I 0

0 0 IH10 Hm H

10 H2 H

10 H1

is a block companion matrix


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MIMO systems

Uncertain multivariable systems are modeled

by uncertain polynomial matrices

P(s, q) =P0(s)+qP1(s)+q2P2(s)+ +q

mPm(s)

where p0(s) = det P0(s) is a stable polynomial

We can apply the scalar procedure to the

determinant polynomial

det P(s, q) =p0(s)+qp1(s)+q2p2(s)+ +q

rpr(s)

Example

MIMO design on the plant with left MFD

A1(s, q)B(s, q) =

s2 q

q2 + 1 s

1 s + 1 0

q 1

=

s

2

+ s q2

qqs2 (q2 + 1)s (q2 + 1) s2

s3q2q

with uncertain parameter q [0, 1]


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MIMO systems: example

Using some design method, we obtain a

controller with right MFD

Y(s)X1(s) = 94 51s 1 8 + 1 7s55 100 55 + s 171 18 + s Closed-loop system with characteristic

denominator polynomial matrix

D(s, q) = A(s, q)X(s) + B(s, q)Y(s)= D0(s) + qD1(s) + q

2D2(s)

Nominal system poles: roots of det D0(s)

Applying theeigenvalue criterionon det D(s, q)

yields the stability interval

q ] 0.93, 1.17[ [0, 1]

so the closed-loop system is robustly stable


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Independent uncertainty

So far we have studied polynomials affected bya single uncertain parameter

p(s, q) = (6 + q) + ( 4 + q)s + ( 2 + q)s2

However in practiceseveralparameters can beuncertain, such as in

p(s, q) = (6 + q0) + ( 4 + q1)s + ( 2 + q2)s2

Independent uncertainty structure: eachcomponent qi enters into only one coefficient

Intervaluncertainty: independent structure anduncertain parameter vectorqbelongs to a givenbox, i.e. qi [q

i , q

+i ]

Example

Uncertain polynomial

(6 + q0) + ( 4 + q1)s + ( 2 + q2)s2, |qi| 1

has interval uncertainty, also denoted as

[5, 7] + [3, 5]s + [1, 3]s2

Some coefficients can be fixed, e.g.

6 + [3, 5]s + 2s2


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Kharitonovs polynomials

Associated with the interval polynomial

p(s, q) =n

i=0

[qi , q+i ]s

i

are four Kharitonovs polynomials

p(s) = q0 + q1 s + q

+2 s

2 + q+3 s3 + q4 s

4 + q5 s5 +

p+(s) = q0 + q+1s + q

+2 s

2 + q3 s3 + q4 s

4 + q+5s5 +

p+(s) = q+0 + q1 s + q

2 s

2 + q+3 s3 + q+4 s

4 + q5 s5 +

p++(s) = q+0 + q+1s + q

2 s

2 + q3 s3 + q+4 s

4 + q+5 s5 +

where we assume q

n >0 and q

+

n >0

Example

Interval polynomial

p(s, q) = [1, 2 ] + [ 3, 4]s + [5, 6]s2 + [7, 8]s3

Kharitonovs polynomials

p(s) = 1 + 3s + 6s2 + 8s3p+(s) = 1 + 4s + 6s2 + 7s3

p+(s) = 2 + 3s + 5s2 + 8s3

p++(s) = 2 + 4s + 5s2 + 7s3


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Kharitonovs theorem

In 1978 the Russian researcher Vladimr Kharitonov

proved the following fundamental result

A continuous-time interval polynomialis robustly stable iff its

four Kharitonov polynomials are stable

Instead of checking stability of an infinite

number of polynomials we just have to check

stability offourpolynomials, which can be done

using the classical Hurwitz criterion

Peter and Paul fortress in St Petersburg


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Affine uncertainty

Sadly, Kharitonovs theorem is valid only for continuous-time polynomials for independent interval uncertaintyso that we have to use more general tools in practice

When coefficients of an uncertain polynomial p(s, q) or

a rational function n(s, q)/d(s, q) depend affinely onparameter q, such as in

aTq+ b

we speak about affine uncertainty

x(s)

n(s,q)

d(s,q)

y(s)

The above feedback interconnection

n(s, q)x(s)d(s, q)x(s) + n(s, q)y(s)

preserves the affine uncertainty structure of the plant


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Polytopes of polynomials

A family of polynomials p(s, q), q Q is said to

be a polytope of polynomials if

p(s, q) has an affine uncertainty structure

Q is a polytope

There is a natural isomorphism between

a polytope of polynomials and its

set of coefficients

Example

p(s, q) = (2q1 q2+ 5) + (4q1+ 3q2+ 2)s + s2, |qi| 1Uncertainty polytope has 4 generating vertices

q1 = [1, 1] q2 = [1, 1]q3 = [1, 1] q4 = [1, 1]

Uncertain polynomial family has 4 generating vertices

p(s, q1) = 4 5s + s2 p(s, q2) = 2 + s + s2

p(s, q3) = 8 + 3s + s2 p(s, q4) = 6 + 9s + s2

Any polynomial in the family can be written as

p(s, q) =

4i=1

ip(s, qi),

4i=1

i= 1, i 0


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Interval polynomials

Interval polynomials are a special case of

polytopic polynomials

p(s, q) =n

i=0

[qi , q+i ]s

i

with at most 2n+1 generating vertices

p(s, qk) =n

i=0

qki si, qki =

qior

q+i

1k2n+1

Example

The interval polynomialp(s, q) = [5, 6] + [3, 4]s + 5s2 + [7, 8]s3 + s4

can be generated by the 23 = 8 vertex polynomials

p(s, q1) = 5 + 3s + 5s2 + 7s3 + s4

p(s, q2) = 6 + 3s + 5s2 + 7s3 + s4

p(s, q3) = 5 + 4s + 5s2 + 7s3 + s4

p(s, q4) = 6 + 4s + 5s2 + 7s3 + s4

p(s, q5) = 5 + 3s + 5s2 + 8s3 + s4p(s, q6) = 6 + 3s + 5s2 + 8s3 + s4

p(s, q7) = 5 + 4s + 5s2 + 8s3 + s4

p(s, q8) = 6 + 4s + 5s2 + 8s3 + s4


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The edge theorem

Let p(s, q), qQ be a polynomial with

invariant degree over polytopic set Q

Polynomial p(s, q) is robustly stableover the whole uncertainty polytope Q

iff p(s, q) is stable along the edges of Q

In other words, it is enough to check robust

stability of the single parameter polynomial

p(s, qi1) + ( 1 )p(s, qi2), [0, 1]

for each pair of vertices qi1 and qi2 of Q

This can be done with the eigenvalue criterion


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More about uncertainty structure

In typical applications, uncertainty structure is

more complicated than interval or affine

Usually, uncertainty enters highly non-linearly

in the closed-loop characteristic polynomial

We distinguish between

multilinear uncertainty, when each uncertain

parameter qi is linear when other parameters

qj, i=j are fixed polynomic uncertainty, when coefficients are

multivariable polynomials in parameters qi

We can define the following hierarchy on the

uncertainty structures

interval affine multilinear polynomic


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Examples of uncertainty structures

Examples

The uncertain polynomial

(5q1 q2+ 5) + (4q1+ q2+ q3)s + s2

has affine uncertainty structure


(5q1 q2+ 5) + (4q1q3 6q1q3+ q3)s + s2

has multilinear uncertainty structure


(5q1 q2+ 5) + (4q1 6q1 q23)s + s

2

has polynomic (here quadratic) uncertainty

structure


(5q1 q2+ 5) + (4q1 6q1q23+ q3)s + s

2

has polynomic uncertainty structure


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Polynomial stability analysis: summary

Checking robust stability can be

easy (polynomial-time algorithms) or more

difficult (exponential complexity)

depending namely on the uncertainty model

We focused on polytopic uncertainty:

Interval scalar polynomials

Kharitonovs theorem (ct only)

Polytope of scalar polynomials(affine polynomial families)

Edge theorem

Interval matrix polynomials

(multiaffine polynomial families)

Mapping theorem

Polytopes of matrix polynomials(polynomic polynomial families)


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Lessons from robust analysis:

lack of extreme point results

Ensuring robust stability of the parametrized

polynomial

p(s, q) =p0(s) + qp1(s)q [qmin, qmax]

amounts to ensuring robust stability of the

whole segment of polynomials

p(s, qmin) + ( 1 )p(s, qmax)

= qmaxqqmaxqmin[0, 1]

A natural question arises: does stability of twovertices imply stability of the segment ?

Unfortunately, the answer is no

Example

First vertex: 0.5 7 + 6s + s2

+ 10s3

stableSecond vertex: 1.5 7 + 8s + 2s2 + 10s3 stable

But middle of segment:

1.0 7 + 7s + 1.50s2 + 10s3 unstable


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Lessons from robust analysis:

lack of edge results

In the same way there is lack of vertex results

for affine uncertainty, there is a lack of edge

results for multilinear uncertainty

ExampleConsider the uncertain polynomial

p(s, q) = (4.032q1q2+ 3.773q1+ 1.985q2+ 1.853)+(1.06q1q2+ 4.841q1+ 1.561q2+ 3.164)s+(q1q2+ 2.06q1+ 1.561q2+ 2.871)s2

+(q1+ q2+ 2.56)s3 + s4

with multilinear uncertainty over the polytope

q1 [0, 1], q2 [0, 3], corresponding to the

state-space interval matrix

p(s, q) = det(sI

[1.5, 0.5] 12.06 0.06 00.25 0.03 1 0.5

0.25 4 1.03 00 0.5 0 [4, 1]

)

The four edges of the uncertainty bounding

set arestable, however for q1= 0.5 and q2= 1

polynomial p(s, q) is unstable..


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Non-convexity of stability domain

Main problem: the stability domain in the space

of polynomial coefficients pi is non-convex in

general

0 1 2 3 4 5 61

0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

UNSTABLE

STABLE

q1

q2

Discrete-time stability domain in (q1, q2) plane for poly-

nomial p(z, q) = (0.825 + 0.225q1+ 0.1q2) + (0.895 +

0.025q1+ 0.09q2)z+ (2.4 7 5 + 0.675q1+ 0.3q2)z2

+ z3

How can weovercomethe non-convexity of the

stability conditions in the coefficient space ?


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Handling non-convexity

Basically, we can pursue two approaches:

we canapproximatethe non-convex stabilitydomain with a convex domain (segment, polytope,sphere, ellipsoid, LMI)

we can address the non-convexity with thehelp of non-convex optimization (global or localoptimization)


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Stability polytopes

Largesthyper-rectangle around a nominally

stable polynomial

p(s) + rn

i=0

[i, i]si

obtained with the eigenvalue criterion appliedon the 4 Kharitonov polynomials

In general, there is no systematic way to

obtain more general stability polytopes, namely

because of computational complexity

(no analytic formula for the volume of a polytope)

Well-known candidates:

ct LHP: outer approximation(necessary stab cond)positive cone pi >0

dt unit disk: inner approximation(sufficient stab cond)diamond |p0| + |p1| + + |pn1|


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Stability region (second degree)

Necessary stab cond in dt: convex hull of

stability domain is a polytope whose n + 1vertices are polynomials with roots +1 or -1

Example

When n= 2: triangle with vertices(z+ 1)(z+ 1) = 1 + 2z+ z2

(z+ 1)(z 1) = 1 + z2

(z 1)(z 1) = 1 2z+ z2


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Stability hyper-spheres

Largest hyper-sphere around a nominally

stable polynomial

p(s) +n

i=0

qisi, q r

has radius

rmax= min{|p0|, |pn|, inf>0

(Rep(j))2

1 + w4 + w8..+

(Imp(j))2

w2 + w6 + ..

Example(2 + q0) + ( 1.4 + q1)s + (1.5 + q2)s2 + ( 1 + q3)s3, q r

1.15 1.152 1.154 1.156 1.158 1.16 1.162 1.164 1.166 1.168 1.171

1.2

1.4

1.6

1.8

2

2.2x 10

3

f()

rmax = min{2, 1, inf>0 f(w)}=1.08 103


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Stability ellipsoids

A weighted androtated hyper-sphere is

anellipsoid

We are interested in inner ellipsoidal

approximations of stability domains

E={p : (p p)P(p p)1}

where

p coef vector of polynomial p(s)p center of ellipsoidP positive definite matrix


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Hermite stability criterion

Charles Hermite (1822 Dieuze - 1901 Paris)

The polynomial p(s) =p0+ p1s + +pnsn is

stable if and only if

H(x) =

i

jpipjHij 0

where matrices Hij are given and depend onthe root clustering region only

Examples for n= 3:continuous-time stability

H(p) = 2p0p1 0 2p0p3

0 2p1p2 2p0p3 02p0p3 0 2p2p3

discrete-time stability

H(p) =

p23 p

20 p2p3 p0p1 p1p3 p0p2

p2p3 p0p1 p22+ p23 p

20 p

21 p2p3 p0p1

p1p3 p0p2 p2p3 p0p1 p23 p20


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Inner ellipsoidal appromixation

Our objective is then to findp andP such that

the ellipsoid

E={p: (p p)P(p p)1}

is a convex inner approximation of the actual

non-convex stability region

S={p:H(p)0}

that is to say

ES

Naturally, we will try to enlarge the volume of

the ellipsoid as much as we can

Using the Hermite matrix formulation, we can

derive (details omitted) a suboptimal

LMIformulation (not given here) with decision

variables p and P


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Example

Discrete-time second degree polynomial

p(z) =p0+ p1z+ z2

We solve the LMI problem and we obtain

P =

1.5625 0

0 1.2501

p=

0.2000

0

which describes an ellipse E inscribed in the

exact triangular stability domain S

1.5 1 0.5 0 0.5 1 1.52.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

p0

p1


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ExampleDiscrete-time third degree polynomial

p(z) =p0+ p1z+ p2z2 + z3

We solve the LMI problem and we obtain

P =

2.3378 0 0.53970 2.1368 0

0.5397 0 1.7552

x=

00.1235

0

which describes aconvexellipse E inscribed in the exactstability domain S delimited by the non-convexhyperbolic paraboloid

1

0

11

0 1

2 3

3

2

1

0

1

2

3

A

D

p1

Q

C

B

p0

p2

Very simple scalar convex sufficient stability condition

2.4166p20+ 2.2088p21+ 1.8143p

22 0.5458p1+ 1.1158p0p2 1


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Fixed-order robust design:

a difficult problem

In this last part of course, we study robust

stabilization with a fixed-order controller

affected by parametric uncertainty

A difficult problem in general because fixed-order controller means non-convexityof the design space

parametric uncertainty meanshighly structured uncertainty and exponential

(combinatorial) complexity

In the literature: a lot of analysis results,

but very few design results..

Low-order controllers are required in embedded devices


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Once more: uncertainty models

We can consider several uncertainty models, in

decreasing order of complexity:

Polytopic uncertainty, where plant (numer-ator and/or denominator) polynomial p(s) be-longs to a polytope with given vertices

Ex. p(s) =1(s + 1)3 + 2(s + 1)(s + 2)2 with 1+ 2 = 1

Very general, but often leads to intractable

analysis/design problems

Interval uncertainty, where each of the plantparameters are assumed to vary independently

in given intervalsEx. p(s) = 1 + [2, 3]s + [4, 1]s2 + s3

Leads to nice analysis results (Kharitonov)

but intractable design problems

Ellipsoidal uncertainty, also called rank-one(LFT), or norm-bounded

Ex. p(s) =p0+ p1s +p2s2 + s3 with 3p20+ p0p1+ 2p21+ p22 1Less structured, but more realistic, naturally arises in the context

of parameter estimation for process identification,

ellipsoid = covariance matrix


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Existing results

H design methods, state-space techniques Critical direction (Nyquist), convexoptimization with cutting-plane algorithms

Infinite-dimensional Youla-Kucera parametriza-tion generally leading to high-order controllers

Use of linear programming with polytopicsufficient conditions for stability

In this course

Use of polynomial techniques Use of LMIoptimization

Controllers offixed (hence low) order


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Nominal Pole placement

We consider the SISO feedback system

b

a

x

y

+

Closed-loop transfer function

bx

ax + byIn the absence of hidden modes (a and b coprime

polynomials),pole placementamounts to finding

polynomials x and y solving the Diophantine

equation (from Diophantus of Alexandria 200-284)

ax + by =c

where c is a given closed-loop characteristic

polynomial capturing the desired system poles


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Pole placement: numerical aspects

The polynomial Diophantine equation

ax + by =c

is linear in unknowns x and y, and denotinga(s) =a0+ a1s + + ada sdax(s) =x0+ x1s + + xdbxdb

etc..

we can identify powers of the indeterminate s to builda linear system of equations

a0 b0

a1. . . b1

. . .... a0

... b0ada a1 bdb b1

. . . ...

. . . ...

ada bdb

x0x1...

xdxy0y1...

ydy

=

c0c1...

cdc

The above matrix is called Sylvester matrix, it has aspecial Toeplitz banded structure that can be exploitedwhen solving the equation

James J Sylvester Otto Toeplitz(1814 London - 1897 London) (1881 Breslau - 1940 Jerusalem)


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Pole placement for MIMO systems

Pole placement can be performed similarly for

a plant left MFD

A1(s)B(s)

with a controller right MFD

Y(s)X1(s)

The Diophantine equation to be solved is now

over polynomial matrices

A(s)X(s) + B(s)Y(s) =C(s)

and right hand-side matrix C(s) captures

information on invariant polynomials and

eigenstructure

For example C(s) may contain H2 or Hoptimal dynamics (obtained with spectral

factorization)


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Robust pole placement

Now assume that the plant transfer function

b(q)

a(q)

contains some uncertain parameter q

The problem ofrobust pole placementwill thenconsists in finding a controller

y

xsuch that the uncertain closed-loop charact.

polynomial

a(q)x + b(q)y =c(q)

is robustly stable

How can we find x, y to ensure robust stability

of c(q) for all admissible uncertainty q ?

Coefficients of c are linear in x and y, but we

saw that stability conditions arenon-linearand

highly non-convex in c..


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Robust pole placement

One possible remedy is a suitable

Convex approximation ofthe stability region

Then we can perform design with

linear programming (polytopes) quadratic programming (spheres, ellipsoids) semidefinite programming (LMIs)

Complexity of design algorithm increases

Conservatism of control law decreases


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Robust design via polytopic approximation

MIMO plant with right MFD

B(s)A1(s) =

b 00 0

s + 1 0

0 s + 1

1with uncertainty in parameter

b [0.5, 1.5]We seek a proper first order controller

X1(s)Y(s) = s + x1 x2

0 1 1

= y1s + y2 y3s + y4

0 y5 assigning robustly the closed-loop polynomial matrixC(s) =

s2 + s + (s)

0 s +

whose coefficients live in the polytope

14 1 016 2 02 1 0

0 0 10 0 1

>

196564

214

These specifications amounts to assigning the poles within the disk|s + 8|


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Robust design via polytopic approximation (2)

Equating powers of indeterminate s in the polynomialmatrix Diophantine equation

X(s)A(s) + Y(s)B(s) =C(s)

we obtain the design inequalities

13 7 0.514 8 11 1 0.5

13

21 1.5

14 24 31 3 1.5

x1y1

y2 >

182402

182402

characterizing all parameters x1, y1 and y2 of admissiblerobust controllers

5

0

5

10

15

20

10

0

10

20

30

0

50

100

150

200

250

300

350

400

y1

x1

y2

Corresponding polytope with 9 vertices


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Robust design via ellipsoidal approximation

Closed-loop characteristic polynomial

c(s) = a(s)x(s) + b(s)y(s)

= c0+ c1s + + cd1sd1 + sdwhose coefficients are given by the LSE

c0c1...cd1

=

y0 x0

y1.

. . x1.

. .... y0... x0

ym1 y1 xm1 x1ym

... 1 ...

. . .ym1. . .xm1

ym 1

b0b

1...bn1

a0a1...

an1

+

00...0

x0x1...

xm1

c=S(x, y)p + e(x)

It is assumed that uncertain plant parameters belong totheellipsoid

Ep = {p : (p p)P(p p) 1}where p is a givennominalplant vector and P is a givenpositive definite covariance matrix

Robust control problem:

Find controller coefficients x, yrobustly stabilizing plant a, b subject toellipsoidal uncertainty p Ep


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Ellipsoidal robust design: example

We consider the two mixing tanks arranged in

cascade with recycle stream

P

Fa

Fa Fb

Fa+FbTa

Tb

The controller must be designed to maintain

the temperature Tb of the second tank at a

desired set point by manipulating the power P

delivered by the heater located in the first tank

The only available measurement is

temperature Tb


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Ellipsoidal robust design: example (2)

The identification of the nominal plant model

is carried out using a standard least-squares

method

A second-order discrete-time model

p(z) = b0+ b1za0+ a1z+ z2

is obtained with nominal plant vector

p=

0.0038 0.0028 0.2087 1.1871

The positive definite matrix P characterizingthe uncertainty ellipsoid

Ep = {p : (p p)P(p p) 1}is readily available as a by-product of the

identification technique

P = 105

2.4179 0.0568 0.0069 00.0568 2.4121 0.0045 0.00620.0069 0.0045 0.0015 0.0014

0 0.0062 0.0014 0.0015


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Ellipsoidal robust design: example (3)

Solving the LMI analysis problem we obtain first aninner ellipsoidal approximation

Eq = {q : (q q)Q(q q) 1}of the non-convex stability region, with

Q= 2.3378 0 0.5397

0 2.1368 00.5397 0 1.7552 q =

0

0.12350

Then we solve the design LMI to obtainthe first-order robustly stabilizing controller

y(z)x(z)

= 0.3377+166.0z0.6212+z

Robust closed-loop root-locus forrandom admissible ellipsoidal uncertainty


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Strict positive realness

Let

D = {s :

1s

a bb c

H

1s


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Stability and strict positive realness

Consider two square polynomial matrices of

size n and degree d

N(s) = N0+ N1s + + NdsdD(s) = D0+ D1s + + Ddsd

Polynomial matrix N(s) is stable iff there is astable polynomial D(s) such that rational

matrix N(s)D1(s) is strictly positive real

Proof

From the definition of SPRness, N(s)D1(s)SPR with D(s) stable implies N(s) stable

Conversely, if N(s) is stable then the choice

D(s) =N(s) makes rational matrix

N(s)D1

(s) =I obviously SPR

It turns out that this condition can be

characterized by an LMI..


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LMI condition for design

Given a stable polynomial matrix D(s), poly-

nomial matrixN(s) is stable if there is a matrix

P satisfying the LMI

DN+ ND

H(P)

0

(it follows then that P 0)

New convex inner approximationof stability domain

Shape described by an LMI

Depends on the particular choice of D(s) More general than polytopes and ellipsoids

Useful for design because linear in N

Polynomial D(s) will be referred to as the

central polynomial


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LMI condition for analysis

The LMI condition of SPRness can be used

also for design if we interchange the respective

roles played by N(s) and D(s)

If there is a matrix P and a polynomial matrix

D(s) satisfying the LMI

DN+ ND H(P) 0P 0

then polynomial matrix N(s) is stable

(it follows that D(s) is stable as well)

Useful for (robust)analysisbecause linear in D

Note that, in contrast with the design LMI, we

have here to enforce P 0


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Second-degree discrete-time polynomial

Consider the discrete polynomial

n(z) =n0+ n1z+ z2

We will study the shape of the LMI stability

region for the following central polynomial

d(z) =z2

We can show that non-strict feasibility of the

LMI is equivalent to existence of a matrix Psatisfying

p00+ p11+ p22 = 1p10+ p01+ p21+ p22 = n1

p20+ p02 = n0P

0

which is an LMI in the primal SDP form


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Second-degree discrete-time polynomial (2)

Infeasibility of primal LMI is equivalent to the

existence of a vector satisfying the dual LMI

y0+ n1y1+ n0y20 0 i 1


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Second-degree discrete-time polynomial (3)

The implicit equation of the envelope is

(2n0 1)2 + (

2

2 n1)

2 = 1

a scaled circle

The LMI stability region is then the union ofthe interior of the circle with the interior of the

triangle delimited by the two lines

n0 n1+ 1 = 0tangent to the circle, with vertices [

1, 0],

[1/3, 4/3] and [1/3,4/3]

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

2

1.5

1

0.5

0

0.5

1

1.5

2

p0

p1


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Application to robust stability analysis

Assume that N(s, ) is a polynomial matrix

with multi-linear dependence in a parameter

vector belonging to a polytope

Denote by Ni(s) the vertices obtained by

enumerating each vertex in

Polytopic polynomial matrix N(s, ) isrobustly

stable if there exists a matrix D and matrices

Pi satisfying the LMI

DNi+ NiD H(Pi) 0

Pi 0 iProof

Since the LMI is linear in D - matrix of coefficientsof polynomial matrix D(s) - it is enough to check thevertices to prove stability in the whole polytope


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Robust stability of polynomial matricesExample

Consider the following mechanical system

x

x

u

m

m

d

d

2

1 2

1

2

1c1

c12

c2

It is described by the polynomial MFDm1s2 + d1s + c1+ c12 c12

c12 m2s2 + d2s + c2+ c12

x1(s)x2(s)

=

0

u(s)

System parameters = [ m1 d1 c1 m2 d2 c2]belong to the uncertainty hyper-rectangle = [1, 3] [0.5, 2] [1, 2] [2, 5] [0.5, 2] [2, 4]and we set c12 = 1

This mechanical system is passive so it must be open-loop stable (when u(s) = 0) independently of the valuesof the masses, springs, and dampers


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Polytope of polynomials

We can also check robust stability ofpolytopes

of polynomialswithout using the edge theorem

or the graphical value set

Example

Continuous-time polytope of degree 3 with 3 vertices

n1(s) = 28.3820 + 34.7667s + 8.3273s2 + s3

n2(s) = 0.2985 + 1.6491s + 2.6567s2 + s3

n3(s) = 4.0421 + 9.3039s + 5.5741s2 + s3

The LMI problem is feasible, so the polytope is robustlystable see robust root locus below

4 3.5 3 2.5 2 1.5 1 0.5 04

3

2

1

0

1

2

3

4

Real


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Interval polynomial matrices

Similarly, we can assess robust stability of intervalpolynomial matrices, a difficult problem in general

Example

Continuous-time interval polynomial matrix of degree 2with 23 = 8 vertices

[7.7 2.3s + 4.3s2,

3.7 + 2.7s + 4.3s

2

]

[3.1 6s 2.2s2,4.1 7s 2.2s

2

]3.6 + 6.4s + 4.3s2

[3.2 + 1 1s + 8.2s2,16 + 12s + 8.2s2]

LMI is feasible so the matrix is robustly stableSee robust root locus below


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State-space systems

One advantage of our approach is that state-

space results can be obtained as simple by-

products, since stability of a constant matrix

A is equivalent to stability of the pencil matrix

N(s) =sI A

Matrix A is stable iff there exists a matrix F

and a matrix P solving the LMI

FA + AF aP A F bPA F bP 2I cP 0P 0

Proof

Just take D(s) = sI F and notice that theLMI can be also written more explicitly asF

I

A I

+

A

I

F I

aP bPbP cP

>0


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Robust stability of state-space systems

We recover the LMI stability conditionsobtained by Geromel and de Oliveira in 1999

Nice decoupling between Lyapunov matrix Pand additional variable F allows for construc-tion of parameter-dependent Lyapunov matrix

Assume that uncertain matrix A() has multi-linear dependence on polytopic uncertain pa-rameter and denote by Ai the correspondingvertices

Matrix A() is robustly stable if there exists amatrix F and matrices Pi solving the LMI

FAi+ A

i F aPi Ai F bPi

Ai F bPi 2I cPi

0

Pi 0 i

Proof

Consider the parameter-dependent Lyapunovmatrix P() built from vertices Pi


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Robust design

Assume now that system matrix C(s, ) comes

from a polynomial Diophantine equation

C(s, ) =A(s, )X(s) + B(s, )Y(s)

where system matrices A and B are subject to

multi-linear polytopic uncertainty

In order to ensure robust SPRness of the ra-

tional matrix D1(s)C(s, ) central polynomialD(s) must becloseto the nominal closed-loop

matrix, such as

D(s) =C(s, 0)

where 0 is the nominal parameter vector

A sensible simple choice of D(s) is thereforethe nominal closed-loop denominator polyno-

mial matrix, obtained by any standard design

method (pole assignment, LQ, H)


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Reactor

Consider the stirred tank reactor model described bynon-linear state-space equations

1 = (2+ 0.5)exp(E1/(1+ 2)) (2 + u)(1+ 0.25)2 = 0.5 2 (2+ 0.5)exp(E1/(1+ 2))

where E is a parameter related to the activation energy

During the life of the reactor, some representative valuesof E are 20, 25 and 30

Using only 1 for feedback we obtain a polytopic systemwith 3 linearized vertex transfer functions

b1(s)/a1(s) = (0.5

0.25s)/(11

5s + s2)

b2(s)/a2(s) = (0.5 0.25s)/(2.25 2.25s + s2)b3(s)/a3(s) = (0.5 0.25s)/(3.5 3.5s + s2).

Our objective is to stabilize the whole polytopic systemwith asingle static output feedback controller y(s)/x(s)


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Reactor

For the choice

d(s) = (s + 1)2

as a central polynomial, solving the LMI yields therobustly stabilizing controller

y(s)/x(s) =k = 21.4409

With the eigenvalue criterion we can check that k issimultaneous stabilizing iff

22< k < 20Note however that the problem solved here is moredifficult since we must stabilize the whole polytope,not only its vertices

Edges of root locus


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Oblique wing aircraft

We consider the model of an experimental oblique wingaircraft

The linearized transfer function

[90, 166] + [54, 74]s

[0.1, 0.1] + [30.1, 33.9]s + [50.4, 80.8]s2 + [2.8, 4.6]s3 + s4

features 6 uncertain parameters, and must be stabilizedwith a PI controller

y(s)

x(s)=Kp+

Ki

s

One can check easily that the choice Kp = 1 and Ki = 1stabilizes the vertex plant

9 0 + 5 4s

0.1 + 3 0s + 50s2 + 2.8s3 + s4


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Oblique wing aircraft (2)

With the choice

d(s) =sa1(s) + ( 1 + s)b1(s)

as the central polynomial, the LMI problem is infeasible

But when considering another vertex plant

b2(s)

a2(s)

= 9 0 + 5 4s

0.1 + 3 0s + 50s2

+ 4.6s3

+ s4

the choice

d(s) =sa2(s) + ( 1 + s)b2(s)

now proves successful, the LMI is solved and we obtainthe robustly stabilizing PI controller

y(s)

x(s)= 0.8634 +

0.6454

s


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Satellite

We consider asatellitecontrol problem where the satel-lite model is two masses with the same inertia connectedby a spring with torque constant q1 and viscous dampingconstant q2

The transfer function between the control torque andthe satellite angle is given by

b(s, q)

a(s, q)=

q1+ q2s + s2

s2(2q1+ 2q2s + s2)

where uncertain parameters are assumed to vary withinthe bounds

q1 [0.09, 4], q2 [0.04q110, 0.2q210]With the choice

d(s) = (s + 1)6

as a central polynomial, the LMI design method cannotstabilize the whole polytope


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Satellite (2)

However we can stabilize each vertex ai(s), bi(s)individuallywith controller polynomials xi(s), yi(s)

The roots of the corresponding closed-loop polynomialsci(s) =ai(s)xi(s) + bi(s)yi(s) are given below

i roots of ci(s)1 0.4520j1.8129,0.0365j0.8954,0.0019 j0.25332 0.6161j1.3829,0.2190j0.6745,0.0237 j0.21363 1.0484,0.0917 j2.1598,0.0425 j0.5019,0.00044 0.2220j2.0695,0.1695j0.5354,0.0709 j0.0045

The third vertex has polesnearest to the imaginary axis,and with the choice

d(s) =c3(s)

as a central polynomial, the LMI is found feasible andwe obtain the robustly stabilizing controller

y(s)

x(s)=

0.0181 + 0.0170s 1.4048s23.6082 + 1.0237s + s2

Sometimes it can be tricky to be find the central poly-nomial (provided one exists = the system is robustlystabilizable)..


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Robot

We consider the problem of designing a robust controllerfor the approximate ARMAX model of a PUMA roboticdisk grinding process

From the results of identification and because of the

nonlinearity of the robot, the coefficients of the numer-ator of the plant transfer function change for differentpositions of the robot arm. We consider variations ofup to 20% around the nominal value of the parameters

The fourth-order discrete-time model is given by

b(z1, q)

a(z1, q) = (0.0257 + q1) + (0.0764 + q2)z1

+(

0.1619 + q3)z2 + (

0.1688 + q4)z3 1 1.914z1 + 1.779z2 1.0265z3 + 0.2508z4

where

|q1| 0.00514, |q2| 0.01528, |q3| 0.03238, |q4| 0.03376


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Robot

Closed-loop polynomial

d(z, q) =z12[(1z1)a(z1, q)x(z1)+z5b(z1, q)y(z1)]where the term 1 z1 is introduced in the controllerdenominator to maintain the steady state error to zerowhen parameters are changed

With the input central polynomial d(z) = z19 the LMI

returns the seventh-order robust controller

y(z1)x(z1)

=

0.2863 + 0.2928z1 + 0.0221z20.1558z3 + 0.0809z4 + 0.1420z5

0.1254z6 + 0.0281z7

1 + 1.1590z1 + 0.9428z2+0.4996z3 + 0.3044z4 + 0.4881z5

+0.4003z6 + 0.3660z7


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Second-order systems

Second-order linear system

(A0+ A1s + A2s2)x = Bu

y = Cx

to be controlled by PD output-feedback

controller

u= (F0+ F1s)yApplications: large flexible space structures, earthquake engineer-

ing, mechanical multi-body systems, damped gyroscopic systems,

robotics control, vibration in structural dynamics, flows in fluid me-

chanics, electrical circuits

320m long Millenium footbridge over river Thames in London


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PD controller

Closed-loop system behavior captured by zeros

of quadratic polynomial matrix

N(s) = (A0+ BF0C) + (A1+ BF1C)s + A2s2

Zeros ofN(s) must be located in somestability

region

D characterized as before by matrix H

Uncertainty can affect A0 (stiffness)

A1 (damping) and A2 (mass)

Given A0, A1, A2, B, C

find F0, F1ensuring robust pole placement


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Robust LMI stability condition

Norm-bounded (unstructured) uncertaintyN(s) + M(s) max()

LMI robust stability condition on N(s)

DN+ ND H(P) DD M

M I

0

Polytopic (structured) uncertaintyN(s) =

i

iNi(s) i

i = 1 i 0

Vertex LMI robust stability conditions:

DNi + (Ni)D

H(Pi)

0, i= 1, 2, . . .

Parameter-dependent Lyapunov matrix

P() =

i iPi


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Robust design

Once central polynomial matrix D(s) is fixed,

robust stability condition is LMI in N(s), so

extension to design is straightforward

Easy incorporation of structural constraints

on controller coefficient matrices F0, F1:

minimization of 2-norm (SOCP) enforcing some entries to zero (LP)

maximization of uncertainty radius (SDP)

Key point is choice ofcentral polynomial matrix

Good policy: set D(s) to some nominal sys-

tem matrix obtained by some standard designmethod (pole placement, LQ,H2 orH), thentry to optimize around D(s)


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Example: five masses

Five masses linked by elastic springscontrolled by two external forces

1

2

3

4

5

A0 =

2.565 1.080 0 0 1.0890.6038 0.8206 0.4766 0 0

0 0.6009 1.504 0.4808 00 0 0.4300 1.114 0.5131

0.6190 0 0 0.4626 0.8352

B =

0 1.9640 00 00 0

1.116 0

A1 = 05A2=I5C=I5

Purely imaginary open-loop poles i1.783, i1.380, i1.145i0.5675 andi0.3507Nominal PD controller F00 , F

01 obtained with LQ design

Resulting central polynomial matrix

D(s) = (A0+ BF00 C) + (A1+ BF0

1 C)s + A2s2

Stability regionD = {s: Re s < 0.1}


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Five mass example (2)

Minimizing the norm of feedback matrices F0,

F1 over the design LMI yields

[F0 F1] =0.7537< [F00 F01 ] = 1.462

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