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COURSE ON LMIPART II.1

LMIs IN SYSTEMS CONTROL

STATE-SPACE METHODS

STABILITY ANALYSIS

Didier HENRION

www.laas.fr/∼henrion

October 2006

http://www.laas.fr/~henrion

State-space methods

Developed 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

Linear systems stability

The continuous-time linear time invariant (LTI)system

ẋ(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) = x?Px such that

V (x(t)) > 0V̇ (x(t)) < 0

along system trajectories• equivalently, matrix A satisfies

maxi real λi(A) < 0

Lyapunov stability

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

Since V̇ (x) = ẋ?Px + x?P ẋ = x?A?Px + x?PAx positivityof V (x) and negativity of V̇ (x) along system trajectoriescan be expressed as an LMI

A?P + PA ≺ 0 P � 0

Matrices P satisfying Lyapunov’s LMI with A =[

0 1−1 −2

]

Lyapunov equation

The Lyapunov LMI can be written equivalently

as the Lyapunov equation

A?P + PA + Q = 0

where Q � 0

The following statements are equivalent

• the system ẋ = 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 solvingthe Lyapunov equation satisfies P � 0

The Lyapunov LMI can be solved numerically

without IP methods since solving the above

equation amounts to solving a linear system of

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

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) =

[−A?P − PA 0

0 P

]� 0

is the existence of a nonzero matrix

Z =

[Z1 00 Z2

]� 0

such that

Z1A? + AZ1 − Z2 = 0

Alternative to Lyapunov LMI (proof)

Suppose that there exists such a matrix Z 6= 0and extract Cholesky factor

Z1 = UU?

Since Z1A? + AZ1 � 0 we must have

AUU? = USU?

where 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 λi(S) ≥ 0

Conversely, suppose λi(A) = σ + jω with σ ≥ 0for some i with eigenvector v

Then rank-one matrices

Z1 = vv? Z2 = 2σvv

?

solve the alternative LMI

Discrete-time Lyapunov LMI

Similarly, the discrete-time LTI system

xk+1 = Axk

is asymptotically stable iff

• there exists a quadratic Lyapunov functionV (x) = x?Px such that

V (xk) > 0V (xk+1)− V (xk) < 0

along system trajectories

• equivalently, matrix A satisfies

maxi |λi(A)| < 1

Here too this can be expressed as an LMI

A?PA− P ≺ 0 P � 0

More general stability regions

Let

D = {s ∈ C :[

1s

]? [d0 d1d?1 d2

] [1s

]< 0}

with d0, d1, d2 ∈ C3 be a region of the complexplane (half-plane or disk)

Matrix A is said D-stable when its spectrumσ(A) = {λi(A)} belongs to region D

Equivalent to generalized Lyapunov LMI

[IA

]? [d0P d1Pd?1P d2P

] [IA

]≺ 0 P � 0

LMI stability regions

We can consider D-stability in LMI regions

D = {s ∈ C : D(s) = D0 + D1s + D?1s? ≺ 0}such as

D dynamicsreal(s) < −α dominant behaviorreal(s) < −α, |s| < r oscillationsα1 < real(s) < α2 bandwidth|imag(s)| < α horizontal stripreal(s) tan θ < −|imag(s)| damping cone

or intersections thereof

Example for the cone

D0 = 0 D1 =

[sin θ cos θcos θ sin θ

]

Lyapunov LMI for LMI stability regions

Matrix A has all its eigenvalues in the region

D = {s ∈ C : D0 + D1s + D?1s? ≺ 0}

if and only if the following LMI is feasible

D0 ⊗ P + D1 ⊗AP + D?1 ⊗ 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 + D?1s? + D2s?s ≺ 0}

parabolae, hyperbolae, ellipses etc

convex (D2 � 0) or not

Uncertain systems and robustness

When modeling systems we face several sourcesof 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 (l2)• 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 thewhole range of admissible uncertainty (can bedesigned off-line, usually cheap)

Uncertainty modeling

Consider the continuous-time LTI system

ẋ(t) = Ax(t) A ∈ A

where matrix A belongs to an uncertainty setA

For unstructured uncertainties we considernorm-bounded matrices

A = {A + B∆C : ‖∆‖2 ≤ µ}

For structured uncertainties we considerpolytopic matrices

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

There are other more sophisticated uncertaintymodels not covered here

Uncertainty modeling is an important anddifficult step in control system design !

Robust stability

The continuous-time LTI system

ẋ(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

]

Symmetry

If dynamic systems were symmetric, i.e

A = A?

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

A + A? ≺ 0

and discrete-time stability maxi |λi(A)| < 1 to

A?A ≺ I ⇐⇒[−I AA? −I

]≺ 0

which are both LMIs !

We can show that stability of a symmetric linear systemcan be always proven with the Lyapunov matrix P = I

Fortunately, the world is not symmetric !

Robust and quadratic stability

Because of non-convexity of the cone of stable

matrices, robust stability is sometimes difficult

to check numerically, meaning that

computational cost is an exponential functionof the number of system parameters

Remedy:

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

is quadratically stable if its robust stability can

be guaranteed with the same quadratic

Lyapunov 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

Quadratic stability for polytopic uncertainty

The system with polytopic uncertainty

ẋ(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

λi(ATi P + PAi) = A

T (λ)P + PA(λ) ≺ 0

for all λi ≥ 0 such that∑

i λi = 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

Quadratic and robust stability: example

Consider the uncertain system matrix

A(δ) =

[−4 4−5 0

]+ δ

[−2 2−1 4

]

with real parameter δ such that |δ| ≤ µ= polytope with vertices A(−µ) and A(µ)

stability maxµquadratic 0.7526robust 1.6666

Quadratic stability for norm-bounded

uncertainty

The system with norm-bounded uncertainty

ẋ(t) = (A + B∆C)x(t) ‖∆‖2 ≤ µ


P solving the LMI

[A?P + PA + C?C PB

B?P −γ2I

]≺ 0 P � 0

with γ−1 = µ

This is called the bounded-real lemma

proved next with the S-procedure

We can maximize the level of

allowed uncertainty by minimizing scalar γ

Norm-bounded uncertainty as feedback

Uncertain system

ẋ = (A + B∆C)x

can be written as the feedback system

ẋ = Ax + Bwz = Cxw = ∆z

.x = Ax+Bw

∆

w z

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

V̇ (x) = 2x?P ẋ= 2x?P (Ax + Bw)= x?(A?P + PA)x + 2x?PBw

=

[xw

]? [A?P + PA PB

B?P 0

] [xw

]

Norm-bounded uncertainty as feedback (2)

Since ∆?∆ � µ2I it follows that

w?w = z?∆?∆z � µ2z?z⇐⇒

w?w − µ2z?z =[

xw

]? [ −C?C 00 γ2I

] [xw

]≤ 0

Combining with the quadratic inequality

V̇ (x) =

[xw

]? [A?P + PA PB

B?P 0

] [xw

]< 0

and using the S-procedure we obtain[A?P + PA PB

B?P 0

]≺

[−C?C 0

0 γ2I

]or equivalently

[A?P + PA + C?C PB

B?P −γ2I

]≺ 0 P � 0

Norm-bounded uncertainty: generalization

Now consider the feedback system

ẋ = Ax + Bwz = Cx+Dww = ∆z

with additional feedthrough term Dw

We assume that matrix I−∆D is non-singular= well-posedness of feedback interconnection

so that we can write

w = ∆z = ∆(Cx + Dw)(I −∆D)w = ∆Cx

w = (I −∆D)−1∆Cxand derive the linear fractional transformation

(LFT) uncertainty description

ẋ = Ax + Bw = (A + B(I −∆D)−1∆C)x

Norm-bounded LFT uncertainty

The system with norm-bounded

LFT uncertainty

ẋ =(A + B(I −∆D)−1

)x ‖∆‖2 ≤ µ


P solving the LMI

[A?P + PA + C?C PB + C?D

B?P + D?C D?D − γ2I

]≺ 0 P � 0

Notice the lower right block D?D − γ2I ≺ 0which ensures non-singularity of I −∆D hencewell-posedness

LFT modeling can be used more generally to

cope with rational functions of uncertain pa-

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

Sector-bounded uncertainty

Consider the feedback system

ẋ = Ax + Bwz = Cx + Dww = f(z)

where vector function f(z) satisfies

z?f(z) ≤ 0 f(0) = 0

which is a sector condition

f(z)

z

f(z) can also be considered as an uncertaintybut also as a non-linearity

Quadratic stability for sector-bounded

uncertainty

We want to establish quadratic stability with

the quadratic Lyapunov matrix V (x) = x?Px

whose derivative

V̇ (x) = 2x?P (Ax + Bf(z))

=

[x

f(z)

]? [A?P + PA PB

B?P 0

] [x

f(z)

]must be negative when

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

=

[x

f(z)

]? [0 C?

C D + D?

] [x

f(z)

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

to derive the LMI

[A?P + PA PB − C?B?P − C −D −D?

]≺ 0 P � 0

This is called the positive-real lemma

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

ẋ(t) = A(λ(t))x(t) = (A0 +∑N

i=1 λi(t)Ai)x(t)

where

λ ∈ Λ = {λi ∈ [λi, λi]}we may consider parameter-dependentLyapunov matrices, such as

P (λ(t)) = P0 +∑

i λi(t)Pi

Polytopic Lyapunov certificate

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

P (λ) � 0 ∀λ ∈ Λand

A?(λ)P (λ) + P (λ)A?(λ) + Ṗ (λ) ≺ 0 ∀λ ∈ Λ

We have to solve parametrized LMIs

Parametrized LMIs feature non-linear terms in λ soit is not enough to check vertices of Λ, denoted by vertΛ

λ21 − λ1 + λ2 ≥ 0 on vert ∆but not everywhere on ∆ = [0, 1]× [0, 1]

Parametrized LMIs

Central problem in robustness analysis:

find x such that

F (x, λ) =∑

α λαFα(x) � 0, ∀λ ∈ Λ

where Λ 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 various hierarchies of LMIs

are available:

• Pólya’s theorem• Schmüdgen’s representation• Putinar representationSee EJC 2006 survey by Carsten Scherer



STATE-SPACE METHODS

CONTROL DESIGN

Didier HENRION


October 2006


H2 norm

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

‖G‖2 =(∫ ∞

0h?(t)h(t)dt

)1/2=

(1

2π

∫ ∞−∞

H(jω)H?(jω)dω

)1/2

For a continuous-time system

ẋ = Ax+Buy = Cx+Du

with transfer function G(s) = C(sI−A)−1B+D we mustassume D = 0 to have ‖G‖2 finite

Computing the H2 norm

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

the impulse response of G, then

‖G‖22 =∑i

‖hi‖22

=∑i

∫ ∞0

B?i eA?tC?CeAtBidt

= traceB?(∫ ∞

0eA

?tC?CeAtdt

)B

Matrix

Po =∫ ∞

0eA

?tC?CeAtdt

is the observability Grammian solution to theLyapunov equation

A?Po + PoA+ C?C = 0

and hence

‖G‖22 = traceB?PoB

If (A,C) observable then Po � 0

Dual and LMI computation of the H2 norm

Defining the controllability Grammian

Pc =∫ ∞

0eAtBB?eA

?tdt

solution to the Lyapunov equation

APc + PcA? +BB? = 0

we have a dual expression

‖G‖22 = trace CPcC?

Dual lyapunov equations formulated as LMIs

‖G‖22 = min traceB?PB

s.t. A?P + PA+ C?C � 0P � 0

‖G‖22 = min trace CQC?

s.t. AQ+QA? +BB? � 0Q � 0

H∞ norm

The H∞ norm is the induced energy gain(l2 to l2)

‖G‖∞ = sup‖x‖2=1

‖Gx‖2 = supω‖G(jω)‖

It is the worst case gain

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 function G(s) = C(sI−A)−1B+Dwe have ‖G(s)‖∞ < γ iff R = γ2I − D?D � 0and the Hamiltonian matrix

[A+BR−1D?C BR−1B?

−C?(I +DR−1D?)C −(A+BR−1D?C)?

]

has no eigenvalues on the imaginary axis

We can then design a bisection algorithm

with guaranteed quadratic convergence

to find the minimum value of γ such that

the Hamiltonian has no imaginary eigenvalues

LMI computation of the H∞ norm

Refer to the part of the course on norm-boundeduncertainty

sup‖z‖2=1

‖w‖ = ‖∆‖ < γ−1

to prove that for the continuous-time system


with transfer function G(s) = C(sI−A)−1B+Dwe have ‖G(s)‖∞ < γ iff there exists a matrixP solving the LMI

[A?P + PA+ C?C PB + C?D

B?P +D?C D?D − γ2I

]≺ 0 P � 0

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

A?P + PA PB C?B?P −γI D?C D −γI

≺ 0 P � 0

Linear systems design

Open-loop continuous-time LTI system

ẋ = Ax+Bu

with state-feedback controller

u = Kx

produces closed-loop system

ẋ = (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 !

State-feedback design:

linearizing change of variables

Project BMI onto P−1 � 0

(A+BK)?P + P (A+BK) ≺ 0⇐⇒

P−1 [(A+BK)?P + P (A+BK)]P−1 ≺ 0⇐⇒

P−1A? + P−1K?B? +AP−1 +BKP−1 ≺ 0Denoting

Q = P−1 Y = KP−1

we derive a state-feedback design LMI

AQ+QA? +BY + Y ?B? ≺ 0 Q � 0

We obtained an LMI thanks to a one-to-one

linearizing change of variables

Simple but very useful trick..

Finsler’s lemma

A very useful trick in robust control..

The following statements are equivalent

1. x?Ax > 0 for all x 6= 0 s.t. Bx = 02. B̃?AB̃ > 0 where BB̃ = 03. A+ λB?B > 0 for some scalar λ4. A+XB +B?X? > 0 for some matrix X

Paul Finsler(1894 Heilbronn - 1970 Zurich)

State-feedback design: null-space projection

We can also use item 2 of Finsler’s lemma,projecting onto the (full column rank)null-space B̃ of B?

B?B̃ = 0

so that BMI

A?P + PA+K?B?P + PBK ≺ 0is equivalent to the projected LMI

B̃?(AQ+QA?)B̃ ≺ 0 Q � 0

Feedback K can be recovered fromLyapunov matrix Q as

K = −λB?Q−1

where λ is a suitably large scalar

Here we obtained an LMI thanks to aprojection onto a null-space

State-feedback design: Riccati inequality

We can also use item 3 of Finsler’s lemma toconvert BMI

A?P + PA+K?B?P + PBK ≺ 0

into

A?P + PA− λPBB?P ≺ 0

where λ ≥ 0 is an unknown scalar

Now replacing P with λP we get

A?P + PA− PBB?P ≺ 0

which is equivalent to the Riccati equation

A?P + PA− PBB?P +Q = 0

for some matrix Q � 0

Shows equivalence between state-feedback LMIstabilizability and the linear quadratic regulator(LQR) problem

Robust state-feedback design

for polytopic uncertainty

Open-loop system ẋ = Ax+Bu with polytopic

uncertainty

(A,B) ∈ conv {(A1, B1), . . . , (AN , BN)}

and robust 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 the quadratic stabilizability LMI

AiQ+QA?i +BiY + Y

?B?i ≺ 0 ∀i Q � 0

with the linearizing change of variables

Q = P−1 Y = KP−1

State-feedback H2 control

Continuous-time LTI open-loop system

ẋ = Ax+Bww +Buuz = Czx+Dzww +Dzuu

with state-feedback controller

u = Kx

yields closed-loop system

ẋ = (A+BuK)x+Bwwz = (Cz +DzuK)x+Dzww

with transfer function

G(s) = Dzw+ (Cz +DzuK)(sI−A−BuK)−1Bwbetween performance signals w and z

H2 performance specification

‖G(s)‖2 < γ

We must have Dzw = 0 (finite gain)

H2 design LMIs

As usual, start with analysis condition:there exists K such that ‖G(s)‖2 < γ iff

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

The trace inequality can be written as

trace(Cz+DzuK)Q(Cz+DzuK)? < traceW < γ

for some matrix W such that[W (Cz +DzuK)Q

Q(Cz +DzuK)? Q

]� 0

We obtain the overall LMI formulation

traceW < γ[W CzQ+DzuY

QC?z + Y?D?zu Q

]� 0

AQ+QA? +BuY + Y ?B?u +BwB?w ≺ 0

with resulting H2 suboptimal state-feedback

K = Y Q−1

State-feedback H∞ control

Similarly, with closed-loop system

ẋ = (A+BuK)x+Bwwz = (Cz +DzuK)x+Dzww

and H∞ performance specification

‖G(s)‖∞ < γ

on transfer function between w and z we obtain

the design LMI

[AQ+QA? +BuY + Y ?B?u +BwB

?w ?

CzQ+DzuY +DzwB?w DzwD?zw − γ2I

]≺ 0

Q � 0

with resulting H∞ suboptimal state-feedback

K = Y Q−1

Optimal H∞ control: minimize γ

Mixed H2/H∞ control

State-feedback controller systemwith two performance channels

ẋ = (A+BuK)x+Bwwz∞ = (C∞ +D∞uK)x+D∞wwz2 = (C2 +D2uK)x

and mixed performance specifications

‖G∞(s)‖∞ < γ∞ ‖G2(s)‖2 < γ2

on transfer functions from w to z∞ and z2 respectively

Formulation of H∞ constraint[AQ∞ +BuKQ∞ + (?) +BwB?w ?C∞Q∞ +D∞uKQ∞ +D∞wB?w D∞wD

?∞w − γ2∞I

]≺ 0

Q∞ � 0BMI formulation of H2 constraint

traceW < γ2[W C2Q2 +D2uKQ2? Q2

]� 0

AQ2 +BuKQ2 + (?) +BwB?w ≺ 0

Problem:

We cannot linearize simultaneouslyterms KQ∞ and KQ2 !

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 + (?) +BwB?w ?C∞Q+D∞uY +D∞wB?w D∞wD

?∞w − γ2∞I

]≺ 0

traceW < γ2[W C2Q+D2uY? Q

]� 0

AQ+BuY + (?) +BwB?w ≺ 0

Guaranteed cost mixed H2/H∞:given γ∞ minimize γ2

Can be used for H2 design of uncertain systems

with norm-bounded uncertainty

Mixed H2/H∞ control: example

Active suspension system (Weiland)

m2q̈2 + b2(q̇2 − q̇1) + k2(q2 − q1) + F = 0m1q̈1 + b2(q̇1 − q̇2) + k2(q1 − q2)

+k1(q1 − q0) + b1(q̇1 − q̇0) + F = 0

z =

[q1 − q0Fq̈2

q2 − q1

]y =

[q̈2

q2 − q1

]w = q0 u = F

G∞(s) from q0 to [q1 − q0 F ]G2(s) from q0 to [q̈2 q2 − q1]

Trade-off between ‖G∞‖∞ ≤ γ1 and ‖G2‖2 ≤ γ2

Dynamic output-feedback

Continuous-time LTI open-loop system

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

with dynamic output-feedback controller

ẋc = Acxc +Bcyu = Ccxc +Dcy

Denote closed-loop system as

˙̃x = Ãx̃+ B̃wz = C̃x̃+ D̃w

with x̃ =

[xxc

]and

Ã =

[A+BuDcCy BuCc

BcCy Ac

]B̃ =

[Bw +BuDcDyw

BcDyw

]C̃ =

[Cz +DzuDcCy DzuCc

]D̃ = Dzw +DzuDcDyw

Affine expressions on controller matrices

H2 output feedback design

H2 design conditions

traceW < γ[W C̃Q̃? Q̃

]� 0[

ÃQ̃+ Q̃Ã? B̃B̃? −I

]≺ 0

can be linearized with a specificchange of variables

Denote

Q̃ =

[Q Q̄?

Q̄ ×

]P̃ = Q̃−1 =

[P P̄P̄ ? ×

]

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

PQ+ P̄ Q̄ = I

Always possible when controller has same orderthan the open-loop plant

Linearizing change of variablesfor H2 output-feedback design

Then define[X UY V

]=

[P̄ PBu0 I

] [Ac BcCc Dc

] [Q̄ 0CyQ I

]+

[P0

]A[Q 0

]which is a one-to-one affine relation with converse[Ac BcCc Dc

]=

[P̄−1 −P̄−1PBu

0 I

] [X − PAQ U

Y V

] [Q̄−1 0

−CyQQ̄−1 I

]We derive the following H2 design LMI

traceW < γDzw +DzuV Dyw = 0 W CzQ+DzuY Cz +DzuV Cy? Q I

? ? P

� 0 AQ+BuY + (?) A+BuV Cy +X? Bw +BuV Dyw? PA+ UCy + (?) PBw + UDyw? ? −I

≺ 0in decision variables Q,P ,W (Lyapunov)and X,Y , U, V (controller)

Controller matrices are obtained via the relation

PQ+ P̄ Q̄ = I

(tedious but straightforward linear algebra)

H∞ output-feedback design

Similarly two-step procedure for full-order H∞output-feedback design:• solve LMI for Lyapunov variables Q,P ,W andcontroller variables X,Y , U, V• retrieve controller matrices via linear algebra

For reduced-order controller of order nc < nthere exists a solution P̄ ,Q̄ to the equation

PQ+ P̄ Q̄ = I

iff

rank (PQ− I) = nc⇐⇒

rank

[Q II P

]= n+ nc

Static output feedback iff PQ = I

Difficult rank constrained LMI !

H∞ output-feedback design

Alternative LMI formulation via projection ontonull-spaces (recall Finsler’s lemma)

N?

AQ+QA? QC?z Bw? −γI Dzw? ? −γI

N ≺ 0

M?

A?P + PA PBw C?z? −γI D?zw? ? −γI

M ≺ 0[Q II P

]� 0

where N and M are null-space basis[B?u D

?zu 0

]N = 0

[Cu Dyw 0

]M = 0

Controller coefficients retrieved afterwardswith a similar change of variables

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



ROBUSTNESS ANALYSIS

POLYNOMIAL METHODS

Didier HENRION


October 2006


Polynomial methods

Based on the algebra of polynomials andpolynomial matrices, typically involve• linear Diophantine equations• quadratic spectral factorization

Pioneered in central Europe during the 70smainly by Vladiḿır Kučera from the formerCzechoslovak 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 developedduring the 60s most notably by Rudolf Kalmanin the USA, rather based on• linear Lyapunov equations• quadratic Riccati equations

http://www.utia.cas.cz/europoly

Ratio of polynomials

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

ExampleConsider the mechanical system

k2

1k

mu

y

• y displacement • u external force

• k1 viscous friction coeff • k2 spring constant

• m mass

Neglecting static and Coloumb frictions, weobtain the linear transfer function

G(s) =y(s)

u(s)=

1

ms2 + k1s + k2

Ratio of polynomial matrices

Similarly, a MIMO transfer function can beviewed as the ratio of polynomial matrices

G(s) = NR(s)D−1R (s) = D

−1L (s)NL(s)

the so-called matrix fraction description (MFD)

Lightly damped structures such as oil derricks,regional power models, earthquakes models,mechanical multi-body systems, damped gyro-scopic systems are most naturally representedby second order polynomial MFDs

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

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

D(s) =

[121 18.9 15.90 2.7 0.145

11.9 3.64 15.5

]+

[7.66 2.45 2.10.23 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

First-order polynomial MFD

Example

RCL network

C

R

L

u y y21

• y1 voltage through inductor

• y2 current through inductor

• u voltage

Applying Kirchoff’s laws and Laplace transform we get[1 −Ls

Cs 1 + RCs

] [y1(s)y2(s)

]=

[0

Cs

]u(s)

and thus the first-order left system MFD

G(s) =

[1 −Ls

Cs 1 + RCs

]−1 [0

Cs

].

Second-order polynomial MFD

Examplemass-spring system

Vibration of system governed by 2nd-order differential

equation Mẍ + Cẋ + Kx = 0 where e.g. n = 250, mi =

1, κi = 5, τi = 10 except κ1 = κn = 10 and τ1 = τn = 20

Quadratic matrix polynomial

D(s) = Ms2 + Cs + K

with

M = IC = tridiag(−10,30,−10)K = tridiag(−5,15,−5).

Another second-order polynomial MFD

Example

Inverted pendulum on a cart

Linearization around the upper verticalposition yields the left polynomial MFD[

(M + m)s2 + bs lms2

lms2 (J + l2m)s2 + ks− lmg

] [x(s)φ(s)

]=

[10

]f(s)

With J = mL2/12, l = L/2 and g = 9.8, M =2, m = 0.35, l = 0.7, b = 4, k = 1, we obtainthe denominator polynomial matrix

D(s) =

[5s + 3s2 0.35s2

0.35s2 −3.4 + s + 0.16s2

]

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 Higham’s homepage at

www.ma.man.ac.uk/∼higham

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

Stability analysis for polynomials

Well established theory - LMIs are of no use here !

Given a continuous-time polynomial

p(s) = p0 + p1s + · · ·+ pn−1sn−1 + pnsn

with pn > 0 we define its n× n Hurwitz matrix

H(p) =

pn−1 pn−3 0 0pn pn−2

... ...0 pn−1

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

Hurwitz stability criterion: Polynomial p(s) is stable iffall principal minors of H(p) are > 0

Adolf Hurwitz(Hanover 1859 - Zürich 1919)

Robust stability analysis for polynomials

Analyzing stability robustness of polynomials is

a little bit more interesting..

Here too computational complexity depends on

the uncertainty model

In increasing order of complexity, we will

distinguish between

• single parameter uncertainty q ∈ [qmin, qmax]• interval uncertainty qi ∈ [qimin, qimax]• polytopic uncertainty λ1q1 + · · ·+ λNqN• multilinear uncertainty q0 + q1 · q2 · q3

LMIs will not show up very soon..

..just basic linear algebra

Single parameter uncertaintyand 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 deg p1(s) < deg p0(s)

The largest stability interval

q ∈]qmin, qmax[

such that p(s, q) is robustly stable is given by

qmax = 1/λ+max(−H−10 H1)

qmin = 1/λ−min(−H

−10 H1)

where λ+max is the max positive real eigenvalueλ−min is the min negative real eigenvalueHi is the Hurwitz matrix of pi(s)

Higher powers of a single parameter

Now consider the continuous-time polynomial

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

with p0(s) stable and deg p0(s) > deg pi(s)

Using the zeros (roots of determinant) of the polynomialHurwitz matrix

H(p) = H(p0) + qH(p1) + q2H(p2) + · · ·+ qmH(pm)

we can show that

qmin = 1/λ−min(M)

qmax = 1/λ+max(M)

where

M =

0 0 0... . . . ... ...0 I 00 0 I

−H−10 Hm · · · −H−10 H2 −H

−10 H1

is a block companion matrix

MIMO systems

Uncertain multivariable systems are modeled

by uncertain polynomial matrices

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

where p0(s) = detP0(s) is a stable polynomial

We can apply the scalar procedure to the

determinant polynomial

detP (s, q) = p0(s)+qp1(s)+q2p2(s)+· · ·+qrpr(s)

Example

MIMO design on the plant with left MFD

A−1(s, q)B(s, q) =

[s2 q

q2 + 1 s

]−1 [s + 1 0

q 1

]

=

[s2 + s− q2 −q

qs2 − (q2 + 1)s− (q2 + 1) s2

]s3−q2−q

with uncertain parameter q ∈ [0,1]

MIMO systems: example

Using some design method, we obtain a

controller with right MFD

Y (s)X−1(s) =[

94− 51s −18 + 17s−55 100

] [55 + s −17−1 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 detD0(s)

Applying the eigenvalue criterion on detD(s, q)

yields the stability interval

q ∈]− 0.93, 1.17[ ⊃ [0, 1]

so the closed-loop system is robustly stable

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 practice several parameters 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

Interval uncertainty: independent structure anduncertain parameter vector q belongs to a givenbox, i.e. qi ∈ [q−i , q

+i ]

ExampleUncertain 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

Kharitonov’s polynomials

Associated with the interval polynomial

p(s, q) =n∑

i=0

[q−i , q+i ]s

i

are four Kharitonov’s polynomials

p−−(s) = q−0 + q−1 s + q

+2 s

2 + q+3 s3 + q−4 s

4 + q−5 s5 + · · ·

p−+(s) = q−0 + q+1 s + q

+2 s

2 + q−3 s3 + q−4 s

4 + q+5 s5 + · · ·

p+−(s) = q+0 + q−1 s + q

−2 s

2 + q+3 s3 + q+4 s

4 + q−5 s5 + · · ·

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

−2 s

2 + q−3 s3 + q+4 s

4 + q+5 s5 + · · ·

where we assume q−n > 0 and q+n > 0

ExampleInterval polynomial

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

Kharitonov’s polynomials

p−−(s) = 1 + 3s + 6s2 + 8s3

p−+(s) = 1 + 4s + 6s2 + 7s3

p+−(s) = 2 + 3s + 5s2 + 8s3

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

Kharitonov’s theorem

In 1978 the Russian researcher Vladiḿır 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 of four polynomials, which can be done

using the classical Hurwitz criterion

Peter and Paul fortress in St Petersburg

Affine uncertainty

Sadly, Kharitonov’s 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) ora 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

Polytopes of polynomials

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

Examplep(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 + s2p(s, q3) = 8 + 3s + s2 p(s, q4) = 6 + 9s + s2

Any polynomial in the family can be written as

p(s, q) =4∑

i=1

λip(s, qi),

4∑i=1

λi = 1, λi ≥ 0

Interval polynomials

Interval polynomials are a special case ofpolytopic polynomials

p(s, q) =n∑

i=0

[q−i , q+i ]s

i

with at most 2n+1 generating vertices

p(s, qk) =n∑

i=0

qki si, qki =

q−ior

q+i

1 ≤ k ≤ 2n+1

ExampleThe interval polynomial

p(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 + s4

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

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

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

The edge theorem

Let p(s, q), q ∈ Q be a polynomial withinvariant degree over polytopic set Q

Polynomial p(s, q) is robustly stableover the whole uncertainty polytope Qiff p(s, q) is stable along the edges of Q

In other words, it is enough to check robuststability of the single parameter polynomial

λp(s, qi1) + (1− λ)p(s, qi2), λ ∈ [0, 1]

for each pair of vertices qi1 and qi2 of Q

This can be done with the eigenvalue criterion

Interval feedback systemExampleWe consider the interval control system

Kn(s,q)d(s,q)

with n(s, q) = [6, 8]s2 + [9.5, 10.5], d(s, q) =s(s2 + [14, 18]) and characteristic polynomial

K[9.5, 10.5] + [14, 18]s + K[6, 8]s2 + s3

For K = 1 we draw the 12 edges of its root set

−6 −5 −4 −3 −2 −1 0−3

−2

−1

0

1

2

3

Real

Imag

The closed-loop system is robustly stable

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 uncertainparameter qi is linear when other parameters

qj, i 6= j are fixed• polynomic uncertainty, when coefficients aremultivariable polynomials in parameters qi

We can define the following hierarchy on the

uncertainty structures

interval ⊂ affine ⊂ multilinear ⊂ polynomic

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 + s2

has polynomic (here quadratic) uncertaintystructure


(5q1 − q2 + 5) + (4q1 − 6q1q23 + q3)s + s2

has polynomic uncertainty structure

Multilinear uncertainty

We will focus on multilinear uncertainty because it arisesin a wide variety of system models such as:• multiloop systems

G G G

H

H

1

1

2

2

3

u y+ + +−−

−

Closed-loop transfer functiony

u=

G1G2G3

1 + G1G2H1 + G2G3H2 + G1G2G3• state-space models with rank-one uncertainty

ẋ = A(q)x, A(q) =n∑

i=1

qiAi, rank Ai = 1

and characteristic polynomial

p(s, q) = det(sI −A(q))• polynomial MFDs with MIMO interval uncertainty

G(s) = A−1(s, q)B(s, q), C(s) = Y (s)X−1(s)

and closed-loop characteristic polynomial

p(s, q) = det(A(s, q)X(s) + B(s, q)Y (s))

Robust stability analysis for multilinear and

polynomic uncertainty

Unfortunately, there is no systematic

computational tractable necessary and

sufficient robust stability condition

On the one hand, sufficient condition through

polynomial value sets, the zero exclusion

condition and the mapping theorem

On the other hand, brute-force method:

intensive parameter gridding, expensive in

general

No easy trade-off between computational

complexity and conservatism

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 polynomialsKharitonov’s 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)

Lessons from robust analysis:lack of extreme point results

Ensuring robust stability of the parametrizedpolynomial

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

amounts to ensuring robust stability of thewhole segment of polynomials

λp(s, qmin) + (1− λ)p(s, qmax)λ = qmax−qqmax−qmin

∈ [0, 1]

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

Unfortunately, the answer is no

ExampleFirst vertex: 0.57 + 6s + s2 + 10s3 stableSecond vertex: 1.57 + 8s + 2s2 + 10s3 stableBut middle of segment:1.07 + 7s + 1.50s2 + 10s3 unstable

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 thestate-space interval matrix

p(s, q) = det(sI −

[−1.5, −0.5]−12.06−0.06 0−0.25 −0.03 1 0.50.25 −4 −1.03 00 0.5 0 [−4, 1]

)The four edges of the uncertainty bounding

set are stable, however for q1 = 0.5 and q2 = 1

polynomial p(s, q) is unstable..

Non-convexity of stability domain

Main problem: the stability domain in the spaceof polynomial coefficients pi is non-convex ingeneral

0 1 2 3 4 5 6−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

UNSTABLE

STABLE

q1

q 2

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 + (−2.475 + 0.675q1 + 0.3q2)z2 + z3

How can we overcome the non-convexity of thestability conditions in the coefficient space ?

Handling non-convexity

Basically, we can pursue two approaches:

• we can approximate the 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)

Stability polytopes

Largest hyper-rectangle around a nominallystable polynomial

p(s) + rn∑

i=0

[−εi, εi]si

obtained with the eigenvalue criterion appliedon the 4 Kharitonov polynomials

In general, there is no systematic way toobtain more general stability polytopes, namelybecause 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|+ · · ·+ |pn−1| < 1

Stability region (second degree)

Necessary stab cond in dt: convex hull ofstability domain is a polytope whose n + 1vertices are polynomials with roots +1 or -1

ExampleWhen 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

Stability region (third degree)

ExampleThird degree dt polynomial: two hyperplanes and anon-convex hyperbolic paraboloid with a saddle pointat p(z) = p0 + p1z + p2z2 + z3 = z(1 + z2)

(z + 1)(z + 1)(z + 1) = 1 + 3z + 3z2 + z3

(z + 1)(z + 1)(z − 1) = −1− z + z2 + z3(z + 1)(z − 1)(z − 1) = 1− z − z2 + z3(z − 1)(z − 1)(z − 1) = −1 + 3z − 3z2 + z3

Stability hyper-spheres

Largest hyper-sphere around a nominallystable polynomial

p(s) +n∑

i=0

qisi, ‖q‖ ≤ r

has radius

rmax = min{|p0|, |pn|, infω>0

√(Re p(jω))2

1 + w4 + w8..+

(Im p(jω))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

ω

f(ω

)

rmax = min{2, 1, infω>0 f(w)} = 1.08 · 10−3

Stability ellipsoids

A weighted and rotated hyper-sphere is

an ellipsoid

We are interested in inner ellipsoidal

approximations of stability domains

E = {p : (p− p̄)?P (p− p̄) ≤ 1}

where

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

Hermite stability criterion

Charles Hermite (1822 Dieuze - 1901 Paris)

The polynomial p(s) = p0 + p1s + · · ·+ pnsn isstable if and only if

H(x) =∑

i∑

j pipjHij � 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 − p20 p2p3 − p0p1 p1p3 − p0p2

p2p3 − p0p1 p22 + p23 − p20 − p21 p2p3 − p0p1p1p3 − p0p2 p2p3 − p0p1 p23 − p20

]

Inner ellipsoidal appromixation

Our objective is then to find p̄ and P such that

the ellipsoid

E = {p : (p− p̄)?P (p− p̄) ≤ 1}

is a convex inner approximation of the actual

non-convex stability region

S = {p : H(p) � 0}

that is to say

E ⊂ S

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

LMI formulation (not given here) with decision

variables p̄ and P


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

]p̄ =

[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 1.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

p0

p 1


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 00.5397 0 1.7552

x̄ = 00.1235

0

which describes a convex ellipse E inscribed in the exactstability domain S delimited by the non-convexhyperbolic paraboloid

−1

0

1−1

01

23

−3

−2

−1

0

1

2

3

A

D

p1

Q

C

B

p0

p 2

Very simple scalar convex sufficient stability condition

2.4166p20 + 2.2088p21 + 1.8143p

22 − 0.5458p1 + 1.1158p0p2 ≤ 1

Volume of stability ellipsoid

In the discrete-time case, the well-known sufficientstability condition defines a diamondD = {p : |p0|+ |p1|+ · · ·+ |pn−1| < 1}

For different values of degree n, we compared volumesof exact stability domain S, ellipsoid E and diamond D

n = 2 n = 3 n = 4 n = 5Stability domain S 4.0000 5.3333 7.1111 7.5852

Ellipsoid E 2.2479 1.4677 0.7770 0.3171Diamond D 2.0000 1.3333 0.6667 0.2667

E is “larger” than D, yet very small wrt S

In the last part of this course, we will propose betterLMI inner approximations of the stability domain

−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

p 1



ROBUST CONTROL DESIGN

POLYNOMIAL METHODS

Didier HENRION


October 2006


Fixed-order robust design:a difficult problem

In this last part of course, we study robuststabilization with a fixed-order controlleraffected 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

Once more: uncertainty models

We can consider several uncertainty models, indecreasing order of complexity:

• Polytopic uncertainty, where plant (numer-ator and/or denominator) polynomial p(s) be-longs to a polytope with given verticesEx. p(s) = λ1(s+ 1)3 + λ2(s+ 1)(s+ 2)2 with λ1 + λ2 = 1

Very general, but often leads to intractable

analysis/design problems

• Interval uncertainty, where each of the plantparameters are assumed to vary independentlyin 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-boundedEx. p(s) = p0 + p1s+ p2s2 + s3 with 3p20 + p0p1 + 2p

21 + p

22 ≤ 1

Less structured, but more realistic, naturally arises in the context

of parameter estimation for process identification,

ellipsoid = covariance matrix

Existing results

• H∞ design methods, state-space techniques

• Critical direction (Nyquist), convexoptimization with cutting-plane algorithms

• Infinite-dimensional Youla-Kučera 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 LMI optimization

• Controllers of fixed (hence low) order

Nominal Pole placement

We consider the SISO feedback system

ba

x

y

+

−

Closed-loop transfer function

bx

ax+ by

In the absence of hidden modes (a and b coprimepolynomials), pole placement amounts to findingpolynomials x and y solving the Diophantineequation (from Diophantus of Alexandria 200-284)

ax+ by = c

where c is a given closed-loop characteristicpolynomial capturing the desired system poles

Pole placement: numerical aspectsThe polynomial Diophantine equation

ax+ by = c

is linear in unknowns x and y, and denotinga(s) = a0 + a1s+ · · ·+ adasdax(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)

Pole placement for MIMO systems

Pole placement can be performed similarly for

a plant left MFD

A−1(s)B(s)

with a controller right MFD

Y (s)X−1(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 H∞optimal dynamics (obtained with spectral

factorization)

Robust pole placement

Now assume that the plant transfer function

b(q)

a(q)

contains some uncertain parameter q

The problem of robust pole placement will 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 stabilityof c(q) for all admissible uncertainty q ?

Coefficients of c are linear in x and y, but wesaw that stability conditions are non-linear andhighly non-convex in c..

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

Robust design via polytopic approximationMIMO plant with right MFD

B(s)A−1(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

X−1(s)Y (s) =[s+ x1 x2

0 1

]−1=[y1s+ y2 y3s+ y4

0 y5

]assigning robustly the closed-loop polynomial matrix

C(s) =[s2 + αs+ β δ(s)

0 s+ γ

]whose coefficients live in the polytope −14 1 016 −2 0−2 1 0

0 0 −10 0 1

[ αβγ

]>

−19656−42−14

These specifications amounts to assigning the poles within the disk

|s+ 8| < 6

−15 −10 −5 0

−6

−4

−2

0

2

4

6

Real

Imag

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 −1−1 −1 0.5−13 −21 1.514 24 −3−1 −3 1.5

[ x1y1y2]>

−182

40−2−182

40−2

characterizing all parameters x1, y1 and y2 of admissiblerobust controllers

−5

0

5

10

15

20

−10

0

10

20

300

50

100

150

200

250

300

350

400

y1

x1

y2

Corresponding polytope with 9 vertices

Robust design via ellipsoidal approximation

Closed-loop characteristic polynomial

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

= c0 + c1s+ · · ·+ cd−1sd−1 + sd

whose coefficients are given by the LSE

c0c1...cd−1

=

y0 x0

y1. . . x1

. . .... y0

... x0ym−1 y1 xm−1 x1

ym... 1

.... . .ym−1

. . .xm−1ym 1

b0b1...

bn−1a0a1...

an−1

+

00...0x0x1...

xm−1

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

It is assumed that uncertain plant parameters belong tothe ellipsoid

Ep = {p : (p− p̄)?P (p− p̄) ≤ 1}where p̄ is a given nominal plant 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

Robust design via ellipsoidal approximation

Recall that by solving an LMI we could approxi-

mate from the interior the non-convex stability

region with an ellipsoid

Eq = {q : (q − q̄)?Q(q − q̄) ≤ 1}

In other words, q ∈ Eq implies q(s) stable

Using conditions for inclusion of an ellipsoid

into another we can show that finding x and

y such that q ∈ Eq for all p ∈ Ep amounts tosolving another LMI problem (not given here)

Coefficients x, y are such that the controller

y(s)/x(s) robustly stabilizes plant b(s)/a(s)

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

Ellipsoidal robust design: example (2)

The identification of the nominal plant modelis carried out using a standard least-squaresmethod

A second-order discrete-time model

p(z) =b0 + b1z

a0 + a1z + z2

is obtained with nominal plant vector

p̄ =[

0.0038 0.0028 0.2087 −1.1871]?

The positive definite matrix P characterizingthe uncertainty ellipsoid

Ep = {p : (p− p̄)?P (p− p̄) ≤ 1}is readily available as a by-product of theidentification 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

Ellipsoidal robust design: example (3)

Solving the LMI analysis problem we obtain first aninner ellipsoidal approximation

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

Q =

2.3378 0 0.53970 2.1368 00.5397 0 1.7552

q̄ = 00.1235

0

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

Strict positive realness

Let

D = {s :[

1s

]? [a bb? c

]︸︷︷︸

H

[1s

]< 0}

be a stability region in the complex plane where

Hermitian matrix H has inertia (1,0,1)

Let ∂D denote the 1-D boundary of D

Standard choices are

H =

[0 11 0

]H =

[−1 00 1

]for the left half-plane and the unit disk resp.

We say that a rational matrix G(s) is strictly

positive real (SPR for short) when

ReG(s) � 0 for all s ∈ ∂D

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 a

stable polynomial D(s) such that rational

matrix N(s)D−1(s) is strictly positive real

Proof

From the definition of SPRness, N(s)D−1(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)D−1(s) = I obviously SPR

It turns out that this condition can be

characterized by an LMI..

SPRness as an LMI

Let N = [N0 N1 · · · Nd], D = [D0 D1 · · · Dd]and

Π =

I 0.. . ...

I 00 I... . . .0 I

Given a stable D(s), N(s) ensures SPRness ofN(s)D−1(s) iff there exists a matrix P = P ? ofsize dn such that

D?N +N?D −H(P ) � 0

where

H(P ) = Π?(S ⊗ P )Π = Π?[aP bPb?P cP

]Π

Proof

Similar to the proof on positivity of a polynomial, basedon the decomposition as a sum-of-squares with liftingmatrix P

LMI condition for design

Given a stable polynomial matrix D(s), poly-nomial matrix N(s) is stable if there is a matrixP satisfying the LMI

D?N +N?D −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

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

D?N +N?D −H(P ) � 0P � 0

then polynomial matrix N(s) is stable

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

Useful for (robust) analysis because linear in D

Note that, in contrast with the design LMI, we

have here to enforce P � 0

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 P

satisfying

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

p20 + p02 = n0P � 0

which is an LMI in the primal SDP form

Second-degree discrete-time polynomial (2)

Infeasibility of primal LMI is equivalent to the

existence of a vector satisfying the dual LMI

y0 + n1y1 + n0y2 < 0

Y =

y0 y1 y2y1 y0 y1y2 y1 y0

� 0The eigenvalues of Toeplitz matrix Y are

y0 − y2 and (2y0 + y2 ±√y22 + 8y

21)/2

so it is positive definite iff y1 and y2 belong to

the interior of a bounded parabola scaled by y0

The corresponding values of n0 and n1 belong

to the interior of the envelope generated by

the curve

(2λ2−1)n0+(2λ1−1)√λ2n1+1 > 0 0 ≤ λi ≤ 1

Second-degree discrete-time polynomial (3)

The implicit equation of the envelope is

(2n0 − 1)2 + (√

2

2n1)

2 = 1

a scaled circle

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

p 1

Application to robust stability analysis

Assume that N(s, λ) is a polynomial matrix

with multi-linear dependence in a parameter

vector λ belonging to a polytope Λ

Denote by Ni(s) the vertices obtained by

enumerating each vertex in Λ

Polytopic polynomial matrix N(s, λ) is robustly

stable if there exists a matrix D and matrices

Pi satisfying the LMI

D?Ni +N?i D −H(Pi) � 0

Pi � 0 ∀i

Proof

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

Robust stability of polynomial matrices

ExampleConsider the following mechanical system

xx

u

m

m

d

d

2

1 2

1

2

1c1

c12

c2

It is described by the polynomial MFD[m1s2 + d1s+ c1 + c12 −c12

−c12 m2s2 + d2s+ c2 + c12

] [x1(s)x2(s)

]=[

0u(s)

]System parameters λ = [m1 d1 c1 m2 d2 c2 ]belong to the uncertainty hyper-rectangleΛ = [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

Robust stability of polynomial matrices

However, it is a non-trivial task to know whether theopen-loop system is robustly D-stable in some stabilityregion D ensuring a certain damping. Here we choosethe disk of radius 12 centered at -12

D = {s : (s+ 12)2 < 122}The robust stability analysis problem amounts to assess-ing whether the second degree polynomial matrix in theMFD has its zeros in D for all admissible uncertainty ina polytope with m = 26 = 64 vertices

LMI problem is feasible – vertex zeros shown below

Polytope of polynomials

We can also check robust stability of polytopes

of polynomials without using the edge theorem

or the graphical value set

ExampleContinuous-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 0−4

−3

−2

−1

0

1

2

3

4

Real

Interval polynomial matrices

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

ExampleContinuous-time interval polynomial matrix of degree 2with 23 = 8 vertices [7.7− 2.3s+ 4.3s2,3.7 + 2.7s+ 4.3s2] [−3.1− 6s− 2.2s2,−4.1− 7s− 2.2s2]

3.6 + 6.4s+ 4.3s2[3.2 + 11s+ 8.2s2,

16 + 12s+ 8.2s2]

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

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

[F ?A+A?F − aP −A? − F ? − bP−A− F − b?P 2I − cP

]� 0

P � 0

Proof

Just take D(s) = sI − F and notice that theLMI can be also written more explicitly as[−F ?I

] [−A I

]+

[−A?I

] [−F I

]−[aP bPb?P cP

]> 0

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(λ) has multi-linear dependence on polytopic uncertain pa-rameter λ and denote by Ai the correspondingvertices

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

F ?Ai +A?iF − aPi −A

?i − F

? − bPi−Ai − F − b?Pi 2I − cPi

]� 0

Pi � 0 ∀i

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

Robust design

Assume now that system matrix C(s, λ) comes

from a polynomial Diophantine equation

C(s, λ) = A(s, λ)X(s) +B(s, λ)Y (s)

where system matrices A and B are subject to

multi-linear polytopic uncertainty λ

In order to ensure robust SPRness of the ra-

tional matrix D−1(s)C(s, λ) central polynomialD(s) must be close to the nominal closed-loop

matrix, such as

D(s) = C(s, λ0)

where λ0 is the nominal parameter vector

A sensible simple choice of D(s) is therefore

the nominal closed-loop denominator polyno-

mial matrix, obtained by any standard design

method (pole assignment, LQ, H∞)

Reactor

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

ξ1 = (ξ2 + 0.5)exp(Eξ1/(ξ1 + 2))− (2 + u)(ξ1 + 0.25)ξ2 = 0.5− ξ2 − (ξ2 + 0.5)exp(Eξ1/(ξ1 + 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 ξ1 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 a single static output feedback controller y(s)/x(s)

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

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

90 + 54s

−0.1 + 30s+ 50s2 + 2.8s3 + s4

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)=

90 + 54s

−0.1 + 30s+ 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

Satellite

We consider a satellite control 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 ∈ [0.04√q110, 0.2

√q210]

With the choice

d(s) = (s+ 1)6

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

Satellite (2)

However we can stabilize each vertex ai(s), bi(s)individually with 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 −0.4520± j1.8129, −0.0365± j0.8954, −0.0019± j0.25332 −0.6161± j1.3829, −0.2190± j0.6745, −0.0237± j0.21363 −1.0484, −0.0917± j2.1598, −0.0425± j0.5019, −0.00044 −0.2220± j2.0695, −0.1695± j0.5354, −0.0709± j0.0045

The third vertex has poles nearest 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− 1.4048s2

3.6082 + 1.0237s+ s2

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

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 thenonlinearity 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(z−1, q)

a(z−1, q)=

((0.0257 + q1) + (−0.0764 + q2)z−1

+(−0.1619 + q3)z−2 + (−0.1688 + q4)z−3)

1− 1.914z−1 + 1.779z−2 − 1.0265z−3 + 0.2508z−4

where

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

Robot

Closed-loop polynomial

d(z, q) = z12[(1−z−1)a(z−1, q)x(z−1)+z−5b(z−1, q)y(z−1)]where the term 1 − z−1 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 LMIreturns the seventh-order robust controller

y(z−1)

x(z−1)=

(−0.2863 + 0.2928z−1 + 0.0221z−2−0.1558z−3 + 0.0809z−4 + 0.1420z−5

−0.1254z−6 + 0.0281z−7

)(

1 + 1.1590z−1 + 0.9428z−2

+0.4996z−3 + 0.3044z−4 + 0.4881z−5

+0.4003z−6 + 0.3660z−7

)

Second-order systems

Second-order linear system

(A0 +A1s+A2s2)x = Buy = Cx

to be controlled by PD output-feedback

controller

u = −(F0 + F1s)y

Applications: 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

PD controller

Closed-loop system behavior captured by zeros

of quadratic polynomial matrix

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

Zeros of N(s) must be located in some stability

region D characterized as before by matrix H

Uncertainty can affect A0 (stiffness)

A1 (damping) and A2 (mass)

Given A0, A1, A2, B, Cfind F0, F1

ensuring robust pole placement

Robust LMI stability condition

• Norm-bounded (unstructured) uncertainty

N(s) + ∆M(s) σmax(∆) ≤ δ

LMI robust stability condition on N(s)

[D?N +N?D −H(P )− γD?D δM?

δM γI

]� 0

• Polytopic (structured) uncertainty

N(s) =∑i

λiN i(s)∑i

λi = 1 λi ≥ 0

Vertex LMI robust stability conditions:

DN i + (N i)?D −H(P i) � 0, i = 1,2, . . .

Parameter-dependent Lyapunov matrix

P (λ) =∑i λiP i

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 design

method (pole placement, LQ, H2 or H∞), thentry to optimize around D(s)

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 00 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, ±i1.145,±i0.5675 and ±i0.3507

Nominal PD controller F 00 , F01 obtained with LQ design

Resulting central polynomial matrix

D(s) = (A0 +BF 00C) + (A1 +BF01C)s+A2s

2

Stability region D = {s : Re s < −0.1}

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

Closed-loop pseudospectrum of the five masses example

lmis in systems control state-space methods …of the number of system parameters remedy: the...

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