AN LMI OPTIMIZATION APPROACH FOR GUARANTEED COST

CONTROL OF SYSTEMS WITH STATE AND INPUT DELAYS

O.I. Kosmidou, Y.S. Boutalis and Ch. Hatzis

Department of Electrical and Computer Engineering Democritus University of Thrace

67100 Xanthi, Greece

Keywords:

Uncertain systems, time-delays, optimal control, guaranteed cost control, LMI.

Abstract:

The robust control problem for linear systems with parameter uncertainties and time-varying delays is exam-

ined. By using an appropriate uncertainty description, a linear state feedback control law is found ensuring

the closed-loop system’s stability and a performance measure, in terms of the guaranteed cost. An LMI ob-

jective minimization approach allows to determine the ”optimal” choice of free parameters in the uncertainty

description, leading to the minimal guaranteed cost.

1 INTRODUCTION

Model uncertainties and time-delays are frequently

encountered in physical processes and may cause per-

formance degradation and even instability of control

systems (Malek-Zavarei and Jamshidi, 1987), (Mah-

moud, 2000), (Niculescu, 2001). Hence, stability

analysis and robust control problems of uncertain dy-

namic systems with delayed states have been studied

in recent control systems literature; for details and

references see e.g. (Niculescu et al., 1998), (Kol-

manovskii et al., 1999), (Richard, 2003). Since con-

trol input delays are often imposed by process design

demands as in the case of transmission lines in hy-

draulic or electric networks, it is necessary to consider

uncertain systems with both, state and input time-

delays. Moreover, when the delays are imperfectly

known, one has to consider uncertainty on the delay

terms, as well. In recent years, LMIs are used to solve

complex analysis and control problems for uncertain

delay systems (e.g. (Li and de Souza, 1997), (Li et al.,

1998), (Tarbouriech and da Silva Jr., 2000), (Bliman,

2001), (Kim, 2001), and related references).

The purpose of the present paper is to design con-

trol laws for systems with uncertain parameters and

uncertain time-delays affecting the state and the con-

trol input. Although the uncertainties are assumed

to enter linearly in the system description, it is well

known that they may be time-varying and nonlinear

in nature, in most physical systems. Consequently,

the closed-loop system’s stability has to be studied

in the Lyapunov-Krasovskii framework; the notion

of quadratic stability is then extended to the class of

time-delay systems (Barmish, 1985),(Malek-Zavarei

and Jamshidi, 1987). On the other hand, it is desir-

able to ensure some performance measure despite un-

certainty and time-delay, in terms of guaranteed up-

per bounds of the performance index associated with

the dynamic system. The latter speciﬁcation leads to

the guaranteed cost control (Chang and Peng, 1972),

(Kosmidou and Bertrand, 1987).

In the proposed approach the uncertain parameters af-

fecting the state, input, and delay matrices are allowed

to vary into a pre-speciﬁed range. They enter into

the system description in terms of the so-called uncer-

tainty matrices which have a given structure. Differ-

ent unity rank decompositions of the uncertainty ma-

trices are possible, by means of appropriate scaling.

This description is convenient for many physical sys-

tem representations (Barmish, 1994). An LMI opti-

mization solution (Boyd et al., 1994)is then sought in

order to determine the appropriate uncertainty decom-

position; the resulting guaranteed cost control law

ensures the minimal upper bound. The closed-loop

system’s quadratic stability follows as a direct conse-

quence.

The paper is organized as follows: The problem for-

mulation and basic notions are given in Section 2.

Computation of the solution in the LMI framework is

presented in Section 3. Section 4 presents a numerical

example. Finally, conclusions are given in Section 5.

230

Kosmidou O., Boutalis Y. and Hatzis C. (2004).

AN LMI OPTIMIZATION APPROACH FOR GUARANTEED COST CONTROL OF SYSTEMS WITH STATE AND INPUT DELAYS.

In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 230-236

DOI: 10.5220/0001145202300236

 SciTePress

2 PROBLEM STATEMENT AND

DEFINITIONS

Consider the uncertain time-delay system described

in state-space form,

˙x(t) = [A

+ ∆A

(t)]x(t) +

+ ∆A

(t)]x(t − d

(t)) + [B

+ ∆B

(t)]u(t)

+[B

+ ∆B

(t)]u(t − d

(t)) (1)

for t ∈ [0, ∞) and with x(t) = φ(t) for t < 0.

In the above description, x(t) ∈ R

is the state

vector, u(t) ∈ R

is the control vector and A

, A

∈ R

n×n

, B

∈ R

n×m

are constant matrices.

The model uncertainty is introduced in terms

∆A

(t) =

i=1

(t), | r

(t) |≤ ¯r

∆A

(t) =

i=1

(t), | r

(t) |≤ ¯r

∆B

(t) =

i=1

(t), | p

(t) |≤ ¯p

∆B

(t) =

i=1

(t), | p

(t) |≤ ¯p

(2)

where A

, A

, B

are given matrices with con-

stant elements determining the uncertainty structure

in the state, input, and delay terms.

The uncertain parameters r

, r

, p

are

Lebesgue measurable functions, possibly time-

varying, that belong into pre-speciﬁed bounded

ranges (2), where ¯r

, ¯r

, ¯p

are positive scalars;

since their values can be taken into account by the

respective uncertainty matrices, it is assumed that

¯r

= ¯r

= ¯p

= 1, without loss of generality.

Moreover, the advantage of the afﬁne type uncertainty

description (2) is that it allows the uncertainty matri-

ces to have unity rank and thus to be written in form

of vector products of appropriate dimensions,

= d

, i = 1, ..., k

= d

, i = 1, ..., k

= f

, i = 1, ..., l

= f

, i = 1, ..., l

(3)

Obviously, the above decomposition is not unique;

hence, the designer has several degrees of freedom in

choosing the vector products, in order to achieve the

design objectives. By using the vectors in (3), deﬁne

the matrices,

:= [d

...d

], E

:= [e

...e

]

:= [d

...d

], E

:= [e

...e

]

:= [f

...f

], G

:= [g

...g

]

:= [f

...f

], G

:= [g

...g

] (4)

which will be useful in the proposed guaranteed cost

approach.

The time delays in (1) are such that,

0 ≤ d

(t) ≤

< ∞,

(t) ≤ β

< 1

0 ≤ d

(t) ≤

< ∞,

(t) ≤ β

< 1 (5)

∀t ≥ 0.

Associated with system (1) is the quadratic cost

function

J(x(t), t) =

∞

(t)Qx(t)+u

(t)Ru(t)]dt (6)

with Q > 0, R > 0, which is to be minimized for a

linear constant gain feedback control law of the form

u(t) = Kx(t) (7)

by assuming (A

, B

) stabilizable, (Q

1/2

, A

)

detectable and the full state vector x(t) available for

feedback.

In the absence of uncertainty and time-delay, the

above formulation reduces to the optimal quadratic

regulator problem (Anderson and Moore, 1990).

Since uncertainties and time-delays are to be taken

into account, the notions of quadratic stability and

guaranteed cost control have to be considered. The

following deﬁnitions are given.

Deﬁnition 2.1

The uncertain time-delay system (1)-(5) is quadrat-

ically stabilizable independent of delay, if there ex-

ists a static linear feedback control of the form of

(7), a constant θ > 0 and positive deﬁnite matrices

P ∈ R

n×n

, R

∈ R

n×n

and R

∈ R

m×m

, such

that the time derivative of the Lyapunov-Krasovskii

functional

L(x(t), t) = x

(t)P x(t) +

t−d

(t)

(τ)R

x(τ)dτ +

t−d

(t)

(τ)R

u(τ)dτ (8)

satisﬁes the condition

L(x(t), t) =

dL(x(t), t)

≤ −θ k x(t) k

(9)

AN LMI OPTIMIZATION APPROACH FOR GUARANTEED COST CONTROL OF SYSTEMS WITH STATE AND

INPUT DELAYS

231

along solutions x(t) of (1) with u(t) = Kx(t), for

all x(t) and for all admissible uncertainties and time-

delays, i.e. consistent with (2), (3) and (5), respec-

tively.

The resulting closed-loop system is called quadrati-

cally stable and u(t) = Kx(t) is a quadratically sta-

bilizing control law.

Deﬁnition 2.2

Given the uncertain time-delay system (1)-(5) with

quadratic cost (6), a control law of the form of (7)

is called a guaranteed cost control, if there exists a

positive number V(x(0), φ(−

), φ(−

)), such that

J(x(t), t) ≤ V(x(0), φ(−

), φ(−

)) (10)

for all x(t) and for all admissible uncertainties and

time-delays. The upper bound V(.) is then called a

guaranteed cost.

The following Proposition provides a sufﬁcient

condition for quadratic stability and guaranteed cost

control.

Proposition 2.3

Consider the uncertain time-delay system (1)-(5) with

quadratic cost (6). Let a control law of the form

of (7) be such that the derivative of the Lyapunov-

Krasovskii functional (8) satisﬁes the condition

L(x(t), t) ≤ −x

(t)[Q + K

RK]x(t) (11)

for all x(t) and for all admissible uncertainties and

time-delays. Then, (7) is a guaranteed cost control

law and

V(x(0), φ(−

), φ(−

)) = x

(0)P x(0) +

−

(τ)R

φ(τ)dτ +

−

(τ)K

Kφ(τ )dτ (12)

is a guaranteed cost for (6). Moreover, the closed-loop

system is quadratically stable.

Proof

By integrating both sides of (11) one obtains

L(x(t), t)dt ≤ −

(t)Qx(t)+u

(t)Ru(t)]dt

(13)

and thus

L(x(T ), T ) − L(x(0), 0) = x

(T )P x(T ) +

T −d

(t)

(τ)R

x(τ)dτ +

T −d

(t)

(τ)K

Kx(τ )dτ −

(0)P x(0) −

−d

(t)

(τ)R

x(τ)dτ −

−d

(t)

(τ)K

Kx(τ )dτ ≤

−

(t)Qx(t) + u

(t)Ru(t)]dt (14)

Since (11) is satisﬁed for L > 0, it follows from Def-

inition 2.1 that (9) is also satisﬁed for θ = λ

min

(Q +

RK). Consequently u(t) = Kx(t) is a quadrati-

cally stabilizing control law and thus L(x(t), t) → 0

as T → ∞ along solutions x(.) of system (1). Conse-

quently, the above inequality yields

∞

(t)Qx(t) + u

(t)Ru(t)]dt ≤

(0)P x(0) +

−

(τ)R

φ(τ)dτ +

−

(τ)K

Kφ(τ )dτ (15)

and thus

J(x(t), t) ≤ V(x(0), φ(−

), φ(−

)) (16)

for V(x(0), φ(−

), φ(−

)) given by (12).

Remark 2.4

The condition resulting from Proposition 2.3 is a

delay-independent condition. It is well known (Li and

de Souza, 1997), (Kolmanovskii et al., 1999)that con-

trol laws arising from delay-independent conditions

are likely to be conservative. Besides, quadratic sta-

bility and guaranteed cost approaches provide conser-

vative solutions, as well. A means to reduce conser-

vatism consists in minimizing the upper bound of the

quadratic performance index by ﬁnding the appropri-

ate guaranteed cost control law. Such a solution will

be sought in the next Section.

3 SOLUTION IN THE LMI

FRAMEWORK

In order to determine the unity rank uncertainty de-

composition in an ”optimal” way such that the guar-

anteed cost control law minimizes the corresponding

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

232

guaranteed cost bound, an LMI objective minimiza-

tion problem will be solved. The uncertainty decom-

position (3) can be written as (Fishman et al., 1996),

= (s

1/2

)(s

−1/2

)

i = 1, ..., k

= (s

1/2

)(s

−1/2

)

i = 1, ..., k

= (t

1/2

)(t

−1/2

)

i = 1, ..., l

= (t

1/2

)(t

−1/2

)

i = 1, ..., l

(17)

where s

, t

are positive scalars to be determined

during the design procedure. For this purpose, diago-

nal positive deﬁnite matrices are deﬁned,

:= diag(s

, ..., s

)

:= diag(s

, ..., s

)

:= diag(t

, ..., t

)

:= diag(t

, ..., t

) (18)

The existence of a solution to the guaranteed cost con-

trol problem is obtained by solving an LMI feasibility

problem. The following Theorem is presented:

Theorem 3.1

Consider the uncertain time-delay system (1)-(5)

with quadratic cost (6). Suppose there exist positive

deﬁnite matrices S

, S

, T

, W ,

, such

that the LMI

Λ(.) W E

W E

−1

W −S

0 0

W 0 −S

−1

0 0 −T

−1

0 0 0

−1

0 0 0

W 0 0 0

−1

W W

0 0 0 0

−T

0 0 0

0 −

0 0

0 0 −

0 0 0 −

< 0 (19)

where

Λ(.) = A

W + W A

− B

−1

+ A

+ D

+ F

+ B

−1

(20)

and

Q = (I

+ Q)

−1

(21)

= R

−1

(22)

= R

−1

(23)

admits a feasible solution (S

, S

, T

, W ,

). Then, the control law

u(t) = −R

−1

P x(t) (24)

with

P := W

−1

(25)

is a guaranteed cost control law. The corresponding

guaranteed cost is

V(x(0), φ(−

), φ(−

)) = x

(0)P x(0) +

−

(τ)R

φ(τ)dτ +

−

(τ)P B

−1

P φ(τ)dτ (26)

Proof

The time derivative of the Lyapunov-Krasovskii func-

tional is

L(x(t), t) = 2x

(t)P ˙x(t) + x

(t)R

x(t) −

(1 −

(t))x

(t − d

(t))R

x(t − d

(t)) +

(t)K

Kx(t) − (1 −

(t))

(t − d

(t))K

Kx(t − d

(t)) (27)

By using (1), (5) and (17), one obtains the inequality

L(x(t), t) ≤ 2x

(t)P {[A

+ ∆A

(t)]x(t) +

+ ∆A

(t)]x(t − d

(t)) −

+ ∆B

(t)]R

−1

P x(t) −

+ ∆B

(t)]R

−1

P x(t − d

(t))} +

(t)R

x(t) − (1 − β

(t − d

(t))R

x(t − d

(t)) +

(t)P B

−1

P x(t) − (1 − β

)

(t − d

(t))P B

−1

P x(t − d

(t)) (28)

which is to be veriﬁed for all x(.) ∈ R

. Further-

more, by using the identity 2 | ab |≤ a

+ b

, for

any a, b, ∈ R

, as well as (2)-(4), (17) and (18), the

following quadratic upper bounding functions are de-

AN LMI OPTIMIZATION APPROACH FOR GUARANTEED COST CONTROL OF SYSTEMS WITH STATE AND

INPUT DELAYS

233

rived,

(t)P ∆A

(t)x(t) ≤| 2x

(t)P ∆A

(t)x(t) |≤

i=1

| 2x

(t)P A

(t)x(t) |≤

i=1

| 2x

(t)P (d

1/2

)(e

−1/2

)

x(t) |≤

(t)P

i=1

P x(t) +

(t)

i=1

−1

x(t)

= x

(t)P D

P x(t) +

(t)E

−1

x(t) (29)

∀x(.) ∈ R

. In a similar way one obtains

(t)P A

x(t − d

(t)) ≤ x

(t)P A

P x(t)

(t − d

(t))x(t − d

(t)) (30)

(t)P ∆A

(t)x(t − d

(t)) ≤

(t)P D

P x(t) +

(t − d

(t))E

−1

x(t − d

(t)) (31)

−2x

(t)P ∆B

(t)R

−1

P x(t) ≤

(t)P F

P x(t) +

(t)P B

−1

P x(t)(32)

−2x

(t)P B

−1

P x(t − d

(t)) ≤

(t)P B

−1

P x(t) +

(t − d

(t))P B

−1

P x(t − d

(t))(33)

−2x

(t)P ∆B

(t)R

−1

P x(t − d

(t)) ≤

(t)P F

P x(t) +

(t − d

(t))P B

−1

x(t − d

(t)) (34)

The above inequalities are true ∀x(.) ∈ R

. By us-

ing these quadratic upper bounding functions, it is

straightforward to show that the guaranteed cost con-

dition (11) is satisﬁed if

+ A

P − P B

−1

P + Q +

P [D

+ F

+ D

+ A

+ B

]P +

P B

−1

+ G

−1

+ β

−1

P + I

−1

+ E

−1

+ β

< 0 (35)

By multiplying every term of the above matrix

inequality on the left and on the right by P

−1

:= W

and by applying Shur complements, the LMI form

(19) is obtained.

In the sequel, a minimum value of the guaran-

teed cost is sought, for an ”optimal” choice of the

rank-1 uncertainty decomposition. For this purpose,

the minimization of the guaranteed cost is obtained

by solving an objective minimization LMI problem.

Theorem 3.2

Consider the uncertain time-delay system (1)-(5) with

quadratic cost (6). Suppose there exist positive deﬁ-

nite matrices S

, S

, T

, W ,

, M

, such that the following LMI objective minimiza-

tion problem

min

J = min

(T r(M

) + T r(M

))

(36)

X = (S

, S

, T

, W,

, M

), with

LMI constraints (19) and

−M

−W

< 0 (37)

−M

−

< 0 (38)

−M

−

< 0 (39)

has a solution X (.). Then, the control law

u(t) = −R

−1

P x(t) (40)

with

P := W

−1

(41)

is a guaranteed cost control law. The corresponding

guaranteed cost

V(x(0), φ(−

), φ(−

)) = x

(0)P x(0) +

−

(τ)R

φ(τ)dτ +

−

(τ)P B

−1

P φ(τ)dτ (42)

is minimized over all possible solutions.

Proof

According to Theorem (3.1), the guaranteed cost (42)

for the uncertain time-delay system (1)-(5) is ensured

by any feasible solution (S

, S

, T

, W,

)

of the convex set deﬁned by (19). Furthermore, by

taking the Shur complement of (37) one has −M

−1

< 0 =⇒ M

> W

−1

= P =⇒ T race(M

) >

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

234

T race(P ). Consequently, minimization of the trace

of M

implies minimization of the trace of P . In

a similar way it can be shown that minimization of

the traces of M

and M

implies minimization of the

traces of R

and R

, respectively. Thus, minimiza-

tion of

J in (36) implies minimization of the guaran-

teed cost of the uncertain time-delay system (1)-(5)

with performance index (6). The optimality of the so-

lution of the optimization problem (36) follows from

the convexity of the objective function and of the con-

straints.

4 EXAMPLE

Consider the second order system of the form of

equation (1)with nominal matrices,

−2 1

0 1

, A

−0.2 0.1

0 0.1

, B

0.1

The uncertainty matrices on the state and con-

trol as well as on the delay matrices are according to

(2),

∆A

0 0

0.1 0.1

, ∆A

0 0

0.03 0.03

∆B

0.1

, ∆B

0.03

and hence, the rank-1 decomposition may be

chosen such that,

0.1

, D

0.03

, E

= E

0.1

, F

0.03

, G

= G

= 1

The state and control weighting matrices are,

respectively, Q = I

, R = 4. It is β

= 0.3,

= 0.2.

By solving the LMI objective minimization problem

one obtains the guaranteed cost

J = 25.1417.

The optimal rank-1 decomposition is obtained

with s

= 0.6664, s

= 2.2212, t

= 2.5000,

= 8.3333. Finally, the corresponding control gain

is K = [−0.2470 −6.0400].

5 CONCLUSIONS

In the previous sections, the problem of guaranteed

cost control has been studied for the class of uncer-

tain linear systems with state and input time varying

delays. A constant gain linear state feedback control

law has been obtained by solving an LMI feasibility

problem. The closed-loop system is then quadrati-

cally stable and preserves acceptable performance for

all parameter uncertainties and time varying delays

of a given class. The system performance deviates

from the optimal one, in the sense of LQR design of

the nominal system, due to uncertainties and time de-

lays. However, the performance deterioration is lim-

ited and this is expressed in terms of a performance

upper bound, namely the guaranteed cost. In order to

make the GC as small as possible, one has to solve

an LMI minimization problem. The minimal upper

bound corresponds to the ”optimal” rank-1 decompo-

sition of the uncertainty matrices.

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