Optimal Guaranteed-Cost Control with Pole Placement for Uncertain Systems: An LMI Approach

Copyright © IFAC Robust Control Design, Budapest, Hungary, 1997

OPTIMAL GUARANTEED-COST CONTROL WITH POLE PLACEMENT FOR UNCERTAIN SYSTEMS: AN LMI APPROACH S. H. Esfahani*

S. O. R. Moheimani* I. R. Petersen*

-Department of Electrical Engineering, Australian Defence Force Academy, Canberra, ACT 2600, Av.stralia.

Abstract. The paper presents a linear matrix inequality (LMI) approach to the problem of pole placement in a specific stability region while minimizing a guaranteed cost bound for a class of uncertain systems. The uncertainty is assumed to be norm bounded and the uncertain parameters are allowed to enter both the state and the input matrices. The stability region is the intersection of a vertical strip and a disk in the open left-half complex plane. First, a state feedback controller is designed to place the closed loop poles of the uncertain system in the given stability region via any feasible solution of a corresponding LMI. Then, linear quadratic (LQ) and H2 measures are considered for the closed loop performance of the uncertain system. Finally, an LMI Eigenvalue Problem is proposed to select the optimal guaranteed cost controller that also satisfies the pole placement requirements. In two examples, our results are compared with the existing results. Key Words. Uncertain Systems, Norm Bounded Uncertainty, Linear Matrix Inequality, Quadratic V-stabilization, Guaranteed Cost.

ever, the solution to this problem is not unique and it is desirable to select a controller that minimizes a cost function as well.

1. INTRODUCTION

During the recent years, the problem of quadratic stabilization of uncertain linear systems with norm-bounded uncertainty has received considerable attention in the Control Community; e.g. see (Barmish, 1983; Petersen 1987; Khargonekar, et al., 1990). It has also been proved that this problem can be stated in terms of a linear matrix inequality (LMI) problem, see (Boyd, et al., 1994). Using any feasible solution of the corresponding LMI problem, a robust state feedback controller is constructed that quadratically stabilizes the uncertain system.

In this paper, the problem of optimal guaranteedcost controller design with pole placement in a region which is the intersection of a disk and a vertical strip is considered. This problem for a linear quadratic (LQ) performance measure and H2 performance measure with pole placement in a disk have been considered in (Moheimani and Petersen, 1996; Garcia, et al., 1995). However, it will be shown that the approach in this paper leads to a less conservative results than the existing results.

One approach to including issues of control system performance in the quadratically stabilizing controller design problem is to place the closedloop poles of the uncertain system in a specific region in the open left-half complex plane, see (Moheimani and Petersen, 1996; Garcia, et al., 1995; Chilali and Gahinet, 1996; Garcia and Berussou, 1995). In the literature, this problem is called the quadratic V-stabilization problem. It has also been observed that problems of pole placement for a wide class of stability regions can be formulated as LMI problems, see (Chilali and Gahinet, 1996) for more details.

The approach in (Moheimani and Petersen, 1996) is based on the stabilizing solution of a parameterdependent Riccati equation which is required to satisfy an additional constraint. The Riccati equation presents pole placement requirements in a disk. This equation also contains weighting matrices related to the LQ cost. The optimal controller is found by searching over parameter values to minimises an upper bound on the LQ cost function. An LMI formulation for this problem is presented in (Garcia, et al., 1995) for an H2 cost. The LMI guarantees the pole placement condition. The optimal solution is found by solving a convex optimization problem. In our approach, pole positioning and the upper bound minimization are

Pole placement in a given stability region can be used to guarantee the desired behaviour of the transient response of the closed loop system. How-

189

out uncertainty. For this system, the eigenvalues of A lie in V = Dl n D 2 , if and only if, the LMIs

stated as different LMIs. Thus, with suitable selection of the corresponding LMI related to each cost, the upper bound minimization is solved as an LMI Eigenvalue Problem, (Boyd, et al., 1994).

QA' +AQ + 20 l Q QA' +AQ + 20 2 Q

The paper is organized as follows. In Section 2, the necessary and sufficient LMI constraints for quadratic V-stabilization of uncertain systems with norm bounded uncertainty are presented. The stable region under consideration is the intersection of a disk and a vertical strip in the open left-half complex plane. The optimal guaranteed cost formulation for LQ and H2 costs are stated in Section 3. Finally, in Section 4, two examples are presented which illustrate our approach and compare our results with the results of (Moheimani and Petersen, 1996; Garcia, et al., 1995).

-rQ [ Q(A + ql)'

(A+ql)Q] -rQ

< 0, > 0, < 0,

have a symmetric solution Q > 0, see (Chilali and Gahinet, 1996) for a proof. Now we extend this notion to the quadratically stable system (1). Definition 2.2 The uncertain system (1) is said to be quadratically V-stable with respect to the region Dl n D2 if there exists symmetric Q > such that:

°

2. QUADRATIC V-STABILITY AND V-STABILIZATION In this section, we first consider the problem of quadratic stability with pole location constraints for uncertain systems with norm bounded uncertainty. The uncertain system under consideration is described by the state equation:

x(t)

= (A + D6(t)El )x(t)

(

(1)

-rQ

where x(t) E Rn is the state and 6(t) E Rpxq is a time-varying matrix of uncertain parameters satisfying the bound 6' 6 ~ I. All other matrices are real constant matrices with appropriate dimenSlOns.

for all 6 satisfying 6' 6

~

I.

Theorem 1 The uncertain system (1) is quadratically V-stable if and only if the LMIs

Definition 2.1 The uncertain system (1) is said to be quadratically stable, if there exists a positive definite symmetric matrix Q such that

Q(A + D6Ed + (A + D6EI)Q <

~

In the next theorem, we present necessary and sufficient conditions for quadratic V-stability of the uncertain system (1) .

We first present a standard definition of quadratic stability, see (Khargonekar, et al., 1990).

for all 6 : 6' 6

(A+D6El +ql)Q

[

QA'+AQ+ElDD'+20lQ ElQ

QE'1 ] < 0, (2)

-Ell

°

I.

We wish to add some pole location constraints to a quadratically stable system (I). In this paper, we consider two regions in the complex plane. The first one is the disk with centre (-c,O) and radius

[

r:

-rQ + E3DD' Q(A + ql)'

o

(A + ql)Q -rQ EIQ

(4)

have a positive definite symmetric solution Q for some positive values of the constants El, E2 and E3. Proof The proof is given in the complete version of the paper. 0

The second region is the vertical strip defined by:

D2

= {x + iy E C : -02 < X

=

<

-01

< O}. Remark 2.1 It should be pointed out that the LMIs (2)-(4) are also satisfied for fi = 1, i = 1, 2, 3, when the system is quadratically V-stable.

=°

We first assume that D El in (1) . Therefore we have a linear time-invariant system with190

We now extend the result of Theorem 1 to the synthesis case. Let us consider the uncertain system

x(t) = (A + D~(t)Edx(t) +(B l + D~(t)E2)U(t),x(0) = Xo

considered in (Moheimani and Petersen, 1996), where V is a disk in the open left-half complex plane. A similar idea for H2 guaranteed cost control is considered in (Garcia, et al., 1995). However, we propose a different approach which results in less conservative results from those obtained in (Moheimani and Petersen, 1996; Garcia, et al., 1995).

(5)

where u(t) E R m is the input vector. For this system, we wish to find a state feedback u(t) = Kx(t), such that the closed loop system

x(t)

= (A + B1K + D~(t)(El + E2K))x(t),

We first consider a problem of H2 guaranteed cost minimization over the set 1jJ. The uncertain system under consideration is described by the state equation

(6)

is quadratically V-stable. Theorem 2 The uncertain system (5) is quadratically V-stabilizable with state feedback control u(t) = Kx(t) if and only if the LMIs

(

QA' +AQ +BIY + Y'B~

) (EIQ+

+EIDD' + 2alQ

x(t) = (A + D~(t)El)X(t) +(Bl + D~(t)E2)U(t) + B2W(t), z(t) = Cx(t) + Hu(t).

~Y)' 1< 0, (7)

(10)

where w(t) E RI is the disturbance vector and z(t) E Rq is the output vector.

[ (E I Q+E2 y)

QA' +AQ

(

+BIY+Y'B~ -E 2 DD'

[ (EIQ

+ 2a 2Q

+ E 2 y)

For any admissible~: tion w(t) to z(t) is:

-Ell

) (EIQ

+ E2Y)'

1>

HA(S) = (C + HK) (sI - A - B1K - D~(El

0, (8)

[

I, the transfer func-

+ E 2K))-1 B 2·

The H2 norm of HA(s) is given by

E21

IIHA(s)lI~

-rQ+€3DD' Q(A + qll + Y'B~

~' ~ ~

(A + ql)Q + BIY -rQ (EIQ +E2Y)

1~Y)' 1

(EIQ -€31

where,

= trace{(C + HK)Lc(~)(C + HK)'}

Lc(~)

is the controllability gramian which

is the solution of the Lyapunov equation

[A + B1K + D~(El + E2K)lLc(~) +Lc(~)[A + B1K + D~(El + E 2K)1'

(9)

+B2B~

have a solution Q = Q' > 0 and Y for some positive values of Cl, c2 and C3, where Y KQ.

= o.

(11)

Lemma 1 Assume that the uncertain system (10)

=

is quadratically V-stabilizable, then for any pair (Q,Y) E 1jJ, the LMI

Proof The proof is similar to the proof of Theorem 1 for closed loop system (6) by using the new variable Y = KQ. 0 Now we define the set

(

AQ+QA' ) +BIY +Y'B~ +4DD' +B2B~

[

= Q' > 0, (Q, Y) satisfies (7), (8), (9) for some El > 0 C2 > 0 and C3 > o}.

IjJ = {(Q, Y) : Q

(EIQ + E 2 y)

is satisfied for some

We note that the set IjJ is convex and for any pair (Q, Y) E 1jJ, the pair (J.LQ, 11'y), J.L > 0 also belongs to the set 1jJ. Theorem 2 implies that the uncertain system (5) is quadratically V-stabilizable, if and only if the set IjJ is non-empty.

(EIQ

+ E 2 Y)'

1

< 0,(12)

-4 1 C4

> O. Moreover,

IIHA(s)lI~

< trace{(CQ + HY)Q-l(CQ + HY)'} for all

~

satisfying

~' ~ ~

(13)

I.

Proof The proof is given in the complete version of the paper. 0

3. OPTIMAL GUARANTEED COST CONTROL

Theorem 2 states that the convex set IjJ contains all the quadratically V-stabilizing controllers for uncertain system (10) . On the other hand, Lemma 1 implies that the solution set IjJ also satisfies LMI

In this section we consider a problem of guaranteed cost control and quadratic V-stabilization. This problem for an LQ performance measure is 191

(12) for some value of E4 > O. Furthermore, the corresponding upper bound (13) for the H2 cost is a convex function of Q and Y, (Khargonekar and Rotea, 1991). Thus, the optimal controller can be achieved via a convex optimization problem. This problem is stated in the next theorem.

bound (16) is a convex function that is defined on the set IP. Theorem 4 The optimal LQ guaranteed cost quadratically V-stabilizing controller is the solution of the Eigenvalue Problem:

minimize 'Y subject to

Theorem 3 The optimal H2 guaranteed cost quadratically V-stabilizing controller is the solution of the LMI Eigenvalue Problem:

Q

minimize trace(X) subject to Q

= Q' > 0, El X

[ (CQ + HY)'

[

= Q' > 0, El > 0, E2 > 0, E3 > 0, E4 'Y X'] Xo Q > 0, (7), (8), (9), (15).

>0

> 0, E2 > 0, E3 > 0, E4 > 0 Proof The proof is a direct consequence of Theorem 2 and Lemma 2. 0

CQ+HY] Q > 0, (7), (8), (9),

(12). 4. ILLUSTRATIVE EXAMPLES In this section, we present two examples to illustrate the optimal guaranteed cost approach developed in sections 2-3. Also we compare the results obtained with our approach to those obtained using the approaches of (Moheimani and Petersen, 1996; Garcia, et al., 1995).

Proof The proof is a direct consequence of Theorem 2 and Lemma 1. 0 In the next section, we compare our results with the results of (Garcia, et al., 1995) for the same example. It is shown that the above approach results a less conservative value for the upper bound on the H2 cost.

Example 1 As the first example, we consider the example presented in (Garcia, et al., 1995). We have the uncertain system (10) with parameters:

We now consider the LQ cost function J

= l°O[X(t)'RlX(t) +U(t)'R2U(t)] dt,

A=[

(14)

RI > 0, R2 > 0, associated with the uncertain system (5) . D

Lemma 2 Assume that the uncertain system (5) is quadratically V-stabilizable, then for any pair (Q,Y) E lP, the LMI AQ+QA'+BIY+Y'B~

(EIQ + E 2Y) QRi/2

[

0

o o

is satisfied for some

0

-1

0

< 0 (15)

0-1 E4

> O.

Moreover,

Q=

(16) for all

fj"

satisfying fj,,' fj"

~

o

o

0.8 ] -~25 ,El

90.24 ] -11.10 ,Bl -250

= [0.1

0.1

=[

-91.24] 0 , 250

1] ,E2

= 1.25,

fj,,(t) is a scalar uncertain parameter subject to the bound Ifj,,(t) I ~ 1. We wish to find the optimal H2 guaranteed cost that places the poles of the closed loop system in the disk with centre (-5,0) and radius 4.5. Using the minimization approach in Theorem 3, we obtain the following result:

+ E2Y)' QRi/2 Y'~/21

-E4 1

17.76 -0.75

+E4DD'

y~/2

(EIQ

=[

-0.82 0.17

0.3859 -0.0332 [ 0.0074

-0.0332 0.0118 0.0035

0.0074] 0.0035 , E3 0.0031

= 0.05,

1. y

Proof The proof is similar to the the proof of Lemma 1. 0

= [0.0074

0.0034

0.0031] , E4

= 0.16.

The optimal controller is K = [0.0022 0.0156 0.9517]. This controller implies that the upper bound related to the square

It should be noted that the corresponding upper 192

of H2 norm of the transfer function from w(t) to z(t) is 0.4039. In (Garcia, et al. , 1995), the value is reported as 0.7968. Fig.l shows a plot of the square of H2 upper bound and the value of the square H2 cost as a function of the uncertain parameter 1::1. It should be noted that in this plot, only constant values for 1::1 have been considered. In this figure, the position of the closed loop poles of the uncertain system is also plotted for different values of 1::1.

ance. We wish to construct the optimal quadratic guaranteed cost controller which places the closed loop poles of the uncertain system in the disk with center (-2,0) and radius 2 and also minimizes the upper bound on the cost function

In this example, the value of the initial condition Xo is assumed to be random. Thus, in the optimization procedure given in Theorem 4, we need to minimise trace( Q-l) . The following results are obtained:

Q _ [ 0.2311 ~~o------~----~

-0.5

t!.

0

0.5

y

Fig. 1. Closed loop poles and H? upper bound of the uncertain system (10) for different values of the uncertain parameter i:!J..

-0.1045

= [0.4103

-0.1045 ] 0.0858 '

-0.1165],

€3

= 0.14,

€4

= 0.16.

Therefore the optimal state feedback controller is K [2.5860 1.7922] and the corresponding upper bound on the cost function is 35.6089. The upper bound given in (Moheimani and Petersen, 1996) is 54.9322. Thus, our approach leads to a less conservative value for this bound. Fig.3 shows a plot of the upper bound and the closed loop value of cost J as a function of the uncertain parameter 1::1. The closed loop poles are plotted for different values of 1::1 as well. Now we find

=

Now we wish to find the optimal controller that also moves the poles of the closed loop system to the left-half of the given disk. Therefore, we have Ql = 10 in LMI (7). The resulting feedback controller is K = [0.0063 0.0375 0.9335] and the corresponding bound ofthe square H2 norm is 1.0522. In Fig.2 the corresponding plots for this case are shown.

, O~----~--~~~

-1

X-axis

Fig. 2. Closed loop poles and H? upper bound of the uncertain system (10) for different values of the uncertain parameter i:!J..

=

the example presented in (Moheimani and Petersen, 1996). The uncertain system is described by the state equation

+[

-;.1 ]

u(t) , x(O)

0.5

the optimal controller that places the poles of the closed loop system inside the given disk and so that the real part larger than -2.5. Therefore, we have Q2 = 5 in LMI (8). The resulting feedback controller is K [2.2350 0.6562] and the corresponding bound of the cost is 66.7793. Fig.4 shows the results for this case.

Example 2 For the second example, we consider

= [-2.811::1(t) :~]

t!.

Fig. 3. Closed loop poles and LQ upper bound of the uncertain system (5) for different values of the uncertain parameter i:!J..

~.70------~_5~-----:0

x(t)

-0.5

x(t)

5. CONCLUSIONS In this paper, the problem of guaranteed cost controller design with pole placement in the intersection of a vertical strip and a disk has been considered. Necessary and sufficient conditions are derived in terms of feasibility of certain LMIs.

= xo,

with 11::1 (t) 1 :::; 1. The initial condition Xo is a zero mean Gaussian random vector with unit covari193

Petersen, I.R. (1987) . A Stabilization Algorithm for a Class of Uncertain Linear Systems. Systems and Control Letters, 8, 351-357. Petersen, I.R. and C.V. Roliot (1986). A Riccati Equation Approach to the Stabilization of Uncertain Linear Systems, Automatica, 22, 397411. -3

-2 X-axis

-1

-0.5

a

0

0.5

Fig. 4. Closed loop poles and LQ upper bound of the uncertain system (5) for different values of the uncertain parameter ~.

These conditions guarantee the pole location for the closed loop uncertain system in the stability region. On the set of all quadratically 1)stabilizing controllers, the problem of minimizing the guaranteed LQ and H2 costs are defined. An LMI Eigenvalue Problem is presented to find the optimal guaranteed cost controller. In two examples, it is shown that our approach leads to less conservative value for each cost with respect to the existing results.

REFERENCES Barmish, B.R. (1983). Stabilization of Uncertain Systems via Linear Control, IEEE Transaction on Automatic Control, 28, 848-850. Boyd, S.P., L. El Ghaoui, E. Feron and V. Balakrishnan (1994) Linear Matrix Inequalities in Systems and Control Theory, SIAM. Chilali, M. and P. Gahinet (1996) . Hoo Design with Pole Placement Constraints: An LMI Approach. IEEE Transaction on Automatic Control, 41, 358-367. El Ghaoui, L., F. Delebecque and R. Nikoukhah (1995). LMITOOL: A User Friendly Interface for LMI Optimization. Garcia, G. and J. Berussou (1995). Pole Assignment for Uncertain Systems in a Specified Disk by State Feedback. IEEE Transaction on Automatic Control, 40, 184-190. Garcia, G., J. Berussou and D. Arzelier (1995). A LMI solution for Disk Pole Location with H2 Guaranteed Cost. In Proc. of 3rd E.E.C. Conference, Rome, Italy, 3728-3733. Khargonekar, P.P., I.R. Petersen and K . Zhou (1990). Robust Stabilization of Uncertain Systems and Hoo Optimal Control. IEEE Transaction on Automatic Control, 35, 356-361. Khargonekar, P.P. and M.A. Rotea (1991). Mixed H2/ Hoo Control: A Convex Optimization Approach. IEEE Transaction on Automatic Control, 36, 824-837. Moheimani, S.O.R. and I.R. Petersen (1996). Quadratic Guaranteed Cost Control with Robust Pole-Placement in a Disk. lEE Proc. Control Theory Appl., 143, 37-43. 194