H2 Robust Control with Pole Placement

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

H2 ROBUST CONTROL WITH POLE PLACEMENT J.C. Geromel*, G. Garcia** and J. Bernussou** *LAC-DT, Faculty of Electrical Engineering, UNICAMP, Brazil **LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulose Cedex, France

abstract. Conditions are given for pole assignability in particular regions of the complex plane such as circular region, vertical strip, and circular sector. The conditions are formulated in terms of semi positive definite matrices which constitute a parametrization of the feedback controls. The malO pOint IS that this parametrization is made through convex parameter domains so that the control determination can be carned by solving convex parametric optimization problems. The f{, transfer matriX norm IS used to define the cost function for these optimization problems so that the optimal solutIOn provides a guaranteed bound for the 1-l2 norm. Both certain and uncertain linear time invariant systems are considered. Numerical experiments illustrate the usefulness of the method. K e y Words. Lyapunov Methods, Quadratic Stabilizability, Uncertain Linear systems,

1

which do give a feedback whenever the corresponding class is not empty. Moreover, the approach is easily extended to uncertai n linear systems defining necessary and sufficient conditions for quadratic assignability. The auxiliary parametric optimization problem is defined in such a way that its optimal value is an upper bound for the ?-l2 norm of a given transfer function defined on the linear system. With this property, the results are reinforced by providing some results about guaranteed cost control.

Introduction

A natural way to get satisfactory performances in terms of speed and damping for linear time invariant systems is to put some constraints on the poles location for the closed loop transfer matrix. Practically, it is often sufficient to define particular regions of the left hand complex plane. This is for instance, the only way for un certain systems for which, of course, exact pole location is impossible. In this paper, we adress the problem of controller design under regional pole constraints, a problem which has been quite often investigated.

The plan of the paper is as follows . In the next section, some previous results [6], [7J are given on a control parametrization which enables to develop a new method for the minimum ?-l2 norm control problem (in fact, an L.Q . problem). The idea is extended in the following sections to provide the parametrization associated with feedback gains assigning the poles of linear time invariant systems in given regions of the complex plane. Vertical strip, circular and combined circular-strip regions are considered in the following 3, 4 and 5 sections. Necessary and sufficient conditions for pole assignability are developped. In section 6, quadrati c pole assignability for linear uncertain systems is introduced, together with the related necessary and sufficient conditions. All the conditions defining convex sets in the parameter space of unknowns, an ?-l2 norm type optimization problem is stated for all regional cases in section 7. The section 8 presents some numerical experiments to illustrate the interest of the proposed approach.

Among numerous works in analysis of root clustering, some of them have derived extended Lyapunov equations that provide necessary and sufficient conditions for a given matrix to have all its eigenvalues in regions of the complex plane like elliptical regions, vertical strip, sectors ... [1], [2J. Adaptation of such conditions in design procedures is valuable since it then enables to control in a quite understandable way, the performances of the controlled systems . Some papers have been dealing with this problem [3], [4], [5J but the results therein can hardly been considered as const ructive way for region assignment control synt hesis. The conditions are given in terms of necessary conditions related to auxiliar parametrical optimization problems which do not present any strong mathematical properties. In this paper, the assignability conditions for circular regions, vertical strip and combined regions are given with respect to convex parametrical domains which makes a full parametrization of the classes of feed backs that assign the modes in one of the above regions. The proposed method for feedback synthesis consists in solving a convex optimization problem associated with sufficient and sometimes necessary conditions for regional pole assignability which is a constructive design method

2

Preliminaries

Let us consider the li near time invariant system (LTI ):

Ax

+

ex + -J(x

73

B1w Du

+

B 2u

(1 )

The 'H 2 norm can be computed using the so-called controllability and observability grammians Le, Lo respectively which are given by the solutions of two Lyapunov equations:

where x E ~n, U E ~m, W E ~/, z E ~q are respectively the state, control, disturbance and output vectors. The matrices in (1) are known and of appropriate dimension. We assume that C'D = 0 with D of full rank. We also introduce the extended matrices F E ~p.p, P = n + m and C E ~p.m

Ae/Le + LeA~/ + BIB; = 0 { A~/Lo + LoAcI + C~Pcl = 0

where AcI = A - B 2J(, then:

(2) and the symmetric matrix W E

~p.p

(6)

IIHII~ = Tr[B;LoBd = Tr[CclLeC;d

partitionned as:

(7)

The problem (3)

Min IIHII~ J(EK

with W I > 0 E ~nn, W 2 E ~n . m , W3 E ~mm (' means transposition). N will denote the null space of C'. Let K the set of all stabilizing gains for (1) and denote 0(W) = FW + WF' + Q, where Q ~ 0 E ~p.p.

possesses, in fact, a very complex structure when considered in the J( parameter space since neither the cost function nor the K domain are convex. (8) can be solved using convex programming methods.

The formulation (2), (3) enables a parametrization of K which is apparent in the next lemma ([8]).

Let us define the parametrical optimization problem:

Lemma 1 The set C defined by:

c

(8)

P

= {W ~ 0, v'.G(W).v:::: 0, V v EN}

{Min Tr(W R) oWE C

(9)

with:

is convex and:

Problem Po is obviously a convex problem since C convex with respect to Wand the cost is linear. Proof: The set of non negative definite matrices is a convex cone and G(W) is linear with respect to W, then C is convex. Now, let J( E K, then there exist P = P' > 0, QI ~ 0 such that V x E ~n

o~

x'.[AP - B 2J( P

+ PA' -

P J(' B;

IS

Theorem 1 [8} Let W = Arg Min{ Tr(W R), W E C }. Then J( = W;WI- I solves the minimum 1-{2 norm p1'Oblem (8).

+ QI].X

Proof: First, from WE C, one gets:

o ~ v'.G(W).v

AWl

where the last inequality is fulfilled by:

+ WIA' -

B2W~ - W2B;

+ BIB;

:::: 0

which is written AeWI

+ WIA~ + BIB;

:::: 0, with J( = W~WI-I

By comparison with (6), one gets: and consequently W E C. Conversely, if WEe and vEN:

Le:::: W I

Then, O?: v'.8(W).v?: x'.[(A - B,W;W,-')W, + WdA - B,W;W,-')' + Q,J.x

which shows that W 2W I- I E K since W 1

IIH(s)lI~

> o.

Lemma 1 shows that the non convex set of stabilizing gains for a given LT! system (1) can be generated from the elements of a convex set C. The pair (A , B 2 ) is stabilizable iff C i- 0 [9]. This convexity property is indeed very interesting since it enables to state associated parametrical optimization problems for gain determination. Some results for uncertain linear systems stabilization have been derived from this [10], [9]. Recently, a new result has been given which, by a proper definition of the convex optimization problem provides a new way to solve the optimal 'H 2 norm (or L.Q.) optimization problem [8]. Defining the matrices Ad = A - B 2 f(, CeI = C - DJ(. The closed loop transfer function from w to z is such that :

Since W ~ 0, W3 ~ W;WI- I W2 :

IIH(s )II~

Tr[H*(jw)H(jw)] dw

(10)

For (10), the equality holds with W given by : W

-

(Le J( Le

LJ(') J( LJ(

which obviously belongs to the set C. Let us now consider the case of uncertain linear systems with A and B; and hence F uncertain matrix. The uncertainty defined directly on the extented F matrix is introduced through a poly topic convex set:

00

-00

= Tr(W R)

where P is the definite positive symmetric matrix solution of the algebraic Riccati equation: A'P + PA - PB2(D'Dt l B;P + C'C = 0

For stable systems, the 'H 2 norm can be defined in the frequency domain by:

1+

:::: Tr[WI-IC'C + W3D'D]

Under the given assumptions, the solution of (8) is given by: J( = (D' D)-I B;P

(4)

1 IIHII~ = 211"

Tr[Ce/LeC;/] :::: Tr[CclWIC~/] Tr[(C - Df\)Wl(C - DJ()'] Tr[WIC'C + DW;WI- I W2D']

(5)

74

N

'OF

two inequalities in (13). This constraint is given to enable the determination of a feedback satisfying the pole location specification, and its interest will become clear when dealing with pole location for uncertain systems. It can be shown that for a controllable system a single P solution exists for (13). Conditions (13) involving two matrices J( and P cannot easily be used for synthesis purpose. We shall use the same type of parametrization for the non convex set K, of the feed backs assigning alAe) to 'O(o,{3). Denote now:

N

= {F E 'iRPP,F = LJl.jFj,Jl.j ~ O,LJl.j = 1} i=l

(11)

1=1

Now, we define:

8 j(W)

= FjW + WF: + Q,

i

= 1...N

and the set:

Cu(W) ~r { W ~ 0, v' .8j(W).v :S 0 V v EN and Vi = 1 ... N } (12)

8".(W) = F"W + WF~ + BIB; 8.a,(W) = -F.a W - WFb + BIB;

Theorem 2 [S}. Let W = Arg Min{Tr(W R), WE

Cu(W) }. Then: • J(

Theorem 4 The set C, defined by:

= W~WI-I stabilizes (1)

C,

• IIH(s)lI~ :S Tr(W R), VF E 'OF.

=

{W ~ 0, v'.8",(W).v:S 0, v l 8JJ,(W).v:S 0, Vv EN} (14)

is convex and:

The parametrical optimization is still a convex one, thus the theorem 2 provides practical conditions for the determination of a stabilizing feedback for uncertain linear systems while giving a guaranteed cost bound. This kind of feedback parametrization is followed in the next sections devoted to pole placement.

3

Proof:

A.>cWI + WIA~e + BIB; :S 0 -A.aeWI - WIAPe + BIB; :S 0

Pole placement in a vertical strip : a necessary and sufficient condition

which from (13) obviously imply that W~WI-I E K,. For the converse, given a gain J( E K. and the associated P from (13), we construct a W matrix:

In the complex plane, we now consider the region shown in figure 1, a vertical strip:

'O(o,{3)

=

Condition (14) is equivalent to:

p

W

{z E C , -{3:S Re(z) :S -0 }

= ( [( P

PI(,)

[( P [('

which is easily shown to satisfy to (14). Finally, the determination of a W matrix satisfying (14) and hence the determination of a feedback assigning the modes of the controlled matrix in the vertical strip can be found as the optimal solution of a convex parametrical optimization problem which, moreover, provides a guaranteed bound for the gi ven closed loop transfer function (section 7).

4

Figure 1: vertical strip

Pole placement in a circular region: a necessary and sufficient condition

Let the circular region 'O(r, 0) be depicted in figure 2. Im

Theorem 3 The system (l) is assignable with respect to '0(0, {3) (i. e. the modes of Ac belong to V( 0, {3)) if there exists a matrix J( E Rm.n and a matrix P = pi > 0

R.

such that:

Figure 2: Circular region The notations are the same as for the previous section.

Proof: The sufficiency follows in a straighforward manner since that, with the assumption P > 0 and invoking the Lyapunov theorem one gets:

Theorem 5 [11] The system (l) is assignable with respect to the region 'O(r, 0) iff there exist a gain J( E Rm.n and a mal1'ix P = pi > 0 E 'iR n .n such that:

ReIA(A" - B 2 J()] = ReIA(A - B 2 J()] + 0 :S 0 -ReIA(A JJ - B 2 J()] = -ReIA(A - B 2 J()] - {3 :S 0

Ba(P)

The lack of necessity in theorem 3 commes from the fact that a single P matrix is to be found as solution for the

75

= A"c P + PA~c + ~A"cPA~c + BIB; r

:S 0 (15)

Theorem 8 The set Cas defined by :

The condition (15) can hardly been considered as giving a constructive procedure for control determination. In [11], an auxiliary minimization problem is stated and the necessary optimality conditions are derived in the J( parameter space. Because of the lack of strong geometrical properties for the overall parametrical optimization problem, it can be said that the given condition are merely useless for synthesis purpose.

Ca, =

K a,

K a , is the set of feed backs assigning the modes of (1) in Q, (3). Proof:

= F"W + WF~ + ~F"WF~ + BIB; r

= {W

~

By simple use of that of theorems 4 and 7.

Remark:

We have: }i..~ V(r,

0, v'.8 a(W).v S 0, "Iv EN}

(16)

= V(Q, (3)

C, as r approaches infinity. When (3 ~ = V(r,Q) and Ca, = Ca so that the results of the section encompass the previous ones. and Ca,

6 Proof: Convexity follows by the same reasons as for lemma 1. Since v = (x' O)',x E ~n, developping (16) one gets: AaWI - B2W~

+ WIA~

.....

+ ~[AaWIA~ - AaW;B~­ -B2W;A~ + B2W3Bi] + BIBi :5 0

+ WIA~c + .!.AacWIA~c + .!.B 2 (W3 r

W2Wl-1W2)B~ + BIB~ :5 0

-

(17)

r

where J( = W~WI-I. Since W3 - W;WI- I W 2 ~ 0, W > 0, A"e is a stable matrix with spectrum belonging to V(r,Q), then I( = W~WI-l E Ka.

Definition 1 The uncertain system (l) is quadratically assignable with respect to V(r, Q) (V(Q, (3) or V(r, Q, (3)) iJJ there exist J( E ~mXn and P = P' > such that:

°

Conversely let J( E Ka , then from (15) the matrix: W

=(

P J(P

"IF E V p , (15) ((13) or (I8)) holds

PJ(') J(PJ('

We want to stress that this definition pertains to the same philosophy as in the concept of quadratic stabilizability [12J, which has been used for stabilizability purpose of uncertain linear systems. The quadratic approaches amounts to extend the necessary and sufficient conditions written for N = 1 (certain case) to the uncertain one (N > 1) using a unique Lyapunov matrix. With the above definition, we are now able to state necessary and sufficient conditions and the associated parametrical optimization problems for assignability in the different regions of the complex plane.

belongs to Ca.

Pole placement in a circular sector: a necessary and sufficient condition

We now want to assign the modes of system (1) in the shaded area of figure 3 defined as the intersection: V(r,Q)

Pole placement for linear uncertain systems

We consider the system (1) where the (A, B 2 ) matrices are uncertain ones according to the assumption formulated in (11) which is related to the convexity of parameter uncertainty domain (convex polytope). This assumption is in fact a realistic one suited for instance to the case of interval matrices or more generally to matrices whose elements are multilinear functions of interval parameters.

- W;B~

After a few algebraic manipulations (16) is written:

5

Q, (3)

Q + r, V(r,Q,(3)

is convex and:

Ao.:W)

::2 { W;W1- I , W E Ca. }.

V(r,

Theorem 6 The set Ca defined by: Ca

(19)

is convex and:

Let Ka be the set of all gains J( that assigns u(Ae) in V(r,Q). Denote: 8 a(W)

{ W ~ 0, v'.8 a(W).v S 0, v'.8 {3s(W).v S 0, v EN}

n V(Q,(3) ~c V(r,Q,(3)

We denote by K ua , Kus> K ua , the sets of all feedback gains that quadratically assigns the modes of (1) to respectively V(r,Q), V(Q,(3),V(r,Q,(3). Let:

lm D(r.a,~)

R.

Theorem 9 The set Cua defined by:

Cua

Figure 3: Circular sector The circular region for pole assignability was said of practical value in [l1J, since it enables to bound the damping ratios and the natural undamped and damped frequencies. This is, of course even more true for the circular sector D(r, Q, (3) with its three free parameters for definition.

= i .. . N

}

is convex and:

Proof: Cua is the intersection of a finite number of convex sets, so convexity holds . For the second proposition, (20) is equivalently written as:

°

sO

{W ~ 0, v'.8 ai .v SO, V v E N, VI

(20)

Theorem 7 The system (1) is assignable with respect to D(r, Q, (3) if there exists a gain K E ~m.n and a P = P' > such that: A"eP + PA~e + ~A"ePA~e + BIB; { -A{3e P - PA~e + BIB; sO

=

A"icWI

(18)

76

+ WIA~ic + ~AOicWIA~ + BIB,' <- 0 r le

(21)

7

1t2 synthesis with pole constraints

In the preceeding sections, all the necessary and sufficient conditions for region assignability gives a way of parametrization of the assigning feedback gains by means of parametric convex domains. This permits to define convex optimization problem on which mathematical programming methods are applied to end at a constructive approach for feedback synthesis. Following the results surveyed in section 2, one way is to use the cost function tr(W R). Such a choice is likely since in complementation to pole assignability a guaranteed bound for the 1-f.2 transfer matrix norm can be exh ibited.

N

Since with W ~ 0, LPjA"jcW1A~jc ~ A"cW1A~c' we

,=1

have:

then I< = W;W1- 1 E Kua. Conversely let I< E K ua , then (15) holds for F i.e. :

=F

j,

Vi,

(24) Let us choose:

In the following K denote either K a, K" Ka, and C either

Ca, C., Ca •. We first deal with the certain case (N=I).

P PI<) W = ( K'P I<'PI<

Theorem 12 Let W = Arg Min {Tr(W R), W E C}.

W > 0 and satisfies (21) since with the particular choise of W3 (W3 = W;W1- 1W2 ), (22) reduces exactly to (21) .

Then : • I< = W;W1-

Let us now move to the vertical strip V( a, {3). We define:

E K

• Min{IIH(s)II~, K E K} S Tr(W R)

0"j,(W) = F"jW + WF~j + B1B; 0 Il j.(W) = -FlljW - W Fpj + B1B;

Proof: The proof is given for K = Ka and C = Ca· The other cases are proved in the same way. The first proposition is a direct consequence of theorem 6. For the second proposition , WE Ca means that (see (17)):

and accordingly 0". for F", 01l. for FIl , VF E VF · Theorem 10 The set Cu • defined by:

AeW1 + W1A~ + B1B; + Q S 0 1 , 1 , -1 Q = 2aWj + -A"eW1A"e + -Bz(W3 - WZ W1 Wz)

s 0,

v'.0Ilj.(W).v SO, Vi = 1 . . . N and V v E .N } (25)

v' .0"j,(W),v

1

°

r

Q > since W j > I<, W 1 > Le. But:

is convex and:

°

r

and W ~ O. Then for a given

Ku. = { W~W1-1, W E Cu• } Proof: The first proposition follows immediately from the affine pattern with respect to W of the relations N

(25). Then VF

=L

1=1

N

pjFj, L p j 0"j.(W)

since the optimal solution under W ~ 0 must satisfy W3 = W;W1- 1WZ ' Then:

= 0".(W) and

Tr(WR) = Tr(WdC'C

i=1

N

LPj 0 Il j.(W) = 01l.(W), i.e conditions (13) is fulfilled.

with I<

':=1

°

=F

We now move to the uncertain case and state a general theorem covering the three cases corresponding to the three regions of assignability. As before, Ku will denote either K ua , K u" K ua , and Cu either Cua, Cu " C ua, respectively.

j ,

Theorem 13 Let W = Arg Min {Tr(W R) , WE Cu }.

With:

Then:

P PI<) W = ( K'P I<'PI< conditions (25) are satisfied since W is indeed non negative and the development of ( v'.0"j,(W),v 0, v' .0Ilj.(W) .v 0) gives exactly (26) .

s

s

• Min{IIH(s)II~, I< E Ku} S Tr(WR), V FE V F

8

The development for the circular sector is done in much the same way as before.

Numerical examples

As stressed several times before, a parametrical auxiliary optimization problem is in the different W domains (C and Cu ) convex problem for which efficient mathematical programming methods work. However, expressed in terms of the entries of W, the relations defining the C domains are highly non linear and not suitable for a practical use. A way to overcome this difficulty has already been proposed in [8] for similar problems. It consists in the use of the cutting plane technique and linear programming. The given example is borrowed from [13J. The A, B2 matrices are uncertain, A belonging to a 4 vertices polytope and Ez to a 2 vertices one. The vertices are given below. The uncertainties on A and B2 are assumed independant so that DF is a N = 8 vertices polytope.

Theorem 11 The set Cua. defined by:

Cua.

But:

Tr(WR) ~ Tr(Le(C'C+I
The eigenvalues of Ac belong to V(a, {3), i.e. I< E Ku.· Conversely, given a I< E Ku .. (14) must hold for F V i. There exists a P = P' ~ such that:

= W~W1-1.

+ l{D'DK ))

{ W ~ 0, v'.0aj·v S 0, v' .0 Ilj.(W).v, V i = 1 ... N and V v E .N }

is convex and: Kua. = { W~W1-1 , WE Cua. } The proof is contained in the ones of previous theorems and since: Cua. = Cua n Cu• Kua. = Kua n Ku.

77

=

A,

(

17.4100 -0.8512 0

-0.9896 0.2g48

A,

=(

-0.9896 0.0820 0

A,

=(

-0.6606 0.2648 0

96.1500 ) -1\.3900 -250.0000

17.4100 -0.6586 0

96.1500 ) -10.8100 -250.0000

18.1100 -0.8512 0

84.3400 ) -1\.3900 -250.0000

-0.6606

18.1100

0.0~20

-0.g586

A. = (

7798

0

B" = ( -97

The circular sector with \.4285 W = 10-4.

9

0

250.0000

1

0

0

1

For pole assignability in the vertical strip V{ et, (3), the results are given below in terms of Wand J( = W~W1-1 matrix for et = 1,(3 = 10. 3.7809 -0.4717 0.3368 -0.3217

(

-0.4717 0.2978 0.0548 -0.0755

K = (-0.0115

0.3368 0.0548 0.5239 -0.5119

-0.0952

-0.3217) -0.0755 -0.5119 0.5041

-0.9598)

-0.0087

0.0122 0.0051

0.0033 -0.0033

-0.0119 ) -0.0051 -0.0033 0.0032

-0.9697)

Conclusion

10

On figure 4 are plotted the 24 modes corresponding to the Fi matrices (vertices of V F ), they all lie in the pre· scri bed region.

..."

....

~r-~--~----.---~--~----~-----'-'

Il

-0.1443 0.0332 0.0051 -0.0051

We have presented a constructive approach for feedback determination under pole regional constraints. By con· structive, it is meant that the method and its associated numerical tools work on necessary and sufficient conditions and do converge. Although, the considered regions are circular, vertical strip and mixed, it is thought that the results are general enough since such constraints en· able to control the response speed and damping. The particular choice of the convex parametrical optimization problem moreover leads to kind of guaranteed minimal 'H 2 norm problem, a response also to exogeneous perturbation rejection .

( 1o 0 0) 0

= 4.5, et = I, (3 = 5 has given:

The plotted modes are all located in the given region as seen in figure 6.

84.3400)

The Bll C and D matrices are given by : B, =

0.0122 -0.0 119

K = (-0.0009

B22 = ( -85 1002 )

)

-0.1443

. (

~2~008;0~0

250.0000

T

.~.

·s ·10

'.

10

·Il

.W L-.~\4~--.I~'--~.170--~.'~--~ .'~--47---~.,~~

Figure 6: modes for the vertices: circular sector ·s

References

·10

I1] A.G. Mazco. The lyapunov matrix equation for a certain class of regions bounded by algebraic curves. Soviet. Automatic Con irol, 1980. [2] A. Abdul-Amir A. Abdul-Wahab. Lyapunov bounds for rool clustering in the presence of uncertainly. 1nl. J. Sy.stems Sci., 2l{12}, 1990.

Figure 4: modes for the vertices: vertical strip

[3) F. Jabbari. Robustness bounds for linear systems wit.h Uncertainty: keeping eigenvalues in specified regions. In ProceedingJ of the 29th Conferena on Decision and Control, Honolulu, Ha.wa.i , 1990.

For the circular region, et = 1, T = 4.5. This region is included in the vertical strip. We obtain: 2.3400 -0.2319 0.0181 -0.0175

W = (

K

=

-0.2319 0.0640 0.0163 -0.0161

(-0.0014

00181

-00175 )

00163

-00161

00129 -00126

-00126 00123

-0.0137

[4] Y.T. Juang Z. C. Hong Y.T. Wang. Robustness of pole assignment of linear s ystems with structured uncertainty. I. E.E.E. TranJ. on Aut. Cont., 34(7).1989. [5] Y.T. Juang. Robust stability and robust pole assignment of linear systems with structured uncertainty. I.E. E.E. TranJ. on A ut. Cont., 36(5), 199\.

-0.9571) [6] P.L.D.Peres J .C.Geromel. An alternate numerical solution to the linear quadratic problem. Submitted t o I.E.E.E. Trans. on Aut. Control, 1992.

Figure 5 shows the location of the modes for the ver· tices. We can note that with this prescribed region, the damping factor of the modes is improved .

[7] G.Garcia J.Bernussou D.Arzelier. Pole assignment of linear uncertain systems in a sector by state and dynamic output feedback: A quadratic stabilizabilityapproach. Submitted to S.I.A .M. Journal of Cont. a.nd Opt.. 1992. (8] J.C.Geromel P.L.D.Peres S.R.Souza. H2 guaranteed cost control for unce rtain continuous-time linear systems. Sy.!tems an.! Control letter.!, 19, 1992.

~.----------------------------, Il

(9] J.C.Ceromel P.L.O.Peres J.Bernussou. On a convex parameter space method for linear control design of uncertain syst.ems. S.I.A.M. Cant. and Op! .. 29(2). 1991.

10

c·~:·:~ -----

i ·s

[10] J.Bernussou P.L.D.Peres J .C.Ce romel. A linear programming oriented procedure for quadratic stabili:1.lt.tion of uncertain systems. Systems and Control LetieT.t, 13, 1989. [11] W.M.Haddad D.S.BernsLein. Controller design with regional pole constraints. I.E.E.E. Tran.!. on Aut. Cont., 37(1), 1992.

·10 ·Il .~

·14

."

·10

..

[12] B.R.Barrnish. Necessary and sufficient conditions for quadratic stabilizability of an uncertain system. Journal Optimi. Thtory Appl., 46(4), 1985.

.,

[13] W.E. SchmiLendorf. Designing stabilizing controllers for uncertain systems using the riccati equation approach. I.E. E. £. Trans. on Aut. Control. 33(4), 1988.

Figure 5: modes for the vertices: circular region

78