Numerical methods for robust control design for distributed parameter systems

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WP11 17:20

NUMERICAL METHODS FOR ROBUST CONTROL DESIGN FOR DISTRIBUTED PARAMETER SYSTEMS

R.H. Fabian0

A. J. Kurdila C. Kim

Department of Mathematics Texas A&M University College Station, Texas 77843

Department of Aerospace Engineering Texas A&M University College Station, Texas 77843

ABSTRACT In this paper we discuss a numerical method for constructing feedback control laws which are robust with respect to disturbances or structured uncertainties. The idea of treating the control and disturbance as competing players in a differential game is well knowii (see [2] and [3], for example) and leads to a non standard algebraic Riccati equation. We show that known convergence results for the standard linear quadratic regulator problein may be implemented and used as the basis for a numerical inethod for constructing control laws. For the case of structured uncertainties, we show that recent results of Speyer and Rliee ([16])for the finite dimensional case can be extended to infinite dimensions. Their approach is to take advantage of the factorization of the structured uiicertainty so that the uncertainty is treated as a disturbance. Then the differential game framework is applied. We provide numerical examples to illustrate each case. I. INTRODUCTION In recent years there has been considerable interest in the problem of designing control laws which are robust with respect to model uncertainties and/or disturbances. The H , methods provide one approach to solving these types of problems. Another approach is to make use of differential game theory ([13]). For example, in the case of an unknown disturbance, the control and disturbance can be treated as two competing players in a so-called 'soft-constrained' differential game ([2], [3]). In the case of structured uncertainty Rhee and Speyer ([lS]) have recent results in which the uncertainty is treated as a disturbance, and then the game theory approach is applied. The results of [16] can be extended to distributed parameter systems (see f71). In both cases the solution is given in terms of a nonstandard algebraic Riccati equation in a Hilbert space. In this paper we consider the problem of constructing approximation schemes for the solution of these infinite dimensional differential games. The method is based on applying known LQR approximation resid ts ([11,[ 8 ] ,[9],[lo], [111 [121) to justify convergence of solutions of finite diniensioiial algebraic Riccati equations.

CH3229-2/92/0000-1172$1 .OO 0 1992 IEEE

CONTROL DESIGN IN THE PRESENCE OF DISTURBANCE Let H , U, and W be separable Hilbert spaces and suppose that B E L(U,H ) , @ E L( W,H ) and C E L(H, H ) are bounded operators. Consider the following abstract Cauchy problem on 11.

H: ? ( t )= h ( t ) + BU(t) + h ( t )

(2.1)

z(0) = 20 y(t) = Cz(t).

Here A is the infinitesimal generator of a strongly continuous semigroup T ( t )on H . The control problem to be considered is an infinite climensioiial version of the soft-constrained differential game discussed in [2]. We first define the 'disturbance augmented' cost functional

6

where y is a fixed positive coiistant and R E L ( U , U ) , M L(W,W). Define the spaces U = L2(0,00;U) and W = L'(0,oo; W ) . The differential game or rninmax problem is to find inf sup J ( u , w ) (2.3) UEU

wew

su1,jec.t to dynamics governed by (2.1). A solution ( u 0 , w o )is called a saddle point of J if and only if J ( u o ,w ) 5 J ( u 0 ,W O ) 5 J(u,W O ) for all ( U , w ) E U x W . The following conditions will be sufficient to guarantee that there exists a unique saddle point solution to (2.3). Vu E U such that ( R U , ~2) ~ dllulg , such that ( M w , w j w 2 dz1wI~ Vw E W (H2) BR-'B* - y Z @ M - ' @ *2 0

(Hl) 3 4 > 0

1172

3dz > 0

The following is true (see [2] for finite dimensions, [3] for infinite dimensions).

THEOREhI 1. If ( H l ) and ( H 2 ) hold, then there ezz.qt,q a gzven zn feedback unzque Jaddle poznt solutzon (U'. w') E l?x f o r m by u O ( t )= - R - ' B * n x ( t )

where y(x, t ) is tlie transverse displacement , E I is the bending iiiodulus, p is the density, c is the damping coefficient a i d IL is tlie thickness of the bean. The distributed coiitrol influence is relm~sentedby the fuiiction b ( x ) , while the structured uncertainty is given by a(.). The control input is u ( t ) , and w(t) is tlir tlisturbance signal. In this example, the disturbance @(.E) i-an 1)einterpreted either as a region of delamination in the distrilmted control, or as a region in which the authority of the distributed control has evolved due to usage. For tliis model problem, tlie coiitrol influence b ( x ) has been selected to be a linear function of the beam length

(2.4)

ul'(t) = y2,v-1 @*rI.E(t). Here II as a p o d z v e definzte solutzon of the algebraic Riccata equatzon

f o r all x , y E d o m -4, where

b(x) =

L

Remark. The assuInption (H2) is used in [3]to make a c o n whereas the disturbance is given by nectioii between the miiiiriax problem and a related LQR minimization problem. This same observation allows us to apply 5x known approximation results (for the LQR problem) to the niinniax prohleni. We suinniarize tliis approach next (see [6]). @('I =

-J

The tlistrilmtetl control influence and end tip disturbance are depic.tetl in figiirr 1.

be a family of finite dimensional subspaws Let { H" } of H , and let P" lie the orthogoiial projection from H to H N . Ass~umethat P" converges strongly to tlie identity 01,erator. In addition. assuine that there are operators A" E L(H","), 0''' E L ( H " , H " ) , C" E L ( H ~ , H " ) and . that T " ( t ) is the semigroup generated by A". Our numerical method is based upon considering the following finite dimensional versioii of tlie algebraic Ricc-ati equation (2.5): ~N'nh'

NAN - ~

N

~

N

~

+I

cN V' c N ~

N=

0

(2.7)

Tlie following conditions are typical of those found in the LQR approximation literature and are sufficient for the convergence result that we desire. (H3) For each .e E H . T N ( t ) P " r -+ T ( t ) sand T N * ( t ) P N x i T * ( t ) s .and the c-onve~genc-ei5 uniform in t in I~ounded t-intervals. (H4) For each s E H , R"P"r + Ox, C'l"x 4 C x and

FIGURE 1

C1V-PN.x + c * x .

(HS) The family of pairs (A", !d' ) and ( A N C , N )are uniformly stabilizal,le and uniformly tletectablr. respectively.

Figures 2 tliroiigli 9 depict the transient results obtained for several discretizatioiis of the governing equations for both a liiirar quadratic regulator and the game theoretic controller. In all the simulations, the value y z is fixed to be

Tlie following result is true (see[lO]; for related LQR a p proximation results, see [l] ,[8],[9],[ 1I],[121).

yz = 11264.4

THEOREM 2. If (HS)-(HS) hold, then the unique nonnegative solution nN of (2.7) converges skrongly to the nonnegative solution of (2.5).

This value was selected by iteratively calculating the maxiniuni value of y for which it was nuinerically possible (i.e. no ill-c.oiiditioiiiiiff) to solve the game theoretic algebraic Riccati cquation . Roughly speaking, the value of y has been numerically clioseii to represent the inaxiinuin allowable disturbance. We conducted two distinct sets of numerical experiments. 111 all tlie following examples, no external forces are applied to the beam. In the first set of simulations, the "worst-case" initial condition { y o ,y o } is calculated as the orthonormalizetl cigeiivector of the solution to the game theoretic Riccati equation [4]. Figures 2 and 3 plot the tip displacement as a function of time for a 20 element model of the beam. Likewise, figures 4 and 5 illustrate tlie response for a 30 element model. Figures ? ailtl 4 show that the response is stable in the presence of the

n

Thus a numerical algorithm consists of constructing an approximation scllemr satisfying (H3)-(H5), and then solving a matrix representation of (2.7) 011 a computer. We use this method in the following example.

Exam& As a first example of the disturbance rejection propeities for structured uncertainty, a simple Bernoulli-Euler beam with Kelvin-Voigt damping is considered. Tlie equations governing the inotioii of the beam are 1173

unstructured tip disturbance in all cases using the game theoretic controller. When the feedback gains are calculated using the LQR controller in figures 3 and 5, the presence of the disturbance degrades the performance in all cases. Again, it should be eniphasized that the LQR response is for the worst case [4] initial condition with the worst case disturbance w o ( t ) injected through the disturbance function O ( x ) .

Tip Displacement of beam30-model(MMXLQR)

o' 6 -I

0.41

I

0.2

0 Tip Displacement of beamZO-modcl(MMXLQR) 1

0.8

i -0.2

-0.4

0.6 0.4

-0.6'

0

0.2

I SO

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500

300

350

400

450

500

T i S e C )

0

FIGURE 4

-0.2 -0.4 -0.6

i

-0.8 1' 0

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I

500

Time(sec)

0.2

F1GUR.E 2 0 Tip Displacement of beamZO-modcl(LQR) 1

-0.2

-0.4

-0.6 I 0

50

100

150

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250

I

Time(sec)

FIGURE 5 -0.6.

'

-0.8 .

The results of the siinulatioiis are depicted in figures 6 through 9 for nioclels with 20 and 30 elements. As in the previous set the performance of the game theoretic controller is superior to that of the linear quadratic controller with injected disturbance. It is important to note that the superior performance of the game theoretic controller is predicted analytically from the results derived earlier. Recalling that the control-disturbance pair (u'(t),w'(t)) is an actual saddle point, one can write

Time(sec)

FIGURE 3

In the second set of of numerical experiments, a much simpler initial condition has been employed while the same worst case disturbance w o ( t ) is applied in all cases. The initial condition has been selected to be

J

ill f

(U0,WO)

which iinplies that

1174

=

sup - J(W) U E V U E W

Tip Displacement of beam30-model(MMXLQR)

Expanding the right hand iiiequality, oiie ol)taiii\

I

01,

But .since the identical %signal u ! " ( t )is employed in both simiilatioiis, this inequality reduces to

In other words, it is exprctecl that the transient response of tlw game theoretic controller should decay more rapidly tliaii tliat of the L Q R controller with th.P injected worst diqturhon,cc.

-0 1 1 0

50

100

150

250

200

300

350

400

450

500

Timeisec)

FIGURE 8

Tip Displacement of benm2l)-model(MMXLQRl 02r

,

I

I

I Tip Displacement of team30-model(LQR)

0 I5

0.08,

01

,

I

0 06

0 05

004

0

0 02

-0 05

0 -0 02

-0 1

-0 04

-0 15

-0.21 -0.25

0

50

-0 06

1

I 100

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41x)

450

.O'O8i -0.1

S(X)

Time1wc I

I

FIGURE F

Time(5ec)

FIGURE 9 Tip Displacement of beam20-modeliLQR) 0.3

CONTROL DESIGN IN THE PRESENCE O F STRUCTURED UNCERTAINTY We now timi to tlistrilmtetl parameter systems with structiired unr-ertainty. Coiisitler the following abstract Cauchy p r o l h i i in a Hilbert s p a r e H : 111.

0.2

0.1

i ( t ) = Az(t) J ( 0 ) = -U".

0

(3.1)

Here B E L( U , H ) and -4is the infinitesimal generator of a strongly contiiiuous semigroup T ( t ) in H . We associate with (3.1) the cost functional

-0.1

-0.2

Jo(u) = -0.3 0

+ Bu(t)

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100

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300

350

400

450

lm { + 4 lCs(t)l',

(RU,

dt,

(3.2)

where C E L ( H . H ) and R is a positive definite selfadjoiiit lx)uii