Guaranteed cost control of discrete time uncertain systems

Guaranteed cost control of discrete time uncertain systems

GUARANTEED COST CONTROL OF DISCRETE TIME UNCERTAIN... Copyright © 1999 IFAC 14th Triennial World Congress, Beijing~ F.R. 14th World Congress ofIFAC...

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GUARANTEED COST CONTROL OF DISCRETE TIME UNCERTAIN...

Copyright © 1999 IFAC 14th Triennial World Congress,

Beijing~ F.R.

14th World Congress ofIFAC

G-2e-20-5 China

GUARAl'rrEED COST CONTROL OF DISCRETE TIME UNCERTAIN SYSTEMS

R .. Tzoneva

I. Popchev*

Depart111ent ofElectrical Engineering, Peninsula TechnikOll, Modderdant Road, Bellville 7535, South Africa, Email:[email protected] *Jnstitute 0..( Inforrnation Technologies, Acad. G.Bonchev SIr. b1.2, Sofia 1113 Bulgaria, popchev@bgcict. acad. bg J

Abstract. Guaranteed cost control problem is considered and the conventional linear quadratic optimal state feedback design method is applied to find the robust regulator for linear discrete time systems with structured uncertainties in the kind of linear combination in the parameters. The obtained control is a robust optimal linear state feedback \vhich guarantees global asymptotic stability of the closed loop system and

minimizes the maximal perfonnance bound corresponding to a used quadratic criterion. Copyright© 1999 IFAC

Key Words. Discrete-time systems~ Quadratic performance index, Uncertainty, Robust control, Riccati equations, Lyapunov function

1. INTRODUCTION The linear quadratic design method is well known for its attractive feature of robust stability and also for its practical difficulty with the choice of design paralneters. "fhe robustness of obtained regulators is however guaranteed only in tenns of the stability margin and hence \vith respect to unstructured uncertainty. This does not necessary means that the regulator possesses stability robustness to parameter uncertainty as well as such as quadratic stability. In fact it has been sho\vn that LQ regulators may suffer from poor robustness to parameter uncertainties in spite of well-kno\-vn large stability margins. It is also desirable to design a control system that is not only stable but also guarantees an adequate level of perfonnance. From this point a view, it is desirable to extend the LQ method in such a way that stability robustness to parameter uncertainty can also be taken into consideration in the design process. One approach to the robust linear quadratic control problem is the guaranteed cost approach of (Barrnish, 1985; Chang, and Pcng~ 1972; Douglas, and Athans. 1992; Luo, and Van DenBosch, 1993; Neto, et al.~

1992; Petersen, and McFarlane, 1993). In these papers a continuous time linear quadratic optimal control problem is considered and a state feedback controller is designed to minimize a certain upper bound of the quadratic cost functional. The robust control for discrete time systems has received a lack of an attention (Bertsekas, and Rhodes, 1973; Ezzille, 1995; Kabamba, and Hara, 1995~ Lyashevskiy, 1995; Madana, and Zak, 1988; Petersen, at aI., 1993; Savkin, and Petersen, 1995)~ Most Tesearches have been concentrated on the stability analysis and robust control design based on the second method of Lyapunov (Ezzine, 1995; Kabamba, and Hara, 1995; Madana, and Zak, 1988) on the theory of the Dynamic Programming (Bertsekas, and Rhodes, 1973; Lyashevskiy, 1995), or on both (Savkin, and Petersen, 1995). In this paper we consider a guaranteed cost control problem and apply the conventional linear quadratic optimal state feedback design method to find the robust regulator for linear discrete time systems with structured uncertainties in the kind of linear combination of the parameters in both state and control matrices . The main result of

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GUARANTEED COST CONTROL OF DISCRETE TIME UNCERTAIN...

this paper extends the result of (Luo, and Van DenBosch, 1993) to the discrete time case.

14th World Congress ofIFAC

OH I &(k) == 0,

OH I a(k)

= p(k) ,

OH / cp(k + 1) = x(k + 1) ,

2" PROBLEM FORJ.'1ULATION

Ru(k)+B T p(k+l)

The following problem for linear quadratic regulator synthesis is considered~ find the regulator matrix W in such a way that the control u(k) := W(k)x(k) , (1)

=:

0,

Qx(k) + AT p(k + 1) :: p(k) , x(k + 1) = Ax(k) + Bu(k) ,

(8)

the regulator matrix is WJI (k) == -R -18 T (A r )-1 [GIJ{k) - Q],

(9)

minimizes the quadratic cost functional K-I

J

= (1/ l){ll x(K)lr: + L£llx(k)ll~ + l1u(k)I':]}) (2) ~o;..O

and satisfies the constraints x(k + 1):= A(a)x(k) + B(P)u(k)

G u(K) = S., (3)

x(O)

(4)

Xo , q

,.

La,A

l1,.;\(a) =

p

t1B(fJ) =

L,BjB J , ]':=-1

i~1

ai.llllU

sa,

:$

IT

(10)

=:

[A + ~A(a)]x(k) + [B + B(f})]u(k), :::=

where G (k) > 0 is a solution of the Riccati equatjon G;.l(k+l)= A[GII(k)-Q]-lA T -BR-tB r ,

a,.l1ux,P'.min S;~. ~ ~.tl1ax

When there are uncertainties in the state and control matrices the closed loop system can be unstable. It is necessary to account for the uncertainties. One approach is of the guaranteed cost control where the bound values of the uncertainties are used in Riccati equation computing.

4. GUARANTEED COST CONTROL

where x A

E

R">:II ,B

Q 2. 0, Q

E

Rill are state and control vectors,

E R'/'01J/

are the state and control matrices,

RH, U

E

E

R,m: ~ R > 0, R

G.,f3

tnatrices,

E

RI/IX"1

are

the

weight

are the time varying uncertain

j

parameters, \vhich are independent of the system states~ A, ~ B j are the constant matrices determined

The following definition is introduced: Definition 1. Suppose there exists a linear feedback control law u(k) == Wx(k), which determines a number z ~ 0 as the upper bound of the maximal cost, such that J[x(O), u(k), a,PJ < z for all

a

E0, pEr , Then z

is called a guaranteed cost and

by the structure of the uncertainties. Each uncertain paranleter is assumed to be in a bounded range (5).

u is called a guaranteed cost control. To determine the kind of the guaranteed cost contTol

Therefore the uncertain vectors belong to tw"o rectangular shaped closed regions a En E R ~P Er E RI" , which include the origin. It is supposed that all states are available for feedback control.

the following Theorem is proved:

t ,

Theorem. Let G:= G T > 0 be a positive definite matrix that satisfies the difference Riccati equation G-t(k + 1) = A[G(k)-Ql 1 AT _BR-tB T + U, G(K):::S,

(11)

3. NOMINAL SYSTEM SOLUTION

where U is defined as The nominal systenl has the form x(k + 1) = Ax(k)+ Bu(k), x(O)

=: X o

(6)

U==A[G(k)-Q]-I~T

- ABR -lB T

In this case the control synthesis problem is solved on the basis of functional of Hamilton

-

+

~[G(k)-Q]-IAT ~

BR-lAB/" +

+~A[G(k)-Q]-1~7'_~BR-I~BT.,

(12)

After giving an account of (5) the matrix U is: ""-1

I1

= L {(Ii !)[lIx(k)Jf~ + l{u(k)II:] +

u>

~:::O

+ p(k + l)T[Ax(k) + Bu(k)]} ~

q

(7)

> Laj{A[G(k)-Q]-IA; + A.[G(k)-Q]-lA T };=-1

p(k) E R are the conjugate variables. .t\ccording to the necessary and sufficient conditions for optimality

\vhere

II

- t,Bj{BR-'B; +BjR-'B T } ,

I

(13)

j=1

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fJ

if

La,2 A ;[G(k)-Q]-1 A~ - LPj2B)R-IB~,' > 0

in the above inequality, gives

j-

for

)=1

i=l

14th World Congress ofIFAC

xT(k + l)G(k + l)x(k + 1) +xT(k)G(k)x(k)-

any a E Q E R q ~j3 E r E R Then the value of J defined by (2) has an upper bound of the form F



T

[x (k)A(a) +

UT (k)B('p)]G(k

+ l)x(k + 1)-

T

x (k + l)G(k + l)[A(a)x(k) + B(P)u(k)]-

J[x(O), u,a,p} < (1/2)x T (O)G(O)x(O), q

a En E R ,ji E r E R" ,

xT(k)Qx(k)-uT(k)Ru(k) > 0,

(14)

-this can be vlritten in the form where the linear feedback control is detennined by u( k) =- R -1 B T (A4. r ) -1 [G( k) - Q ]x( k) , ( 15)

x t (k)Qx(k) + u r (k )Ru(k) < - ~[xT(k)G(k)x(k)J

which is the guaranteed cost control and where (A r ) -1 exists and T

z == (1 i 2)x (O)G(O)x(O)

-summation of (19) from 0 to K

(16)

1.:-1

ilx (K)II: + L [\lx(k~l~ + Ilu(k)ll:l <

Proof: The proof is based on some algebraic

where G(k) is the positive definite solution of (] 1). Use of(19) gives

tranSf01111atioIl

-pre- and post-multiplication of (11) by x r (k + l)G(k + 1), G(k + l)x(k + 1) yields

2~V ~ X T (k

x r (k + l)G(k + l)A[G(k) - Q]-I ATG(k + l)x(k + 1) +

+ u (k)Ru(k)]

XT

(k)G(k)x(k) <

~ -x (k) {Q + T

As Amin {Q+ (18)

[G(k) - Q]T A -lBR 1B T (AT

)-1

[G(k) - Q]}

= 2a

~

0

(23) is the minimum eigenvalue of the

-adding of

where Amin {.} matrix {.}. it is obtained

± x1" (k + l)G(k + l)A[G(k) - Q]-l AT G(k + l)x(k + 1) and of ± x T (k + l)G(k + l)BR -lBTG(k + l)x(k + 1),

(24)

and substitution of (13), (3) - (5) in (18), gives

and the closed loop system is asymptotically stable.

This completes the proof.

(k + 1)G( k + 1)x( k + 1) -

x r (k + l)G(k + l)... \.[G(k) -

i

[G(k) - Q]T A-1BR -IB T (AT )-I[G(k) - Q]}x(k) .(22)

x" (k + l)G(k + l)BR -'B "'G(k + l)x(k + 1)-

Xt

+ l)G(k + l)x(k + 1) -

- [XT (k)Qx(k)

+ l)G(k + I)G -I (k + l)G(k + l)x(k + 1)-

= 0,

T

(20)

is asymptotically stable.

T

x (O)G(O)x(O) ~

To prove asymptotic stability of the closed loop system (17) the Lyapunov's functional is taken in the form V = (1 f 2)x T (k)G(k)x(k) , (21)

T

x (k + l)G(k + l)UG(k + l)x(k + 1)

with

,1;",.0

= A(a)-B{p)R- Bl'(A )-l[G(k)-Q]}x(k), (17)

X T (k

(19)

G(K) =S gives

is the guaranteed cost. The closed loop system x(k+l)= t

= -26V(k).

QJ- AT (a)G(k + l)x(k + 1)l

+ l)G(k + l)A(a)[G(k) ~ Q]-l ATG(k + l)x(k + 1) +

5. MINIMAX GUARANTEED COST CONTROL

X 7' (k

+ l)G(k + l)A[G(k) - Q]_1 ATG(k + l)x(k + 1)-

X T (k

+ l)G(k + l)BR -lRR -lBrG(k + l)x(k + 1) +

Then the controller is constructed to minirnize the cost bound (16) for the closed loop uncertain system. However the bound depends on the initial condition. This dependence may be removed either by averaging the cost (2) when the initial condition is a random variable or by computing the worst case cost

X T (k

xT(k + I)G(k + l)BR -IB T (fJ)G(k + l)x(k + 1) + X

I'

(k + l)G(k + 1)B(,8)R -IBTG(k + l)x(k + 1) > 0

-substitution of the expressions obtained

from the necessary and sufficient conditions for the op tinlali ty u(k) :;;: -R -'B T G(k + l)x(k + 1) x T (k + l)G(k + l)A ~~I"G(k +

= xT(k)[G(k) - Q]

l)x(k + 1) = (G(k) - Q]x(k)

when the initial condition is unkno\\rn but bounded (Petersen, at al., 1993; VV? ang~ et al., 1987). The problem here is how to choose an upper bound function U to compile the influence of Wlcertainties in the Riccati equation (5). It is of practical interest to find the best U that leads to the minimal value of the worst case of the guaranteed cost or a minimax value of(2).

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Definition 2: Suppose there exists a Jinear feedback control law u+(k) = W·x(k), which determines a number z· ~ 0 ~ as the minimum upper bound of the maximal (worst case) cost, such that J[x(O)) u· (k),a,pl < z· ~ z for all

a

E

n

R q ,p

E

ErE

R'- , where z is the guaranteed

cost of any other constant linear feedback control. 1~hen

z~

~

0 is called a minimax guaranteed cost and u k) is called a minimax guaranteed cost control. To use the worst case cost the spectral norm of the io (

matrix G (Lyashevskiy, 1995) -

!!GII = [A

max

2

(G )f/2 is

used, where ;tm~x(G) is the maximum eigenvalue of matrix and the functional to be minimized is ]

=

T

max x (O)G(O)x(O)

~I;{OH""I

= I}G(O)11

(25)

The obtained control u * (k)

= -R -lD T (AT )-\ [G· (k) -

QJx(k),

lie· (k)" ~ J)G(k)jJ ~

(26)

is the nlinimax guaranteed cost controL The functional (25) is minimized following the solution of this problem for the continuous time case given in (Luo, and Van DenBosch, 1993). '

6.CONCLUSION The: method for existence and calculation of the guaranteed control and of the best upper bound function of the guaranteed cost control is proposed for the discrete time systems. The obtained control leads to a robust optimal linear state feedback that guarantees global asymptotic stability of the closed loop system and minimizes the maximal performance bound corresponding to a used quadratic criterion.

7. ACKNO\VLEDGEMENTS The research was supported partly by Foundation for Research Development - FRD of South Africa under the grant No: 204 2812 and by Bulgarian National Scientific Research Foundation under the grand No: 1-805/98.

REFERENCES Barmish~ B.

(1985). Necessary and sufficient conditions for quadratic stabilizability of an uncertain systems. J. Optimiz. Theory Appl. 46~

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Trans. on AUtOl1t ContrDI, AC-18, No 4 117123. ' Chang S. and T.Peng T. (1972). Adaptive guaranteed cost control of systems with uncertain parameters. IEEE Trans on Autom Control, AC17, No 4, 474-483. Douglas J. and Athans M. (1992). Robust linear quadratic design with respect to parameter

uncertainty. Proc. Alnerican Control Conference 2905-2910, , Ezzine J. (1995). Robust stability bounds for sampled-data systems: The unstrucrnred perturbation case. Prac. of Anxerican Control Conference, 5, 3353-3357. Kabamba P. and Hara S. (1995). Performance robustness analysis of sampled data systems against control input matrix uncertainties. ProG'. of the American Control Conference~ 2, 14911492. Lankaster P. (1969). Theory oj·matrices, Academic Press, New York. Luo J. and Van DenBosch P. (1993). Minimax guaranteed cost control for linear systems with large parameter uncertainties-Riccati equation approach. Proc. of the XII lFAC World Congress, 8,51-54. Lyashevskiy s. (1995). Synthesis of robust controllers for uncertain discrete systems. Proc of the American Control Conference) 3, 19881989. Magana M. and Zak S. (1988). Robust output feedback stabilization of discrete time uncertain dynamical systems. IEEE Trans on Autonl. Control, AC-33, No 11, 1082-1085. Neto A., J. Dion J. and Dugard L. (1992). Robustness bounds for LQ regulators. IEEE Trans. Aut. Control AC-37, 1373-1377. Petersen I. and McFarlane D. (1992). Optimal guaranteed cost control 0 f uncertain linear systems. Proc. of the Alnerican Control Conference, 2929-2930. Petersen I., McFarlane D. and Rotea M. (1993). Optimal guaranteed cost control of discrete time uncertain linear systems. Proc. of the Xl/-th IFAC World Congress, 1,407-410. Savkin A. and Petersen 1. (1995). Optimal guaranteed cost control of discrete time uncertain nonIinear systems. Proc. of the IFAC Conference on

Systems) Structure and Control, 213-218. Wang S., Kuo T., Lin Y., Hsu C. and Juang Y. (1987). Robust control design for linear systems with uncertain parameters. 1nl J Control. 46, No 5,1557-1567.

399-408~

Bertsekas D. and Rhodes I. (1973). Sufficiently informative functions and the minimax feedback control of uncertain dynamic systems. IEEE

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