Unmatched Uncertainties in Robust LQ Control

Copyright@ IFAC Robust Control Design, Prague, Czech Republic, 2000

UNMATCHED UNCERTAINTIES IN ROBUST LQ CONTROL DuSan Krokavec, Anna Filasova

Technical University of Kosice, Faculty of Electrical Engineering and Informatics Department of Cybernetics and Artificial Intelligence Letna 91B, 04£ 00 Kosice, Slovak Republic e-mail: [email protected]@ccsun.tuke.sk

Abstract: In this note a robustification procedure for the LQ state feedback design is presented. A quadratic performance index together with a set of criterions for choosing the state and input weighting matrices according to the system unmatched uncertainties as well as system input uncertainties are considered. The preferred approach translates the robust control problem into an optimal control with emphasis on selection of cost function weighting matrices. The solution of that robust control problem is a solution of the optimal control problem where the unmatched uncertainties are compensated and additionally, the stability of the closed-loop system is guaranteed by the satisfaction of an appropriate Lyapunov function . Copyright ©£OOO IFAC Keywords: Robust control, uncertain linear systems, LQR control methods.

1. INTRODUCTION

trollers accepts some information about system parameter uncertainties for properly choosing of performance index weighting matrices.

For linear systems, the optimal control problem reduces to a linear quadratic regulator (LQ) problem, whose solution can be obtained by solving an algebraic Riccati equation. It is well-known fact that linear quadratic optimal control yields a stable closed-loop system and the minimal value of the performance index. The disadvantage, however, is that the performance of this controllers is degraded in the presence of system parameter deviations.

In the last years many significant results have spurred interest in problem of robust control. It was shown that robust stabilization can be achieved by a state feedback based on a properly chosen Lyapunov function, which is independent of uncertainties, if the so-called matching condition is satisfied. The matching condition requires that the uncertainty in the nominal state space model be in the range of the nominal input matrix; e.g. see (Neto, at al., 199£) and references therein. This approach was generalized in (Amato, at al., 1996), (Barmish, at al., 1983), (Petersen and Hollot, 1986), (Zak, 1990). If the matching condition is not satisfied an augmented control is used to deal with the unmatched uncertainty (Lin, at al., 1992).

While a given model leads to a controller design, such a design must possess a sufficient degree of robustness with respect to the plant uncertainty. Once the uncertainties are characterized in some mathematical form, they can be used to analyze properties of controllers designed using nominal plant models but applied to plants with uncertainties and to synthesize "robust" controller. One approach to the problem of finding the robust con-

The approach, to be presented in this paper, translates the robust control problem into an op-

329

timal control where the emphasis is on the recovery of the un-nominal closed-loop performance which is determined by the selection of a cost function weighting matrices. The solution to the optimal control problem is then a solution to the robust control problem if the matching condition is not fullfiled and additionally, the stability of the closed-loop system with parameter uncertainties in the system matrix as well as in the input matrix is guaranteed by the satisfaction of an appropriate Lyapunov function. An interesting point is that presented robust LQ control has the essential structure of a standard LQ controller. An example is presented to demonstrate the role of the bounds in the optimization procedure.

where

Vet) = Bd: ~A(t).

(9)

With the control laws

= -K(t)x(t)

(10)

vet) = -L(t)x(t)

(11)

u(t)

the optimal control design task is, in general, for a nominally stabilizable system described by (8) , (2) to determine such (10), (11) that minimize the quadratic cost function JT

!

= XT(T)Q-X(T)+

T

2. PROBLEM STATEMENT

+

where T is finite, Q E IRnxn and Q- E JRnxn are symmetric positive semi definite matrices, R.. E JRrxr, Rv E IRPxp are symmetric positive definite matrices, and K(i) E JRnxr, L(i) E JRnxp are the control gain matrices.

x(t) = (A + ~A(t»x(t) + B(D + ~D(t»u(t) (1)

with system measurements given by

(2)

where x(t) E JRn, u(t) E JR r , y(y) E IR m , matrices A E IR nxn , B E IRnxr, C E IRmxn are finite valued, and ~A(t), ~B(t) are unknown matrices which represent time-varying parametric uncertainties. It is assumed that considered uncertainty matrices to be of the form ~A(t) ~D(t)

= A(p(t» - A

(3)

= D(p(t» - D = DW(t)

(4)

(12)

o

The system under consideration is an uncertain continuous-time linear dynamic system

yet) = Cx(t)

(x T (t)Q~(~): uT(t)R..u(t)+

Making the assumption that V(x(t»

= xT(t)P(t)x(t)

(13)

is a Lyapunov function for (8), where pet) E JRnxn is a symmetric positive definite matrix, the Lyapunov function derivative is given by V(x(t» = xT(t)P(t)x(t)+ +xT (t)p(t)x(t) + x T (t)P(t)x(t)

where

(14)

I

Wet) = D-1D(t) - I ~ 0

and the Lyapunov function at the time point yields

(5)

= XT(T)P(T)X(T)

T

and pet) is an uncertain parameter vector.

V(X(T»

In order to solve the robust control problem the uncertainty ~A(t) may be decomposed into the sum of a matched component and an unmatched component using projection of ~A(t) into the range of Bd, that is

Assuming that the control laws (10), (11) are applied to the system of (8) another form of the Lyapunov function at T is

~A(t) = BdBd: ~A(t)

V(X(T»=! V(X(t»dt=! xT(t)JtI(t»x(t)dt (16)

+ (I -

T

BdBd:)~A(t) (6)

o

where

T

0

where

(7)

Jv(t) = P(t)+ +[A - BdK(t) - r-l(1 - BdBd:)L(t)+ +BdV(t) - BdW(t)K(t)+ +(1 - BdBd: )~A(t)lTp(t)+ (17) +P(t)[A - BdK(t) - r-l(1 - BdBd:)L(t)+ +BdV(t) - BdW(t)K(t)+ +(1 - BdBd: )~A(t)l

is the Penrose inverse of matrix Bd. The system augmented with symbolical additional inputs is given by x(t) = Ax(t) + r-1(1 - BdBd:)v(t)+ +BdU(t) + BdV(t)X(t) + BdW(t)U(t)+ +(1 - BdBd:)~A(t)x(t)

- xT(O)P(O)x(O) (15)

(8)

330

and the performance index (12) is given by

J

yields

J r = XT(T)Q·X(T)

+

= R~1Brp(t)

K(t)

T

xT(t)Jj(t)x(t)dt

(18)

L(t)

= r-lR;l(I -

BdBd})Tp(t)

(28) (29)

o By applying the identity rule

where

XTy

+ yTX

~ XTX

+ yTy

(30)

first uncertain part of (25) is bounded by conditions

At all time points one may add (16) to and subtract (15) from (18) to get r

= xT(O)p(O)x(O) +

JT

J

xT(t)J(t)x(t)dt, (20)

o

where P(T) = Q-

(21)

J(t) = Jj(t) + Jv(t) = P(t)+ +Q + KT(t)RuK(t) + LT(t)Rv(t)L(t)+ +[A - BdK(t) - r- 1(I - BdBd; )L(t)+ +BdV(t) - BdW(t)K(t)+ (22) +(1 - Bc/Bd;).6.A(t)fp(t)+ +P(t)[A - BdK(t) - r-1(1 - BdBd;)L(t)+ +BdV(t) - BdW(t)K(t)+ +(1 - Bc/B d;).6.A(t)]

However, since matrix Ruk is positive definite, sufficient condition for matrix Qv is Qv=

=

+ Ruw

p

Boundary condition of the second uncertain part of (25) is given by

~P(t)(1 - B dB d;)R;!R1.6.A(t)+

Now assume that new weighting matrices Qc, Ruk can actually be written as

Ruk = Ru

inf{Q~ : Q~~ .6.AT(t)BdpTRukBd; .6.A(t)} (33)

r

1

1

+~.6.AT(t)RJR;2 (I - BB;lfp(t) ~

r ~ r2 .6.AT (t)Rv.6.A(t)+ (34) +r- 2p (t) (1- BdB )R;l (I - BdB d;) Tp( t) ~ ~ Qn

d;

(24)

and note, that if the J(t) has been positive semidefinite, then

and

J(t) = = P(t) + ATp(t) + P(t)A + Qc-

Sufficient condition to select the matrix Qn is

-P(t)BdK(t) - P(t)r- 1 (I - BdBd; )L(t)-P(t)B dW(t)K(t)-K T (t)W T (t)Brp(t)_KT (t)RuwK(t)+ (25) +KT(t)(RukK(t) - Brp(t))+ +LT(t)(RvL(t) - r-1(1 - BdBd;fp(t))-Qv + P(t)Bc/V(t) + VT(t)Brp(t)-Qn + P(t)(1 - BB;l).6.A(t)-.6.A T (t)(1 - BB;l )Tp(t) ~ 0

Qn=

= inf{Q~ : Q~ ~ r2.6.AT(t)Rv.6.A(t)}

(36)

p

since matrix Rv is positive definite. Applying inequality (30) on the last uncertainty part of (25) implies _KT(t)RuwK(t)1.

where Qc, Qn, Qv are positive semidefinite matrices and Rub Ruw are positive definite matrices.

_l.

-(P(t)B dR.1Ru 2W(t)K(t)+

+KT(t)WT(t)R!R;~Brp(t)) ~

(37)

~ _KT(t)RuwK(t) + (P(t)BdR;;-lBrp(t)+

+KT(t)WT(t)Ru W(t)K(t)) = = KT(t)(Ru-Ruw+WT(t)Ru W(t))K(t)

3. QUADRATIC BOUNDS Defining that 0= KT(t)(RukK(t) - Brp(t))

and Ruw can be selected as

(26)

Ruw = = inf{R:w: R:w ~WT (t)Ru W(t) + Ru} p

0= LT(t)(RvL(t) - r-l(1 - BdBd;fp(t)) (27)

331

(38)

By noting that

Substituting (8) into (44) yields

0= pet) + ATp(t) + P(t)A + Qc-P(t)BdK(t) - r-1P(t)(1 _ BdBd;)L(t)

2 T(t)"R (39)

U

8V(x(t)) &x(t) _

.L"Uk

= 2uT (t)"O

the Riccati equation can be expressed as a function of the nominal system parameters only

+ &x(t) &u(t)+ 8V(x(t)) B = 0 8x(t)

.L"Uk

(46)

d

and (8) into (45) yields (40)

2vT (t)Rv

+ 8V(x(t)) &x(t)

=

&x(t)

&vet) (47) =2vT(t)Rv + 8V(x(t))r-~I_B B- 1 ) = 0 &x(t) d dp

and in steady-state the algebraic Riccati equation is given by 0= -Qc - ATp - PA + p(Bd R ;:-lBI + +r- 2 (1 - BdBd;)R~l(1 - BdBd;f)P.

The Hamilton - Jacobi - Bellman equation gives

(41)

min(xT (t)Qcx(t) u{t)

+v

4. CLOSED-LOOP SYSTEM STABILITY

T()"O ()' ;, 8V(x(t)) dx(t)) _ t .L...,U t + " &x(t) dt -

8V(x(t)) 8x(t) (Ax(t)

+r- (1 - BdBd;)v(t)) = = _xT (t)Qcx(t) - x T (t)KTRukKx(t)-

Solving for V(x(t)) to system (8) without additional inputs at an un-nominal condition one can obtain

Proof : The minimal value of the cost function of the optimal control for the nominal system parameters can be used as a Lyapunov function of the closed-loop system, i.e.

V(x(t))

u{t)

T

+(1 - BdBdp1)LlA(t)x(t))-xT(t)Qcx(t) - xT(t)KTRukKx(t)_xT (t)LTRvLx(t)-

(42)

Since matrices Q c a Q* are symmetric positive semidefinite and R tL , RtL are symmetric positive definite matrices immediately follows that

(43)

+ uT(t)RukU(t)+ (44)

uU t

+v

T()R t

8~~~;;)),r-l(1 -

(50)

BdBd;)V(t)

and substitution (46), (47) into (50) yields

Pontryagin minimum principle implies, if there is no bounds on u(t) and vet), that minimizing u(t) and vet) must be such, that ) (xT(t)Qcx(t)

= 8V(x(t)) x(t) =

&x(t) 8V(x(t)) = 8x(t) (BdV(t)X(t) + BdW(t)U(t)+

V(x(t)) = min{xT(r)Q*x(r)+

~ 8(

(49)

_xT (t)LTRvLx(t).

when r > O.

V(x(t)) { > 0, for all x(t) :j:. 0, = 0, for x(t) = O.

(48)

+ BdU(t)+

1

>0

+ /(xT(t)Q cX(t) + uT(t)Ruu(t)+ o +v T (t)Rv v(t))dt}

o.

and for the optimal control at the nominal system parameters yields

Theorem 1. The LQ control of the uncertain system (8) with the control law (10) and the cost function (1 2) is asymptotically stable for all positive definite matrices Ru and Rv such that (33), (36) and (38) are satisfied and for positive semidefinite matrix Q such that Q - 2LTRvL

+ uT (t)Ruku(t)+

() 8V(x(t)) dx(t)) - 0 vv t + !:l ( ) , ux t dt

8

-;:;--() (xT(t)Qcx(t) + uT(t)RukU(t)+ uv t (45) +vT(t)R vet) + 8V(x(t)) dx(t)) = o. v &x(t) dt

V(x(t)) = -2uT (t)Ruk V(t)x(t)-2uT (t)Ruk W(t)u(t)T -2v (t)RvrLlA(t)x(t) - xT(t)Qcx(t)-xT(t)KTRukKx(t)_xT (t)LTRvLx(t)+ +2vT (t)Rvv(t)

(51)

V(x(t)) = 2xT(t)K T Ruk V(t)x(t)+ +2xT (t)LTRvrLlA(t)x(t)-xT(t)Qcx(t) - xT(t)KTRukKx(t)_xT (t)LTRvLx(t)_2xT(t)KT Ruk W(t)Kx(t)+ +2xT (t)LTRvLx(t)

(52)

respectively.

332

To use presented technique let D = 1, W(t) 0.0693s(t), s(t) E< 0,1 >, Bd = B and

Using (32) , (35) one can express elements of in terms of the following set of inequalities

V(x (t))

_xT (t)KTRukKx(t)+ +2xT (t)KTRuk V(t)x(t) =

Bdp

= (B T B)-IB T = [ -0.0013

=

0 0.0035]

1

= -xT(t)(R~k(K -

V(t)fx

1

X(R~k(K - V(t))x(t)+ +xT(t)VT(t)Ruk V(t)x(t) :::; :::; XT (t)VT(t)RukV(t)X(t) :::; xT(t)Qvx(t)

Using r = 0.1, Ru = 5 the additive weighting matrices are

(53)

0.0002 0.0005 -0.0081] 0.0005 0.0010 -0.0172 .10- 3 [ -0.0001 -0.0172 0.2903

=

Qv

_XT (t)LTRvLx(t)+ +2xT(t)LTRvrAA(t)x(t) =

= _xT (t)(R! (L -

rAA(t)f x

1

=

Qn

(54)

x (RJ (L - rAA(t))x(t)+ +xT (t)r 2 AAT (t)RvAA(t)x(t) :::; :::; xT(t )r2AAT(t )RvAA(t) :::; xT(t)Qnx(t)

Ruw = 5.0240 and for Rv = 1

In view of (23) , (53 ), (54) , the Lyapunov function derivative is given by V(x(t) ) :::; -xT(t )Q cx(t) + xT(t)Qvx(t)+ +xT(t)Qnx(t) + 2xT(t)LTRuLx(t)_2xT(t)KTRuk W(t)Kx(t) :::; :::; -xT(t)(Q - 2LTRvL)x(t) .

0.0004 0.0005 -0.0100] 0.0005 0.0013 -0.0204 [ -0.0100 -0.0204 0.3495

RI

= BR~1BT =

(55)

R2 =

r-

0.8341 0 -2.2805]

0

0

[ -2.2805 0

0

.103

6.2350

2(I - BBi;)R;I(I - BBi;f =

352.7209320.4607232.2602]

= [ 320.4607 300.0000 244.0597

The term (55) be certainly negative, i.e.

1/ (x(t))

<0

232.2602 244.0597 276.3199

(56)

With Q = I, S = 0 the lliccati equation (41) has positive definite solution

if matrix (Q - 2LTRvL) be positive definite, that is (57)

P

Therefore, closed-loop system is stable for all admissible uncertainties and control law is a solution to the robust LQ control.

Then, from (28), optimal robust control law is u(t) = -Kx(t) with K

5. ILLUSTRATIVE EXAMPLE The dynamic system of a fighter aircraft in a longitudal mode (Schmitendorf, 1988) for defined range of operating condition is given by the equation (1), (3) and (4) where

A =

B

=

=

=

-0.0603 0.0361]

aAA(t),bAB(t);

a,b= -1,0,1

are given in Table 1.

-0.1645 -0.3500 5.9050] 0.0914 -0.0963 -0.2900 [ o 0 0

-91.4400] 0 , AB(t) [ 250.0000

= [ -0.0306

The eigenvalues of the nominal closed-loop system are Al = -2.6795x10 2 , >'2,3, = (-0.0111 ± 0.0l59i)x10 2 and the variances of the dominant eigenvalue for

-0.8251 17.7600 90.2450] 0.1734 -0.7549 -11.1000 [ o 0 -250.0000

AA(t)

0.0449 -0.0051 0.0152] 0.9800 -0.0043 0.0152 -0.0043 0.0081

= [ -0.0051

Table 1. Dominant Eigenvalue Variances (x 10- 2 )

I -0.0116+±O0.0102i

[-6.3400] 0 17.3250

-0 -0.0105 ± 0.0200i

0-0.0109 ± 0.0147i

Autonomous nominal system is unstable with system matrix eigenvalues >'1 = -2.5452, A2 = 0.9652, >'3 = -250 .

I 333

-0.0108-; 0.0209i

-0.0103 -,; 0.0192i

I

ACKNOWLEDGEMENTS

For LQ control given by 00

J = j(XT(t)Q cX(t)

The work presented in this paper was supported by Grant Agency of Ministry of Education and Academy of Science of Slovak Republic VEGA under Grant No. 1/6270/99.

+ uT(t)Ruku(t))dt

o

the solution of the algebraic Riccati equation and the gain matrix are

7. REFERENCES

0.1733 0.2926 0.0494] P = 0.2926 1.1271 0.0587 [ 0.0494 0.0587 0.0179

Amato, F. , A. Pironti, and S. Scala (1996) . Necessary and sufficient conditions for quadratic stability and stabilizability of uncertain linear time-varying systems, IEEE Trans. Automat. Contr., 41, pp. 125-128. Anderson, B.D.O. and J.B. Moore (1990). Optimal Control. Linear Quadratic Methods. Prentice Hall, Englewood Cliffs. Barmish, B.R. , M~ Corless and G. Leitmann (1983). A new ~lass of stabilizing controllers for uncertain dynamical systems, SIAM J. Control Optim., 21, pp. 246-255. Filasova, A. (1997). Robust controller design for large-scale uncertain dynamic systems. In: Preprints of the 2nd IFAC Workshop on New Trends in Design of Control Systems (Eds. Koz6.k, S., Huba, M.) . Smolenice, Slovak Republic, pp. 427-432. Green, M. and D.J.N. Limebeer (1995). Linear Robust Control, Prentice Hall, Englewood Cliffs. Jabbari, F. and W .E . Schmitendorf (1990). A noniterative method for the design of linear robust controllers, IEEE Trans. Automat. Contr., 35, pp. 954-957. Krokavec, D. (1998). Robust state estimation for structured noise uncertainty. In: Proceedings of the 12th Conference Process Control '99 (Eds . Mikles j J., Dvoran, J., Krejci, S., Fikar, M.). Tatranske Matliare, Slovak Republic, pp. 235-239. Krokavec, D. and A. Filasova (1999). Optimal Stochastic Systems. Elfa, Kosice (in Slovak) . Lin, F., R. Brandt and J. Sun (1992). Robust control of nonlinear systems: Compensating for uncertainty, Int. J. Control, 56, pp. 14531459. Neto, A.T ., J .M. Dion and L. Dugard (1992). Robustness bounds for LQ regulators. IEEE Trans. Automat. Contr., 37, pp. 1373-1377. Petersen, I.R. and C.V. Hollot (1986). A Riccati equation approach to the stabilization of uncertain linear systems. Automatica, 22, pp. 397-411. Schmitendorf, W.E. (1988). Designing stabilizing controllers for uncertain systems using Riccati equation approach. IEEE Trans. Automat. Contr., 33, pp. 376-379. Zak, S.H. (1990). On the stabilization and observation of nonlinear uncertain dynamic systems, IEEE Trans . Automat. Contr., 35, pp. 604-607.

K = [ -0 .2328 -0.8026 -0.0024 ]

The eigenvalues of the nominal closed-loop system are .Ai = -2 .621Ox10 2 , .A 2,3 = (-0.0509 ± 0.0479i) x 10 2 and t he variances of the dominant eigenvalue are given in Table 2. Table 2. Dominant Eigenvalue Variances ( x 10- 2 ) +0 -0 .0467 ± 0.0503i

0-

0

I _0.0537 :

-0 -0 .0552 ± 0.0438i

0. 0483 i

-0.0480 ± 0.0473i

+-0.0442 ± 0.0491i -0 .0520 ± 0.0439i

6. CONCLUDING REMARKS

The note presents an algorithm to solve the LQ control for systems with parameter uncertainties in the system matrix as well as in the input matrix. The exposed problems was reduced to a standard formul ation but with the necessary modifications of th e performance index weighting matrices where the robustness of the closed-loop system is given in t he dependency on matrices Q , R and the stabilit y of the closed-loop system is guaranteed by the satisfaction of an appropriate Lyapunov functi on. This results in a performance (state response, unit step response) which is close to the performance of the nominal system. The emphasis here was on a state output feedback for linear systems with uncertainties if the matching conditions were not fullfiled. Presented applications can be considered as a task concerned the class of quadratic optimization problems where the optimal and sub-optimal solutions were formulated and the principle was demonstrated on one illustrative example. Although the interest is usually in the state transition system matrix uncertainty, here a unified approach for both system and input matrix uncertainties has been presented for the design of LQ controller which is capable of working with uncertainty at the system input as well.

334