ntrol of sampled data systems

F.M. AI-Sun ni S . H. AI-Amer

Indexing terms: Sampled data systems, Controller design, Robust stability bounds, Numerical approach

Abstract: The authors present robust stability bounds for sampled data systems. The bounds are derived for the general case of additive perturbation in a system, and the control gain matrices for continuous time systems under discrete state feedback control. They then present a numerical controller design algorithm based on the derived bounds. Examples are used for demonstration.

List of symbols

o(A), d A ) = maximum and minimum singular values of A p(A) = logarithmic norm X(A) = maximum eigenvalue of A

1 Introduction

Physical systems often exhibit some forms of uncer- tainty. The stability of systems with uncertainty has been of considerable interest for the last 15 years. There are three major approaches to represent uncer- tainty. One approach is to express the uncertainty in terms of bounds on the coefficients of the characteristic equation. Kharitonov’s theorem and related work take such an approach [l]. A second approach is to express the uncertainty as frequency dependent additive or multiplicative perturbation on the plant transfer func- tion. H- and p-synthesis are used in the analysis and the controller design of such problems [2]. A third approach is the time domain approach, which is fol- lowed in this paper. The main idea here is to express the true state space matrices as the sum of known nom- inal matrices and unknown additive perturbations. In [3-91 the Lyapunov equation is used to come up with bounds on the perturbation matrices with which the system remains stable.

For continuous time systems, Lyapunov equations have been used to come up with bounds on the struc- tured and unstructured perturbations. A summary of existing literature of robust stability bounds for contin- uous time systems is given in [4]. Analogous results were obtained for discrete time systems [ S , 61. In a sam-

0 IEE, 1998 IEE Proceedings online no. 1998 1848 Paper first received 11th September 1996 and in revised form 14th November 1997 The authors are with the Department of Systems Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

pled data control system a continuous time process is controlled by a discrete time controller. Bounds for robust stability of sampled data systems are more involved than continuous or pure discrete time systems. Bernstein and Hollot [7] proposed a procedure that exploits the exponential-like uncertainty structure in sampled data control systems to obtain conditions for robust stability. Ezzine [3], provided robust stability bounds for both structured and unstructured perturba- tion for open-loop systems. In this paper we extend the work in [3] to handle perturbations for closed-loop sys- tem. Based on the developed robustness bounds, an optimisation-based controller design scheme is formu- lated.

2 Statement of the problem

Consider the linear continuous time model { A , B, C}, k ( t ) = A(t)rc(t) + B(t)u(t)

Y(t) = C( t )X( t ) (1) Assume that the true model in eqn. 1 is represented as:

{A(& WL W)) = {An(t) f bA( t ) ,Bn( t ) -k b B ( t ) , cn(t)> (2)

where An(& B,(t), C,(t) are known nominal matrices and SA, 6, are unknown perturbations of comparable dimensions. The discrete-time equivalent for the system {A,(t), B,(t), C,(t)} is given by:

z ( k + 1) = A n ( k ) z ( k ) + B,(k)u(k)

~ ( k ) C n ( k ) z ( k ) ( 3 ) where

A&) = Q(to + ( k + 1)T, t o + kT)

B,(k) =

B , ( k ) = C n ( k T )

t o + ( k + l ) T

Q(to + ( k + 1)T, s )Bn(s )ds to+kT

(4)

s Y is the state transition matrix and T is the sample period. For time-invariant systems, the above defini- tions reduces to:

A -,&T n -

T B, = 1 eAnSBnds

The process in eqn. 1 is to be controlled by a discrete time state feedback controller:

c, = c,

u(kT) = -Kz (kT) (5) Classical design techniques such as pole placement or LQ design may lead to state feedback gain K that

IEE Psoc.-Control Theory Appl., Vol 14S, No. 2, Masch 1998 236

stabilises the nominal model but does not give any guarantee about the stability of the perturbed model.

In [3], additive perturbation in the A matrix only is allowed and it was possible to express the allowable uncertainty region in terms of bounds on 6, as a func- tion of A,,, B,, and T. In this paper we derive robust stability bounds for the closed-loop system, so both the A and the B matrices may be perturbed and, as expected, the robust stability test is more complicated. Based on the developed tests, a controller design prob- lem is then formulated as an optimisation problem, over the controller gain K, of a function in two varia- bles, namely, a(6,) and @(6A).

3 Preliminaries

In this Section, important results needed throughout this paper are summarised.

3. I Logarithmic norm The logarithmic norm of a matrix A , &A) is defined as:

3.2 Results on robust stability Lemma 1 [3 ] Consider the stable discrete time system:

The perturbed system:

is stable if:

~ ( k + 1) = Az(lc) (7)

~ ( k + 1) = ( A + D ) z ( k )

w d P ) + o(Q)] 1/2

(1 + w).(P)

@ 2 = [ (1 + w).(P) I 1/2

with P = PT > 0, the solution of the Lyapunov equa- tion:

(1 + w)ATPA - P = -Q (9) and U ) > 0 is selected so that the eigenvalues of A lie within a disc of radius (1 + o)-~ '~ .

4 Main results

In this Section we provide robust stability tests for closed-loop sampled data systems with uncertainty in both the A and the B matrices. Given the continuous time nominal model {A, , B,, C,,}, the discrete time equivalent is given by:

Using the discrete state feedback gain K, the nominal closed loop system matrix is given by:

and the true closed loop system matrix is:

where

(A, - BnK) (10)

(A - B K ) (11)

{ A , B ) C)

The difference between the true and nominal models D, is given by:

Using Lemma 1, robust stability is guaranteed if B(D) 5 @. The next theorem uses the above result to derive several bounds on the allowed continuous uncertainty. Theorem 1 Consider the sampled-data system consisting of the continuous time process in eqns. 1 and 2 and the con- trol law (eqn. 5). A sufficient condition for the stability of the closed-loop system for all perturbations 6, and 6, is:

D = (A - B K ) - (An - B,K) (14

Pro08 D from eqn. 12 can be expressed as:

D = e ( A T b + h ~ ) T - &,T

- I T e ( A n + 6 A ) " d ~ ( B n + ~ B ) K

+ lT eAnSdsB ,K (14)

Taking the singular value of both sides and using the property (eqn. 6), we get:

@(D) 5 e P ( A n ) T [ e @ ( 6 A ) T - 1 1 + 1' ,[P1(An)+8(sA)lsds~(B,K)

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Using Lemma 1 and rearranging produces the results in the theorem. 0

The bounds can be simplified for the cases where 6, = 0, or SA = 0 as in the following corollaries. Corollary I : If 6, = 0, the closed loop system (eqn. 11) is stable if:

3(6B) < ~ T @ ( K ) for ,u(A,) = 0 and

( - T e - P ( A y b ) T - T + 2 ( l - e - ' ( x " ) * ) ) C(B,K) p ( A n )

T(1+ e - P ( A n ) T ) S ( K ) +

for p(A,) # 0. Proofi Direct substitution of 6, = 0 in Theorem 1, and

U Corollary 2; If 6, = 0, then the closed loop system (eqn. 11) is stable if:

simplifying, lead to the above results.

for p(A,) f 0. Prooj Using 6, = 0, &A,) = 0, inequality (eqn. 15) simplifies to:

from which the first result is obtained directly. For &A,) # 0, evaluate the integral in eqn. 15, rearrange

0 Now we present possibly a more conservative bound

and solve for B(aA) to get the above result.

yet more appealing for computation purposes. Theorem 2 The closed loop system (eqn. 11) is stable for all pertur- bations SA and 6, satisfying:

e D ( & A ) T ( l + To(B,K) + T @ ( K ) P ( b B ) ) < 1 + Q, + B(BnK) (17)

for ,u(A,) = 0 and

e a ( b A ) T ( l + TC(B,K) + T C ( K ) C ( G ~ ) )

evaluating the integral, and imposing eqn. 8 we get the first result.

Evaluating the integral in eqn. 15 we get: Case 2: &A,) f 0

a ( D ) <eP(An)T+' (GA)T - eP(An)T

e ~ ( A n ) T + a ( 6 ~ ) T - 1

I*(An) + C ( b A ) + a(BnK)

from condition (eqn. 8), we have:

further manipulation gives the results in the theorem. Corollary 3: If 6, = 0, the closed loop system (eqn. 11) is stable if

CP S ( 6 B ) <

( K ) ci (J,' e Ansds)

Proof For 6, = 0, eqn. 14 reduces to T

D = - 1 eAnsdsbBK

Taking the singular value of both sides and imposing condition (eqn. S), we get the above results. Remark 1: If no feedback is used (i.e. K = 0), the results of Theorems 1 and 2 reduce to the result reported in [3]. Observe that, if K = 0, the perturbation in B is not going to affect the stability of the system. Corollary 4: For the case 6, = 0, the bounds in Theo- rem 2 reduce to:

1 for ,u(A,) = 0, and

@(bA)

for p(A,) # 0. Proof Direct substitution of 6, = 0 in Theorem 2 and simplifying, leads to the above result. Remark 2: The bounds in Theorems 1 and 2 and the corollaries can be solved for a(6,) to give inequalities of the form:

Q, + U + bC(B,K) + c C ( 6 ~ ) d + e * ( B , K ) + f b ( b ~ )

with different values of a, b, c, d, e and f for different bounds.

238 IEE Proc.-Control Theory Appl., Vol 145, No. 2, March 1998

5 Controller design

The bounds given in Theorem 1, Theorem 2 and all corollaries define uncertainty regions that are robustly stabilised. Based on the material of Section 4, four con- troller design strategies are proposed. Assuming 6, = 0, one can select the controller that maximises the bound on a(&) given by Corollary 1 or Corollary 3. Two other controllers are obtained by maximising the bounds on (7(6,) with 6, = 0 using Corollaries 2 or 4. In maximising the bounds, stability needs to be assured. Now the controller design problem can be converted to a constrained optimisation problem.

In this Section we develop a controller design scheme in which neither 6, nor 6, are assumed to be zero. Instead, the algorithms maximises the area of the robust stability region in terms of (7(6,) and 8(aB).

Examining the bounds in Theorem 2, one can see that it is possible to express them in the following form:

with S ( 6 B ) < ae-c(6A)T - y (19)

B(6A)-axes. So the area is given by:

for y(A,) z 0, and 1 a = [@ + 1 + Ta(B,K)] Ta (B,K)

for y(A,) = 0, and 1

T a ( K ) y = ~ [1+ Ta(B,K)]

for both cases. Inequality (eqn. 19) defines an area in the 0(6A)-8(6B) plane. This region defines a robust stability region, which can be found by integrating B(6B) from 0 to the point of intersection with the

Table 1: Results for example 1

which results in:

A robust controller can be obtained by maximising the area over all possible controller gain K. The maximisa- tion of the area is highly nonlinear and nonconvex problem, so local optimal solutions may be obtained. To solve the optimisation problem, the ‘constr’ routine in MATLAB is used. The constrained optimisation routine ‘constr’ uses sequential quadratic programming method to find the constrained minimum of a scalar function of several variables [lo].

6 Examples

In this Section we give two examples to illustrate the robust sampled data state-feedback controller design methods developed in this paper. In the two examples, the five control design schemes discussed in Section 5 are obtained. In all cases, the design problem is con- verted to a constrained optimisation problem which are then solved using the ‘constr’ routine.

6.1 Example I Consider the nominal system with A and B given by:

- -

The system is discretised with a sample period T = 0,l. A discrete-time optimal linear quadratic regulator KDLQR is designed with Q = I and R = 1. The MAT- LAB function ‘dlqr’ is used to obtain the controller

Method Optimal gain KO Bound on uncertainty for KO Bound on uncertainty for KDLQR

Corollary 1 [1.1982 1.54251 ~ ( 6 8 ) < 0.18369 a ( 6 ~ ) < 0.1567

Corollary 3 L1.1982 1.54261 a(&) < 0.18866 a(&) < 0.1499

Corollary 2 14.71 12 3.87671 ~ ( C S A ) < 0.46334 a ( 6 ~ ) < 0.3153

Corollary 4 [2.3440 2.48631 a ( 6 ~ ) < 0.26729 a(6A) < 0.201 1

Area [1.6035 1.91121 a(&) < 5.1355e-0.10(6A)- 5.0084 a(&) < 5.0617~0.10(sA)-4.9609

Table 2: Results for example 2

Method Optimal gain KO

r10.2139 1 Corollary 1

L 0 10.658d

Corollary 3 Po.,,, 70.:57J

Corollary 2

[,,’~lg 10.:06d

[,,,,,, 0 10.6580 O 1 [$:it: 10.;34d

Corollary 4

Area

Bound on uncertainty for

(actual bounds)

Bound on uncertainty for KO (actual bounds) KDLQR

No robustness is a (6~) < 2.80783 ( o ( ~ A ) < 8.66) guaranteed

~ ( 6 ~ ) < 2.5729e- o . l D ( s ~ ) - 1.9316 guaranteed

(a (6~) < 1.255) No robustness is

(Area = 0.325)

IEE ProcContro2 Theory Appl., Vol. 145, No. 2, March 1998 239

gain KDLQR = [1.2278 2.20601. The five digital control- lers discussed in Section 5 are designed using KDLQR as an initial guess. The obtained controllers and the corre- sponding bounds are shown in Table 1. The bounds for KDLQR are included for comparison.

6.2 Example 2 Consider the following system which was studied in [8] under continuous-time state feedback. The A and B matrices are given by:

1 0 1 0 A = [ o 21 13= [ o 11

The system is discretised with T = 0.1 as a sampling time. The optimal discrete time linear quadratic regula- tor with Q = R = I is obtained as:

K D L Q R = 2.2534 0 3.8093 “ I This gain is used as an initial guess for the design of the other controllers proposed in Section 5. In Table 2, the controllers and the corresponding bounds are given. In addition, an estimate of the true bounds for the obtained controllers are generated by Monte-Carlo simulation and they are included in the table for com- parison. The last column in the Table shows the bounds when KDLQR is used.

0.8 m 0.7 L \ L

I \ \

actual instablity region - 0.5

0 1 2 3 4 5 6 7 8 9

Fig. 1 - theorem 1

theorem 2 Cao

Comparison of predicted robust stability bounds for KO -~~ _ _ _ _

In Fig. 1, we show a comparison of the guaranteed stability regions provided by Theorems 1 and 2 for the proposed controller. The following observations about the numerical values are discussed here. (i) The bounds have been improved considerably for all designed controllers. (ii) The bounds are conservative. For the schemes based on the area and Corollary 4, we see that no degree of robustness is guaranteed for KDLQR using the proposed bounds, while the actual bound 1.255 for Corollary 4 and the actual area is 0.325. This is also evident from the relative values of the bounds (i.e. the obtained and the actual) for KDLeR gain. Also, when we examine the relative values of the bounds of the optimal gain KO, we see that the bounds on B(6,) are less conservative compared to that of @(SA). (iii) From Fig. 1, the actual area of the robust stability region of the proposed controller is approximately equal to 3.6 which is more than ten times as large as

the area corresponding to the KDLQR. In [9], Wu and Mizukam presented an approach for the design of robust linear state feedback controller for uncertain linear systems. The approach is based on the stabilisa- tion of a nominal system by making use of the Lyapu- nov stability criteria. The result of [8] is a modification of that in [9].

Assume that the bounds PI and p2 of the maximum singular values of 6A and 6B are known. Then the con- tinuous controller:

k U = --BTPz

2 where P is the solution of the ARE:

( A + G Y I ) ~ P + P ( A + a I ) - kPBBTP = -2yQ stabilises the system. And, if we can find a set of scalar parameters (a, k ) which satisfies:

(81 + ~ b l l ~ T P l l ) IlPll < y L n ( Q ) + QL, , (P)

then the above controller stabilises the uncertain system.

Applying this method, the uncertainty which can be accommodated by the optimal linear quadratic control- ler Q = I and R = I for example 2 is given by:

The stability region guaranteed by the continuous-time controller developed by Cao [8] is also shown in Fig. 1.

7 Conclusions

a ( b ~ ) + 8.06825(6~) < 3.55

In this paper stability bounds for sampled data systems were presented. Both A and B are assumed to be uncer- tain. A graphical interpretation of the results is also presented, and a controller design method based on it is developed. The bounds and the controller design schemes were illustrated using two examples. As expected in using the singular values, the bounds are found to be conservative. However, significant improvements of the robustness are obtained by the proposed controller design scheme.

8 Acknowledgment

The authors acknowledge King Fahd University of Petroleum and Minerals for its support.

9 References

1 CHAPELLAT, H., and BHATTACHARYYA, S.P.: ‘A generali- zation of Kharitonov’s theorem: robust stability of interval plants’, IEEE Trans., 1989, AC--3, pp. 306-311

2 DOYLE, J.C.: ‘Advances in multivariable control. Lecture note’,- ONR/Honeywell Workshop, 1984, (Minneapolis)

3 EZZINE, J.: ‘Robust stability bounds for sample data systems’, Int. J. Syst Sei., 1995, 26, (10) pp. 1951-1966

4 YEDAVALLI, R.K.: ‘On measures of stability for linear state space systems with real parameter perturbation: a perspective’ in YEDAVALLI, R.K., and DORATO, P.: ‘Recent advances in robust control’ (IEEE Press, 1990), pp. 109-1 11

5 KOLLA, S.R., YEDAVALLI, R.K., and FARISON, J.B.: ‘Robust stability hounds on time-varying perturbations for state- space models of linear discrete-time systems’, h t . J. Contr., 1989, 26, pp. 113-716

6 YAZ, E.: ‘Deterministic and stochastic robustness measures for discrete systems’, IEEE Trans., 1988, AC--3, pp. 952-955

7 BERNSTEIN, D.S., and HOLLOT, C.V.: ‘Robust stability of sampled data control systems’, Syst. Contr. Lett., 1989, 13, pp. 217-226

8 CAO, D.Q.: ‘Comments on robust stabilization of uncertain lin- ear dynamical systems’, Int. J. Syst. Sei., 1989, 50, pp. 151-159

9 WU, H., and MIZUKAM, K.: ‘Robust stabilization of uncertain linear dynamical systems’, Int. J. Syst. Sei., 1993, 24, pp. 265-216

10 GRACE, A.: ‘Optimization toolbox’ (The Math Works Inc., 1992)

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