Design of Sampled-Data Controller Based on Approximation Model of Nonlinear Systems

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DESIGN OF SAMPLED-DATA CONTROLLER BASED ON APPROXIMATION MODEL OF NONLINEAR SYSTEMS Yufan Zheng**' * Zhiroing Wang ** * Department of Electrical and Electronic Engineering,

University of Melbourne, Victoria 3010, Australia ** Department of Mathematics, East China Normal University,

Shanghai 20062, China Abstract: In this work we consider the design of nonlinear sampled-data state feedback controller with a fixed sampling period T to stabilize a continuous-time plant. When one has a sampled-data state feedback controller, which is designed according to an approximate discrete-time model and exponentially stabilizes the approximate discrete-time model, our result gives a computable sufficient condition to check if the sampled-data state feedback controller also exponentially stabilizes the continuoustime plant. Copyright © 2001 [FAC Keywords: Nonlinear control systems, Sampled-data control, Exponentially stable, Lyapunov method. 1. INTRODUCTION

it is assumed that the vector field f(x , u) is analytic on some open connected areas of state and input space with 1(0, 0) = 0. Where x E nn, u E n, the state measurements x( k) := x( kT) are available only at sampling instant kT, k E N+ := {O, 1, 2" , .}, where T(> 0) is sampling period. The control is assumed to be implemented via a sampler and a zero-order holder: u(k) := u(t) = u(kT), "It E [kT, (k + l)T) , k E N+.

There are several approaches to design a digitally implemented controller for a continuoustime plant. One way is to design a controller in continuous-time, then discretize the controller using a sampler and holder. It has been shown in (Chen, 1991) for linear systems and in (Teel, 1998; Kazantzis, 1997) for nonlinear systems, that if the continuous-time controller is stabilizing and the sampling frequency is fast enough, then the closed-loop sampled-data system is also stable. However, in many applications, the actuator bandwidth is limited and for these applications one should design a controller in discrete time. The study of sample-data controller for nonlinear continuous-time control systems is in a very primary stage. Thus, our attention is only paid to some very fundamental problems relevant to the study.

When the sampling period T is given, like linear systems, there exists the exact discrete-time model of nonlinear systems (1). Assume that the f(x,u) is an analytic vector field , then its exact discrete-time model can be formulated by Taylor expansion in terms of Lie derivatives of f [2]. The Taylor expansion is described by infinite series both with respect to the sampling period T and the control u. Therefore, the exact discrete-time model of (1) is in general very difficult to he described concisely. In most cases only approximate discrete-time models ((Kotta2, 1994; Teel, 1998; Kazantzis, 1997; Nesic, 2000; Nesic1, 1999)) are

Given a nonlinear continuous-time plant

x = f(x,u)

(1)

493

Let t = (k + l)T in (2), an approximate discretetime model of (1) with fixed sampling period T is constructed by

available for designing the sampled-date controller of system (1). Consequently the following problem arises naturally. Under what conditions the sampled-data controller, which is designed according to an approximate discrete-time model of system (1) and stabilizes the approximate discrete-time model, can also work well for the continuous-time system (1)?

xa(k + 1)

= xa(k)+

Ti L~1 I(xa(k), u(k»"1

N

L

(4)

t.

i=l

where Xa (k) is the state vector of the approximate discrete-time model at t = kT. It is noticed that when N = 1, (4) leads to the Ewer approximate approximation discrete-time model.

In (Nesic, 2000; Nesicl, 1999) the two controlled models are parameterized with the sampling period T . To describe the accuracy of approximate model a notion called higher order approximation is defined. The accuracy of approximate model is improved by tuning the sampling time T. Under our consideration the accuracy of the approximate model is improved not just by tuning the sampling time T. We will give some conditions to restrain the up-bound of the T under a mathematical consideration, which is relevant to system behaviors. When T is given, the accuracy of approximate model is improved at the expense of constructing more complicated approximate model, for example, to take more terms from Taylor expansion of the exact solution of system (1).

The approximate discrete-time control model is described as xa(k

+ 1)

= tPa«k + l)T,xa(k));u) =Fa(xa(k),u)

(5)

Under a sampled-data state feedback control, u(t) = u(kT) = a(xa(k», 'Vt E [kT, (k+1)T), k E N+, the closed loop approximate discrete-time model of system (5) is described by xa(k + 1) = Fa(xa(k),a(xa(k») =Fa(xa(k))

(6)

3. CONTINUOUS-TIME PROTOTYPE OF DISCRETE-TIME MODEL 2. CONSTRUCTION OF APPROXIMATE DISCRETE-TIME MODEL

Definition 3.1. Given a discrete-time dynamics xd(k + 1) = F(Xd(k)) with sampling period T, a continuous-time dynamics

Consider the nonlinear control system (1) with u being constant over a sampling period, then

x = I(x, t)

formally one can discretize the system as follows :

is called a prototype of the discrete-time dynamics xd(k + 1) = F(Xd(k» if the initial condition of (7) satisfies that x(O) = Xd(O), then its trajectory (solution) x(t) for t 2:: 0 satisfies that x(kT) = xd(k) for each k E N+.

(H1)T

x(k + 1)

= x(k) +

J

(7)

I(x(t),u(k»dt

kT

An approximate solution of (1) starting from x(k) is obtained as follows .

where L,h denotes the Lie derivative of h along I, i.e. L,h = 12':::1 ~ k

It is assumed that the trajectories under consideration are remaining in some open connected subset B., of 'Rn and all controls are taken from a bounded connected subset B.. of 'R. The closure of B." denoted by B"" is bounded, that is B", C B R := {x E 'Rn; IIxll ~ R}. The B", and B.. could be large, but they must be bounded.

When a sampled-data controller u( k) = a( x( k» is applied to (2), the approximate solution starting from x(k) is given by

We give the following assumption on the approximate solution if;a(t,x) = tPa(t,x;a(x». Let T* be the up-bound of sampling time T .

tPa(t, x(k), u(k)) N

= x(k)

+L

.

L}-1 I(x(k), u(k)) (t -

if;a(t,x(k)) = x(k)+ N LL}-1 I(x(k),a(x«k») (t i=1

.~T)'

(2)

~.

i=1

Assumption 1: For each i: E B"" u E B.. if the sampling period T < T*, then there exists a local homomorphism defined on J/6 (i:) x [0, Tj as follows .

.

-.~T)·

(3)

t.

lJIa(x,t) :J/6 (i:) x [O,T] --+ Bx x [O,Tj;

Remark: When the state x is near the origin, if there exists x such that if;a (l, x) = 0 for 0 < l < T, then let if;a(t,x) = 0 for l ~ t ~ T.

(

494

~) ~

( if;a

(!'

x) )

(8)

where N'6(X) := {x E Bz; Ilx neighborhood of i:.

- xII <

4. MAIN THEOREM

&}) is an

We assume that the system (1) is (exponentiaily) stabilizable and called the Exact Continuous-time Model (ECM) of a plant. With sampling time period T the Exact Discrete-time control Model (EDM) of (1) is denoted by

Under Assumption 1 we have the following equivalent statement. Lemma 3.1. For each x E B z , t E [0, T] there exists N'.(x), an neighborhood of x, such that

(13)

(9)

Assumed that the state feedback for k u(t)

defines a local homomorphism. Where ii>; 1 (t, x) is inverse map of ii>1l(t, x) subject that t is a constant.

xe(k

+ 1) = Fe(xll(k),a(xe(k)))

(10)

When the same controller applies to the ECM (1) we have the closed-loop EDM of (1) described by x(t)

Let {(s) = ii>;l(S,X), then the ii>a(t,{(s)) is a trajectory of (10) passing through x at time instance

= r(x(t))(:= f(x(t),a(x(kT)))), + I)T), k ~ O.

By Proposition 3.1, Ia(o, t) = 0 and the compactness of Bz there exists a positive number 11:() > 0 such that for (10) and x E Bz

Recall the formula (3), we have that for x E Bz and t E [O,T)

(t ) ._ dii>ll(t, x) 'I'll ,x.dt

.1,

IIr(x,t)1I ~

+ LLif(x,a(x))

.,

IIxll(k)1I ::::; K;2I1xll (0)lIe-')'kT

Lemma 3.2. Under Assumption 1 assume that (10) is a prototype of (6), then it is valid that for any x E Ba and s ~ 0

01/JIl(t, x) 11 ox

(17)

IT (10) is exponentially stable, then (6) is exponentially stable by the definition. IT (6) is exponentially stable, then there exist positive numbers K;1,), such that any trajectory of (6) satisfies that, for k ~ 1, \fx(k) E B z ,

~.

By (11) we have the following important estimation of 8[~~,t), which plays the role of Lipschitz constant of III (x, t).

< SUP{tE[O,T),zEB.} 11

II:()lIxll

(t _ kT)i (11)

i=l

lIolll(x,s) 11 < L ox -

(16)

Our work concentrates on finding the conditions under which the exponential stability of (6) implies that (16) is also exponentiaily stable.

= s.

= f(x,a(x))

(15)

By Proposition 3.1 there exists a continuous prototype model (10) of (6) and (6) is the Exact Discrete-time Model of (10).

where t E [kT, (k

.

(14)

Correspondingly, (5) and (6) are called the Approximate Discrete-time Model (ADM) and the closed loop ADM of (1), respectively.

In general, the prototype (10) of system (6) does not uniquely exist.

N

t E [kT, (k + I)T)

= Fe(xe(k))

Proposition 3.1. The discrete-time dynamics (6) has a T-period, continuous-time prototype, which is described by

t

0

is continuous, i.e. a{) : Bz -+ Bu is continuous map, with a(O) = o. Under (14) the closed loop exact discrete-time system model of system (1) is described by

Let the approximate discrete-time control model (5) be obtained by approximate solution lPa(t, x; u) of system (1) for t E [0, T). We design a sample data controller u(k) = a(x(k)) under which the closed loop system (6) of (5) is exponentialiy stable. Assume that Assumption 1 holds for the ii>1l(t,X) := lPll(t,x,a(x)), then we have the following proposition.

x = r(x,t)

= a(x(k)),

~

(18)

Proposition 4.1. IT (6) is exponentiaily stable, then (10) is exponentiaily stable. Conventional Converse Theorem of Lyapunov stability ( (Khalil, 1996; Vidyasagar, 1993)) is given for continuously differentiable dynamical systems.

(12)

< 00

495

where

Assume (10) is an exponentially stable prototype. Adopting the notations defined in section 3 we construct a Lyapunov function of (10) for x E B:z;

f

1I
> 0,

is an arbitrary small positive number.

This work is in a very preliminary stage. Some estimation given in this paper can be further modified. Especially, the estimation of the Lipschitz constant of ja(x, t). The estimation of L in Lemma 3.2 is conservative. H the L is larger, then the d is larger. Then lid becomes a small number. Furthermore, it requires the t:.j be small. Certainly it makes the sufficient condition too strong. On the other hand, the calculation of (25) or (26) can be done off-line by numerical computation. However, a better computation method on it should be investigated further.

HT

V(t,x):=

f

(19)

t

where T(> 0) should be chosen properly. By (18) and Lemma (3.2) one has that (see p.150 of (Khalil, 1996))

Ilxlle-L(T-t) :::; II
2L (1- e- 2LT )IIxIl 2 :::; V(t,x) (20)

2

:::; ~ (1 - e- 2'Y(T -t))lIxIl 2

5. REFERENCES Tongwen Chen. Francis, Bruce A. (1991).Inputoutput stability of sampled-data systems. IEEE Trans. Automat. Control 36, no. 1,5058. Kotta, U. (1995).Inversion method in the discrete-time nonlinear control systems synthesis problems, Lecture Notes in Control and Information Sciences 205, Springer. Kotta, U. (1994). Discrete-time models of a nonlinear continuous-time system, Proc. Estonian Acad. Sci. Phys. Math., v.43, 64-78. Tee!, A., D. Nesic and P. Kokotovic, (1998).A note on input-to-state stability of sampleddata nonlinear systems, Proceedings of the 37th IEEE Conference on Decision and Control, Piscataway, NJ, USA, 2473-8. Kazantzis, N., C. Kravaris, (1997). Systemtheoretic properties of smapled-data representations of nonlinear system obtained via Taylor-Lie series, Int. J. Control, 67, 9971020. Nesic, D., A. Tee! and P. Kokotovic, (2000). On the design of a controller based on the discrete-time approximation of nonlinear plant model, Systems and Control Letter, to appear. Nesic, D., A. Teel and P. Kokotovic, (1999). Sufficient conditions for stabilization of sampleddata nonlinear systems via discrete-time approximations, System and Control Letter 38, 259-270. Nesic,D., A. Teel and E. Sontag, Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems, Systems and Control Letters 38, p.49-60. Khalil, Hassan K., (1996). Nonlinear systems, Second Edition, Prentice Hall. Vidyasagar, M. Nonlinear Systems Analysis, Prentice-Hall International Editions.

Along the trajectory x(t) (see (Khalil, 1996)) we have (21)

Furthermore, by (12) TH

av 11 ax 11

= 11

f 2
t

211:2 [1 _ e-b-L)T]lIxll

:::;

,-L

Let , for example,T> In(21t~) and d 21'

e- b -

L)

1.. (2K~) 2.,.

],

= 'Y-L 2!i2...[1 -

then

II~~II :::;dllxll

(23)

- av V· +aV - j-a -<--2111 x 112 at ax

(24)

Now we give the main result of this paper. Theorem 4.1. Given sampled-data feedback controller (14) under which (6) is exponentially stable and (18) holds, then under (14), (16) is exponentially stable if t:.J(t, x(t)) := je(x(t))- r(t, x(t)), where the x(t) indicates the exact trajectory of (16), satisfies that T

f

IIt:.j(s,x(s))lIds

o

<

1 2d

T

f

IIx(s) lids (25)

0

An alternative, but conservative condition is for E [O,T)

8

-

lIt:.j(s,x(s))1I

I-f

< u"x(s)"

(26)

496