Hanyong Shao, Jianrong Zhao, and Dan Zhang

Abstract—This paper is concerned with a novel Lyapunovlike functional approach to the stability of sampled-data systems with variable sampling periods. The Lyapunov-like functional has four striking characters compared to usual ones. First, it is time-dependent. Second, it may be discontinuous. Third, not every term of it is required to be positive definite. Fourth, the Lyapunov functional includes not only the state and the sampled state but also the integral of the state. By using a recently reported inequality to estimate the derivative of this Lyapunov functional, a sampled-interval-dependent stability criterion with reduced conservatism is obtained. The stability criterion is further extended to sampled-data systems with polytopic uncertainties.Finally,three examples are given to illustrate the reduced conservatism of the stability criteria.

I. INTRODUCTION

SAMPLED-DATA systems have received substantial attention over the last two decades due to their wide applications in digital control systems and networked control systems[1]-[7]. Stability of sampled-data systems is an especially interesting topic for many researchers [8]. In the literature there are mainly four approaches to the stability of sampleddata systems. The first is the discrete-time system method,which transforms sampled-data systems into discrete-time systems and then applies the classical system theory to stability analysis [9]. However, the method encounters difficulties for systems with variable sampling periods or uncertainties. The second approach to stability of sampled-data systems is the impulsive system method [10], [11]. As indicated in [10], the method requires the sampled-data system to be represented in the form of an impulsive model, and stability conditions are derived by constructing a time-dependent discontinuous Lyapunov functional.The third method is the input delay approach[12]-[14] by which sampled-data systems are formulated as continuous-time systems with a time-varying delay, and the time-dependent Lyapunov functional method is employed to study the stability of the continuous-time systems [15]-[18].As shown in [10]-[14] the time-dependent Lyapunov functional can lead to a stability condition that determines an upper bound of the time-varying delay, namely the size of the sampling interval.It is well known that both computational burden and data transmission rate of the sampled-data systems are decreased as the sampling interval increases. Therefore,the second and third methods are of significance in obtaining a possibly larger sampling period that ensures the stability of sampled-data systems. However, the Lyapunov functional involved in the latter two methods is too restrictive. The fourth is the Lyapunov-like functional method. It does not involve model transformation. Moreover, the functional is not necessarily positive definite [19]-[22]. Recently stability criteria of sampled-data system were provided by the fourth method in [19]. A further improved stability criterion was obtained in [20] by using a new inequality to estimate the derivative of the Lyapunov-like functional.Very recently those stability results have been extended to sampled-data systems with state quantization [21]. Note that the existing Lyapunovlike functional does not include the integral of the state; there is still room for the functional to improve.

In this paper we further investigate the stability of a sampled-data system with variable sampling periods. Novel sampling-interval-dependent stability criteria are derived by a new Lyapunov-like functional approach that does not involve model transformation.Compared with existing ones,the Lyapunov functional makes use of the integral of the state as well as the sampled state. It is time-dependent, may be discontinuous, and not every term of it is necessarily positive definite. It is illustrated by examples that the stability criteria derived are less conservative than some existing ones.

Throughout this paperIrefers to an identity matrix with appropriate dimensions. For real symmetric matricesXandY, the notationX ＞Y(respectively,X ≥Y) means that the matrixX-Yis positive definite (respectively, positive semidefinite). TheSym(X) stands forX+XT. In symmetric block matrices, we use an asterisk*to represent a term that is induced by symmetry. The smallest and the largest eigenvalues of a real symmetric matrixXare denoted byλmin(X) andλmax(X), respectively.| · |is the Euclidean norm for a vector while‖·‖is the induced matrix norm. We writeMatrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

II. PROBLEM FORMULATION

Consider the linear system

wherex(t)∈Rnis the state,A ∈Rn×n,B ∈Rn×mare known real constant matrices,u(t)∈Rmis the sampled-data inputu(t)=ud(tk),t ∈[tk,tk+1), with sampling instantstksatisfying 0=t0＜t1＜···＜tk ＜···, and

For a state-feedback controller in the form of

the corresponding closed-loop system is

whereAd=BK.

The purpose of this paper is to study the stability problem for system(3)subject to(2),that is,for a givenK,to establish some sampling-interval-dependent stability conditions such that the system is asymptotically stable. In the following we give a lemma and a proposition that play a crucial role in studying the stability problem.

Lemma 1[20]:For a given matrixR ＞0, the following inequality holds for all continuously differentiable functionωin [a,b]→Rn

To study the stability problem mentioned above,the following proposition is also needed.

Proposition 1:Consider the following sampled-data system described by

where the sampling interval satisfies (2),f(0,0)=0, and fory(t),y(tk)∈Rn

whereL1＞0 andL2＞0 are known constants. Forc1＞0,c2＞0 and a solutionx(t) to the system, suppose that there exist a continuous functionalVa(x(t)) and a piecewise continuous functionalsatisfying

1)

2)

3)

Then the trivial solution of system(4)is asymptotically stable.

Proof:From 3) it follows:

Noting thatVa(x(t)) is continuous at sampling instant, it is seen from 2) that

Therefore, from (i) we havex(tk)→0,k →∞.

On the other hand, from the system (4) it follows that fort ∈[tk,tk+1)

Thus

Apply Grownwall-Bellman lemma to obtain

Now it can be concluded that the system (4) is asymptotically stable.

Remark 1:Proposition 1 provides a general stability result for a class of nonlinear systems which covers the system (3)subject to (2) as a special case.

It is noted thatV(x(t),t) is not the same as a usual Lyapunov functional becauseVb(x(t),t)may be discontinuous at sampling instants and it is not required to be positive definite. In the following we referV(x(t),t) to as a kind of Lyapunov-like functional.

III. STABILITY CRITERIA

For system (3) subject to (2), we construct a Lyapunov-like functional on [tk,tk+1) as follows:

where

with

Remark 2:Note that

This implies Lyapunov functional(5)is a 2-order function int,and it includes the integral of the state as well as the sampled state. As seen fromV4(x(t),t),

So the Lyapunov functional may be discontinuous at the sampling instants. In the following we will see that not every term of Lyapunov functional (5) is required to be positive definite when employed to derive the following samplinginterval-dependent stability result.

Theorem 1:For givenandsystem (3) subject to (2) is asymptotically stable if there exist symmetric matricesP ＞0,

Q ＞0,Q1＞0,Q2,Q3,S ∈Rn×nand matricesNα ∈R4n×n,Lα,α= 1,2,3,Mj ∈Rn×n(j= 1,2,...,6), such that for

where

Proof:Firstly, for the Lyapunov functional (5) we have

Therefore,On the other hand,Letc1=λmin(P),c2=λmax(P), and thenThat is to say the Lyapunov functional (5) satisfies 1) and 2)of Proposition 1.

In the following we will show it also satisfies 3) of Proposition 1. Define

Integrating both sides of system (3) subject to (2) leads to

So there existsN3∈R4n×nsuch that

Employing Lemma 1 we have

As per [20] there existN1andN2∈R4n×nsuch that

Using Jensen inequality [18] gives

Now from (8)-(14) it is derived that

where

On the other hand, from (6) and (7) it is concluded that for

By (16), it follows thatW(tk)＜0 andW(tk+1)＜0. SinceW(t) is linear int,W(t)＜0 fort ∈(tk,tk+1). By (15),

Now it is shown under (6) and (7) the Lyapunov functional(5) also satisfies 3) of Proposition 1. By Proposition 1 system(3) subject to (2) is asymptotically stable.

When, we have a sampling-intervaldependent stability result for the periodic sampling case in the following corollary.

Corollary 1:For, the system (3) subject to(2) is asymptotically stable if there exist symmetric matricesand matrices,j= 1,2,...,6 such that (6) and (7) hold.

Remark 3:Theorem 1 and Corollary 1 provide samplinginterval-dependent stability criteria for system (3) subject to(2), by which we can compute the admissible upper bound of sampling intervals that ensures the system to be asymptotically stable. Recently, sampling-interval-dependent stability for system(3)subject to(2)was also studied in[10],[12],[14],and [17]-[21] by employing Lyapunov functional methods.However, as a whole, Lyapunov functionalV(x(t),t) in (5)is different from those in that it is second order with respect to timet, and possibly discontinuous at the sampling points;it involves not only the sampled state but also the integral of the state, and not every term of it is positive definite.Moreover, different from [10],[12],[14] and [17]-[19],[21],when estimatingthis paper employs Lemma 1 and the integral equation (9) to take advantage of the integral of the stateas well as the statex(t). This method is expected to result in less conservative stability results, as illustrated in Section IV.

When system (3) subject to (2) involves polytopic uncertainties, by Theorem 1 we have the stability result stated as follows.

Theorem 2:Assume that the matricesAandAdin system(3) subject to (2) belong to a polytope: [A Ad]∈Θ, Θ =with...,land

Remark 4:It is not difficult to present a stability analysis result using Theorem 1 for sampled-data systems with normbounded parameter uncertainties. Based on Theorem 1 one can also consider the stabilization problem, which is omitted here, given that the objective of this paper is to focus on the stability problem.

IV. NUMERICAL EXAMPLES

In this section,we give three examples to show the reduced conservatism of our stability criteria.

Example 1:Consider the sampled-data control system in Fig.1.

The physical plant is given bywith

and the controller is given byu(t) =Kx(t) withK=[3.75 11.5].Since the controller is connected with the physical plant via the sampler and the zero-order hold ZOH,the closedloop system is in a form of (3) with

which was employed in [10],[12],[14],[17],[19] and [21].

Firstly we intend to find the admissible upper boundhon the periodic sampling, which guarantees the asymptotic stability of the system. Using Corollary 1 in this paper and some existing stability results we can compute the admissible upper bound, which are listed in Table I.

Methods [10] [14] [19] [21] Corollary 1 h 1.32 1.69 1.723 1.7239 1.7294

From the table above, it is seen that the stability result(Corollary 1) in this paper can provide a larger admissible upper boundhon periodic sampling than the corresponding ones in [10],[14],[19] and [21]. In this sense the stability result(Corollary 1)in this paper is less conservative,compared with those in [10],[14],[19] and [21].

In this example, choose the initial conditionx0=[2-1.8]and the periodh= 1.7294, and then the state responses of the system can be obtained as in Fig.2.

As shown by the Fig.2, the state trajectory of the closedloop system converges to zero.

Secondly for the case of variable sampling, with the lower boundwe attempt to compute the admissible upper boundwhich is given in Table II.

Methods [17] [19] [12] Theorem 1 h 1.36 1.721 1.723 1.729

As shown from the table, for the variable sampling case,the admissible upper boundin this paper is also larger than the ones in [12],[17] and [19]. Therefore, the stability result Theorem 1 in this paper is less conservative than those in[12],[17] and [19].

Example 2:Consider a 3rd order system described by (3)with

The objective is to fnid admissible upper boundfor given lower bound= 0 on the variable sampling such that the closed-loop system is stable. By the methods in [19],[20] and this paper, the comparison results are given in Table III.

Methods [19] [20] Theorem 1 h 1.9637 2.3724 3.0887

It is clearly shown our method has the least conservatism.

Example 3:Consider the uncertain system that was employed in [10],[19],[23] and [24], with parameters

where|g1| ≤0.1 and|g2| ≤0.3. Obviously the system can be formulated as one with parameters from a polytope,the vertices of which are

To guarantee the asymptotic stability of the uncertain system, by the stability result Theorem 2 and those in[10],[19],[23] and [24], we can find the admissible upper boundhon the periodic sampling in Table IV.

Methods [10] [19] [23] [24] Theorem 2 h 0.4610 0.6674 0.7255 0.7310 0.7354

It is obvious that the stability result Theorem 2 is less conservative than those in [10],[19],[23] and [24].

V. CONCLUSION

The stability of sampled-data systems with variable sampling periods has been investigated by constructing a new Lyapunov-like functional. Compared to existing ones the Lyapunov functional is more generalized in the sense of being second order with respect to time t, possibly discontinuous at the sampling instants, including the integral of the state as well as the sampled state. Moreover, not every term of it is required to be positive definite. Some new samplinginterval-dependent stability criteria have been obtained for the sampled-data systems with or without uncertainties.It has been illustrated that the stability criteria are less conservative than some existing ones.