#### Abstract

A novel decentralized adaptive integral sliding mode control law is proposed for a class of nonlinear uncertain large-scale systems subject to quantization mismatch between quantizer sensitivity parameters. Firstly, by applying linear matrix inequality techniques, integral-type sliding surface functions are derived for ensuring the stability of the whole sliding mode dynamics and obtaining the prescribed bounded gain performance. Secondly, the decentralized adaptive sliding mode control law is developed to ensure the reachability of the sliding manifolds in the presence of quantization mismatch, interconnected model uncertainties, and external disturbances. Finally, an example is shown to verify the validity of theoretical results.

#### 1. Introduction

It is well known that quantization phenomena are frequently encountered in numerous practical engineering systems; as a result, the control design considering signal quantization has received much attention since 1998. Numerous well-known results have been issued, such as stability analysis of linear and nonlinear continuous-time systems [1, 2], distributed coordination of multiagent systems [3], event-triggered control, or stabilization for networked control systems [4–6], filter design for linear continuous-time systems [7], robust or fault-tolerant control using sliding mode control technique [8–12], and control of continuous-time linear systems [13] subject to input quantization and matched external disturbances.

In this paper, the input quantization problem will be considered. Though similar discussions are also investigated in existing results, such as in [13–15], it should be pointed out that the research works there and in [1, 2, 4, 7, 16] are all built on the assumption that quantizer sensitivity parameters, which are cultivated separately at the coder and decoder sides, are identical all the time or the ratio of them keeps unchanged [17–19]. This assumption actually requires that the adjustment synchronization of sensitivity parameters is enforced at every time step in practical engineering applications; thus it might not be implementably induced by hardware imperfections of coder and decoder. For coping with this problem, a time-varying ratio model with known lower and upper boundaries is first introduced in [20]. By utilizing robust control technique, the control design for a class of uncertain systems subject to input quantization mismatch is investigated there.

On the other hand, the large-scale systems are widely used in modern industrial systems, and thus decentralized control design has been well studied in the past two decades; see [21–23] and the references therein. In [21, 22], based on sliding mode control technique, the decentralized output feedback of nonlinear interconnected systems is well investigated. Specially, since the large-scale systems usually are linked via networks, the decentralized control involving signal quantization has aroused the attention of scholars [24, 25]. However, to the best of our knowledge, no results have considered the decentralized control design via sliding mode control technique for large-scale systems subject to input quantization mismatch.

Motivated by all the mentioned above, we will address the problems of quantized feedback decentralized adaptive integral sliding mode control design for a class of nonlinear uncertain large-scale systems with input quantization mismatch. The purpose is to contribute to the development of quantized feedback sliding mode control theory and the bounded gain performance analysis for nonlinear large-scale uncertain system. The main contribution includes two aspects. First, applying linear matrix inequality technique, sufficient conditions are derived for guaranteeing the robust stability of sliding mode dynamics with bounded gain performance. Second, considering the established* boundary-unknown* time varying ratio relation model of the quantization sensitivity parameters, the decentralized adaptive integral sliding mode control strategy is proposed to eliminate the effect of the mismatch of the quantization sensitivity parameters, model uncertainties, and external disturbances.

This paper is organized as follows. Section 2 provides a system description, relation model for the mismatch of the quantization sensitivity parameters, and some preliminary results. The main results are presented in Section 3. In Section 4, an example is given to illustrate the results and this paper is concluded in Section 5 finally.

The following notations are used in this paper. denotes the -dimensional Euclidean space; denotes the transpose of matrix ; and and represent the identity matrix and a zero matrix in appropriate dimension, respectively. () means that is real symmetric and positive definite (semipositive definite). The symbol represents . denotes the -norm of the vector ; that is, . When , . Specially, the notation denotes the standard Euclidean norm of a vector, or the induced norm of a matrix, respectively. In symmetric block matrices, we use a notation to represent a term that is inferred by symmetry.

#### 2. Problem Statement and Preliminaries

In this paper, the following class of nonlinear uncertain large-scale systems with interconnected subsystems is considered:where , , , and are the state, the control input, quantized control input, and controlled output of the th subsystem, respectively; . , belonging to , is the exogenous input of the th subsystem. is the unknown mismatched time varying model uncertainty in the th subsystem. describes the nonlinear interconnected term affecting the th subsystem, where and is uncertain system parameter. Some assumptions are made in the following.

*Assumption 1. *Time varying uncertainties satisfywhere and are known real constant matrices with appropriate dimensions, is an unknown time varying matrix function.

*Assumption 2. *The interconnected term satisfieswhere , , and are unknown positive constants.

Now we show the general description of the quantizer . For the variable , the quantizer operator is defined by a mathematical function that rounds the elements of towards the nearest integers; namely,where and are the quantization sensitivity parameters at the coder and decoder sides of the th subsystem, respectively. During the operation process, the information of quantized measurement is generated at the coder side of th subsystem; then it is sent to the decoder side over a communication channel. While at the decoder side of th subsystem, under the assumption that the channel is ideal, the value of quantized measurement is received, and the quantization sensitivity parameter of the quantizer is adopted; then the decoder of th subsystem generates the signal , that is, the quantization operator .

Usually as done in [1, 2], it is assumed that the quantization parameters and are equal. In [17], the robustness of quantized control systems with respect to the mismatch of quantizer parameters is firstly presented. However, the adjustments of and are required to be synchronized at each instant. That is to say, , where is a time-invariant parameter. In actual control engineering, both of the requirements are obviously quite strict and hard to be implemented due to the hardware imperfections. For this, we establish the following* time-varying* ratio parameter model for the th subsystem:where , , and , satisfying , are unknown positive scalars.

Compared with our existing result in our previous result [20], the established model is more general since the lower and upper boundaries cannot be known.

*Remark 3. *Usually, besides quantization, signals transmitted over the channels also assume discretization on time, such as in [4, 14]. To focus on the control design for the mismatch problem of quantizer sensitivity, only signal quantization is considered in this paper. Similar discusses were also made in [13, 15, 19].

The main aim of this paper is to form decentralized adaptive integral sliding mode control strategy with performance for the nonlinear large-scale system (1) subject to the quantization mismatch as described in (6).

Now considering in (1) and the ratio in (6), we havewhere and . And satisfies with ; is the dimension of the vector .

*Definition 4. ** Gain Performance*. Given positive scalars , the nonlinear large-scale system in (1)-(2) with is said to be robust stable with a bounded gain performance; if it is robust stable for , and, under zero initial condition, for nonzero , it holds that

*Remark 5. *Denote that and letting , one can easily obtain thatThis is the usually considered gain performance for simple linear system as in [26].

For the considered problem, the following lemma will be used.

Lemma 6 (see [27]). *Given a symmetric matrix and matrices , with appropriate dimensions, then **for all satisfying , if and only if there exists a scalar such that the following inequality holds: *

#### 3. Main Results

In this paper, the following integral-type sliding surface function is considered:where and are system matrices defined in large-scale system (1). and are real matrices to be designed. In particular, the matrix is selected such that is nonsingular. Without loss of generality, as done in [28, 29], is designed to be , the pseudoinverse of for the convenience of the proof.

By the theory of sliding mode control [30, 31], when the trajectory of large-scale system (1) is kept on the sliding dynamics, it has and . Thus, by taking , one can obtain the equivalent control for the th subsystem of large-scale system (1):Substituting (15) into (1), one can get thatDenoting , , , and , one can see that (16) can be represented aswhere .

We have the following theorem for the sliding mode dynamics in (17).

Theorem 7. *Given scalars , the sliding mode dynamics in (17) is quadratically stable with bounded gain performance , if there exist a symmetric positive definite matrix , a general matrix , and a positive scalar such that the following linear matrix inequality holds:**Moreover, if the LMI condition has a feasible solution, then the matrices in (14) can be derived by *

*Proof. *Consider a Lyapunov function candidate aswhere . Along the solution of sliding mode dynamics (17), one can getTherefore,where andwith , , and .

Suppose that , we haveIntegrating both sides of the inequality (24) with respect to time from to yieldsUnder the zero initial conditions, we have and ; one can see thatTaking the limit as on both sides of (26) gives rise to (9).

Performing a congruence transformation to (23) by matrix , we haveDefining and , we havewhere . Furthermore, according to Lemma 6, one can see that (28) is equivalent towhere . Applying Schur complement technique, we can further get (18).

This completes the proof.

In what follows, let us denote that and for convenience. The following decentralized adaptive integral sliding mode control law is proposed:where , . ; . The parameters , , and are the estimations of , , and , respectively. And they are derived from the designed adaptive laws:with the initial values , , and , and the parameters , , and are selected positive scalars; . and will be given later.

*Remark 8. *The controller form as (31)–(33) can be often seen in existing theory results, for instance, [23, 32]. However, it is hard to expect to be exactly zero in practice engineering; thus how to solve the gain increase problem in theory completely is a very interesting research topic. In addition, one can see from the simulation part later, by choosing sufficiently large parameters , , and , the gains cannot increase indefinitely in a finite time. So they can be used in some engineering applications.

*Remark 9. *The solution for systems with discontinuous right-hand side as (30) can be interpreted in the sense of [33–35].

To clarify the proof of the main result, a lemma is first summarized in the following.

Lemma 10. *For the quantized control input satisfying the mismatched relation shown in (6), the control law in (30) guarantees the establishment of the inequality:*

*Proof. *It is obvious that the above inequality (34) holds when . And when , we have . Combining with the mismatched relation in (6), the definition of quantization error, one can see thatBy the design of the parameters, and , one can see thatCombining (35) with (36), we haveSince , it is not difficult to get thatSimilarly, when , we have . Combining with the mismatched relation in (6), the definition of quantization error again, one can see that According to the design of the parameters, and , one can see thatCombining (39) with (40), we haveFurthermore, we haveTherefore, the following inequality holds:The proof has been achieved completely.

The main result of this paper is summarized as follows.

Theorem 11. *Consider the nonlinear large-scale system (1) subject to Assumptions 1-2 with the mismatch between the quantization sensitivity parameters; if the control law is designed as shown in (30) with adaptive laws in (31)–(33), then the trajectories of system (1) can be driven onto the sliding surfaces asymptotically.*

*Proof. *Let us first consider , then the time derivative along the system trajectories isAccording to the design of the switching vector , one can see that the above equation can be rewritten to beBy Assumptions 1-2, we haveApplying the basic inequality , , , one can see thatSince , , and are unknown parameters, for convenience, similar to what is done in [23], we denote that unknown parameters and satisfy then (47) can be rewritten asTake Lyapunov function candidate as where , , and . Thus we have According to the design of the adaptive laws (31)–(33) and combining with , , and , we have According to Lemma 10, one can see that Furthermore, noticing that , , and , , one can easily obtain that Integrating both sides of the above equation from to gives rise toOne can easily obtain thatTaking the limit as on both sides of (56) yields According to Barbalat’s lemma, one can see thatSince , , and are positive increasing functions in terms of the adaptive control laws in (31) and (32), thus we have as .

Hence the proof is obtained completely.

#### 4. Illustrative Example

Consider the nonlinear uncertain large-scale system (1) with the following data:

In addition, the interconnected terms and and the exogenous disturbance terms and are selected as

Our aim here is to verify the effectiveness of the proposed theoretical results in the previous sections. Combining with the selection of parameters and , and solving LMI condition in Theorem 7, one can obtain that and . Moreover, the integral-type sliding surface functions in (14) can be computed as

In order to illustrate the effect caused by the mismatch of the quantization sensitivity parameters, the evolutions of and , , are selected to be of the following form: