Quantized Feedback Control Design of Nonlinear Large-Scale Systems via Decentralized Adaptive Integral Sliding Mode Control

A novel decentralized adaptive integral sliding mode control law is proposed for a class of nonlinear uncertain large-scale systems subject to quantizationmismatch between quantizer sensitivity parameters. Firstly, by applying linearmatrix inequality techniques, integral-type sliding surface functions are derived for ensuring the stability of the whole sliding mode dynamics and obtaining the prescribed boundedL 2 gain performance. Secondly, the decentralized adaptive slidingmode 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.


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][5][6],  ∞ filter design for linear continuous-time systems [7], robust or fault-tolerant control using sliding mode control technique [8][9][10][11][12], and  2 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][14][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][18][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 timevarying ratio model with known lower and upper boundaries is first introduced in [20].By utilizing robust control technique, the  2 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][22][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 2 Mathematical Problems in Engineering 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 L 2 gain performance analysis for nonlinear largescale 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 L 2 gain performance.Second, considering the established boundaryunknown 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.R  denotes the -dimensional Euclidean space;   denotes the transpose of matrix ; and  and 0 represent the identity matrix and a zero matrix in appropriate dimension, respectively. > 0 ( ≥ 0) means that  is real symmetric and positive definite (semipositive definite).The symbol He() represents  +   .||  denotes the -norm of the vector ; that is, 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.

Problem Statement and Preliminaries
In this paper, the following class of nonlinear uncertain large-scale systems with  interconnected subsystems is considered: where where   and   are known real constant matrices with appropriate dimensions, Δ  () is an unknown time varying matrix function.
where    ,  1  , and  2  are unknown positive constants.Now we show the general description of the quantizer (⋅).For the variable   ∈ R   , the quantizer operator   (  ) is defined by a mathematical function round(  ) 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 timevarying ratio parameter model for the th subsystem: where   () ∈ ( min ,  max ),  min , and  max , satisfying  max ≥  min , are unknown positive scalars.

Mathematical Problems in Engineering 3
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 L 2 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 have where with ∇  = √  /2;   is the dimension of the vector   .
Definition 4. L 2 Gain Performance.Given positive scalars   , the nonlinear large-scale system in (1)-( 2) with   () = 0 is said to be robust stable with a bounded L 2 gain performance; if it is robust stable for   () = 0, and, under zero initial condition, for nonzero Remark 5. Denote that and letting   = ,  = 1, 2, . . ., , one can easily obtain that This is the usually considered L 2 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 Θ and matrices ,  with appropriate dimensions, then for all Ξ() satisfying Ξ  ()Ξ() ≤ , if and only if there exists a scalar  > 0 such that the following inequality holds:

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).  ∈ R   ×  and   ∈ R   ×  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   () = 0 and σ  () = 0. Thus, by taking σ  () = 0, one can obtain the equivalent control for the th subsystem of large-scale system (1): Substituting ( 15) into (1), one can get that Denoting , and   =     , one can see that ( 16) can be represented as where We have the following theorem for the sliding mode dynamics in (17).
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  1 ,  2 , and  3 , 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 righthand side as (30) can be interpreted in the sense of [33][34][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: 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   () = (1/2)   ()  (), then the time derivative along the system trajectories is According to the design of the switching vector   =  +  , one can see that the above equation can be rewritten to be By Assumptions 1-2, we have Since  0  , According to the design of the adaptive laws ( 31)-( 33) and combining with l =   − l , ρ =   − ρ , and γ = γ−1  −   , we have Since   > 1, l > 0, and ρ > 0 are positive increasing functions in terms of the adaptive control laws in ( 31) and ( 32), thus we have   () → 0 as  → +∞.
Hence the proof is obtained completely.

Conclusions
This paper has been concerned with the problem of decentralized adaptive integral sliding mode control design for a class of nonlinear uncertain large-scale systems subject to quantization mismatch.The LMI conditions are derived for the design of the switching surfaces of each subsystem

Figure 1 :
Figure 1: The response curves of system states.