Model Predictive Control of Linear Systems over Networks with State and Input Quantizations

Abstract

Although there have been a lot of works about the synthesis and analysis of networked control systems (NCSs) with data quantization, most of the results are developed for the case of considering the quantizer only existing in one of the transmission links (either from the sensor to the controller link or from the controller to the actuator link). This paper investigates the synthesis approaches of model predictive control (MPC) for NCS subject to data quantizations in both links. Firstly, a novel model to describe the state and input quantizations of the NCS is addressed by extending the sector bound approach. Further, from the new model, two synthesis approaches of MPC are developed: one parameterizes the infinite horizon control moves into a single state feedback law and the other into a free control move followed by the single state feedback law. Finally, the stability results that explicitly consider the satisfaction of input and state constraints are presented. A numerical example is given to illustrate the effectiveness of the proposed MPC.

1. Introduction

Model predictive control (MPC) commonly refers to a class of computer control algorithms that use an explicit process model to forecast the future response of a plant [1]. In practical applications, MPC can now be found in a wide variety of industrial fields, such as chemical industries, food processing, and automotive applications, since it has advantages including good tracking performance, physical constraints handling, and extension to nonlinear systems. Not only that, in recent decades, it also has a great improvement in theoretical research; various methodologies have been addressed and developed, such as dynamic matrix control (DMC), generalized predictive control (GPC), and robust MPC. From the 1990s, the main stream of the theoretical research about MPC has been turned to the synthesis approach, which means that the MPC controller is synthesized so that the closed-loop system is stable whenever the optimization problem is feasible. A simple procedure of synthesis approach of MPC is to parameterize the infinite horizon control moves into a state feedback control law at each sampling time. In this field, [2] introduces the linear matrix inequality (LMI) technique to solve the optimization problem of minimizing an upper bound on “worst case” value of the infinite horizon objective function and reduces the problem to a convex optimization problem involving a set of LMIs. Reference [3] applies the technique in [2] off-line, and a sequence of control laws corresponding to a sequence of asymptotically stable invariant ellipsoids is constructed. The real-time state feedback gain is chosen from the designed control laws. In this way, the on-line computational burden can be significantly reduced. Another procedure of synthesis approach of MPC is to add free control moves before the linear state feedback law, instead of a single state feedback. In this field, the related works can be found in [4, 5]. Reference [4] proposes a “quasi-min-max” MPC algorithm for linear parameter varying systems by adding one free control move, and the conservatism and feasibility in [2] are reduced and improved. Reference [5] extends the results in [4] to 𝑁>1 free control moves, which further improves the optimality and feasibility.

Networked control systems (NCSs) are distributed systems in which the sensors, the controllers, and the actuators are spatially distributed and interconnected through communication networks [6–12]. Data quantization is inevitable in NCSs, which is caused by 𝐴/𝐷 and 𝐷/𝐴 conversions and has an undesirable effect on system performance or even stability. The issue of mitigating the effect of data quantization has been considered by many researchers [13–20]. For static and time-invariant quantizer, [13] first points out that the coarsest quantizer should follow a logarithmic law in quadratic stabilization of single-input-single-output (SISO) linear time-invariant (LTI) systems. Reference [14] generalizes the results of [13] to the multi-input-multi-output (MIMO) systems and the output feedback control and converts the quantized quadratic stabilization problem into the robust control problem by using the sector bound method. By adopting a quantization error dependent Lyapunov function, [15] proposes a new approach to the problem of analysis and synthesis for quantized feedback system based on the results in [14], which offers less conservative results. Based on the results in [14], a robust MPC strategy is proposed to deal with the stabilization of NCS with quantized control input in [16] and a simple dynamic scaling method for a logarithmic quantizer based output feedback controller is given in [17]. However, all the mentioned papers consider only one quantizer existing in one communication link, either in the sensor to controller (S-C) link or the controller to actuator (C-A) link. The quantized stabilization problem in both links has not received much attention than what it deserves. In the existing results, to the best of our knowledge, there are only a small amount of results [18–20] about the quantization problem when both the state and the input are quantized, and all of them do not consider the synthesis approach of MPC. This situation motivates our study presented here.

The synthesis approach of MPC is known as a control algorithm that can significantly improve the control performance, especially when system uncertainties exist; see [2, 3]. Therefore, it is of theoretical and practical significance to extend these approaches to the networked control. However, very few works have focused on the synthesis approaches of MPC for NCS with quantizations. The goal of this paper is to give synthesis approaches of MPC for discrete LTI MIMO system over networks with state and input quantizations. Firstly, by applying the sector bound approach in [14], a novel model which describes the NCS with quantizers in both S-C and C-A links is established. The result converts the quantized stabilization problem into robust control problem. Secondly, based on the developed new model, two synthesis approaches of MPC, without free control move and with one free control move, are derived, respectively, by extending the results in [2, 4]. Finally, the stability results of the closed-loop system are given by applying the Lyapunov method.

Notation. 𝐼is the identity matrix with appropriate dimensions. For any vector 𝑥 and matrix 𝑊, ‖𝑥‖²_𝑊∶=𝑥^T𝑊𝑥. The superscript T denotes the transpose for vectors or matrices. 𝑥(𝑘+𝑖∣𝑘) is the value of vector 𝑥 at a future time 𝑘+𝑖 predicted at time 𝑘.

2. System Description

Consider the quantized feedback control system in Figure 1. The plant to be controlled is modeled by 𝑥(𝑘+1)=𝐴(𝑘)𝑥(𝑘)+𝐵(𝑘)𝑢(𝑘),𝑘≥0, (1) where 𝑢(𝑘)∈ℜ^𝑚 and 𝑥(𝑘)∈ℜ^𝑛 are the control input and measurable state of the plant, respectively. 𝑤(𝑘)∈ℜ^𝑛 and 𝑣(𝑘)∈ℜ^𝑚 are input and output of the controller, respectively. We assume that [𝐴(𝑘)∣𝐵(𝑘)] varies inside a corresponding polytope Ω whose vertices consist of 𝐿 local systems matrices; that is, [𝐴(𝑘)∣𝐵(𝑘)]∈Ω,𝑘≥0, (2) with Ω=Co𝐴₁∣𝐵₁,𝐴₂∣𝐵₂,…,𝐴_𝐿∣𝐵_𝐿, (3) where Co{⋅} denotes the convex hull. The input and state constraints are −𝑢≤𝑢(𝑘+𝑖)≤𝑢,−𝜓≤Ψ𝑥(𝑘+𝑖+1)≤𝜓,∀𝑖≥0, (4) where 𝑢∶=[𝑢₁,𝑢₂,…,𝑢_𝑚]^T, 𝑢∶=[𝑢₁,𝑢₂,…,𝑢_𝑚]^T, 𝑢_𝑗>0, 𝑢_𝑗>0, 𝑗∈{1,2,…,𝑚}; 𝜓∶=[𝜓₁,𝜓₂,…,𝜓_𝑞]^T, 𝜓∶=[𝜓₁,𝜓₂,…,𝜓_𝑞]^T, 𝜓_𝑠>0, 𝜓_𝑠>0, 𝑠∈{1,2,…,𝑞}; Ψ∈ℜ^𝑞×𝑛.

Figure 1 

Structure of NCS with quantizations.

The quantized state feedback controller is given by 𝑢(𝑘)(𝑣(𝑘,𝑣(𝑘)(𝑘)(𝑥(𝑘,=𝑓))=𝐾𝑤=𝐾𝑔)) (5) where 𝐾∈ℜ^𝑚×𝑛 is the feedback gain to be designed and 𝑓(⋅) and 𝑔(⋅) are two static logarithmic quantizers which are defined as 𝑓(𝑣(𝑘))=𝑓𝑣₁(𝑘)𝑓𝑣₂(𝑘)⋯𝑓𝑣_𝑚(𝑘)^T,𝑔(𝑥(𝑘))=𝑔𝑥₁(𝑘)𝑔𝑥₂(𝑘)⋯𝑔𝑥_𝑛(𝑘)^T, (6) where 𝑓(⋅) and 𝑔(⋅) are assumed to be symmetric, that is, 𝑓(−𝑣_𝑗(𝑘))=−𝑓(𝑣_𝑗(𝑘)), 𝑗∈ℳ₁∶={1,2,…,𝑚}, and 𝑔(−𝑥_𝑖(𝑘))=−𝑔(𝑥_𝑖(𝑘)), 𝑖∈𝒩₁∶={1,2,…,𝑛}, and 𝑣_𝑗(𝑘) (𝑥_𝑖(𝑘)) is the 𝑗th (𝑖th) component of 𝑣(𝑘) (𝑥(𝑘)).

In terms of [14], the set of quantization levels for logarithmic quantizers 𝑓(⋅) and 𝑔(⋅) is described by 𝒰_𝜁=±𝑢_𝑠∶𝑢_𝑠=𝜌^𝑠_𝜁𝑢₀,𝑠=0,±1,±2,…∪{0},0<𝜌_𝜁<1,𝑢₀>0, (7) where 𝜁 is 𝑓 or 𝑔. The associated quantizer 𝜁 is defined as follows: 𝜁(𝜃)=𝑢_𝑠1,if1+𝛿_𝜁𝑢_𝑠1<𝜃≤1−𝛿_𝜁𝑢_𝑠(),𝜃>0,0,if𝜃=0,−𝜁−𝜃,if𝜃<0, (8) with 𝛿_𝜁=1−𝜌_𝜁1+𝜌_𝜁. (9) Denote by ♯𝑔[𝜀] the number of quantization levels in the interval [𝜀,1/𝜀]. The density of the quantizer 𝜁 is 𝜗_𝜁=limsup_𝜀→0♯𝑔_𝜁[𝜀].−ln𝜀 (10) A small 𝜌_𝜁 means coarse quantization and a large 𝜌_𝜁 implies dense quantization. In the following, we will call 𝜌_𝜁 the quantization density of the quantizer 𝜁, instead of 𝜗_𝜁, and assume that 𝜌_𝜁 is a known number and, hence, 𝛿_𝜁 is known and satisfies 0<𝛿_𝜁<1.

Applying the sector bound approach in [14], 𝑓(𝑣(𝑘)) and 𝑔(𝑥(𝑘)) can be written as 𝑓(𝑣(𝑘))=𝐼+Δ_𝑓(𝑘𝑣(𝑘),𝑔(𝑥(𝑘=)))𝐼+Δ_𝑔(𝑘)𝑥(𝑘), (11) where Δ_𝑓(𝑘)=diagΔ_𝑓1(𝑘),Δ_𝑓2(𝑘),…,Δ_𝑓𝑚(𝑘,Δ_𝑔)(𝑘)=diagΔ_𝑔1(𝑘),Δ_𝑔2(𝑘),…,Δ_𝑔𝑛(𝑘) (12) with |Δ_𝑓𝑗(𝑘)|≤𝛿_𝑓, 𝑗∈ℳ₁, and |Δ_𝑔𝑖(𝑘)|≤𝛿_𝑔, 𝑖∈𝒩₁.

Let ℱ be the set of 𝑚×𝑚 diagonal matrices whose diagonal elements are either −𝛿_𝑓 or 𝛿_𝑓. There are 2^𝑚 elements in ℱ. Suppose that each element in ℱ is labeled as 𝔽_𝑗, 𝑗∈ℳ₂∶={1,2,…,2^𝑚}. Then, ℱ={𝔽_𝑗∶𝑗∈ℳ₂}. Let 𝒢 be the set of 𝑛×𝑛 diagonal matrices whose diagonal elements are either −𝛿_𝑔 or 𝛿_𝑔. There are 2^𝑛 elements in 𝒢. Suppose that each element in 𝒢 is labeled as 𝔾_𝑖, 𝑖∈𝒩₂∶={1,2,…,2^𝑛}. Then, 𝒢={𝔾_𝑖∶𝑖∈𝒩₂}. It is easy to show that Δ_𝑓(𝑘)∈Ω^𝑓=Co𝔽_𝑗∶𝑗∈ℳ₂,Δ_𝑔(𝑘)∈Ω^𝑔=Co𝔾_𝑖∶𝑖∈𝒩₂. (13) This means that Δ_𝑓(𝑘)=^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝔽_𝑗,Δ_𝑔(𝑘)=^2𝑛_𝑖=1𝛽_𝑖(𝑘)𝔾_𝑖, (14) where ∑^2𝑚_𝑗=1𝛼_𝑗=1, ∑^2𝑛_𝑖=1𝛽_𝑖=1, 𝛼_𝑗≥0, 𝛽_𝑖≥0.

Taking into account (11), the closed-loop system becomes 𝑥()(𝑘)𝑥(𝑘)(𝑘)𝑘+1=𝐴+𝐵𝐼+Δ_𝑓(𝑘)𝐾𝐼+Δ_𝑔(𝑘)𝑥(𝑘). (15) It is shown in [14] that the system (15) and system (1), (2), (5) are equivalent for stability analysis.

3. Stabilization Using Networked MPC

3.1. Optimization Problem for Networked MPC

As in the nonnetworked MPC (see, e.g., [2, 4]), the state feedback 𝑣=𝐾w will be utilized, where 𝐾 is to be optimized. We will give two different techniques: case 1 that is without free control move; case 2 that is with one free control move. For simplicity, denote 𝐼+Δ_𝑓(𝑘)𝐾(𝑘)𝐼+Δ_𝑔(𝑘)∶=𝐾(𝑘)+Δ𝐾(𝑘). (16) For case 1, the predicted control input becomes 𝑢()𝑔(𝑥(=(𝐾(𝑘)(𝑘𝑥()𝑘+𝑙∣𝑘=𝑓𝑘+𝑙∣𝑘))+Δ𝐾))𝑘+𝑙∣𝑘,𝑙≥0. (17) The closed-loop state prediction satisfies 𝑥()()𝑥()()×(𝐾(𝑘)(𝑘𝑥()𝑘+𝑙+1∣𝑘=𝐴𝑘+𝑙𝑘+𝑙∣𝑘+𝐵𝑘+𝑙+Δ𝐾))𝑘+𝑙∣𝑘,𝑙≥0, (18) where 𝑥(𝑘∣𝑘)∈𝜕(𝑘)𝑤(𝑘) (19) with 𝜕(𝑘)=diag𝜕₁(𝑘),𝜕₂(𝑘),…,𝜕_𝑛(𝑘),1+𝛿_𝑔⁻¹≤𝜕_𝑖(𝑘)≤1−𝛿_𝑔⁻¹,𝑖∈𝒩₁. (20) Let 𝒞={ℂ_ℎ∶ℎ∈𝒩₂} be the set of 𝑛×𝑛 diagonal matrices whose diagonal elements are either (1+𝛿_𝑔)⁻¹ or (1−𝛿_𝑔)⁻¹. Then, it is shown that 𝜕(𝑘)∈Ω^𝑐=Co{ℂ_ℎ∶ℎ∈𝒩₂}.

The objective of case 1 is to design 𝐾(𝑘) which drives (18) to the equilibrium point 𝑥=0, by minimizing the infinite horizon worst case performance index at each sampling instant: min_𝐾(𝑘)max_⋆𝐽(𝑘)=^∞_𝑙=0‖𝑥(𝑘+𝑙∣𝑘)‖²_𝒬+‖𝑢(𝑘+𝑙∣𝑘)‖²_ℛs.t.(4),(18),(19), (21) where ⋆={𝜕(𝑘)∈Ω^𝑐,Δ_𝑓(𝑘)∈Ω^𝑓,Δ_𝑔(𝑘)∈Ω^𝑔,[𝐴(𝑘+𝑙)∣𝐵(𝑘+𝑙)]∈Ω,𝑙≥0}; 𝒬 and ℛ are symmetric positive-definite weight matrices.

For case 2, the output of the controller achieves 𝑣()(𝑘)𝑣()=(𝑘)𝑘∣𝑘=𝑣,𝑙=0,𝑘+𝑙∣𝑘𝐼+Δ_𝑔(𝑘)𝑥(𝑘+𝑙∣𝑘),𝑙≥1, (22) where 𝑣(𝑘) is freedom for optimization. The predicted control input becomes 𝑢(𝑘∣𝑘)=𝑓(𝑣(𝑘∣𝑘))=𝐼+Δ_𝑓(𝑘𝑣(𝑘)𝑢()=(𝐾(𝑘)(𝑘𝑥()),𝑙=0,𝑘+𝑙∣𝑘+Δ𝐾))𝑘+𝑙∣𝑘,𝑙≥1. (23) Hence, the closed-loop state prediction satisfies 𝑥()(𝑘)𝑥()(𝑘)𝑘+1∣𝑘=𝐴𝑘∣𝑘+𝐵𝐼+Δ_𝑓(𝑘)𝑣(𝑘)𝑥()()𝑥()()×(𝐾(𝑘)(𝑘𝑥(),𝑙=0,𝑘+𝑙+1∣𝑘=𝐴𝑘+𝑙𝑘+𝑙∣𝑘+𝐵𝑘+𝑙+Δ𝐾))𝑘+𝑙∣𝑘,𝑙≥1. (24) The objective of case 2 is to design {𝑣(𝑘),𝐾(𝑘)} which drives (24) to the equilibrium point 𝑥=0, by minimizing the infinite horizon worst case performance index at each sampling instant: min_𝐾(𝑘)max_⋆𝐽(𝑘),s.t.(4),(19),(24). (25) Problems (21) and (25) are the “min-max” optimization problem, for which the techniques of LMI can be efficiently utilized.

3.2. Networked MPC without Free Control Move

In order to derive an upper bound on max_⋆𝐽(𝑗_𝑙) and solve problem (21), let us impose the following stability constraint: 𝑉(𝑥((𝑥(‖𝑥(𝑘+𝑙+1∣𝑘))−𝑉𝑘+𝑙∣𝑘))≤−𝑘+𝑙∣𝑘)‖²_𝒬−‖𝑢(𝑘+𝑙∣𝑘)‖²_ℛ,∀𝑙≥0, (26) where 𝑉(𝑥(𝑘+𝑙∣𝑘))=‖𝑥(𝑘+𝑙∣𝑘)‖²_𝑃. For stable closed-loop system, lim_𝑙→∞𝑉(𝑥(𝑘+𝑙∣𝑘))=0. Hence, by summing (26) from 𝑙=0 to 𝑙=∞, we can obtain 𝑉(𝑥(𝑘∣𝑘))≥𝐽(𝑘). Define 𝑄=𝛾𝑃⁻¹>0, where 𝛾 is a scalar. By minimizing 𝛾 satisfying 𝑉(𝑥(𝑘∣𝑘))≤𝛾, the upper bound on performance cost is minimized. By using the Schur complement, 𝑉(𝑥(𝑘∣𝑘))≤𝛾 is guaranteed by 𝑥()𝑄1∗𝑘∣𝑘≥0. (27) Considering (19) and (14), this is equivalent to ℂ1∗_ℎ𝑤(𝑘)𝑄≥0,ℎ∈𝒩₂. (28) Substituting 𝑄=𝛾𝑃⁻¹ and using Schur complement, it is shown that (26) is equivalent to 𝑄⁻¹𝐴()(𝐾(𝑘)(𝑘𝒬∗∗∗𝑘+𝑙+𝐵𝑘+𝑙)(+Δ𝐾))𝑄∗∗_1/2ℛ0𝛾𝐼∗_1/2(𝐾(𝑘)+Δ𝐾(𝑘))00𝛾𝐼≥0. (29) Denote 𝐾(𝑘)=𝑌𝐺⁻¹ where 𝐺 is nonsingular matrix. Performing congruence transformation to (29) by diag{𝐺,𝐼,𝐼,𝐼}, utilizing the fact that 𝐺𝑄⁻¹𝐺^T≥𝐺+𝐺^T−𝑄 and the convexity of polytopic description, it is shown that (29) can be guaranteed by 𝐺+𝐺^T𝐴−𝑄∗∗∗_𝑠+𝐵_𝑠𝑌+𝔽_𝑗𝑌+𝑌𝔾_𝑖+𝔽_𝑗𝑌𝔾_𝑖𝒬𝑄∗∗_1/2ℛ𝐺0𝛾𝐼∗_1/2𝑌+𝔽_𝑗𝑌+𝑌𝔾_𝑖+𝔽_𝑗𝑌𝔾_𝑖{}00𝛾𝐼≥0,𝑠∈1,2,…,𝐿,𝑗∈ℳ₂,𝑖∈𝒩₂. (30) Then, we consider handling of the input constraint. Define 𝜉_𝑠 as the 𝑠th row of the 𝑚-ordered identity matrix. We get max_𝑙≥0𝜉_𝑠𝑢(𝑘+𝑙∣𝑘)²=max_𝑙≥0𝜉_𝑠(𝐾(𝑘)+Δ𝐾(𝑘))𝑥(𝑘+𝑙∣𝑘)²≤max_𝑙≥0𝜉_𝑠(𝐾(𝑘)+Δ𝐾(𝑘))𝑄^1/22𝑄^−(1/2)𝑥(𝑘+𝑙∣𝑘)²≤max_𝑙≥0𝜉_𝑠(𝐾(𝑘)+Δ𝐾(𝑘))𝑄^1/22. (31) If there exists a symmetric matrix 𝑍, 𝑍_𝑠𝑠≤𝑢²_𝑠,inf, 𝑠∈𝐦, such that max_𝑙≥0(𝐾(𝑘)+Δ𝐾(𝑘))𝑄(𝐾(𝑘)+Δ𝐾(𝑘))^T≤𝑍, (32) then |𝜉_𝑠𝑢(𝑘∣𝑗_𝑙)|≤𝑢_𝑠,inf, 𝑠∈𝐦. It is easy to show that (32) is equivalent to 𝐾(𝑘)+𝔽_𝑗𝐾(𝑘)+𝐾(𝑘)𝔾_𝑖+𝔽_𝑗𝐾(𝑘)𝔾_𝑖×𝑄𝐾(𝑘)+𝔽_𝑗𝐾(𝑘)+𝐾(𝑘)𝔾_𝑖+𝔽_𝑗𝐾(𝑘)𝔾_𝑖^T≤𝑍,𝑗∈ℳ₂,𝑖∈𝒩₂. (33) By performing congruence transformation to (33) by diag{𝐺,𝐼}, utilizing the fact that 𝐺𝑄⁻¹𝐺^T≥𝐺+𝐺^T−𝑄, and applying the Schur complement, it is shown that (33) can be guaranteed by 𝐺+𝐺^T−𝑄∗𝑌+𝔽_𝑗𝑌+𝑌𝔾_𝑖+𝔽_𝑗𝑌𝔾_𝑖𝑍≥0,𝑗∈ℳ₂,𝑖∈𝒩₂,𝑍_𝑠𝑠≤𝑢²_𝑠,inf,𝑠∈𝐦. (34) Lastly, consider the state constraint. Define 𝜉_𝑠 as the 𝑠th row of the 𝑞-ordered identity matrix. Then, we have max_𝑙≥0𝜉_𝑠Ψ𝑥(𝑘+𝑙+1∣𝑘)²=max_𝑙≥0𝜉_𝑠Ψ[⋆]𝑥(𝑘+𝑙∣𝑘)²≤max_𝑙≥0𝜉_𝑠Ψ[⋆]𝑄^1/22𝑄^−(1/2)𝑥(𝑘+𝑙∣𝑘)²≤max_𝑙≥0𝜉_𝑠Ψ[⋆]𝑄^1/22, (35) where ⋆=𝐴(𝑘+𝑙)+𝐵(𝑘+𝑙)(𝐾(𝑘)+Δ𝐾(𝑘)). If there exists a symmetric matrix Γ, Γ_𝑠𝑠≤𝜓²_𝑠,inf, 𝑠∈𝐪, such that max_𝑙≥0Ψ⋆𝑄⋆^TΨ^T≤Γ, (36) then, by performing congruence transformation to (36) by diag{𝐺,𝐼}, utilizing the fact that 𝐺𝑄⁻¹𝐺^T≥𝐺+𝐺^T−𝑄 and the convexity of polytopic description, and applying the Schur complement, it is easy to show that (36) is guaranteed by 𝐺+𝐺^TΨ𝐴−𝑄∗_𝑠𝐺+𝐵_𝑠𝑌+𝔽_𝑗𝑌+𝑌𝔾_𝑖+𝔽_𝑗𝑌𝔾_𝑖Γ{}≥0,𝑠∈1,2,…,𝐿,𝑗∈ℳ₂,𝑖∈𝒩₂,Γ_𝑠𝑠≤𝜓²_𝑠,inf,𝑠∈𝐪. (37)

Therefore, the “min-max” optimization problem (21) for the proposed quantized state feedback MPC can be developed as min_{𝛾,𝑄,𝐺,𝑌,Γ,𝑍}𝛾s.t.(28),(30),(34),and(37). (38)

Theorem 1. If the optimization problem (38) is feasible at time 𝑘, then it is feasible for all times 𝑡>𝑘, and the receding horizon quantized state feedback control move 𝑢(𝑘)=𝑓(𝐾(𝑘)𝑔(𝑥(𝑘))) asymptotically stabilizes the closed-loop system.

Proof. Firstly, we consider the recursive feasibility. Assume that the optimization problem (38) is feasible at time 𝑘. The only LMI in this problem which depends on the measured state of the system is (28). Hence, we only need to prove that (28) is feasible for future measured state 𝑤(𝑘+𝑖), 𝑖>0. From [2], when the constraints (28) and (30) are satisfied, it must have 𝑥(𝑘+𝑙∣𝑘)^T𝑄⁻¹𝑥(𝑘+𝑙∣𝑘)<1,𝑙≥1. (39) At time 𝑘, the state prediction for 𝑘+1 is 𝑥(𝑘+1∣𝑘)=𝐴_𝑠+𝐵_𝑠𝐾(𝑘)+𝔽_𝑗𝐾(𝑘)+𝐾(𝑘)𝔾_𝑖+𝔽_𝑗𝐾(𝑘)𝔾_𝑖ℂ_ℎ𝑤(𝑘),𝑠∈{1,2,…,𝐿},𝑗∈ℳ₂,𝑖,ℎ∈𝒩₂. (40) At time 𝑘+1, there exist [𝐴∣𝐵]∈Ω, 𝔽∈Ω^𝑓, 𝔾∈Ω^𝑔, and ℂ∈Ω^𝑐 such that 𝑥()=((𝐾(𝑘)(𝑘)(𝑘)(𝑘)𝔾(𝑘).𝑘+1∣𝑘+1𝐴+𝐵+𝔽𝐾+𝐾𝔾+𝔽𝐾))ℂ𝑤 (41) From (40) and (41), it is easy to show that 𝑥(𝑘+1∣𝑘+1) is contained in 𝑥(𝑘+1∣𝑘). Taking into account (39), we get 𝑥(𝑘+𝑙∣𝑘+1)^T𝑄⁻¹𝑥(𝑘+𝑙∣𝑘+1)<1, which is equivalent to ℂ1∗_ℎ𝑤(𝑘+1)𝑄≥0,ℎ∈𝒩₂. (42) Thus, the optimization problem is also feasible at time 𝑘+1, and the same results can be achieved at times 𝑘+2,𝑘+3,… by analogy. Hence, the recursive feasibility of optimization problem (38) is guaranteed.
Secondly, we prove the asymptotic stability. Assume that the optimization problem (38) is feasible and the solution is depicted as “*”. By applying stability constraints, we have ‖𝑥(𝑘+1∣𝑘)‖²_𝑃^∗_(𝑘)−‖𝑥(𝑘∣𝑘)‖²_𝑃^∗_(𝑘)≤−‖𝑥(𝑘∣𝑘)‖²_𝒬−𝑢^∗(𝑘∣𝑘)²_ℛ. (43) Since 𝑥(𝑘+1∣𝑘+1) is contained in 𝑥(𝑘+1∣𝑘), the following holds: ‖𝑥(𝑘+1∣𝑘+1)‖²_𝑃^∗_(𝑘)−‖𝑥(𝑘∣𝑘)‖²_𝑃^∗_(𝑘)≤−‖𝑥(𝑘∣𝑘)‖²_𝒬−𝑢^∗(𝑘∣𝑘)²_ℛ. (44) By noting ‖𝑥(𝑘∣𝑘)‖²_𝑃^∗_(𝑘)≤𝛾^∗(𝑘),‖𝑥(𝑘+1∣𝑘+1)‖²_𝑃(𝑘+1)≤𝛾(𝑘+1), (45) it is admissible to choose 𝛾(𝑘+1)=𝛾^∗(𝑘)−‖𝑥(𝑘∣𝑘)‖²_𝒬−𝑢^∗(𝑘∣𝑘)²_ℛ. (46) Solving the optimization problem (38) at time 𝑘+1, it leads to 𝛾^∗(𝑘+1)≤𝛾(𝑘+1). (47) It must have 𝛾^∗(𝑘+1)≤𝛾^∗(𝑘)−‖𝑥(𝑘∣𝑘)‖²_𝒬−𝑢^∗(𝑘∣𝑘)²_ℛ. (48) Therefore, 𝛾^∗(𝑘) decreases monotonously and can be chosen as the Lyapunov function for proving the stability of the closed-loop system.

3.3. Networked MPC with One Free Control Move

In Section 3.2, we developed the synthesis approach of networked MPC by parameterizing the infinite horizon control moves into a single state feedback law. This part, in order to reduce the conservatism, provides a synthesis approach of MPC by adding one free control move, which is easier to be feasible and can obtain more optimal control moves.

The optimization problem is shown in (25). Similar to Section 3.2, let us impose the stability constraint (26) for all 𝑙≥1. For stable closed-loop system, lim_𝑙→∞𝑉(𝑥(𝑘+𝑙∣𝑘))=0. By summing (26) from 𝑙=1 to 𝑙=∞, we get 𝐽(𝑘)≤𝑉(𝑥(𝑘+1∣𝑘))+‖𝑥(𝑘∣𝑘)‖²_𝒬+‖𝑢(𝑘∣𝑘)‖²_ℛ. Let 𝑉(𝑥(‖𝑢(𝑘+1∣𝑘))≤𝛾,𝑘∣𝑘)‖²_ℛ≤𝛾₁, (49)(50) where 𝛾₁ is a scalar. By defining 𝑄=𝛾𝑃⁻¹>0 and using Schur complement, it is shown that (49) is guaranteed by 𝑥()𝑄1∗𝑘+1∣𝑘≥0. (51) Considering (14), (19) and (24), this is equivalent to 𝐴1∗_𝑠𝐶_ℎ𝑤(𝑘)+𝐵_𝑠𝐼+𝔽_𝑗𝑣(𝑘)𝑄{}≥0,𝑠∈1,2,…,𝐿,𝑗∈ℳ₂,ℎ∈𝒩₂. (52) By considering (24) and (14), and applying Schur complement, it is shown that (50) is guaranteed by ℛ1∗_1/2𝐼+𝔽_𝑗𝑣(𝑘)𝛾₁𝐼≥0,𝑗∈ℳ₂. (53)

Lemma 2. Suppose there exist a scalar 𝛾, a vector 𝑣(𝑘), symmetric matrices Γ, 𝑍, and 𝑄, and any matrices 𝐺,𝑌 such that (30), (34), (37), (52), and the following are satisfied: −𝑢≤𝑣(𝑘)≤𝑢,−𝜓≤Ψ𝑥(𝑘+1)≤𝜓. (54)(55) Then, (4) is satisfied.

Proof. According to the deductions in Section 3.2, if (30) and (52) are satisfied, then (4) for 𝑙≥1 is guaranteed by (34) and (37). Equations (54) and (55) guarantee the satisfaction of (4) for 𝑙=0. This completes the proof.

According to (24), (55) is equivalent to −𝜓≤𝐴(𝑘)𝑥(𝑘∣𝑘)+𝐵(𝑘)𝐼+Δ_𝑓(𝑘)𝑣(𝑘)≤𝜓. (56) Considering (19) and the convexity of the polytopic description, it is shown that (56) can be guaranteed by −𝜓≤𝐴_𝑠𝐶_ℎ𝑤(𝑘)+𝐵_𝑠𝐼+𝔽_𝑗𝑣(𝑘)≤{}𝜓,𝑠∈1,2,…,𝐿,𝑗∈ℳ₂,ℎ∈𝒩₂. (57)

Therefore, the “min-max” optimization problem (25) for the proposed quantized networked MPC with one free control move can be approximated by min_{𝛾,𝛾1,𝑣(𝑘),𝑄,𝐺,𝑌,Γ,𝑍}𝛾₁(),(),(),()(),().+𝛾s.t.30343752--5457 (58)

Theorem 3. If the optimization problem (58) is feasible at time 𝑘, then it is feasible for all times 𝑡>𝑘, and the receding horizon quantized state feedback control move 𝑢(𝑘)=𝑓(𝑣(𝑥(𝑘))) asymptotically stabilizes the closed-loop system.

Proof. Assume that the optimization problem (58) is feasible and the optimal solutions are depicted as “*.”At time 𝑘+1, the feasible solution can be chosen as 𝑣(𝑘+1)=𝑣^∗(𝑘+1∣𝑘)=𝐾^∗(𝑘)𝑤(𝑘+1). Denote 𝐿(𝑘)=‖𝑥(𝑘+1∣𝑘)‖²_𝑃(𝑘)+‖𝑥(𝑘∣𝑘)‖²_𝒬+‖𝑢(𝑘∣𝑘)‖²_ℛ. (59) We can take 𝐿(𝑘+1)=𝑥^∗(𝑘+1∣𝑘)²_𝒬+𝑢^∗(𝑘+1∣𝑘)²_ℛ+𝑥^∗(𝑘+2∣𝑘)²_{𝑃∗(𝑘)}. (60) It is easy to show 𝐽(𝑘+1)≤𝐿(𝑘+1). (61) Since stability constraint (26) is satisfied, it must have 𝑥^∗(𝑘+2∣𝑘)²_{𝑃∗(𝑘)}−𝑥^∗(𝑘+1∣𝑘)²_{𝑃∗(𝑘)}≤−𝑥^∗(𝑘+1∣𝑘)²_𝒬−𝑢^∗(𝑘+1∣𝑘)²_ℛ. (62) Substituting (62) into (61), we get 𝐿(𝑘+1)≤𝑥^∗(𝑘+1∣𝑘)²_{𝑃∗(𝑘)}≤𝐿^∗(𝑘). (63) At time 𝑘+1, the optimization is performed again, and it leads to 𝐿^∗(𝑘+1)≤𝐿(𝑘+1)<𝐿^∗(𝑘). (64) Hence, 𝐿^∗(𝑘) is decreased monotonously, and we can choose it as the Lyapunov function for proving the stability of the closed-loop system. This completes the proof.

4. Numerical Example

In this section, the effectiveness of the proposed MPC is illustrated by a numerical example. Consider the following linear parameter varying system: 𝑥()=(𝑘)𝑥(𝑘)+𝑢(𝑘).𝑘+10.872−0.0623𝜅0.09350.9970.09350.00478 (65) We assume that the uncertain parameter 𝜅(𝑘) lies in the area 𝜅(𝑘)∈[1,5] and is generated by 𝜅(𝑘)=1+4|sin(0.01𝑘)|. The initial state is assumed as 𝑥(0)=−1.5−0.2^T. The input constraint is |𝑢|≤1. Moreover, for the networked environment, the quantizers 𝑓(⋅) and 𝑔(⋅) are chosen as logarithmic with quantization density 𝜌_𝑓=𝜌_𝑔=0.8091 and the quantizer parameter 𝑢₀ is selected as 𝑢₀=15. It is easy to show that the set of quantization is able to cover the state trajectories according to the evaluation of the initial condition. The parameters of the cost function are ℛ=1,𝒬=𝐼.

Let 𝐽_true=∑^∞_𝑘=0(𝑥(𝑘)^T𝑥(𝑘)+𝑢(𝑘)²) and denote 𝑇_ave as the average computation time for each 𝑘. By solving (38) and (58), the resultant state trajectories, the control input, and the evolution of 𝛾(𝑘) (𝐿(𝑘)) are shown in Figures 2, 3, and 4, respectively, in dash lines (solid lines), with 𝐽_true=19.005 and 𝑇_ave=0.2078 s (𝐽_true=18.586 and 𝑇_ave=0.2954 s). It is shown that the closed-loop system is stable by applying the two techniques we proposed. Compared with the networked MPC without free control move, by adding one free control move one can achieve faster state responses and more aggressive control moves. Moreover, 𝛾(𝑘) and 𝐿(𝑘) are monotonic with the evolution of time.

Figure 2 

The state responses of the closed-loop system by applying (38) and (58).

Figure 3 

The control input by applying (38) and (58).

Figure 4 

The evolutions of 𝛾(𝑘) and 𝐿(𝑘) by applying (38) and (58), respectively.

5. Conclusions

In this paper, the synthesis approaches of MPC have been introduced to deal with the problem of NCS with input and state quantizations. The issue of quantized stabilization of this system is transformed into the robust control problem by applying the sector bound approach. Two synthesis approaches of MPC are formulated and the stability results for the closed-loop system are given by taking the presented MPC which explicitly consider the input and state constraints. The numerical example illustrates the effectiveness of the proposed techniques.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61102145, 61374093, and 61304197), the National Science and Technology Major Project (2013ZX03005005), the National Special Funds Program of Development of Internet of Things (2011.353), the Program of Development of Information Industry of Chongqing (2010.626), the Research Project of Chongqing Science and Technology Commission (cstc2013jcyjA40014), the Youth Science and Technology Innovation Talents Project of Chongqing Science and Technology Commission (cstc2013kjrc-qnrc40005), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ130506, KJ130516), and the Doctoral Start-Up Fund of Chongqing University of Posts and Telecommunications (A2013-14).