# Output Feedback Model Predictive Control for NCSs with Input Quantization

Authors: Hongchun Qu; Yu Li; Wei Liu
Journal: Complexity
 (2022)

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## Abstract

This paper addresses the robust output feedback model predictive control (MPC) schemes for networked control systems (NCSs) with input quantization. The logarithmic quantizer is considered in this paper, and the sector bound approach is applied, which appropriately treats the quantization error as a sector-bounded uncertainty. The presented method involves an offline designed state observer using linear matrix inequality (LMI) and online robust output feedback MPC algorithms which optimize one free control move followed by the output feedback using the estimated state. Moreover, due to the uncertainty of estimation error, a technique of refreshing the bound of estimation error which involves the quantization error is provided so as to guarantee the recursive feasibility of the optimization problem. The proposed MPC schemes inherit the characteristics of the synthesis approach of MPC, guaranteeing the recursive feasibility of the optimization problem and the stability of a closed-loop system, and explicitly account for quantization error. Two simulation examples are given to illustrate the effectiveness of the proposed methods.


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## Body

## 1. Introduction

Networked control systems (NCSs) are systems where the feedback loop is closed via a communication network in that information, including control inputs and plant outputs, is transmitted through the communication channel. NCSs are encountered in practice frequently because their structures are suitable and flexible [1]. Compared with traditional control systems, NCSs have a lot of advantages, such as lower installation costs, easy system maintenance, and enhanced reliability. However, some defects exist in NCSs, for example, data dropout, quantization, and network-induced delay, which may be caused by poor network traffic and make it more difficult to analyze the stability and design of the controller. In the past few decades, NCSs have received increasing attention and have been investigated by many researchers. Three foregoing common phenomena have been studied, and the methods to deal with these problems have been found by Li et al. [2]; Wu et al. [3]; and Yang and Zheng [4]. Moreover, system modeling and stability analysis have been studied by Liu and Yu [5] and Rouamel et al. [6], and the system controller design problem has also been investigated by Jiang and Zhang [7]; Fan [8]; and Deng and Cui [9]. In addition, reducing the burden of network communication in the network control system is also one of the hot research issues. An event-triggered mechanism with relative thresholds was introduced by Liang et al. [10, 11], and a new algorithm was proposed.As we all know, quantization is inevitable for NCSs when the information is transmitted. Sixty years ago, the research on quantization had already been carried out by Kalma [12]. Among various quantizers, two types of quantizers are often used. One is memoryless and static quantizers, and the other is dynamic and time-varying quantizers [13–15]. For the first type, the signal is quantized in the status of offline and its parameters are consistent. Logarithmic quantizer is one of this type. In many literature studies [16, 17], logarithmic quantizers have been investigated and it was also proved that they are the best quantizers to guarantee the quadratic stability [18]. Besides, the logarithmic quantizer can model the quantization errors, so we can use robust control methods to handle the errors easily. For the second type, parameters of quantizers would change as time goes on. State feedback controller and dynamic output feedback H∞ controller were designed based on dynamic and time-varying quantizers by Che and Yang [19, 20]. In addition, strategies to guarantee the asymptotic stability of the closed-loop system were proposed by Fu and Xie [21]. Undoubtedly, many perfect results have been achieved based on quantization for NCSs, but there are few papers that take model predictive control (MPC) into consideration, which is an effective technique and widely used in the industrial process.Actually, MPC is a controller that is used to deal with the optimal control problem of a constraint system with multiple variables. The idea of MPC is based on online repeated optimization of the sliding time domain to obtain the approximate solution of the optimal control problem with infinite time constraints. The optimal control sequence of the system is achieved by optimizing an objective function with some physical constraints. But, only the first control in this sequence is carried out. At the next sampling time, by repeating the calculation process based on the new measured data, we can get the control law [22]. In the past decades, it is reported that MPC has solved many constrained optimal control problems in various fields, such as advanced manufacturing, oil fractionation, environment, aerospace, and medical treatment [23]. In previous works of MPC, two main paradigms, polytopic paradigm and structured feedback uncertainty, are often mentioned in many papers 24, 25 Furthermore, one free control move had been introduced to solve the MPC problem, which was a very significant way to improve the performance of the system [26, 27]. Due to its ability to handle the input and output constraints, MPC is often used to cope with the problem of NCSs currently. In [28, 29], the state feedback MPC problem for NCSs has been studied. Especially in the latter, the authors present a new way for the analysis and synthesis of discrete-time linear systems with control input quantization over a digital network. The sector bound approach was also used which was put forward by Fu and Xie [31] so as to deal with the quantization density conveniently.However, as a matter of fact, the state of the closed-loop system is always not fully available in industrial systems which is often related to the physical constraints, and the methods of state feedback for NCSs in [30] is infeasible. Thus, in [33] and [34], the uncertain nonlinear system was studied based on the high gain (HG) observer. These studies showed that it is an essential matter to make further research on output feedback MPC for NCSs based on unmeasurable state.Motivated by the above discussion, this paper aims to undertake the robust output feedback MPC for NCSs with input quantization. (1) The control input is quantized by a logarithmic quantizer before transmitting, and the quantization error is represented by sector-bound uncertainty using the sector-bound approach, which can be easily dealt with by applying the robust control theory. (2) Two output feedback MPC techniques, which optimize the one free control move followed by output feedback, are presented, based on an offline design of a robust state observer. (3) A technique for refreshing the bound of estimation error, which involves data quantization, is provided and it plays a key role in guaranteeing the recursive feasibility of the whole optimization problem.Notation:I is the identity matrix with appropriate dimension. Rn and Rn×m denote n-dimensional Euclidean space and the set of n×m real matrices, respectively. For any vector x, xk+i|k is the value of vector x at the future time k+i predicted at time k, where i≥0 represents the prediction time. The symbol ∗ stands for a symmetric element or submatrix. Q⋅ denotes the quantizer, and V⋅ denotes the quadratic function.


## 2. Problem Formulation

Consider a linear time-varying (LTV) system:(1)xk+1=Akxk+Bkuk,yk=Ckxk,k≥0,Ak,Bk,Ck∈Ω,where xk∈Rnx is the state, uk∈Rnu is the input, yk∈Rny is the output, and(2)Ω=CoA1,B1,C1,A2,B2,C2,…,Am,Bm,Cm.Here, Co represents the convex hell. If A,B,C∈Ω, then the following expression can be obtained:(3)A,B,C=∑r=1mλrAr,Br,Cr,where λ1,λ2,…,λm are nonnegative and m=1 corresponds to the nominal linear time-invariant (LTI) system.The input constraint is also considered:(4)−u˜≤u≤u˜,where u˜=u˜1u˜2⋯u˜nuT and u˜s>0,s∈1,…,nu.The logarithmic quantizer is denoted asQ⋅=Q1⋅,Q2⋅,…,Qnu⋅, and the set of quantization level is presented as follows:(5)ϑ=±ηj:ηj=çjη0,j=0,±1,±2,…∪0,η0>0,where ç∈0,1 is a scalar and each quantization level ηj corresponds to a segment such that the quantizer maps the whole segment.As in [31], the logarithmic quantizer, shown in Figure 1, is defined as follows:(6)Qiθ=çijη0i,if11+δiçijη0i<θ<11−δiçijη0i,θ>00,ifθ=0,−Qi−θ,ifθ<0,where δi=1−çi/1+çi and δ=diagδ1,δ2,…,δnu is quantization density. It is observed that the quantizer Qi⋅, i=1,2,…,nu, is symmetric, namely, Qi−θ=−Qiθ, and time invariant.Figure 1
Logarithmic quantizer.In this paper, we consider the observer-based quantized feedback control, as shown in Figure2. The offline designed state observer is applied, and the estimated state x^k is obtained. Then, vk, which is determined by estimated state x^k and feedback gain Fk, is quantized by the quantizer. From the system, the output yk can be achieved.Figure 2
Block diagram of observer-based quantized feedback control.The input quantizer is modeled byuk=Qvk. Using the sector-bound approach in [31], the control input can be expressed as(7)uk=Qvk=I+Δkvk,where(8)Δk=diagΔ1k,Δ2k,…,Δnuk,Δik≤δi,Δk∈−δ,δ.


## 3. Problem Statement for Output Feedback MPC of NCSs with Input Quantization

### 3.1. Offline Design of the State Observer

In this paper, the state is unmeasurable and, hence, we take the following state observer to estimate the uncertain state:(9)x^k+1=Akx^k+Bkuk+Lpyk−Ckx^k=Akx^k+BkI+Δkvk+Lpyk−Ckx^k,where x^k and Lp are the estimated state and observer gain, respectively. Define the estimation error as ek=xk−x^k. Thus, from system dynamics and state observer (9), the dynamics of e can be derived as(10)ek+1=Ak−LpCkek.By combining (9) and (10), we can obtain the following augmented closed-loop system:(11)x^k+1ek+1=AkLpCk0Ak−LpCkx^kek+Bk0I+Δkvk,k≥0.In this paper, first of all, we should get the observer gainLp in terms of LMI. Thus, a quadratic function of the estimation error is defined.(12)Eek=ekTPekek,Pek>0.For the quadratic function, we need to guarantee its stability. It means thatek⟶0 when k⟶∞. Then, Eek must satisfy the following condition:(13)Eek+1−ρEek≤0,where ρ0<ρ<1 is the decay rate determined by the user, which is used in designing of the observer. Then, the observer gain Lp is presented by the following theorem.Theorem 1.
LetHek=PekLp, if there exists Pek>0 satisfying the following LMI constraint:(14)ρPek∗PekAs−HekCsPek≥0,s=1,2,…,m.
Then, the stability of error dynamics (10) is guaranteed, with the observer gain Lp given by Lp=Pe−1kHek.Proof.
Substituting the quadratic functionEek into (13), then we can obtain the following inequality:(15)ρPek−(Ak−LpCk)TPekAk−LpCk≥0.
The previous inequality is equivalent to(16)ρPek−PekAk−LpCkTPe−1kPekAk−LpCk≥0.
By utilizing Schur complement, (16) can be transformed into(17)ρPek∗PekAk−LpCkPek≥0.
Because ofHek=PekLp, (17) is equivalent to(18)ρPek∗PekAk−HekCkPek≥0.
Inequality (18) is affine in Ak,Ck. Thus, it is satisfied for all Ak,Ck∈Ω=CoA1,C1,A2,C2,…,Am,Cm, if and only if (14) is established and the proof is completed.AlthoughLp=Pe−1kHek is time invariant, Pek and Hek are time varying. Thus, we can choose the feasible values of Lp, Pe0, and He0 by means of Theorem 1 which satisfy Hek=KkHe0 and Pek=KkPe0, where Kk>0 is a time-varying scalar.


### 3.2. Output Feedback MPC Problem for NCS with Input Quantization

In the MPC scheme with quantized inputs, the control inputuk|k is the first sequence that is to be computed, i.e., uk=uk|k=I+Δkvk|k. The rest of the future control sequence uk+i|k is computed as follows:(19)uk+i|k=I+Δkvk+i|k=I+ΔkFkx^k+i|k,i≥1,where Fk is the feedback gain determined at sampling instant k. For an optimization problem of output feedback MPC, uk+i|k should satisfy the following constraint:(20)−u˜≤uk+i|k≤u˜,i≥0,so as to guarantee (4) by receding horizon implementation. Like (11), the augmented prediction model is obtained as follows:(21)x^k+1+i|kek+1+|k=Ak+i|k+Bk+i|kI+ΔkFk0LpCk+i|kAk+i|k−LpCk+i|kx^k+i|kek+i|k,i≥0.In this paper, the quasi-min-max optimization problem of the infinite horizon output feedback MPC for NCSs is as follows:(22)minuk+i|kmaxΔk,Ak,Bk,Ck∈ΩJ01k+J1∞k,J01k=x^k|kTLx^k|k+I+Δkvk|kTR(I+Δk)vk|k,J1∞k=∑i=1∞x^k+i|kTLx^k+i|k+(I+Δk)vk+i|kTR(I+Δk)vk+i|k,where L>0 and R>0 are weighting matrices of performance objective function.The main issue of MPC is to obtain the optimal solution of (22). First, define the following quadratic function:(23)Vx^k+i|k,ek+i|k=x^k+i|kek+i|kTP^k00Pekx^k+i|kek+i|k,where P^k>0andPek>0. At sampling instant k, we impose the following constraint:(24)Vx^k+1+i|k,ek+1+i|k−Vx^k+i|k,ek+i|k≤−x^k+i|kTLx^k+i|k+(I+Δk)vk+i|kTR(I+Δk)vk+i|k,for any i>0. For asymptotic stability, it has x^∞|k=0 and Vx^∞|k=0. By summing (24) from i=1 to ∞, we have(25)maxΔk,Ak,Bk,Ck∈ΩJ1∞≤Vx^k+1|k,ek+1|k.Let the upper boundγk satisfy(26)x^k|kTLx^k|k+I+Δkvk|kTRI+Δkvk|k+x^k+1|kTP^kx^k+1|k+ek+1||kTPekek+1|k≤γk.Here, γk is a nonnegative variable to be minimized; then, the problem of output feedback MPC for NCS with input quantization (22) is transformed into(27)minuk|k,P^k,Fkγk.This is subjected to (19-21), (24), and (26).


## 3.1. Offline Design of the State Observer

In this paper, the state is unmeasurable and, hence, we take the following state observer to estimate the uncertain state:(9)x^k+1=Akx^k+Bkuk+Lpyk−Ckx^k=Akx^k+BkI+Δkvk+Lpyk−Ckx^k,where x^k and Lp are the estimated state and observer gain, respectively. Define the estimation error as ek=xk−x^k. Thus, from system dynamics and state observer (9), the dynamics of e can be derived as(10)ek+1=Ak−LpCkek.By combining (9) and (10), we can obtain the following augmented closed-loop system:(11)x^k+1ek+1=AkLpCk0Ak−LpCkx^kek+Bk0I+Δkvk,k≥0.In this paper, first of all, we should get the observer gainLp in terms of LMI. Thus, a quadratic function of the estimation error is defined.(12)Eek=ekTPekek,Pek>0.For the quadratic function, we need to guarantee its stability. It means thatek⟶0 when k⟶∞. Then, Eek must satisfy the following condition:(13)Eek+1−ρEek≤0,where ρ0<ρ<1 is the decay rate determined by the user, which is used in designing of the observer. Then, the observer gain Lp is presented by the following theorem.Theorem 1.
LetHek=PekLp, if there exists Pek>0 satisfying the following LMI constraint:(14)ρPek∗PekAs−HekCsPek≥0,s=1,2,…,m.
Then, the stability of error dynamics (10) is guaranteed, with the observer gain Lp given by Lp=Pe−1kHek.Proof.
Substituting the quadratic functionEek into (13), then we can obtain the following inequality:(15)ρPek−(Ak−LpCk)TPekAk−LpCk≥0.
The previous inequality is equivalent to(16)ρPek−PekAk−LpCkTPe−1kPekAk−LpCk≥0.
By utilizing Schur complement, (16) can be transformed into(17)ρPek∗PekAk−LpCkPek≥0.
Because ofHek=PekLp, (17) is equivalent to(18)ρPek∗PekAk−HekCkPek≥0.
Inequality (18) is affine in Ak,Ck. Thus, it is satisfied for all Ak,Ck∈Ω=CoA1,C1,A2,C2,…,Am,Cm, if and only if (14) is established and the proof is completed.AlthoughLp=Pe−1kHek is time invariant, Pek and Hek are time varying. Thus, we can choose the feasible values of Lp, Pe0, and He0 by means of Theorem 1 which satisfy Hek=KkHe0 and Pek=KkPe0, where Kk>0 is a time-varying scalar.


## 3.2. Output Feedback MPC Problem for NCS with Input Quantization

In the MPC scheme with quantized inputs, the control inputuk|k is the first sequence that is to be computed, i.e., uk=uk|k=I+Δkvk|k. The rest of the future control sequence uk+i|k is computed as follows:(19)uk+i|k=I+Δkvk+i|k=I+ΔkFkx^k+i|k,i≥1,where Fk is the feedback gain determined at sampling instant k. For an optimization problem of output feedback MPC, uk+i|k should satisfy the following constraint:(20)−u˜≤uk+i|k≤u˜,i≥0,so as to guarantee (4) by receding horizon implementation. Like (11), the augmented prediction model is obtained as follows:(21)x^k+1+i|kek+1+|k=Ak+i|k+Bk+i|kI+ΔkFk0LpCk+i|kAk+i|k−LpCk+i|kx^k+i|kek+i|k,i≥0.In this paper, the quasi-min-max optimization problem of the infinite horizon output feedback MPC for NCSs is as follows:(22)minuk+i|kmaxΔk,Ak,Bk,Ck∈ΩJ01k+J1∞k,J01k=x^k|kTLx^k|k+I+Δkvk|kTR(I+Δk)vk|k,J1∞k=∑i=1∞x^k+i|kTLx^k+i|k+(I+Δk)vk+i|kTR(I+Δk)vk+i|k,where L>0 and R>0 are weighting matrices of performance objective function.The main issue of MPC is to obtain the optimal solution of (22). First, define the following quadratic function:(23)Vx^k+i|k,ek+i|k=x^k+i|kek+i|kTP^k00Pekx^k+i|kek+i|k,where P^k>0andPek>0. At sampling instant k, we impose the following constraint:(24)Vx^k+1+i|k,ek+1+i|k−Vx^k+i|k,ek+i|k≤−x^k+i|kTLx^k+i|k+(I+Δk)vk+i|kTR(I+Δk)vk+i|k,for any i>0. For asymptotic stability, it has x^∞|k=0 and Vx^∞|k=0. By summing (24) from i=1 to ∞, we have(25)maxΔk,Ak,Bk,Ck∈ΩJ1∞≤Vx^k+1|k,ek+1|k.Let the upper boundγk satisfy(26)x^k|kTLx^k|k+I+Δkvk|kTRI+Δkvk|k+x^k+1|kTP^kx^k+1|k+ek+1||kTPekek+1|k≤γk.Here, γk is a nonnegative variable to be minimized; then, the problem of output feedback MPC for NCS with input quantization (22) is transformed into(27)minuk|k,P^k,Fkγk.This is subjected to (19-21), (24), and (26).


## 4. Online Synthesis Approaches of Output Feedback MPC for NCSs with Input Quantization

Next, two LMI approaches for quasi-min-max optimization problem (27) of the infinite-horizon MPC scheme are described.Before proceeding, we introduce the following lemma [35].Lemma 1.
For anyd,z∈Rn and any positive definite matrix w∈Rn×n, the following inequality holds:(28)2dTz≤dTw−1d+zTwz.Theorem 2.
Letx^k be the estimated state of (9) at sampling instant k. With the given logarithmic quantizer, optimization problem (27) to find control input uk|k and feedback gain Fk can be solved by(29)minMk,Sk,Zk,Yk,Nk,uk|kγk.Subject to (4), (30) and (31) are obtained:(30)αk−1∗∗∗∗βkζk∗∗∗L1/2x^k0−γkI∗∗R1/2Fkx^kηk0ϑk∗Fkx^k000−Sk<0,s=1,2,…,m,where βk=Asx^k+BsFkx^k+LpCsek, ζk=−Qk+BsδSkδBsT, ηk=R1/2δSkδBsT, and ϑk=−γkI+R1/2δSkδR1/2.(31)−Qk∗∗∗∗∗0ℓ¯k∗∗∗∗κ¯kLpCsϖ¯k∗∗∗L1/2Q00−γkI∗∗R1/2Gk0Λ¯k0τk∗Gk0000−Mk<0,s=1,2,…,m.,where ℓ¯k=αk/Tk+1−Pe0+As−LpCsTPe0As−LpCs, κ¯k=AsQk+BsGk, ϖ¯k=−Qk+BsδMkδBsT, τk=−γkI+R1/2δMkδR1/2, and Λ¯k=R1/2δMkδBsT.If there exist matricesGk>0,Qk>0, control input uk|k, and a scalar γk>0, the feedback control gain Fk is obtained by(32)Fk=GkQ−1k.At each sampling instantk≥1, the bound of ek+1 is presented as follows:(33)ek+1TPe0ek+1≤Tk+1,Tk+1=minT¯k+1Kk−1,λTk.T¯k+1=γk−1−x^k−1TLx^k−1+I+Δkvk−1TRI+Δkvk−1−x^kTLx^k−I+Δkvk|k−1TRI+Δkvk|k−1−x¯k+1|k−1TP^k−1x¯k+1|k−1,where(34)x¯k+1|k−1=Ak+BkI+ΔkFk−1x^k+Lpyk−Ckx^k.Proof.
LetΔ∗k:=argmaxΔ∈−δ,δJ01k+J1∞, ℵk=αkγk/Tk+1, and Pek=Pe0/Tk+1αkγk. Due to x^k|k=x^k and ek|k=ek, from (9), (26) can be rewritten as(35)x^kTLx^k+I+Δ∗kvk|kTRI+Δ∗kvk|k+Akx^k+BkI+Δ∗kFkx^k+LpC(k)ekTP^k×[Akx^k+BkI+Δ∗kFkx^k+LpCkek+Ak−LpCkekTPekAk−LpCkek]≤γk,where vk|k=Fkx^k|k=Fkx^k. It is noted that ek+1=ek+1|k=Ak−LpCkek. Because of Pek=Pe0/Tk+1αkγk and ek+1TPe0ek+1≤Tk+1, (35) can be written as(36)x^kTLx^k+(I+Δ∗k)vk|kTR(I+Δ∗k)vk|k+Akx^k+BkI+Δ∗k+LpCkekTP^k×Akx^k+BkI+Δ∗k+LpCkek≤1−αkγk.
By multiplyingγk−1, we can obtain the following inequality:(37)αk−1+γk−1x^kTLx^k+γk−1(I+Δ∗k)vk|kTR(I+Δ∗k)vk|k+γk−1Akx^k+BkI+Δ∗k+LpCkekTP^k×Akx^k+BkI+Δ∗k+LpCkek≤0.
By takingP^k=γkQk−1, we have(38)αk−1∗ℑk−Qk+L1/2x^k0R1/2I+ΔkFkx^k0T×γkI∗0γkIL1/2x^k0R1/2I+ΔkFkx^k0<0,where ℑk=Akx^k+BkI+ΔkFkx^k+Lpyk−Ckx^k. By Schur complement, we can get(39)αk−1∗∗∗χk−Qk∗∗L1/2x^k0−γkI∗R1/2Fkx^k00−γkI+0∗∗∗BkΔ∗kFkx^k0∗∗000∗R1/2Δ∗kFkx^k000<0,where χk=Akx^k+BkFkx^k+LpCkek. (39) is equivalent to the following inequality:(40)αk−1∗∗∗χk−Qk∗∗L1/2x^k0−γkI∗R1/2Fkx^k00−γkI+0Bk0R1/2Δ∗kFkx^k000+(Fkx^k)T000Δ∗k0BTk0R1/2<0.
According to Lemma1, we have(41)0Bk0R1/2Δ∗kFkx^k000+(Fkx^k)T000Δ∗k0BTk0R1/2≤0Bk0R1/2δSkδ0BTk0R1/2+(Fkx^k)T000S−1kFkx^k000,where Sk is a positive matrix. Condition (40) holds if(42)αk−1∗∗∗χk−Qk∗∗L1/2x^k0−γkI∗R1/2Fkx^k00−γkI+0Bk0R1/2δSkδ0BTk0R1/2+(Fkx^k)T000S−1kFkx^k000<0,
Because inequality (42) is affine in Ak,Bk,Ck, it is satisfied for all Ak+i|k,Bk+i|k,Ck+i|k∈Ω=CoA1,B1,C1,A2,B2,C2,…,Am,Bm,Cm, by Schur complement, if and only if (30) is established.
Then, we need to prove that (31) is satisfied, which denotes the robust stability constraint for system (1). According to (9), (23), and (24), we have(43)x^Tk+i|k−P^k+Lx^k+i|k+x^Tk+i|kFTkI+ΔkTRI+Δk×Fkx^k+i|k+[Ak+i|kx^k+i|k+Bk+i|kI+ΔkFkx^k+i|k+LpCk+i|kek+i|k]TP^k[Ak+|kx^k+i|k+Bk+i|kI+ΔkFk×x^k+i|k+LpCk+i|kek+i|k]+ek+i|kT[Ak+i|k−LpCk+i|kT×PekAk+i|k−LpCk+i|k−Pek]ek+i|k<0.
Constraint (43) is satisfied for all admissible x^k+i|k and ek+i|k, if and only if(44)−P^k+ℏk∗0−Pek+ℶkLpCk+i|k0Ak+i|k−LpCk+|kT×P^k00PekℶkLpCk+i|k0Ak+i|k−LpCk+i|k<0,where ℏk=L+FkT(I+Δk)TRI+ΔkFk and ℶk=Ak+i|k+Bk+i|kI+ΔkFk. (44) can be transformed into(45)−P^k+ℏk∗0ℒk+ℶkLpCk+i|kTP^kℶkLpCk+i|k<0,where ℐk=−Pek+Ak+i|k−LpCk+i|kTPekAk+i|k−LpCk+i|k, ℶk=Ak+i|k+Bk+i|kI+ΔkFk, and P^k is nonsingular. Considering Qk=γkPk−1, multiplying by γk−1, we can obtain(46)−Qk−1+γk−1ℏk∗0γk−1Jk+JkLpCk+i|kT×Qk−1JkLpCk+i|k<0.
Because ofℵk=αkγk/Tk+1, Pek=Pe0/Tk+1αkγk, (47) is achieved:(47)−Qk−1+γk−1ℏk∗∗0ℓk∗ℶkLpCk+i|k−Qk<0,where ℓk=αk/Tk+1−Pe0+Ak+i|k−LpCk+i|kTPe0Ak+i|k−LpCk+i|k.
Premultiplying and postmultiplying bydiagQk,I,I, (47) becomes(48)−Qk+Qkγk−1ℏkQk∗∗0ℓk∗ℶkQkLpCk+i|k−Qk<0.
We setFkQk=Gk; then, we have(49)−Qk∗∗0ℓk∗Ak+i|kQk+Bk+i|kI+ΔkGkLpCk+i|k−Qk+L1/2Qk00R1/2I+ΔkGk00Tγk−1I00γk−1IL1/2Qk00R1/2I+ΔkGk00<0.
By Schur complement, (49) becomes(50)−Qk∗∗∗∗0ℓk∗∗∗ΓkLpCk+i|k−Qk∗∗L1/2Qk00−γkI∗R1/2I+ΔkGk000−γkI<0,where Γk=Ak+i|kQk+Bk+i|kI+ΔkGk.
(50) can be decomposed into the following form:(51)−Qk∗∗∗∗0ℓk∗∗∗Ak+i|kQk+Bk+i|kGkLpCk+i|k−Qk∗∗L1/2Qk00−γkI∗R1/2Gk000−γkI+00Bk+i|k0R1/2ΔkGk0000+GTk0000Δk00BTk+i0R1/2<0.
Because ofΔk≤δ, from Lemma 1, we have(52)00Bk+i|k0R1/2ΔkGk0000+GTk0000Δk00BTk+i|k0R1/2≤00Bk+i|k0R1/2δMkδ00BTk+i|k0R1/2+GTk0000M−1kGk0000,where Mk is a positive definite matrix. Condition (51) holds if(53)−Qk∗∗∗∗0ℓk∗∗∗Ak+i|kQk+Bk+i|kGkLpCk+i|k−Qk∗∗L1/2Qk00−γkI∗R1/2Gk000−γkI+00Bk+i|k0R1/2δMkδ00BTk+i|k0R1/2+GTk0000M−1kGk0000<0.
By using Schur complement, we have(54)−Qk∗∗∗∗∗0ℓk∗∗∗∗κkLpCk+i|kϖk∗∗∗L1/2Q00−γkI∗∗R1/2Gk0Λk0τk∗Gk0000−Mk<0,where ℓk=αk/Tk+1−Pe0+Ak+i|k−LpCk+i|kTPe0Ak+i|k−LpCk+i|k, κk=Ak+i|kQk+BkGk, ϖk=−Qk+Bk+i|kδMkδBTk+i|k, τk=−γkI+R1/2δMkδR1/2, and Λk=R1/2δMkδBTk+i|k.
Inequality (54) is affine in Ak+i|k,Bk+i|k,Ck+i|k. Thus, it is satisfied for all Ak+i|k,Bk+i|k,Ck+i|k∈Ω=CoA1,B1,C1,A2,B2,C2,…,Am,Bm,Cm if and only if(55)−Qk∗∗∗∗∗0ℓ¯k∗∗∗∗κ¯kLpCsϖ¯k∗∗∗L1/2Q00−γkI∗∗R1/2Gk0Λ¯k0τk∗Gk0000−Mk<0.
Hence, (31) is achieved and the feedback control gain is given by Fk=GkQ−1k, and the proof is completed. □Remark 1.
Inequality (31) denotes the robust stability constraint for the NCSs. According to Theorem 2, if there exist Mk>0,Sk>0,Gk>0,andWk>0 such that (29) is satisfied, the closed-loop system is asymptotically stable. We also can get the control input uk and feedback gain Fk.
Next, we provide another theorem, which would further reduce the conservativeness.Theorem 3.
Letx^k be the estimated state of (9) at sampling instant k. With the given logarithmic quantizer, optimization problem (27) to find control input uk|k and feedback gain Fk, which minimize the worst-case upper bound γk, can be solved by the following optimization problem:(56)minXk,Ok,Mk,Sk,uk|kγk.
Subject to (4), (30) becomes the following equation:(57)ϱk∗∗∗∗∗0ℓ¯k∗∗∗∗∗φkLpCsϖk∗∗∗L1/2Xk00−γkI∗∗R1/2Ok0Λk0τk∗Ok0000−Mk<0,s=1,2,…,m.,where ℓ¯k=αk/Tk+1−Pe0+As−LpCsTPe0As−LpCs, ϱk=−Xk−XTk+Qk, φk=AsXk+BsOk, ϖ¯k=−Qk+BsδMkδBsT, τk=−γkI+R1/2δMkδR1/2, and Λ¯k=R1/2δMkδBsT.
If there exist matricesSk>0,Xk>0,Ok>0,Mk>0,andWk>0, control input uk|k, and a scalar γk>0, the feedback control gain Fk is obtained by(58)Fk=OkX−1k.Proof.
In order to prove (57) (implying (31)), first, we can see that the element (1, 1) of (57) implies −Xk−XTk+Qk<0. For Xk>0, it has Xk−QkT(−Qk)−1Xk−Qk≤0, which means(59)−XkTQk−1Xk≤−Xk−XTk+Qk.
According to (59), we can obtain the following inequality:(60)−XkTQk−1Xk∗∗∗∗∗0ℓ¯k∗∗∗∗φkLpCsϖ¯k∗∗∗L1/2Xk00−γkI∗∗R1/2Ok0Λ¯k0τ¯k∗Ok0000−Mk<0.
Premultiplying bydiagQkXk−T,I,I,I,I,I and postmultiplying by diagX−1kQk,I,I,I,I,I, we have(61)−Qx∗∗∗∗∗0ℓ¯k∗∗∗∗ℸkLpCsϖ¯k∗∗∗L1/2Qk00−γkI∗∗R1/2FkQk0Λ¯k0τ¯k∗FkQk0000−Mk<0,where ℸk=AsQk+BsFkQk.
DefiningGk=FkQk, (61) can be transformed into (31), and the proof is completed. □Remark 2.
Similar to Theorem2, (57) guarantees the asymptotical stability of closed loop (1). In essence, (31) can be transformed into (57) by choosing Xk=XTk=Gk>0Xk=XTk=Gk>0; that is to say, the constraints of Theorem 3 are less conservative than those in Theorem 2.
The following assumption is made to give the bound ofek:Assumption 1. e1TPe0e1≤T1, where T1 is a constant and given by users.Assumption 2. If (4), (30), (31), and (57) are satisfied, then ek+1TPe0ek+1≤Tk+1 for all k≥1, where Tk+1 is prespecified according to (33) and (34).Remark 3.
Assumption 1 combined with refreshing formulas (33) and (34) gives the bound of estimation error at each sampling instant k. The estimation error bound is closely related to the recursive feasibility of the optimization problem of output feedback MPC, which can be seen in the proof of Theorem 4 (Feasibility and Stability Analysis of MPC for NCSs).Proof.
According to (23) and (24), the following inequality is obtained:(62)x^k+1|kuk+1|kTL00Rx^k+1|kuk+1|k+x^k+2|kek+2|kTP^k00Pek×x^k+2|kek+2|k≤x^k+1|kek+1|kTP^k00Pekx^k+1|kek+1|k,where uk+1|k=I+Δkvk=I+ΔkFkx^k. Let γ˜k+1=γk−x^kTLx^k−((I+Δk)Fkx^k)TRI+ΔkFkx^k; if (30) is satisfied at sampling instant k, then applying (35) yields(63)Akx^k+BkI+ΔkFkx^k+LpCkekek+1TP^k00Pek×Akx^k+BkI+ΔkFkx^k+LpCkekek+1≤γ˜k+1.
Due tox^k+1|k=Akx^k+BkI+ΔkFkx^k+LpCkek, (63) can be written as(64)x^k+1|ke(k+1))TP^k00Pekx^k+1|kek+1≤γ˜k+1.
At sampling instantk, if (4), (30), (31), and (57) are satisfied, then applying (62) and (64) and taking ek+1|k=ek+1 and x^k+1|k=x^k+1 into account, the following inequality is achieved:(65)ek+2|kTPekek+2|k≤γ˜k+1−x^k+1TLx^k+1−I+ΔkFkx^kTRI+ΔkFkx^k−x^k+2|kTP^kx^k+2|k.
In the prediction ofek+2|k=Ak+1|k−LpCk+1|kek+1|k and ek+2=ek+2|k+1=Ak+1−LpCk+1ek+1, we can conclude that all the possible values of ek+2 are included in ek+2|k. Thus, (65) can be transformed into(66)ek+2TPekek+2≤γ˜k+1−x^k+1TLx^k+1−I+ΔkFkx^kTRI+ΔkFkx^k−x^k+2|kTP^kx^k+2|k.
In addition, consider the following conditions:(67)x^k+2|k=Ak+1+Bk+1I+ΔkFkx^k+1+LpCk+1ek+1x¯k+2|k=Ak+1+Bk+1I+ΔkFkx^k+1+LpCk+1ek+1.
(66) can be transformed into(68)ek+2TPekek+2≤T¯k+2=γ˜k+1−x^k+1TLx^k+1−x^k+2|kTP^kx^k+2|k−I+ΔkFkx^kTRI+ΔkFkx^k%,where T¯k+2 is given according to (33). Taking Pek=ℵkPe0 into (68), the following inequality is obtained:(69)ek+2TPe0ek+2≤T¯k+2Kk.
When applying Theorem2, the following inequality must be satisfied at sampling instant k+1:(70)ek+2TPe0ek+2≤λTk+1.
According to (69) and (70), we have(71)Tk+1=minT¯k+1Nk−1,λTk.
Based on the above analysis, it is apparent that, for allk≥0, Tk+1 can be prespecified on the basis of (33) and (34), so that ek+1TPe0ek+1≤Tk+1. Thus, the proof is completed. □The algorithm of output feedback model predictive control for NCSs with input quantization can be summarized as follows:Step 1. Select the parameter ρ∈0,1, calculate the optimization problem described by Theorem 1, and get the values of Pe and LpStep 2. Set parameters L, R, x0, x^0, u˜, δ, and T1Step 3. At the sampling time k≥0, the optimization problem of Theorem 3 is solved online and the control input uk is obtainedStep 4. Calculate Tk+1 to estimate the states x^k+1|k and x^k+2|k)Step 5. Update the sampling time to k+1, and then return to Step 3


## 5. Feasibility and Stability Analysis of MPC for NCSs

### 5.1. The Analysis of Feasibility of MPC for NCSs

Theorem 4.
Consider LTV system (1). At each sampling instant k≥0, (29) or (56) is solved, and for all k≥1, Tk is prespecified according to (33) and (34). If a feasible solution can be determined at sampling instant k, then (29) or (56) is feasible for all future sampling instants.Proof.
In this theorem, we just need to prove that the solution is obtained from optimization problem at sampling instantk; a feasible solution also can be achieved at sampling instant k+1. We set that uk+1|k+1=uk+1|k=I+ΔkFkx^k+1|k and αk+1=Tk+2/Tk+1αk. The only thing we need to do is to check (4) and (30).
As depicted in Fu and Xie [31], for the linear time-varying system, the method which we use to deal with quantization density is also suitable for the output feedback MPC problem. It is obvious that uk+1|k+1=uk+1|k=I+ΔkFkx^k+1|k, and we can conclude that (4) is feasible at sampling instant k+1. Then, we check (30) and use the relevant result of Assumption 2. Considering Pek=Pe0/Tk+1αkγk and (33), (67) can be written as(72)Tk+2≤Tk+1αkγk[γ˜k+1−x^k+1TLx^k+1−uk+1|kTRuk+1|k−x¯k+2|kTP^kx¯k+2|k].
If we takeuk+1=uk+1|k=I+Δk+1Fkx^k+1 and Fk+1=Fk, thus x¯k+2|k=x^k+2|k+1=ℑk+1 and (72) becomes(73)Tk+2≤Tk+1αkγk[γ˜k+1−x^k+1TLx^k+1−(I+Δk+1Fk+1×x^k+1)TRI+Δk+1Fk+1x^k+1−ik+1TP^kik+1].
We can obtainP^k+1≤P^k if we take γk+1=γ¯k+1 and Qk+1=Qk at sampling instant k+1, and (73) guarantees that(74)Tk+2≤Tk+1αkγk[γk+1−x^k+1TLx^k+1−(I+Δk+1Fk+1×x^k+1)TRI+Δk+1Fk+1x^k+1−ℑk+1TP^k+1ℑk+1].
If we chooseαk+1/Tk+2=αk/Tk+1 at sampling instant k+1, (74) can be turned into(75)x^k+1TLx^k+1+ℑk+1TP^k+1ℑk+1+(I+Δ(k+1Fk+1×x^k+1)TRI+Δk+1Fk+1x^k+1≤γk+1−αk+1γk.
Because ofγk+1≤γk, (75) can be presented as(76)x^k+1TLx^k+1+ℑk+1TP^k+1ℑk+1+(I+Δk+1Fk+1×x^k+1)TRI+Δk+1Fk+1x^k+1≤1−αk+1γk+1.
By using Lemma1 and Schur complement, we can find that (76) can be transformed into (30) with k replaced by k+1. Based on the above analysis, when (30) is feasible at sampling instant k, it is also feasible at instant k+1; in other words, the recursive feasibility of (29) or (56) is guaranteed and the proof is completed.


### 5.2. The Analysis of Stability of MPC for NCSs

Theorem 5.
At sampling instantk≥0, optimization problem (29) or (56) can be solved for the linear time-varying system. If (29) or (56) is feasible at sampling instant k=0, the system is asymptotically stable, i.e., xk⟶0 as k⟶∞.Proof.
As depicted in Fu and Xie [31], for linear time-varying systems, the method which we use to deal with quantization density is also suitable for output feedback MPC problem. At sampling instant k=0, if (29) or (56) is feasible according to Theorem 2 and Theorem 3, for all sampling instant k>0, it is also feasible. Thus, γk becomes a Lyapunov function of augmented closed-loop system (11). We can obtain the following inequality:(77)γk+1−γk≤−x^kTLx^k−((I+Δk)Fkx^k)TRI+ΔkFkx^k.
Thus, ifx^k≠0, γk will decrease with the speed γk+1−γk≤−x^kTLx^k until x^k=0 and γk=0. The estimation error ek evolves according to (11) so that ek⟶0 as k⟶∞. Hence, xk⟶0 as k⟶∞, and the proof is completed.


## 5.1. The Analysis of Feasibility of MPC for NCSs

Theorem 4.
Consider LTV system (1). At each sampling instant k≥0, (29) or (56) is solved, and for all k≥1, Tk is prespecified according to (33) and (34). If a feasible solution can be determined at sampling instant k, then (29) or (56) is feasible for all future sampling instants.Proof.
In this theorem, we just need to prove that the solution is obtained from optimization problem at sampling instantk; a feasible solution also can be achieved at sampling instant k+1. We set that uk+1|k+1=uk+1|k=I+ΔkFkx^k+1|k and αk+1=Tk+2/Tk+1αk. The only thing we need to do is to check (4) and (30).
As depicted in Fu and Xie [31], for the linear time-varying system, the method which we use to deal with quantization density is also suitable for the output feedback MPC problem. It is obvious that uk+1|k+1=uk+1|k=I+ΔkFkx^k+1|k, and we can conclude that (4) is feasible at sampling instant k+1. Then, we check (30) and use the relevant result of Assumption 2. Considering Pek=Pe0/Tk+1αkγk and (33), (67) can be written as(72)Tk+2≤Tk+1αkγk[γ˜k+1−x^k+1TLx^k+1−uk+1|kTRuk+1|k−x¯k+2|kTP^kx¯k+2|k].
If we takeuk+1=uk+1|k=I+Δk+1Fkx^k+1 and Fk+1=Fk, thus x¯k+2|k=x^k+2|k+1=ℑk+1 and (72) becomes(73)Tk+2≤Tk+1αkγk[γ˜k+1−x^k+1TLx^k+1−(I+Δk+1Fk+1×x^k+1)TRI+Δk+1Fk+1x^k+1−ik+1TP^kik+1].
We can obtainP^k+1≤P^k if we take γk+1=γ¯k+1 and Qk+1=Qk at sampling instant k+1, and (73) guarantees that(74)Tk+2≤Tk+1αkγk[γk+1−x^k+1TLx^k+1−(I+Δk+1Fk+1×x^k+1)TRI+Δk+1Fk+1x^k+1−ℑk+1TP^k+1ℑk+1].
If we chooseαk+1/Tk+2=αk/Tk+1 at sampling instant k+1, (74) can be turned into(75)x^k+1TLx^k+1+ℑk+1TP^k+1ℑk+1+(I+Δ(k+1Fk+1×x^k+1)TRI+Δk+1Fk+1x^k+1≤γk+1−αk+1γk.
Because ofγk+1≤γk, (75) can be presented as(76)x^k+1TLx^k+1+ℑk+1TP^k+1ℑk+1+(I+Δk+1Fk+1×x^k+1)TRI+Δk+1Fk+1x^k+1≤1−αk+1γk+1.
By using Lemma1 and Schur complement, we can find that (76) can be transformed into (30) with k replaced by k+1. Based on the above analysis, when (30) is feasible at sampling instant k, it is also feasible at instant k+1; in other words, the recursive feasibility of (29) or (56) is guaranteed and the proof is completed.


## 5.2. The Analysis of Stability of MPC for NCSs

Theorem 5.
At sampling instantk≥0, optimization problem (29) or (56) can be solved for the linear time-varying system. If (29) or (56) is feasible at sampling instant k=0, the system is asymptotically stable, i.e., xk⟶0 as k⟶∞.Proof.
As depicted in Fu and Xie [31], for linear time-varying systems, the method which we use to deal with quantization density is also suitable for output feedback MPC problem. At sampling instant k=0, if (29) or (56) is feasible according to Theorem 2 and Theorem 3, for all sampling instant k>0, it is also feasible. Thus, γk becomes a Lyapunov function of augmented closed-loop system (11). We can obtain the following inequality:(77)γk+1−γk≤−x^kTLx^k−((I+Δk)Fkx^k)TRI+ΔkFkx^k.
Thus, ifx^k≠0, γk will decrease with the speed γk+1−γk≤−x^kTLx^k until x^k=0 and γk=0. The estimation error ek evolves according to (11) so that ek⟶0 as k⟶∞. Hence, xk⟶0 as k⟶∞, and the proof is completed.


## 6. Illustrative Example

In this section, two examples are provided to illustrate the effectiveness of the proposed controller design methods.Example 1.
Consider a system with the weighting matricesR=1I and L=1I,(78)xk+1=θθ.=Akxk+Buk=0.850.242αk0.82xk+00.0787uk,yk=Cxk=10xk.
The parameterαk is an uncertain parameter, satisfying αk≤0.05; we conclude that Ak∈ΩA1,A2, where(79)A1=0.0850.24−0.10.82,A2=0.0850.240.10.82.
Thus, the uncertainty setΩ is a polytope as in (2). In order to undertake the simulation study, given an initial state of x0=0.79.5T, an initial estimated state is x^k=0.50.7T. The control inputs satisfy uk+i|k≤10. In addition, choose a logarithmic quantizer Q.. The control law is generated by minimizing the nominal unconstrained infinite horizon.
Table1 shows the values of quantization density. We take two different values of quantization density in each theorem. Figures 3 to 6 show the information of the state trajectory, estimated state trajectory, γk trajectory, and the control input signals. The major outcomes are illustrated with state and estimated state trajectories of NCS and the upper bound of the performance index under the design robust MPC.
As we can see from Figures3 and 4, when k⟶∞, state trajectory, estimated state trajectory, and γk trajectory converge to zero. It indicates that both theorems can get a good control performance index γγ⟶0 and the effectiveness and the stability of the system are guaranteed. Comparing Theorem 3 with Theorem 2, when k⟶∞, the speed of the state trajectory and estimated state trajectory of Theorem 2 converging to zero is slower than that of Theorem 3. It reflects that Theorem 3 is more effective than Theorem 2. In other words, when using Theorem 3, the system has better performance. From Figure 5, it is worthwhile mentioning that the smaller δ is, the faster it converges to 0. When there is a known disturbance ωk=0.1sink in the system, the simulation results are shown in Figures 7–10.Table 1
Values of quantization density.
Methodδ1ρ1δ2ρ2By Theorem20.675 6(0.193 6)0.393 6(0.435 1)By Theorem30.675 6(0.193 6)0.393 6(0.435 1)Figure 3
State and estimated state withδ=0.6756.Figure 4
State and estimated state withδ=0.3936.Figure 5
γk trajectory.Figure 6
Control input signals.Figure 7
State and estimated state withδ=0.6756.Figure 8
State and estimated state withδ=0.3936.Figure 9
γk trajectory.Figure 10
Control input signals.Example 2.
As given by Wie, Dennis, and Bernstein [36] and Wan and Kothare [37], consider the two-mass-spring systems shown in Figure 11. The system is represented in state-space form as(80)xk+1=x1k+1x2k+1x3k+1x4k+1=00100001−Km1Km100Km2−Km200x1kx2kx3kx4k+001m10u,yk=x2k,where m1 and m2 are the masses of the two bodies, x1, and x2 are the positions of body 1 and body 2, x3 and x4 are their velocities, respectively, and y is the sensor output. The parameter K is the spring constant and also an uncertain parameter satisfying 0.5≤K≤1.2, and we conclude that Ak∈ΩA1,A2, where(81)A1=00100001−0.50.500−0.50.500,A2=00100001−1.21.2001.21.200.
Setm1=m2=1 with appropriate units for simplicity. The control input constraints must satisfy |u|≤1. We choose the initial values as x0=−2.79.5−0.77.5T and x^k=−2.57.5−0.55.5T. Let R=1I and L=1I, and take the same values of quantization density as example 1. The state trajectory and its estimated state trajectory of the system are shown in Figures 11 to 15. γk trajectory and control input signals are shown in Figures 16 and 17. When there is a known disturbance ωk=0.1sink in the system, the simulation results are shown in Figures 18–20.
From Figures11 to 17, we can get the same conclusions as Example 1:(i)
For the same quantization density, Theorem3 is better than Theorem 2 according to state trajectory, estimated state trajectory, and control input singles.(ii)
For the same theorem, the speed of the state trajectory and estimated trajectory converging to zero would be fast ifδ becomes smaller.
Example2 shows the advantages of proposed methods in practical application in terms of stabilisation results. We generalize that, for the robust performance of the system, we should take small δ into Theorem 3 to settle the output feedback MPC problem for NCSs with input quantization in this paper.Figure 11
Coupled spring-mass system.Figure 12
State trajectory withδ=0.6756.Figure 13
Estimated state trajectory withδ=0.6756.Figure 14
State trajectory withδ=0.3936.Figure 15
Estimated state trajectory withδ=0.3936.Figure 16
γk trajectory.Figure 17
Control input signals.Figure 18
State trajectory withδ=0.6756.Figure 19
State trajectory withδ=0.3936.Figure 20
Control input signals.


## 7. Conclusion

For the NCSs, the output feedback MPC algorithm with input quantization has been proposed in this paper. The presented strategy not only can be used to handle the quantization error in networked control systems but also is suitable for more practical application situations, such as unmeasurable state variables and controller with one free control move. In the process of handling optimization problem, the sector bound approach, which involves quantization error, and the technique of refreshing the bound of estimation error are employed. In addition, the most important feature of our provided method is that it inherits the general merits of a synthesis approach of output feedback MPC, i.e., recursive feasibility and closed-loop stability, and it is significant to improve the spectrum of the synthesis approaches of output feedback MPC and the complex networked control systems. Simultaneously, there are some limitations here. On the one hand, the limiting condition in this article is no disturbance. If the disturbance is added, more optimization conditions need to be added. On the other hand, fault-tolerant control with additive and multiplicative actuator faults is more appropriate for practical applications. Furthermore, the network-induced time delay is also a hot topic in the networked control system, and we will pay more attention to the synthesis approach of model predictive control for a networked control system with a network-induced time delay. [32].



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