# A Two-Stage Scheduling RPC Based on Time-Varying Coefficient Information of State-Dependent ARX Model

Authors: Feng Zhou; Peidong Zhu; Minghua Xie; Jun Wu; Lihua Cao
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A two-stage scheduling robust predictive control (RPC) algorithm, which is based on the time-varying coefficient information of the state-dependent ARX (SD-ARX) model, is designed for the output tracking control of a class of nonlinear systems. First, by using the parameter variation range information of the SD-ARX, a strategy for constructing the system’s polytopic model is designed. To further reduce the conservativeness of the convex polytopic sets which are designed to wrap the system’s future dynamics, the variation range information of the SD-ARX model’s parameters is also considered and compressed. In this method, the polytopic state-space model of the system is constructed directly based on the special structure of the SD-ARX model itself, and there is no need to make such assumption that the bounds on the parameter’s variation range in the system model are known or measurable. And then, a two-stage scheduling RPC algorithm is designed for the output tracking control. A numerical example is presented to demonstrate the effectiveness of the proposed RPC strategy.


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## Body

## 1. Introduction

Model predictive control (MPC), which directly uses the mathematical model to predict the future behavior of the system, has been widely studied in the past decade [1]. Considering that the actual systems are always nonlinear and the working conditions often undergo uncertainties, which may have significant impact on the system [2], how to deal with the system uncertainties, whether in the form of external disturbance or parameter estimation error, is still the key issue that needs to be solved in the MPCs [3]. Considering that models with polytopic description can be effectively used to include the uncertainties of the nonlinear systems, the MPC strategies based on these types of models have been widely studied. For the polytopic LPV model, Kothare et al. [4] proposed an RPC algorithm and used the linear matrix inequalities (LMIs) to settle the control gains in the robust controller. Based on the min-max principle, Lu and Arkun [5, 6] designed a state feedback predictive control strategy for an uncertain LPV system without external disturbance. Mayne et al. [7] proposed an RPC strategy for linear systems with external bounded disturbances. He et al. [8] designed an RPC algorithm for constrained continuous-time nonlinear systems with bounded disturbances. A neural network model-based RPC strategy considering the uncertain but bounded disturbances is proposed for constrained discrete-time nonlinear systems [9]. For constrained nonlinear systems, He et al. [10] designed a quasi-min-max RPC algorithm. Although many achievements have been achieved in the research of RPC algorithms for systems with polytopic description [4–10], they mainly focused on the state adjustment or output tracking problems based on certain restrictive assumptions; for example, the steady-state information of the system is accurately known or measured [11]. In these methods, to derive the system’s LPV model, the polynomial model of the system usually needs to be linearized at the operating point of the system, and the obtained LPV model may be different when the operating point changed [11]. Considering that there also exist many systems in practice with unmeasured steady-state knowledge due to uncertainty or disturbance [11, 12], it is meaningful to develop new RPC strategies which can overcome the above difficulties [13].Some researchers have paid their attention to the output feedback RPCs based on the state estimators. For discrete-time systems with bounded disturbances, Mayne et al. [14] proposed an output feedback controller design method. Based on the min-max principle, Park et al. [12] proposed a robust output feedback MPC scheme for the LPV systems. Ding [15] designed an output feedback RPC for the LPV system with external disturbance. Ding et al. [16] also designed an output feedback RPC for the LPV models with norm-bounded disturbances. Based on a high-gain observer, Du et al. [17] studied a robust output feedback MPC for dynamically positioned ships with uncertain parameters. However, considering that the state estimation errors exist in the state estimators [12–15], the synthesis procedures of the output feedback RPCs may be more complicated than the state feedback MPCs [4–8]. Moreover, to ensure the system’s variable constraints, the synthesis process of the output feedback RPCs may be further complicated [11, 18]. In general, the complex nonlinear systems are more accessible to be modeled as the input-output formal equations. So, the RPC strategies directly based on the system’s input-output model are valuable [18, 19]. Directly based on the system’s input-output model, Ding and Zou [18] designed an output feedback RPC synthesis approach. Abbas et al. [19] proposed an RPC strategy for the system described by the LPV model in input-output form. In this paper, an RPC strategy directly based on the data-driven state-dependent (SD) autoregressive (SD-ARX) model with exogenous variable is studied. It distinguishes from the RPC strategies under the assumption that the state of the system is measurable [4–8] or can be obtained by the state estimator [12–15], and the steady-state information is never used in the designing process of the robust predictive controller.The SD-ARX model is a quasi-LPV model with functional coefficients and is often used in nonlinear system modeling [20]. Using RBF to approximate the state-dependent coefficients of the SD-ARX model generates the RBF-ARX model [21, 22]. These type model-based MPC strategies have been successfully applied to many systems, such as the industrial drying process [23, 24], the NOx decomposition process [21, 25], the water tank system [26], and the ship’s tracking control [27]. But, most of these literatures [21–26] are limited to nominal conditions, and the stabilities of these MPCs are also difficult to guarantee. Based on the RBF-ARX model, Peng et al. [28] designed a min-max MPC algorithm for nonlinear systems without external disturbance. Zhou et al. [29] proposed a one-stage scheduling quasi-min-max MPC algorithm, in which the external bounded disturbance is considered. What is more, Zhou et al. [30] further studied a two-stage scheduling quasi-min-max MPC algorithm based on the RBF-ARX model. Although some results have been achieved, the current literatures [28–30] are mainly focused on using the parameter bound information of the SD-ARX model to wrap the system’s future dynamics, and these types of methods have serious conservatism when constructing the system’s state-space model. To reduce the conservativeness of these type of methods, the variation range information of the SD-ARX model’s parameters is further considered in this paper to compress the size of the convex polytopic sets which are designed to wrap the system’s future dynamics. And then, the control performance of the subsequent RPC algorithm may be improved. Moreover, it is no need to make such assumption that the bounds on the parameter’s variation range in the LPV model are available [31–34]. The polytopic state-space model of the system is constructed directly based on the special structure of the SD-ARX model itself.The rest of the paper is as follows: by using the SD-ARX model’s parameter variation range information, the construction process of the system’s polytopic LPV state-space model is introduced in Section2. And then, a two-stage scheduling robust predictive control (RPC) algorithm based on the constructed LPV state-space model is designed in Section 3. An example is presented to illustrate the design procedure in Section 4. The conclusions are drawn in Section 5.


## 2. Construction of the System’s Polytopic State-Space Model

The class of nonlinear systems, which can be depicted with the following autoregressive model [21], is considered:(1)yt+1=fyt,…,yt−kα+1,ut−kd,…,ut−kβ−kd+1+ςt+1,where yt∈ℜ and ut∈ℜ are the system’s output and input; ςt+1∈ℜ is the white noise interference term; and kα and kβ are the variable orders, and kd is the variable time delay. If the nonlinear mapping f⋅ is continuously differentiable at any working state, model (1) can be represented by the following SD-ARX model [21]:(2)yt+1=φ0st+∑i=0kα−1φα,istyt−i+∑i=kdkβ+kd−1φβ,istut−i+ςt+1,φ0st=κ00+∑h=1mκh0e−σhαst−νhα22,φj,ist=κi,0j+∑h=1mκi,hje−σhjst−νhj22,νhj=νh,1j,νh,2j,…,νh,djT,j=α,β,where m and d are the model orders; st=st,1,st,2,…,st,dT is called the state vector, and it may include delayed inputs or outputs, or other signals that can be measured; φ0st, φα,ist, and φβ,ist are the state-dependent functional coefficients; σhj and νhj are the widths and centers of the neural network, respectively; κ00, κh0, κi,0j, and κi,hj are the linear weights of the neural network; and ⋅2 represents the vector 2-norm. The nonlinear parameters θN=σhj,νhj and linear parameters θL=κ00,κh0,κi,0j,κi,hj in model (2) are optimized by the regularized SNPOM [22]. The details of the regularized SNPOM algorithm can be found in Ref. [22], which is omitted here.To obtain a more general system model, first, the term representing the bounded unknown disturbance is added into the SD-ARX model (2), thus obtaining(3)yt+1=φ0st+∑i=0kα−1φα,istyt−i+∑i=kdkβ+kd−1φβ,istut−i+ζt+1,where δ>0 is a constant and ζt+1≤δ represents the bounded unknown disturbance including the modeling error. And then, model (3) can be rewritten as follows:(4)yt+1=φ0,t+∑i=0kn−1αi+1,tyt−i+∑i=0kn−1βi+1,tut−i+ζt+1,φ0,t=φ0st,αi+1,t=κi,0α+∑h=1mκi,hαe−σhαst−νhα22,0≤i≤kα−1,0,else,βi+1,t=κi,0β+∑h=1mκi,hβe−σhβst−νhβ22,kd≤i≤kβ+kd−1,0,else,kn=maxkα,kβ+kd,ζt+1≤δ.Next, define the system deviation inputs and outputs as(5)u¯t+j1=ut+j1−ut+j1−1,j1=0,−1,−2,…,y¯t+j2=yt+j2−yr,t+j2,j2=1,−1,−2,…,yr,t+1=ηyt+1−ηyr,0≤η<1,yr,t+j2=yr,j2<1,where yr is the expected output and the expected output sequence yr,t+j2 is designed to be an exponential decay curve. Based on models (4) and (5), the output deviation of the one-step-forward prediction y¯t+1t of the model (4) is derived as follows:(6)y¯t+1t=∑i=0kn−1αi+1,ty¯t−i+∑i=0kn−1βi+1,tu¯t−i+ϑt,(7)ϑt=∑i=0kn−1αi+1,tyr,t−i+∑i=0kn−1βi+1,tut−i−1−yr,t+1+φ0,t+ζt+1t=ϕt+ζt+1t.Here,ϑt is selected as an indicator of whether the system enters into the steady state because ϑt should be zero if the system input ut is perfect and the system output yt is stable at the expected output yr under the steady state. To make ϑt+jt,j≥1 be zero according to (6) and (7), one can design a sequence of “perfect” inputs u¯tt,u¯t+1t,… and the output deviations of the multistep-forward prediction y¯t+j+1t,j≥1 of the model (4) satisfy(8)y¯t+j+1t=∑i=0kn−1αi+1,t+jy¯t+j−it+∑i=0kn−1βi+1,t+ju¯t+j−it,where if j is less than or equal to i, then y¯t+j−it is equal to y¯t+j−i and u¯t+j−it is equal to u¯t+j−i. In this paper, model (8) is designed to satisfy the condition that the indicators ϑt+jt,j≥1 are equal to zero. Next, to construct the system’s state-space models based on the polynomial models (6) and (8), the state vector Xt+jt,j=0,1,2,… is selected as follows:(9)Xt+jt=x1,t+jt,x2,t+jt,…,xkn,t+jtT,x1,t+jt=y¯t+jt,xk,t+jt=∑i=1kn+1−kαi+k−1,t+j−1y¯t+j−it+∑i=1kn+1−kβi+k−1,t+j−1u¯t+j−it,k=2,3,…,kn.Then, the system’s state-space models can be obtained:(10)Xt+1t=AtXtt+Btu¯tt+Ξt,At=α1,t10⋯0α2,t01⋯⋮⋮⋮⋮⋱0αkn−1,t00⋯1αkn,t00⋯0,Bt=β1,tβ2,t⋮βkn,t,Ξt=ϑt0⋮0,(11)Xt+j+1t=At+jtXt+jt+Bt+jtu¯t+jt,At+jt=α1,t+jt10⋯0α2,t+jt01⋯⋮⋮⋮⋮⋱0αkn−1,t+jt00⋯1αkn,t+jt00⋯0,Bt+jt=β1,t+jtβ2,t+jt⋮βkn,t+jt,j≥1.Since the statest is known at time t, the matrices At and Bt in (10) can be obtained; similarly, based on the system’s history data, the state vector Xtt in (10) can be obtained by (9). Considering that the vector Ξt in (10) contains a bounded unknown disturbance term ζt+1t, it is obvious that the exact value of Ξt cannot be obtained at time t, but its variation range can be obtained. Here, the polytope ΩΞ that Ξt in (10) belongs to is constructed as follows:(12)ΩΞ:Ξt=∑τ=12λτ,tΞΞt,τ,∑τ=12λτ,tΞ=1,λτ,tΞ≥0,where the vertices Ξt,1=ϕt−δ0⋯0T and Ξt,2=ϕt+δ0⋯0T and ϕt is defined as in (7).Remark 1.
The coefficientsαi+1,t+jt,βi+1,t+jt|i=0,…,kn−1;j≥1 in (11) is established on the future working states st+jt. So, the future matrices At+jt,Bt+jt|j≥1 cannot be obtained. But, one can see that, if the varying region of every exponential term in αi+1,t+jt or βi+1,t+jt can be found out, a polytope set can be built to cover the variation scope of the matrices At+jt or Bt+jt in (11).
Next, based on the time-varying coefficient information of the model (4), a method for constructing the convex polytope sets, which are used to cover the variation scope of the matrices At+jt,Bt+jt|j≥1, is designed.


### 2.1. Based on Parameter Upper and Lower Boundary Information

If only considering the boundary information of the parameters of the SD-ARX model (4), the convex polytopic sets can be constructed as follows:(13)ΩA:At+jt=∑a=12mλa,t+jtαAa,∑a=12mλa,t+jtα=1,λa,t+jtα≥0,ΩB:Bt+jt=∑b=12mλb,t+jtβBb,∑b=12mλb,t+jtβ=1,λb,t+jtβ≥0,where Aa and Bb are the vertices of ΩA and ΩB and(14)Aa=κ0,0α+∑h=1mκ0,hαε¯α,hε¯α,h10⋯0κ1,0α+∑h=1mκ1,hαε¯α,hε¯α,h01⋯0⋮⋮⋮⋱0κkα−1,0α+∑h=1mκkα−1,hαε¯α,hε¯α,h⋮⋮⋯⋮0⋮⋮⋯0⋮⋮⋮⋱0000⋯1000⋯0kn×kn,(15)Bb=0⋮0κkd,0β+∑h=1mκkd,hβε¯β,hε¯β,h⋮κkβ+kd−1,0β+∑h=1mκkβ+kd−1,hβε¯β,hε¯β,h0⋮0kn×1,where κi,0α,κi,hα|h=1,…,m;i=0,…,kα−1 and κi,0β,κi,hβ|h=1,…,m;i=kd,…,kβ+kd−1 are the linear parameters as in (4), the mark ε¯β,h|ε¯β,h indicates ε¯α,h or ε¯α,h, the mark ε¯β,h|ε¯β,h indicates ε¯β,h or ε¯β,h, and they are defined as follows:(16)εhαt=e−σhαst−νhα22,εhβt=e−σhβst−νhβ22,ε¯α,h=maxεhαt,∀st,ε¯α,h=minεhαt,∀st,ε¯β,h=maxεhβt,∀st,ε¯β,h=minεhβt,∀st,where σhα,σhβ,νhα,νhβ|h=1,…,m are the widths and centers of the neural networks as in (4) and st is the historical state data. From the boundary ε¯α,h,ε¯α,h or ε¯β,h,ε¯β,h in (16) and the coefficients of regression structure with m exponential terms as in (4), 2m vertices Aa or Bb of the set ΩA or ΩB can be obtained from (14) or (15). So, the polytopic LPV model (11) can be constructed.Remark 2.
Considering that the matricesAt+1t,Bt+1t in (11) may not reach to the boundary of the sets ΩA,ΩB directly in only one step, in this paper, the variation range information of the SD-ARX model’s parameters is further considered to compress the size of the convex polytopic sets ΩA,ΩB. Thus, the size of ΩA,ΩB, which is built to wrap the matrices At+1t,Bt+1t in (11), can be greatly compressed. Moreover, it may reduce the conservativeness of subsequent robust predictive controller and improve its control performance.


### 2.2. Based on Parameter Variation Rate Information

By using the parameter variation rate information of the SD-ARX model (4), the convex polytopic sets used to wrap the state matrices At+1t,Bt+1t are designed as(17)ΩC:At+1t=∑c=12mλc,t+1tαCc,∑c=12mλc,t+1tα=1,λc,t+1tα≥0,ΩD:Bt+1t=∑d=12mλd,t+1tβDd,∑d=12mλd,t+1tβ=1,λd,t+1tβ≥0,where Cc and Dd are the vertices of ΩC and ΩD and the basic structure of Cc or Dd is the same as that of Aa or Bb in (14) or (15), except that the calculation method for the item ε¯α,h|ε¯α,h or ε¯β,h|ε¯β,h in (14) or (15) is different. In order to facilitate the difference, here we redefine the item ε¯α,h|ε¯α,h or ε¯β,h|ε¯β,h of the vertex Cc or Dd as ε↔α,h|ε↔α,h or ε↔β,h|ε↔β,h. In this paper, the item ε↔α,h|ε↔α,h or ε↔β,h|ε↔β,h in the vertex Cc or Dd is calculated by the following strategy:(18)ε↔α,hε↔α,h=ε↔α,horε↔α,h,ε↔α,h=minε¯α,h,εhαt+Δε¯α,h,ε↔α,h=maxε¯α,h,εhαt+Δε¯α,h,Δε¯α,h=max−2σhαst−νhαTεhαtΔst,∀st,Δε¯α,h=min−2σhαst−νhαTεhαtΔst,∀st,(19)ε↔β,hε↔β,h=ε↔β,horε↔β,h,ε↔β,h=minε¯β,h,εhβt+Δε¯β,h,ε↔β,h=maxε¯β,h,εhβt+Δε¯β,h,Δε¯β,h=max−2σhβst−νhβTεhβtΔst,∀st,Δε¯β,h=min−2σhβst−νhβTεhβtΔst,∀st,where the marks ε¯j,h,ε¯j,h,εhjt,σhj,νhj|h=1,…,m;j=α,β have been defined in (16) and st is the historical state data, and Δst is the variation rate of st. From the boundary ε↔α,h,ε↔α,h or ε↔β,h,ε↔β,h in (18) or (19) and the coefficients of regression structure with m exponential terms as in (4), 2m vertices Cc or Dd can be obtained. And then, the convex polytopic set ΩC or ΩD, which is used to wrap state matrix At+1t or Bt+1t, can be built.Remark 3.
Based on the time-varying coefficient information of the SD-ARX model, the state-space model (10) is built to represent the system’s current behavior, and the polytopic state-space model (11) is constructed to wrap the future dynamics of the system. Note that, there is no need to make such assumption that the bounds on the parameter’s variation range in the system’s model are available. In this paper, the polytopic state-space model is constructed directly based on the special structure of the SD-ARX model itself.
Next, based on the established models (10) and (11), a two-stage scheduling robust controller is designed.


## 2.1. Based on Parameter Upper and Lower Boundary Information

If only considering the boundary information of the parameters of the SD-ARX model (4), the convex polytopic sets can be constructed as follows:(13)ΩA:At+jt=∑a=12mλa,t+jtαAa,∑a=12mλa,t+jtα=1,λa,t+jtα≥0,ΩB:Bt+jt=∑b=12mλb,t+jtβBb,∑b=12mλb,t+jtβ=1,λb,t+jtβ≥0,where Aa and Bb are the vertices of ΩA and ΩB and(14)Aa=κ0,0α+∑h=1mκ0,hαε¯α,hε¯α,h10⋯0κ1,0α+∑h=1mκ1,hαε¯α,hε¯α,h01⋯0⋮⋮⋮⋱0κkα−1,0α+∑h=1mκkα−1,hαε¯α,hε¯α,h⋮⋮⋯⋮0⋮⋮⋯0⋮⋮⋮⋱0000⋯1000⋯0kn×kn,(15)Bb=0⋮0κkd,0β+∑h=1mκkd,hβε¯β,hε¯β,h⋮κkβ+kd−1,0β+∑h=1mκkβ+kd−1,hβε¯β,hε¯β,h0⋮0kn×1,where κi,0α,κi,hα|h=1,…,m;i=0,…,kα−1 and κi,0β,κi,hβ|h=1,…,m;i=kd,…,kβ+kd−1 are the linear parameters as in (4), the mark ε¯β,h|ε¯β,h indicates ε¯α,h or ε¯α,h, the mark ε¯β,h|ε¯β,h indicates ε¯β,h or ε¯β,h, and they are defined as follows:(16)εhαt=e−σhαst−νhα22,εhβt=e−σhβst−νhβ22,ε¯α,h=maxεhαt,∀st,ε¯α,h=minεhαt,∀st,ε¯β,h=maxεhβt,∀st,ε¯β,h=minεhβt,∀st,where σhα,σhβ,νhα,νhβ|h=1,…,m are the widths and centers of the neural networks as in (4) and st is the historical state data. From the boundary ε¯α,h,ε¯α,h or ε¯β,h,ε¯β,h in (16) and the coefficients of regression structure with m exponential terms as in (4), 2m vertices Aa or Bb of the set ΩA or ΩB can be obtained from (14) or (15). So, the polytopic LPV model (11) can be constructed.Remark 2.
Considering that the matricesAt+1t,Bt+1t in (11) may not reach to the boundary of the sets ΩA,ΩB directly in only one step, in this paper, the variation range information of the SD-ARX model’s parameters is further considered to compress the size of the convex polytopic sets ΩA,ΩB. Thus, the size of ΩA,ΩB, which is built to wrap the matrices At+1t,Bt+1t in (11), can be greatly compressed. Moreover, it may reduce the conservativeness of subsequent robust predictive controller and improve its control performance.


## 2.2. Based on Parameter Variation Rate Information

By using the parameter variation rate information of the SD-ARX model (4), the convex polytopic sets used to wrap the state matrices At+1t,Bt+1t are designed as(17)ΩC:At+1t=∑c=12mλc,t+1tαCc,∑c=12mλc,t+1tα=1,λc,t+1tα≥0,ΩD:Bt+1t=∑d=12mλd,t+1tβDd,∑d=12mλd,t+1tβ=1,λd,t+1tβ≥0,where Cc and Dd are the vertices of ΩC and ΩD and the basic structure of Cc or Dd is the same as that of Aa or Bb in (14) or (15), except that the calculation method for the item ε¯α,h|ε¯α,h or ε¯β,h|ε¯β,h in (14) or (15) is different. In order to facilitate the difference, here we redefine the item ε¯α,h|ε¯α,h or ε¯β,h|ε¯β,h of the vertex Cc or Dd as ε↔α,h|ε↔α,h or ε↔β,h|ε↔β,h. In this paper, the item ε↔α,h|ε↔α,h or ε↔β,h|ε↔β,h in the vertex Cc or Dd is calculated by the following strategy:(18)ε↔α,hε↔α,h=ε↔α,horε↔α,h,ε↔α,h=minε¯α,h,εhαt+Δε¯α,h,ε↔α,h=maxε¯α,h,εhαt+Δε¯α,h,Δε¯α,h=max−2σhαst−νhαTεhαtΔst,∀st,Δε¯α,h=min−2σhαst−νhαTεhαtΔst,∀st,(19)ε↔β,hε↔β,h=ε↔β,horε↔β,h,ε↔β,h=minε¯β,h,εhβt+Δε¯β,h,ε↔β,h=maxε¯β,h,εhβt+Δε¯β,h,Δε¯β,h=max−2σhβst−νhβTεhβtΔst,∀st,Δε¯β,h=min−2σhβst−νhβTεhβtΔst,∀st,where the marks ε¯j,h,ε¯j,h,εhjt,σhj,νhj|h=1,…,m;j=α,β have been defined in (16) and st is the historical state data, and Δst is the variation rate of st. From the boundary ε↔α,h,ε↔α,h or ε↔β,h,ε↔β,h in (18) or (19) and the coefficients of regression structure with m exponential terms as in (4), 2m vertices Cc or Dd can be obtained. And then, the convex polytopic set ΩC or ΩD, which is used to wrap state matrix At+1t or Bt+1t, can be built.Remark 3.
Based on the time-varying coefficient information of the SD-ARX model, the state-space model (10) is built to represent the system’s current behavior, and the polytopic state-space model (11) is constructed to wrap the future dynamics of the system. Note that, there is no need to make such assumption that the bounds on the parameter’s variation range in the system’s model are available. In this paper, the polytopic state-space model is constructed directly based on the special structure of the SD-ARX model itself.
Next, based on the established models (10) and (11), a two-stage scheduling robust controller is designed.


## 3. Robust Predictive Controller Design Process

MPC is a model-based computer optimization control method, which makes prediction about the system’s future behavior using dynamic model. In this subsection, based on models (10) and (11), which the system’s dynamics is forced in, an RPC algorithm is designed. The objective function of the RBF-ARX model-based RPC (RBF-ARX-RPC) algorithm is designed as follows:(20)minu¯t+jt,j=0,1,2,…maxΞt∈ΩΞ,At+1t∈ΩC,Bt+1t∈ΩD,At+jt∈ΩA,Bt+jt∈ΩB,j≥2.J0∞t=∑j=0∞Xt+jtTWXt+jt+u¯t+jtTRu¯t+jt,s.t.10,11,u¯t+jt≤u¯max,j≥0;umin≤utt≤umax;y¯t+it≤y¯max,i≥1,where R=RT>0 and W=WT≥0 are the weight coefficients; u¯t+jt is the predicted input increment at t+j; u¯t+jt and y¯t+it are imposed with constraints u¯t+jt≤u¯max and y¯t+it≤y¯max, respectively; and utt is the control law at time t and is constrained by umin≤utt≤umax.To design the two-stage scheduling RPC algorithm, first,J0∞t in (20) is divided into three parts: J0∞t=J01t+J12t+J2∞t, where J01t=XttTWXtt+u¯ttTRu¯tt, J12t=Xt+1tTWXt+1t+u¯t+1tTRu¯t+1t, and J2∞t is the remaining part of the objective function J0∞t. And, in the objective function (20), u¯tt and u¯t+1t are free decision variables, and the input increments u¯t+jt|j≥2 are calculated by u¯t+jt=FtXt+jt, in which the matrix Ft can be determined at time t. And then, the LMI synthesis method is used to settle the infinite horizon optimization problem (20).First, the quadratic function of state vector is defined as follows:(21)ϒj,t=Xt+jtTPj,tXt+jt,j≥2,where Pj,t is a positive definite matrix. And, for any At+jt∈ΩA and Bt+jt∈ΩB, j≥2, we suppose ϒj,t at time t satisfies(22)ϒj+1,t−ϒj,t≤−Xt+jtTWXt+jt+u¯t+jtTRu¯t+jt.Next, one can get the following inequality by summing (22) from j=2 to ∞:(23)maxAt+jt∈ΩA,Bt+jt∈ΩB,j≥2J2∞t<ϒ2,t.And then, optimization problem (20) can be converted as follows:(24)minu¯tt,u¯t+1t,P2,t.J01t+J12t+ϒ2,t,s.t.10,11,25,u¯t+jt≤u¯max,j≥0;umin≤utt≤umax;y¯t+it≤y¯max,i≥1.In this paper, the Lyapunov matricesPab|a,b=1,2,…,2m are used to build the time-varying Lyapunov matrices Pj,t=∑a=12mλa,t+jtα∑b=12mλb,t+jtβPab for j≥2. If the Lyapunov matrices Pab exist, the minimization problem (24) can then be settled by Theorem 1.Theorem 1.
The min-max optimization problem (20), which is designed to find u¯tt, u¯t+1t, and the feedback gain Ft, can be settled by the following semidefinite programming:(25)minγ,u¯tt,u¯t+1t,Qab,Y,G,Z,Kγ,subject to(26)1∗∗∗∗∗∗∗CcAtXtt+Btu¯tt+Ξt,τ+Ddu¯t+1tQab∗∗∗∗∗∗00I∗∗∗∗∗000I∗∗∗∗W1/2Xtt000γI∗∗∗R1/2u¯tt0000γI∗∗W1/2AtXtt+Btu¯tt+Ξt,τ00000γI∗R1/2u¯t+1t000000γI≥0,(27)G+GT−Qab∗∗∗AaG+BbYQef∗∗W1/2G0γI∗R1/2Y00γI≥0,and with the following variable constraints,(28)u¯tt≤u¯max,u¯t+1t≤u¯max,umin≤utt≤umax,ZYYTG+GT−Qab≥0,Zii≤u¯max2;(29)CAtXtt+Btu¯tt+Ξt,τ≤y¯max,CCcAtXtt+Btu¯tt+Ξt,τ+Ddu¯t+1t≤y¯max,KCAaG+BbYAaG+BbYTCTG+GT−Qab≥0,Kjj≤y¯max2,where a,b,c,d,e,f=1,2,…,2m; τ=1,2; the symbol ∗ indicates a symmetric structure; Qab=γPab−1>0; the feedback gain matrix is defined as Ft=YG−1; Z and K are the symmetric matrices, Zii is the i-th diagonal element of the symmetric matrix Z, and Kjj is the j-th diagonal element of the symmetric matrix K. In the above (26)–(29), At, Bt, Xtt, Aa, Bb, Cc, Dd, and Ξt,τ can be calculated by model (4) at time t and the historical data of the system.Proof.
Substituting the state-space model (11), u¯t+jt=FtXt+jt and (21) into (22), one can get the following inequality:(30)At+jt+Bt+jtFtTPj+1,tAt+jt+Bt+jtFt−Pj,t+FtTRFt+W≤0.
Next, it can be verified that the inequality (30) is satisfied if and only if the Lyapunov matrices Pab|a,b=1,2,…,2m, which are used to build the time-varying Lyapunov matrices Pj,t=∑a=12mλa,t+jtα∑b=12mλb,t+j|tβPab for j≥2, exist, and(31)Aa+BbFtTPefAa+BbFt−Pab+FtTRFt+W≤0.
By using the Schur complement [35] and defining Qab=γPab−1, Qef=γPef−1, Ft=YG−1, the inequality (31) can then be converted to the inequality (27).
Therefore, the optimization problem (24) can be converted as follows:(32)minγ,u¯tt,u¯t+1t,Qab,Y,Gγ,s.t.10,11,29,J01t+J12t+ϒ2,t≤γ,u¯t+jt≤u¯max,j≥0;umin≤utt≤umax;y¯t+it≤y¯max,i≥1,where γ is a suitable nonnegative scalar.
According to (24) and introducing (10) and (11) into (32), one can obtain that(33)J01t+J12t+Xt+2tTPabXt+2t≤γ,where the state Xt+1t in J12t is defined in (10) and the vector Ξt belongs to ΩΞ; the state Xt+2t is defined in (11), and the state matrix At+1t or Bt+1t belongs to ΩC or ΩD, respectively. And then, by using the Schur complement [35], inequality (33) can be finally converted to the LMIs (26).
The variable constraints in (24) can also be converted to the LMIs. First, separate the variables into three parts: U¯0∞=u¯tt,u¯t+1t,U¯2∞ and Y¯1∞=y¯t+1t,y¯t+2t,Y¯3∞. Then each constraint is imposed on each part. Similar to Ref. [35], the incremental constraints contained in (36) can ultimately be represented as LMI (28) and (29).
Hence, the two-stage scheduling RPC optimization problem established based on (10), (11), and (20) is finally converted to minγ,u¯tt,u¯t+1t,Qab,Y,G,Z,Kγ, subject to the LMIs (26)–(29).Remark 4.
Although bothu¯tt and u¯t+1t are computed at each sampling time, the control action implemented on the controlled system is only ut=u¯tt+ut−1, and this calculation process is repeated at the next sampling time.
According to the conclusion shown in Theorem1, the result similar to Theorem 3 in Ref. [6] can be obtained:Theorem 2.
When Theorem1 is executed in a receding horizon fashion, the feasibility of the minimization problem (25) guarantees stability.Remark 5.
The main idea behind the two-stage scheduling RPC algorithm is to separate the second-stage costJ12t in addition to the first-stage cost J01t from the objective function J0∞t, so as to relax more control moves from the constant feedback control law.Remark 6.
It is different to the one-stage scheduling RPC algorithms [28, 29], in which only the first control increment u¯tt is separated from the future control series, and all other future control actions are in feedback forms with constant feedback gains. In the two-stage scheduling RPC algorithm, in addition to u¯tt, the control increment u¯t+1t is also separated from the future control series and is acted as a free control variable to increase the degree of freedom of the control.


## 4. Case Study

The modeling and control of a water tank system is provided in this section to illustrate the performance of the two-stage scheduling RPC algorithm. Level control of the water tank system is often used in literatures [36, 37] for comparison study. The mass and energy balance equations for the water tank system [36] are as follows:(34)dH1dt=−A1S12gH1+π1μS1U,dH2dt=A1S22gH1−A2S22gH2,where U is the control voltage of the pump and H1 and H2 are the liquid levels of the two tanks; the related parameters of the water tank system are shown in Table 1. In this case study, the control problem is to control the level H2t of the lower tank by adjusting the input voltage Ut of the pump.Table 1
Parameters of the water tank system.
Cross section of Tank 1
S
1
=
500
cm
2
Cross section of Tank 2
S
2
=
360
cm
2
Scaling factor
π
1
=
0.05
cm
3
/
s
−
v
Water density
μ
=
0.001
kg
/
cm
3
Outlet hole section of Tank 1
A
1
=
9
cm
2
Outlet hole section of Tank 2
A
2
=
8.5
cm
2
Gravity constant
g
=
980
cm
/
s
2


### 4.1. Modeling for Water Tank System

The modeling method in Section2 is used to identify the characteristics of the water tank system. First, the basic SD-ARX model is designed as follows:(35)H2t+1=φ0st+∑i=0kα−1φα,istH2t−i+∑i=kdkβ+kd−1φβ,istUt−i+ξt+1,φ0st=κ00+∑h=1mκh0e−σhαst−νhα22,φj,ist=κi,0j+∑h=1mκi,hje−σhjst−νhj22,νhj=νh,1j,νh,2j,…,νh,djT,j=α,β,where the input voltage Ut and liquid level H2t are the input and output, respectively; kα, kβ, kd, m, and d are the orders; the state vector of the system is designed as st=H2t,…,H2t−d+1T because the nonlinearity of the system is mainly caused by the change of the liquid level of the Tank 2; and the linear parameters θL=κ00,κh0,κi,0j,κi,hj and nonlinear parameters θN=σhj,νhj in model (35) are optimized by the R-SNPOM [22]. The detailed optimization process of the nonlinear parameters and linear parameters can be found in Ref. [22], which is omitted here.In identification modeling, to fully excite the dynamics of the water tank system, the modeling data of the system is sampled under a PID controller. The period of the identification data collection in this modeling process is 0.2 s. And, the identification data of the water tank system are finally selected as shown in Figure1, in which the first 2500 data are used for training and the next 2500 are used for testing. In this paper, the AIC value is selected as the main evaluation standard to determine the orders of the SD-ARX model (35). The orders of the final suitable SD-ARX model are selected as kα = 5, kβ = 4, kd = 1, m = 1, and d = 2.Figure 1
Output and input data of the water tank system.Figures2 and 3 show the comparison between the actual output and the predicted output of the model (35), predicted error, and its histogram for the training data and testing data, respectively. From Figures 2 and 3, it is clearly seen that the predicted errors of the model (35) are only within the range of −0.004 cm to +0.004 cm and they are almost Gaussian distribution. It indicates that the identified SD-ARX model (35) has good modeling accuracy.Figure 2
Actual and predicted output, predicted error, and its histogram for training data.Figure 3
Actual and predicted output, predicted error, and its histogram for testing data.Next, by defining the system deviation variables (5) and the state vector (9), the model (10) of the water tank system at each sampling instant can be obtained just as described in Section 2. Figure 4 shows the poles of the state-space model (10) varying with the state st. The different color in Figure 4 denotes the different pole.Figure 4
State-space model’ poles varying with the statest.
(a)
(b)From Figure4, one can see that the dynamics of the water tank system vary obviously with the state st. So, based on the variation range information of the state st in the SD-ARX model (35), the polytopic state-space models (10) and (11) can be designed to wrap the future dynamics of the water tank system. In this paper, based on the parameters’ boundary information of the identified model (35), the vertices of convex polytopic sets ΩA,ΩB, i.e., Aa and Bb, can be obtained according to the (14) and (15), where ε¯α,h, ε¯α,h, ε¯β,h, and ε¯β,h are calculated by (16) on the basis of the identification data shown in Figure 1. Moreover, based on the parameters’ variation rate information of the identified model (35), the vertices of convex polytopic sets ΩC,ΩD, i.e., Cc and Dd, can be obtained according to (18) and (19), where ε↔α,h, ε↔α,h, ε↔β,h, and ε↔β,h are calculated by (18) and (19). So, the state-space models (10) and (11) can be finally obtained.


### 4.2. Control Results and Analysis

The trajectory tracking and disturbance rejection experiments are carried out on the water tank system in this subsection. The simulation experiments are carried out in Matlab 2017b software, the LMI control toolbox in Simulink is used to settle the robust optimization problem, and the sampling period of the control system is 1.5 second. For comparison, two other algorithms, which are the well-tuned PID with multiple sets of parameters (PID-M) and the RBF-ARX model-based RPC algorithm proposed in Ref. [30] (RBF-ARX-TRPC), are also carried out.To obtain better PID control performance, we use three different PIDs to control three step responses of the system (steps from 60 cm to 100 cm, 100 cm to 30 cm, and 30 cm to 70 cm, respectively). In this paper, the well-tuned parameters of each PID, which are selected through trial and error, are shown in Table2. For fair comparison, the RBF-ARX-TRPC algorithm and the proposed RBF-ARX-RPC algorithm are all based on the same model, which is identified in Section 4.1. The controller parameters and variable constraints of the three algorithms are all given in Table 2.Table 2
Controller parameters.
Algorithms
PID-M
RBF-ARX-TRPC and RBF-ARX-RPC
Controller parameters
1:Kp=40,Ki=0.1,Kd=20;
R
=
0.001
,
W
=
1.
2:Kp=80,Ki=0.03,Kd=20;
3:Kp=60,Ki=0.08,Kd=20.
Variable constraints
u
min
=
2
,
u
max
=
100.
u
min
=
2
,
u
max
=
100
,
u
¯
max
=
150.Figure5 shows the trajectory tracking performances of the three algorithms (PID-M, RBF-ARX-TRPC [30], and RBF-ARX-RPC proposed in this paper). In Figure 5, the pink dotted line denotes the desired trajectory of the liquid level of the lower tank H2t, and it increases from 60 cm to 100 cm at the 20th second, decreases from 100 cm to 30 cm at the 130th second, and then increases from 30 cm to 70 cm at the 210th second.Figure 5
Trajectory tracking performances without external disturbance.From Figure5, one can see that the trajectory tracking performance of the PID-M has greater overshoot and longer adjustment time than the RBF-ARX-TRPC [30] and RBF-ARX-RPC, whether in the ascending or descending phase. Meanwhile, it is clearly seen that the trajectory tracking performance of the RBF-ARX-RPC proposed in this paper has smaller overshoot and shorter adjustment time than the other two algorithms.To verify the algorithms’ disturbance rejection ability, the anti-jamming experiments are carried out. Figures6 and 7 show the disturbance rejection performances of the three algorithms with forward or reverse square wave disturbance. In Figure 6, a forward square wave disturbance with an amplitude of 3 cm and a duration of 15 seconds is added into the liquid level of the lower tank H2t at the 80-th second, the 170-th second, and the 260-th second, respectively. Similarly, in Figure 7, a reverse square wave disturbance with an amplitude of −3 cm and a duration of 15 seconds is added into the liquid level of the lower tank H2t at the 80-th second, the 170-th second, and the 260-th second, respectively.Figure 6
Disturbance rejection performances with forward square wave disturbance.Figure 7
Disturbance rejection performances with reverse square wave disturbance.From Figures6 and 7, one can see that the anti-jamming result of the PID-M is the worst. The RBF-ARX-TRPC [30] has a certain degree of anti-jamming ability, and the result is better than the PID-M. The result of the RBF-ARX-RPC is the best, especially when there is external interference. As shown in Figure 6, one can clearly see that the lower tank-level fluctuation caused by the forward square wave disturbance during 80-th second and 95-th second can be suppressed within 6.6 seconds, when it is controlled by the RBF-ARX-RPC algorithm. However, the RBF-ARX-TRPC [30] and the PID-M require more than 13.1 seconds and 15 seconds, respectively. From Figure 7, one can see that, when the reverse square wave disturbance vanishes, the RBF-ARX-RPC algorithm can stabilize the output H2t on the expected output faster than that of the other algorithms. In general, it is clearly seen that the proposed RPC algorithm has best trajectory tracking and disturbance rejection performance.One drawback of general RPC algorithms is their relatively large online computational burden, which may limit their applicability. The real-time computation times of the two RPC algorithms in the step response experiment whose control results are shown in Figure6 are shown in Figure 8. The software environment of the control implementation is based on the Matlab 2017b on Windows 10-based PC (Intel Core i7-9700 16GB-RAM), and the two RPC algorithms are all written in M language. From Figure 8, one can see that the RBF-ARX-RPC costs more computation time than the RBF-ARX-TRPC. But, the maximum computation time (1.41 second) of the RBF-ARX-RPC is less than the sampling period (1.5 second) of the control system. Therefore, the proposed RPC strategy can complete all operations during the sampling period. In order to further reduce the computation time of the proposed RPC strategy, we will try to study the partially offline computing and partially online synthesis RPC strategy based on the SD-ARX model in the future research.Figure 8
Real-time computation times of the two RPC algorithms.


## 4.1. Modeling for Water Tank System

The modeling method in Section2 is used to identify the characteristics of the water tank system. First, the basic SD-ARX model is designed as follows:(35)H2t+1=φ0st+∑i=0kα−1φα,istH2t−i+∑i=kdkβ+kd−1φβ,istUt−i+ξt+1,φ0st=κ00+∑h=1mκh0e−σhαst−νhα22,φj,ist=κi,0j+∑h=1mκi,hje−σhjst−νhj22,νhj=νh,1j,νh,2j,…,νh,djT,j=α,β,where the input voltage Ut and liquid level H2t are the input and output, respectively; kα, kβ, kd, m, and d are the orders; the state vector of the system is designed as st=H2t,…,H2t−d+1T because the nonlinearity of the system is mainly caused by the change of the liquid level of the Tank 2; and the linear parameters θL=κ00,κh0,κi,0j,κi,hj and nonlinear parameters θN=σhj,νhj in model (35) are optimized by the R-SNPOM [22]. The detailed optimization process of the nonlinear parameters and linear parameters can be found in Ref. [22], which is omitted here.In identification modeling, to fully excite the dynamics of the water tank system, the modeling data of the system is sampled under a PID controller. The period of the identification data collection in this modeling process is 0.2 s. And, the identification data of the water tank system are finally selected as shown in Figure1, in which the first 2500 data are used for training and the next 2500 are used for testing. In this paper, the AIC value is selected as the main evaluation standard to determine the orders of the SD-ARX model (35). The orders of the final suitable SD-ARX model are selected as kα = 5, kβ = 4, kd = 1, m = 1, and d = 2.Figure 1
Output and input data of the water tank system.Figures2 and 3 show the comparison between the actual output and the predicted output of the model (35), predicted error, and its histogram for the training data and testing data, respectively. From Figures 2 and 3, it is clearly seen that the predicted errors of the model (35) are only within the range of −0.004 cm to +0.004 cm and they are almost Gaussian distribution. It indicates that the identified SD-ARX model (35) has good modeling accuracy.Figure 2
Actual and predicted output, predicted error, and its histogram for training data.Figure 3
Actual and predicted output, predicted error, and its histogram for testing data.Next, by defining the system deviation variables (5) and the state vector (9), the model (10) of the water tank system at each sampling instant can be obtained just as described in Section 2. Figure 4 shows the poles of the state-space model (10) varying with the state st. The different color in Figure 4 denotes the different pole.Figure 4
State-space model’ poles varying with the statest.
(a)
(b)From Figure4, one can see that the dynamics of the water tank system vary obviously with the state st. So, based on the variation range information of the state st in the SD-ARX model (35), the polytopic state-space models (10) and (11) can be designed to wrap the future dynamics of the water tank system. In this paper, based on the parameters’ boundary information of the identified model (35), the vertices of convex polytopic sets ΩA,ΩB, i.e., Aa and Bb, can be obtained according to the (14) and (15), where ε¯α,h, ε¯α,h, ε¯β,h, and ε¯β,h are calculated by (16) on the basis of the identification data shown in Figure 1. Moreover, based on the parameters’ variation rate information of the identified model (35), the vertices of convex polytopic sets ΩC,ΩD, i.e., Cc and Dd, can be obtained according to (18) and (19), where ε↔α,h, ε↔α,h, ε↔β,h, and ε↔β,h are calculated by (18) and (19). So, the state-space models (10) and (11) can be finally obtained.


## 4.2. Control Results and Analysis

The trajectory tracking and disturbance rejection experiments are carried out on the water tank system in this subsection. The simulation experiments are carried out in Matlab 2017b software, the LMI control toolbox in Simulink is used to settle the robust optimization problem, and the sampling period of the control system is 1.5 second. For comparison, two other algorithms, which are the well-tuned PID with multiple sets of parameters (PID-M) and the RBF-ARX model-based RPC algorithm proposed in Ref. [30] (RBF-ARX-TRPC), are also carried out.To obtain better PID control performance, we use three different PIDs to control three step responses of the system (steps from 60 cm to 100 cm, 100 cm to 30 cm, and 30 cm to 70 cm, respectively). In this paper, the well-tuned parameters of each PID, which are selected through trial and error, are shown in Table2. For fair comparison, the RBF-ARX-TRPC algorithm and the proposed RBF-ARX-RPC algorithm are all based on the same model, which is identified in Section 4.1. The controller parameters and variable constraints of the three algorithms are all given in Table 2.Table 2
Controller parameters.
Algorithms
PID-M
RBF-ARX-TRPC and RBF-ARX-RPC
Controller parameters
1:Kp=40,Ki=0.1,Kd=20;
R
=
0.001
,
W
=
1.
2:Kp=80,Ki=0.03,Kd=20;
3:Kp=60,Ki=0.08,Kd=20.
Variable constraints
u
min
=
2
,
u
max
=
100.
u
min
=
2
,
u
max
=
100
,
u
¯
max
=
150.Figure5 shows the trajectory tracking performances of the three algorithms (PID-M, RBF-ARX-TRPC [30], and RBF-ARX-RPC proposed in this paper). In Figure 5, the pink dotted line denotes the desired trajectory of the liquid level of the lower tank H2t, and it increases from 60 cm to 100 cm at the 20th second, decreases from 100 cm to 30 cm at the 130th second, and then increases from 30 cm to 70 cm at the 210th second.Figure 5
Trajectory tracking performances without external disturbance.From Figure5, one can see that the trajectory tracking performance of the PID-M has greater overshoot and longer adjustment time than the RBF-ARX-TRPC [30] and RBF-ARX-RPC, whether in the ascending or descending phase. Meanwhile, it is clearly seen that the trajectory tracking performance of the RBF-ARX-RPC proposed in this paper has smaller overshoot and shorter adjustment time than the other two algorithms.To verify the algorithms’ disturbance rejection ability, the anti-jamming experiments are carried out. Figures6 and 7 show the disturbance rejection performances of the three algorithms with forward or reverse square wave disturbance. In Figure 6, a forward square wave disturbance with an amplitude of 3 cm and a duration of 15 seconds is added into the liquid level of the lower tank H2t at the 80-th second, the 170-th second, and the 260-th second, respectively. Similarly, in Figure 7, a reverse square wave disturbance with an amplitude of −3 cm and a duration of 15 seconds is added into the liquid level of the lower tank H2t at the 80-th second, the 170-th second, and the 260-th second, respectively.Figure 6
Disturbance rejection performances with forward square wave disturbance.Figure 7
Disturbance rejection performances with reverse square wave disturbance.From Figures6 and 7, one can see that the anti-jamming result of the PID-M is the worst. The RBF-ARX-TRPC [30] has a certain degree of anti-jamming ability, and the result is better than the PID-M. The result of the RBF-ARX-RPC is the best, especially when there is external interference. As shown in Figure 6, one can clearly see that the lower tank-level fluctuation caused by the forward square wave disturbance during 80-th second and 95-th second can be suppressed within 6.6 seconds, when it is controlled by the RBF-ARX-RPC algorithm. However, the RBF-ARX-TRPC [30] and the PID-M require more than 13.1 seconds and 15 seconds, respectively. From Figure 7, one can see that, when the reverse square wave disturbance vanishes, the RBF-ARX-RPC algorithm can stabilize the output H2t on the expected output faster than that of the other algorithms. In general, it is clearly seen that the proposed RPC algorithm has best trajectory tracking and disturbance rejection performance.One drawback of general RPC algorithms is their relatively large online computational burden, which may limit their applicability. The real-time computation times of the two RPC algorithms in the step response experiment whose control results are shown in Figure6 are shown in Figure 8. The software environment of the control implementation is based on the Matlab 2017b on Windows 10-based PC (Intel Core i7-9700 16GB-RAM), and the two RPC algorithms are all written in M language. From Figure 8, one can see that the RBF-ARX-RPC costs more computation time than the RBF-ARX-TRPC. But, the maximum computation time (1.41 second) of the RBF-ARX-RPC is less than the sampling period (1.5 second) of the control system. Therefore, the proposed RPC strategy can complete all operations during the sampling period. In order to further reduce the computation time of the proposed RPC strategy, we will try to study the partially offline computing and partially online synthesis RPC strategy based on the SD-ARX model in the future research.Figure 8
Real-time computation times of the two RPC algorithms.


## 5. Conclusions

Based on the time-varying coefficient information of the SD-ARX model, three polytopic state-space models are designed to wrap the dynamics of a class of nonlinear systems. Based on the established state-space models, a two-stage scheduling RPC without using the system’s steady state information is proposed. Considering that the state of the system may not reach to the boundary of the convex polytopic sets, which are constructed using only the parameter boundary information of the SD-ARX model, directly in only one step, in this paper, the parameter variation rate information of the model is considered to further compress the size of the convex polytopic sets. From the simulation results and comparison of the water tank system, one can see that the proposed RPC algorithm improved the trajectory tracking and disturbance rejection performance.



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*Source: 5319408-2020-03-07.xml*