# A Quasi-ARX Model for Multivariable Decoupling Control of Nonlinear MIMO System

Authors: Lan Wang; Yu Cheng; Jinglu Hu
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper proposes a multiinput and multioutput (MIMO) quasi-autoregressive eXogenous (ARX) model and a multivariable-decoupling proportional integral differential (PID) controller for MIMO nonlinear systems based on the proposed model. The proposed MIMO quasi-ARX model improves the performance of ordinary quasi-ARX model. The proposed controller consists of a traditional PID controller with a decoupling compensator and a feed-forward compensator for the nonlinear dynamics based on the MIMO quasi-ARX model. Then an adaptive control algorithm is presented using the MIMO quasi-ARX radial basis function network (RBFN) prediction model and some stability analysis of control system is shown. Simulation results show the effectiveness of the proposed control method.


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## Body

## 1. Introduction

Nonlinear system control has become a considerable topic in the field of control engineering [1, 2]. Many control results have been obtained for nonlinear single-input and single-output (SISO) systems based on the black box models, such as neural networks (NNs), wavelet networks (WNs), neurofuzzy networks (NFNs), and radial basis function networks (RBFNs), because of their abilities to approximate arbitrary mapping to any desired accuracy [3–9]. These black box models have been directly used to identify and control nonlinear dynamical systemsDue to the complexity of nonlinear Multi-Input and Multi-Output (MIMO) systems, most of the control techniques developed for SISO systems cannot be extended directly for MIMO systems. One of the main difficulties in MIMO nonlinear system control is coupling problem. As such, it is important to investigate the realization of decoupling control. Many adaptive decoupling control algorithms have been proposed to deal with coupling in nonlinear system based on linear methods and nonlinear networks [10–14]. Some decoupling control methods of them are difficult not only to achieve accurate requirement and stability but also to be implemented in industrial applications. On the other hand, PID controller has been widely applied in controlling the SISO system because of its simple structure and relatively easy industrial application [15, 16]. However, PID controller cannot be directly used for MIMO model. Lang et al. [17] proposed a multivariable decoupling PID controller for MIMO linear systems based on the linear PID control and generalized minimum variance control law. What's more, Zhai & Chai [18] presented a multivariable PID control method using neural network to deal with nonlinear multivariable processes. In this control system, the nonlinear unmodeled part estimated by neural network is considered as a black box. The initial weights of neural network, local minima, and overfitting are the problems which need to be resolved.In our previous work, a quasi-autoregressive exogenous (ARX) model with an ARX-like macromodel part and a kernel part was proposed, and a controller was designed for SISO systems [4, 19–21]. The kernel part is an ordinary network model, but it is used to parameterize the nonlinear coefficients of macromodel. As we know, RBFNs have played an important role in control engineering, especially in nonlinear system control because of their simple topological structure and precision in nonlinear approximation [22, 23]. Especially, RBFNs can be regarded as nonlinear models which are linear in parameters when fixing the nonlinear parameters by a priori knowledge [24, 25]. Incorporating the network models with this property, the quasi-ARX models become linear in parameters. Therefore, the RBFNs are chosen to replace the NNs as in [4].The SISO model and control methods based on quasi-ARX model cannot directly be applied to MIMO nonlinear systems. Motivated by the above discussions, an MIMO quasi-ARX model is first proposed for MIMO nonlinear systems and then a nonlinear multivariable decoupling PID controller is proposed based on the MIMO quasi-ARX model, which consists of a traditional PID controller with a decoupling compensator and a feed-forward compensator for the nonlinear dynamics based on the MIMO quasi-ARX model. Then an adaptive controller is presented using the MIMO quasi-ARX RBFN prediction model. The parameters of such controller are selected based on the generalized minimum control variance. In this paper, quasi-ARX RBFN model is divided into two parts: the linear part is used to guarantee the stability and decoupling, and the nonlinear part is used to improve the accuracy.The paper is organized as follows: in Section2 the nonlinear MIMO system considered is first described, and then a hybrid system expression is obtained and an MIMO quasi-ARX RBFN model is proposed. In Section 3, a multivariable decoupling PID controller is developed based on the proposed model and generalized minimum variance control law. Then an adaptive control algorithm is presented using the MIMO quasi-ARX RBFN prediction model, and the corresponding parameter estimation methods are proposed in Section 4. Section 5 carries out numerical simulations to show the effectiveness of the proposed control method. Finally, Section 6 presents the conclusions.


## 2. An MIMO Quasi-ARX Model

### 2.1. Systems

Consider an MIMO nonlinear dynamical system with input-output relation as(2.1)y(t+d)=f(φ(t)),φ(t)=[y(t+d-1)T,…,y(t+d-ny)T,u(t)T,…,u(t-nu+1)T]T,
where y=[y1,…,yn]T∈Rn and u=[u1,…,un]T∈Rn are system input and output vectors, respectively, d the known integer time delay, φ(t) the regression vector, and ny,nu the system orders. f(·)=[f1(·),…,fn(·)]T is a vector-valued nonlinear function, and, at a small region around φ(t)=0(0=[0,…,0]T), they are C∞ continuous. The origin is an equilibrium point, then f(0)=0. The system is controllable, in which a reasonable unknown controller may be expressed by u(t)=ρ(ξ(t)), where ξ(t) is defined in Section 2.4.


### 2.2. ARX-Like Expression

Under the continuous condition, the unknown nonlinear functionfk(φ(t)),(i=1,…,n) can be performed Taylor expansion on a small region around φ(t)=0:(2.2)yk(t+d)=fk′(0)φ(t)+12φT(t)fk′′(0)φ(t)+⋯,
where the prime denotes differentiation with respect to φ(t). Then the following notations are introduced:(2.3)(fk′(0)+12φT(t)fk′′(0)+⋯)T=[a1,t1,k⋯any,t1,k⋯any,tn,kb1,t1,k⋯bnu,t1,k⋯bnu,tn,k]T,
where ai,tl,k=ail,k(φ(t))(i=1,…,ny) and bj,tl,k=bjl,k(φ(t))(j=0,…,nu-1) are nonlinear functions of φ(t).However, we need to gety(t+d) by using the input-output data up to time t in a model. The coefficients ai,tl,k and bj,tl,k need to be calculable using the input-output data up to time t. To do so, let us iteratively replace y(t+l) in the expressions of ai,tl,k and bj,tl,k with functions:(2.4)y(t+s)⟹g(φ̃(t+s)),s=1,…,d-1,
where φ̃(t+s) is φ(t+s) whose elements y(t+m),s+1<m≤d-s are replaced by (2.4), and define the new expressions of the coefficients by(2.5)ai,tl,k=ãi,tl,k=ãil,k(ϕ(t)),bj,tl,k=b̃j,tl,k=b̃jl,k(ϕ(t)),
where ϕ(t) is a vector:(2.6)ϕ(t)=[y(t)T⋯y(t-ny+1)Tu(t)T⋯u(t-nu-d+2)T]T.Now, introduce two polynomial matricesA(q-1,ϕ(t)) and B(q-1,ϕ(t)) based on the coefficients, defined by(2.7)A(q-1,ϕ(t))=I-a1,tq-1-⋯-any,tq-ny,B(q-1,ϕ(t))=b0,t+⋯+bnu-1,tq-nu+1,
where ai,t=(ai,tl,k)N×N,i=1,…,ny and bj,t=(bj,tl,k)N×N,j=1,…,nu. Then, the nonlinear system (2.1) can be equivalently represented as the following ARX-like expression:(2.8)A(q-1,ϕ(t))y(t+d)=B(q-1,ϕ(t))u(t).By (2.8), let y(t+d) satisfies the following equation:(2.9)y(t+d)=A(q-1,ϕ(t))y(t)+B(q-1,ϕ(t))u(t),
where (2.10)A(q-1,ϕ(t))=A0,t+A1,tq-1+⋯+Any-1,tq-ny+1,B(q-1,ϕ(t))=F(q-1,ϕ(t))B(q-1,ϕ(t)),=B0,t+B1,tq-1+⋯+Bnu+d-2,tq-nu-d+2,Ai,t(i=0,…,ny-1) and Bj,t(j=0,…,nu+d-2) are coefficient matrices. And G(q-1,ϕ(t)), F(q-1,ϕ(t)) are unique polynomials satisfying (2.11)F(q-1,ϕ(t))A(q-1,ϕ(t))=I-A(q-1,ϕ(t))q-d.


### 2.3. Hybrid Expression

The coefficients matricesAi,t(i=0,…,ny-1) and Bj,t(j=0,…,nu+d-2) can be considered as a summation of two parts: the constant part Ail and Bjl and the nonlinear function part on ϕ(t) which are denoted Ai,tn and Bi,tn. Then, the expression of system in the predictor form (2.9) can be described by (2.12)y(t+d)=Al(q-1)y(t)+Bl(q-1)u(t)+An(q-1,ϕ(t))y(t)+Bn(q-1,ϕ(t))u(t),
where (2.13)Al(q-1)=A0l+A1lq-1+⋯+Any-1lq-ny+1,An(q-1,ϕ(t))=A0,tl+A1,tlq-1+⋯+Any-1,tlq-ny+1,Bl(q-1)=B0l+B1lq-1+⋯+Bny-d+2lq-nu+d-2,Bn(q-1,ϕ(t))=B0,tl+B1,tlq-1+⋯+Bny-d+2,tlq-nu+d-2.Similar with [18], the linear polynomial matrix ℬl(q-1) can be expressed as ℬl(q-1)=ℬ¯l(q-1)+ℬ̿l(q-1) with ℬ¯l(q-1) being diagonal and ℬ̿l(q-1) being a polynomial matrix with zero diagonal elements.Then, the linear and nonlinear expression of system (2.12) can be obtained as (2.14)y(t+d)=Al(q-1)y(t)+B¯l(q-1)u(t)+B̿l(q-1)u(t)+An(q-1,ϕ(t))y(t)+Bn(q-1,ϕ(t))u(t).


### 2.4. Quasi-ARX RBFN Model

Now, we will propose an MIMO quasi-ARX RBFN model. However, thev(ϕ(t)) are based on Ψ(t) whose elements contain u(t). To solve this problem, an extravariablex(t) Obviously, in a control system, the reference signal y*(t+d) can be used as the extra variable x(t+d), is introduced, and an unknown nonlinear function ρ(ξ(t)) is used to replace the variable u(t) in ϕ(t). Under the assumption of the system is controllable in Section 2.1, the function ρ(ξ(t)) exists. Define(2.15)ξ(t)=[y(t)T⋯y(t-n1)Tx(t+d)T⋯x(t-n3+d)Tu(t-1)T⋯u(t-n2)T]T,
including the extra variable x(t+d) as an element. A typical choice for n1,n2, and n3 in ξ(t) is n1=ny-1, n2=nu+d-2, and n3=0. We can express (2.14) by(2.16)y(t+d)=ψT(t)Ω0+(t)θξn,
where ψT(t)=φ(t-d). The elements of θξn are unknown nonlinear function of ξ(t), which can be parameterized by NN or RBFN. In this paper, the RBFN is used which has local property:
(2.17)θξn=∑j=1MΩjRj(pj,ξ(t)),
where M is the number of RBFs. Ωj=[Ωj,1,…,Ωj,n] is the coefficient matrix with Ωj,i=[ωj,i1,…,ωj,iN]T, j=1,…,M. And ℛj(ξ(t),Ωj) the RBFs defined by(2.18)Rj(pj,ξ(t))=e-λj‖ξ(t)-Zj‖2,j=1,2,…,M,
where pj={λj,Zj} is the parameters set of the RBFN; Zj is the center vector of RBF and λj are the scaling parameters; ∥•∥2 denotes the vector two norm. Then we can express the quasi-ARX RBFN prediction model for (2.16) in a form of(2.19)y(t+d)=ψT(t)Ω0+ξT(t)∑j=1MΩjRj(pj,ξ(t)).


## 2.1. Systems

Consider an MIMO nonlinear dynamical system with input-output relation as(2.1)y(t+d)=f(φ(t)),φ(t)=[y(t+d-1)T,…,y(t+d-ny)T,u(t)T,…,u(t-nu+1)T]T,
where y=[y1,…,yn]T∈Rn and u=[u1,…,un]T∈Rn are system input and output vectors, respectively, d the known integer time delay, φ(t) the regression vector, and ny,nu the system orders. f(·)=[f1(·),…,fn(·)]T is a vector-valued nonlinear function, and, at a small region around φ(t)=0(0=[0,…,0]T), they are C∞ continuous. The origin is an equilibrium point, then f(0)=0. The system is controllable, in which a reasonable unknown controller may be expressed by u(t)=ρ(ξ(t)), where ξ(t) is defined in Section 2.4.


## 2.2. ARX-Like Expression

Under the continuous condition, the unknown nonlinear functionfk(φ(t)),(i=1,…,n) can be performed Taylor expansion on a small region around φ(t)=0:(2.2)yk(t+d)=fk′(0)φ(t)+12φT(t)fk′′(0)φ(t)+⋯,
where the prime denotes differentiation with respect to φ(t). Then the following notations are introduced:(2.3)(fk′(0)+12φT(t)fk′′(0)+⋯)T=[a1,t1,k⋯any,t1,k⋯any,tn,kb1,t1,k⋯bnu,t1,k⋯bnu,tn,k]T,
where ai,tl,k=ail,k(φ(t))(i=1,…,ny) and bj,tl,k=bjl,k(φ(t))(j=0,…,nu-1) are nonlinear functions of φ(t).However, we need to gety(t+d) by using the input-output data up to time t in a model. The coefficients ai,tl,k and bj,tl,k need to be calculable using the input-output data up to time t. To do so, let us iteratively replace y(t+l) in the expressions of ai,tl,k and bj,tl,k with functions:(2.4)y(t+s)⟹g(φ̃(t+s)),s=1,…,d-1,
where φ̃(t+s) is φ(t+s) whose elements y(t+m),s+1<m≤d-s are replaced by (2.4), and define the new expressions of the coefficients by(2.5)ai,tl,k=ãi,tl,k=ãil,k(ϕ(t)),bj,tl,k=b̃j,tl,k=b̃jl,k(ϕ(t)),
where ϕ(t) is a vector:(2.6)ϕ(t)=[y(t)T⋯y(t-ny+1)Tu(t)T⋯u(t-nu-d+2)T]T.Now, introduce two polynomial matricesA(q-1,ϕ(t)) and B(q-1,ϕ(t)) based on the coefficients, defined by(2.7)A(q-1,ϕ(t))=I-a1,tq-1-⋯-any,tq-ny,B(q-1,ϕ(t))=b0,t+⋯+bnu-1,tq-nu+1,
where ai,t=(ai,tl,k)N×N,i=1,…,ny and bj,t=(bj,tl,k)N×N,j=1,…,nu. Then, the nonlinear system (2.1) can be equivalently represented as the following ARX-like expression:(2.8)A(q-1,ϕ(t))y(t+d)=B(q-1,ϕ(t))u(t).By (2.8), let y(t+d) satisfies the following equation:(2.9)y(t+d)=A(q-1,ϕ(t))y(t)+B(q-1,ϕ(t))u(t),
where (2.10)A(q-1,ϕ(t))=A0,t+A1,tq-1+⋯+Any-1,tq-ny+1,B(q-1,ϕ(t))=F(q-1,ϕ(t))B(q-1,ϕ(t)),=B0,t+B1,tq-1+⋯+Bnu+d-2,tq-nu-d+2,Ai,t(i=0,…,ny-1) and Bj,t(j=0,…,nu+d-2) are coefficient matrices. And G(q-1,ϕ(t)), F(q-1,ϕ(t)) are unique polynomials satisfying (2.11)F(q-1,ϕ(t))A(q-1,ϕ(t))=I-A(q-1,ϕ(t))q-d.


## 2.3. Hybrid Expression

The coefficients matricesAi,t(i=0,…,ny-1) and Bj,t(j=0,…,nu+d-2) can be considered as a summation of two parts: the constant part Ail and Bjl and the nonlinear function part on ϕ(t) which are denoted Ai,tn and Bi,tn. Then, the expression of system in the predictor form (2.9) can be described by (2.12)y(t+d)=Al(q-1)y(t)+Bl(q-1)u(t)+An(q-1,ϕ(t))y(t)+Bn(q-1,ϕ(t))u(t),
where (2.13)Al(q-1)=A0l+A1lq-1+⋯+Any-1lq-ny+1,An(q-1,ϕ(t))=A0,tl+A1,tlq-1+⋯+Any-1,tlq-ny+1,Bl(q-1)=B0l+B1lq-1+⋯+Bny-d+2lq-nu+d-2,Bn(q-1,ϕ(t))=B0,tl+B1,tlq-1+⋯+Bny-d+2,tlq-nu+d-2.Similar with [18], the linear polynomial matrix ℬl(q-1) can be expressed as ℬl(q-1)=ℬ¯l(q-1)+ℬ̿l(q-1) with ℬ¯l(q-1) being diagonal and ℬ̿l(q-1) being a polynomial matrix with zero diagonal elements.Then, the linear and nonlinear expression of system (2.12) can be obtained as (2.14)y(t+d)=Al(q-1)y(t)+B¯l(q-1)u(t)+B̿l(q-1)u(t)+An(q-1,ϕ(t))y(t)+Bn(q-1,ϕ(t))u(t).


## 2.4. Quasi-ARX RBFN Model

Now, we will propose an MIMO quasi-ARX RBFN model. However, thev(ϕ(t)) are based on Ψ(t) whose elements contain u(t). To solve this problem, an extravariablex(t) Obviously, in a control system, the reference signal y*(t+d) can be used as the extra variable x(t+d), is introduced, and an unknown nonlinear function ρ(ξ(t)) is used to replace the variable u(t) in ϕ(t). Under the assumption of the system is controllable in Section 2.1, the function ρ(ξ(t)) exists. Define(2.15)ξ(t)=[y(t)T⋯y(t-n1)Tx(t+d)T⋯x(t-n3+d)Tu(t-1)T⋯u(t-n2)T]T,
including the extra variable x(t+d) as an element. A typical choice for n1,n2, and n3 in ξ(t) is n1=ny-1, n2=nu+d-2, and n3=0. We can express (2.14) by(2.16)y(t+d)=ψT(t)Ω0+(t)θξn,
where ψT(t)=φ(t-d). The elements of θξn are unknown nonlinear function of ξ(t), which can be parameterized by NN or RBFN. In this paper, the RBFN is used which has local property:
(2.17)θξn=∑j=1MΩjRj(pj,ξ(t)),
where M is the number of RBFs. Ωj=[Ωj,1,…,Ωj,n] is the coefficient matrix with Ωj,i=[ωj,i1,…,ωj,iN]T, j=1,…,M. And ℛj(ξ(t),Ωj) the RBFs defined by(2.18)Rj(pj,ξ(t))=e-λj‖ξ(t)-Zj‖2,j=1,2,…,M,
where pj={λj,Zj} is the parameters set of the RBFN; Zj is the center vector of RBF and λj are the scaling parameters; ∥•∥2 denotes the vector two norm. Then we can express the quasi-ARX RBFN prediction model for (2.16) in a form of(2.19)y(t+d)=ψT(t)Ω0+ξT(t)∑j=1MΩjRj(pj,ξ(t)).


## 3. Controller Design

### 3.1. Nonlinear Multivariable Decoupling PID Controller

Introduce the following performance index:(3.1)M(t+d)=‖y(t+d)-R(q-1)y*(t+d)+S(q-1)u(t)+Q(q-1)u(t)‖,
where R and S are the diagonal weighting polynomial matrices, and Q is a weighting polynomial matrix with diagonal elements.The optimal control law minimizing (3.1) is (3.2)y(t+d)-R(q-1)y*(t+d)+S(q-1)u(t)+Q(q-1)u(t)=0.
Substituting (2.14) into (3.2), the following equation is obtained:(3.3)(B¯l(q-1)+Q(q-1))u(t)=R(q-1)y*(t+d)-Al(q-1)y(t)-(B̿l(q-1)+S(q-1))u(t)-(Bn(q-1,ϕ(t))u(t)+An(q-1,ϕ(t))y(t)),
where ℬ¯l(q-1)+Q(q-1)=λ-1H¯(q-1), with λ=diag{λ1,…λn} and H¯(q-1)=(1-q-1)·I. By introducing R(q-1)=𝒜l(q-1) and ℬ¯l(q-1)S(q-1)=Q(q-1)ℬ̿l(q-1), when ny-1≤2, a nonlinear decoupling PID controller is obtained, similar to a traditional PID controller: (3.4)H¯(q-1)u(t)=λAl(q-1)e(t)-H̿(q-1)u(t)-v(ϕ(t)),
where H̿(q-1)=λ(ℬ̿l(q-1)+S(q-1)) and v(ϕ(t))=λ(ℬn(q-1,ϕ(t))u(t)+𝒜n(q-1,ϕ(t))y(t)). e(t)=y*(t+d)-y(t).The controller (3.4) is substituted into the system (3.2), the obtained closed-loop system which is shown in Figure 1 will be stable, and the decoupling control effect and tracking errors can be eliminated.Figure 1
The multivariable decoupling PID control system based on MIMO quasi-ARX model.A velocity-type form of the PID controller is given:(3.5)H¯(q-1)u(t)=Kp(e(t)-e(t-1))+KIe(t)+KD(e(t)-2e(t-1)+e(t-2))-H̿(q-1)u(t)-v(ϕ(t)).
The gain can be selected as (3.6)Kp=-λ(2A2+A1),KI=λ(A0+A1+A2),KD=λA2,
where when ny=1, A1=A2=0 and when ny=2, A2=0.


### 3.2. Parameter Estimation

#### 3.2.1. A Simple Strategy for Determiningpj

Now let us initializepj, denoted as follows: (3.7)pj=[z¯1jz¯2j⋯z¯Nj,λj]T(j=1,…,M),
where N=dim(ξ(t)). Since pj is associated with partition of ξ(t), the bounds of ξ(t) can be used to determine a fairly good initial value. It will not be discussed here, and the interested readers are referred to [26].


#### 3.2.2. Estimation of Parameter VectorΩ0

If the process is known,Ω0 is obtained by using Taylor expansion at its equilibrium; otherwise, it will be replaced by its estimation Ω̂0.


#### 3.2.3. Estimation of Parameter VectorΩj

Parameter vectorΩj(j=1,…,M) can be estimated by a simplified multivariable least-squares algorithm as in [27]. By introducing the notations: (3.8)Ω=[Ω1T,…,ΩMT]T,Φ(t)=[ξ(t)T⊗ΨRT(t)]T,
where the symbol ⊗ denotes Kronecker production, then ΨℛT(t)=[ℛj(pj,ξ(t)),j=1,…,M], the MIMO quasi-ARX model (2.12) can be expressed in a regression form:(3.9)y(t+d)=ψT(t)Ω0+ΦT(t)Ω.The parameterΩ is updated by an LS algorithm while fixing pj and Ω0:
(3.10)Ω̂(t)=Ω̂(t-d)+p(t)Φ(t-d)e(t)1+Φ(t-d)Tp(t)Φ(t-d),
where Ω̂(t) is the estimate of Ω at time instant t. e(t) is the error vector of MIMO quasi-ARX model, defined by(3.11)e(t)=y(t)-ψT(t)Ω0-Φ(t-d)TΩ̂(t-d),P(t)=P(t-d)-PT(t-d)Φ(t-d)TΦ(t-d)P(t-d)1+Φ(t-d)TP(t)Φ(t-d).Remark 3.1.
Comparing with [18], there are three improvements: the unmodeled part is modeled in this paper by quasi-ARX model, RBFN is used to replace NN, and some priori knowledge can be used to determine the parameters.


## 3.1. Nonlinear Multivariable Decoupling PID Controller

Introduce the following performance index:(3.1)M(t+d)=‖y(t+d)-R(q-1)y*(t+d)+S(q-1)u(t)+Q(q-1)u(t)‖,
where R and S are the diagonal weighting polynomial matrices, and Q is a weighting polynomial matrix with diagonal elements.The optimal control law minimizing (3.1) is (3.2)y(t+d)-R(q-1)y*(t+d)+S(q-1)u(t)+Q(q-1)u(t)=0.
Substituting (2.14) into (3.2), the following equation is obtained:(3.3)(B¯l(q-1)+Q(q-1))u(t)=R(q-1)y*(t+d)-Al(q-1)y(t)-(B̿l(q-1)+S(q-1))u(t)-(Bn(q-1,ϕ(t))u(t)+An(q-1,ϕ(t))y(t)),
where ℬ¯l(q-1)+Q(q-1)=λ-1H¯(q-1), with λ=diag{λ1,…λn} and H¯(q-1)=(1-q-1)·I. By introducing R(q-1)=𝒜l(q-1) and ℬ¯l(q-1)S(q-1)=Q(q-1)ℬ̿l(q-1), when ny-1≤2, a nonlinear decoupling PID controller is obtained, similar to a traditional PID controller: (3.4)H¯(q-1)u(t)=λAl(q-1)e(t)-H̿(q-1)u(t)-v(ϕ(t)),
where H̿(q-1)=λ(ℬ̿l(q-1)+S(q-1)) and v(ϕ(t))=λ(ℬn(q-1,ϕ(t))u(t)+𝒜n(q-1,ϕ(t))y(t)). e(t)=y*(t+d)-y(t).The controller (3.4) is substituted into the system (3.2), the obtained closed-loop system which is shown in Figure 1 will be stable, and the decoupling control effect and tracking errors can be eliminated.Figure 1
The multivariable decoupling PID control system based on MIMO quasi-ARX model.A velocity-type form of the PID controller is given:(3.5)H¯(q-1)u(t)=Kp(e(t)-e(t-1))+KIe(t)+KD(e(t)-2e(t-1)+e(t-2))-H̿(q-1)u(t)-v(ϕ(t)).
The gain can be selected as (3.6)Kp=-λ(2A2+A1),KI=λ(A0+A1+A2),KD=λA2,
where when ny=1, A1=A2=0 and when ny=2, A2=0.


## 3.2. Parameter Estimation

### 3.2.1. A Simple Strategy for Determiningpj

Now let us initializepj, denoted as follows: (3.7)pj=[z¯1jz¯2j⋯z¯Nj,λj]T(j=1,…,M),
where N=dim(ξ(t)). Since pj is associated with partition of ξ(t), the bounds of ξ(t) can be used to determine a fairly good initial value. It will not be discussed here, and the interested readers are referred to [26].


### 3.2.2. Estimation of Parameter VectorΩ0

If the process is known,Ω0 is obtained by using Taylor expansion at its equilibrium; otherwise, it will be replaced by its estimation Ω̂0.


### 3.2.3. Estimation of Parameter VectorΩj

Parameter vectorΩj(j=1,…,M) can be estimated by a simplified multivariable least-squares algorithm as in [27]. By introducing the notations: (3.8)Ω=[Ω1T,…,ΩMT]T,Φ(t)=[ξ(t)T⊗ΨRT(t)]T,
where the symbol ⊗ denotes Kronecker production, then ΨℛT(t)=[ℛj(pj,ξ(t)),j=1,…,M], the MIMO quasi-ARX model (2.12) can be expressed in a regression form:(3.9)y(t+d)=ψT(t)Ω0+ΦT(t)Ω.The parameterΩ is updated by an LS algorithm while fixing pj and Ω0:
(3.10)Ω̂(t)=Ω̂(t-d)+p(t)Φ(t-d)e(t)1+Φ(t-d)Tp(t)Φ(t-d),
where Ω̂(t) is the estimate of Ω at time instant t. e(t) is the error vector of MIMO quasi-ARX model, defined by(3.11)e(t)=y(t)-ψT(t)Ω0-Φ(t-d)TΩ̂(t-d),P(t)=P(t-d)-PT(t-d)Φ(t-d)TΦ(t-d)P(t-d)1+Φ(t-d)TP(t)Φ(t-d).Remark 3.1.
Comparing with [18], there are three improvements: the unmodeled part is modeled in this paper by quasi-ARX model, RBFN is used to replace NN, and some priori knowledge can be used to determine the parameters.


## 3.2.1. A Simple Strategy for Determiningpj

Now let us initializepj, denoted as follows: (3.7)pj=[z¯1jz¯2j⋯z¯Nj,λj]T(j=1,…,M),
where N=dim(ξ(t)). Since pj is associated with partition of ξ(t), the bounds of ξ(t) can be used to determine a fairly good initial value. It will not be discussed here, and the interested readers are referred to [26].


## 3.2.2. Estimation of Parameter VectorΩ0

If the process is known,Ω0 is obtained by using Taylor expansion at its equilibrium; otherwise, it will be replaced by its estimation Ω̂0.


## 3.2.3. Estimation of Parameter VectorΩj

Parameter vectorΩj(j=1,…,M) can be estimated by a simplified multivariable least-squares algorithm as in [27]. By introducing the notations: (3.8)Ω=[Ω1T,…,ΩMT]T,Φ(t)=[ξ(t)T⊗ΨRT(t)]T,
where the symbol ⊗ denotes Kronecker production, then ΨℛT(t)=[ℛj(pj,ξ(t)),j=1,…,M], the MIMO quasi-ARX model (2.12) can be expressed in a regression form:(3.9)y(t+d)=ψT(t)Ω0+ΦT(t)Ω.The parameterΩ is updated by an LS algorithm while fixing pj and Ω0:
(3.10)Ω̂(t)=Ω̂(t-d)+p(t)Φ(t-d)e(t)1+Φ(t-d)Tp(t)Φ(t-d),
where Ω̂(t) is the estimate of Ω at time instant t. e(t) is the error vector of MIMO quasi-ARX model, defined by(3.11)e(t)=y(t)-ψT(t)Ω0-Φ(t-d)TΩ̂(t-d),P(t)=P(t-d)-PT(t-d)Φ(t-d)TΦ(t-d)P(t-d)1+Φ(t-d)TP(t)Φ(t-d).Remark 3.1.
Comparing with [18], there are three improvements: the unmodeled part is modeled in this paper by quasi-ARX model, RBFN is used to replace NN, and some priori knowledge can be used to determine the parameters.


## 4. Stability Analysis

There are some assumption made.Assumption 1.
(i)y*(t) is a bounded deterministic sequence; (ii) v(ϕ(t)) is globally bounded, |v(ϕ(t))|≤Δ, where the boundary Δ is known; (iii) the choices of λ and S(q-1) are such that det{H̃(q-1)A(q-1)+q-dB̃(q-1)λ𝒜l(q-1)}≠0.Theorem 4.1.
For the MIMO nonlinear (2.1) with the controller (3.5), together with the parameters of the controller selected by Section 3.2, all the signals in the closed-loop system described above can be bounded, and the tracking error can be made less than any specified constant δ over a compact set by properly choosing the structures and parameters of quasi-ARX RBFN model, that is, limt→∞∥y(t+d)-y*(t+d)∥≤ɛ.Proof.
The nonlinear part estimation error vector can be described by(4.1)ɛ(t)=v(ϕ(t+d))-ξT(t+d)∑j=1MΩ̂(t+d)jRj(pj,ξ(t+d)).
We can see that, if the nonlinear decoupling PID controller (3.5) is used to the system (2.14), the following input-output dynamics are obtained as in [18]:
(4.2)(H̃(q-1)A(q-1)+q-dB̃(q-1)λAl(q-1))y(t+d)=B̃(q-1)λAl(q-1)y*(t+d)+H̃(q-1)v(ϕ(t+d))-B̃(q-1)v̂(ϕ(t+d)),(A(q-1)H(q-1)+q-dλA(q-1)Al(q-1))u(t+d)=Ã(q-1)λAl(q-1)y*(t+d)-q-dλAl(q-1)v(ϕ(t+d))-A(q-1)v̂(ϕ(t+d)).
Substitute (4.1) into (4.2), the equations are given as follows:
(4.3)(H̃(q-1)A(q-1)+q-dB̃(q-1)λG(q-1))y(t+d)=B̃(q-1)λG(q-1)y*(t+d)+(H̃(q-1)-B̃(q-1))v(ϕ(t+d))+B̃(q-1)ɛ(t),(A(q-1)H(q-1)+q-dλA(q-1)Al(q-1))u(t+d)=Ã(q-1)λAl(q-1)y*(t+d)-(q-dλAl(q-1)+A(q-1))v(ϕ(t+d))-A(q-1)ɛ(t).
From (4.3) and Assumption 1, there exist constants C1,C2,C3,C4 satisfying
(4.4)‖y(t+d)‖≤C1+C2max⁡0≤τ≤t‖ɛ(t)‖,‖u(t)‖≤C3+C4max⁡0≤τ≤t‖ɛ(t)‖.
Because of the universal approximations of the RBFNs, the estimation error ɛ(t) can be achieved less than any constant ζ over a compact set by properly choosing their structures and parameters. It can be got that
(4.5)‖φ(t+d)‖≤C5+C6max⁡0≤τ≤t‖ɛ(t)‖≤C7+C8ζ≤C9.
where C5,C6,C7,C8,C9 are constants.
Then, the boundness of all the signals in the closed-loop system is got.The tracking error of the system is obtained as(4.6)e=lim⁡t→∞‖y(t+d)-y*(t+d)‖≤C,
where C>0 is a constant.


## 5. Numerical Simulations

In order to study the behavior of the proposed control method, a numerical simulation is described in this section. The MIMO nonlinear system to be controlled is described by(5.1)y1(t+1)=0.9y1(t)-0.3y1(t-1)1+y22(t-1)+0.4sin⁡⁡(u1(t))+0.7u1(t-1)+0.3u2(t)-0.5u2(t-1),y2(t+1)=-0.4sin⁡⁡(y22(t))-0.1y2(t-1)+u2(t-1)-0.3sin⁡⁡(u1(t))+0.2u1(t-1+0.8sin⁡⁡(u2(t)))+0.5u22(t-1),fort∈[0,150),y1(t+1)=0.6y1(t)-0.4y1(t-1)1+y22(t-1)+0.4sin⁡⁡(u1(t))+0.6u1(t-1)+0.4u2(t)-0.5u2(t-1),y2(t+1)=-0.5sin⁡⁡(y22(t))-0.1y2(t-1)+u2(t-1)-0.3sin⁡⁡(u1(t))+0.3u1(t-1)+0.9sin⁡⁡(u2(t))+0.5u22(t-1),fort∈[150,∞).In this example, a system disturbance appears whent=150. The desired output of system is given y1*(t)=sign(sin(πt/50)) and y2*(t)=0.7.In this example, the proposed control method in Sections3 and 4 is illustrated effective in the control stability and robustness. The order is chosen as ny=nu=2, and time delay d=1. The regression φ(t)=[y1(t-1)y2(t-1)y1(t-2)y2(t-2)u1(t-1)u2(t-1)u1(t-2)u2(t-2)]T and ξ(t)=[y1(t-1)y2(t-1)y1(t-2)y2(t-2)y1*(t)y1*(t)y2*(t)u1(t-2)u2(t-2)]T. Based on the priori acknowledge, we choose Zmax=[22224141] and Zmin=[-2-2-2-2-4-1-4-1]. The parameters pj can be determined by the proposed method in Section 3.2.Under the same simulation conditions and with the same parameters value, the control output results by a typical PID controller are given for comparison, where the PID controller has neither the decoupling compensator nor the nonlinear part. The control outputs are shown in Figure2, the solid red line is the desired outputs, the dashed blue line is the typical PID control outputs, and the dotted green line is the proposed method control outputs. The corresponding control inputs u1(t) and u2(t) are given in Figures 3 and 4. We can see that our proposed method is nearly consistent with the desired output at most of the time which is better than typical PID control method when t∈[0,150). Obviously, the control performance of our proposed method is much better than typical PID control method when the system has disturbance when t=150. The input signals have small fluctuation as shown in Figure 4.Control outputs.
(a)(b)Corresponding control inputs of the PID control method.
(a)(b)Corresponding control inputs of the proposed control method.
(a)(b)Table1 gives the comparison results of the errors. Obviously, the mean and variance of errors of the proposed method are smaller than the typical PID control method.Table 1
Comparison results of errors based on two control method.
Mean of errorsVariance of errorsy1(t) typical method0.01320.1350: proposed method−0.00900.0668y2(t) typical method−0.00670.0157: proposed method−0.00390.0098


## 6. Conclusions

In this paper, an MIMO quasi-ARX model is first introduced, and a nonlinear multivariable decoupling PID controller is proposed based on the proposed model for MIMO nonlinear systems. The proposed controller consists of a traditional PID controller with a decoupling compensator and a feed-forward compensator for the nonlinear dynamics based on the MIMO quasi-ARX model. And an adaptive control system is presented using the MIMO quasi-ARX RBFN prediction model. The parameters of such controller are selected based on the generalized minimum control variance. The proposed control method has more simplicity structures and better control performance. The nonlinear part is not a black box whose parameters can be determined bya priori acknowledge. Simulation results show the effectiveness of the proposed method on control accuracy and robustness when a disturbance appears in the system. Because the PID controller can be realized on standard DCS/PLC modules, the algorithm is more useful for industrial process control. Otherwise, because the parameters of controller are chosen from the generalized minimum variance control law, it is easier for engineers and process operators to relate the parameter settings.



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*Source: 570498-2011-11-02.xml*