# Design of Explicit Fuzzy Prediction Controller for Constrained Nonlinear Systems

Authors: Yan Yan; Baili Su
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

This paper presents an explicit fuzzy predictive control method for a class of nonlinear systems with constrained inputs. The main idea is to construct a terminal invariant set and explicit predictive controller with affine input on the basis of T-S fuzzy model. This method need not compute the complex nonconvex nonlinear programming problem of earlier nonlinear predictive control methods and decreases the number of optimization variables and guarantees stability of the closed-loop system. The simulation results on a numerical example show the validity of the method presented in this paper.


---

## Body

## 1. Introduction

Almost all of the industrial processes have strong nonlinearity. Such strongly nonlinear industrial process is difficult to be modeled and controlled. It has attracted much attention in industry and academia by referring to [1, 2]. There are many constraints due to physical conditions to limit the flexible performance of the closed-loop systems, which need high skills in control system design.Model predictive control (MPC), as an efficient control strategy to handle constraints within an optimal control setting, has received much attention in the past decades (e.g., [3–5]). The nonlinearity of the nonlinear systems makes the optimization problems nonconvex and thus leads to heavy calculation. It makes the parameters difficult to be adjusted online. In literature [6], using norm-bounded linear differential inclusion (LDI) of nonlinear system, a kind of predictive control scheme was put forward based on terminal domain optimization by solving linear matrix inequalities optimization problem. In [7] a robust model predictive control strategy was presented on the basis of polyhedral description systems for discrete-time nonlinear systems with bounded persistent disturbances. Both literatures [6, 7] used linearization of the original nonlinear model approximately. However the methods produce large errors. So these methods can only be applied in weakly nonlinear systems. Literature [8] combined the robust method and hybrid method to design the MPC for constrained piecewise linear (PWL) systems with structured uncertainty. For the proposed approach, as the system model is known at current time, a free control move is optimized to be the current control input. Paper [9] investigated the problem of predictive control for constrained control systems, in which the measurement signal may be multiply missing. However their applications are limited by their linearity.T-S fuzzy models have become an important tool for researching control problems of the nonlinear system because of their universal approximation capability by referring to [10]. And then fuzzy predictive control scheme is developed. Literature [11] gives a fuzzy multistep linear predictive control strategy taking advantage of T-S model as predictive model and achieves better effect. But it also increases the online optimization computation. In literature [12] a nonlinear predictive control algorithm is proposed based on subsection Lyapunov function and T-S model for the input and output constrained Hammerstein-Wiener nonlinear system. Most of the study results can not obtain explicit expression of controller that is easy to adjust (e.g., [13–15]).In this paper, we present a model predictive control algorithm for a class of constrained nonlinear system based on T-S fuzzy model. Firstly, the feedback correction is introduced after we establish T-S fuzzy model. For this corrected T-S fuzzy model, the terminal invariant set is designed, and in the terminal set the linear feedback control law is proposed to satisfy constraints and guarantee closed-loop stability. Secondly, we design predictive controller with affine control outside the terminal set which satisfies the constraints as well as making the system states eventually enter the terminal set. Meanwhile, the procedure guarantees closed-loop stability and reduces the amount of computation burden. At last we show the validity of this method by a simulation example.In literatures [16], filters are given when the systems’ states are immeasurable. Inspired by them, we will consider the cases with immeasurable states and stochastic disturbance in the future research. We hope to be given more tips.Notations introduction: for vectorx and positive definite matrix W, xW2=xTWx; symbol ∗ represents symmetric structure H∗TR=HTTTR, where H and R are symmetric matrices.


## 2. Problem Statement

Consider the following constrained nonlinear system:(1)x(k+1)=fxk,uksubject to control constraints(2)u_≤uk≤u-,where xk∈Rn and uk∈Rm are state and control vectors, f is a nonlinear function in Rn×Rm, and f(0,0)=0, u_=[u_1,…,u_m]T, and u-=u-1,…,u-mT, so control condition (2) can also be expressed as u_i≤ui(k)≤u-i; ui(k), u_i, and u-i are the ith element of the u(k), u_, and u-, respectively, i=1,…,m.The objective of this paper is to design a predictive controller that can make the following performance index reach to optimization and ensure the stability of the closed-loop system:(3)min⁡u(k+i),i≥0J(k)=∑i=0∞(x(k+i∣k)Rx2+u(k+i∣k)Ru2);x(k+i∣k) and u(k+i∣k) denote predicted values of the states and input vectors. Here we choose T-S fuzzy model as the predictive model. Rx∈Rn×n and Ru∈Rm×m are positive definite, symmetric weighting matrices.


## 3. Identification of T-S Model

As T-S fuzzy model can approximate nonlinear systems with arbitrary precision, we use T-S fuzzy model to approximate the nonlinear system. The rules of T-S fuzzy model are(4)Rj:ifx1isM1j,x2isM2j,…,xnisMnj,iiiiiu1isMn+1j,…,umisMn+mj,thenx(k+1∣k)=Ajxk+Bjuk,where j=1,…,c, c is the number of fuzzy rules and Mij are fuzzy sets, for all i=1,…,n+m. Aj∈Rn×n and Bj∈Rn×m.Many kinds of the membership functions can be chosen, such as triangular membership function, trapezoidal membership function, and Gaussian membership function. The selection of membership functions depends on the expert experience. In general, the membership functions of fuzzy sets with sharp curve shape have high resolution and high control sensitivity; on the contrary, the membership function with gentle curve has relatively smooth control performance and good stable performance. So, in this paper, we choose the Gaussian membership functions; namely, the membership ofxi belonging to the set Mij is Gij(xi)=exp⁡(-(xi-vij)2/2σij2), where vij, σij2 denote the center and the variance of the function.G-K fuzzy clustering algorithm is chosen to determine the premise parameters combined with the least square method to complete the identification of the consequent parameters of T-S fuzzy model by referring to [12].The objective function of G-K algorithm is(5)min⁡J(Z,V,U)=∑j=1c∑r=1Nμrjmd2zr,vj,s.t.μrj∈0,1,∑j=1curj=1,0<∑r=1Nurj<N,where Z=[z1,z2,…,zN] is the data set, U=[μrj]N×c is the membership matrix, V=[v1,v2,…,vc] is the clustering center and also the center of membership function, the number of clustering is c (which is also the number of fuzzy rules), N is the sample size, m is the fuzzy exponent, μrj is the membership of the rth data relative to the jth clustering center, and d(zr,vj) denotes the distance norm of the jth clustering relative to the rth data:(6)d2zr,vj=zr-vjEj2=zr-vjTEjzr-vj,where(7)Ej=det⁡Fj1/nFj-1,Fj=∑r=1Nμrjm(zr-vj)zr-vjT∑r=1Nμrjm.The necessary conditions obtained by Lagrange multipliers that make the objective function minimum are(8)μrj=d2zr,vj-1/m∑j=1cd2zr,vj-1/m,vj=∑r=1Nμrjmzr∑r=1Nμrjm.The variance of the Gaussian membership function isσij2=∑t=1Nμtj(xit-vit)/∑t=1Nμtj.By the weighted least squares, the amount of the parameters to be identified is(n2+n×m)×c; make D=(A1,B1,A2,B2,…,Ac,Bc)T, where(9)Aj=a11j⋯a1nj⋮⋱⋮an1j⋯annj,Bj=b11j⋯b1nj⋮⋱⋮bn1j⋯bnmj.For identification, defineAij=(Ai1j,…,Ainj) and Bij=(Bi1j,…,Bimj) which are the ith line of Aj and Bj, N is assumed to be the number of data, and zr={xr(k+1),xr(k),ur(k)}, r=1,…,N. Construct the following matrix:(10)Ψ=h1x1Th1u1T⋯hcx1Thcu1T⋮⋮⋱⋮⋮h1xNTh1uNT⋯hcxNThcuNT.Then by the least squares we get the coefficient matrices(11)D^i=ΨiTΨi-1ΨiTxk+1,where(12)hj=∏i=1nμij(xi)∑j=1c∏i=1nμij(xi),j=1,…,c.The proposed T-S fuzzy model parameters identification algorithm can now be specified as follows.Step 1. Select the number of fuzzy rule c, fuzzy exponent m, and standard termination ε>0.Step 2. Generate fuzzy matrix U0 randomly.Step 3. Update fuzzy matrix, denote U1, and calculate clustering center by (8).Step 4. Calculate distance norm by (6).Step 5. If U1-U0≤ε, stop; otherwise, repeat Step 3.Step 6. Calculate consequent parameters of the T-S model according to (11); that is to say, work out the coefficient matrices Aj=A^j,  Bj=B^j.The rules of the T-S model we get by G-K clustering algorithm are(13)Rj:ifzr={xr(k),ur(k)}∈Mj,thenxk+1=A^jxk+B^juk.The expression of T-S fuzzy model by the above algorithm is(14)xk+1=∑j=1chjA^jxk+B^juk.


## 4. Prediction Control Strategy on the Basis of T-S Fuzzy Models

In this paper, T-S fuzzy model (14) is selected as prediction model. Due to stochastic disturbance, modeling error, and so on, there must be error between predicted values and the actual state values x(k). We assume the state error at sample time k is e(k)=x(k)-xm(k), where xm(k) is the predictive model state values being gotten by predictive model (14). For the sake of eliminating the error of prediction values caused by several reasons, revising xm(k+i∣k) by e(k), we obtain(15)xpc(k+i∣k)=xm(k+i∣k)+e(k),where xm(k+i∣k) is the predicted values of state variables and xpc(k+i∣k) is the predicted values achieved from the revised T-S model as follows. Substituting (15) into (14), we can obtain(16)xpc(k+i∣k)=∑j=1chjAjxpc(k+i-1∣k)hhhhhhh+Bju(k+i-1∣k)+(I-Aj)e(k).For the revised model, solve the following optimal problem:(17)min⁡u(k+i),i≥0J(k)=∑i=0∞(x(k+i∣k)Rx2+u(k+i∣k)Ru2)s.t.(2),(16),x(k∣k)=x(k),x(k+p∣k)∈Ω,where p is a prediction horizon (for simplicity of exposition, the control and prediction horizons are chosen to have identical values in this paper) and Ω denotes terminal region.Now we first design terminal variant set and linear feedback control law which meet the condition of constraints as well as guaranteeing the closed-loop stability of the system in the terminal variant set.Theorem 1.
Suppose there exist symmetric positive definite matricesS∈Rn×n, P∈Rm×n, such that the following equations are satisfied:(18)S0SAjT-PTBjTPTS∗0I-Aj00∗∗S00∗∗∗Ru-10∗∗∗∗Rx-1≥0,∀j=1,…,c(19)u^i2∗PTeiS≥0,∀i=1,…,m.Then there exists a terminal variant set Ω=x∈Rn∣xTS-1x≤1, such that the controller u=-PS-1xdefined in Ω meets the constraints and the following inequality holds:(20)xpc(k+i+1∣k)S-12-xpc(k+i∣k)S-12≤-xpc(k+i∣k)Rx2-u(k+i∣k)Ru2,∀i≥0.Proof.
Give the setΩ=x∈Rn∣xTS-1x≤1 and candidate Lyapunov function V(x)=xTS-1x.
Then (20) is equivalent to (21) based on (16) and u=-PS-1x in Ω. Consider(21)ξTkAj-BjPS-1I-AjTS-1hhhhhhAj-BjPS-1I-AjT×Aj-BjPS-1I-Aj+Φξ(k)≤0,where ξk=xTkeTkT, Φ=diag{-S-1+Rx+(PS-1)TRuPS-1,0}.
Based on (21) and the Schur complement, inequation (18) is obtained.
Asu_≤u(k)≤u-, it should hold ui2k≤u^i2, i=1,…,m, where u^i=max⁡{u_i,u-i}. Hence we can obtain the following inequality by ui(k)=-eiT(PS-1)x(k) (where ei is the ith item in standard vector base of the input space Rm):(22)xTkPS-1TeieiTPS-1xk≤u^i2.Then due to xTS-1x≤1, we can obtain a sufficient condition of (22):(23)xTkPS-1TeieiTPS-1u^i2xk≤xTkS-1xk.The above inequality can also be expressed as follows:(24)S-1-PS-1TeieiTPSu^i2≥0.Thus constraints (2) can be turned into (19) according to (24) and the Schur complement.Remark 2.
The stability of the system in the terminal setΩ and the invariance of Ω are guaranteed by (20).Remark 3.
ξ
k
=
x
T
k
e
T
k
T is designed to deal with the difficulty getting the results brought by considering error.Remark 4.
In the terminal setΩ, the controller is chosen as u=-PS-1x to be convenient to adjust.Solve the following optimization problem forP, S:(25)min⁡S,Ptrace(S)s.t.1819.In the terminal variant setΩ, feedback control law u=-PS-1x can ensure the stability of the closed-loop system. However with this control law outside the set Ω, the system performance will be worse. Therefore define the following affine input structure (referring to [13]):(26)u(k+i∣k)=-PS-1xpc(k+i∣k)+d(k+i∣k),i=1,…,p-1,-PS-1xpc(k+i∣k),i=p,….The perturbation d(k+i∣k) is optimization variables.Remark 5.
Here we can translate the optimization aboutu(k+i∣k) into the optimization about d(k+i∣k) and thus reduce the computational burden. zk=xpcTkdTkT denotes augmented state variables, where dk=[dT(k∣k)⋯dT(k+p-1∣k)]T; then we get the augmented state space model:(27)zk+1=Γjzk+Gjek,j=1,…,c,where(28)Γj=Aj-BjPS-1BjD0F,F=0(p-1)m×mI(p-1)m×(p-1)m0m×m0m×(p-1)m,D=Ip-1×p-1,0,…,0,Gj=I-AjT,0,…,0T.For augmented system (27), the augmented feasible set is denoted by Ω-={z∈Rn+pm∣zTS~-1z≤1}, where S~ is symmetric positive definite matrix. Then, from Theorem 1, inequality (30) holds for arbitrary z∈Ω- if the following conditions are satisfied:(29)S~0S~TΓjTS~TP~TS~∗0GjT00∗∗S~00∗∗∗Ru-10∗∗∗∗ETRx-1E≥0,∀j=1,…,c,u^i2∗PS-1,I,…,0TeiS~≥0,∀i=1,…,m,where E=I0n×(n+p), P~=[PS-1,I,…,0], and(30)z(k+i+1∣k)S~-12-z(k+i∣k)S~-12≤-x(k+i∣k)Rx2-u(k+i∣k)Ru2.Adding each side of (30) from i=0 to i=∞, we get J(k)≤zT(k)S~-1z(k), where S~ can be obtained by solving the following convex optimal problem:(31)min⁡S~trace(S~)s.t.29.As a consequence, optimal problem (17) can be transformed as follows:(32)min⁡d(k)zT(k)S~-1z(k)s.t.zT(k)S~-1z(k)≤1.Then d(k) can be obtained by solving optimal problem (32).Theorem 6.
In the case of the existence of optimal problem (32), the system is Lyapunov stable under input control (26) in the augmented feasible set Ω-.Proof.
The optimal solution at timek is denoted by d∗(k)={d∗(k∣k),…,d∗(k+p-1∣k)}; meanwhile the feasible sequence at time k+1 is d′(k+1)={d∗(k+1∣k),…,d∗(k+p-1∣k),0}. As a result the cost function at time k+1 is(33)V′zk+1=z′Tk+1S~-1z′k+1=zk∗TΓjTS~-1Γjzk∗.According to (30), we can obtain ΓjTS~-1Γj-S~-1≤0, so V′(z(k+1))≤V∗(z(k)).
And the performance index with optimal solution must not be larger than that with feasible solution. Namely,V∗(z(k+1))≤V′(z(k+1)). Thus we can get V∗(z(k+1))≤V∗(z(k)). In conclusion, V∗(z(k)) is a decreasing Lyapunov function of the system. That is to say, the system is closed-loop stable.Now, we give the nonlinear predictive control strategy on the basis of T-S fuzzy model.Step 1. Get T-S model (4) of nonlinear system (1) by T-S fuzzy model parameters identification algorithm in Section 3.Step 2. Compute S, P by optimal problem (25), and obtain the terminal variant set Ω as well as linear feedback control law.Step 3. Use affine input structure (26), and solve optimal problem (31) for S~. Consequently, determine the variant set Ω- of the augmented system and set initial state x0 that should be in Ω-.Step 4. Measure the current state x(k). If x(k)∈Ω, adopt the control law u=-PS-1x. Otherwise, solve optimal problem (32) for d(k) at the current time.Step 5. Apply feedback control u(k∣k)=-PS-1x(k)+d(k∣k). Let k=k+1 and go to Step 4.


## 5. Simulation Example

Consider the problem of balancing and swing-up of an inverted pendulum on a cart from [12]. The equations of motion for the pendulum are(34)x˙1x˙2=x2gsinx1-am~lx22sin(2x1)/2-acos⁡x1u4l/3-am~lcos⁡2x1,where x1 denotes the angle (in radians) of the pendulum from the vertical and x2 denotes the angular velocity. g=9.8m/s2 is the gravity constant, m~ is the mass of the pendulum, M is the mass of the cart, 2l is the length of the pendulum, and u is the force applied to the cart. We choose m~=2 kg, M=8 kg, and l=0.5 m.Discretize the system by forward difference(xk+1-xk)/T=x˙, and here T=0.01 s.Identify the above model using the scheme in Section2. Let m=2 and ε=0.02; we obtain the T-S fuzzy model as follows.R1: if x1 is A1, x2 is B1, and u is C1, then(35)x1k+1x2k+1=10.010.10341.002x1kx2k+0-0.004uk.R2: if x1 is A2, x2 is B2, and u is C2, then(36)x1k+1x2k+1=10.010.01621.0006x1kx2k+0-0.0912uk.R3: if x1 is A3,  x2 is B3, and u is C3, then(37)x1k+1x2k+1=10.01-0.06591x1kx2k+00.002uk.Membership functions are as Figure1.Figure 1
Membership functions.The comparison between actual measure output and T-S model output is as Figure2. The result shows that the simulation acquires good fitting effect.Figure 2
Comparison between actual output and T-S model output.Compute the control law by the algorithm in Section4. Set the algorithm parameters as follows: prediction horizon p=5, control constraint umax⁡=1000 N, weight matrix of optimal performance index R=0.01, and Rx=diag{100,1}.Solving the optimal problem (25), we obtain(38)S=0.2028-0.0572-0.05720.4332,P=0.0258-0.0311.Set the initial state valuex0=[2,-3]T, and for facilitating comparison we adopt the approach in this paper and [12] for state value and control law, respectively. We can obtain the simulation results showed by Figure 3, Figure 4 and Figure 5. From the figures we can see the control law designed by this paper converges to origin faster than that by [12]. The system reaches the stability state at last and the effect is better than that in [12].Figure 3
State trajectoryx1 (the solid lines are states obtained by the approach in this paper and the dotted linear are states obtained by approach in the paper [12]).Figure 4
State trajectoryx2 (the solid lines are states obtained by the approach in this paper and the dotted linear are states obtained by approach in the paper [12]).Figure 5
Input profiles (the solid line is the input obtained by the approach in this paper and the dotted line is the input obtained by the approach in [12]).


## 6. Conclusions

In this paper, we considered the design and stability problem of predictive controller based on T-S fuzzy models for a class of nonlinear system with constrained inputs. We approximated the original nonlinear model by the T-S model on the basis of G-K clustering algorithm and least squares method. And then we designed predictive controller with affine input based on T-S models. Meanwhile the closed-loop stability of the above-mentioned approach was proved by Lyapunov theory and the validity of the approach was showed by a simulation example.



---

*Source: 857546-2015-05-04.xml*