Design of Explicit Fuzzy Prediction Controller for Constrained Nonlinear Systems

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.


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][4][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][14][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 vector  and positive definite matrix , ‖‖

Problem Statement
Consider the following constrained nonlinear system: subject to control constraints where () ∈   and () ∈   are state and control vectors,  is a nonlinear function in   ×   , and (0, 0) = 0,  = [ 1 , . . .,   ]  , and  = [ 1 , . . .,   ]  , so control condition (2) can also be expressed as   ≤   () ≤   ;   (),   , and   are the th element of the (), , and , respectively,  = 1, . . ., .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 closedloop system: min

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 where  = 1, . . ., ,  is the number of fuzzy rules and    are fuzzy sets, for all  = 1, . . .,  + .  ∈  × and   ∈  × .
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 of   belonging to the set 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 where is the clustering center and also the center of membership function, the number of clustering is  (which is also the number of fuzzy rules),  is the sample size,  is the fuzzy exponent,   is the membership of the th data relative to the th clustering center, and (  , V  ) denotes the distance norm of the th clustering relative to the th data: where The necessary conditions obtained by Lagrange multipliers that make the objective function minimum are The variance of the Gaussian membership function is By the weighted least squares, the amount of the parameters to be identified is ( 2 +  × ) × ; make  = ( 1 ,  1 ,  2 ,  2 , . . .,   ,   )  , where For identification, define Then by the least squares we get the coefficient matrices where The proposed T-S fuzzy model parameters identification algorithm can now be specified as follows.
Step 1. Select the number of fuzzy rule , fuzzy exponent , and standard termination  > 0.
Step 6. Calculate consequent parameters of the T-S model according to (11); that is to say, work out the coefficient matrices   = Â ,   = B .
The rules of the T-S model we get by G-K clustering algorithm are The expression of T-S fuzzy model by the above algorithm is

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 ().We assume the state error at sample time  is () = () −   (), where   () 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   ( +  | ) by (), we obtain where   ( +  | ) is the predicted values of state variables and  pc ( +  | ) is the predicted values achieved from the revised T-S model as follows.Substituting ( 15) into ( 14), we can obtain For the revised model, solve the following optimal problem: min where  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 matrices  ∈  × ,  ∈  × , such that the following equations are satisfied: where Based on (21) and the Schur complement, inequation (18) is obtained.
Remark 2. The stability of the system in the terminal set Ω and the invariance of Ω are guaranteed by (20).
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 ,  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  0 that should be in Ω.

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 where  1 denotes the angle (in radians) of the pendulum from the vertical and  2 denotes the angular velocity. = 9.8 m/s 2 is the gravity constant, m is the mass of the pendulum,  is the mass of the cart, 2 is the length of the pendulum, and  is the force applied to the cart.We choose m = 2 kg,  = 8 kg, and  = 0.5 m.
Identify the above model using the scheme in Section 2. Let  = 2 and  = 0.02; we obtain the T-S fuzzy model as follows.(36) Membership functions are as Figure 1.
The comparison between actual measure output and T-S model output is as Figure 2. The result shows that the simulation acquires good fitting effect.
Compute the control law by the algorithm in Section 4. Set the algorithm parameters as follows: prediction horizon  = 5, control constraint  max = 1000 N, weight matrix of optimal performance index  = 0.01, and   = diag{100, 1}.
Set the initial state value  0 = [2,−3]  , 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].: 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]).

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.

Figure 2 :
Figure 2: Comparison between actual output and T-S model output.

Figure 3 :Figure 4 :Figure 5
Figure3: State trajectory  1 (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]).