Robust multi-criteria optimal fuzzy control of discrete-time nonlinear systems

This paper presents a novel fuzzy control design of discrete-time nonlinear systems with multiple performance criteria. The purpose behind this work is to improve the traditional fuzzy controller performance to satisfy several performance criteria simultaneously to secure quadratic optimality with inherent stability property together with dissipativity type of disturbance reduction. The Takagi-Sugeno type fuzzy model is used in our control system design. By solving a linear matrix inequality at each time step, the optimal control solution can be found to satisfy mixed performance criteria. The effectiveness of the proposed technique is demonstrated by simulation of the control of the inverted pendulum system on a cart.


I. INTRODUCTION
Fuzzy control systems have recently shown growing popularity in nonlinear system control applications [1]- [6].A fuzzy control system is essentially an effective way to decompose the task of nonlinear system control into a group of local linear controls based on a set of design-specific model rules.Fuzzy control also provides a mechanism to blend these local linear control problems all together to achieve overall control of the original nonlinear system.In this regard, fuzzy control technique has its unique advantage over other kinds of nonlinear control techniques.Latest research on fuzzy control systems design is aimed to improve the optimality and robustness of the controller performance by combining the advantage of modern control theory with the Takagi-Sugeno fuzzy model [1]- [6].
In this paper, we address the nonlinear state feedback control design of discrete-time nonlinear fuzzy control systems using the Linear Matrix Inequality (LMI) approach.We characterize the solution of the nonlinear discrete-time control problem with the LMI, which provides a sufficient condition for satisfying various performance criteria.A preliminary investigation into the LMI approach to nonlinear fuzzy control systems can be found in [1]- [3].The purpose behind this novel approach is to convert a nonlinear system control problem into a convex optimization problem which is solved by a LMI at each time step.The recent development in convex optimization provides efficient algorithms for solving LMIs.If a solution can be expressed in a LMI form, then there exist optimization algorithms providing efficient global numerical solutions [7].Therefore if the LMI is Xin Wang and Edwin E. Yaz and are with the Electrical and Computer Engineering Department, Marquette University, Milwaukee, WI 53201, USA.(E-mail: {xin.wang;edwin.yaz}@marquette.edu)feasible, then LMI control technique provides globally stable solutions satisfying the corresponding mixed performance criteria at each time step [8]- [11].We further propose to employ mixed performance criteria to design the controller guaranteeing the quadratic sub-optimality with inherent stability property in combination with dissipativity type of disturbance attenuation.
In the following section, we first describe the Takagi-Sugeno fuzzy model.We then introduce the mixed performance criteria in section III.Then, the LMI control solution is derived to characterize the optimal and robust fuzzy control of nonlinear systems.Finally, an inverted pendulum on a cart control problem is used as an illustrative example.The following notation is used in this work: denotes n-dimensional real vector with norm x .

II. TAKAGI-SUGENO SYSTEM MODEL
The importance of the Takagi-Sugeno fuzzy system model is that it provides an effective way to decompose a complicated nonlinear system into local dynamical relations and express those local dynamics of each fuzzy implication rule by a linear system model.The overall fuzzy nonlinear system model is achieved by fuzzy "blending" of the linear system models, so that the overall nonlinear control performance is achieved.At time step k , the th i rule of the Takagi-Sugeno fuzzy model can be expressed by the following forms: THEN, the input-affine discrete-time fuzzy system equation is: ( where It is assumed that the state feedback is available and the nonlinear state feedback control input is given by Substituting this into the system and performance output equation, we have then the system equation becomes

III. PERFORMANCE INDEX
Consider the quadratic Lyapunov function ( ) ( ) ( ) ( ) 0 Note that upon summation over k , (17) yields By properly specifying the value of weighing matrices , , , , , mixed performance criteria can be used in nonlinear control design, which yields a mixed Nonlinear Quadratic Regulator (NLQR) in combination with dissipativity type performance index with disturbance reduction capability.For example, if we take 0 , 0 , 1 γ can be minimized to achieve a smaller 2 2 l l − or H ∞ gain for the closed loop system.Other possible performance criteria which can be used in this framework with various design parameters are given in Table .1.Design coefficients α and γ can be maximized or minimized to optimize the controller behavior.It should also be noted that the satisfaction of any of the criteria in Table 1 will also guarantee asymptotic stability of the controlled system.The following theorem summarizes the main results of the paper: Theorem -Given the closed loop system Eqn.(14), performance output Eqn.(15), if there exist matrices

Proof
The inequality (17) can be explicitly written as (25) which can be written, after collecting terms, as (28) By applying Schur complement [7], we obtain 11 12 Equivalently, the following inequality holds By factoring out the ∑∑ term, we have 11  The inverted pendulum on a cart problem is a benchmark used widely to test control algorithms.A pendulum beam attached to the cart can rotate freely in the vertical 2dimensional plane.The angle of the beam with respect to the vertical direction is denoted at angle θ.The external force u is desired to set angle of the beam θ and angular velocity θ  all to zero, while satisfying the mixed performance criteria.A model of the inverted pendulum problem is given by: [2] [14] ( ) ( ) x is close to zero, THEN The following values are used in our simulation:  This paper presents a novel discrete time nonlinear system fuzzy control approach based on the LMI solutions.We have first applied the Takagi-Sugeno fuzzy model to decompose the nonlinear system.Mixed performance criteria have been used to design the controller and the relative weighting matrices of these criteria can be achieved by choosing different coefficient matrices.The optimal control can be obtained by solving LMI at each time step.The benchmark inverted pendulum on a cart problem has been used as an example to demonstrate its effectiveness.The simulation studies show that the proposed method provides a satisfactory alternative to the existing nonlinear control approaches.
a positive semi-definite matrix. 2 l is the space of infinite sequences of finite dimensional random vectors with finite energy: 32)By applying Schur complement one more time, we have the LMI (36) holds, inequality (17) is satisfied.This concludes the proof of Theorem.■ V. INVERTED PENDULUM FUZZY LMI CONTROL USING WITH MIXED PERFORMANCE CRITERIA

Fig. 4 .
Fig.4.Control input applied to the inverted pendulum