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

This paper presents a novel fuzzy control design of continuous-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 an LMI at each time, the 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 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 system 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 continuous-time nonlinear fuzzy control systems using the Linear Matrix Inequality (LMI) approach. We characterize the solution of the nonlinear continuoustime control system 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 numerical techniques for convex optimization provides efficient algorithms for solving LMIs. If a solution can be expressed in an LMI form, then there exist optimization algorithms providing efficient global numerical solutions [11]. Therefore if the LMI is feasible, Xin  then LMI control technique provides globally stable solutions satisfying the corresponding mixed performance criteria at each time step [8]- [10]. We further propose to employ mixed performance criteria to design the controller guaranteeing the quadratic 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, the inverted pendulum on a cart control problem is used as an illustrative example. The following notation is used in this work: symmetric matrix denotes a positive semi-definite matrix. 2 L is the space of infinite sequences of finite dimensional random vectors with finite energy:

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.
The th i rule of the Takagi-Sugeno fuzzy model can be expressed by the following forms: THEN, the input-affine continuous-time fuzzy system equation is: total number of the model rules ij M fuzzy set 1 known premise variables which can be functions of state variables, external disturbance and time It is assumed that the premises are not the function of the input vector ) (t u , which is needed to avoid the defuzzification process of fuzzy controller [1]. If we use ) (t It is assumed that the state is available for feedback and the nonlinear state feedback control input is given by (9) Substituting this into the system and performance output equations, we have then the system equation becomes Note that upon integration over time from 0 to f T , (17) By properly specifying the value of the 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 , . Other possible performance criteria which can be used in this framework with various design parameters γ β α , , are given in Table.1.

Remark:
For the chosen performance criterion among those in Table 1, the LMI (20) needs to be solved at each time to find matrices i S and i M . Then by using relation (22), the necessary feedback control gains can be found to satisfy the chosen criterion.

V. SIMULATION STUDIES
The inverted pendulum on a cart problem is a benchmark used widely to test control algorithms. A pendulum beam attached at one end 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  to zero while satisfying the mixed performance criteria. A model of the inverted pendulum on a cart problem is given by [ , ε ε weighting coefficients of the disturbance Due to the system nonlinearity, we approximate the system using the following two-rule fuzzy model: x is close to zero, The following values are used in our simulation:  The following design parameters are chosen to satisfy mixed NLQR-∞ H criteria: [ ] The mixed criteria control performance results are shown in the Figs.2-3. From these figures, we find that the novel fuzzy LMI control has satisfactory performance. The new technique controls the inverted pendulum very well under the effect of finite energy disturbances. It should also be noted that the LMI fuzzy control with mixed performance criteria satisfies global asymptotic stability [14][15].

VI. CONCLUSION
This paper presents a novel fuzzy control approach for continuous time nonlinear systems based on LMI solutions. The Takagi-Sugeno fuzzy model is applied to decompose the nonlinear system. Multiple performance criteria are 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 an LMI at each time. The inverted pendulum on a cart is 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.