Modified descend curvature based fixed form fuzzy optimal control of nonlinear dynamical systems

https://doi.org/10.1016/j.compchemeng.2005.12.012Get rights and content

Abstract

In this study, a fuzzy rule-based optimal controller is designed for nonlinear dynamical systems. The direct second order method (or direct-descend-curvature algorithm) with a modification called “modified descend controller (MDC)” is used for calculating the parameters of the fuzzy feedback controller. The optimal control problem defined here has dynamic constraints of nonlinear system states and static constraint of a known form of fuzzy controller. The form used here is a standard fuzzy system that uses singleton fuzzifier, product inference engine, center average defuzzifier, and with the Gaussian membership functions of the system states to be controlled. The design is developed by minimizing a quadratic performance index selected for the desired operating conditions. Successful simulation results of controlling the temperature of a continuous stirred tank reactor (CSTR) and a bioreactor are given.

Introduction

Two main advantages of fuzzy systems for the control and modeling applications are (i) fuzzy systems are useful for uncertain or approximate reasoning, especially for the system with a mathematical model that is difficult to derive and (ii) fuzzy logic allows decision making with the estimated values under incomplete or uncertain information (Zadeh, 1975). For static systems, especially for function approximation, there are lots of methods to obtain fuzzy system parameters such in Jang (1993), Wang (1997) and Wang and Mendel, 1992a, Wang and Mendel, 1992b. But determination of the fuzzy system parameters of the dynamical systems for control and modeling applications is not always easy due to the complexity and nonlinearity. This study proposes to design fuzzy controllers for nonlinear dynamical systems using optimal control algorithms. So the term “fuzzy optimal control (FOC)” comes from this idea (Wang, 1998). There are many applications of optimal control for fuzzy controller designs of dynamical systems. In Heckenthaler and Engell (1994), a fuzzy controller was designed for a two-tank system, which is a strongly nonlinear plant due to the characteristics of the valves. The rules of the fuzzy controller designs were derived from the approximate time-optimal control law. A number of stable and optimal fuzzy controllers were developed for linear systems by using the Pontryagin minimum principle with quadratic cost function and the application of the optimal fuzzy controller to the ball-and-beam system were presented in Wang (1998). Chen, Tseng, and Uang (1999) introduced a fuzzy linear control design method for nonlinear systems with a fuzzy linear model that provides rough control to approximate the nonlinear control system, and an optimal H-infinity scheme that provides precise control to achieve the optimal robustness performance. Moreover, Wu and Lin, 2000a, Wu and Lin, 2000b presented local and global approaches of optimal and stable fuzzy controller design methods for both continuous- and discrete-time fuzzy systems under both finite and infinite horizons by applying traditional linear optimal control theory. Another approach, intelligent optimal control problem is considered as a nonlinear optimization with dynamic equality constraints, and dynamic fuzzy network (DFN) and dynamic neural network (DNN) as a control trajectory priming system. The resulting algorithm operates as an auto-trainer for DNN (a self-learning structure) and generates optimal feed-forward control trajectories in a significantly smaller number of iterations (Becerikli, 1998; Becerikli, Konar, & Samad, 2003; Becerikli, Oysal, & Konar, 2004).

Our study differs from these studies for the usage of second order gradient information of an optimal control method for fuzzy controller designs of nonlinear dynamical systems. The direct second order method (or direct-descend-curvature algorithm) that is one of the best algorithms of optimal control is used for calculating the parameters of the fuzzy feedback controller with a modification called as modified descent controller (MDC) algorithm. The optimal control problem defined here has dynamic constraints of nonlinear system states and static constraint of a known form of fuzzy controller.

The organization of this paper is as follows. In Section 2, the description of a standard fuzzy system will be given. Section 3, the optimal fuzzy control problem will be introduced. The solution algorithm of this problem will be given in Section 4. In Section 5, some of the useful formula for a quadratic performance index is derived. Finally, in Section 6, simulation results for a CSTR and a bioreactor will be presented.

Section snippets

Standard fuzzy system

A fuzzy system consists of linguistic IF-THEN rules that have fuzzy antecedent and consequent parts. It is a static nonlinear mapping from the input space to the output space. The inputs and outputs are crisp real numbers and not fuzzy sets. The fuzzification block converts the crisp inputs to fuzzy sets, and then the inference mechanism uses the fuzzy rules in the rule-base to produce fuzzy conclusions or fuzzy aggregations, and finally the defuzzification block converts these fuzzy

Fuzzy optimal control (FOC) problem

In a traditional optimal control problem such as a tracking problem, a performance criteria or index is selected such that whenever it is minimized, the states of the system of our interest will track the desired trajectories. In other words, the cost or the penalty of the performance index will be minimum around the desired trajectory demands. In this study, an optimal control form known as Bolza problem (Lewis, 1992) is selected with the following performance index:J=ϕ(x(tf))+t0tfL(x(t),u(t))

Solution of the FOC problem

For solving the FOC problem or for calculating the time-varying fuzzy system parameter vector θ(t), a second order direct performance index minimization method can be used (Mitter, 1966). This method is based on the Taylor series expansion that uses similar solution techniques as Newton. The solution steps of the known form FOC problem with the second order direct minimization method can be summarized as follows (Oysal (2002), see Appendix A):

  • (1)

    Select the initial values of the stopping criteria

Simulation results

Our design of FOC algorithm will be demonstrated with two examples of two-benchmark chemical process control. Although in this algorithm only output center parameters are calculated while other parameters of the fuzzy system are kept fixed, it is aimed to show the effectiveness of the FOC algorithm in the control of nonlinear dynamical systems. Our first set of experiments examine the fitness of this issue by utilizing a continuously stirred tank reactor (CSTR) process (Ray, 1981), and the

Conclusions and future works

In this work, we developed an algorithm to find the parameters of a feedback fuzzy controller for nonlinear dynamical systems. We have used a second order (curvature) direct descent algorithm with a modification to generate optimal time-varying fuzzy system output parameters. This algorithm is fast and robustly convergent if the initially suitable parameters are chosen. Successful simulation results have been obtained for highly nonlinear chemical processes such as a CSTR (Ray, 1981) and a

References (22)

  • Y. Becerikli et al.

    Intelligent optimal control with dynamic neural networks

    Neural Networks

    (2003)
  • Becerikli, Y. (1998). Neuro-optimal control, PhD dissertation. Sakarya University, Sakarya,...
  • Y. Becerikli et al.

    Trajectory priming with dynamic fuzzy networks in nonlinear optimal control

    IEEE Transaction on Neural Networks

    (2004)
  • A.E. Bryson et al.

    Applied Optimal Control

    (1975)
  • T.E. Bullock

    Computation of optimal controls by a method based on second variations, vol. 297

    (December 1966)
  • B.S. Chen et al.

    Robustness design of nonlinear dynamic systems via fuzzy linear control

    IEEE Transaction on Fuzzy Systems

    (1999)
  • T. Heckenthaler et al.

    Approximately time-optimal fuzzy control of a 2-tank system

    IEEE Control System Magazine

    (1994)
  • J.-S.R. Jang

    ANFIS: adaptive-network-based fuzzy inference systems

    IEEE Transactions on Systems Man and Cybernetics

    (1993)
  • L. Lapidus et al.

    Optimal Control of Engineering Processes

    (1966)
  • F.L. Lewis

    Applied optimal control and estimation

    (1992)
  • Mitter, S. K. (1996). Successive approximation methods for the solution of optimal control problems. In Automatica,...
  • Cited by (12)

    View all citing articles on Scopus
    View full text