Stanislaw H. Zak

Stabilizing controller design for uncertain nonlinear systems using fuzzy models

A Lyapunov-based stabilizing control design method for uncertain nonlinear dynamical systems using fuzzy models is proposed. The controller is constructed using a design model of the dynamical process to be controlled. The design model is obtained from the truth model using a fuzzy modeling approach. The truth model represents a detailed description of the process dynamics. The truth model is used in a simulation experiment to evaluate the performance of the controller design. A method for generating local models that constitute the design model is proposed. Sufficient conditions for stability and stabilizability of fuzzy models using fuzzy state feedback controllers are given. The results obtained are illustrated with a numerical example involving a four-dimensional nonlinear model of a stick balancer.

Combined observer-controller synthesis for uncertain dynamical systems with applications

Control of a class of nonlinear/uncertain systems is discussed using a variable-structure systems approach. Observations of the states of such systems is also considered. The natural extension to an observer-controller design is illustrated using a computer simulation example of a theta -r manipulator. Next, the problem of path planning is addressed using a combined observer-controller strategy. The aspects of hardware implementation of the proposed observer-controller are then analyzed. >

Application of feedforward neural networks to dynamical system identification and control

Methods for identification and control of dynamical systems by adalines, two-layer, and three-layer feedforward neural networks (FNNs) using generalized weight adaptation algorithms are discussed. The FNNs considered contain odd nonlinear operators in both the neurons and the weight adaptation algorithms. Two application examples, each involving a nonlinear dynamical system, are considered. The first is identification of the systems forward and inverse dynamics. The second is control of the system using coordination of feedforward and feedback control combined with inverse system dynamics identification. Simulation results are used to verify the methods feasibility and to examine the effect of ENN parameter changes. Specifically the effect that the type of nonlinear activation functions present in the neurons and the type of nonlinear functions present in the weight adaptation algorithms have on FNN system dynamics identification performance is investigated. >

On variable structure output feedback controllers for uncertain dynamic systems

A class of variable structure output feedback controllers for uncertain dynamic systems with bounded uncertainties is proposed. No statistical information about the uncertain elements is assumed. A variable structure systems approach and a geometric approach to the analysis and synthesis of system zeros are employed in the synthesis of the proposed controllers. >

On solving constrained optimization problems with neural networks: a penalty method approach

Deals with the use of neural networks to solve linear and nonlinear programming problems. The dynamics of these networks are analyzed. In particular, the dynamics of the canonical nonlinear programming circuit are analyzed. The circuit is shown to be a gradient system that seeks to minimize an unconstrained energy function that can be viewed as a penalty method approximation of the original problem. Next, the implementations that correspond to the dynamical canonical nonlinear programming circuit are examined. It is shown that the energy function that the system seeks to minimize is different than that of the canonical circuit, due to the saturation limits of op-amps in the circuit. It is also noted that this difference can cause the circuit to converge to a different state than the dynamical canonical circuit. To remedy this problem, a new circuit implementation is proposed.

Dynamical analysis of the brain-state-in-a-box (BSB) neural models

A stability analysis is performed for the brain-state-in-a-box (BSB) neural models with weight matrices that need not be symmetric. The implementation of associative memories using the analyzed class of neural models is also addressed. In particular, the authors modify the BSB model so that they can better control the extent of the domains of attraction of stored patterns. Generalizations of the results obtained for the BSB models to a class of cellular neural networks are also discussed.

An analysis of a class of neural networks for solving linear programming problems

A class of neural networks that solve linear programming problems is analyzed. The neural networks considered are modeled by dynamic gradient systems that are constructed using a parametric family of exact (nondifferentiable) penalty functions. It is proved that for a given linear programming problem and sufficiently large penalty parameters, any trajectory of the neural network converges in finite time to its solution set. For the analysis, Lyapunov-type theorems are developed for finite time convergence of nonsmooth sliding mode dynamic systems to invariant sets. The results are illustrated via numerical simulation examples.

On the stabilization and observation of nonlinear/uncertain dynamic systems

The problem of stabilization and observation of uncertain and/or nonlinear dynamic systems for which matching conditions are not satisfied is investigated. No statistical description of uncertain elements is assumed. The uncertain and/or nonlinear quantities are described only in terms of bounds on their possible sizes. Stabilizing feedback controllers and observers are proposed whose design is based on the constructive use of Lyapunov functions and the Bellman-Gronwall lemma. >

Solving linear programming problems with neural networks: a comparative study

In this paper we study three different classes of neural network models for solving linear programming problems. We investigate the following characteristics of each model: model complexity, complexity of individual neurons, and accuracy of solutions. Simulation examples are given to illustrate the dynamical behavior of each model.

Stabilizing fuzzy system models using linear controllers

A Lyapunov-based approach is used to derive a sufficiency condition for stabilizing a class of fuzzy system models using linear controllers. A design algorithm for constructing stabilizing linear controllers is given. The results obtained are illustrated with a design of a linear stabilizing controller for a system consisting of an inverted pendulum mounted on a cart. The controller is designed using a fuzzy model of the system and tested on the original system model.