Control Of Nonlinear Dynamical Systems Research Articles

Multi-Layer Neural Networks for the Optimal Control of Nonlinear Dynamic Systems

An N-stage optimal control problem formulated in very general terms is transformed into a nonlinear programming problem by constraining the closed-loop strategies to take on the structure of multi-layer feed-forward neural networks. A nonlinear programming problem results from substitution of the neural strategies in the process cost, which becomes a function of the synaptic weights and of the initial state. The dependence on the initial state is removed by averaging the cost function with respect to the initial state, which is assumed to be a stochastic vector uniformly distributed on a given region. The gradient method is then applied to optimize the synaptic weights by using the stochastic approximation concept. Recursive equations to compute the gradient components in a distributed way are presented, which generalize the adjoint system equations. Simulation results for nonlinear non-quadratic problems show that the method performs efficiently.

Nonlinear operator equations and applications to modelling

The main purpose of this paper is to treat the problem of numerical solutions of Lyapunov matrix equations and nonlinear matrix equations such as the Riccati type which occur in many areas of applications such as in control theory, robotics, mathematical modelling and computer simulation, etc. The main approach is to obtain solutions of such type equations in a more direct manner as in contrast to completely interactive methods. Parallel processing of algorithms is made use of in the computation of solutions of such nonlinear matrix equations associated with nonlinear matrix differential equations arising in the study of nonlinear dynamical control systems.

Identification and control of dynamical systems using neural networks

It is demonstrated that neural networks can be used effectively for the identification and control of nonlinear dynamical systems. The emphasis is on models for both identification and control. Static and dynamic backpropagation methods for the adjustment of parameters are discussed. In the models that are introduced, multilayer and recurrent networks are interconnected in novel configurations, and hence there is a real need to study them in a unified fashion. Simulation results reveal that the identification and adaptive control schemes suggested are practically feasible. Basic concepts and definitions are introduced throughout, and theoretical questions that have to be addressed are also described.

An Eclectic Approach to the State Feedback Control of Nonlinear Dynamical Systems

This paper examines the problem of robust state-feedback stabilization of a class of nonlinear multi-input dynamical systems. Four approaches to the problem are investigated: the variable structure control (VSC) method, the high-gain feedback technique, the feedback linearization algorithm, and finally the deterministic approach to the control of uncertain systems. It is shown that each design method can lead to a controller such that the closed-loop system exhibits a sliding mode property. The sliding mode is a desirable property since it results in a robust control. The analysis is illustrated by means of a simple numerical example.

AN ADAPTIVE MODEL REFERENCE CONTROLLER WITH BOUNDED GAINS

It has been established that currently available adaptive control algorithms may become unstable in the presence of high frequency unmodeled dynamics and additive sinusoidal disturbances. In such nonlinear dynamic adaptive control system ,there exist two infinite-gain operators which can cause the loop gains to increase without bound. Such unbounded parameters would likely cause instability. These problems can be alleviated by using a proposed model reference adaptive algorithm with bounded gains. In this paper, the mechanism of disturbance instabilities is investigated. The modified algorithm uses a limiter for the controller gains to prevent the unbounded drift. The limiting values of the proposed limiter, can be estimated analytically by applying stability criteria for linear control systems,to the linearized model of the controlled plant. The proposed adaptive controller is tested by simulation. Simulation results show the algorithm to be capable of ensuring stability of the adaptive control system , in the presence of sinusoidal disturbances.

Neurocontroller design via supervised and unsupervised learning

In this paper we study the role of supervised and unsupervised neural learning schemes in the adaptive control of nonlinear dynamic systems. We suggest and demonstrate that the teacher's knowledge in the supervised learning mode includes a-priori plant sturctural knowledge which may be employed in the design of exploratory schedules during learning that results in an unsupervised learning scheme. We further demonstrate that neurocontrollers may realize both linear and nonlinear control laws that are given explicitly in an automated teacher or implicitly through a human operator and that their robustness may be superior to that of a model based controller. Examples of both learning schemes are provided in the adaptive control of robot manipulators and a cart-pole system.

Suboptimal control of nonlinear dynamic systems via moving model

Suboptimal feedback control with a quadratic performance index is considered for nonlinear dynamic systems. The systems are represented by a moving model which is derived by using block pulse functions. This approach permits the linear feedback control law to be applied to nonlinear systems. The suboptimal feedback control of a nonisothermal CSTR with control constraints is presented to illustrate the considerable promise that the method exhibits.

Optimal Regulation of Nonlinear Dynamical Systems on a Finite Interval

In this paper the optimal control of nonlinear dynamical systems on a finite time interval is considered. The free end-point problem as well as the fixed end-point problem is studied. The existence of a solution is proved and a power series solution of both the problems is constructed.

Suboptimality bounds and stability in the control of nonlinear dynamic systems

Certain results paralleling those of Bailey and Ramapriyan [1] for linear dynamic systems are presented for nonlinear systems. For weakly perturbed systems certain suboptimal controls are shown to be stabilizing and upper and lower bounds on the suboptimality are investigated.

Comments on "Parallel Processing Algorithms for the Optimal Control of Nonlinear Dynamic Systems"

Comments on "Parallel Processing Algorithms for the Optimal Control of Nonlinear Dynamic Systems"

Parallel Processing Algorithms for the Optimal Control of Nonlinear Dynamic Systems

This paper describes the recent development of parallel processing algorithms for solving optimal control problems for nonlinear dynamic systems. Both the deterministic and stochastic cases are considered. The resulting algorithms are applicable to a large range of parallel architecture computers, including Iliac IV, associative processors, and potential designs based on integrated circuit technology.

Approximation in control of nonlinear dynamic systems

AbstractThe type of system approximation in which the system state space x is mapped into a scalar domain V by a quadratic transformation where Q is appropriately determined, is used to develop a suboptimal control procedure for unconstrained lumped parameter dynamic systems via the application of Pontryagin's Maximum Principle. The optimization problem in the scalar domain becomes an initial value problem when the scalar adjoint variable is held constant through out the course of control. The resulting computational scheme includes an effective and simple way to construct the transformation matrix Q and a straightforward minimum seeking approach to locate the best constant overall average scalar adjoint parameter. For the class of problems with quadratic performance index, system equation approximation further reduces the determination of Q to the solution of a matrix Riccati equation.The application of the proposed suboptimal control procedure to four chemical engineering systems shows that the procedure is simple direct, and efficient and works particularly well for problems where the final time is large.

A controllability theory for nonlinear systems

A Lyapunov-like approach to the controllability of nonlinear dynamic systems is presented. A theory is developed which yields sufficient conditions for complete controllability for some classes of nonlinear systems; feedback controllers which drive the systems to desired terminal conditions, at a specified final time, are also obtained. Well-known controllability conditions for linear dynamic systems are derived using this general controllability theory. Elliptical regions are found which contain (bound) the trajectories of a class of systems controlled according to these methods. These regions are used in synthesizing controllers for nonlinear systems and for a class of state-variable inequality constrained problems. An uncontrollability theorem, based also upon Lyapunov-like notions, is presented; this yields sufficiency conditions for uncontrollability for some types of nonlinear systems. Relationships of the theories to other nonlinear controllability approaches are indicated.

Stochastic optimal control for non-linear dynamical systems under noisy observations

In this paper, an approximate technique based on stochastic linearization is developed which permits us to design sub-optimal state estimates and feedback controls for non-linear dynamical systems with state-independent noise. First, a method of establishing approximate filter dynamics is briefly outlined. Secondly, by using the filter dynamics obtained, an approximate technique for finding sub-optimal controls is presented for the quadratic cost functional. Finally, detailed discussions are given by a numerical example, including quantitative aspects of sample paths behavior of state estimate and sub-optimal control signals.

An approximate method of stochastic terminal control for nonlinear dynamical systems

The purpose of this paper is to establish an approximate method of stochastic terminal control for nonlinear dynamical systems with state-independent noise. Guided by the state variable representation concept in control theory, we describe approximately mathematical models for both the dynamical system and the observation mechanism by the nonlinear vector stochastic differential equations of Itô-type. First, for the purpose of overcoming difficulties caused by nonlinear characteristics, a method of stochastic linearization is introduced. By using the stochastic linearization technique presented here, an approximate method is described for solving state estimation problems of nonlinear dynamical systems in a Markovian framework. Secondly, by using the estimator dynamics obtained, an approximation to the optimal control is presented for a terminal cost functional. Finally, detailed discussions are given by an example, including quantitative aspects of sample paths behavior of state estimation and optimal control signal.

状態依存性雑音をうける非線形力学系の近似最適制御

状態依存性雑音をうける非線形力学系の近似最適制御

On Adaptive Optimal Control of Nonlinear Processes

The paper concerns an approach to adaptive optimal control of nonlinear dynamic systems which has been introduced by Kulikowski. In this approach, the required identification is carried out at each stage of constructing a sequence of inputs (xn(t)), tε(0, T) converging to a relative extremum of a given performance functional. The major contributions of this paper relate to the identification problem and its incorporation into the optimal control formulation.