Stephen P. Banks

Nonlinear optimal tracking control with application to super-tankers for autopilot design

A new method is introduced to design optimal tracking controllers for a general class of nonlinear systems. A recently developed recursive approximation theory is applied to solve the nonlinear optimal tracking control problem explicitly by classical means. This reduces the nonlinear problem to a sequence of linear-quadratic and time-varying approximating problems which, under very mild conditions, globally converge in the limit to the nonlinear systems considered. The converged control input from the approximating sequence is then applied to the nonlinear system. The method is used to design an autopilot for the ESSO 190,000-dwt oil tanker. This multi-input-multi-output nonlinear super-tanker model is well established in the literature and represents a challenging problem for control design, where the design requirement is to follow a commanded maneuver at a desired speed. The performance index is selected so as to minimize: (a) the tracking error for a desired course heading, and (b) the rudder deflection angle to ensure that actuators operate within their operating limits. This will present a trade-off between accurate tracking and reduced actuator usage (fuel consumption) as they are both mutually dependent on each other. Simulations of the nonlinear super-tanker control model are conducted to illustrate the effectiveness of the nonlinear tracking controller.

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

Abstract Optimal control of general nonlinear nonaffine controlled systems with nonquadratic performance criteria (that permit state- and control-dependent time-varying weighting parameters), is solved classically using a sequence of linear- quadratic and time-varying problems. The proposed method introduces an “approximating sequence of Riccati equations” (ASRE) to explicitly construct nonlinear time-varying optimal state-feedback controllers for such nonlinear systems. Under very mild conditions of local Lipschitz continuity, the sequences converge (globally) to nonlinear optimal stabilizing feedback controls. The computational simplicity and effectiveness of the ASRE algorithm is an appealing alternative to the tedious and laborious task of solving the Hamilton–Jacobi–Bellman partial differential equation. So the optimality of the ASRE control is studied by considering the original nonlinear-nonquadratic optimization problem and the corresponding necessary conditions for optimality, derived from Pontryagins maximum principle. Global optimal stabilizing state-feedback control laws are then constructed. This is compared with the optimality of the ASRE control by considering a nonlinear fighter aircraft control system, which is nonaffine in the control. Numerical simulations are used to illustrate the application of the ASRE methodology, which demonstrate its superior performance and optimality.

Sliding Mode Control with Optimal Sliding Surfaces for Missile Autopilot Design

A new method is introduced to design sliding mode control with optimally selected sliding surfaces for a class of nonlinear systems. The nonlinear systems are recursively approximated as linear time-varying systems, and corresponding time-varyingsliding surfaces are designedfor each approximatedsystem so thatagivenoptimization criterion isminimized.The control input,which is designedbyusing an approximatedsystem, is then applied to the nonlinear system. The method is used to design an autopilot for a missile where the design requirement is to follow a given acceleration command. The sliding surface is selected such that a performance index formed as a function of angle of attack, pitch rate, and velocity error is minimized. It is shown that the response of the approximating sequence of linear time-varying systems converges to the response of the missile.

Approximate Optimal Control and Stability of Nonlinear Finite- and Infinite-Dimensional Systems

AbstractWe consider first nonlinear systems of the formx=A(x)x+B(x)u together with a standard quadratic cost functional and replace the system by a sequence of time-varying approximations for which the optimal control problem can be solved explicitly. We then show that the sequence converges. Although it may not converge to a global optimal control of the nonlinear system, we also consider a similar approximation sequence for the equation given by the necessary conditions of the maximum principle and we shall see that the first method gives solutions very close to the optimal solution in many cases. We shall also extend the results to parabolic PDEs which can be written in the above form on some Hilbert space.

CHAOS IN A THREE-DIMENSIONAL CANCER MODEL

In this study, we develop a new dynamical model of cancer growth, which includes the interactions between tumour cells, healthy tissue cells, and activated immune system cells, clearly leading to c...

Linear approximations to nonlinear dynamical systems with applications to stability and spectral theory

There are many approaches to the study of nonlinear dynamical systems, including local linearizations in phase space (Perko, 1991), global linear representations involving the Lie series solution (Banks & Iddir, 1992; Banks, 1992; Banks et al., 1996), Lie algebraic methods (Banks, 2001) and global results based on topological indices (McCaffrey & Banks, 2002; Perko, 1991). Linear systems, on the other hand are very well understood and there is, of course, a vast literature on the subject (see, for example, Banks, 1986). The simplicity of linear mathematics relative to nonlinear theory is evident and forms the basis of much of classical mathematics and physics. It is therefore attractive to try to attack nonlinear problems by linear methods, which are not local in their applicability. In this paper we study a recently introduced approach to nonlinear dynamical systems based on a representation of the system as the limit of a sequence of linear, time-varying approximations which converge in the space of continuous functions to the solution of the nonlinear system, under a very mild local Lipschitz condition. This approach has already been used in optimal control theory (Banks & Dinesh, 2000), in the theory of nonlinear delay systems (Banks, 2002) and in the theory of chaos (Banks & McCaffrey, 1998). In these papers, however, only a local (in time) proof of convergence was given—here we shall give a global proof for dynamical systems which do not have finite escape time. We shall also study two further applications of the method: to the study of stability of nonlinear systems and the definition of a spectral theory for nonlinear systems. In the former case, we shall define a generalized Lyapunov function, the existence of which is equivalent to the stability of the system. In the second case we first study the spectral theory of a linear, time-varying system as the perturbation of a time-invariant system and then use the approximation scheme to extend it to nonlinear systems.

Structure and control of piecewise-linear systems

Elementary algebric topology is used in the study of the structure and optimal control of piecewise-linear and nonlinear systems.

LIE ALGEBRAS, STRUCTURE OF NONLINEAR SYSTEMS AND CHAOTIC MOTION

The structure theory of Lie algebras is used to classify nonlinear systems according to a Levi decomposition and the solvable and semisimple parts of a certain Lie algebra associated with the system. An approximation theory is developed and a new class of chaotic systems is introduced, based on the structure theory of Lie algebras.

Observer design for nonlinear systems using linear approximations

There exist several approaches to the design of observers for nonlinear systems, including the separation of the nonlinear system into a linear part and a nonlinear perturbation of the system with a bounded condition [2], the use of Lie derivatives and the inversion of the Jacobian of a coordinate transformation to obtain the gain of the nonlinear observer [10] and the use of a Lyapunov equation to design the observer for a nonlinear system represented in a special canonical form [8]. The design of observers for linear systems is better understood (see [11], [14]) since in nonlinear theory there is a necessity to use more complex mathematics. Therefore, there exist interest in the development of simpler and general methods to solve the problem of nonlinear state reconstruction. This paper deals with the design of observers for nonlinear systems by using a recent technique in which the nonlinear dynamical system is represented as the limit of a sequence of linear time-varying approximations that converge to the solution of the nonlinear system under a local Lipschitz condition.

A note on non-linear observers

A general non-linear exponential observer is presented for non-linearly perturbed nonlinear systems. This is done by using the non-linear variation of constants formula.