Witold Respondek

Almost disturbance decoupling for single-input single-output nonlinear systems

The almost disturbance decoupling problem for nonlinear single-input single-output systems is addressed by using singular perturbation methods and high-gain feedback. Sufficient conditions and the explicit high-gain nonlinear state feedback in solvable cases are given. They generalize both previous almost results for linear systems and exact ones for nonlinear systems. The necessity of the conditions is discussed; in particular, an example is given where the main structural condition is not satisfied and the high-gain control designed on the basis of linear approximations fails to achieve almost disturbance decoupling for the original system. >

Nonlinear H ∞ almost disturbance decoupling

The L2-gain almost disturbance decoupling problem for SISO nonlinear systems is formulated. Sufficient conditions are identified for the existence of a parametrized state feedback controller such that the L2-gain from disturbances to output can be made arbitrarily small by increasing its gain. The controller is explicity constructed using a Lyapunov-based recursive scheme. Sufficient conditions for the solvability of the H∞ almost disturbance decoupling problem and the explicit construction of teh controller are given for a more restrictive class of nonlinear systems.

On decomposition of nonlinear control systems

In this paper the problem is studied when the system ẋ =ƒ( x ) + ∑ i k=1 u t g t (x) , x (0) = x 0 , defined on an n -dimensional analytic manifold, can be decomposed into either independent subsystems or a cascade of subsystems. Necessary and sufficient conditions for the existence of such local decompositions are given in terms of Lie algebras generated by the system.

Partial and robust linearization by feedback

It is argued that if the nonlinearities in a system are mild, and the controller is sufficiently stabilizing, the inaccuracies of a linear model, which is often taken to be sufficient for a controller design, can be safely neglected. For systems with severe nonlinearity a linearizing technique is described, based on the change of state coordinates and nonlinear feedback; in the total context of a stable feedback design the linearization technique is considered robust. Furthermore, attention is paid to partial linearization using the same transformations. It is found that there always exist maximally linearizing transformations which are not necessarily unique.

Exact linearization of nonlinear systems with outputs

This paper discusses the problem of using feedback and coordinates transformation in order to transform a given nonlinear system with outputs into a controllable and observable linear one. We discuss separately the effect of change of coordinates and, successively, the effect of both change of coordinates and feedback transformation. One of the main results of the paper is to show what extra conditions are needed, in addition to those required for input-output-wise linearization, in order to achieve full linearity of both state-space equations and output map.

Global Aspects of Linearization, Equivalence to Polynomial Forms and Decomposition of Nonlinear Control Systems

In the last fifteen years a theory for nonlinear control systems has been developed using differential geometric methods. Many problems have been treated in this fashion and interesting results have been obtained for nonlinear equivalence, decomposition, controllability, observability, optimality, synthesis of control (with desired properties: decoupling or noninteracting), linearization and many others. We refer the reader to Sussmann [28] for a survey and bibliography. We want to emphasize only that in most of the papers devoted to nonlinear control systems (using geometric methods) only a local viewpoint is presented. This is due to two kinds of obstructions: singularities of the studied objects (functions, vector fields, distributions) and topological obstructions for the global existence of the sought solutions.

Decoupling via dynamic compensation for nonlinear control systems

The purpose of this paper is to determine in an analytic fashion whether or not we can achieve input-output decoupling of a nonlinear control system via dynamic precompensation. The solution of this problem is based on an algorithm which determines at each step the maximal number of input-output channels that can be decoupled. We also discuss some interesting relations with differential geometric methods and we show that for systems with two inputs and two outputs our necessary and sufficient conditions can also be formulated and proved in a geometric way.

Feedback classification of nonlinear control systems on 3-manifolds

We consider nonlinear control-affine systems with two inputs evolving on three-dimensional manifolds. We study their local classification under static state feedback. Under the assumption that the control vector fields are independent we give complete classification of generic systems. We prove that out of a “singular” smooth curve a generic control system is either structurally stable and thus equivalent to one of six canonical forms (models) or finitely determined and thus equivalent to one of two canonical forms with real parameters. Moreover, we show that at points of the “singular” curve the system is not finitely determined and we give normal forms containing functional moduli. We also study geometry of singularities, i.e., we describe surfaces, curves, and isolated points where the system admits its canonical forms.

Disturbance Decoupling Via Dynamic Feedback

In the last twenty years there have been extensive studies of geometric methods in control problems. Geometric linear control theory started in the beginning of seventies and is based on the concept of controlled invariant subspaces introduced by Basile and Marro [BM] and by Wonham and Morse [WM]. In the beginning of eighties the nonlinear generalization of the controlled invariant subspace, namely the controlled invariant distribution, was introduced by Isidori et al [IKGM1] and by Hirschorn [H]. This concept has been successfully used in such nonlinear control synthesis problems like disturbance decoupling, noninteracting, invertibility and many others (compare [I] and [NS3]).

Dynamic Controllability Distributions in Nonlinear Systems

Abstract We introduce the notion of dynamic controllability distributions for nonlinear control systems. we show the usefulness of this concept by applying it. to study the left and right invertibility problem