Ülle Kotta

Linearization of Discrete-Time Systems

The algebraic formalism developed in this paper unifies the study of the accessibility problem and various notions of feedback linearizability for discrete-time nonlinear systems. The accessibility problem for nonlinear discrete-time systems is shown to be easy to tackle by means of standard linear algebraic tools, whereas this is not the case for nonlinear continuous-time systems, in which case the most suitable approach is provided by differential geometry. The feedback linearization problem for discrete-time systems is recasted through the language of differential forms. In the event that a system is not feedback linearizable, the largest feedback linearizable subsystem is characterized within the same formalism using the notion of derived flag of a Pfaffian system. A discrete-time system may be linearizable by dynamic state feedback, though it is not linearizable by static state feedback. Necessary and sufficient conditions are given for the existence of a so-called linearizing output, which in turn is a sufficient condition for dynamic state feedback linearizability.

Irreducibility, reduction and transfer equivalence of nonlinear input-output equations on homogeneous time scales

The purpose of this paper is to present a necessary and sufficient condition for irreducibility of nonlinear input–output delta differential equations. The condition is presented in terms of the common left divisor of two differential polynomials describing the behaviour of the system defined on a homogenous time scale. The concept of reduction is explained. Subsequently, the definition of transfer equivalence based upon the notion of an irreducible differential form of the system is introduced, inspired by the analogous definition for continuous-time systems.

Submersive rational difference systems and their accessibility

The paper describes an algebraic construction of the inversive difference field associated with a discrete-time rational nonlinear control system under the assumption that the system is submersive. We prove that a system is submersive iff its associated difference ideal is proper, prime and reflexive. Next, we show that Kähler differentials of the above inversive field define a module over the corresponding ring of Ore operators, and relate its torsion submodule to the vector space of autonomous one-forms, introduced elsewhere. The above results allow us to check accessibility property and simplify transfer functions with computer algebra techniques.

Transfer Function Approach to the Model Matching Problem of Nonlinear Systems

Abstract The mainstream for the analysis and synthesis of nonlinear control systems is the so-called state space approach. The Laplace transform of a nonlinear differential equation is non tracktable and any transfer function approach was not developed until recently. Herein, we show that one may use such mathematical tools to recast and solve the model matching problem. Note that the latter was originaly stated for linear time invariant systems, in terms of equality of the transfer function of both the model and the compensated system.

Transfer Equivalence and Realization of Nonlinear Input-Output Delta-Differential Equations on Homogeneous Time Scales

Nonlinear control systems on homogeneous time scales are studied. First the concepts of reduction and irreducibility are extended to higher order delta-differential input-output equations. Subsequently, a definition of system equivalence is introduced which generalizes the notion of transfer equivalence in the linear case. Finally, the necessary and sufficient conditions are given for the existence of a state-space realization of a nonlinear input-output delta-differential equation.

Pseudo-linear algebra: A powerful tool in unification of the study of nonlinear control systems

Abstract This paper introduces methods of pseudo-linear algebra to unify the algebraic formalism of one-forms and the related polynomial approach for both continuous and discrete-time nonlinear control systems. Given approach covers also differene, q-shift and q-difference operators whereby this algebraic formalism is not only unified but also extended to wide class of nonlinear control systems. Also the notion of transfer function of nonlinear control system is defined and some basic properties are shown. Transfer function definiton is based on skew polynomial ring which can be embedded into its quotient field.

Control systems on regular time scales and their differential rings

The paper describes an algebraic construction of the inversive differential ring, associated with a nonlinear control system, defined on a nonhomogeneous but regular time scale. The ring of meromorphic functions in system variables is constructed under the assumption that the system is submersive, and equipped with three operators (delta- and nabla-derivatives, and the forward shift operator) whose properties are studied. The formalism developed unifies the existing theories for continuous- and discrete-time nonlinear systems, and accommodates also the case of non-uniformly sampled systems. Compared with the homogeneous case the main difficulties are noncommutativity of delta (nabla) derivative and shift operators and the fact that the additional time variable t appears in the definition of the differential ring. The latter yields that the new variables of the inversive closure, depending on t, have to be chosen to be smooth at each dense point t of the time scale.

Equivalence of Different Realization Methods for Higher Order Nonlinear Input–Output Differential Equations*

Several state space realizability conditions and realization algorithm for nonlinear single-input single-output higher order input–output differential equation are compared. Three different necessary and sufficient state space realizability conditions, respectively, given in terms of integrability of the subspaces of one-forms, involutivity of the conditionally invariant distributions, and commutativity of iterative Lie brackets of vector fields, are proved to be equivalent. Moreover, the algorithm-based solutions are demonstrated for computing the integrable basis of the subspaces of oneforms. Finally, explicit formulas are provided for calculation of the differentials of the state coordinates which, in case the necessary and sufficient realizability conditions are satisfied, can be integrated to obtain the state coordinates.

Neural Networks Based ANARX Structure for Identification and Model Based Control

This article is devoted to the training and application of neural networks based additive nonlinear autoregressive exogenous (NN-based ANARX) model. Training of NN-based ANARX model with MATLAB is discussed in detail and illustrated by examples. Dynamic state feedback linearization control algorithm is then applied for control of unknown nonlinear system

Classical state space realizability of input-output bilinear models

This paper studies the realizability property of bilinear input-output (i/o) models in the classical state space form. Constraints on the parameters of the bilinear i/o model are suggested that lead to realizable models. The complete list of 2nd and 3rd order realizable input-output bilinear models together with the corresponding state equations is given. In the general case some subclasses of realizable bilinear models together with their state-space realizations are presented, including the diagonal bilinear model and the special subclass of the superdiagonal bilinear model.