Palle Kotta

Irreducibility Conditions for Continuous-time Multi-input Multi-output Nonlinear Systems

The purpose of this paper is to present necessary and sufficient condition for irreducibility of continuous-time nonlinear multi-input multi-output system. The condition is presented in terms of the greatest common left divisor of two polynomial matrices related to the input-output equations of the system. The basic difference is that unlike the linear case the elements of the polynomial matrices belong to a non-commutative polynomial ring. This condition provides a basis for finding the equivalent minimal irreducible representation of the I/O equations which is a suitable starting point for constructing an observable and accessible state space realization

State space realization of bilinear continuous-time input-output equations

This paper studies the realizability property of continuous-time bilinear input–output (i/o) equations in the classical state space form. Constraints on the parameters of the bilinear i/o model are suggested that lead to realizable models. The paper proves that the 2nd order bilinear i/o differential equation, unlike the discrete-time case, is always realizable in the classical state space form. The complete list of 3rd and 4th order realizable i/o bilinear models is given and two subclasses of realizable i/o bilinear systems are suggested. Our conditions rely basically upon the property that certain combinations of coefficients of the i/o equations are zero or not zero. We provide explicit state equations for all realizable 2nd and 3rd order bilinear i/o equations, and for one realizable subclass of bilinear i/o equations of arbitrary order.

Irreducibility and reduction of nonlinear control systems: Unification and extension via pseudo-linear algebra

The paper applies the pseudo-linear algebra to unify the study of irreducibility and reduction problems for continuous- and discrete-time multi-input multi-output nonlinear control systems. The necessary and sufficient condition for irreducibility of the set of nonlinear input-output equations is presented in terms of the greatest common left divisor of two polynomial matrices describing the behaviour of the system. The basic difference is that, unlike the linear case, the elements of the polynomial matrices belong to a non-commutative skew polynomial ring. The suggested condition provides a bases for finding the (transfer) equivalent minimal irreducible representation of the set of input-output equations, which is a suitable starting point for constructing an observable and accessible state space realization. Besides unification, the tools of pseudolinear algebra allow to extend the results for systems defined by difference, q-shift and q-difference operators.

Practical polynomial formulas in MIMO nonlinear realization problem

The explicit and simple polynomial formulas are given for computation of the (differentials) of state coordinates from the set of nonlinear input-output (i/o) equations under assumption that the set of i/o equations is realizable in the state-space form. The paper addresses a very wide class of nonlinear control systems, that accommodates continuous- as well as two discrete-time cases, based either on the shift or difference operator. Two types of polynomial formulas are suggested, based either on polynomial left division or on the application of the concept of adjoint polynomials and implemented in Mathematica-based software.

Minimal realisation of bilinear and quadratic input–output difference equations in state-space form

This article studies the realisability property of discrete-time bilinear and quadratic input–output (i/o) equations in the classical state-space form. Constraints on the parameters of the i/o model are suggested that lead to realisable models. Using new formulae for computing basis vectors of certain vector spaces of differential one-forms, we present in this article the complete list of the third- and fourth-order realisable i/o bilinear models, and a new realisable subclass of an arbitrary order is suggested. Moreover, we provide the sufficient conditions of the second- and third-order realisable i/o quadratic models, respectively. All the developed theory and algorithms are illustrated by means of several examples.

On classical state space realisability of quadratic input–output differential equations

This article studies the realisability property of continuous-time quadratic input–output (i/o) equations in the classical state space form. Constraints on the parameters of the quadratic i/o model are suggested that lead to realisable models. The complete list of second- and third-order realisable i/o quadratic models is given and two subclasses of the n-th order realisable i/o quadratic systems are suggested. Our conditions rely basically upon the property that certain combinations of coefficients of the i/o equations are zero or not zero. We provide explicit state equations for realisable second-order quadratic i/o equations, and for one realisable subclass of quadratic i/o equations of arbitrary order.

State-space realization of nonlinear input-output equations: Unification and extension via pseudo-linear algebra

The paper focuses on unification of the study of continuous- and discrete-time nonlinear control systems using the mathematical tools of pseudo-linear algebra. The realization problem is studied through the language of differential forms. Necessary and sufficient conditions are given for the existence of the state space realization of the nonlinear input-output equation defined in terms of the pseudo-linear operator. The sufficiency part of the proof gives a constructive procedure (up to integration of the one-forms) for obtaining the observable state equations. The special cases for continuous- and discrete-time systems, described either in terms of the shift or difference operator, follow from the general result as the special cases. Moreover, the result accommodates also the systems described via q-shift or q-difference operators.