Miroslav Halás

Design and implementation of fractional-order PID controllers for a fluid tank system

In this paper, we investigate the practical problems of design and digital implementation of fractional-order PID controllers used for fluid level control in a system of coupled tanks. We present a method for obtaining the PIλDμ controller parameters and describe the steps necessary to obtain a digital implementation of the controller. A real laboratory plant is used for the experiments, and a hardware realization of the controller fit for use in embedded applications is proposed and studied. The majority of tasks is carried out by means of the FOMCON (“Fractional-order Modeling and Control”) toolbox running in the MATLAB computing environment.

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.

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.

Symbolic Computation for Nonlinear Systems Using Quotients over Skew Polynomial Ring

Nonlinear control systems are more difficult to handle than linear, since the associativity is not valid. Laplace transforms and transfer functions, which form a symbolic computation for linear systems, are, therefore, disabled. A modern development of nonlinear control systems is thus related mainly to the systematic use of differential algebraic methods. However, such methods allow us to introduce similar symbolic computation also for nonlinear systems. To provide a basis for such a symbolic computation the theory of non-commutative polynomials over the field of meromorphic functions is introduced. In that respect, differential operators, which act on one-forms, play a key role. They form the left skew polynomial ring. Quotients of such polynomials stand for transfer functions of nonlinear systems. In other words, presented paper tries to show that there is no reason to believe well known dogma saying that nonlinear systems have no transfer functions

A transfer function approach to the realisation problem of nonlinear systems

This article studies the nonlinear realisation problem, i.e. the problem of finding the state equations of a nonlinear system from the transfer function representation being easily computable from the higher order input–output differential equation. The realisation in both observer and controller canonical forms is studied. The results demonstrate a clear connection with those from linear theory. In the solution the concept of adjoint polynomials, adjoint transfer function and right factorisation of the transfer function play a key role. Finally, the results are applied for system linearisation up to input–output injection used in the observer design.

Realization problem of SISO nonlinear systems: A transfer function approach

This paper studies the nonlinear realization problem, i.e. the problem of finding an observable state space representation of a SISO nonlinear system described by an input-output differential equation, within an alternative polynomial approach. To find the solution the so-called adjoint polynomials and adjoint transfer functions are introduced for nonlinear systems. Thereby the clear connection to the solution known from linear systems is established.

Nonlinear Controllers for a Fluid Tank System

This chapter deals with an application of the algebraic formalism in nonlinear control systems on the nonlinear controller design for a fluid tank system. A nonlinear discrete-time model of a fluid tank system and its transfer function are derived. Then, nonlinear continuous- and discrete-time controllers are designed using transfer function formalism for nonlinear systems which was developed recently. Verification on the real plant is also included and it suggested the modification of the original model of one-tank system to achieve better performance on the real plant.

Definition of Eigenvalues for a Nonlinear System

In this paper the concept of eigenvalues and eigenvectors of nonlinear systems, both continuous- and discrete-time, is suggested. It represents a generalization of the concept known from linear control theory. Some basic properties, like invariance of eigenvalues under a (nonlinear) change of coordinates, possibility to transform the system to the diagonal form and, respectively, to the feedforward form are then shown.

When retarded nonlinear time-delay systems admit an input-output representation of neutral type

The paper shows that nonlinear retarded time-delay systems can admit an input-output representation of neutral type. This behaviour represents a strictly nonlinear phenomenon, for it cannot happen in the linear time-delay case where retarded systems always admit an input-output representation of retarded type. A necessary and sufficient condition for a nonlinear system to exhibit this behaviour is given, and a strategy for finding an input-output representation of retarded type is outlined. Some open problems that arise consequently are discussed as well. All the systems considered in this work are time-invariant and have commensurable delays.