Arvo Kaldmäe

• Researcher at Tallinn University of Technology

The paper addresses the problem of transforming observable, time-reversible and multi-input single-output discrete-time state equations into the extended observer form which comprises a linear observable component and a nonlinear injection term depending on the inputs, output, and a finite number of their past values. The intrinsic necessary and su...

In the paper accessibility of an analytic discrete-time control system given by a set of difference equations is studied. Ideals of germs of analytic functions and modules of germs of one-forms, associated with the considered system and invariant with respect to shift operators defined by the state equations, are used to present sufficient and nece...

The problem of finding a minimal realization of a single-input single-output nonlinear retarded time-delay input-output equation with constant commensurable delays is addressed. First, an algorithm is given to find an observable realization of dimension $n$ of a $n$ -th order input-output equation. Then, this algorithm is modified under two tec...

The paper investigates some properties of recently defined forward and backward shifts of vector fields. The main purpose of the paper is to show that the forward and backward shifts of vector fields commute with the Lie bracket operator and with some commonly used system transformations. The latter include, for example, classical and parametrised...

In this paper, a novel solution is proposed for verifying difference flatness of a nonlinear discrete-time control system. A necessary and sufficient condition is proved under which a discrete-time control system is flat. The condition is based on a sequence of vector spaces of differential 1-forms. A more general definition of flatness is consider...

The article studies possibilities to find realization of nonlinear retarded control system with constant commensurable delays, described by higher order input‐output equation, in a set of first‐order equations, which is in the so‐called feedforward form. Necessary and sufficient conditions are found for solvability of the problem when the realizati...

The paper addresses the problem of checking the accessibility property for nonlinear single-input single-output retarded time-delay systems with constant commensurable delays, described by higher order input output functional differential equations. Necessary and sufficient conditions are found to check the accessibility property, and sufficient co...

The paper presents a brief overview of the most popular disturbance estimation techniques together with their application to flatness-based control. Two disturbance estimation approaches, the basic disturbance observer and the extended state observer, are described in a tutorial manner. Positive and negative aspects of both approaches are pointed o...

The problem of trajectory tracking for nonlinear differentially flat systems is addressed in the paper. A novel event-based control methodology is developed for systems depending on unmeasurable disturbances. A standard flatness-based feedforward control is applied when the flat outputs are close to their desired values. If the error becomes too la...

The paper addresses the problem of transforming single-output continuous and discrete-time nonlinear dynamical systems into their respective extended observer forms. The output injection term in the extended observer form of degree I depends, besides on the output also on its first I derivatives (in the continuous-time case) or first I past values...

In the comments we claim that the disturbance observer suggested in the commented paper is not applicable, since it depends implicitly on the disturbance it should estimate.

In this paper necessary and sufficient conditions are given which guarantee that there exists a realization of a set of nonlinear higher order differential input-output equations in the controller canonical form. Two cases are studied, corresponding respectively to linear and nonlinear output functions. The conditions are formulated in terms of cer...

In this paper the problem of transforming a nonlinear multi-input multi-output time-delay control system, described by a set of higher order functional differential equations relating system inputs and outputs, into a set of first order functional differential equations, is studied. The globally linearized system equations are described in terms of...

The paper studies the problem of disturbance decoupling of nonlinear hybrid systems. The hybrid systems under consideration are switched systems that consist of finite automaton, which defines the switching rule, a set of nonlinear discrete-time systems and the so-called mode activator, that defines an input for the automaton. Such systems allow to...

The paper addresses the problem of transforming a single-input single-output nonlinear retarded time-delay system, described by an input–output equation, in the traditional observable state space form. The solution is generalized from the delay-free case and depends on integrability of certain submodule of differential 1-forms. The integrability co...

The paper studies the possibility of constructing observer©based residuals to detect faults in a nonlinear discrete©time system. The residuals are generated in such a manner that they detect one specific fault and are not affected by other faults and disturbances. Thus, a bank of residuals has been found to detect and isolate different faults in th...

This paper studies the flatness-based control of a nonlinear multi-input multi-output heating, ventilating and air conditioning (HVAC) system. First, a novel method for checking flatness property is used to verify that the given HVAC model is flat and that the flat outputs are the temperature and the humidity ratio of the thermal space. Since the f...

The paper describes the Mathematica-based software for studying nonlinear control systems. The software relies on an algebraic method, called functions’ algebra. The advantage of this approach, over well-known linear algebraic and differential geometric methods is that it is applicable to certain non-smooth systems. The drawback is that the computa...

The algebraic approach known as functions’ algebra is used to develop the necessary and sufficient conditions for the existence of state transformation and static state feedback that linearize the system equations. The advantage of this method is that it allows considering also non-smooth systems. The main object in functions’ algebra is the set of...

The paper gives an overview of an algebraic approach based on differential 1-forms, developed for the study of nonlinear control systems. The purpose of the paper is to describe the approach, comment on the necessary assumptions made, and demonstrate the effectiveness and limitations of the approach. Two very important aspects of the approach are a...

This paper introduces a method for the construction of fiat outputs of nonlinear discrete-time systems which is based on the idea of finding state- and input transformations which split the system into a smaller-dimensional subsystem and an endogenous dynamic feedback. If such a transformation exists, the complete system is flat if and only if the...

In this note, the notion of integrability is defined for 1-forms defined in the time-delay context. While in the delay-free case, a set of 1-forms defines a vector space, it is shown that 1-forms computed for time-delay systems have to be viewed as elements of a module over a certain non-commutative polynomial ring. Two notions of integrability are...

In this paper the integrability of differential one-forms is studied in case time-delays are present. Two notions, weak and strong integrability, are defined and the difference with delay-free case is explained. After giving the algorithms to check if some one-forms are strongly or weakly integrable, based on examples, the differences of the two no...

The paper addresses the problem on sensor location regarding the solvability of the plant reconfiguration problem by dynamic measurement feedback. Two methods are given to find a measured output, which guarantee the solution to the problem.

The disturbance decoupling problem by dynamic measurement feedback (DDDPM) for discrete-time nonlinear control systems is considered in this paper. The mathematical approach known under the name “the algebra of functions” is used to derive an algorithm that finds, whenever possible, a measurement feedback which solves the DDDPM. Finally, the applic...

In this paper the integrability of differential one-forms is studied in case of timedelays are present. Two notions, weak and strong integrability, are defined and the difference with delay-free case is explained. After giving the algorithms to check if some one-forms are strongly or weakly integrable, based on examples, the differences of the t...

A necessary and sufficient condition is derived for the solvability of the disturbance decoupling problem by a pure shift dynamic compensator. It corrects previously obtained results for multi-input multi-output (MIMO) systems. Also, necessary and sufficient conditions are given for the solvability of the problem by a dynamic compensator for single...

The paper addresses the disturbance decoupling problem for MIMO discrete-time nonlinear systems. A sufficient conditions are derived to solve the problem by dynamic measurement feedback, i.e. the feedback that depends on measurable outputs only. The solution to the disturbance decoupling problem, described in this paper, is based on the input-outpu...

The paper addresses the input–output decoupling problem for discrete-time nonlinear systems. The algebraic method based on difference algebra and differential forms is used to solve the problem by measurement feedback, i.e., by feedback depending only on some measured functions of state variables. Constructive necessary and sufficient solvability c...

The paper addresses the input-output linearization problem by dynamic output feedback for multi-input multi-output nonlinear systems, described by a set of higher order difference equations. Necessary and sufficient solvability conditions are given together with the constructive procedure to check the conditions and compute the feedback.

The paper addresses the problem on sensor location regarding the solvability of the disturbance decoupling problem by dynamic measurement feedback. Three methods are given to find a measured output, which guarantees the solution to the problem. It is also shown that the results for linear structured systems on sensor location when applied to infini...

Solution to the problem of fault accommodation in dynamic systems described by nonlinear discrete-time models is based on the control law which provides full decoupling with respect to fault effects. Techniques from the algebra of functions are used to solve fault decoupling problem. Solvability conditions are formulated and calculating relations a...

The paper addresses the problem of dynamic feedback linearization of discrete-time nonlinear control systems. Analogously to the continuous-time case, necessary and sufficient conditions for flatness property are obtained and showed to be equivalent to previously known results on feedback linearizability by endogenous dynamic feedback. An example i...

The paper studies the disturbance decoupling problem by the dynamic measurement feedback for discrete-time nonlinear control systems. To address the problem the algebraic approach, called the algebra of functions, is applied, which allows the system description also depend on non-differentiable functions. A necessary and sufficient condition is giv...

The problem of input-output linearization by dynamic output feedback of discrete-time multi-input multi-output nonlinear control systems is considered in this paper. The system is described by the set of higher order input-output difference equations. First, an algorithm is given to find a set of functions, which are then used to provide sufficient...

The paper describes the key elements of the recently developed symbolic software for nonlinear control systems, based on algebraic lattice theory. The software is incorporated into the package NLControl that assists the solution of various modeling, analysis and synthesis problems and using the tools of webMathematica made available for researches...

In this paper, we discuss the possible applications of algebraic framework, based on the theory of differential one-forms, to analysis of the real-life processes. The possibilities of this approach are illustrated on two common Heating, Ventilating, and Air Conditioning (HVAC) models. Algebraic formalism provides a unified framework to handle vario...

In this paper, the problem of linearization of discrete-time nonlinear control systems by dynamic state feedback is considered. Two different methods for finding certain polynomial matrix are compared. This polynomial matrix is necessary to compute linearizing (or flat) outputs of the system. Finally, a necessary and sufficient condition to check u...

The paper describes the first attempt to develop the software for computing the so-called flat output of the discrete-time nonlinear control system. The functions are implemented in NLControl, a package for nonlinear control systems, developed within the computer algebra system Mathematica. The implemented algorithm is quite elaborate. It involves...

The problem of disturbance decoupling of discrete-time nonlinear control systems by dynamic measurement feedback is addressed in this paper. New input-output linearization conditions are derived to provide weaker sufficient conditions for disturbance decoupling problem than existing ones. Additionally, it is shown that these conditions are necessar...

This paper addresses the disturbance decoupling problem (DDP) for nonlinear systems, extending the results for continuous-time systems into the discrete-time case. Sufficient conditions are given for the solvability of the problem. The notion of the rank of a one-form is used to find the static measurement feedback that solves the DDP whenever poss...