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1
Content available remote Adaptive control for a jump linear system with quadratic cost
100%
Czornik A.
Control and Cybernetics
|
2004
|
tom Vol. 33, no 1
51-69
EN
The adaptive control problem for a jump linear system with quadratic cost functional on infinite time interval is solved in this paper. It is assumed that the coefficients of the state equation are unknown but a compact set that contains the parameters is known. A diminishing excitation accompanies the adaptive control signal, to ensure the strong consistency of the weighted least squares algorithm.
2
Adaptive conyrol of discrete time-varying LQG
100%
Czornik A.
Roczniki Polskiego Towarzystwa Matematycznego. Seria 3: Matematyka Stosowana
|
1999
|
tom [Z] 41
21-33
EN
The adaptive version of the discrete time-varying linear quadratic control is considered under the assumption that the coefficients have limits as time tends to infinity sufficiently fast in certain sense and the limiting system is observable and stabilizable. It is proved that time invariant LS estimator can be used to estimate the limits of the coefficients and that it is strongly consistent under some conditions well known from the time invariant case. The estimator of the parameters is used define an adaptive control law and it is shown that the control law is optimal.
3
On control problems for jump linear systems
100%
Czornik A.
Zeszyty Naukowe. Automatyka / Politechnika Śląska
|
2003
|
tom z. 137
5-132
EN
In this book we consider the problems of controllability, stability and optimal control with quadratic index for discrete-time linear systems with randomly jumping parameters. In the analyzed model the parameters are functions of a Markov chain with finite state space. First we study various concepts of controllability and deliberately illustrate the relationships between them. For all proposed types of controllability we present necessary and sufficient conditions as well as several methods of synthesis of control law that ensures reaching of required goal. A first impression, when we consider the problem of controllability for jump linear systems, may be to reduce it to a problem of controllability of linear systems with time-varying parameters. However, one important problem arise in this approach. When we consider deterministic time-varying systems and we want to find a control that drives certain initial conditions to a final state in given time then starting from the first moment we know values of all the parameters up to the final moment. Whereas for jump linear systems in each moment we know only the past values of coefficients and the future values could be predicated with given probability. This causes that for jump linear systems quite different approach must be used. The presented results significantly extend and complete the existing knowledge in the fild of controllability of jump linear systems. Stability of jump linear systems is the next subject discussed in this book is. We focus on two types of stability: moment stability and almost sure stability. For one dimensional systems we present full description of both types of stability together with relationships between them. Such complete solution is nevertheless available only for this class of systems. Next we present results on mean square stability. This special case of moment stability deserves special attention from the following two reasons. First, it is the only case of moment stability for which the necessary and sufficient conditions are known. Secondly, mean square stability is closely related to linear quadratic problem which is one of the most important optimization problems. It is also interesting that conditions for mean square stability can be expressed in terms of solutions of properly definite set of matrix linear equations. This set of equation called coupled Lyapunov equation is also investigated. Regarding almost sure stability, which is the most desirable from practical point of view, only partial results are available. We present several sufficient conditions, however only for special commuting structure of the matrix coefficients we can present necessary and sufficient conditions. Similar situation occurs for moment stability, i.e. in general, only sufficient conditions are known and some more specific results can be formulated under additional assumptions about commuting structure. We also discuss the Lyapunov exponent approach to stability problem. However, these results are purely theoretical unless methods for determining the sign of the Lyapunov exponent are developed. The last problem discussed in this book is the problem of minimizing quadratic cost functional. It is called JLQ problem. The important difference between the results from the literature and those presented here is that we consider the situation when the coefficient of the systems depend also on time. We start with the JLQ problem on finite time interval. In this case the optimal control is given in the form of linear feedback with the feedback matrices depending on time and the state of Marków chain (the mode). The optimal feedback is given by a solution of a set of quadratic recurrent matrix equations. This set of equations is called recurrent coupled Riccati equation. Next we consider the situation of an infinite time interval. In the case the solution does not always exists. The existence of solution depends on the existence of a global and bounded solution of recurrent coupled Riccati equation. Therefore, next we investigate properties of this equation. If we consider the case when the coefficients of the system and cost functional does not explicitly depend on time the recurrent coupled Riccati equation changes into a set of algebraic quadratic matrix equations called coupled algebraic Riccati equation. Properties of this equation together with numerical algorithm of solving are also presented. We end our considerations with JLQ problem for jump linear system with additive disturbance. This problem is called noise JLQ problem. It is interesting that noise JLQ problem may have more than one solution. Basing on this property we show that for certain class of time varying systems the optimal control can be realized in the time invariant feedback form.
PL
W pracy omawia się zagadnienia sterowalności, stabilności i sterowania optymalnego z kwadratowym funkcjonałem kosztów dla dyskretnych układów liniowych ze skokowo zmieniającymi się parametrami. W rozdziale 1 zebrano istniejące koncepcje sterowalności takich układów i zaproponowano pewne nowe definicje sterowalności. Rozważa się zarówno sterowalność w ustalonym czasie, jak i sterowalność w czasie losowym. Następnie przedyskutowano zależności między różnymi typami sterowalności i dla każdego z nich podano metody syntezy prawa sterowania zapewniającego osiągnięcie wymaganego celu. Wyniki tego rozdziału w pełni rozwiązują problem sterowalności dyskretnych układów liniowych ze skokowo zmieniającymi się parametrami. Rozdział 2 poświęcony jest stabilności. Rozdział ten rozpoczyna się od wprowadzenia różnych typów sterowalności i dyskusji prostszych relacji między nimi. Następnie dla układów jednowymiarowych podane są warunki konieczne i wystarczające dla każdego typu stabilności i dokładny opis relacji między nimi. Jest to jedyna klasa układów, dla której taki kompletny opis udało się uzyskać. Stabilność średniokwadratowa została szczególnie wnikliwie opisana z dwóch powodów. Po pierwsze jest ona ściśle związana z jednym z najważniejszych zagadnień sterowania optymalnego, a mianowicie z problemem liniowo kwadratowym. Po drugie jest to jedyny typ stabilności, dla którego znane są efektywne warunki konieczne i wystarczające. Z punktu widzenia praktyki najbardziej pożądana jest stabilność z prawdopodobieństwem jeden. Niestety otrzymane wyniki nie rozwiązują w pełni tego problemu. Rozdział 3 poświęcony jest problemowi sterowania optymalnego z kwadratowym wskaźnikiem jakości. W pierwszej części tego rozdziału przedstawiono znane w literaturze wyniki dotyczące przypadku sterowania na skończonym przedziale czasowym. Następnie przedstawiono nowe wyniki dotyczące nieskończonego horyzontu czasowego. Istotną nowością w porównaniu ze znanymi pracami jest rozpatrywanie sytuacji, w której zarówno współczynniki modelu, jak i wskaźnika jakości zależą od czasu. Rezulataty te zostały osiągnięte poprzez analizę układu stowarzyszonych równań różnicowych Riccatiego.
4
Dynamika układów hybrydowych
100%
Czornik A.
Zeszyty Naukowe. Automatyka / Politechnika Śląska
|
2008
|
tom z. 151
31-36
PL
Badanie dynamiki układów hybrydowych jest źródłem wielu interesujących i trudnych problemów matematycznych. Celem tej pracy jest zreferowanie postępów w badaniach własności dynamiki układów hybrydowych ze szczególnym uwzględnieniem problemu stabilności. Z problemem stabilności ściśle związane są problemy eksponencjalnego wzrostu i twierdzenia odwrotne do twierdzenia Lapunowa. Oba te zagadnienia zostaną zreferowane. W pracy zostanie zasygnalizowanych również kilka otwartych problemów.
EN
The study of the hybrid systems dynamics gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to review progress made in research of hybrid systems dynamics. We concentrate our attention on stability. Closely related to the concept of stability are the notations of exponential growth and converse Lyapunov theorems, both of which are discussed. We also point out some problems that remain open.
5
Content available remote Continuity of Solutions of Riccati Equations for the Discrete-Time Jlqp
63%
Czornik A. , Świerniak A.
International Journal of Applied Mathematics and Computer Science
|
2002
|
tom Vol. 12, no 4
539-543
EN
The continuity of the solutions of difference and algebraic coupled Riccati equations for the discrete-time Markovian jump linear quadratic control problem as a function of coefficients is verified. The line of reasoning goes through the use of the minimum property formulated analogously to the one for coupled continuous Riccati equations presented by Wonham and a set of comparison theorems.
6
Content available remote On the discrete time-varying JLQG problem
63%
Czornik A. , Świerniak A.
International Journal of Applied Mathematics and Computer Science
|
2002
|
tom Vol. 12, no 2
203-207
EN
In the present paper optimal time-invariant state feedback controllers are designed for a class of discrete time-varying control systems with Markov jumping parameter and quadratic performance index. We assume that the coefficients have limits as time tends to infinity and the boundary system is absolutely observable and stabilizable. Moreover, following the same line of reasoning, an adaptive controller is proposed in the case when system parameters are unknown but their strongly consistent estimators are available.
7
Content available On descriptor systems and related linear quadratic problem
63%
Gessing R. , Czornik A.
Control and Cybernetics
|
2000
|
tom Vol. 29, no 1
79-89
EN
It is shown that a descriptor system under the condition of impulse controllability, Cobb (1984), may be converted, by means of linear transformations, to a system described in a state space and composed of state and output equations. The transfomations determine one to one correspondence between the solutions of both the systems. It is noted that the control in a feedback form may not determine a unique solution of the descriptor system what is often overlooked in many previous papers. It is also shown that the LCD problem formulated in a descriptor space for the impulse observable system, Cobb (1984); may be converted by means of linear transformations to the usual LQ problem formulated in the state space. It is stressed that the second problem may be regular even then, when the weighting matrix of the control, in the cost functional of the first problem, is singular. The proposed approach simplifies the calculations related to the LQ problem solution significantly.
8
Controllability of continuous - time jump linear systems
63%
Czornik A. , Świerniak A.
Zeszyty Naukowe. Automatyka / Politechnika Śląska
|
2000
|
tom z. 132
21-26
EN
In this paper we consider a problem of controllability of continous time linear system with randomly jumping parameters which can be described by finite state Markov chain. Different kinds for controllability of such a system are analysed and the sufficient and neccesery conditions for them are established.
PL
W pracy rozważa się problem sterowalności ciągłych liniowych układów ze skokowo zmieniającymi się parametrami które mogą być opisane jednorodnym łańcuchem Markowa o skończonej liczbie stanów. Analizowane są różne koncepcje sterowalności takich układów oraz są wyprowadzone dla nich konieczne i wystarczające warunki sterowalności.
9
Content available remote On the bounds on the solutions of the algebraic Lyapunov and Riccati equations
63%
Czornik A. , Nawrat A.
Archives of Control Sciences
|
2000
|
tom Vol. 10, no. 3/4
197-244
EN
Different types of bounds for solutions of continuous and discrete Lyapunov and Riccati equations obtained up to now are summarized in this paper. Some new bounds are also presented. The efficiency of each bound is illustrated with three numerical examples. A discussion and comparison are given as well. The results may be particularly convenient to get the ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds.
10
Content available remote Some properties of the spectral radius of a set of matrices
63%
Czornik A. , Jurgaś P.
International Journal of Applied Mathematics and Computer Science
|
2006
|
tom Vol. 16, no 2
183-188
EN
In this paper we show new formulas for the spectral radius and the spectral subradius of a set of matrices. The advantage of our results is that we express the spectral radius of any set of matrices by the spectral radius of a set of symmetric positive definite matrices. In particular, in one of our formulas the spectral radius is expressed by singular eigenvalues of matrices, whereas in the existing results it is expressed by eigenvalues.
11
Content available remote On Adaptive Control for the Continuous Time-varying JLQG Problem
63%
Czornik A. , Świerniak A.
International Journal of Applied Mathematics and Computer Science
|
2005
|
tom Vol. 15, no 1
53-62
EN
In this paper the adaptive control problem for a continuous infinite time-varying stochastic control system with jumps in parameters and quadratic cost is investigated. It is assumed that the unknown coefficients of the system have limits as time tends to infinity and the boundary system is absolutely observable and stabilizable. Under these assumptions it is shown that the optimal value of the quadratic cost can be reached based only on the values of these limits, which, in turn, can be estimated through strongly consistent estimators.
12
O stabilności dyskretnych układów liniowych
63%
Czornik A. , Jurgaś P.
Zeszyty Naukowe. Automatyka / Politechnika Śląska
|
2006
|
tom z. 145
63-68
PL
Rozważamy tutaj skończone zbiory macierzy [sigma] = {A1, A2,…An} takie, że promień spektralny każdej macierzy takiego zbioru jest mniejszy niż 1. Następnie formułujemy twierdzenie pokazujące, że dla takiego zbioru możemy zawsze skonstruować zbiór [sigma]' = {A1(l1), A2(l2),…An(ln)} (l1, l2,…,ln) taki, że promień spektralny tego zbioru jest mniejszy niż l ([ro]( [sigma]') < l).
EN
We consider finite sets of matrices [sigma] = {A1, A2,…An} such that spectral radius of each matrix that belongs to such set is less than 1. Then we formulate the theorem that shows that for such set we can always construct the set [sigma]' = {A1(l1), A2(l2),…An(ln)} (l1, l2,…,ln) such that spectral radius o this set is less than 1 ([ro]( [sigma]') < l).
13
Content available remote Falseness of the finiteness property of the spectral subradius
63%
Czornik A. , Jurgaś P.
|
|
tom Vol. 17, no 2
173-178
EN
We prove that there exist infinitely may values of the real parameter α for which the exact value of the spectral subradius of the set of two matrices (one matrix with ones above and on the diagonal and zeros elsewhere, and one matrix with α below and on the diagonal and zeros elsewhere, both matrices having two rows and two columns) cannot be calculated in a finite number of steps. Our proof uses only elementary facts from the theory of formal languages and from linear algebra, but it is not constructive because we do not show any explicit value of α that has described property. The problem of finding such values is still open.
14
Content available On a continuity of characteristic exponents of linear discrete time-varying systems
51%
Czornik A. , Mokry P. , Niezabitowski M.
Archives of Control Sciences
|
2012
|
tom Vol. 22, no. 1
17-27
EN
In this paper we present a sufficient condition for continuity of Lyapunov exponents of discrete time-varying linear system. Basing on this result we show that Lyapunov exponents of time-invariant systems depend continuously on the time-varying perturbations.
15
Content available Assignability of numerical characteristics of time-varying systems
51%
Babiarz A. , Czornik A. , Klamka J.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2019
|
tom Vol. 67, nr 6
1007--1022
EN
The main aim of this article is to survey and discuss the existing state of art concerning the assignability by a feedback of numerical characteristics of linear continuous and discrete time-varying systems. Most of the results present necessary or sufficient conditions for different formulation of the Lyapunov spectrum assignability problem. These conditions are expressed in terms of various controllability types and optimalizability of the controlled systems and certain properties of the free system such as: regularity, diagonalizability, boundness away, integral separation and reducibility.
16
Content available remote On the stability of one dimensional discrete-time jump linear systems
51%
Czornik A. , Nawrat A. , Rabsztyn S.
Archives of Control Sciences
|
2003
|
tom Vol. 13, no. 1
81-93
EN
In this paper we present necessary and sufficient conditions for "delta" - moment stability and almost sure stability of one-dimensional discrete-time linear systems subject is equivalent to "delta"-moment stability for certain "delta">0.
17
Content available Stability and controllability of switched systems
51%
Klamka J. , Czornik A. , Niezabitowski M.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2013
|
tom Vol. 61, nr 3
547--555
EN
The study of properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. This paper aims to briefly survey recent results on stability and controllability of switched linear systems. First, the stability analysis for switched systems is reviewed. We focus on the stability analysis for switched linear systems under arbitrary switching, and we highlight necessary and sufficient conditions for asymptotic stability. After that, we review the controllability results.
18
Content available The selected problems of controllability of discrete-time switched linear systems with constrained switching rule
51%
Babiarz A. , Czornik A. , Klamka J. , Niezabitowski M.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2015
|
tom Vol. 63, nr 3
657--666
EN
In this paper the controllability problem for discrete-time linear switched systems is considered. The main goal is to find a control signal that steers any initial state to a given final state independently of the switching signal. In the paper, it is assumed that there are some constraints posed on the switching signal. Moreover, we present a necessary and sufficient conditions of some kinds of controllability. Three types of controllability, namely: from zero initial state to any final state, from any initial state to zero final state and from any initial state to any final state are considered. Finally, three illustrative examples are shown.
19
Content available Variation of constant formulas for fractional difference equations
45%
Anh P. T. , Babiarz A. , Czornik A. , Niezabitowski M. , Siegmund S.
Archives of Control Sciences
|
2018
|
tom Vol. 28, No. 4
617--633
EN
In this paper, we establish variation of constant formulas for both Caputo and Riemann-Liouville fractional difference equations. The main technique is the Z-transform. As an application, we prove a lower bound on the separation between two different solutions of a class of nonlinear scalar fractional difference equations.
20
Content available Asymptotic properties of discrete linear fractional equations
45%
Anh P. T. , Babiarz A. , Czornik A. , Niezabitowski M. , Siegmund S.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2019
|
tom Vol. 67, nr 4
749--759
EN
In this paper we study the dynamical behavior of linear discrete-time fractional systems. The first main result is that the norm of the difference of two different solutions of a time-varying discrete-time Caputo equation tends to zero not faster than polynomially. The second main result is a complete description of the decay to zero of the trajectories of one-dimensional time-invariant stable Caputo and Riemann-Liouville equations. Moreover, we present Volterra convolution equations, that are equivalent to Caputo and Riemann-Liouvile equations and we also show an explicit formula for the solution of systems of time-invariant Caputo equations.
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