On control problems for jump linear systems

• Autorzy: Czornik A.

Warianty tytułu

PL

O problemach sterowania układami liniowymi ze skokowo zmieniającymi się parametrami

• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

linear system; discrete-time system; controllability; stability; optimal control; Riccati equation; quadratic cost functional

PL

układ liniowy; układ dyskretny; sterowalność układu; stabilność układu; sterowanie optymalne; równanie Riccatiego; kwadratowy wskaźnik jakości

• Wydawca: Wydawnictwo Politechniki Śląskiej
• Czasopismo: Zeszyty Naukowe. Automatyka / Politechnika Śląska
• Rocznik: 2003
• Tom: z. 137
• Strony: 5--132
• Opis fizyczny: Bibliogr. 102 poz.

Twórcy

autor

Czornik A.

• Instytut Automatyki Politechniki Śląskiej, 44-100 Gliwice, ul. Akademicka 16, tel. [phone],

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