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1

Extended homogeneous balance conditions in the sub-equation method

Song Chenwei, Liu Yinping

Journal of Applied Analysis

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2022

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Vol. 28, nr 1

165--179

EN

The sub-equation method is a kind of straightforward algebraic method to construct exact solutions of nonlinear evolution equations. In this paper, the sub-equation method is improved by proposing some extended homogeneous balance conditions. By applying them to several examples, it can be seen that new solutions could indeed be obtained.

2

Suboptimal control of nonlinear continuous-time locally positive systems using input-state linearization and SDRE approach

Bernat J., Kołota J., Stępień S., Superczyńska P.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2018

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Vol. 66, nr 1

17--21

EN

The infinite time suboptimal control problem for continuous-time nonlinear positive systems is formulated and solved. A solution to the problem using input-state linearization and state-dependent Riccati equation method (SDRE) is established, a procedure for solving the problem is proposed and illustrated with a numerical example.

3

Determination of optimal controllers. Comparison of two methods for electric network chain

Górecki H., Zaczyk M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2018

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Vol. 66, nr 3

267--273

EN

In the paper the comparison of two methods for calculation optimal gains is considered. One method using a Kalman procedure and one using a Riccati equation are compared. It is proved that a Kalman procedure is much better.

4

On one oscillatory criterion for the second order linear ordinary differential equations

Grigorian G. A.

Opuscula Mathematica

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2016

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Vol. 36, no. 5

589--601

EN

The Riccati equation method is used to establish an oscillatory criterion for second order linear ordinary differential equations. An oscillatory condition is obtained for the generalized Hill's equation. By means of examples the obtained result is compared with some known oscillatory criteria.

5

The simulations of sequential of estimators for objects with a serial structure

Kwater T., Krutys P.

Information Systems in Management

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2016

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Vol. 5, No. 3

347--357

EN

The paper concerns the results of simulation of a certain approach, in which estimation of object state associated with the decentralization of calculations. It is realized by dividing the optimization problem into sub-problems of smaller dimensionality. The main idea of the method is to assign individual quality indicators to each sub-object and to carry out the synthesis of estimators in a sequential manner, starting with the last sub-object. Implementation of the estimation process requires knowledge about the measurements of the individual sub-objects. The parameters of the filter sequential gains are calculated based on Riccati equation solutions for subobjects and certain bilinear equations for cross-linkage connections. In the simulation tests the influence of types of connection between the sub-objects, the intensity disturbances of measurements and system on the values of coefficients of gains, as well as the estimation errors is presented.

6

A Bolza optimal synthesis problem for singular estimate control systems

Lasiecka I., Tuffaha A.

Control and Cybernetics

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2009

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Vol. 38, no 4B

1429-1460

EN

Bolza problem governed by PDE control systems with unbounded controls is considered. The motivating example is fluid structure interaction model with boundary-interface controls. The aim of the work is to provide optimal feedback synthesis associated with well denned gain operator constructed from the Riccati equation. The dynamics considered is of mixed parabolic-hyperbolic type which prevents applicability of tools developed earlier for analytic semigroups. It is shown, however, that the control operator along with the generator of the semigroup under consideration satisfy singular estimate referred to as Revisited Singular Estimate (RSE). This estimate, which measures "unboundedness" of control actions, is a generalization and a weaker form of Singular Estimate (SE) treated in the past literature. The main result of the paper provides Riccati theory developed for this new class of control systems labeled as RSECS (Revisited Singular Estimate Control Systems). The important feature is that the gain operator, constructed via Riccati operator, is consistent with the optimal feedback synthesis. The gain operator, though unbounded, has a controlled algebraically singularity at the terminal point. This enables one to establish well-posedness of the Riccati solutions and of the optimal feedback representation. An application of the theoretical framework to boundary control of a fluid-structure interaction model is given.

7

An algebraic approach to linear-quadratic optimization of second order dynamical systems

Respondek J.

Archives of Control Sciences

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2007

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Vol. 17, no. 1

105-116

8

Analityczna zależność do wyznaczenia ciśnienia w hydrostatycznym łożysku poprzecznym ślizgowym

Kyureghyan K., Kornacki A., Piekarski W.

Inżynieria Rolnicza

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2006

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R. 10, nr 6(81)

7-14

PL

Praca przedstawia analityczne rozwiązanie równania Reynoldsa służące do wyznaczenia teoretycznego rozkładu ciśnienia oleju wewnątrz łożyska poprzecznego ślizgowego. Ogólne równanie Reynoldsa (przy zadanych warunkach brzegowych oraz funkcji zużycia łożyska ) sprowadzono do postaci uproszczonej (równanie Riccatiego), podano ostateczne rozwiązanie analityczne.

EN

The paper presents analytic solution of the Reynolds equation intended to determine the theoretical oil pressure distribution inside a lateral friction bearing. The generalised Reynolds equation (with given boundary conditions and the function of bearing wear) has been simplified (Riccati equation), and ultimate analytic solution has been given.

9

LQR and predictive control tuned via BDU

Ramos C., Sanchis J., Martinez M., Herrero J. M.

Systems Science

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2005

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Vol. 31, no 1

15-25

EN

This work presents the BDU technique (Bounded Data Uncertainties) and the tuning of the linear quadratic regulator (LQR) via this technique, which considers models with bounded uncertainties. The BDU method is based on constrained game-type formulations, and allows the designer to explicitly incorporate a priori information about bounds on the sizes of the uncertainties into the problem statement. Thus, on the one hand, the uncertainty effect is not over-emphasized, avoiding an overly conservative design and, on the other hand, the uncertainty effect is not under-emphasized, avoiding an overly sensitive to errors design. A feature of this technique consists of its geometric interpretation. The structure of the paper is the following, in the first section, some problems about the least-squares method in the presence of uncertainty are introduced. The BDU technique is shown in the second section and the LQR controller in the third. After that a new guided way of tuning the LQR is offered, taking into account the uncertainties bounds via the BDU. The consequence of this method is that both recursive and algebraic Riccati equations are modified. Finally, some examples are shown and the main conclusions and future work are commented.

10

Delay-dependent Asymptotic Stabilitzation for Uncertain Time-delay Systems with Saturating Actuators

Liu P. L.

International Journal of Applied Mathematics and Computer Science

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2005

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Vol. 15, no 1

45-51

EN

This paper concerns the issue of robust asymptotic stabilization for uncertain time-delay systems with saturating actuators. Delay-dependent criteria for robust stabilization via linear memoryless state feedback have been obtained. The resulting upper bound on the delay time is given in terms of the solution to a Riccati equation subject to model transformation. Finally, examples are presented to show the effectiveness of our result.

11

Optimality properties of controls with bang-bang components in problems with semilinear state equation

Felgenhauer U.

Control and Cybernetics

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2005

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Vol. 34, no 3

763-785

EN

In this paper we study optimal control problems with bang-bang solution behavior for a special class of semilinear dynamics. Generalizing a former result for linear systems, optimlity conditions are derived by a duality based approach. The results apply for scalar as well as for vector control functions and, in particular, for the case of the so-called multiple switches, too. Further, an iterative procedure for determining switching points is proposed, and convergence results are provided.

12

On control problems for jump linear systems

Czornik A.

Zeszyty Naukowe. Automatyka / Politechnika Śląska

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2003

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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.

13

Periodic solutions of the Riccati equation in Banach spaces

Baris J., Buraczewski A.

Demonstratio Mathematica

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2003

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Vol. 36, nr 2

343--353

EN

In this paper we study the problem of the existence and the construction of periodic solutions of the Riccati equation with continuous periodic coefficients denned on the real line with values in Banach space.

14

Numerically robust synthesis of discrete-time H[infinity] estimators based on dual J-lossless factorisations

Suchomski P.

Control and Cybernetics

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2003

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Vol. 32, no 4

761-802

EN

An approach to the numerically reliable synthesis of the H[infinity] suboptimal state estimators for discretised continuous-time processes is presented. The approach is based on suitable dual J-lossless factorisations of chain-scattering representations of estimated processes. It is demonstrated that for a sufficiently small sampling period the standard forward shift operator techniques may become ill-conditioned and numerical robustness of the design procedures can be significantly improved by employing the so-called delta operator models of the process. State-space models of all H[infinity] sub-optimal estimators are obtained by considering the suitable delta-domain algebraic Riccati equation and the corresponding generalised eigenproblem formulation. A relative condition number of this equation is used as a measure of its numerical conditioning. Both regular problems concerning models having no zeros on the boundary of the delta-domain stability region and irregular (non-standard) problems of models with such zeros are examined. For the first case, an approach based on a dual J-lossless factorisation is proposed while in the second case an extended dual J-lossless factorisation based on a zero compensator technique s required. Two numerical examples are given to illustrate some properties of the considered delta-domain approach.

15

J-energy preserving well-posed linear systems

Staffans O. J.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 6

1361-1378

EN

The following is a short survey of the notion of a well-posed linear system. We start by describing the most basic concepts, proceed to discuss dissipative and conservative systems, and finally introduce J-energy-preserving systems, i.e., systems that preserve energy with respect to some generalized inner products (possibly semi-definite or indefinite) in the input, state and output spaces. The class of well-posed linear systems contains most linear time-independent distributed parameter systems: internal or boundary control of PDE's, integral equations, delay equations, etc. These systems have existed in an implicit form in the mathematics literature for a long time, and they are closely connected to the scattering theory by Lax and Phillips and to the model theory by Sz.-Nagy and Foias. The theory has been developed independently by many different schools, and it is only recently that these different approaches have begun to converge. One of the most interesting objects of the present study is the Riccati equation theory for this class of infinite-dimensional systems (H2- and Hinfty-theories).

16

On Fast State-Space Algorithms for Predictive Control

Błachuta M. J.

International Journal of Applied Mathematics and Computer Science

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1999

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Vol. 9, no 1

149-160

EN

A Riccati-equation-based solution to a class of receding-horizon predictive control problems for an explicit-delay state-space model of an ARMAX system is found and the corresponding vector Chandrasekhar-type equations are derived for both filter and controller gains to improve the computational efficiency.

17

Zera i bieguny linii długiej RLCG ze sprzężeniami zwrotnymi Riccatiego

Langmann H.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

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1998

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T. 2, z. 2

143--147

PL

W pracy przedstawiono model matematyczny linii długiej objętej lokalnymi sprzężeniami zwrotnymi w każdym ogniwie. Dla takiego przypadku dokonano dekompozycji linii długiej na układy drugiego rzędu znajdując zera i bieguny transmisji analizowanego układu.

EN

In the paper the mathematical model of the network chain with local feedbacks in each element is presented. For this case is was made the decomposition of the network chain to the sum of second order system.

18

Różniczkowe równanie Riccatiego

Langmann H.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

|

1998

|

T. 2, z. 2

139--142

PL

W pracy zamieszczono procedurę rozwiązania różniczkowego równania Riccatiego. Przedstawiono transformację przekształcającą macierz Hamiltona do postaci blokowo-diagonalnej. Na tej podstawie uzyskano wzór analityczny przedstawiający rozwiązanie równania Riccatiego w funkcji rozwiązania algebraicznego równania Riccatiego.

EN

In the paper the analytical formula for the solution of the differential Riccati equation is presented.

19

Adaptive control of continuous time linear stochastic system with quadratic cost functional

Czornik Adam

Matematyka Stosowana : matematyka dla społeczeństwa

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1996

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Nr 39

17--40

EN

An adaptive control problem for linear, continuous time stochastic system is described and solved in this paper. The unknown parameters in the model appear affinely in the drift term of the stochastic differential equation. The parameter estimates given by the maximum likelihood method are used to define the feedback gain. It is proved that the parameter estimates are strongly consistent and the cost functional reaches its minimum, i.e. the adaptive control is optimal. In this paper the continuity of the solution of the algebraic Riccati equation as a function of coefficient is also verified. The continuity is important for applications to problems in adaptive control.

PL

Praca składa się z czterech części. W części pierwszej sformułowano i podano rozwiązanie zagadnienia sterowania optymalnego w liniowym układzie stochastycznym z kwadratowym funkcjonałem kosztów na skończonym i nieskończonym przedziale czasowym. Twierdzenie 1, podające postać sterowania optymalnego na skończonym przedziale czasowym, jest dobrze znane ([l], [5]), natomiast twierdzenie 2 jest uogólnieniem znanych rezultatów. Zwykle formułuje się je przy założeniach gwarantujących istnienie i jedyność rozwiązania algebraicznego równania Riccatiego ([5], [4]). W tym sformułowaniu w jakim znajduje się w pracy można je znaleźć w [16] ale dla układu deterministycznego. W części drugiej zbadano własności algebraicznego równania Riccatiego. Algebraiczne równanie Riccatiego odgrywa pierwszoplanową rolę w konstrukcji sterowania optymalnego i poświęcono mu wiele uwagi w pracach [2], [4], [13], [15], Twierdzenie 5 pokazuje na jakie trudności możemy natrafić w procedurze adaptacyjnego sterowania, gdy nieznane współczynniki równania Riccatiego będziemy zastępować ich ocenami. Problem ten obszerniej omówiono w [4] i [8]. Głównym wynikiem tej części pracy jest twierdzenie 6, które odgrywa zasadniczą rolę w konstrukcji i dowodzie optymalności sterowania adaptacyjnego. W części trzeciej skonstruowano ocenę największego prawdopodobieństwa dla macierzy liniowej transformacji stanu. Estymator ten pojawił się po raz pierwszy w zagadnieniu sterowania optymalnego w pracy [12]. Wreszcie w czwartej, głównej części pracy podano algorytm sterowania adaptacyjnego oraz dowód jego optymalności (twierdzenie 10). Podany algorytm i dowód jego optymalności są modyfikacją wyników podanych w [6] i [7], Obejmują one ogólniejsze przypadki niż w tych pracach, gdzie zakłada się znajomość domkniętego, spójnego i ograniczonego zbioru, do którego należy oceniany parametr, niemniej uzyskane rezultaty są jeszcze dalekie od analogicznych wyników uzyskanych w pracy [3] dla czasu dyskretnego.