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1
Content available An information based approach to stochastic control problems
Bania Piotr
International Journal of Applied Mathematics and Computer Science
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2020
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Vol. 30, no. 1
23--34
EN
An information based method for solving stochastic control problems with partial observation is proposed. First, information-theoretic lower bounds of the cost function are analysed. It is shown, under rather weak assumptions, that reduction in the expected cost with closed-loop control compared with the best open-loop strategy is upper bounded by a non-decreasing function of mutual information between control variables and the state trajectory. On the basis of this result, an information based control (IBC) method is developed. The main idea of IBC consists in replacing the original control task by a sequence of control problems that are relatively easy to solve and such that information about the system state is actively generated. Two examples of the IBC operation are given. It is shown that the method is able to find an optimal solution without using dynamic programming at least in these examples. Hence the computational complexity of IBC is substantially smaller than that of dynamic programming, which is the main advantage of the proposed method.
2
Content available Time-series analysis and modelling to predict aviation safety performance index
Lališ A.
Transport Problems
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2017
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T. 12, z. 3
51--58
EN
Safety performance index is a tool with the potential to grasp the intangible domain of aviation safety, based on quantification of meaningful aviation safety system properties. The tool itself was developed in the form of Aerospace Performance Factor and is already available for the aviation industry. However, the tool turned out to be rather unsuccessful as its potential was not fully recognised by the industry. This paper introduces performed analysis on the potential and it outlines new features, utilising time-series analysis, which can improve both the recognition of the index by the industry as well as the motivations to further research and develop methodologies to evaluate overall aviation safety performance using its quantified system properties. This paper discusses not only the features but also their embedding into the existing approach for the development of aviation safety, highlighting possible deficiencies to overcome and relating the scientific work already performed in the domain. Various types of appropriate time-series methodologies are addressed and key specifications of their use with respect to the discussed issue concerning safety performance index are stated.
3
Content available A direct approach to linear-quadratic stochastic control
Duncan T. E., Pasik-Duncan B.
Opuscula Mathematica
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2017
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Vol. 37, no. 6
821--827
EN
A direct approach is used to solve some linear-quadratic stochastic control problems for Brownian motion and other noise processes. This direct method does not require solving Hamilton-Jacobi-Bellman partial differential equations or backward stochastic differential equations with a stochastic maximum principle or the use of a dynamic programming principle. The appropriate Riccati equation is obtained as part of the optimization problem. The noise processes can be fairly general including the family of fractional Brownian motions.
4
Content available Optimal and Suboptimal Control of a Standard Brownian Motion
Lefebvre M.
Archives of Control Sciences
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2016
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Vol. 26, no. 3
383--394
EN
The problem of optimally controlling a standard Brownian motion until a fixed final time is considered in the case when the final cost function is an even function. Two particular problems are solved explicitly. Moreover, the best constant control as well as the best linear control are also obtained in these two particular cases.
5
Content available Optimal consumption problem in the Vasicek model
Trybuła J.
Opuscula Mathematica
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2015
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Vol. 35, no. 4
547--560
EN
We consider the problem of an optimal consumption strategy on the infinite time horizon based on the hyperbolic absolute risk aversion utility when the interest rate is an Ornstein-Uhlenbeck process. Using the method of subsolution and supersolution we obtain the existence of solutions of the dynamic programming equation. We illustrate the paper with a numerical example of the optimal consumption strategy and the value function.
6
Content available remote Optimal dividend policy in discrete time
Marciniak E., Trybuła J.
Demonstratio Mathematica
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2014
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Vol. 47, nr 1
258--270
EN
A problem of optimal dividend policy for a firm with a bank loan is considered. A regularity of a value function is established. A numerical example of calculating value function is given.
7
Stochastyczny model obciążenia sieci dla sterowania obliczeniami rozproszonymi
Onderka Z.
Studia Informatica
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2012
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Vol. 33, nr 3A
79-90
PL
Artykuł prezentuje modele: pierwszy dla aplikacji typu manager-worker i dla heterogenicznej sieci komputerowej oraz drugi model opisujący dynamikę zmian obciążenia sieci komunikacyjnej. Dla tych modeli przedstawiono problem stochastycznego sterowania alokacją zadań. Sformułowano również problem MDP (aag. Markov Decision Control Problem) dła optymalnej alokacji zadań w sieci komputerowej.
EN
The paper preseated the model describing the dynamics of the background load for the corninunication network. The problem of stochastic control for the task allocation is formulated. Than the Markov Decision Control Problem of optimal task distribution in eomputer network is formulated. The idea of the open-loop and closed-loop control based on the stochastic forecast are presented.
8
Content available remote Incremental value of information for discrete-time partially observed stochastic systems
Banek T.
Control and Cybernetics
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2010
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Vol. 39, no 3
769-781
EN
A discrete-time stochastic control problem for general (nonlinear in state, control, observation and noise) models is considered. The same noise can enter into the state and into the observation equations, and the state/observation does not need to be affine with respect to the noise. Under mild assumptions the joint distribution function of the state/observation processes is obtained and used for computing the Gateaux and Frechet derivatives of the cost function. Under partial observation the control actions are restricted by the measurability requirement and we compute the Lagrange multiplier associated with this "information constraint". The multiplier is called a "dual", or "shadow" price, and in the literature of the subject is interpreted as an incremental value of information . The present and the future are two factors appearing in the multiplier and we study how they are balanced as time goes on. An algorithm for computing extremal controls in the spirit of R. Rishel (1985) is also obtained.
9
Content available remote Compensation of the scan-period irregularities in LQG control systems
Cvejn J.
Archives of Control Sciences
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2008
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Vol. 18, no. 3
277-288
EN
Computer-based control applications, especially if they run under general-purpose opera\-ting systems, often exhibit variance of the scan period of processing inputs and outputs. Although this phenomenon is usually neglected when discrete control algorithms are used, it can cause worse performance of the control loop in comparison to theoretical case. In this paper we describe a modified discrete LQG control algorithm that takes disturbances of the scan period into account and partially compensates their effect. This modification concerns both the state estimation and generating the control output. We also show that such a controller can be implemented even on relatively simple hardware platforms if the system dynamics is time-invariant.
10
Content available remote Two-level stochastic control for a linear system with nonclassical information
Duda Z., Brandys W.
International Journal of Applied Mathematics and Computer Science
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2004
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Vol. 14, no 2
161-166
EN
A problem of control law design for large scale stochastic systems is discussed. Nonclassical information pattern is considered. A two-level hierarchical control structure with a coordinator on the upper level and local controllers on the lower level is proposed. A suboptimal algorithm with a partial decomposition of calculations and decentralized local control is obtained. A simple example is presented to illustrate the proposed approach.
11
Content available remote 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.
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