This paper studies two linear methods for linear and non-linear stochastic
optimal control of partially observable problem (SOCPP). At first, it
introduces the general form of a SOCPP and states it as a functional matrix.
A SOCPP has a payoff function which should be minimized. It also has two
dynamic processes: state and observation. In this study, it is presented a
deterministic method to find the control factor which has named feedback
control and stated a modified complete proof of control optimality in a
general SOCPP. After finding the optimal control factor, it should be
substituted in the state process to make the partially observable system.
Next, it introduces a linear filtering method to solve the related partially
observable system with complete details. Finally, it is presented a
heuristic method in discrete form for estimating non-linear SOCPPs and it is
stated some examples to evaluate the performance of introducing methods.
Summary
In this paper, two spectral methods are presented to solve a stochastic optimal control problem of a partially observable system. These two methods work together to solve such problems. In fact, solving such problems involves two cases: obtaining the control function and simulating the partially observable system. At first, a spectral linear filter is defined as a function of time to obtain an appropriate solution for a partially observable system. This linear filter is equipped with an orthogonal basis and it is made to predict the future behavior of this system. In this method, the goal is to approximate the trend of the partially observable system. The second method is suggested to achieve the optimal control corresponding to each sample path. In this method, the spectral Fourier transform is used. These two methods are used together to solve linear and nonlinear cases. In fact, the innovative contents of this paper are both the spectral linear filter and the suggested spectral optimal control method.
In this paper we studied stochastic optimal control problem based on partially observable systems (SOCPP) with a control factor on the diffusion term. A SOCPP has state and observation processes. This kind of problem has also a minimum payoff function. The payoff function should be minimized according to the partially observable systems consist of the state and observation processes. In this regard, the filtering method is used to evaluat this kind of problem and express full consideration of it. Finally, presented estimation methods are used to simulate the solution of a partially observable system corresponding to the control factor of this problem. These methods are numerically used to solve linear and nonlinear cases.
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