In this paper, a novel algorithm based on fully probabilistic design (FPD) is proposed for a class of linear stochastic dynamic processes with multiplicative noise. Compared with the traditional FPD, the new procedure is presented to deal with multiplicative noise and the system parameters are estimated online by the linear optimisation. The performance index is characterised by the Kullback-Leibler divergence (KLD). The generalised probabilistic control law is obtained by solving the Riccatti equation while taking the multiplicative noise into consideration. To demonstrate the effectiveness of the proposed method, a numerical example is given in comparison with the traditional FPD.

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This paper develops a novel probabilistic framework for stochastic nonlinear and uncertain control problems. The proposed framework exploits the Kullback-Leibler divergence to measure the divergence between the distribution of the closed-loop behavior of a dynamical system and a predefined ideal distribution. To facilitate the derivation of the analytic solution of the randomized controllers for nonlinear systems, transformation methods are applied such that the dynamics of the controlled system becomes affine in the state and control input. Additionally, knowledge of uncertainty is taken into consideration in the derivation of the randomized controller. The derived analytic solution of the randomized controller is shown to be obtained from a generalized state-dependent Riccati solution that takes into consideration the stateand control-dependent functional uncertainty of the controlled system. The proposed framework is demonstrated on an inverted pendulum on a cart problem, and the results are obtained.