Generalised Fully Probabilistic Controller Design for Nonlinear Affine Systems

Fully probabilistic control for uncertain nonlinear stochastic systems

“…The efficiency of the proposed method is demonstrated using the inverted pendulum on a cart problem [22, 23]. The performance of the proposed method is evaluated against the generalized SDRE derived by Zafar and Herzallah [24] that does not account for functional uncertainty. The original inverted pendulum system equation that is used in Wang et al [23] does not involve any noise which is not the case in real‐world situations.…”

“…The consideration of functional uncertainty in this paper means that the covariance matrix is dependent on the state and control input which was taken into account in the derivation of the proposed method. On the contrary, the method implemented in this paper for comparison, namely, the generalized SDRE [24], does not consider the functional uncertainty of the system dynamics. There, the ideal covariance matrix $$$ {\Sigma}_2 $$$ is assumed to be the same as the covariance matrix of the actual noise.…”

“…There, the ideal covariance matrix $$$ {\Sigma}_2 $$$ is assumed to be the same as the covariance matrix of the actual noise. Furthermore, the method in Zafar and Herzallah [24] requires the evaluation of a SDRE which is the same as Equation (). The equation of cautiousness given in () as well as the additional linear term () in the control law do not exist in this method.…”

“…To clarify, two sets of experiments were conducted. In the first experiment, the conventional generalized SDRE [24] is used to derive the optimal randomized controller. In this experiment, the covariance matrix of the ideal distribution of the system state is taken to be equal to its actual covariance matrix, that is, $$$ {\Sigma}_2=0.001I $$$, where $$$ I $$$ is the identity matrix.…”

“…For both, the conventional FPD and the proposed method, the initial state of the pendulum is taken to be, $$$ {x}_0=\left[-1\kern0.5em 2.4\kern0.5em 0.2\kern0.5em -0.2\right] $$$, and the control objective is to bring the four states of the pendulum from their initial values to zero. The results of both experiments are shown in Figures 2–5 from which it can be seen that the states of the pendulum () have converged to zero for both generalized SDRE [24] and the proposed method. However, compared with the generalized SDRE [24], the states converge faster and with less oscillations using the proposed method in this paper which accounts for functional uncertainty and input‐dependent noises.…”