Generalised Probabilistic Control Design for Uncertain Stochastic Control Systems

Abstract

In this paper a novel generalised fully probabilistic controller design for the minimisation of the Kullback-Leibler divergence between the actual joint probability density function (pdf) of the closed loop control system, and an ideal joint pdf is presented for a linear Gaussian uncertain class of stochastic systems. A single layer neural network is used to approximate the probability density function of the system dynamics. The generalised probabilistic control law is obtained by solving the recurrence equation of dynamic programming to the fully probabilistic design control problem while taking into consideration the dependency of the parameters of the estimated probability density function of the system dynamics on the input values. It is shown to be of the class of cautious type controllers which accurately minimises the value of the Kullback-Leibler divergence without disregarding the variance of the model prediction as an element to be minimised. Comparison of theoretical and numerical results obtained from the F-16 fighter aircraft application with existing state-of-the-art demonstrates the effectiveness of the proposed method.

Model‐Reference Adaptive Moment Control of Uncertain Nonlinear Stochastic Systems

“…where w r is considered the zero mean white Gaussian noise with covariance 0.1. Now, using (16), the state-space model of moments for the stochastic reference model (41) is established where Γ raug (t) ∈ ℝ 9 × 1 and V raug (t) ∈ ℝ 9 × 1 are determined based on (13) and (14) and W aug ≜ [0 0.1] T . Considering a 1 = 5, a 2 = 6, a 3 = 7 and b 1 = b 2 = b 3 = 1 to have a stable and controllable reference model (41), the following matrices A aug ∈ ℝ 9 × 9 , B aug ∈ ℝ 9 × 9 and G aug ∈ ℝ 9 × 2 are obtained as:…”

“…where w r ¼ w 1r w 2r ½ T is considered the zero mean white Gaussian noise with covariance 0.1. Now, using (16), the statespace model of moments for the stochastic reference model (48) is established where Γ raug (t) ∈ ℝ 9 × 1 and V raug (t) ∈ ℝ 5 × 1 are determined based on (13) and (14) and W aug ≜ 0 0 0:1 0 0:1 ½ T . Considering a 1 = 9, a 2 = 6, a 3 = 7 and b 1 = b 2 = 5 to have a stable and controllable reference model (59), the following matrices A aug ∈ ℝ 9 × 9 , B aug ∈ ℝ 9 × 5 and G aug ∈ ℝ 9 × 5 are obtained as: ; also C aug is considered in order to control the first and the third A 1 , B 1 ), which is defined in (20), is controllable, by considering the closed-loop poles at -3, -3.25, -3.5, -3.75, -4, -4.25, -4.5, -5.5, -5.75, -6, -6.25, -6.5, -6.75, -7, using the place(A 1 , B 1 , ploes) function in Matlab, K c and K I are obtained and then V raug (t) is determined.…”

“…Using these augmented vectors (13)(14)(15) and moments equations (11) and (12), a new linear state-space model is established, which represents the state expected and covariance dynamics:…”

“…It was a covariance control method that controlled the second moment of linear system states. Since the 1970s, many studies on moment control of linear stochastic systems have been conducted [8][9][10][11][12][13][14], while the literature on moment control of nonlinear systems is more limited. The moment control of nonlinear stochastic systems is much more difficult than that related to linear systems.…”

Fully Probabilistic Design for Stochastic Discrete System with Multiplicative Noise

“…The quadratic distance has been used in many papers to characterise the distance between the actual pdf and the desire pdf [10]- [12]. As many distributions can be expressed by an exponential function, Kullback-Leibler divergence (KLD) can be considered as a convenient scale to characterise the distance between two distributions [13]- [15]. Therefore, Fully Probabilistic Design (FPD) ( [13]- [16]), inspired by Bayesian approach to adaptive control design [17], which is proposed based on KLD, is applied in this paper.…”