A moment-based approach for nonlinear stochastic tracking control

Abstract

This paper describes a new stochastic control methodology for nonlinear affine systems subject to bounded parametric and functional uncertainties. The primary objective of this method is to control the statistical nature of the state of a nonlinear system to designed (attainable) statistical properties (e.g., moments). This methodology involves a constrained optimization problem for obtaining the undetermined control parameters, where the norm of the error between the desired and actual stationary moments of state responses is minimized subject to constraints on moments corresponding to a stationary distribution. To overcome the difficulties in solving the associated Fokker-Planck equation, generally experienced in nonlinear stochastic control and filtering problems, an approximation using the direct quadrature method of moments is proposed. In this approach, the state probability density function is expressed in terms of a finite collection of Dirac delta functions, and the partial differential equation can be converted to a set of ordinary differential equations. In addition to the above mentioned advantages, the state process can be nonGaussian. The effectiveness of the method is demonstrated in an example including robustness with respect to predefined uncertainties and able to achieve specified stationary moments of the state probability density function.

Model‐Reference Adaptive Moment Control of Uncertain Nonlinear Stochastic Systems

“…According to the moment control method, system (17) is established. In order to achieve first and second moment tracking, the poles of the closed-loop system (19) should be placed at the left-half of the S-plane. In order to have stable closed loop system (19) with fast dynamics the closed-loop poles are considered at -5, -5.5, -5.8, -6, -6.5, -7, -7.5, -8, -9, -11, -12, -12.5, -13, -13.2 -13.5, -14, -14.5, -15, and then the following feedback gains are obtained using place(A 1 , B 1 , ploes) function in Matlab:…”

“…These methods lead to inaccurate results. The second relates to the methods that control the nonlinear system's moments by obtaining the probability density function of the response . These methods lead to the solution of a set of complicated differential equations based on the Fokker‐Planck‐Kolmogorov differential equation.…”

Stochastic trajectory optimization for mechanical systems with parametric uncertainties

“…In this paper, we will deal with the discrete-time counterpart of (2) by discretizing the corresponding cost functions and dynamic equations. Moreover, in sections III, IV, we will utilize Polynomial Chaos theory to transform the stochastic optimal control problem in (2) (usually referred to as momentbased stochastic control problem; see for example [33], [34]) into a purely deterministic one.…”

Gaussian sum approximation filter for nonlinear dynamic time-delay system

“…Up to present, many methods have been proposed to deal with the filtering problem of the nonlinear timedelay systems, classically including the state augmentation, robust H-infinity filter [3,4], and particle filter [5,6]. However, it is well known that these methods inevitably face enormous computation burdens owing to the augmentation of the present state with the delayed states in the state augmentation method, to the hard computation of the matrix inequality in H-infinity filter, and to a large number of stochastic sampling particles for fulfilling the estimation accuracy in particle filter.…”