Lyapunov-based Stochastic Nonlinear Model Predictive Control: Shaping the State Probability Density Functions

TL;DR

This paper introduces a stochastic model predictive control method for nonlinear systems that shapes the probability density functions of states, ensuring stability and satisfying chance constraints through the Fokker-Planck equation.

Contribution

It develops a novel control approach that explicitly shapes state probability densities and guarantees stability using a stochastic Lyapunov function.

Findings

Successfully applied to a continuous stirred-tank reactor

Demonstrates effective density shaping and constraint satisfaction

Ensures probabilistic stability in stochastic nonlinear systems

Abstract

Stochastic uncertainties in complex dynamical systems lead to variability of system states, which can in turn degrade the closed-loop performance. This paper presents a stochastic model predictive control approach for a class of nonlinear systems with unbounded stochastic uncertainties. The control approach aims to shape probability density function of the stochastic states, while satisfying input and joint state chance constraints. Closed-loop stability is ensured by designing a stability constraint in terms of a stochastic control Lyapunov function, which explicitly characterizes stability in a probabilistic sense. The Fokker-Planck equation is used for describing the dynamic evolution of the states' probability density functions. Complete characterization of probability density functions using the Fokker-Planck equation allows for shaping the states' density functions as well as direct computation of joint state chance constraints. The closed-loop performance of the stochastic control approach is demonstrated using a continuous stirred-tank reactor.