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.
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