  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.
