Optimal control of dynamical systems withtime-invariant probabilistic parametric uncertainties
2018
The importance of taking model uncertainties into
account during controller design is well established. Although this
theory is well developed and quite mature, the worst-case
uncertainty descriptions assumed in robust control formulations are
incompatible with the uncertainty descriptions generated by
commercial model identification software that produces
time-invariant parameter uncertainties typically in the form of
probability distribution functions. This doctoral thesis derives
rigorous theory and algorithms for the optimal control of dynamical
systems with time-invariant probabilistic uncertainties. The main
contribution of this thesis is new feedback control design
algorithms for linear time-invariant systems with time-invariant
probabilistic parametric uncertainties and stochastic noise. The
originally stochastic system of equations is transformed into an
equivalent deterministic system of equations using polynomial chaos
(PC) theory. In addition, the H2- and H[infinity symbol]-norms
commonly used to describe the effect of stochastic noise on output
are transformed such that the eventual closed-loop performance is
insensitive to parametric uncertainties. A robustifying constant is
used to enforce the closed-loop stability of the original system of
equations. This approach results in the first PC-based feedback
control algorithm with proven closed-loop stability, and the first
PC-based feedback control formulation that is applicable to the
design of fixed-order state and output feedback control designs.
The numerical algorithm for the control design is formulated as
optimization over bilinear matrix inequality (BMI) constraints, for
which commercial software is available. The effectiveness of the
approach is demonstrated in two case studies that include a
continuous pharmaceutical manufacturing process. In addition to
model uncertainties, chemical processes must operate within
constraints, such as upper and lower bounds on the magnitude and
rate of change of manipulated and/or output variables. The thesis
also demonstrates an optimal feedback control formulation that
explicitly addresses both constraints and time-invariant
probabilistic parameter uncertainties for linear time-invariant
systems. The H2-optimal feedback controllers designed using the BMI
formulations are incorporated into a fast PC-based model predictive
control (MPC) formulation. A numerical case study demonstrates the
improved constraint satisfaction compared to past polynomial
chaos-based formulations for model predictive
control.
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