Polynomial Chaos-Based Controller Design for Uncertain Linear Systems with State and Control Constraints

For linear dynamic systems with uncertain parameters, design of controllers which drive a system from an initial condition to a desired final state, limited by state constraints during the transition is a nontrivial problem. This paper presents a methodology to design a state constrained controller, which is robust to time invariant uncertain variables. Polynomial chaos (PC) expansion, a spectral expansion, is used to parameterize the uncertain variables permitting the evolution of the uncertain states to be written as a polynomial function of the uncertain variables. The coefficients of the truncated PC expansion are determined using the Galerkin projection resulting in a set of deterministic equations. A transformation of PC polynomial space to the Bernstein polynomial space permits determination of bounds on the evolving states of interest. Linear programming (LP) is then used on the deterministic set of equations with constraints on the bounds of the states to determine the controller. Numerical examples are used to illustrate the benefit of the proposed technique for the design of a rest-to-rest controller subject to deformation constraints and which are robust to uncertainties in the stiffness coefficient for the benchmark spring-mass-damper system.