Generalised polynomial chaos expansion approaches to approximate stochastic model predictive control^†

Abstract

This paper considers the model predictive control of dynamic systems subject to stochastic uncertainties due to parametric uncertainties and exogenous disturbance. The effects of uncertainties are quantified using generalised polynomial chaos expansions with an additive Gaussian random process as the exogenous disturbance. With Gaussian approximation of the resulting solution trajectory of a stochastic differential equation using generalised polynomial chaos expansion, convex finite-horizon model predictive control problems are solved that are amenable to online computation of a stochastically robust control policy over the time horizon. Using generalised polynomial chaos expansions combined with convex relaxation methods, the probabilistic constraints are replaced by convex deterministic constraints that approximate the probabilistic violations. This approach to chance-constrained model predictive control provides an explicit way to handle a stochastic system model in the presence of both model uncertainty and exogenous disturbances.

Keywords:

• stochastic model predictive control
• generalised polynomial chaos expansions
• chance constraints
• Boole’s inequality
• convex relaxation
• semidefinite programming
• stochastic differential equations

Notes

1. and β can be time-varying, where the forbidden region might correspond to moving objectives and time-varying β can be used to assign different risk of collision in different time sequences in the predicted motions.

2. Consider the time interval [0, T] in which X _t is a second-moment process.

3. The computation of deterministic constant matrices K _(·) and (or ) can be performed off-line.

4. In particular, the computational complexity using a standard interior-point method (Boyd & Vandenberghe, 2004) is at most O(ℓM ⁴log M) where M = npT (n is the dimension of the state variables, p is the number of basis functions for a gPC expansion, T is the length of prediction horizon), and ℓ denotes the number of probabilistic polyhedral constraints. The average computation time at each sampling instance was ≈0.36 CPU seconds for n = 2, p = 3, and T = 5. This computation time includes the computation of the optimisation data, i.e., the time for computing matrices associated with the objective function and constraints, as well as the time for solving the resulting constrained optimisation. Optimisation is performed by the CVX toolbox (Grant, Boyd, & Ye, 2009) on a MacBook Pro laptop (2.53 GHz Intel Core 2 Duo, 4GB DDR3).

5. To be a convex program, ‖β _t ‖₂ cannot be used, since it results in a concave function term in the objective function in a minimisation problem.

6. By efficiency, it is meant that there is a numerical algorithm whose convergence is guaranteed and that provides an infeasibility certificate. Convex programs are such cases.

7. They applied a method of relaxation called the S-procedure (Yakubovich, 1971). Our case is a special case in which all the quadratic forms are convex, whereas Megretski and Treil (1993) considered more general cases where some of quadratic forms might be nonconvex. They proposed a sufficient condition for the set of quadratic form constraints to be lossless, i.e., the resultant relaxation obtained from the S-procedure gives an exact optimum.