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

2013; Taylor & Francis; Volume: 86; Issue: 8 Linguagem: Inglês

10.1080/00207179.2013.801082

1366-5820

Kwang-Ki K. Kim, Richard D. Braatz,

Control Systems and Identification

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 controlgeneralised polynomial chaos expansionschance constraintsBoole's inequalityconvex relaxationsemidefinite programmingstochastic 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 Boyd, S. and Vandenberghe, L. 2004. Convex optimization, Cambridge, , UK: Cambridge Univ. Press. [Crossref] , [Google Scholar]) 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 Grant, M., Boyd, S. and Ye, Y. 2009. cvx usersÕ guide Technical report, Technical Report Build 711 Available at http://citeseerx.ist.psu.edu/viewdoc/download [Google Scholar]) 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 Yakubovich, V. A. 1971. S-procedure in nonlinear control theory. Vestnik Leningrad University, 2: 62–77. English translation in Vestnik Leningrad Univ. 4:73–93, 1977 [Google Scholar]). Our case is a special case in which all the quadratic forms are convex, whereas Megretski and Treil (1993 Megretski, A. and Treil, S. 1993. Power distribution inequalities in optimization and robustness of uncertain systems. Journal of Mathematical Systems, Estimation and Control, 3: 301–319. [Google Scholar]) 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.