Stochastic Model Predictive Control with Discounted Probabilistic Constraints

S. Yan, P. J. Goulart and M. Cannon

@inproceedings{YGC:2018,
  author = {S. Yan and P. J. Goulart and M. Cannon},
  title = {Stochastic Model Predictive Control with Discounted Probabilistic Constraints},
  booktitle = {European Control Conference},
  year = {2018},
  pages = {1003-1008}
}

This paper considers linear discrete time systems with additive disturbances, and designs a Model Predictive Control (MPC) law to minimise a quadratic cost function subject to a chance constraint. The chance constraint is defined as a discounted sum of violation probabilities on an infinite horizon. By penalising violation probabilities close to initial time and heavily discounting violation probabilities in the far future, this form of constraint enables the feasibility of the online optimisation to be guaranteed without an assumption of boundedness of the disturbance. A computationally convenient MPC optimisation problem is formulated using Chebyshev's inequality and we introduce an online constraint-tightening technique to ensure recursive feasibility based on knowledge of a suboptimal solution. The closed loop system is guaranteed to satisfy both the chance constraint and a quadratic stability condition.