Stochastic Model Predictive Control with Discounted Probabilistic Constraints

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 the initial time and ignoring 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 the chance constraint and a quadratic stability condition.Comment: 6 pages, Conference Proceeding

An Improved Constraint-Tightening Approach for Stochastic MPC

The problem of achieving a good trade-off in Stochastic Model Predictive Control between the competing goals of improving the average performance and reducing conservativeness, while still guaranteeing recursive feasibility and low computational complexity, is addressed. We propose a novel, less restrictive scheme which is based on considering stability and recursive feasibility separately. Through an explicit first step constraint we guarantee recursive feasibility. In particular we guarantee the existence of a feasible input trajectory at each time instant, but we only require that the input sequence computed at time $k$ remains feasible at time $k+1$ for most disturbances but not necessarily for all, which suffices for stability. To overcome the computational complexity of probabilistic constraints, we propose an offline constraint-tightening procedure, which can be efficiently solved via a sampling approach to the desired accuracy. The online computational complexity of the resulting Model Predictive Control (MPC) algorithm is similar to that of a nominal MPC with terminal region. A numerical example, which provides a comparison with classical, recursively feasible Stochastic MPC and Robust MPC, shows the efficacy of the proposed approach.Comment: Paper has been submitted to ACC 201

On the convergence of stochastic MPC to terminal modes of operation

The stability of stochastic Model Predictive Control (MPC) subject to additive disturbances is often demonstrated in the literature by constructing Lyapunov-like inequalities that guarantee closed-loop performance bounds and boundedness of the state, but convergence to a terminal control law is typically not shown. In this work we use results on general state space Markov chains to find conditions that guarantee convergence of disturbed nonlinear systems to terminal modes of operation, so that they converge in probability to a priori known terminal linear feedback laws and achieve time-average performance equal to that of the terminal control law. We discuss implications for the convergence of control laws in stochastic MPC formulations, in particular we prove convergence for two formulations of stochastic MPC

Stochastic MPC for additive and multiplicative uncertainty using sample approximations

© 2019 Institute of Electrical and Electronics Engineers Inc.. All rights reserved. We introduce an approach for model predictive control (MPC) of systems with additive and multiplicative stochastic uncertainty subject to chance constraints. Predicted states are bounded within a tube and the chance constraint is considered in a “one step ahead” manner, with robust constraints applied over the remainder of the horizon. The online optimization is formulated as a chance-constrained program that is solved approximately using sampling. We prove that if the optimization is initially feasible, it remains feasible and the closed-loop system is stable. Applying the chance-constraint only one step ahead allows us to state a confidence bound for satisfaction of the chance constraint in closed-loop. Finally, we demonstrate by example that the resulting controller is only mildly more conservative than scenario MPC approaches that have no feasibility guarantee

Stochastic Model Predictive Control for Linear Systems using Probabilistic Reachable Sets

In this paper we propose a stochastic model predictive control (MPC) algorithm for linear discrete-time systems affected by possibly unbounded additive disturbances and subject to probabilistic constraints. Constraints are treated in analogy to robust MPC using a constraint tightening based on the concept of probabilistic reachable sets, which is shown to provide closed-loop fulfillment of chance constraints under a unimodality assumption on the disturbance distribution. A control scheme reverting to a backup solution from a previous time step in case of infeasibility is proposed, for which an asymptotic average performance bound is derived. Two examples illustrate the approach, highlighting closed-loop chance constraint satisfaction and the benefits of the proposed controller in the presence of unmodeled disturbances.Comment: 57th IEEE Conference on Decision and Control, 201