Stochastic MPC Framework for Controlling the Average Constraint Violation

Abstract

This paper considers linear discrete-time systems with additive, bounded, disturbances subject to hard control input bounds and a stochastic constraint on the amount of state-constraint violation averaged over time. The amount of violations is quantified by a loss function and the averaging can be weighted, corresponding to exponential forgetting of past violations. The freedom in the choice of the loss function makes this formulation highly flexible -for instance, probabilistic constraints or integrated chance constraints can be enforced by an appropriate choice of the loss function. For the type of constraint considered, we develop a recursively feasible receding horizon control scheme exploiting the averaged-over-time nature by explicitly taking into account the amount of past constraint violations when determining the current control input. This leads to a significant reduction in conservatism. As a simple extension of the proposed approach we show how time-varying state-constraints can be handled within our framework. The computational complexity (online as well as offline) is comparable to existing model predictive control schemes. The effectiveness of the proposed methodology is demonstrated by means of a numerical example.

Stochastic Model Predictive Control with Discounted Probabilistic Constraints

“…In this respect the approach is similar to constraint-tightening methods that have previously been applied in the context of stochastic MPC (e.g. [7], [8], [9]) in order to ensure recursive feasibility and constraint satisfaction in closed loop operation. However, each of these methods requires that the disturbances affecting the controlled system are bounded, and they become more conservative as the degree of conservativeness of the assumed disturbance bounds increases.…”

“…Although capable of handling chance constraints, existing stochastic MPC algorithms that ensure constraint satisfaction in closed loop operation typically rely on knowledge of worst case disturbance bounds to obtain such guarantees [6]. For the algorithms proposed in [7], [8], [9] for example, which simultaneously ensure closed loop constraint satisfaction and recursive feasibility of the online MPC optimisation, the degree of conservativeness increases as the disturbance bounds become more conservative. This paper ensures both closed loop satisfaction of chance constraints and recursive feasibility but does not rely on disturbance bounds, instead requiring knowledge of only the first and second moments of the disturbance input.…”

An approach to output-feedback MPC of stochastic linear discrete-time systems

“…In this simplified framework, the following S-MPCc problem can be stated. where J is defined in (16), (17), (18), subject to -the dynamics (6) , with K t =K, and…”

“…The chance-constraints are P{x 2 ≥ 2} ≤ 0.1, P{u ≥ 1} ≤ 0.1, and P{−u ≥ 1} ≤ 0.1. In (16), (17), and (20) we set Q L = Q = I 2 , R = 0.01, and N = 9. In Figure 2 we compare the feasible sets obtained with the different methods presented in Section 4, with different assumptions concerning the noise (namely S-MPCc (1), S-MPCc (2), S-MPCl (1), S-MPCl (2), where (1) denotes the case of Gaussian distribution and (2) denotes the case when the distribution is unknown).…”

Single Neuron Stochastic Predictive PID Control Algorithm for Nonlinear and Non-Gaussian Systems Using the Survival Information Potential Criterion

“…(1) The randomized, or scenario-based approach [20][21][22]: It is a very general methodology that can consider linear or nonlinear systems affected by noise with general distributions characterized by possibly unbounded and nonconvex support. (2) The probabilistic approximation approach [23][24][25][26][27][28]: It is based on the point-wise reformulation of probabilistic or expectation constraints in deterministic terms to be included in the MPC formulation. Reference [29] gave an overview of the main developments in the area of stochastic model predictive control (SMPC) in the past decade.…”