Finite‐time convergence results in robust model predictive control

Abstract

Robust asymptotic stability (asymptotic attractivity and -stability) of equilibrium regions under robust model predictive control (MPC) strategies was extensively studied in the last decades making use of Lyapunov theory in most cases. However, in spite of its potential application benefits, the problem of finite-time convergence under fixed prediction horizon has not received, with some few exceptions, much attention in the literature. Considering the importance in several applications of having finite-time convergence results in the context of fixed horizon MPC controllers and the lack of studies on this matter, this work presents a new set-based robust MPC (RMPC) for which, in addition to traditional stability guarantees, finite-time convergence to a target set is proved, and moreover, an upper bound on the time necessary to reach that set is provided. It is remarkable that the results apply to general nonlinear systems and only require some weak assumptions on the model, cost function, and target set.

I-Steps Closed-Loop Sets for Constrained Linear Systems Under Model Predictive Control

“…The next definitions and properties will be referred to system (1), and the corresponding state and input constraints. Definition 1.…”

“…Although the results on stability of MPC are mainly devoted to prove asymptotic (or exponential) stability, there are a few results on finite-time convergence for a fixed control horizon [1], [6], [7]. In particular, [1] proposes a set-based MPCin which the target/objective is given by an arbitrary small invariant set and its corresponding input set -that ensures finite-time convergence and also provides an upper bound for the time of convergence.…”

“…Based on the upper bound provided by [1], the state space can be partitioned into nested regions, which are called the isteps closed-loop set; i.e., a particular regions from where the closed-loop system reaches the target set in i or less steps. The similarity with the concept of the (open-loop) i-steps controllable sets from the classic literature ( [5], [8]) is evident.…”

Discrete-time MPC for switched systems with applications to biomedical problems

“…The main features of MPC -which make it one of the most employed advanced control technique -are the explicit consideration of a model for prediction, the optimal computation of the control actions, and its ability to handle, easily and effectively, hard constraints on control and states [13,12]. MPC theoretical background has been widely investigated in the last decades, showing how this technique is capable to provide stability, robustness, constraint satisfaction and tractable computation for linear and nonlinear systems [14,15]. Set invariance theory, which is closely related to Lyapunov stability theory [16], has also shown to be a powerful tool for analyzing dynamical systems subject to constraints.…”

A Linear Fractional Transformation Based Approach to Robust Model Predictive Control Design in Uncertain Systems

“…Furthermore, the RMPC has been investigated in the switched linear systems [20], linear systems with disturbances [21], positive systems [22], linear parameter-varying systems [23][24][25], output tracking issues [26], and Markovian jump systems [27,28]. Also, the RMPC is developed using two-stages neural network modeling [29], considering state-dependent uncertainties [30], under partial actuator faults [31], guaranteeing stability and satisfying constraints [32], assuming saturated inputs and randomly occurring uncertainties [33], involving finite-time convergence result [34], and employing collective neuro-dynamic optimization [35]. Although the RMPC synthesis is primarily discussed in the discrete-time system, it is extended to continuous-time representations [36,37].…”