A Discrete-Time Optimization-Based Sliding Mode Control Law for Linear Systems with Input and State Constraints

Abstract

This paper proposes a discrete-time sliding mode control (DSMC) strategy for linear (possibly multi-input) systems with additive bounded disturbances, which guarantees the satisfaction of input and state constraints. The control law is generated by solving a finite-horizon optimal control problem at each sampling instant, aimed at obtaining a control variable that is as close as possible to a reference DSMC law, but at the same time enforces constraint satisfaction for all admissible disturbance values. Contrary to previously-proposed control approaches merging DSMC and model predictive control, our proposal guarantees the satisfaction of all standard properties of DSMC, and in particular the finite-time convergence of the state into a pseudo sliding mode band.

Quasi-SMC based on MPC for a constrained continuous-time nonlinear system with external disturbances

“…Then the proof is divided into three parts. Part 1: To show the feasible state x(t k+1 + T |t k+1 ) ∈ Ω ε under control law (22).…”

“…Part 3: To show the state x(τ |t k+1 ) under the control law (22) satisfies the state constraint (13).…”

“…In [19], [20], considering an unconstrained linear system, MPC controller generates the control law based on SMC reaching law to improve the performance of SMC. The linear system with constraints is further considered in [21], [22], which guarantee the convergence of state to the sliding mode band. In [23], considering the unconstrained nonlinear system, the linear approximation is first used and then applies the same approach in [19].…”

Constrained Nonlinear Discrete-Time Sliding Mode Control Based on a Receding Horizon Approach

“…As an alternative, in [10], the DSMC reaching law of [3] generates a reference for an MPC controller, for the case of unconstrained single-input uncertain linear systems. Also, [11] proposes an MPC law for single-input perturbed linear systems, which guarantees asymptotic convergence of the state to a boundary layer of S. Moreover, [12] presents single-input MPC laws for unperturbed linear and nonlinear systems, where S is used to define the terminal constraint of the MPC problem, while [13] proposes a DSMC law for linear multi-input systems based on the solution of a robust linear MPC problem: the resulting control law guarantees finite-time convergence of the state onto S (in case of vanishing disturbance) or into an apriori determined boundary layer of it (in case of persistent disturbance). DSMC for setpoint tracking in constrained linear systems was studied in [14] and [15]: [14] relies on a dualmode receding horizon DSMC law which exploits the flatness property of suitably defined sliding hyperplanes, while [15] proposes a second-order DSMC scheme to add virtual reference variables to the receding horizon law.…”

“…The approach presented in [13] was based on the linear MPC approach of [21], that can be applied for linear system dynamics and linear inequality constraints. Instead, given the need to satisfy arbitrary nonlinear inequality constraints for nonlinear dynamical systems in the presence of disturbances, in this work constraint satisfaction is achieved based on a tightening approach, relying on Lipschitz continuity.…”