Tube‐enhanced multi‐stage model predictive control for flexible robust control of constrained linear systems with additive and parametric uncertainties

Abstract

The trade‐off between optimality and complexity has been one of the most important challenges in the field of robust model predictive control (MPC). To address the challenge, we propose a flexible robust MPC scheme by synergizing the multi‐stage and tube‐based MPC approaches. The key idea is to exploit the nonconservatism of the multi‐stage MPC and the simplicity of the tube‐based MPC. The proposed scheme provides two options for the user to determine the trade‐off depending on the application: the choice of the robust horizon and the classification of the uncertainties. Beyond the robust horizon, the branching of the scenario‐tree employed in multi‐stage MPC is avoided with the help of tubes. The growth of the problem size with respect to the number of uncertainties is reduced by handling small uncertainties via an invariant tube that can be computed offline. This results in linear growth of the problem size beyond the robust horizon and no growth of the problem size concerning small magnitude uncertainties. The proposed approach helps to achieve a desired trade‐off between optimality and complexity compared to existing robust MPC approaches. We show that the proposed approach is robustly asymptotically stable. Its advantages are demonstrated for a CSTR example.

Model predictive control for linear uncertain systems using integral quadratic constraints

“…k and from right with its transpose for d k = 0, r k = 0. Thus, Finsler's Lemma [29] guarantees the existence of γ > 0 large enough such that (17) holds with Λ = γI, (also for nonzero r k , d k ). Therefore, we know that any Γ γI and γΞ γI satisfy (17) with Λ = diag(Γ, γΞ) γI as well.…”

“…Proof. First, we derive ( 17) from ( 13) by using Finsler's Lemma; second, we use (17) to show that a dissipation inequality holds; and third, summing up this inequality from 0 to t yields (18) and (19).…”

“…, τ max the LMI M M τ independent of ρ. We choose ρ = 0.95 and K = [0.18 − 0.35] and observe that the resulting semi-definite program consisting of the LMI constraints (17), M M τ , Γ 0, γ 0, and X 0, as well as the decision variables P, M, X, Γ, γ is feasible with , γ 1 ≈ 0.0036, γ 2 = 1 satisfy the requirements of Assumption 2. We implement the MPC scheme using CasADi [32] and the solver IPOPT to solve the MPC optimization problem.…”

“…This idea is also used in [16] to develop an MPC scheme for systems subject to parametric uncertainties, which is in [7] extended to a mix of parametric uncertainties and bounded additive disturbances. Recently, in [17] the tubebased approach to parametric uncertainties is combined with a less conservative multi-stage MPC allowing the user to trade off between complexity and conservatism of the MPC scheme. In reality, however, uncertainties might not be parametric but are often more complex and dynamic.…”

“…Further, let P be decomposed into 21 P 22 with P 11 ∈ R n x ×n x and define P 121 = P ⊤ 21 P −1 22 P 21 and P = P 11 − P 121 ≻ 0. Then, there exist Γ ≻ 0 and γ > 0 satisfying(17) and the following inequality holds for any sequences d ∈ ℓ…”