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Deadbeat Robust Model Predictive Control: Robustness without Computing Robust Invariant Sets

G. Schildbach

TL;DR

DRMPC addresses robust control of discrete-time linear systems with additive disturbances by extinguishing disturbance effects within a short deadbeat horizon $M$ through a disturbance-feedback policy based on disturbance-vertex barycentric coordinates. Unlike tube-based RMPC, DRMPC avoids computing robust positively invariant sets, while maintaining recursive feasibility and input-to-state stability for both Online and Offline variants. A convex optimization framework handles tightened input/state constraints derived from deadbeat sequences, with Offline DRMPC further reducing online complexity by precomputing the deadbeat inputs. The approach yields competitive performance in a small numerical example and generalizes readily to linear time-varying and LPV contexts, making it appealing for large-scale or uncertain systems where RPIs are intractable.

Abstract

Deadbeat Robust Model Predictive Control (DRMPC) is introduced as a new approach of Robust Model Predictive Control (RMPC) for linear systems with additive disturbances. Its main idea is to completely extinguish the effect of the disturbances in the predictions within a small number of time steps, called the deadbeat horizon. To this end, explicit deadbeat input sequences are calculated for the vertices of the disturbance set. They generalize to a nonlinear disturbance feedback policy for all disturbances by means of a barycentric function. Similar to previous approaches, this disturbance feedback policy can be either part of the online optimization (Online DRMPC) or pre-calculated during the design phase of the controller (Offline DRMPC). The main advantage over all other RMPC approaches is that no Robust Positive Invariant (RPI) set has to be calculated, which is often intractable for systems with higher dimensions. Nonetheless, for Online DRMPC and Offline DRMPC recursive feasibility and input-to-state stability can be guaranteed. A small numerical example compares the two versions of DRMPC and demonstrates that the performance of DRMPC is competitive with other state-of-the-art RMPC approaches. Its main advantage is its easy extension to linear time-varying (LTV) and linear parameter-varying (LPV) systems.

Deadbeat Robust Model Predictive Control: Robustness without Computing Robust Invariant Sets

TL;DR

DRMPC addresses robust control of discrete-time linear systems with additive disturbances by extinguishing disturbance effects within a short deadbeat horizon through a disturbance-feedback policy based on disturbance-vertex barycentric coordinates. Unlike tube-based RMPC, DRMPC avoids computing robust positively invariant sets, while maintaining recursive feasibility and input-to-state stability for both Online and Offline variants. A convex optimization framework handles tightened input/state constraints derived from deadbeat sequences, with Offline DRMPC further reducing online complexity by precomputing the deadbeat inputs. The approach yields competitive performance in a small numerical example and generalizes readily to linear time-varying and LPV contexts, making it appealing for large-scale or uncertain systems where RPIs are intractable.

Abstract

Deadbeat Robust Model Predictive Control (DRMPC) is introduced as a new approach of Robust Model Predictive Control (RMPC) for linear systems with additive disturbances. Its main idea is to completely extinguish the effect of the disturbances in the predictions within a small number of time steps, called the deadbeat horizon. To this end, explicit deadbeat input sequences are calculated for the vertices of the disturbance set. They generalize to a nonlinear disturbance feedback policy for all disturbances by means of a barycentric function. Similar to previous approaches, this disturbance feedback policy can be either part of the online optimization (Online DRMPC) or pre-calculated during the design phase of the controller (Offline DRMPC). The main advantage over all other RMPC approaches is that no Robust Positive Invariant (RPI) set has to be calculated, which is often intractable for systems with higher dimensions. Nonetheless, for Online DRMPC and Offline DRMPC recursive feasibility and input-to-state stability can be guaranteed. A small numerical example compares the two versions of DRMPC and demonstrates that the performance of DRMPC is competitive with other state-of-the-art RMPC approaches. Its main advantage is its easy extension to linear time-varying (LTV) and linear parameter-varying (LPV) systems.
Paper Structure (18 sections, 4 theorems, 48 equations, 3 figures, 3 tables)

This paper contains 18 sections, 4 theorems, 48 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background and Terminology
  3. Basic Terminology and Notation
  4. Control System
  5. Deadbeat Input Sequence
  6. Vertex Representations
  7. Deadbeat Robust MPC
  8. Constraint Tightening
  9. DRMPC Optimization Problem
  10. Computational Complexity
  11. Offline DRMPC
  12. System-theoretic Guarantees
  13. Recursive Feasibility
  14. Input-to-State Stability
  15. Numerical Example
  16. ...and 3 more sections

Key Result

Theorem 8

If the DRMPC problem Equ:DRMPCProblem is feasible at $k=0$ for $\bar{x}_{0}$, it remains feasible for all future states $x_{1},x_{2},\hdots$ of system under the DRMPC regime.

Figures (3)

  • Figure 1: RoA for $t_{\mathrm{S}}=0.1$ and $N=10$ (dark gray), $N=20$ (medium gray), $N=40$ (light gray). The deadbeat horizon is $M=3$ and the terminal set is the origin in both cases.
  • Figure 2: RoA of Offline DRMPC for $t_{\mathrm{S}}=0.1$ and $N=10$ (dark gray), $N=20$ (medium gray), $N=40$ (light gray).
  • Figure 3: RoA of Tube-RMPC and RMPC-ADF for $t_{\mathrm{S}}=0.1$ and $N=10$ (dark gray), $N=20$ (medium gray), $N=40$ (light gray).

Theorems & Definitions (12)

  • Definition 3: deadbeat input sequence
  • Remark 4
  • Remark 5: terminal condition
  • Remark 6: offline pre-computations
  • Remark 7: alternative terminal conditions
  • Theorem 8: recursive feasibility
  • Definition 9: robust positive invariant set
  • Corollary 10: invariance of the feasible set
  • Definition 12: input-to-state stability
  • Definition 13: ISS Lyapunov function
  • ...and 2 more