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Region-free explicit model predictive control for linear systems on Hilbert spaces

Mikael Kurula, Jukka-Pekka Humaloja, Stevan Dubljevic

TL;DR

This work extends discrete-time explicit model predictive control to linear distributed-parameter (infinite-dimensional) systems by formulating MPC on Hilbert spaces, recasting the online optimization as a finite-dimensional parametric quadratic program (pQP). A Hilbert-space KKT analysis yields an affine dependence of the optimal control on the current state and prior input, enabling a region-free explicit MPC via the dual active-set method QPKWIK. The approach is demonstrated on a port-Hamiltonian Timoshenko beam, showing feasible constraint handling and fast on-line computation despite model mismatch. The results indicate practical viability for explicit MPC in distributed-parameter contexts and provide a bridge between infinite-dimensional control theory and engineering practice.

Abstract

We extend discrete-time explicit model predictive control (MPC) rigorously to linear distributed parameter systems. After formulating an MPC framework and giving a relevant KKT theorem, we realize fast regionless explicit MPC by using the dual active set method QPKWIK. A Timoshenko beam with input and state constraints is used to demonstrate the efficacy of the design at controlling a continuous-time hyperbolic PDE with constraints, using a discrete-time explicit MPC controller.

Region-free explicit model predictive control for linear systems on Hilbert spaces

TL;DR

This work extends discrete-time explicit model predictive control to linear distributed-parameter (infinite-dimensional) systems by formulating MPC on Hilbert spaces, recasting the online optimization as a finite-dimensional parametric quadratic program (pQP). A Hilbert-space KKT analysis yields an affine dependence of the optimal control on the current state and prior input, enabling a region-free explicit MPC via the dual active-set method QPKWIK. The approach is demonstrated on a port-Hamiltonian Timoshenko beam, showing feasible constraint handling and fast on-line computation despite model mismatch. The results indicate practical viability for explicit MPC in distributed-parameter contexts and provide a bridge between infinite-dimensional control theory and engineering practice.

Abstract

We extend discrete-time explicit model predictive control (MPC) rigorously to linear distributed parameter systems. After formulating an MPC framework and giving a relevant KKT theorem, we realize fast regionless explicit MPC by using the dual active set method QPKWIK. A Timoshenko beam with input and state constraints is used to demonstrate the efficacy of the design at controlling a continuous-time hyperbolic PDE with constraints, using a discrete-time explicit MPC controller.
Paper Structure (9 sections, 4 theorems, 44 equations, 3 figures, 1 table)

This paper contains 9 sections, 4 theorems, 44 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The discrete-time MPC formulation
  3. Writing the optimization step as a parametric quadratic program (pQP)
  4. Explicit solution of the pQP
  5. Example: Explicit MPC of a Timoshenko beam
  6. Port-Hamiltonian formulation of Timoshenko beam
  7. Cost functional and conversion to discrete time
  8. Numerical simulation
  9. Acknowledgments

Key Result

Lemma 2

The operator $\widetilde{V}$ is positive semidefinite. If $V_k$ are coercive for all (possibly apart from one) $k \in \{0,1,\ldots,N\}$, then $\widetilde{V}$ is coercive.

Figures (3)

  • Figure 1: Controls and the optimal costs (logarithmic scale on the right).
  • Figure 2: Means of the plant state components and the state constraints.
  • Figure 3: Profile of the plant state component $x_3 = \phi'$.

Theorems & Definitions (8)

  • Remark 1
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof : Proof of Theorem \ref{['thm:affine']}