Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convex NMPC reformulations for a special class of nonlinear multi-input systems with application to rank-one bilinear networks

Manuel Klädtke, Moritz Schulze Darup

TL;DR

This work addresses nonconvex NMPC for a restricted class of discrete-time, multi-input systems by reformulating the problem into a finite collection of convex subproblems. By leveraging a diagonal, input-affine structure and a state-space partition, the nonlinear dynamics are captured via a linearized augmented model with an artificial input, enabling exact global solutions through scenario-based subproblem enumeration. The method extends prior single-input results to multi-input systems and encompasses rank-one bilinear networks, as demonstrated in two numerical examples that show feasibility pruning greatly reduces the online burden. While promising, the approach remains restricted by assumptions on $\boldsymbol{G}(\boldsymbol{x})$, partition construction, and terminal ingredients, motivating future work to broaden applicability and address non-diagonal $\boldsymbol{G}(\boldsymbol{x})$ and alternative stability certificates.

Abstract

We show that a special class of (nonconvex) NMPC problems admits an exact solution by reformulating them as a finite number of convex subproblems, extending previous results to the multi-input case. Our approach is applicable to a special class of input-affine discrete-time systems, which includes a class of bilinear rank-one systems that is considered useful in modeling certain controlled networks. We illustrate our results with two numerical examples, including the aforementioned rank-one bilinear network.

Convex NMPC reformulations for a special class of nonlinear multi-input systems with application to rank-one bilinear networks

TL;DR

This work addresses nonconvex NMPC for a restricted class of discrete-time, multi-input systems by reformulating the problem into a finite collection of convex subproblems. By leveraging a diagonal, input-affine structure and a state-space partition, the nonlinear dynamics are captured via a linearized augmented model with an artificial input, enabling exact global solutions through scenario-based subproblem enumeration. The method extends prior single-input results to multi-input systems and encompasses rank-one bilinear networks, as demonstrated in two numerical examples that show feasibility pruning greatly reduces the online burden. While promising, the approach remains restricted by assumptions on , partition construction, and terminal ingredients, motivating future work to broaden applicability and address non-diagonal and alternative stability certificates.

Abstract

We show that a special class of (nonconvex) NMPC problems admits an exact solution by reformulating them as a finite number of convex subproblems, extending previous results to the multi-input case. Our approach is applicable to a special class of input-affine discrete-time systems, which includes a class of bilinear rank-one systems that is considered useful in modeling certain controlled networks. We illustrate our results with two numerical examples, including the aforementioned rank-one bilinear network.
Paper Structure (11 sections, 2 theorems, 51 equations, 4 figures)

This paper contains 11 sections, 2 theorems, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basics on NMPC, system specification, and system transformation
  3. Nonlinear model predictive control
  4. System specification
  5. System transformation
  6. Convex NMPC reformulations
  7. Constraint transformation
  8. Choosing the stage cost and terminal ingredients
  9. Scenario-based evaluation
  10. Numerical Examples
  11. Summary and Outlook

Key Result

Lemma 1

Let Assumption assum:characterization hold. Then, every subset $\mathcal{Z}_j$ as in eq:Z_without_eq is convex.

Figures (4)

  • Figure 1: A tree visualizing the $s^N$ different constraint scenarios, where in each time step $k$ one of the constraints $\hat{\boldsymbol{x}}(k)^\top\hat{\boldsymbol{v}}(k)^\top^\top\in\mathcal{Z}_j$ must hold.
  • Figure 2: Illustration of the (partially overlapping) feasible sets $\mathcal{F}_\mu$ (in blue) and the terminal set $\mathcal{T}$ (in cyan) for Example \ref{['exmp:multi-input']}. The boundary of the sets $\mathcal{X}_j$ is shown in orange.
  • Figure 3: Illustration of the optimal control law $\boldsymbol{u}^\ast(\boldsymbol{x})$ (top) and the optimal value function $V^\ast(\boldsymbol{x})$ (bottom) for Example \ref{['exmp:multi-input']}.
  • Figure 4: Closed-loop simulation of the bilinear network considered in Example \ref{['exmp:bilinear']} showing the system trajectory in original (top) and shifted (middle) coordinates as well as the applied input (bottom).

Theorems & Definitions (7)

  • Definition 1
  • Lemma 1
  • Proof 1
  • Lemma 2
  • Proof 2
  • Example 1
  • Example 2