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Funnel MPC for nonlinear systems with arbitrary relative degree

Thomas Berger, Dario Dennstädt

TL;DR

This work is able to prove initial and recursive feasibility of the novel Funnel MPC scheme for systems with arbitrary relative degree - without requiring any terminal conditions, a sufficiently long prediction horizon or additional output constraints.

Abstract

The Model Predictive Control (MPC) scheme Funnel MPC enables output tracking of smooth reference signals with prescribed error bounds for nonlinear multi-input multi-output systems with stable internal dynamics. Earlier works achieved the control objective for system with relative degree restricted to one or incorporated additional feasibility constraints in the optimal control problem. Here we resolve these limitations by introducing a modified stage cost function relying on a weighted sum of the tracking error derivatives. The weights need to be sufficiently large and we state explicit lower bounds. Under these assumptions we are able to prove initial and recursive feasibility of the novel Funnel MPC scheme for systems with arbitrary relative degree - without requiring any terminal conditions, a sufficiently long prediction horizon or additional output constraints.

Funnel MPC for nonlinear systems with arbitrary relative degree

TL;DR

This work is able to prove initial and recursive feasibility of the novel Funnel MPC scheme for systems with arbitrary relative degree - without requiring any terminal conditions, a sufficiently long prediction horizon or additional output constraints.

Abstract

The Model Predictive Control (MPC) scheme Funnel MPC enables output tracking of smooth reference signals with prescribed error bounds for nonlinear multi-input multi-output systems with stable internal dynamics. Earlier works achieved the control objective for system with relative degree restricted to one or incorporated additional feasibility constraints in the optimal control problem. Here we resolve these limitations by introducing a modified stage cost function relying on a weighted sum of the tracking error derivatives. The weights need to be sufficiently large and we state explicit lower bounds. Under these assumptions we are able to prove initial and recursive feasibility of the novel Funnel MPC scheme for systems with arbitrary relative degree - without requiring any terminal conditions, a sufficiently long prediction horizon or additional output constraints.
Paper Structure (9 sections, 6 theorems, 57 equations, 4 figures)

This paper contains 9 sections, 6 theorems, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Nomenclature
  3. System class
  4. Control objective
  5. Funnel MPC
  6. Proof of the main result
  7. Proof of \ref{['Th:FunnelMPC']}
  8. Simulations
  9. Conclusion

Key Result

theorem 1

Consider system eq:Sys with $(f,g,\mathbf{T})\in\mathcal{N}^{m,r}$, initial data $y^0 \in \mathcal{C}^{r-1}([-\sigma,0],\mathds{R}^m)$ and let $\tau\ge 0$ be the memory limit of the operator $\mathbf{T}$. Let $y_{\mathop{\mathrm{ref}}\limits}\in W^{r,\infty}(\mathds{R}_{\geq0},\mathds{R}^{m})$ and Furthermore, choose parameters $k_1,\ldots,k_{r-1}$ such that for all $i=2,\ldots,r-1$ we have The

Figures (4)

  • Figure 1: Error evolution in a funnel $\mathcal{F}_{\psi}$ with boundary $\psi(t)$.
  • Figure 2: Mass-on-car system.
  • Figure 3: Simulation of system \ref{['eq:ExampleMassOnCarSystem']} under funnel MPC \ref{['Algo:FMPC']} and funnel MPC from BergDenn22BergDenn21
  • Figure :

Theorems & Definitions (15)

  • definition 1
  • definition 2: System class
  • remark 1
  • remark 2
  • theorem 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • ...and 5 more