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Model Predictive Control for output tracking with prescribed performance

Dario Dennstädt

Abstract

Model Predictive Control (MPC) offers a versatile framework for constraint handling and multi-objective optimisation, yet practical application faces challenges regarding initial and recursive feasibility, robustness against model mismatches, and sampled-data constraints. This thesis develops a novel MPC framework for a class of non-linear continuous-time systems governed by functional differential equations. It targets output tracking within prescribed error bounds while systematically overcoming these challenges. First, we introduce funnel MPC, an algorithm eliminating reliance on terminal conditions or restrictive long prediction horizons. Utilising funnel penalty functions -- state costs penalising tracking error deviations from time-varying boundaries -- this framework ensures feasibility while rigorously enforcing tracking performance guarantees. Next, we unify funnel MPC with model-free funnel feedback into a hybrid architecture. This synergises model-based optimisation with adaptive feedback compensation, achieving prescribed tracking performance despite structural model-plant mismatches, unmodelled dynamics, and unknown disturbances. To further enhance predictive accuracy, we integrate a data-driven learning framework that iteratively refines the model using system measurements, improving long-term performance without compromising robustness. Finally, we formalise the transition to sampled-data implementations. We derive explicit bounds on sampling rates and control effort to guarantee stability under piecewise constant control signals, a critical step for digital hardware deployment. By addressing feasibility, robustness, learning, and sampling, this thesis establishes a cohesive framework for output tracking within prescribed bounds, paving the way for future advances in learning-enhanced and robust MPC.

Model Predictive Control for output tracking with prescribed performance

Abstract

Model Predictive Control (MPC) offers a versatile framework for constraint handling and multi-objective optimisation, yet practical application faces challenges regarding initial and recursive feasibility, robustness against model mismatches, and sampled-data constraints. This thesis develops a novel MPC framework for a class of non-linear continuous-time systems governed by functional differential equations. It targets output tracking within prescribed error bounds while systematically overcoming these challenges. First, we introduce funnel MPC, an algorithm eliminating reliance on terminal conditions or restrictive long prediction horizons. Utilising funnel penalty functions -- state costs penalising tracking error deviations from time-varying boundaries -- this framework ensures feasibility while rigorously enforcing tracking performance guarantees. Next, we unify funnel MPC with model-free funnel feedback into a hybrid architecture. This synergises model-based optimisation with adaptive feedback compensation, achieving prescribed tracking performance despite structural model-plant mismatches, unmodelled dynamics, and unknown disturbances. To further enhance predictive accuracy, we integrate a data-driven learning framework that iteratively refines the model using system measurements, improving long-term performance without compromising robustness. Finally, we formalise the transition to sampled-data implementations. We derive explicit bounds on sampling rates and control effort to guarantee stability under piecewise constant control signals, a critical step for digital hardware deployment. By addressing feasibility, robustness, learning, and sampling, this thesis establishes a cohesive framework for output tracking within prescribed bounds, paving the way for future advances in learning-enhanced and robust MPC.
Paper Structure (37 sections, 40 theorems, 384 equations, 23 figures)

This paper contains 37 sections, 40 theorems, 384 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Control objective
  4. Funnel control
  5. Model predictive control
  6. Difficulties and Drawbacks
  7. Structure of this thesis and previously published results
  8. Funnel Model Predictive Control
  9. Funnel stage cost functions
  10. Model class
  11. MPC with funnel stage costs
  12. The higher relative degree
  13. Feasible control signals
  14. Optimal control problem
  15. The funnel MPC algorithm
  16. ...and 22 more sections

Key Result

Proposition 1.1.2

Let $f:\mathds{R}^m\to\mathds{R}^m$ be a locally Lipschitz continuous function, $\psi\in\mathcal{G}$, $y_{\mathop{\mathrm{ref}}\limits}\in W^{1,\infty}(\mathds{R}_{\geq0},\mathds{R}^{m})$, and $y^0\in\mathds{R}^m$ with $\left\| y^0-y_{\mathop{\mathrm{ref}}\limits}(0) \right\|_{ }<\psi(0)$. Then, the to the system leads to the closed-loop initial value problem which has a solution. Moreover, ever

Figures (23)

  • Figure 1.1: Error evolution in a funnel $\mathcal{F}_{\psi}$ with boundary $\psi(t)$. The figure is based on BergLe18a, adapted to the current setting.
  • Figure 2.1: Simulation of system \ref{['eq:ExampleExothermicReaction']} under the control generated by the funnel MPC \ref{['Algo:FunnelMPC']} and the MPC \ref{['Algo:MPC']} with stage cost \ref{['eq:Ex:QuadraticStageCost']} with parameters $T= 10^{-2}$, $\delta=5\cdot 10^{-4}$, and $\lambda_u=0.1$.
  • Figure 2.2: Simulation of system \ref{['eq:ExampleExothermicReaction']} under the control generated by the funnel MPC \ref{['Algo:FunnelMPC']} and the MPC \ref{['Algo:MPC']} with stage cost \ref{['eq:Ex:QuadraticStageCost']} with parameters $T= 1$, $\delta=0.1$, and $\lambda_u=10^{-4}$.
  • Figure 2.3: Simulation of system \ref{['eq:ExampleExothermicReaction']} under the control generated by the funnel MPC \ref{['Algo:FunnelMPC']} with parameters $T= 1$, $\delta=0.1$, and $\lambda_u=10^{-4}$ and the funnel control law \ref{['eq:Ex:ExothermicReaction:FC']} with a constant step size $h=10^{-3}$.
  • Figure 2.4: Simulation of system \ref{['eq:ExampleExothermicReaction']} under the control generated by the funnel MPC \ref{['Algo:FunnelMPC']} with parameters $T= 1$, $\delta=0.1$, and $\lambda_u=10^{-4}$ and the funnel control law \ref{['eq:Ex:ExothermicReaction:FC']} with a constant step size $h=10^{-4}$.
  • ...and 18 more figures

Theorems & Definitions (140)

  • Remark 1.1.1
  • Proposition 1.1.2
  • Remark 1.1.3
  • Definition 1.1.5: Regulated function
  • Lemma 2.1.1
  • proof
  • Remark 2.1.2
  • Proposition 2.1.3
  • proof
  • Remark 2.1.4
  • ...and 130 more