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Dual MPC for quasi-Linear Parameter Varying systems

Sampath Kumar Mulagaleti, Alberto Bemporad

Abstract

We present a dual Model Predictive Control (MPC) framework for the simultaneous identification and control of quasi-Linear Parameter Varying (qLPV) systems. The framework is composed of an online estimator for the states and parameters of the qLPV system, and a controller that leverages the estimated model to compute inputs with a dual purpose: tracking a reference output while actively exciting the system to enhance parameter estimation. The core of this approach is a robust tube-based MPC scheme that exploits recent developments in polytopic geometry to guarantee recursive feasibility and stability in spite of model uncertainty. The effectiveness of the framework in achieving improved tracking performance while identifying a model of the system is demonstrated through a numerical example.

Dual MPC for quasi-Linear Parameter Varying systems

Abstract

We present a dual Model Predictive Control (MPC) framework for the simultaneous identification and control of quasi-Linear Parameter Varying (qLPV) systems. The framework is composed of an online estimator for the states and parameters of the qLPV system, and a controller that leverages the estimated model to compute inputs with a dual purpose: tracking a reference output while actively exciting the system to enhance parameter estimation. The core of this approach is a robust tube-based MPC scheme that exploits recent developments in polytopic geometry to guarantee recursive feasibility and stability in spite of model uncertainty. The effectiveness of the framework in achieving improved tracking performance while identifying a model of the system is demonstrated through a numerical example.

Paper Structure

This paper contains 11 sections, 5 theorems, 40 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Modelling
  4. Combined state and parameter estimation
  5. Dual MPC
  6. Tube-based MPC
  7. Configuration-constrained TMPC
  8. Recursive feasibility and stability
  9. Active exploration
  10. Numerical example
  11. Conclusions

Key Result

Proposition 1

The polytope $X^{\mathrm{s}} = X(z^{\mathrm{s}},s)$ satisfies the RCI condition in eq:RCI if there exist vectors $v^{\mathrm{s}} \in \mathbb{R}^{n_u}$, $c \in \mathbb{R}^{\mathsf{v} n_u}$ and $q \in\mathbb{R}^{\mathsf{f}}$ verifying the inequalities where we define the disturbance vector and matrices $U_j:=e_j \otimes \mathbb{I}_{\mathsf{v}} \in \mathbb{R}^{n_u \times \mathsf{v} n_u}$.

Figures (1)

  • Figure 1: (Top) Tracking performance of \ref{['eq:closed_loop_dual']}. The dull red, green and blue lines indicate the output trajectory $\hat{y}_t$ of \ref{['eq:model']}, and the gray region indicates constraints; (Middle) Lyapunov cost defined in Theorem \ref{['thm:stability']}, and optimality loss due to projection \ref{['eq:EKF_constraint']}; (Bottom) Variation in estimated parameter $\Delta \hat{\theta}_t=\hat{\theta}_{t+1}-\hat{\theta}_t$.

Theorems & Definitions (11)

  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • ...and 1 more