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Tracking optimal feedback control under uncertain parameters

Philipp A. Guth, Karl Kunisch, Sergio S. Rodrigues

Abstract

Optimal control problems of tracking type for a class of linear systems with uncertain parameters in the dynamics are investigated. An affine tracking feedback control input is obtained by considering the minimization of an energy-like functional depending on a finite ensemble of training/sample parameters. It is computed from the nonnegative definite solution of an associated differential Riccati equation. Simulations are presented showing the tracking performance of the computed input for trained as well as untrained parameters.

Tracking optimal feedback control under uncertain parameters

Abstract

Optimal control problems of tracking type for a class of linear systems with uncertain parameters in the dynamics are investigated. An affine tracking feedback control input is obtained by considering the minimization of an energy-like functional depending on a finite ensemble of training/sample parameters. It is computed from the nonnegative definite solution of an associated differential Riccati equation. Simulations are presented showing the tracking performance of the computed input for trained as well as untrained parameters.
Paper Structure (19 sections, 10 theorems, 114 equations, 8 figures)

This paper contains 19 sections, 10 theorems, 114 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Related literature
  3. Contents and notation
  4. Feedback controls for tracking objectives
  5. Extended system.
  6. Optimal control input for the extended system.
  7. From the extended system to the original one
  8. Order of sequence of training parameters
  9. Optimal controls and costs
  10. Comparing optimal costs
  11. Cost of the proposed performant feedback control
  12. Remarks
  13. Comparing state trajectories
  14. Trajectories associated to \ref{['eq:OPextx']} and \ref{['eq:cllsys']}
  15. Trajectories associated to \ref{['eq:OPextx']} and \ref{['eq:OP1sig']}
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let ${\mathbf \Pi}_\varSigma$ denote the unique solution of eq:extRiccati in $\mathcal{S}$. Then, there exists a unique minimizer $({\mathbf x},u)$ for eq:OPextx. This optimal pair satisfies, for $t\in(0,T)$,

Figures (8)

  • Figure 1: The feedback controls are computed using $\varSigma_{2}^{5}$ and tested on $\varSigma_{4}^{6}$. Left: state trajectories corresponding to the feedback control \ref{['eq:robfbcont']} (left) and the feedback control \ref{['eq:averfbcont']} (middle left). Right: feedback control \ref{['eq:robfbcont']} (middle right) and feedback control \ref{['eq:averfbcont']} (right).
  • Figure 2: Increasing training parameter interval $\varSigma_\ell^5$, and increasing test parameter interval $\varSigma^{6}_{2\ell}$ for $\ell \in \{0,\frac{1}{10},\frac{1}{2},1,\frac{3}{2},2\}$. Left: tracking cost $\frac{1}{2} \|Q(y_\sigma - g)\|^2_{L^2(0,T;Y)}$. Middle: feedback control cost $\frac{1}{2} \|u\|^2_{L^2(0,T;U)}$. Right: terminal tracking cost $\frac{1}{2} \|P(y_\sigma - g)\|^2_{Z}$.
  • Figure 3: Fixed training parameter interval $\varSigma_1^5$, and increasing test parameter interval $\varSigma^{6}_\ell$ for $\ell \in \{\frac{1}{2},1,2,3,4\}$. Left: tracking cost $\frac{1}{2} \|Q(y_\sigma - g)\|^2_{L^2(0,T;Y)}$. Middle: feedback control cost $\frac{1}{2} \|u\|^2_{L^2(0,T;U)}$. Right: terminal tracking cost $\frac{1}{2} \|P(y_\sigma - g)\|^2_{Z}$.
  • Figure 4: Five realizations of the random diffusion coefficient \ref{['eq:lognormal']}. Left: training set. Right: test set.
  • Figure 5: Cost for a set of $N=5$ test parameters. Left: tracking cost $\frac{1}{2} \|Q(y_\sigma - g)\|^2_{L^2(0,T;Y)}$. Middle: feedback control cost $\frac{1}{2} \|u\|^2_{L^2(0,T;U)}$. Right: terminal tracking cost $\frac{1}{2} \|P(y_\sigma - g)\|^2_{Z}$.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • Theorem 4.1
  • ...and 7 more