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Legendre-Moment Transform for Linear Ensemble Control and Computation

Xin Ning, Gong Cheng, Wei Zhang, Jr-Shin Li

TL;DR

A sampling-free ensemble control design algorithm is developed, error analysis for control design using truncated moment systems is conducted and error bounds are derived with respect to the truncation orders, which are illustrated with numerical examples.

Abstract

Ensemble systems, pervasive in diverse scientific and engineering domains, pose challenges to existing control methods due to their massive scale and underactuated nature. This paper presents a dynamic moment approach to addressing theoretical and computational challenges in systems-theoretic analysis and control design for linear ensemble systems. We introduce the Legendre-moments and Legendre-moment transform, which maps an ensemble system defined on the $L^2$-space to a Legendre-moment system defined on the $\ell^2$-space. We show that this pair of systems is of one-to-one correspondence and shares the same controllability property. This equivalence admits the control of an ensemble system through the control of the corresponding Legendre-moment system and inspires a unified control design scheme for linear ensemble systems using structured truncated moment systems. In particular, we develop a sampling-free ensemble control design algorithm, then conduct error analysis for control design using truncated moment systems and derive error bounds with respect to the truncation orders, which are illustrated with numerical examples.

Legendre-Moment Transform for Linear Ensemble Control and Computation

TL;DR

A sampling-free ensemble control design algorithm is developed, error analysis for control design using truncated moment systems is conducted and error bounds are derived with respect to the truncation orders, which are illustrated with numerical examples.

Abstract

Ensemble systems, pervasive in diverse scientific and engineering domains, pose challenges to existing control methods due to their massive scale and underactuated nature. This paper presents a dynamic moment approach to addressing theoretical and computational challenges in systems-theoretic analysis and control design for linear ensemble systems. We introduce the Legendre-moments and Legendre-moment transform, which maps an ensemble system defined on the -space to a Legendre-moment system defined on the -space. We show that this pair of systems is of one-to-one correspondence and shares the same controllability property. This equivalence admits the control of an ensemble system through the control of the corresponding Legendre-moment system and inspires a unified control design scheme for linear ensemble systems using structured truncated moment systems. In particular, we develop a sampling-free ensemble control design algorithm, then conduct error analysis for control design using truncated moment systems and derive error bounds with respect to the truncation orders, which are illustrated with numerical examples.
Paper Structure (14 sections, 11 theorems, 79 equations, 4 figures, 2 algorithms)

This paper contains 14 sections, 11 theorems, 79 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Moment Dynamics of Linear Ensemble Systems
  3. Moment Dynamics under the Legendre Basis
  4. Duality of Linear Ensemble Systems
  5. Ensemble Control Design via Moment Systems
  6. Truncation of Infinite-Dimensional Systems
  7. Control Design through Truncated Moment Systems
  8. Truncation Error Analysis
  9. Estimation of Truncation Error Bounds using Banded Structure
  10. Simulation Results
  11. Conclusion
  12. Shift Operators
  13. Infinite Banded Matrix
  14. Semigroup of Bounded Linear Operators

Key Result

Theorem 2.5

\newlabelprop:controllability.equiv.linear.case0 The linear ensemble in eq:prototype is $L^2$-ensemble controllable if and only if its associated Legendre-moment system in eq:moment.dynamics.prototype.in.example is approximately controllable in $\ell^2(\mathbb{R}^n)$.

Figures (4)

  • Figure 1: An illustration of finite truncation of a banded matrix. $\hat{A}_N$ is an $N$-by-$N$ square truncation of $\hat{A}$.
  • Figure 1: Error decay in the simulation for the scalar linear ensemble in \ref{['ex:scalar.case.control.design']}, where $x(0,\beta)=\sin(0.5\pi\beta)$ and $x_{F}(\beta)=\cos(0.5\pi\beta)$. The right-side figure is a zoomed-in plot of the left-side figure for $N\geqslant 9$.
  • Figure 2: Simulation results for the ensemble of harmonic oscillators in \ref{['ex:oscillator.ensemble_1']}. The figure on the left shows the trajectories of the ensemble from $t=0$ to $t=1$, the upper right plot is the controller obtained using \ref{['alg:a.priori']} at $N=5$, and the bottom right plot shows the error $E$ at each truncation order $N$ for time periods $t\in[0,1]$ and $[0,3.5]$.
  • Figure 3: Simulation results of pattern design in \ref{['eg:pattern_design']}. The dotted blue circle represents the initial profile, and the orange square is the attained final profile. Truncated order $N=17$ is used in this example.

Theorems & Definitions (31)

  • Definition 2.1: Ensemble Controllability
  • Example 2.2
  • Remark 2.3
  • Definition 2.4: Approximate Controllability triggiani1975controllability
  • Theorem 2.5
  • Proof 1
  • Proposition 2.6
  • Proof 2
  • Theorem 2.7
  • Lemma 2.8
  • ...and 21 more