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Time Domain Sensitivity of the Tracking Error

S. O'Neil, S. G. Schirmer, F. C. Langbein, C. A. Weidner, E. Jonckheere

TL;DR

A strictly time-domain formulation of the log-sensitivity of the error signal to structured plant uncertainty is presented and analyzed through simple classical and quantum systems to demonstrate the reduced robustness cost concomitant with high-fidelity quantum control schemes predicated on time-based performance measures.

Abstract

A strictly time-domain formulation of the log-sensitivity of the error signal to structured plant uncertainty is presented and analyzed through simple but representative classical and quantum systems. Results demonstrate that across a wide range of physical systems, maximization of performance (minimization of the error signal) asymptotically or at a specific time comes at the cost of increased log-sensitivity, implying a time-domain constraint analogous to the frequency-domain identity $\mathbf{S(s) + T(s) = I}$. While of limited value in classical problems based on asymptotic stabilization or tracking, such a time-domain formulation is valuable in assessing the reduced robustness cost concomitant with high-fidelity quantum control schemes predicated on time-based performance measures.

Time Domain Sensitivity of the Tracking Error

TL;DR

A strictly time-domain formulation of the log-sensitivity of the error signal to structured plant uncertainty is presented and analyzed through simple classical and quantum systems to demonstrate the reduced robustness cost concomitant with high-fidelity quantum control schemes predicated on time-based performance measures.

Abstract

A strictly time-domain formulation of the log-sensitivity of the error signal to structured plant uncertainty is presented and analyzed through simple but representative classical and quantum systems. Results demonstrate that across a wide range of physical systems, maximization of performance (minimization of the error signal) asymptotically or at a specific time comes at the cost of increased log-sensitivity, implying a time-domain constraint analogous to the frequency-domain identity . While of limited value in classical problems based on asymptotic stabilization or tracking, such a time-domain formulation is valuable in assessing the reduced robustness cost concomitant with high-fidelity quantum control schemes predicated on time-based performance measures.
Paper Structure (18 sections, 5 theorems, 66 equations, 10 figures, 1 table)

This paper contains 18 sections, 5 theorems, 66 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Log Sensitivity of the Error
  4. Classical Example -- Spring-Mass System
  5. Real Dominant Eigenvalue
  6. Complex Dominant Eigenvalue Pair
  7. Classical Example---RLC Circuit
  8. Real Dominant Eigenvalue
  9. Complex Dominant Eigenvalue Pair
  10. Open Quantum Systems Example -- Two Qubits in a Cavity
  11. System Description and Problem Formulation
  12. Log-Sensitivity of the Error
  13. Closed Quantum System Example -- Perfect State Transfer
  14. System Description and Problem Formulation
  15. Log-Sensitivity of the Error
  16. ...and 3 more sections

Key Result

Theorem 1

If $A_0$ is diagonalizable with dominant, real eigenvalue $\lambda_1 \le 0$ with algebraic multiplicity one, then the log-sensitivity $s(\xi_0,t)=\xi_0 \bar{s}_{11} t + R(t)$, where $\lim_{t\to\infty} R(t)$ is finite, i.e., $s(\xi_0,t)$ diverges linearly as $\xi_0 \bar{s}_{11} t$ as $t \rightarrow \

Figures (10)

  • Figure 1: Spring-mass system with $\lambda_1 = -2$, $\lambda_2=-5$, $\xi_0 = 4$, and $\bar{s}_{11} = -1/3$. Note the linear divergence of $\left| s(\xi_0,t) \right|$ with a slope of $4/3$.
  • Figure 2: Divergence of $\left|s(\xi_0,t)\right|$ for the spring-mass system with a complex eigenvalue pair at $s = -1 \pm j \pi/5$. The top plot shows both, $e(t)$ and $\left| s(\xi_0,t)\right|$, on a linear scale, and the bottom plot shows both on a log-scale. Note that $s(\xi_0,t)$ displays local maxima every $\pi/\omega = 5s$ as the error periodically goes to zero.
  • Figure 3: RLC circuit with three states consisting of the two capacitor voltages and single inductor current. The input is a voltage step at $t=0$ and the output is the capacitor voltage $x_1(t)$ in the rightmost branch.
  • Figure 4: Divergence of $\left|s(\xi_0,t)\right|$ for the third order circuit with $\lambda_1 = -1$, $\lambda_2 = -2$, and $\lambda_3 = -4$. As predicted, the log-sensitivity of the error diverges linearly with time.
  • Figure 5: $\left| s(\xi_0,t) \right|$ diverging over time for a third-order circuit with dominant complex eigenvalue pair $\lambda_{1,2} = -2 \pm j \pi/10$. The top panel displays $\left|s(\xi_0,t)\right|$ and $e(t)$ on a linear scale, and the lower panel displays the same on a log-scale. Note the periodic maxima of $\left|s(\xi_0,t)\right|$ and corresponding minima of $e(t)$ with period $10s$.
  • ...and 5 more figures

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Corollary 2
  • Remark 2
  • Theorem 3
  • ...and 2 more