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Comparison of Linear Systems Across Time Domains: Continuous-time vs. Discrete-time

Armin Pirastehzad, Bart Besselink

Abstract

We develop a formal framework for the behavioral comparison of linear systems across different time domains. We accomplish this by introducing the notion of system interpolation, which determines whether the input-state trajectories of a continuous-time system can be realized as piecewise polynomial interpolations of the input-state trajectories of a discrete-time system. In this context, a piecewise polynomial interpolation of a discrete-time signal is characterized as a continuous-time function that coincides with the discrete-time signal at given sampling instants and can be realized as a polynomial of a prescribed degree over intervals between these instants. By representing piecewise polynomial functions as linear combinations of shifted Legendre polynomials, we characterize system interpolation as a subspace inclusion that is completely in terms of system parameters. This therefore allows for a computationally efficient comparison of the input-state behavior of a continuous-time system with that of a discrete-time one. We then exploit this characterization to discretize a given continuous-time system into a discrete-time one. Lastly, given a control specification, we exploit system interpolation to synthesize controllers that ensure satisfaction at each given sampling instant, while they measure the extent of (possible) violation over intervals between these instants.

Comparison of Linear Systems Across Time Domains: Continuous-time vs. Discrete-time

Abstract

We develop a formal framework for the behavioral comparison of linear systems across different time domains. We accomplish this by introducing the notion of system interpolation, which determines whether the input-state trajectories of a continuous-time system can be realized as piecewise polynomial interpolations of the input-state trajectories of a discrete-time system. In this context, a piecewise polynomial interpolation of a discrete-time signal is characterized as a continuous-time function that coincides with the discrete-time signal at given sampling instants and can be realized as a polynomial of a prescribed degree over intervals between these instants. By representing piecewise polynomial functions as linear combinations of shifted Legendre polynomials, we characterize system interpolation as a subspace inclusion that is completely in terms of system parameters. This therefore allows for a computationally efficient comparison of the input-state behavior of a continuous-time system with that of a discrete-time one. We then exploit this characterization to discretize a given continuous-time system into a discrete-time one. Lastly, given a control specification, we exploit system interpolation to synthesize controllers that ensure satisfaction at each given sampling instant, while they measure the extent of (possible) violation over intervals between these instants.
Paper Structure (11 sections, 6 theorems, 129 equations, 3 figures)

This paper contains 11 sections, 6 theorems, 129 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contributions
  4. Notation
  5. Problem Statement
  6. Legendre Polynomials
  7. System Interpolation
  8. Discretization by System Interpolation
  9. System Interpolation for Control Synthesis
  10. Numerical Simulation
  11. Conclusion

Key Result

Proposition 1

For a sampling time $\tau\in\mathbb{R}_{>0}$ and an integer $N\geq 1$, $\bm{\Sigma}_c$ is an $N$-th order interpolator of $\bm{\Sigma}_d$ with respect to the sampling time $\tau$ if and only if for any $x_0\in\mathbb{R}^n$ and any $u_d: \llbracket0,1\rrbracket\rightarrow\mathbb{R}^m$, there exists $

Figures (3)

  • Figure 1: Control synthesis for enforcing intricate specifications is usually split into 1) discretization of the continuous-time system into a simplified discrete-time model (see left side of the figure), 2) planning a trajectory (by designing a suitable discrete control sequence) for the discrete-time model that adheres to the specification (see the lower part of the figure), and 3) execution of a continuous-time control that utilizes an interpolation of the planned discrete-time trajectory to enforce the original system to (approximately) satisfy the specification (see the middle part of the figure).
  • Figure 2: Subject to the continuous-time control $u_c\in\mathcal{U}_c(0,u_d)$ (indicated by solid lines in Figure ), the robot $\bm{\Sigma}_c$ experiences a displacement and a velocity that comply with the requirements specified by the STL formula $\mathscr{S}_c$. By contrast, subject to a continuous-time control $u_c$ obtained by ZOH interpolation (indicated by dashed lines in Figure ), neither the displacement of the robot nor its velocity adhere to the formula $\mathscr{S}_c$.
  • Figure 3: Subject to the continuous-time control $u_c\in\mathcal{U}_c(0,u_d)$ (indicated by solid lines), the robot $\bm{\Sigma}_c$ experiences a displacement $x$ (Figure ) and a velocity $v$ (Figure ) that adhere to the requirements specified by the formula $\mathscr{S}_c$. However, subject to a continuous-time control $u_c$ obtained by ZOH interpolation of the control sequence $u_d$ (indicated by dashed lines), the displacement $x$ of the robot does not comply with formula $\mathscr{S}_c$.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • ...and 12 more