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On Sufficient Richness for Linear Time-Invariant Systems

Marco Borghesi, Simone Baroncini, Guido Carnevale, Alessandro Bosso, Giuseppe Notarstefano

TL;DR

This work investigates how to guarantee persistent excitation (PE) of regressors in linear time-invariant (LTI) systems by using sufficiently rich (SR) input signals. It develops a unified framework for discrete-time and continuous-time settings, introducing the notion of partial PE (PPE) and deriving necessary and sufficient conditions that relate PPE/PE of time-shifted inputs $\mathrm{Q}^k(\mathbf{u})$ or derivatives $\mathrm{D}^k(\mathbf{u})$ to PE of the state $\mathbf{x}$ or the state–input pair $(\mathbf{x},\mathbf{u})$ under asymptotic stability and reachability. For single-input systems, the paper fully characterizes SR sets, showing they are open cones with explicit conditions like $\mathrm{Q}^n(\mathbf{u}) \in \Omega^{\text{D}}_n$ (DT) or $\mathrm{D}^n(\mathbf{u}) \in \Omega^{\text{C}}_{nm}$ (CT). The results unify DT and CT analyses through a shift–derivative correspondence and prove the bounds are tight via numerical counterexamples, informing robust design of data-driven and adaptive schemes for LTI identification and control. The work lays groundwork for extending SR concepts beyond linear dynamics by clarifying the structure and limitations of SR inputs in both time domains.

Abstract

Persistent excitation (PE) is a necessary and sufficient condition for uniform exponential parameter convergence in several adaptive, identification, and learning schemes. In this article, we consider, in the context of multi-input linear time-invariant (LTI) systems, the problem of guaranteeing PE of commonly-used regressors by applying a sufficiently rich (SR) input signal. Exploiting the analogies between time shifts and time derivatives, we state simple necessary and sufficient PE conditions for the discrete- and continuous-time frameworks. Moreover, we characterize the shape of the set of SR input signals for both single-input and multi-input systems. Finally, we show with a numerical example that the derived conditions are tight and cannot be improved without including additional knowledge of the considered LTI system.

On Sufficient Richness for Linear Time-Invariant Systems

TL;DR

This work investigates how to guarantee persistent excitation (PE) of regressors in linear time-invariant (LTI) systems by using sufficiently rich (SR) input signals. It develops a unified framework for discrete-time and continuous-time settings, introducing the notion of partial PE (PPE) and deriving necessary and sufficient conditions that relate PPE/PE of time-shifted inputs or derivatives to PE of the state or the state–input pair under asymptotic stability and reachability. For single-input systems, the paper fully characterizes SR sets, showing they are open cones with explicit conditions like (DT) or (CT). The results unify DT and CT analyses through a shift–derivative correspondence and prove the bounds are tight via numerical counterexamples, informing robust design of data-driven and adaptive schemes for LTI identification and control. The work lays groundwork for extending SR concepts beyond linear dynamics by clarifying the structure and limitations of SR inputs in both time domains.

Abstract

Persistent excitation (PE) is a necessary and sufficient condition for uniform exponential parameter convergence in several adaptive, identification, and learning schemes. In this article, we consider, in the context of multi-input linear time-invariant (LTI) systems, the problem of guaranteeing PE of commonly-used regressors by applying a sufficiently rich (SR) input signal. Exploiting the analogies between time shifts and time derivatives, we state simple necessary and sufficient PE conditions for the discrete- and continuous-time frameworks. Moreover, we characterize the shape of the set of SR input signals for both single-input and multi-input systems. Finally, we show with a numerical example that the derived conditions are tight and cannot be improved without including additional knowledge of the considered LTI system.

Paper Structure

This paper contains 25 sections, 11 theorems, 125 equations, 2 figures.

Table of Contents

  1. Introduction
  2. PE
  3. WPE
  4. Preliminaries and Problem Setup
  5. Preliminaries
  6. Persistency of Excitation and Sufficient Richness
  7. Problem Statement
  8. Main Results
  9. Discrete-Time Framework
  10. Continuous-Time Framework
  11. A discussion on the sets of SR signals
  12. Counterexamples
  13. Sufficient condition
  14. Necessary condition
  15. Conclusions
  16. ...and 10 more sections

Key Result

Lemma 1

$\Omega^{\textup{\tiny D}}_{d}$ (resp., $\Omega^{\textup{\tiny C}}_{d}$) is an open cone in ${\ell_\infty({\mathbb{R}}^{d})}$ (resp., ${\mathcal{C}^\infty_b({\mathbb{R}}^{d })}$).$\square$

Figures (2)

  • Figure 1: Directions spanned in time by the signals $({{\bm{x}}}, {{\bm{u}}})$ and ${\mathrm{Q}}^{n}({{\bm{u}}})$.
  • Figure 2: Directions spanned in time by the signals $({{\bm{x}}}, {{\bm{u}}})$ and ${\mathrm{Q}}^{n+1}({{\bm{u}}})$.

Theorems & Definitions (33)

  • Definition 1: BIBO dynamical system - DT
  • Definition 2: BIBO dynamical system - CT
  • Definition 3: Shift operators
  • Definition 4: Derivative operators
  • Definition 5: Discrete-Time PE bai1985persistency
  • Definition 6: Continuous-Time PE anderson1977exponential
  • Remark 1
  • Lemma 1
  • Remark 2
  • Definition 7: Sufficient Richness
  • ...and 23 more