Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Reachability and Controllability of Temporal Continuous-Time Linear Networks: A Generic Analysis

Yuan Zhang, Yuanqing Xia, Long Wang

TL;DR

It is found that verifying the structural reachability/controllability and structural overall reachability are at least as hard as the structural target controllability verification problem of a single system, implying that finding verifiable conditions for them is hard.

Abstract

Temporal networks are a class of time-varying networks, which change their topology according to a given time-ordered sequence of static networks (known as subsystems). This paper investigates the reachability and controllability of temporal continuous-time linear networks from a generic viewpoint, where only the zero-nonzero patterns of subsystem matrices are known. We demonstrate that the reachability and controllability on a single temporal sequence are generic properties with respect to the parameters of subsystem matrices and the time durations of subsystems. We then give explicit expressions for the minimal subspace that contains the reachable set across all possible temporal sequences (called overall reachable set). It is found that verifying the structural reachability/controllability and structural overall reachability are at least as hard as the structural target controllability verification problem of a single system, implying that finding verifiable conditions for them is hard. Graph-theoretic lower and upper bounds are provided for the generic dimensions of the reachable subspace on a single temporal sequence and of the minimal subspace that contains the overall reachable set. These bounds extend classical concepts in structured system theory, including the dynamic graph and the cactus, to temporal networks, and can be efficiently calculated using graph-theoretic algorithms. Finally, applications of the results to the structural controllability of switched linear systems are discussed.

On the Reachability and Controllability of Temporal Continuous-Time Linear Networks: A Generic Analysis

TL;DR

It is found that verifying the structural reachability/controllability and structural overall reachability are at least as hard as the structural target controllability verification problem of a single system, implying that finding verifiable conditions for them is hard.

Abstract

Temporal networks are a class of time-varying networks, which change their topology according to a given time-ordered sequence of static networks (known as subsystems). This paper investigates the reachability and controllability of temporal continuous-time linear networks from a generic viewpoint, where only the zero-nonzero patterns of subsystem matrices are known. We demonstrate that the reachability and controllability on a single temporal sequence are generic properties with respect to the parameters of subsystem matrices and the time durations of subsystems. We then give explicit expressions for the minimal subspace that contains the reachable set across all possible temporal sequences (called overall reachable set). It is found that verifying the structural reachability/controllability and structural overall reachability are at least as hard as the structural target controllability verification problem of a single system, implying that finding verifiable conditions for them is hard. Graph-theoretic lower and upper bounds are provided for the generic dimensions of the reachable subspace on a single temporal sequence and of the minimal subspace that contains the overall reachable set. These bounds extend classical concepts in structured system theory, including the dynamic graph and the cactus, to temporal networks, and can be efficiently calculated using graph-theoretic algorithms. Finally, applications of the results to the structural controllability of switched linear systems are discussed.
Paper Structure (16 sections, 18 theorems, 56 equations, 4 figures, 1 algorithm)

This paper contains 16 sections, 18 theorems, 56 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model description and definitions
  3. Elementary analysis and genericity justification
  4. Preliminary derivations
  5. Genericity of ${\mathbf{\Omega}}_{\{h_i\}}$
  6. Characterizing the minimal subspace containing ${\mathbf{\Omega}}$
  7. Verifying structural (overall) reachability is 'hard'
  8. Graph-theoretic upper/lower bounds of ${\rm gdim}{\mathbf{\Omega}}_{\{h_i\}}$
  9. Upper bound of ${\rm gdim}{\mathbf{\Omega}}_{\{h_i\}}$ based on the CDG
  10. Lower bound of the generic dimension of ${\bf \Omega}_{\{h_i\}}$
  11. Relations between ${\cal C}$ and ${\cal C}_{\rm low}$: on Ezzine & Haddad's conjecture
  12. Graph-theoretic upper/lower bounds of ${\rm gdim}{\bar{\bf \Omega}}$
  13. Upper bound of the generic dimension of $\bar{\bf \Omega}$
  14. Lower bound of the dimension of $\bar{\bf \Omega}$
  15. Applications to structured switched systems
  16. ...and 1 more sections

Key Result

Proposition 1

Given $(A_i,B_i)|_{i=1}^N$, the dimension of ${\bf \Omega}_{\{h_i\}}$ is the same for almost all $h_1,...,h_N\in {\mathbb R}$,That is, $(h_1,...,h_N)$ can take values from ${\mathbb R}^N$ except for a set of measure zero. Note that we can restrict $h_1,...,h_N$ to be in any continuous interval, incl

Figures (4)

  • Figure 1: A temporal network and the associated graph representations. Dotted red lines represent edges in $\bar{E}_{1,2}$, while bold ones form a $\bar{V}_B-\bar{V}_{A_{22}}$ linking of the maximum size.
  • Figure 2: A temporal network and its associated MDG $\tilde{\cal G}$. Only the first $4$ layers of $\tilde{\cal G}$ are presented. Dotted red lines represent edges between successive layers, and bold ones form a $\tilde{V}_U-\tilde{V}_{X0}$ linking of the maximum size $4$.
  • Figure 3: A temporal network with its subsystem digraphs being ${\cal G}_1$ and ${\cal G}_2$, and the associated ${\bar{\cal G}}_{\rm {sw}}$ (note $v_3^1$ is not input-reachable). Dotted blue lines represent the switching edges. The bold red lines constitute a temporal cactus configuration that covers $\{v_1,v_4,v_3\}$. See Example \ref{['second-example']} for the explanation of ${\bar{\cal G}}'_{\rm {sw}}$.
  • Figure 4: The associated CDGs' to a switched system with subsystem digraphs ${\cal G}_1$ and ${\cal G}_2$ given in Fig. \ref{['first-example']}, under two switching paths $\sigma_1=(1,2)$ and $\sigma_2=(1,2,1)$. Bold lines form a linking with the maximum size in the respective graphs.

Theorems & Definitions (41)

  • Definition 1: Reachable state on $\Pi$
  • Definition 2: Reachable set on $\Pi$
  • Definition 3
  • Definition 4: Overall reachable set
  • Definition 5
  • Example 1
  • Definition 6
  • Proposition 1
  • Proposition 2
  • Definition 7: Structural reachability
  • ...and 31 more