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Controllability and Observability of Heterogeneous Networked Systems with Non-uniform Node Dimensions and Distinct Inner-Coupling Matrices

Aleena Thomas, Abhijith Ajayakumar, Raju K. George

Abstract

In this paper we extend the work in the conference paper 'On the Controllability and Observability of Heterogeneous Networked Systems with distinct node dimensions and inner-coupling matrices' wherein the controllability and observability of a heterogeneous networked system with distinct node dimensions were studied. This paper adds to the conference paper a necessary and sufficient condition for controllability of the networked system. The result demonstrates the dependence of controllability of the network on factors like network topology, inner interactions among nodes and nodal dynamics. The result is formulated by characterizing the left eigenvectors of the network state matrix. Another necessary and sufficient condition for controllability, which is a reformulation of the \textit{Popov-Belevitch-Hautus} controllability condition, a necessary and sufficient condition for observability of the networked system and certain necessary conditions for controllability of the networked system are the other results established in this paper. Variants of these results under certain specific network topologies like path, cycle, star and wheel are also discussed.

Controllability and Observability of Heterogeneous Networked Systems with Non-uniform Node Dimensions and Distinct Inner-Coupling Matrices

Abstract

In this paper we extend the work in the conference paper 'On the Controllability and Observability of Heterogeneous Networked Systems with distinct node dimensions and inner-coupling matrices' wherein the controllability and observability of a heterogeneous networked system with distinct node dimensions were studied. This paper adds to the conference paper a necessary and sufficient condition for controllability of the networked system. The result demonstrates the dependence of controllability of the network on factors like network topology, inner interactions among nodes and nodal dynamics. The result is formulated by characterizing the left eigenvectors of the network state matrix. Another necessary and sufficient condition for controllability, which is a reformulation of the \textit{Popov-Belevitch-Hautus} controllability condition, a necessary and sufficient condition for observability of the networked system and certain necessary conditions for controllability of the networked system are the other results established in this paper. Variants of these results under certain specific network topologies like path, cycle, star and wheel are also discussed.

Paper Structure

This paper contains 11 sections, 21 theorems, 69 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations & Preliminaries
  3. Notations
  4. Control Theory
  5. Model Formulation
  6. Main Results
  7. Necessary and Sufficient Conditions for Controllability of the Networked System
  8. A Necessary and Sufficient Condition for Observability of the Networked System
  9. Some Necessary conditions for Controllability of the Networked System
  10. Conclusion
  11. Specific Network Topologies

Key Result

Theorem 2.1

W.Terrell The LTI system $(A,B)$ is controllable if and only if one of the following conditions is satisfied: The PBH eigenvector test can be equivalently stated as follows:

Figures (10)

  • Figure 1: Controllable networked system with dynamics as in Example \ref{['eg1']}
  • Figure 2: Controllable networked system with dynamics as in Example \ref{['eg4']}
  • Figure 3: Uncontrollable networked system with dynamics as in Example \ref{['eg5']}
  • Figure 4: Uncontrollable networked system with dynamics as in Example \ref{['eg6']}
  • Figure 5: Unobservable networked system with dynamics as in Example \ref{['eg7']} (a)
  • ...and 5 more figures

Theorems & Definitions (45)

  • Definition 2.1: Controllability
  • Definition 2.2: Controllable Pair
  • Theorem 2.1
  • Definition 2.3: Observability
  • Definition 2.4: Observable Pair
  • Theorem 2.2
  • Theorem 4.1
  • proof
  • Example 4.1
  • Corollary 4.1: Xiang et al., Theorem 1, L.Xiang
  • ...and 35 more