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Controllability of networked MIMO systems

Lin Wang, Guanrong Chen, Xiaofan Wang, Wallace K. S. Tang

arXiv:1505.01255v3eess.SY

TL;DR

The paper asks how topology, higher-dimensional node dynamics, external inputs, and inner interactions jointly determine controllability in directed, weighted networked LTI systems. It develops general MIMO analysis and necessary-and-sufficient SISO conditions for special and general topologies, showing that controllability is integrated rather than topology-only or nodewise. The results include structural-controllability counterexamples and criteria useful for assembling large-scale networks.

  • Problem

    The paper addresses how multiple network, node, input, and interaction factors determine controllability when directed weighted networks contain higher-dimensional LTI nodes.

    Source: §1 · p. 2

  • Method

    The paper derives controllability conditions for general MIMO networks and precise necessary and sufficient conditions for SISO higher-dimensional nodes in chains, cycles, and general topologies.

    Source: §1 · p. 2, §3 · p. 5, §5 · p. 14

  • Results

    General network controllability is an integrated result that cannot generally be decoupled into individual-node controllability and topology-only properties; topology-uncontrollable identical-node networks remain uncontrollable even when structurally controllable.

    Source: §2.3 · p. 4, §5 · p. 14

  • Takeaways & Limitations

    The criteria provide efficient tests using properties of lower-dimensional matrices and guidelines for assembling uncontrollable nodes into controllable networked systems.

    Source: §5 · p. 14

    For general networks, controllability and observability of the nodes together may be necessary under some conditions but are not sufficient in general.

    Source: §4 · p. 11

Abstract

from arXiv · show

In this paper, we consider the state controllability of networked systems, where the network topology is directed and weighted and the nodes are higher-dimensional linear time-invariant (LTI) dynamical systems. We investigate how the network topology, the node-system dynamics, the external control inputs, and the inner interactions affect the controllability of a networked system, and show that for a general networked multi-input/multi-output (MIMO) system: 1) the controllability of the overall network is an integrated result of the aforementioned relevant factors, which cannot be decoupled into the controllability of the individual node-system and the properties solely determined by the network topology, quite different from the familiar notion of consensus or formation controllability; 2) if the network topology is uncontrollable by external inputs, then the networked system with identical nodes will be uncontrollable, even if it is structurally controllable; 3) with a controllable network topology, the controllability of each individual node-system is not necessary for the controllability of the networked system. For a networked system with single-input/single-output (SISO) LTI nodes, we present precise necessary and sufficient conditions for the controllability of several typical networked systems with special topologies, such as chains and cycles, as well as for the setting of a general network topology.

1 Introduction

The paper studies controllability in directed, weighted networks of higher-dimensional LTI node-systems, where topology, node dynamics, external inputs, and inner interactions jointly determine the result. It develops criteria for general MIMO systems and precise conditions for SISO higher-dimensional nodes across several topologies.

Motivation

  • Higher-dimensional MIMO nodes and multidimensional communication channels make network controllability substantially different from the one-dimensional SISO setting.Existing one-dimensional results cannot generally be extended directly to MIMO or higher-dimensional-node networks.Source: §1 · p. 2

Objectives

  • The paper develops controllability conditions involving network topology, node dynamics, external control inputs, and inner interactions.These conditions support criteria for determining large-scale networked-system controllability.Source: §1 · p. 2

Contribution

  • Unlike some prior MIMO network studies, general network controllability cannot be decoupled into individual-node controllability and topology-only properties.The paper addresses a setting without requiring every subsystem to have an external control input.Source: §1 · p. 2

Scope

  • The paper derives necessary and sufficient conditions for SISO higher-dimensional nodes in chains, cycles, and general network topologies.The paper is organized around preliminaries and the model, special topologies, general topology, and concluding discussions.Source: §1 · pp. 2–3

Main findings

  • For MIMO networks, controllability of individual nodes is necessary for chain-networks but not cycle-networks, while nontrivial trees of identical nodes are always uncontrollable.The results also show that node controllability and observability together may be necessary but are not sufficient for general networks.Source: §1 · p. 3

2 Preliminaries and the networked system

This section defines controllability and the network model, then uses counter-intuitive examples to show that topology and individual-node properties alone do not determine network controllability. Interactions among node states can reverse controllability outcomes even when topology and node-system matrices are unchanged.

2.1 Preliminaries

  • The section introduces directed-graph concepts including matching, reachability, cycles, trees, roots, and leaves for analyzing network topology.A perfect matching matches all graph nodes.Source: §2.1 · p. 3

2.2 The networked system model

  • The paper models directed, weighted MIMO LTI networks using node states x_i, outputs y_i, topology matrix L, input-selection matrix Δ, and inner-coupling matrix H.External inputs enter node i through δ_iB u_i, while inter-node communication contributes β_ijHCx_j.Source: §2.2 · p. 3

2.3 Some counter-intuitive examples

  • A network of three identical higher-dimensional nodes can be uncontrollable for any A while remaining structurally controllable because each MIMO node contains a self-matched cycle.The corresponding one-dimensional-state network is not structurally controllable with one external input.Source: §2.3 · p. 4
  • For a cycle with a perfect matching, individual-node controllability is neither necessary nor sufficient for controllability of the whole network.One example is uncontrollable even though each node has an independent external control input.Source: §2.3 · p. 4
  • The examples show that identical topology and identical node-system matrix A can produce different controllability outcomes because interactions among node states differ.Those interactions can make controllable nodes form an uncontrollable network or assemble uncontrollable nodes into a controllable one.Source: §2.3 · p. 4

3 Special network topologies

This section derives controllability conditions for special network topologies, including chains, cycles, and tree-based networks, using PBH rank analysis. It also identifies uncontrollability results for representative cycle and branching tree configurations under specified conditions.

3.1 Chains

  • For chain networks, controllability requires both (A, B) and (A, HC) to be controllable.When rank(B) < N, observability of (A, C) is also necessary; for SISO nodes, controllability is equivalent to controllability of (A, B) and (A, H) together with observability of (A, C).Source: §3.1 · pp. 5–6
  • Observability of (A, C) is not necessary for controllability of the chain-network (4)-(7), while controllability of (A, B) and (A, HC) plus observability of (A, C) is not sufficient.In the latter example, rank(6 · I −Φ, Ψ) = 3 < 4, indicating that the chain-network is uncontrollable.Source: §3.1 · p. 5

3.2 Cycles

  • For the networked system, controllability of (A, HC) is necessary, whereas controllability of (A, B) and observability of (A, C) are not necessary.The networked system is controllable when rank(Ψ, ΦΨ, Φ2Ψ, · · · , Φ11Ψ) = 12.Source: §3.2 · p. 7
  • For cycles, if N > rank(B), observability of (A, C) becomes necessary; in the SISO case, (A, H) controllability and (A, C) observability are necessary but may not suffice.Examples show both controllable and uncontrollable networks under these node-level conditions, while Theorem 7 supplies an additional necessary-and-sufficient rank condition.Source: §3.2 · pp. 7–8
  • The cycle network in Example 3.6 is uncontrollable under the stated conditions.Source: §3.2 · p. 9

3.3 Trees

  • A star network with N > 2 is uncontrollable when only its root has an external control input.Such a star has more than one leaf node, and the corresponding networked system is proven uncontrollable.Source: §3.3 · p. 9
  • More generally, a tree with every node reachable from the root and more than one leaf is uncontrollable when only the root has an external control input.A tree with only one leaf is a chain, treated separately in Subsection 3.1.Source: §3.3 · pp. 9–10

4 A general network topology

For general networked systems, controllability depends jointly on topology, node dynamics, external inputs, and inner interactions. The paper derives necessary and sufficient criteria for SISO higher-dimensional nodes and identifies topology- and subsystem-level conditions that constrain controllability.

Necessary conditions

  • For a node without incoming edges, controllability requires both controllability of (A, B) and an external input at that node.Without such an input, the corresponding block row causes a PBH rank deficiency.Source: §4 · p. 10
  • If any node lacks an external input, controllability requires controllability of (A, HC).An uncontrollable (A, HC) supplies a nonzero PBH vector supported on the uncontrolled node.Source: §4 · p. 10
  • When N > m·rank(B), controllability requires observability of (A, C).The condition follows because insufficiently distributed external inputs cannot overcome an unobservable node mode.Source: §4 · p. 10

General topology

  • If the network topology is uncontrollable by external inputs, the networked system is uncontrollable.This includes networks with more leaf nodes than externally controlled nodes and follows from the PBH rank condition.Source: §4 · pp. 10–11

Necessary and sufficient conditions

  • For SISO nodes with |U| < N, controllability is equivalent to controllability of (A, H), observability of (A, C), and two PBH-style rank conditions.The conditions are (A, H) controllable, (A, C) observable, and nonzero left eigenvectors of A must remain coupled through L; the remaining condition is rank(I − Lγ, Δη) = N for s outside σ(A).Source: §4 · pp. 11, 13

Special topologies

  • The general criterion specializes consistently: its coupling condition is automatic for cycles, while its resolvent rank condition reduces to the chain and cycle criteria.For chains, condition (iv) is automatic and condition (iii) is equivalent to controllability of (A, B); for cycles, condition (iii) is automatic because L is invertible.Source: §4 · p. 13

5 Conclusions

The paper shows that controllability in networked MIMO systems is an integrated property of topology, node dynamics, external inputs, and inner interactions. It also gives precise criteria for SISO higher-dimensional nodes across several topologies and connects the framework to interdependent networks.

Main conclusions

  • MIMO network controllability integrates topology L, node-system dynamics (A, B, C), external inputs, and inner interactions H.These factors cannot generally be treated as independent controllability components.Source: §5 · pp. 13–14
  • An uncontrollable topology makes an identical-node MIMO network uncontrollable, while node-level controllability and observability conditions remain insufficient in general.This remains true even for a cycle-network with a perfect matching.Source: §5 · p. 14

SISO results

  • For SISO higher-dimensional nodes, the paper derives necessary and sufficient controllability conditions for chains, cycles, and general network topologies.The criteria use properties of lower-dimensional matrices and can guide assembly of controllable networks from potentially uncontrollable nodes.Source: §5 · p. 14

Broader relevance

  • Viewing each higher-dimensional node as a sub-network places the model within interdependent, interconnected, multilayer, or multiplex network frameworks.The authors state that their results should inform controllability studies of such complex networks.Source: §5 · p. 14
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