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Observability and Controllability of Nonlinear Networks: The Role of Symmetry

Andrew J. Whalen, Sean N. Brennan, Timothy D. Sauer, Steven J. Schiff

arXiv:1307.5478v3q-bio.NCnlin.CDq-bio.QM

TL;DR

The paper addresses how symmetries affect observability and controllability in nonlinear networks, a problem difficult to resolve using theory developed mainly for linear networks. It combines nonlinear differential-embedding measures with group representation theory and numerical analysis, finding that symmetry effects depend on symmetry type, while networks with only rotational symmetries remain observable and controllable.

  • Problem

    Existing observability and controllability theory largely concerns linear, generic networks, leaving the effects of explicit symmetries in nonlinear networks insufficiently explored.

  • Method

    The paper combines nonlinear differential-embedding measures, numerical experiments, and group representation theory to analyze network motifs with nonlinear dynamics and explicit symmetries.

  • Results

    Symmetry can decrease observability and controllability, but networks containing only rotational symmetries remain observable and controllable.

  • Takeaways & Limitations

    The specific symmetry operations, node locations, coupling strength, and time evolution should inform sensor, actuator, observer, and controller design.

Abstract

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Observability and controllability are essential concepts to the design of predictive observer models and feedback controllers of networked systems. For example, noncontrollable mathematical models of real systems have subspaces that influence model behavior, but cannot be controlled by an input. Such subspaces can be difficult to determine in complex nonlinear networks. Since almost all of the present theory was developed for linear networks without symmetries, here we present a numerical and group representational framework, to quantify the observability and controllability of nonlinear networks with explicit symmetries that shows the connection between symmetries and nonlinear measures of observability and controllability. We numerically observe and theoretically predict that not all symmetries have the same effect on network observation and control. Our analysis shows that the presence of symmetry in a network may decrease observability and controllability, although networks containing only rotational symmetries remain controllable and observable. These results alter our view of the nature of observability and controllability in complex networks, change our understanding of structural controllability, and affect the design of mathematical models to observe and control such networks.

I. INTRODUCTION

The introduction frames observability and controllability as central to reconstructing and controlling network dynamics, while noting that existing theory largely assumes linear, generic networks. The paper extends this perspective to nonlinear networks with explicit symmetries using nonlinear measures and group representation theory.

  • Observability asks whether measurements contain enough information to reconstruct full system dynamics, while controllability asks whether inputs can steer the system’s states.
  • Prior structural observability and controllability results rely critically on linear, generic dynamics and essentially random network connections.
  • Symmetry can create dependent system-matrix rows and columns, constraining state differences and producing uncontrollable or unobservable subspaces.
  • The paper extends observability and controllability analysis to nonlinear network motifs with explicit symmetries, varying coupling strength and spatial and temporal effects.
  • Nonlinear observability is quantified through a differential embedding map built from measurements and higher Lie derivatives, with conditioning indicating proximity to singularity.

B. Differential Embeddings and Nonlinear Observability

For nonlinear systems, observability is formulated through a differential embedding map built from a measurement function and its higher Lie derivatives. Its Jacobian forms a state-dependent observability matrix whose singularity or conditioning determines information loss.

  • Nonlinear observability replaces linear matrix powers with the measurement function and its higher Lie derivatives along the nonlinear vector field.
  • The differential embedding map φ collects Lie derivatives from order 0 through n−1, where n is the system order.
  • Taking the Jacobian of φ produces the nonlinear observability matrix used to assess local state reconstruction.
  • Unlike the constant matrix of a linear system, the nonlinear observability matrix depends on the system state and therefore on the trajectory through phase space.
  • The nonlinear construction reduces to the linear observability formulation for linear system representations.

C. Lie Brackets and Nonlinear Controllability

Nonlinear controllability is defined dually through an input-based map of higher Lie brackets, while observability and controllability indices quantify conditioning. The study applies these tools to three-node FitzHugh–Nagumo motifs across topology, coupling, and dynamical regimes.

  • The nonlinear controllability matrix maps inputs to states using the input vector field and its higher Lie brackets with the nonlinear system field.
  • The controllability index uses the same singular-value-based conditioning framework as the observability index, substituting Q for O.
  • A. Fitzhugh-Nagumo System Dynamics: The FitzHugh–Nagumo node model represents an excitable neuronal membrane and can exhibit transients, limit cycles, relaxation oscillations, multiple time scales, and chaos.
  • A. Fitzhugh-Nagumo System Dynamics: The three-node network uses membrane voltage and recovery variables, with observability and controllability matrices depending on the states and trajectory.
  • A. Fitzhugh-Nagumo System Dynamics: A 10% variance noise term is added to coupling distances to introduce heterogeneity and break symmetry.
  • A. Fitzhugh-Nagumo System Dynamics: Square-wave and constant inputs generate distinct limit-cycle and chaotic regimes, enabling exploration of how trajectories influence observability and controllability.

B. Network Motifs and Simulated Data

The study simulates three-node FitzHugh–Nagumo network motifs across dynamical regimes, coupling patterns, measurement or control nodes, and coupling strengths. Results show that symmetry type, structural isolation, heterogeneity, and coupling strength shape observability and controllability.

  • Simulation design: Three-node FitzHugh–Nagumo motifs were simulated under limit-cycle and chaotic regimes with identical or heterogeneous coupling.The full six-dimensional equations were integrated with RK4 from common initial conditions and driving inputs.
  • Motifs with symmetry: Full S3 symmetry in motif 1 and reflection S2 symmetry in motif 3 produce zero observability and controllability for specific symmetric measurements or controls.For motif 1, the indices remain zero across coupling strengths; in motif 3, node 2 is affected.
  • Coupling effects: Heterogeneous coupling partially breaks motif 1 and motif 3 symmetry, yielding nonzero indices whose values depend on coupling strength.In latent structural symmetries, increasing coupling can nonetheless cause substantial losses near critical coupling levels.
  • Motifs with symmetry: Rotational C3 symmetry in motif 7 does not reduce observability or controllability under identical coupling strengths.The symmetric case performs comparably to the symmetry-broken case in the reported calculations.
  • Coupling effects: Observability and controllability indices can rise to an optimum and then decline as coupling strength increases beyond a critical value.In motif 1, stronger coupling is associated with bifurcation-driven trajectory changes, including collapse onto a limit cycle and a later reverse Hopf bifurcation.

A. Symmetric Groups and Representations

The paper represents network symmetries as node permutations and uses group representations to expose structure in the network equations. Reducible representations can be transformed into block-diagonal form, supporting analysis of symmetry-related modes.

  • Symmetric groups: A network symmetry is a node permutation that leaves the network configuration unchanged, with S_n containing all permutations of n symbols.Permutation cycles such as (123) specify how node labels are replaced.
  • Symmetric groups: Reflection symmetries σ_n and cyclic rotations C_n are represented as group operations, with C_n rotating the system by 2π/n while preserving its configuration.Motifs 3 and 7 use S2 and C3 subgroups of S3, respectively.
  • Matrix representations: Permutation operations are represented by monomial matrices D(R), which have one nonzero entry per row and column.For the three-node motifs, D(R) gives a three-dimensional representation of S3; S2 and C3 use corresponding subsets.
  • Matrix representations: A representation preserves group composition: if R1 and R2 correspond to D(R1) and D(R2), then R1R2 corresponds to D(R1)D(R2).A one-to-one correspondence is called a faithful representation.
  • Symmetry and network equations: Symmetry representations commute with the network system matrix, linking permutation structure to the network equations.This conjugacy provides the algebraic connection between group operations and the system matrix.
  • Irreducible representations: Reducible representations can be transformed into block-diagonal form, while irreducible representations provide the constituent blocks used in that decomposition.Characters, defined as representation traces, classify group elements into conjugacy classes and determine representation multiplicities.
  • Irreducible representations: The transformation must reduce every representation matrix D(R) to diagonal form for all group elements R in S_n.The dimensions and multiplicities of irreducible components are obtained from orthogonality and character-based relations.

B. Construction of the Similarity Transform4 α

The similarity transform uses irreducible representations to block-diagonalize symmetric network systems, exposing modes that may be unreachable or unmeasurable. For motif 3, this representation-based inspection identifies symmetry-induced non-controllability and non-observability; motif 1 requires Jordan-form criteria because repeated eigenvalues create degeneracy.

  • Constructing α: For motif 3, the symmetry group S2 has two group-element classes and two one-dimensional irreducible representations.The representation dimensions are l1 = l2 = 1.
  • Constructing α: The unitary transformation α is constructed from normalized linearly independent columns of generating matrices associated with the irreducible representations.These columns form the coordinate transformation used to block-diagonalize the system.
  • Motif 3: The transformed motif 3 system is non-controllable and non-observable from node 2 because the mode associated with Z3 is neither reached by the input nor inferred from the output.This conclusion follows by inspecting the transformed control and measurement matrices.
  • Motif 1: Repeated eigenvalues can make a diagonal-form inspection insufficient, requiring Jordan-form conditions based on the placement and independence of control and measurement matrix entries.For single-input and single-output systems, each repeated eigenvalue must correspond to only one Jordan block.
  • Motif 1: Motif 1 is non-controllable and non-observable because two of its three 1 x 1 Jordan blocks share λ2 = −1, violating the single-input/single-output criterion.The repeated eigenvalue has multiple Jordan blocks, producing the relevant degeneracy.

C. Motif 7 and Networks Containing Only Rotation Groups

The group-representation criterion explains why rotational symmetry differs from reflection or full permutation symmetry. Applying it to motif 7 shows that cyclic C3 symmetry does not create a symmetry-induced uncontrollable or unobservable mode, with a stated caveat for invariant state variables.

  • Representation criterion: For a single-input or single-output network, symmetry-induced non-controllability or non-observability occurs when an irreducible representation satisfies the stated invariant-state condition.The condition must be checked across all irreducible representations appearing in the group representation.
  • Motif 7: In motif 7, only the identity operation leaves each node invariant, so the criterion is nonzero for every node and irreducible representation.The relevant invariant operation is Rr = E for r = 1, 2, 3.
  • Motif 7: Motif 7 therefore cannot be symmetry-induced non-controllable or non-observable and remains controllable and observable from any node.The result extends by corollary to networks with only rotational symmetry, subject to the invariant-state caveat.
  • Caveat: A network with only rotational symmetry can still be non-controllable and non-observable if the input and output couple to a state variable invariant under every group operation.Motif 7 avoids this caveat because it has no such state variable.
  • Implication: The representational results explain the nonlinear findings and show that different symmetry types have different effects on network controllability and observability.The conclusion remains valid with generic diagonal entries chosen to preserve the symmetry.

D. Application to Structurally Controllability (Observability)

The analysis extends structural controllability by showing that symmetry imposes parameter constraints beyond the zero pattern used in classical structural definitions. Rotation-only bud networks therefore provide a case where structural and full controllability coincide.

  • Structural controllability: Structural controllability treats systems with the same zero locations as structurally equivalent and requires only that one parameter realization with that structure be controllable.This framework assumes nonzero entries are generic while zero entries are fixed.
  • Symmetry constraints: A structurally controllable network can nevertheless be uncontrollable for specific entries of its system and input matrices, including entries constrained by symmetry.Such specific entries may be uncertain estimates of practical system parameters.
  • Bud networks: Lin’s stem and bud structures are always structurally controllable, but rotation-only bud networks are additionally fully controllable and never symmetry-induced non-controllable.The corresponding observability statement follows by duality.
  • Conclusion: Thus, symmetry alone does not necessarily destroy controllability in structurally controllable networks.The result applies in particular to networks containing only rotation groups or buds in Lin’s terminology.

VI. DISCUSSION

The discussion finds that symmetry type, node placement, coupling strength, and system evolution jointly shape observability and controllability in nonlinear networks. Group representation analysis supports sensor and actuator placement, while strong coupling can reduce these properties near stable equilibria.

  • Observability and controllability depend on symmetry type, sampled or controlled node location, coupling strength, and time evolution.
  • Group representation theory decomposes symmetric network equations into decoupled controllable and uncontrollable, or observable and unobservable, subspaces.This decomposition can inform controller and observer design.
  • Randomly altering coupling strengths can create substantial observability or controllability absent in the fully symmetric network.
  • Increasing coupling strength can decrease observability or controllability when strong coupling drives the system through a reverse Hopf bifurcation toward a stable equilibrium.The resulting lack of dynamic movement severely decreases these measures.
  • More direct incoming connections generally increase observability, while more outgoing connections generally increase controllability from a node.The authors identify extending these nonlinear results to larger symmetric networks as a future challenge.

Appendix A: Supplemental Information

The appendix constructs nonlinear observability and controllability matrices for a FitzHugh–Nagumo motif using Lie derivatives and Lie brackets. It then analyzes local index distributions, finding lognormal behavior for adequately sampled nonzero values with 95% χ2-test confidence.

  • The motif 1 measurement function observes node 1 through y = Cx(t) = v1.
  • The nonlinear observability matrix is formed from Lie derivatives of the measurement function and depends on the system’s location in phase space.Observability indices are averaged over state trajectories as a qualitative measure.
  • The controllability matrix is constructed from the node-1 input function and its higher Lie brackets with the nonlinear vector field.The internal driving square wave is excluded because it connects to all three nodes and contributes nothing to the Lie-bracket mapping of interest.
  • Log-scaled histograms of local observability and controllability indices along phase-space trajectories are close to lognormal distributions.
  • 95% χ2-test confidence supports lognormal fits for cases with adequate data after zeros are removed.Adequate cases contained over 90% of the data for accurate fitting.

3. Group Representation Analysis of Symmetries in Motif 1

For motif 1 with S3 symmetry, the analysis partitions group elements into three classes and decomposes the representation into two one-dimensional and one two-dimensional irreducible representations. The generating matrix is then assembled from the representations that occur.

  • Motif 1 has S3 symmetry with three group-element classes: the identity, three transpositions, and two three-cycles.
  • Reducing D(R) yields two one-dimensional and one two-dimensional irreducible representations of S3.
  • The irreducible representations occur in D(R) with multiplicities 1, 0, and 2, respectively.
  • The generating matrix α is constructed from linearly independent rows of G, with the absent second representation contributing nothing.
  • The final computations from the two-dimensional representation are normalized to complete the construction.

4. Dilations of the graph of (A,B)

The graph of (A,B) augments the system graph with an origin representing the input and defines dilations through a comparison between a node set and its predecessor set. The supplied table is identified as listing irreducible representations for S3 symmetry.

  • The graph of (A,B) contains n state nodes plus an origin node representing the input.
  • A dilation exists exactly when the predecessor set T(S) has fewer nodes than the selected state-node set S.T(S) contains nodes with directed edges pointing to nodes in S.
  • Table I presents the irreducible representations associated with S3 symmetry.
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