Exact Robust Instability Analysis for Networked Dynamical Systems with Biological Application

Technical Summary: Exact Robust Instability Analysis for Networked Dynamical Systems with Biological Application

Problem Formulation
This paper addresses the problem of robust instability in nominally unstable, uncertain networked dynamical systems. The setting involves $n$ agents, each modeled as a Single-Input-Single-Output (SISO) Linear Time-Invariant (LTI) system sharing an identical nominal dynamics $h(s)$, subject to independent dynamic perturbations $\delta_i$. The interconnection is defined by a constant matrix $A$.

The core objective is to determine the Robust Instability Radius (RIR), denoted $\rho^*(\Sigma)$, which is defined as the minimum $H_\infty$-norm of a stable perturbation $\Delta$ required to internally stabilize the nominally unstable network. This problem is distinct from standard robust stability analysis; mathematically, it is equivalent to a strong stabilization problem with an additional minimization of the controller norm. The authors note that finding exact solutions for general robust instability is notoriously difficult, often requiring the solution of the "Phase Change Rate (PCR) Maximization Problem" or relying on conservative bounds. The paper specifically targets network structures where the RIR can be characterized exactly, motivated by the need to theoretically guarantee the preservation of periodic oscillations in biological systems (e.g., repressilators) despite unmodeled dynamics.

Methodology
The methodology relies on reducing the complex network instability analysis to the analysis of a single representative SISO system. The approach proceeds through the following logical steps:

1. Network Decomposition: The paper analyzes the characteristic equation of the network, which depends on the eigenvalues of the interconnection matrix $A$. For cyclic networks, the eigenvalues are distributed on a circle in the complex plane.
2. Homogeneous Equivalence: A key theoretical step (Proposition 3) establishes that for cyclic networks, the robust instability radius under heterogeneous perturbations ($\Delta \in \Delta_d$) is equal to that under homogeneous perturbations ($\Delta \in \Delta_h$). This allows the problem to be reduced to finding a single scalar perturbation $\delta$ that stabilizes the network.
3. SISO Reduction: The stability of the network is shown to be equivalent to the stability of $n$ independent SISO feedback loops, $g_k(s) = \frac{\lambda_k h(s)}{1 - \lambda_k h(s)}$, where $\lambda_k$ are the eigenvalues of $A$. The RIR is determined by the "worst-case" unstable agent.
4. Marginal Stabilization Conditions: The authors utilize and extend previous results on marginal stabilization for complex rational functions. They derive necessary and sufficient conditions (Theorem 1) for a SISO system to be marginally stabilized by a controller with a specific norm (the inverse of the system's peak gain). These conditions rely on the Phase Change Rate (PCR) at the peak-gain frequency.
5. Geometric Properties: To ensure the exact RIR is achieved, the paper introduces specific geometric properties for the inverse nominal dynamics $\phi(s) = 1/h(s)$:
   • Gain and Phase Monotonicity: The gain and phase of $\phi(j\omega)$ must be monotonically increasing in the low-frequency range.
   • Convexity: The region bounded by the Nyquist plot of $\phi$ must be convex.
   • PCR Condition: A specific inequality involving the derivative of the gain and phase must hold to ensure the critical eigenvalue can be stabilized by a minimum-norm perturbation without destabilizing others.

Key Contributions
The paper identifies three specific classes of network structures where the robust instability problem reduces to a single SISO analysis, allowing for exact characterization of the RIR:

1. Cyclic Networks: For networks with a cyclic interconnection matrix $A$, the authors prove that if the inverse nominal dynamics $\phi(s)$ satisfies specific monotonicity and convexity properties, the RIR is exactly determined by the eigenvalue $\lambda^*$ with the smallest angle (closest to the positive real axis). The exact RIR is given by $\rho^* = 1/\|g^*\|_\infty$, where $g^*$ is the transfer function associated with $\lambda^*$.
2. Rank-One Networks: For networks where the interconnection matrix $A$ has rank one, the problem reduces to a scalar SISO system. The exact RIR is derived as a function of the SISO RIR and the entries of the factorization of $A$.
3. Rank-Two Networks (Complex Conjugate Eigenvalues): For rank-two networks with complex conjugate eigenvalues, the authors show that the RIR is determined by the representative SISO system, provided the system satisfies specific PCR conditions.

The paper also provides a class of nominal dynamics (Hurwitz polynomials with roots in specific regions, denoted as class $H_p$) that naturally satisfy the required monotonicity, convexity, and PCR conditions.

Results

• Exact RIR Characterization: The paper derives sufficient conditions under which the robust instability radius is exactly equal to the inverse of the $H_\infty$-norm of the critical representative SISO system ($\rho^* = 1/\|g^*\|_\infty$). This tightens the gap between lower bounds (like the small-gain theorem bound) and the actual instability radius.
• Simultaneous Stabilization: It is demonstrated that a single perturbation $\delta$ (derived from the critical agent) simultaneously stabilizes all other unstable agents in the network, ensuring the entire network becomes marginally stable.
• Biological Application: The theoretical results are applied to a Genetic Regulatory Network (GRN) with a cyclic structure (specifically a repressilator model with $N=7$). Using parameters derived from biological models, the authors calculate the exact RIR. Simulations confirm that a perturbation with a norm slightly above the calculated RIR stabilizes the system (converting oscillations to a steady state), while a perturbation below the RIR fails to do so.

Significance and Claims
The paper claims to provide a tractable framework for exact robust instability analysis, a problem generally considered intractable due to the difficulty of strong stabilization with minimum norm. By identifying specific network topologies (cyclic, rank-deficient) and agent dynamics (satisfying convexity/monotonicity), the authors show that the complex multi-agent problem can be reduced to a single SISO analysis.

The significance lies in the ability to theoretically guarantee the preservation of oscillatory behaviors in biological systems. Since exact mathematical models of biological systems are difficult to derive, robust instability analysis ensures that desirable oscillations (like those in circadian rhythms or synthetic gene circuits) persist even in the presence of unmodeled dynamics and uncertainties. The paper emphasizes that while the derived conditions are sufficient and not exhaustive, they encompass important models found in biological applications, offering a rigorous tool for the analysis and synthesis of oscillatory networks.