Exponential Stability of Homogeneous Positive Systems of Degree One With Time-Varying Delays

Technical Summary: Exponential Stability of Homogeneous Positive Systems of Degree One With Time-Varying Delays

Problem Statement
While the asymptotic stability of positive linear systems in the presence of bounded time delays is well-established—specifically the property that such systems are delay-independent stable if and only if their delay-free counterparts are stable—the theoretical framework for nonlinear positive systems is less developed. Existing literature often relies on constant delay assumptions or techniques like linear Lyapunov-Krasovskii functionals, which can lead to excessive conservatism or fail to apply when delays are time-varying. Furthermore, while asymptotic stability is known, there is a lack of sharp characterizations regarding how maximum delay bounds affect the exponential decay rate of these systems. This paper addresses the gap in establishing delay-independent exponential stability and quantifying decay rates for a significant class of nonlinear positive systems with bounded, time-varying delays.

Methodology
The paper analyzes two primary classes of systems:

1. Continuous-time systems: Defined by $\dot{x}(t) = f(x(t)) + g(x(t - \tau(t)))$, where $f$ is a cooperative vector field and $g$ is order-preserving. Both are assumed to be homogeneous of degree one.
2. Discrete-time systems: Defined by $x(k+1) = f(x(k)) + g(x(k - d(k)))$, where $f$ and $g$ are homogeneous of degree one and order-preserving.

The authors utilize the properties of cooperative (Metzler Jacobian) and order-preserving (monotone) vector fields. A key methodological step involves the use of a weighted $l_\infty$ norm, $\|x\|_v^\infty = \max_i |x_i|/v_i$, defined by a positive vector $v$. The stability analysis relies on finding a vector $v > 0$ that satisfies specific algebraic inequalities involving the vector fields evaluated at $v$.

For linear systems, the authors transform the stability conditions into optimization problems. By applying a logarithmic change of variables ($z_i = \ln v_i$), they demonstrate that the search for the optimal decay rate and the associated vector $v$ can be formulated as a convex optimization problem. This allows for the efficient computation of the tightest possible decay rate bounds provided by their theoretical framework.

Key Contributions and Results

1. Necessary and Sufficient Conditions for Exponential Stability:

   • Continuous-time: The paper establishes that a homogeneous cooperative system with bounded time-varying delays is globally exponentially stable if and only if there exists a vector $v > 0$ such that $f(v) + g(v) < 0$. This condition is identical to the delay-free case, confirming that exponential stability is delay-independent for this class of systems.
   • Discrete-time: Similarly, for discrete-time homogeneous monotone systems, global exponential stability holds if and only if there exists $v > 0$ such that $f(v) + g(v) < v$.

2. Explicit Decay Rate Bounds:

   • The authors derive explicit expressions to quantify the impact of delays on the decay rate.
   • For continuous-time systems, the decay rate $\eta$ is bounded by the unique positive solution to a transcendental equation involving the delay bound $\tau_{max}$:
     $$\left(\frac{f_i(v)}{v_i}\right) + \left(\frac{g_i(v)}{v_i}\right)e^{\eta_i \tau_{max}} + \eta_i = 0$$
   • For discrete-time systems, the decay factor $\gamma$ is derived from:
     $$\left(\frac{f_i(v)}{v_i}\right) + \left(\frac{g_i(v)}{v_i}\right)\gamma_i^{-d_{max}} = \gamma_i$$
   • The results show that as the delay bound increases, the guaranteed decay rate deteriorates (approaching zero for continuous-time and one for discrete-time).

3. Optimization for Linear Systems:

   • For positive linear systems (where $f(x)=Ax, g(x)=Bx$), the paper demonstrates that the best guaranteed decay rate can be found via convex optimization. The constraints are convex in the transformed variables, allowing for efficient numerical solution.
   • The framework is extended to general linear systems (not necessarily positive) by utilizing the absolute values of the system matrices ($A_M$ and $|B|$), ensuring stability if $(A_M + |B|)v < 0$.

4. Validation:

   • The theoretical results are validated through illustrative examples involving both nonlinear and linear systems (continuous and discrete). Simulations confirm that the derived upper bounds on the decay rate are tight and that the convex optimization approach yields significantly better bounds than using arbitrary feasible vectors.

Significance and Claims
The paper claims to extend the fundamental property of delay-independent stability from positive linear systems to a broader class of nonlinear positive systems (homogeneous cooperative and monotone systems) with time-varying delays. Unlike previous works that focused solely on asymptotic stability or constant delays, this paper provides:

• A rigorous proof that exponential stability for this class of systems is independent of the magnitude of bounded time-varying delays.
• Explicit, quantitative bounds on the decay rate as a function of the delay magnitude.
• A constructive method (convex optimization) to find the optimal decay rate for positive linear systems, addressing the lack of sharp characterizations in existing literature.

The authors note that while the results for homogeneous cooperative systems extend prior work on constant delays, the extension to time-varying delays and the provision of explicit decay rate bounds represent a significant advancement, particularly because standard techniques like linear Lyapunov-Krasovskii functionals are often insufficient or overly conservative in this context.