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

I. INTRODUCTION

• Few studies address nonlinear positive-system stability with time-varying delays, and no sharp characterization relates a maximum delay bound to the guaranteed decay rate.Methods used for constant delays, including linear Lyapunov-Krasovskii functionals, may not apply or may be excessively conservative when delays vary.Source: §I · p. 2
• The paper derives necessary and sufficient exponential-stability conditions for homogeneous, cooperative continuous-time positive systems and corresponding results for discrete-time positive systems.The introduction also specifies homogeneous vector fields of degree one for the continuous-time case and homogeneous order-preserving vector fields for the discrete-time case.Source: §I · pp. 2–3
• For bounded time-varying delays, the paper gives explicit expressions bounding decay rates of the considered systems.The introduction states that these expressions apply when the delays have a known upper bound.Source: §I · p. 2
• The best decay rate that the paper's bound provides for positive linear systems can be found using convex optimization.Source: §I · p. 3
• The results are extended to general linear systems with time-varying delays, with corresponding counterparts for discrete-time positive systems.Source: §I · p. 3

• Vectors use bold lowercase letters, matrices use capitals, and the nonnegative orthant and componentwise order are specified.For matrices, |A| denotes the matrix of absolute entry values; Metzler matrices have nonnegative off-diagonal entries.Source: §II · p. 3
• The weighted l∞ norm is defined for a positive weight vector, and C denotes continuous real-valued functions on an interval with values in R^n.Source: §II · p. 3
• The section introduces cooperative vector fields and order-preserving vector fields as key concepts for the results that follow.Source: §II · pp. 3–4
• Homogeneity of degree α means f(λx) = λ^αf(x) for all x and λ > 0; degree one is the α = 1 case.Source: §II · p. 4

• The analysis assumes homogeneous vector fields, a cooperative f, and an order-preserving g; these properties ensure positivity of the system.The initial function specifies the system's initial state, and the delay is bounded above by τmax.Source: §III · pp. 4–5
• A vector v > 0 with f(v) + g(v) < 0 is equivalent to global exponential stability for all bounded delays under Assumption 1.The theorem states this condition for the homogeneous cooperative system.Source: §III · p. 5
• The guaranteed decay rate decreases as the delay upper bound τmax grows and approaches zero as τmax tends to infinity.Any v satisfying the stability condition can be used to compute the associated rate.Source: §III · p. 6
• The stability condition extends to multiple delays by requiring f(v) plus the sum of the delayed vector fields at v to be negative.The extension assumes f is cooperative and degree-one homogeneous, and each delayed field is homogeneous and order-preserving.Source: §III · p. 6
• For positive linear systems, the necessary and sufficient stability condition is a linear program, while the best guaranteed decay rate is found by convex optimization.The optimization uses z_i = ln(v_i) and is convex in the decay rate and z.Source: §III · pp. 5–7
• The result extends to general linear systems: the stability condition is delay-independent and is equivalent to AM + |B| being Hurwitz.The theorem gives global exponential stability when its stated condition holds.Source: §III · pp. 7–8

IV. DISCRETE-TIME CASE

• Theorem 4 uses a positive vector v satisfying f(v)+g(v)<v as a test for global exponential stability under all bounded delays.The system is assumed homogeneous and order-preserving.Source: §IV · pp. 8–9
• The bound quantifies how increasing the maximum delay slows the guaranteed decay rate: γ_i increases with dmax and approaches one as dmax grows.Source: §IV · p. 9
• For discrete-time positive linear systems, Theorem 5 gives a necessary-and-sufficient condition for global exponential stability under all bounded delays.Source: §IV · p. 9
• A logarithmic change of variables formulates the search for the best guaranteed linear-system decay rate as a convex optimization problem that can be solved globally and efficiently.Source: §IV · p. 10
• The section extends the linear-system result: the stated positive-vector condition implies global exponential stability for the discrete-time linear system.Source: §IV · p. 10

A. Continuous-time Nonlinear Positive System

• The example is globally exponentially stable for all bounded time delays because f(1,1)+g(1,1)<0.Source: A. Continuous-time Nonlinear Positive… · p. 11
• For τ(t)=5+sin(t) and τmax=6, the computed values are η1=0.0825 and η2=0.1705; Fig. 1 compares the actual decay rate with its upper bound.Source: A. Continuous-time Nonlinear Positive… · p. 11, Figure 1 · p. 11
• The cited [14, Theorem 4.1] cannot ascertain stability in this example because its delay is time-varying.Source: A. Continuous-time Nonlinear Positive… · p. 11

B. Continuous-time Linear Positive System

• Because A+B is Hurwitz, the positive linear system is exponentially stable for any bounded time-varying delays.Source: B. Continuous-time Linear Positive… · p. 12
• The example uses the eigenvector of A+B associated with π(A+B) as an initial candidate for v in the decay-rate bound.Source: B. Continuous-time Linear Positive… · pp. 12–13
• The optimized choice guarantees a decay rate of 0.0838, compared with 0.0583 from the eigenvector-based choice, and its bound matches simulations very well.Source: B. Continuous-time Linear Positive… · p. 13

C. Discrete-time Linear Positive System

• Since A and B are non-negative and ρ(A+B)<1, Theorem 5 guarantees exponential stability for the example with an upper delay bound of 5.Source: C. Discrete-time Linear Positive System · p. 14
• The section solves the convex optimization problem to find v⋆ and its guaranteed decay-rate bound for the discrete-time system.Source: C. Discrete-time Linear Positive System · p. 14
• Fig. 3 compares the upper bound with the actual decay rate of x(k) for the discrete-time example.The stated initial condition is φ(k)=v⋆.Source: C. Discrete-time Linear Positive System · p. 14, Figure 3 · p. 15

VI. CONCLUSIONS

• Continuous-time homogeneous cooperative systems and discrete-time homogeneous monotone systems are insensitive to bounded time-varying delays.The paper derives necessary and sufficient conditions for delay-independent exponential stability of these systems.Source: §VI · p. 15
• With a known delay upper bound, explicit expressions bound the decay rate, and the best guaranteed bound for positive linear systems can be found via convex optimization.Source: §VI · p. 15
• The results are extended to general linear systems with time-varying delays.Source: §VI · p. 15

A. Proof of Theorem 1

• For Theorem 1, the proof assumes a positive vector satisfying condition (2) and uses a positive rate associated with equation (3).Equation (3) is stated to have a unique positive solution for each component.Source: A. Proof of Theorem 1 · p. 15
• By contradiction, there are an index m and t1 ≥ 0 with z_i(t) ≤ 0 for every i on [0, t1] and z_m(t1) = 0; at t1, x_m(t1) = ||ϕ||e^(−ηt1)v_m and x(t1) ≤ ||ϕ||e^(−ηt1)v.Source: A. Proof of Theorem 1 · p. 16
• The converse assumes exponential stability for all bounded delays and invokes stability of the delay-free aggregate to obtain the required positive vector.Source: A. Proof of Theorem 1 · p. 17

B. Proof of Theorem 3

• Theorem 3 reuses Theorem 1’s boundary-contact argument with |x_i(t)| in the envelope, yielding the corresponding exponential bound.Source: B. Proof of Theorem 3 · p. 17

C. Proof of Theorem 4

• For Theorem 4, the proof defines a componentwise contraction factor from equation (12), then uses induction to establish the exponential bound.The proof uses homogeneity, order preservation, and the known upper bound on the delay.Source: C. Proof of Theorem 4 · p. 18
• The converse uses global asymptotic stability of the delay-free aggregate f+g to obtain the condition for Theorem 4.The proof cites Propositions 5.2 and 5.6 of [20].Source: C. Proof of Theorem 4 · p. 18

D. Proof of Theorem 6

• Theorem 6’s proof uses induction for the desired bound, with equation (17) providing a unique contraction factor in (0, 1).The remaining argument is described as similar to the proof of Theorem 4.Source: D. Proof of Theorem 6 · pp. 18–19