Positive Linear Systems Control and Stability Analysis

Positive linear systems form a class of dynamical models in which all state variables and outputs remain nonnegative whenever initiated from nonnegative conditions. This intrinsic positivity arises naturally in applications such as population dynamics, chemical reaction networks, epidemiology and resource allocation in engineering networks. Control methodologies for these systems must respect the positivity constraint, leading to specialised design techniques based on copositive Lyapunov functions, linear programming and state-feedback laws that guarantee nonnegativity. Stability analysis often employs monotonicity properties and separable Lyapunov constructions, enabling the derivation of conditions under which trajectories converge to equilibrium points or desired operating regimes. In recent years, research has extended classical results to accommodate time-varying delays, switching between subsystems, impulsive effects and uncertain parameters, all within a positivity framework. Practical implementations have demonstrated how structured controllers can achieve robust stability, finite-time performance and specified gain bounds, ensuring safe and efficient operation in biological reactors, communication networks and logistic supply chains.

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Researchers have investigated robust stability of switched positive systems subject to time-varying delays and interval uncertainties, even when each individual subsystem is unstable. By introducing time-scheduled multiple copositive Lyapunov–Krasovskii functionals together with a mode-dependent dwell-time switching strategy, new delay-dependent conditions have been established that guarantee global uniform asymptotic stability. These criteria tighten previous results and reduce conservatism, while linear programming facilitates practical implementation. Numerical examples illustrate successful regulation of uncertain biochemical network models and time-delay circuits.

Another line of work addresses practical stability for time-varying positive systems with delays. Using max-separable Lyapunov–Krasovskii functionals, sufficient conditions for boundedness and convergence to a neighbourhood of equilibrium have been derived. Building on these analytical foundations, state-feedback controllers—both with and without input delay compensation—have been designed to meet prescribed performance levels. Applications to population models and process control demonstrate the effectiveness of the proposed design, particularly when operating conditions vary over time.

Advances have also been made in the stability and input-output gain analysis of periodic piecewise positive systems with constant delays. By exploiting the periodic structure and employing copositive Lyapunov–Krasovskii functionals, explicit linear inequalities have been formulated to characterise exponential decay rates and the L₁-gain of the closed-loop system. This framework yields computationally tractable criteria that accommodate switching between different operational modes, offering a pathway to guaranteed performance in periodic chemical reactors and networked scheduling systems.

Positive Linear Systems Control and Stability Analysis publication trend

The graph below shows the total number of articles in positive linear systems control and stability analysis across all publications each year (not limited to Nature Index journals).

Technical terms

Positive linear system: A dynamical system whose state and output variables remain nonnegative for nonnegative initial conditions and inputs.

Copositive Lyapunov function: A scalar function that is positive over the nonnegative orthant and decreases along system trajectories, used to certify stability of positive systems.

Lyapunov–Krasovskii functional: An extension of Lyapunov functions incorporating integral terms to handle systems with time delays.

Mode-dependent dwell time: A switching constraint that prescribes a minimum time interval for which each subsystem must remain active to ensure overall stability.

L₁-gain: A measure of the worst-case amplification from input to output, defined by the ratio of signal norms in the L₁ sense.

References

1. Separable Lyapunov functions for monotone systems: Constructions and limitations. Discrete and Continuous Dynamical Systems - B (2015).
2. Global Uniform Asymptotic Stability Criteria for Linear Uncertain Switched Positive Time-Varying Delay Systems with All Unstable Subsystems. Mathematics (2020).
3. Practical stability of time‐varying positive systems with time delay. IET Control Theory and Applications (2021).
4. Stability and $L_{1}$-Gain Analysis of Periodic Piecewise Positive Systems With Constant Time Delay. IEEE Transactions on Automatic Control (2021).