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Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems
Corentin Briat
TL;DR
The paper addresses stability and stabilization of linear positive impulsive systems under varied dwell-time assumptions, including settings with limited prior results. It develops copositive-Lyapunov and lifted linear-programming conditions, computational relaxations, and switched-system extensions. The results provide stability and stabilization conditions across these regimes, with examples comparing computational approaches and showing exact or less conservative estimates in selected cases.
Problem
Linear positive impulsive systems have relatively few existing results, motivating stability and stabilization conditions across arbitrary, constant, minimum, maximum, and range dwell-times.
Method
The paper uses linear copositive Lyapunov functions, clock-variable lifting, infinite-dimensional linear programs, and three computational relaxation methods, then reformulates switched systems as impulsive systems.
Results
The paper obtains stability and stabilization conditions for positive impulsive and switched systems, with examples including an exact minimum dwell-time estimate of 0.2311 and faster SOS than gridding in selected comparisons.
Takeaways & Limitations
The framework supplies design-oriented dwell-time conditions and computational procedures for analyzing and stabilizing linear positive impulsive and switched systems.
Takeaways & Limitations
The paper identifies extensions to input gains, interval observers, polyhedral Lyapunov functions, and positive or monotone nonlinear systems as future work.
Abstract
from arXiv · showhide
Several results regarding the stability and the stabilization of linear impulsive positive systems under arbitrary, constant, minimum, maximum and range dwell-time are obtained. The proposed stability conditions characterize the pointwise decrease of a linear copositive Lyapunov function and are formulated in terms of finite-dimensional or semi-infinite linear programs. To be applicable to uncertain systems and to control design, a lifting approach introducing a clock-variable is then considered in order to make the conditions affine in the matrices of the system. The resulting stability and stabilization conditions are stated as infinite-dimensional linear programs for which three asymptotically exact computational methods are proposed and compared with each other on numerical examples. Similar results are then obtained for linear positive switched systems by exploiting the possibility of reformulating a switched system as an impulsive system. Some existing stability conditions are retrieved and extended to stabilization using the proposed lifting approach. Several examples are finally given for illustration.
1. Introduction
The paper studies stability and stabilization of linear positive impulsive systems across multiple dwell-time regimes, motivated by their applications and limited prior results. It uses copositive Lyapunov functions, linear-programming formulations, lifting, and polynomial-positivity relaxations, then extends the framework to switched systems.
- Linear positive systems model communication, biological, epidemiological, and disease-dynamics networks while also possessing theoretically useful properties.
- Positive impulsive systems represent biochemical, population, epidemiological, sampled-data, and switched-system processes, but have received relatively few prior results.
- The paper derives stability and stabilization conditions for arbitrary, constant, minimum, maximum, and range dwell-times.
- The approach uses linear copositive Lyapunov functions and formulates conditions as finite-dimensional or semi-infinite linear programs.
- Lifting introduces clock-dependent conditions for design and uncertain systems, while Handelman and Putinar polynomial-positivity methods support finite-dimensional computation.
- The paper applies the impulsive-system results to switched systems after outlining preliminary definitions and results.
2. Preliminaries
The preliminaries define the positive impulsive-system model, its dwell-times, and stability equivalences used throughout the paper. They also connect asymptotic and exponential stability through a change of variables.
- The impulsive model evolves continuously as x˙(t)=Ax(t) and resets as x(t+)=Jx(t), with nonnegative state and initial condition.
- Impulse times are strictly increasing and unbounded, excluding accumulation points, and dwell-times are T_k=t_{k+1}−t_k.
- The system is positive exactly when A is Metzler and J is nonnegative.
- Asymptotic stability of the impulsive system is equivalent to stability of its dual and several associated discrete-time systems.
- Equivalent stability representations let the analysis choose a convenient system, although their resulting stability conditions need not be equivalent.
- The change z(t)=e^{αt}x(t), α>0, extends asymptotic-stability results to α-exponential stability, with discrete-time rate sup_k e^{−αT_k}.
3. Stability of linear positive impulsive systems
This section organizes the paper’s stability analysis for positive impulsive systems by dwell-time regime and introduces computational comparisons concerning conservatism.
- The paper analyzes arbitrary, constant, minimum, maximum, and range dwell-time stability in successive sections.
- Computational results accompany the stability analysis and discuss the conservatism of the resulting conditions.
3.1. Stability under arbitrary dwell-time
For arbitrary dwell-times, the paper characterizes stability using positive-system conditions and a common linear copositive Lyapunov function. These conditions guarantee asymptotic stability for every positive dwell-time sequence.
- For Metzler A and nonnegative J, the paper gives equivalent arbitrary-dwell-time stability statements.
- The resulting conditions guarantee asymptotic stability for every impulse sequence with T_k∈(0,∞).
- A linear copositive Lyapunov function V(x)=λᵀx decreases along flows and contracts at jumps under the stated conditions.
- The conditions coincide with common-copositive-Lyapunov stability conditions for a positive switched system with modes A and J−I_n.
- Dual, polyhedral, and persistent-flow formulations provide alternative conditions, but examples show that equivalent stability systems can yield non-equivalent conditions.
3.2. Stability under constant dwell-time
For constant dwell-time, asymptotic stability is characterized equivalently through discrete-time, copositive-Lyapunov, and clock-dependent conditions. The clock-dependent formulations are affine in A and J but require infinite-dimensional linear programming.
- Constant dwell-time: Constant dwell-time means jumps occur periodically with T_k = T̄ for every k.The section develops the positive-systems counterpart of constant-dwell-time results using the discrete-time system induced by one flow interval and one jump.
- Equivalent stability conditions: Theorem 3.5 gives equivalent conditions for asymptotic stability under constant dwell-time T̄.The theorem includes discrete-time Lyapunov and clock-dependent formulations.
- Equivalent stability conditions: Stability is equivalent to Schur stability of the nonnegative matrix Je^{A T̄}.This provides a discrete-time test for the periodic impulsive system.
- Clock-dependent formulation: The clock-dependent conditions are affine in A and J, enabling extensions to uncertain matrices and control design.Only endpoint vectors need positivity, rather than the entire vector-valued functions.
- Computational limitation: The clock-dependent conditions are infinite-dimensional linear programs that may be hard to solve computationally.The paper notes they remain less complex than the corresponding infinite-dimensional semidefinite programs cited there.
3.3. Stability under minimum dwell-time
Under a minimum dwell-time constraint, the paper gives equivalent stability conditions based on discrete-time dynamics, common copositive Lyapunov functions, and clock-dependent formulations. These results also connect impulsive-system stability to switched-system Lyapunov analysis and support affine uncertain-system or design formulations.
- Minimum dwell-time setting: Minimum dwell-time assumes T_k ≥ T̄, together with persistent impulses that exclude an infinite dwell interval.The analysis does not impose an a priori upper bound on dwell-times, but restricts attention to sequences with bounded intervals.
- Equivalent stability conditions: Theorem 3.7 gives equivalent conditions for asymptotic stability under minimum dwell-time T̄.The conditions include discrete-time and clock-dependent Lyapunov formulations.
- Connection to switched systems: The results connect minimum-dwell-time stability of an impulsive system to a common linear copositive Lyapunov function for a two-subsystem positive switched system.The switched matrices are A and Je^{A T̄} − I_n, or A and e^{A T̄}J − I_n for the swapped system.
- Uncertainty and design: The affine clock-dependent conditions can be used for uncertain matrices and design, whereas direct common-Lyapunov conditions are difficult for those purposes.The arbitrary-dwell-time results are recovered by letting T̄ approach zero.
3.4. Stability under maximum dwell-time
For maximum dwell-time, the paper provides equivalent stability conditions and an asymptotic-stability result under T_k ≤ T̄. A positivity requirement simplifies one semi-infinite feasibility problem but imposes a scope boundary on the flow matrix.
- Maximum dwell-time setting: Maximum dwell-time assumes T_k ≤ T̄ for every impulse interval.The result is formulated as a positive-systems counterpart of existing maximum-dwell-time conditions.
- Equivalent stability conditions: Theorem 3.9 gives equivalent stability conditions for the maximum-dwell-time case.The conditions include Lyapunov, clock-dependent, and switched-system formulations.
- Stability conclusion: When the theorem’s conditions hold, the linear positive impulsive system is asymptotically stable under maximum dwell-time T̄.This is the principal stability conclusion for the section.
- Scope and simplification: Requiring positive μ, equivalently a positive λ in the other formulations, is equivalent to −A being Hurwitz stable.This requirement excludes unstable A with stable eigenvalues but converts a semi-infinite feasibility problem into a finite-dimensional one.
- Alternative formulation: If −A is not Hurwitz stable, maximum-dwell-time conditions can still be obtained from the range-dwell-time results.The paper proposes setting the lower dwell-time bound to zero or a very small positive value.
3.5. Stability under range dwell-time
For dwell-times constrained to an interval [T_min, T_max], the paper provides equivalent discrete-time and clock-dependent stability conditions. The range formulation also supplies a route to maximum-dwell-time analysis when −A is not Hurwitz stable.
- Range dwell-time setting: Range dwell-time assumes T_k ∈ [T_min, T_max] with 0 < T_min ≤ T_max < ∞.Both lower and upper dwell-time bounds are prescribed.
- Equivalent stability conditions: Theorem 3.12 gives equivalent stability conditions for the range-dwell-time case.The conditions include a discrete-time Lyapunov formulation and a differentiable clock-dependent vector function.
- Clock-dependent formulation: The clock-dependent inequalities are required for every τ ∈ [0, T̄] and every θ ∈ [T_min, T_max].This expresses stability uniformly over the clock variable and the allowed dwell-time range.
- Relation to maximum dwell-time: The range-dwell-time theorem can yield maximum-dwell-time conditions by setting T_min = 0 or a very small value when −A is not Hurwitz stable.The paper gives 10^-5 as an example of a small positive lower bound.
3.6. Computational considerations
The paper compares three computational relaxations for infinite-dimensional stability conditions: piecewise linear, sum-of-squares, and Handelman-based approaches. Each relaxation yields finite-dimensional programs, with asymptotic exactness established for the considered conditions.
- Computational methods: Three methods are developed to verify infinite-dimensional stability conditions: piecewise linear approximation, sum-of-squares programming, and Handelman-based relaxation.The methods are presented as computational approaches for the conditions associated with Theorem 3.5(e).
- Piecewise linear approach: The piecewise linear approach approximates the functions ζ or ξ and relaxes infinite-dimensional conditions into finite-dimensional linear matrix inequalities.
- Asymptotic exactness: The piecewise linear, sum-of-squares, and Handelman relaxations are asymptotically exact when the original Theorem 3.5(e) conditions hold.Feasibility is guaranteed for sufficiently large discretization order or polynomial degree, respectively.
- Sum-of-squares approach: The sum-of-squares approach represents polynomial nonnegativity through semidefinite programming, with componentwise sum-of-squares constraints for polynomial vectors.SOS feasibility can be formulated as a semidefinite program and solved with standard semidefinite-programming solvers.
- Stability certification: Under suitable polynomial-vector assumptions, the sum-of-squares and Handelman conditions imply asymptotic stability under the specified constant dwell-time.The proof uses nonnegativity over the dwell-time interval to recover the required Lyapunov decrease condition.
- Handelman-based approach: The Handelman-based approach uses nonnegative coefficients of polynomial products over compact polytopes, producing finite-dimensional linear programming conditions.The resulting formulation uses linear programming rather than semidefinite programming.
3.7. Examples
The examples evaluate dwell-time estimates and computational methods across minimum, maximum, and range dwell-time settings. The reported results include exact estimates in several cases and faster SOS solutions than gridded alternatives in the comparisons provided.
- Minimum dwell-time: 0.2311 is the estimated minimum dwell-time in Example 3.22, matching the exact lower bound and improving on the previously reported 1.0986.
- Minimum dwell-time: 0.2443 is obtained for the minimum dwell-time when δ = 1, while for δ = 3 the compared conditions yield 0.4290 and 0.3615.The value 0.3615 also matches the constant-dwell-time result and the quadratic-Lyapunov estimate.
- Maximum dwell-time: 4.6051 is the estimated maximum dwell-time in Example 3.24, matching the constant-dwell-time value and improving on the reported 1.2040.
- Maximum dwell-time: The SOS method is faster than gridding for the maximum-dwell-time example, while gridding is non-exact because it checks only 201 points in the interval.
- Method comparison: Table 1 compares minimum-dwell-time estimates, optimization-problem sizes, and solving times for discretization, SOS, and Handelman methods using SeDuMi.
- Range dwell-time: For the range-dwell-time example, the interval (0.2779, 0.6056) yields a Schur-stable matrix product, and SOS again solves faster than the gridded approach.
4. Stabilization of positive linear impulsive systems
The paper develops stabilization conditions for positive linear impulsive systems across multiple dwell-time regimes, using positivity and clock-dependent lifting to support controller design. The resulting controllers ensure positive, asymptotically stable closed-loop systems under arbitrary, constant, minimum, maximum, and range dwell-time conditions.
- Scope: The stabilization framework covers arbitrary, constant, minimum, maximum, and range dwell-time constraints for positive linear impulsive systems.The continuous and discrete feedback inputs are treated across these dwell-time regimes.
- Arbitrary dwell-time: For arbitrary dwell-time, linear-program conditions in X, Uc, and Ud yield controller gains Kc = UcX−1 and Kd = UdX−1 that ensure positivity and asymptotic stability.The closed-loop conditions are [AX + BcUc]1n < 0 and [JX + BdUd − X]1n < 0.
- Constant dwell-time: For constant dwell-time, clock-dependent continuous gains Kc(τ) and a discrete gain Kd stabilize the system by repeating a finite-time design over successive dwell-time intervals.The clock measures elapsed time since the last impulse and anticipates the next impulse time.
- Minimum dwell-time: For minimum dwell-time, the controller gain becomes constant after the minimum dwell-time threshold while preserving positive, asymptotically stable closed-loop behavior.The gain-locking structure is used over intervals whose duration may exceed the minimum dwell-time.
- Maximum and range dwell-time: The same stabilization pattern is extended to maximum and range dwell-time, with equivalent conditions stated for positive asymptotic stability.The maximum-dwell-time and range-dwell-time results are given as Theorems 4.4 and 4.5.
- Examples: Simulations with randomly generated impulse times satisfying the relevant dwell-time conditions show the designed controllers driving the closed-loop states to zero.The reported validations use Figures 1–3 for different closed-loop designs.
5. Application to linear positive switched systems
The paper reformulates linear positive switched systems as impulsive systems, applying dwell-time stability and stabilization results across arbitrary, minimum, and range dwell-time switching. It retrieves existing arbitrary-switching stability conditions, derives stabilization conditions, and illustrates the results computationally.
- 5.1. Switched systems as impulsive systems: Any linear switched system is reformulated as an impulsive reset system with lifted block-diagonal dynamics and mode-transition jump maps.The lifted state and input stack the mode-specific states and inputs, while jump maps encode transitions between modes.
- 5.2. Stability under arbitrary switching: Arbitrary-switching stability is characterized by equivalent linear copositive conditions that ensure asymptotic stability when all mode matrices are Metzler.The conditions can be written using either row-vector or column-vector forms involving the lifted flow and transition maps.
- 5.3. Minimum dwell-time switching: Minimum-dwell-time conditions provide equivalent stability tests and corresponding controller-design conditions for positive switched systems.The stabilization result uses clock-dependent matrix-valued gains whose closed-loop flow remains Metzler over the dwell-time interval.
- 5.4. Range dwell-time switching: Range-dwell-time conditions establish asymptotic stability and positive stabilization when switching intervals lie within a mode-dependent range.The paper states both an uncontrolled stability result and a controller-existence result for range dwell-time switching.
- 5.5. Examples: Numerical examples show that sum-of-squares conditions approach the minimum-dwell-time values from the linear-program conditions as polynomial degree increases, while SOS is more accurate and faster than gridding in one range-dwell-time example.A simulated minimum-dwell-time trajectory also displays the stabilizing effect of the controller.
6. Conclusion
The paper develops dwell-time stability and stabilization conditions for linear positive impulsive and switched systems, including lifted formulations suitable for design. It closes by identifying extensions to performance analysis, observers, polyhedral Lyapunov functions, and positive or monotone nonlinear systems.
- 6. Conclusion: The paper obtains stability and stabilization conditions for linear positive impulsive and switched systems using dwell-time formulations and lifting-based design conditions.Because the lifted conditions are infinite-dimensional, the paper proposes finite-dimensional relaxation approaches and illustrates them with examples.
- 6. Conclusion: Future work includes input-gain analysis, delayed and Markov jump extensions, interval-observer design, polyhedral Lyapunov functions, and positive or monotone nonlinear systems.The stated extensions include L1-, L2-, and L∞-gain analysis for positive impulsive systems with inputs.