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Stability analysis and stabilization of stochastic linear impulsive, switched and sampled-data systems under dwell-time constraints
Corentin Briat
TL;DR
This paper addresses stability analysis and control for stochastic impulsive systems with noise in both continuous and discrete dynamics, a setting that prior work rarely treated together. It derives clock-dependent LMI conditions for dwell-time stability and extends them to state-feedback design and stochastic sampled-data systems. The paper reports conditions for several dwell-time types and applies the approach to analysis and control of stochastic sampled-data systems.
Problem
Few prior works address stochastic impulsive systems with noise in both continuous and discrete dynamics, despite their relevance to sampled-data systems.
Source: §1 · p. 2Method
The authors derive dwell-time-based mean-square stability conditions as clock-dependent LMIs and extend them to state-feedback stabilization and stochastic sampled-data systems.
Source: §1 · p. 2Results
The paper obtains mean-square stability conditions for constant, ranged, and minimum dwell-times and applies the approach to stochastic sampled-data analysis and control.
Source: §1 · p. 2, §6 · pp. 10–11Takeaways & Limitations
The conditions are convex in system matrices and can be relaxed using discretization or sum-of-squares techniques for analysis and control design.
Source: §1 · p. 2, §6 · p. 11Dynamic output-feedback design, observers, filters, induced-norm performance, other dwell-time concepts, and controller/mode mismatch from decision delays remain possible extensions.
Source: §6 · p. 11
Abstract
from arXiv · showhide
Impulsive systems are a very flexible class of systems that can be used to represent switched and sampled-data systems. We propose to extend here the previously obtained results on deterministic impulsive systems to the stochastic setting. The concepts of mean-square stability and dwell-times are utilized in order to formulate relevant stability conditions for such systems. These conditions are formulated as convex clock-dependent linear matrix inequality conditions that are applicable to robust analysis and control design, and are verifiable using discretization or sum of squares techniques. Stability conditions under various dwell-time conditions are obtained and non-conservatively turned into state-feedback stabilization conditions. The results are finally applied to the analysis and control of stochastic sampled-data systems. Several comparative examples demonstrate the accuracy and the tractability of the approach.
1 Introduction
The paper extends dwell-time stability analysis from deterministic impulsive systems to stochastic impulsive systems with noise in both continuous- and discrete-time dynamics. It develops clock-dependent conditions for stability and control, then applies them to stochastic sampled-data systems.
1 Introduction
- Earlier minimum-dwell-time LMIs were tractable but limited to certain linear time-invariant systems without uncertainty and did not yield state-feedback design conditions.The introduction motivates looped-functionals as a way to address some of these restrictions.Source: §1 · p. 1
- Looped-functionals support uncertain linear and nonlinear systems but lead to infinite-dimensional programs and are limited to stability analysis; clock-dependent conditions also support design.The paper notes that clock-dependent conditions can attain equivalent accuracy at lower computational cost.Source: §1 · pp. 1–2
- The paper studies systems with noise in both the continuous stochastic differential equation and discrete stochastic difference equation, while impulse times remain deterministic.The introduction notes that prior work rarely addressed noise in both parts together.Source: §1 · p. 2
- The authors derive mean-square stability conditions for constant, ranged, and minimum dwell-times, express them as clock-dependent LMIs, and extend them to state-feedback design and stochastic sampled-data systems.The conditions can be relaxed using piecewise-linear approximation or sum-of-squares programming; comparative examples demonstrate accuracy and tractability.Source: §1 · p. 2
2 Preliminaries
The preliminaries specify the stochastic impulsive system’s state, inputs, noise processes, and impulse timing. They define mean-square asymptotic stability and note how stability relates to persistent and nonpersistent impulses.
2 Preliminaries
- The system has continuous and discrete control inputs, and its impulse intervals satisfy T_k ≥ T_min > 0.Its continuous and discrete dynamics are driven by independent Wiener processes and independent, zero-mean, unit-variance random variables, respectively.Source: §2 · p. 2
- For system (1) with ud, uc ≡ 0, mean-square asymptotic stability is defined by E[||x(t)||2^2] tending to zero as t tends to infinity.Source: §2 · pp. 2–3
- If impulses cease after a finite time, asymptotic mean-square stability eventually becomes equivalent to stability of the continuous-time dynamics.The text also states that the result remains valid with state-dependent control inputs.Source: §2 · p. 3
3 Mean-square stability of stochastic linear impulsive systems
This section develops mean-square stability criteria for stochastic impulsive systems, with constant dwell-time conditions presented as equivalent characterizations. It also explains how clock-dependent conditions make the results amenable to tractable relaxations.
3.1 Constant dwell-time
- Under constant dwell-time, mean-square stability is characterized equivalently by a matrix transition condition and clock-dependent matrix-function conditions.The transition matrix is M(T̄) = exp(A T̄)J, with A = A ⊕ A + E_c ⊗ E_c and J = J ⊗ J + E_d ⊗ E_d.Source: §3.1 · p. 3
- Clock-dependent conditions use a matrix-valued function of elapsed time since the last impulse, with the clock reset at each impulse.The conditions are infinite-dimensional LMIs that can be relaxed using approximation techniques.Source: §3.1 · p. 4
- When noise terms are zero, the conditions reduce to the deterministic results; the clock-dependent LMI forms can be made tractable through relaxations.The expectation-based matrix condition is not directly tractable in its stated form.Source: §3.1 · p. 4
3.2 Ranged dwell-time
For ranged dwell-times, the section gives equivalent stability conditions and states that they guarantee mean-square asymptotic stability for every impulse sequence whose intervals stay within the specified range.
3.2 Ranged dwell-time
- If the ranged-dwell-time conditions hold, the system is mean-square asymptotically stable for any impulse sequence with T_k in [T_min, T_max].The conditions include an inequality required for every dwell-time in the interval and a matrix-valued function over [0, T_max].Source: §3.2 · p. 4
3.3 Minimum dwell-time
For minimum dwell-time, the paper gives equivalent stability conditions and concludes that their satisfaction guarantees mean-square asymptotic stability whenever every inter-impulse interval is at least the specified bound.
3.3 Minimum dwell-time
- Theorem 7 gives equivalent conditions for minimum-dwell-time stability, including an LMI condition and a clock-dependent matrix-function condition.The function S is positive at the dwell-time bound and periodic thereafter; the conditions hold over the interval from zero to that bound.Source: §3.3 · p. 5
- If any equivalent condition holds, the system is mean-square asymptotically stable for every impulse sequence with intervals Tk ≥ T̄.The result concerns the system with both control inputs set to zero.Source: §3.3 · p. 5
- The proof of the reverse implication constructs an expected quadratic Lyapunov quantity and uses its nonincreasing behavior to establish stability.The construction starts from f_T̄(θ) = E[Φ(T̄ + θ)^T PΦ(T̄ + θ)].Source: §3.3 · p. 5
3.4 Computational aspects
The stability tests are infinite-dimensional, so the paper presents piecewise-linear and sum-of-squares relaxations that make them tractable while retaining links to the original conditions.
3.4 Computational aspects
- The original stability conditions are infinite-dimensional LMI feasibility problems, motivating two relaxations that turn them into tractable tests.The relaxations are described for Theorem 5(iv), with similar ones available for Theorems 6 and 7.Source: §3.4 · p. 5
3.4.1 Piecewise linear approach
- The piecewise-linear approach restricts the matrix-valued functions to piecewise-linear form, yielding finite-dimensional LMIs checkable at interval vertices.The resulting conditions can be solved with standard SDP solvers, and feasibility is ensured for sufficiently large discretization order when the original conditions are feasible.Source: §3.4.1 · pp. 5–6
3.4.2 Sum of squares programs
- The sum-of-squares approach uses polynomial matrix functions; feasibility certifies mean-square asymptotic stability under constant dwell-time.For sufficiently high polynomial degree, feasibility follows when the original conditions are feasible; SOS is generally more tractable than piecewise-linear approximation requiring large discretization order.Source: §3.4.2 · p. 6
3.5 Application to switched systems
The paper extends its impulsive-system stability results to stochastic switched systems by representing switching through multiple reset maps and applying the minimum-dwell-time conditions.
3.5 Application to switched systems
- The impulsive-system stability results also apply to linear stochastic switched systems represented with multiple jump maps, which act as reset maps.The switched model is reformulated in the impulsive-system form before applying the conditions.Source: §3.5 · p. 6
- Under the stated conditions, the linear stochastic switched system is mean-square asymptotically stable under minimum dwell-time.The result adapts Theorem 7 to the block-diagonal reformulation of the switched model.Source: §3.5 · p. 6
- With a piecewise-linear approximation, the compact condition recovers the conditions of [33, Theorem 1].The paper notes that its condition includes those obtained in.Source: §3.5 · p. 7
3.6 Examples
The examples report accurate dwell-time estimates for one system and compare ranged-dwell-time estimates with a gridded quadratic-stability test for another.
3.6 Examples
- The proposed method is reported as very accurate for the first example, with minimum- and constant-dwell-time results quite close there.The text cautions that the closeness is not a general rule and says deterministic results are retrieved while stability deteriorates as κ and δ increase.Source: §3.6 · p. 7
- For the first example, the tables report estimated smallest constant dwell-times and estimated minimum dwell-times across κ and δ using degree-6 matrix polynomials.The deterministic case corresponds to κ = δ = 0.Source: Table 1 · p. 7, Table 3 · p. 7, §3.6 · p. 7
- For the second example, the ranged-dwell-time analysis estimates Tmax with Tmin = 0.01 and compares it with a gridded quadratic-stability test.The gridded test finds slightly larger Tmax values, but at higher computational cost and with gridding imprecision; its exponential conditions apply only to time-invariant systems, unlike Theorem 6's conditions for time-varying uncertainties or parameters.Source: §3.6 · p. 7, Table 4 · p. 8, Table 5 · p. 8
4 Mean-square stabilization of stochastic linear impulsive systems
The section extends stochastic impulsive-system results to state-feedback stabilization under ranged and minimum dwell-time constraints.
4.1 Stabilization under ranged dwell-time
- Under ranged dwell-time, the stated conditions establish mean-square asymptotic stability and provide a way to compute suitable controller gains.The dwell-time sequence satisfies T_k ∈ [T_min, T_max].Source: §4.1 · p. 8
4.2 Stabilization under minimum dwell-time
- Under minimum dwell-time, the stated conditions likewise establish mean-square asymptotic stability and provide a way to compute suitable controller gains.The minimum dwell-time is T̄, and the continuous-time gain is defined over [0, T̄].Source: §4.2 · p. 9
4.3 Example
- With minimum dwell-time T̄ = 0.1, the example’s closed-loop state-norm trajectory converges to zero, illustrating mean-square asymptotic stability.The controller is obtained using Theorem 14 with polynomials of orders 1 and 2.Source: §4.3 · p. 9
5 Application to aperiodic sampled-data systems
The section applies ranged-dwell-time stabilization to aperiodic stochastic sampled-data systems. The stated result guarantees mean-square asymptotic stability under the considered dwell-time range, and the example reports similar convergence rates for controllers with different input-update times.
5 Application to aperiodic sampled-data systems
- The sampled-data model includes a piecewise-constant state that models the zero-order hold.The description also specifies independent zero-mean Wiener processes and a parameter scaling noise amplitude on the control channel.Source: §5 · p. 10
- The sampled-data result establishes mean-square asymptotic stability for T_k ∈ [T_min, T_max] and gives a formula for computing a suitable controller gain.The application derives a stabilization condition using the ranged-dwell-time result.Source: §5 · pp. 9–10
- The example’s designed controllers yield similar convergence rates despite having different time-scales for control-input updates.The result is obtained using Theorem 15 with polynomials of order 2; Table 6 lists controller gains.Source: §5 · p. 10, Table 6 · p. 10
6 Conclusion
The paper obtains dwell-time stability and stabilization conditions for stochastic impulsive systems. These are infinite-dimensional LMI problems solvable by discretization or sum-of-squares techniques, and include switched and sampled-data systems as particular cases.
6 Conclusion
- Dwell-time stability and stabilization conditions for stochastic impulsive systems are expressed as infinite-dimensional LMIs solvable by discretization or sum-of-squares techniques.The approach also includes switched and sampled-data systems as particular cases.Source: §6 · pp. 10–11
- Fig. 2 tracks state and control norms under ranged dwell-time conditions, with randomly generated impulse sequences and Tmin = 0.001 across several Tmax values.Source: Figure 2 · p. 11, §6 · p. 11
- Switched systems and sampled-data systems are included as particular cases of the approach.Source: §6 · p. 11
- Future work includes dynamic output-feedback controllers, observers, filters, induced-norm performance characterization, other dwell-time concepts, and controller/mode mismatch due to decision delays.Source: §6 · p. 11