Dwell Time Stability Research Articles

Dwell-time stability conditions for infinite dimensional impulsive systems

We consider nonlinear impulsive systems on Banach spaces subjected to disturbances and look for dwell-time conditions guaranteeing the ISS property. In contrary to many existing results our conditions cover the case where both continuous and discrete dynamics can be unstable simultaneously. Lyapunov type methods are used for this purpose. The effectiveness of our approach is illustrated on a rather nontrivial example, which is feedback connection of an ODE and a PDE system.

Continuous-time switched systems with switching frequency constraints: Path-complete stability criteria

We propose a novel Lyapunov construction for continuous-time switched systems relying on a graph theoretical Lyapunov construction. Starting with a finite family of continuously differentiable functions, suitable inequalities involving these functions and the vector fields defining the switched system are encoded in a direct and labeled graph. We then provide sufficient conditions for (asymptotic) stability subject to constrained switching times, by relying on the path-completeness of the chosen graph. The analysis is first carried out under the hypothesis of constant switching frequency. Then, the results are generalized to dwell time setting. In the case of linear dynamics, the graph formalism allows us to interpret the existing results on dwell time stability in a unified language. Some numerical examples illustrate the usefulness of the conditions.

Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers

Linear Parameter-Varying (LPV) systems with jumps and piecewise differentiable parameters is a class of hybrid LPV systems for which no tailored stability analysis and stabilization conditions have been obtained so far.11Except, of course, in the conference version of this work. We fill this gap here by proposing an approach based on a clock- and parameter-dependent Lyapunov function yielding stability conditions under both constant and minimum dwell-times. Interesting adaptations of the latter result consist of a minimum dwell-time stability condition for uncertain LPV systems and LPV switched impulsive systems. The minimum dwell-time stability condition is notably shown to naturally generalize and unify the well-known quadratic and robust stability criteria all together. Those conditions are then adapted to address the stabilization problem via timer-dependent and a timer- and/or parameter-independent (i.e. robust) state-feedback controllers, the latter being obtained from a relaxed minimum dwell-time stability condition involving slack-variables. Finally, the last part addresses the stability of LPV systems with jumps under a range dwell-time condition which is then used to provide stabilization conditions for LPV systems using a sampled-data state-feedback gain-scheduled controller. The obtained stability and stabilization conditions are all formulated as infinite-dimensional semidefinite programming problems which are then solved using sum of squares programming. Examples are given for illustration.

Exponential stability analysis for singular switched positive systems under dwell-time constraints

The article addresses the exponential stability analysis for singular switched positive systems(SSPSs) under dwell-time constraints which include mode-dependent minimum dwell time(MDMDT), mode-dependent constant dwell time(MDCDT) and mode-dependent ranged dwell time(MDRDT) constraints. For SSPSs in both delay-free case and time-varying delay case, a sufficient exponential stability condition is proposed with MDMDT, MDCDT and MDRDT constraints, respectively, and the exponential decay rate can be set as a free parameter based on diverse circumstances. To analyze the dwell-time stability, a novel discretized linear copositive Lyapunov function(DLCLF) approach is introduced in the article, and compared with the general copositive and homogeneous Lyapunov function approach, the main advantage of the DLCLF technique is itself affine dependence on conditions of system matrix, which will be extended and utilized to systems with uncertainties or/and time-varying parameters quite easily. Meanwhile, the proposed condition of exponential stability under MDMDT(or MDCDT or MDRDT) constraint will be degenerated into the one under minimum dwell time(or constant dwell time or ranged dwell time) constraint for some specific situations, which implies that the considered MDMDT(or MDCDT or MDRDT) case is more general and practical. Finally, the validity and significance of the results are illustrated by seven examples.

Unified mode‐dependent average dwell time stability criteria for discrete‐time switched systems

SummaryIn this article, a unified mode‐dependent average dwell time (MDADT) stability result is investigated, which could be applied to switched systems with an arbitrary combination of stable and unstable subsystems. Combined with MDADT analysis method, we classified subsystems into two categories: switching stable subsystems and switching unstable subsystems. State divergence caused by switching unstable subsystems could be compensated by activating switching stable subsystems for a sufficiently long time. Based on the above considerations, a new globally exponentially stability condition was proposed for discrete‐time switched linear systems. Under the premise of not resolving the LMIs, the MDADT boundary of the new stability condition is allowed to be readjusted according to the actual switching signal. Furthermore, the new stability result is a generalization of the previous one, which is more suitable for the case of more unstable subsystems. Some simulation results are given to show the advantages of the theoretic results obtained.

New alternative convex conditions on exponential stability and stabilisation of switched positive linear systems with dwell time

This study is concerned with dwell time stability and stabilisation problems of switched positive linear systems (SPLSs). The dwell time refers to minimum dwell time and constant dwell time. Several stability conditions for primal and transpose SPLSs with dwell time are presented, and the relation between these conditions is illustrated. Some of these conditions are given in terms of infinite-dimensional linear programming (LP), which cannot be solved directly. Then, by utilising the piecewise linear approach, new alternative convex conditions are formulated in terms of finite-dimensional LP. Compared to the existing literature, results with lower or at least the same conservatism can be obtained under the new conditions for the same discretised order. An algorithm is given to reduce the computational cost. Meanwhile, it is proved that there exists a relation between these convex and non-convex conditions if the discretised order is sufficiently large. By utilising the transpose conditions, alternative convex conditions on stabilisation of SPLSs with dwell time are also presented. The controller gain matrices can be computed by solving a set of LP directly. Finally, the correctness and superiority of the results are verified by numerical examples.

Dwell time stability and stabilization of interval discrete-time switched positive linear systems

This paper investigates the dwell time stability and stabilization problems of discrete-time switched positive linear systems (DSPLSs) with interval uncertainties. The dwell time refers to constant dwell time, ranged dwell time and minimum dwell time. At first, some sufficient conditions are presented for the dwell time stability of primal and transpose interval DSPLSs. Meanwhile, the conservatisms of these conditions are compared. Then, equivalent stabilization conditions are presented for interval DSPLSs with dwell time. The controller gain matrices can be computed by solving the convex lifted conditions directly. All the results are given in terms of linear programming (LP). At last, several numerical examples verify the correctness and significance of the results.

Online partitioning method for decentralized control of linear switching large-scale systems

A novel partitioning approach for linear switching large-scale systems is presented. We assume that the modes of the switching system are unknown a priori but can be detected. We propose an online partitioning scheme that can partition the system when the mode switches, thus adapting the partition to the mode. Moreover, after the system has been partitioned, we apply a decentralized state-feedback control scheme to stabilize the system. We also apply a dwell time stability scheme to prove that the closed-loop system remains stable even after both the mode and partition changes. The proposed approach is illustrated by means of an automatic generation control problem related to frequency deviation regulation in a large-scale power network.

Fault detection filtering for nonlinear switched systems via event-triggered communication approach

This paper considers the fault detection problem for networked switched control systems subject to repeated scalar nonlinearities and stochastic disturbance under an event-triggered scheme. A nonlinear fault detection filter is proposed to generate the residual signal and detect system faults, and an event-triggered strategy is applied to limit signal transmission. Sufficient conditions to guarantee an event-based residual system is mean-square exponentially stable with the desired performance are obtained using average dwell time stability theory and a positive diagonally dominant Lyapunov functional approach. The corresponding fault detection mechanism parameters are calculated using the linearization technique. The event-triggered fault-detection filter and controller coordinated design are extended and then applied to cognitive radio networks. Two examples are provided to verify the proposed new design method.

Stability, L1-gain analysis and asynchronous L1-gain control of uncertain discrete-time switched positive linear systems with dwell time

In this paper, the stability, L1-gain analysis and asynchronous L1-gain control problems of uncertain discrete-time switched positive linear systems (DSPLSs) with dwell time are investigated. First, several convex and non-convex conditions on dwell time stability of DSPLSs with interval and polytopic uncertainties are presented, and the relation between these conditions is revealed. Then, via a switched dwell-time-dependent co-positive Lyapunov functions (SDTLFs) approach, convex sufficient conditions on L1-gain analysis and asynchronous L1-gain control of DSPLSs with interval uncertainties are derived. Meanwhile, via the switched parameter-dwell-time-dependent co-positive Lyapunov functions (SPDTLFs) approach, the L1-gain analysis and asynchronous L1-gain controller design problems of DSPLSs with polytopic uncertainties are also solved. The stability and L1-gain analysis results are given in terms of linear programming (LP). The controller design results are presented in terms of bilinear programming (BP), which can be solved with the help of iterative algorithm. At last, both numerical and practical examples are provided to show the effectiveness of the results.

Dwell‐time‐dependent stability results for impulsive systems

This study is concerned with the dwell-time stability for impulsive systems under periodic or aperiodic impulses. A Lyapunov-like functional approach is established to study the stability for impulsive systems. The Lyapunov-like functional is time-varying and decreasing but is not imposed definite positive nor continuous. A specific Lyapunov-like functional is constructed by introducing the integral of state together with the cross-terms of the integral and impulsive states. A tight bounding is obtained for the derivative of this functional with the help of the improved Jensen inequality reported recently and the integral equation of the impulsive system. On the basis of the Lyapunov-like functional approach, new dwell-time-dependent stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time are derived for periodic or aperiodic impulsive systems. The stability results have less conservatism than some existing ones, which are illustrated by numerical examples.

Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

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.

On Dwell Time Minimization for Switched Delay Systems: Time-Scheduled Lyapunov Functions

In the present paper, dwell time stability conditions of the switched delay systems are derived using scheduled Lyapunov-Krasovskii functions. The derivative of the Lyapunov functions are guaranteed to be negative semidefinite using free weighting matrices method. After representing the dwell time in terms of linear matrix inequalities, the upper bound of the dwell time is minimized using a bisection algorithm. Some numerical examples are given to illustrate effectiveness of the proposed method, and its performance is compared with the existing approaches. The yielding values of dwell time via the proposed technique show that the novel approach outperforms the previous ones.

Convex conditions for robust stabilization of uncertain switched systems with guaranteed minimum and mode-dependent dwell-time

Alternative conditions for establishing dwell-time stability properties of linear switched systems are considered. Unlike the hybrid conditions derived in Geromel and Colaneri (2006), the considered ones are affine in the system matrices, allowing then for the consideration of uncertain switched systems with time-varying uncertainties. The low number of decision variables moreover permits to easily derive convex stabilization conditions using a specific class of state-feedback control laws. The resulting conditions are enforced using sum of squares programming which are shown to be less complex numerically that approaches based on piecewise linear functions or looped-functionals previously considered in the literature. The sums of squares conditions are also proven to (1) approximate arbitrarily well the conditions of Geromel and Colaneri (2006); and (2) be invariant with respect to time-scaling, emphasizing that the complexity of the approach does not depend on the size of the dwell-time. Several comparative examples illustrate the efficiency of the approach.

Affine Characterizations of Minimal and Mode-Dependent Dwell-Times for Uncertain Linear Switched Systems

An alternative approach for minimum and mode-dependent dwell-time characterization for switched systems is derived. While minimum-dwell time results require the subsystems to be asymptotically stable, mode-dependent dwell-time results can consider unstable subsystems and dwell-times within a, possibly unbounded, range of values. The proposed approach is related to Lyapunov looped-functionals, a new type of functionals leading to stability conditions affine in the system matrices, unlike standard results for minimum dwell-time. These conditions are expressed as infinite-dimensional LMIs which can be solved using recent polynomial optimization techniques such as sum-of-squares. The specific structure of the conditions is finally utilized in order to derive dwell-time stability results for uncertain switched systems. Several examples illustrate the efficiency of the approach.