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Stability analysis of switched systems under dynamical dwell time control approach
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This article investigates the stability of a class of switched systems using dynamical dwell time approach. First, the condition for stability of switched systems whose subsystems are stable are presented with dynamical dwell time approach, which is shown to be less conservative in switching law design than dwell time approach. Then the proposed approach is extended to the switched systems with both stable and unstable subsystems. Finally, some numerical examples are given to illustrate the effectiveness of the proposed results.
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... For a switched system, it may have unstable subsystems. Studies on stability properties of switched systems which have some unstable subsystems were done in [23,29,35,38]. In order to make this type of switched systems stable, we have to let the activate time of stable systems being sufficiently long. ...
... In order to show asymptotic stability of system (26), we plot the figures of the states of (26) with the switching signal (29) (Figure 3) satisfying (27)- (28) σ(k) = A 1 , k = 5s, 5s + 1, s ∈ N + A 2 , k = 5s + 2, 5s + 3, 5s + 4, s ∈ N + (29) and an initial condition x 0 = (1, 1) . Figure 1. ...
... Figure 1. The trajectory of the state x 1 of system (26) with a given switching signal (29) and an initial condition x 0 = (1, 1) . ...
... In recent years, the stability issue of switched systems with unstable subsystems has been extensively investigated. For instance, in [2][3][4][5][6][7] the researchers have derived some stability results for switched systems with both stable and unstable subsystems. The main strategy of some literature is to ensure that the dwell time of stable subsystems is sufficiently large to compensate for the state divergence caused by unstable subsystems and switching behaviors. ...
... The main strategy of some literature is to ensure that the dwell time of stable subsystems is sufficiently large to compensate for the state divergence caused by unstable subsystems and switching behaviors. Obviously, if there is no stable subsystem to absorb the state divergence, these results proposed in [2][3][4][5][6][7] are disabled. ...
... This section presents the stability criteria for the switched system (1) under the state-dependent switching rule (3). Owing to the Leibniz-Newton formula, we have the following equation ...
This paper studies the stability of linear switched systems with time-varying delays and all unstable subsystems. According to the largest region function strategy, the state-dependent switching rule is designed. By bringing in integral inequality and multiple Lyapunov-Krasovskii functionals, the stability results of delayed switched systems with or without sliding motions under the designed state-dependent switching rule are derived for different assumptions on time delay. Several numerical examples are employed to show the effectiveness and superiority of the proposed results.
... Many meaningful research results on the switched system have been reported due to the potential application in manufacturing systems, network communication, chemical engineering, etc., for example see Borangiu et al. (2015); Wang and Zhao (2015); Wu et al. (2022) and references therein. At first, the control or filtering problem is investigated for switched systems under the dwell time (DT) method (Sano et al., 2000;Xiang & Xiang, 2009). Along with research went through, many results on switched systems have been obtained based on the average dwell time (ADT) approach (Yuan & Wu, 2015;Zhou et al., 2015), which is more flexible and efficient than DT one. ...
This paper considers the problem of event-triggered $ H_{\infty } $ H∞ output tracking control for uncertain networked switched systems with both communication constraints and asynchronous switching. A new sampled delay-dependent error tracking uncertain networked switched system is constructed on the basis of event-triggered communication mechanism (ETCM) and the given reference model. By introducing a parameter-dependent Lyapunov–Krasovskii functional and employing the mode-dependent average dwell time (MDADT) method, results on exponential stability and robust $ H_{\infty } $ H∞ performance are proposed. Furthermore, the gains of the output tracking controller with matched system mode are obtained by solving a set of linear matrix inequalities (LIMs), at the same time, the parallel results with the mismatched case are also obtained. Simulation results show that the proposed design scheme not only guarantees tracking performance but also saves communication resources effectively.
... For a general switched system (11), stability can be guaranteed by satisfying the average dwell time theory [29]. However, since the updated system (12) has been reformed for BT performance with Λ σ (t k ) that related to the switching instant, we propose a synthesis condition that draws on the dynamic dwell time technique [30], in which the dwell time is defined as follows. ...
Since the output signal of the controller is prone to jump suddenly when a switching occurs, the practical system usually suffers from undesired bumps. To address this “bump” phenomenon, a parameters-free bumpless transfer structure based on multiple differentiators and a co-integrator is proposed in this study. The proposed structure is applicable for switching between manual and automatic modes, as well as switching during steady state and transient state. Unlike the previous compensator-based BT methods, our structure exploits the continuous property of the integrator, which prompts optimal performance and preserves original performance. Furthermore, sufficient conditions based on the dynamic dwell time are given to guarantee the stability of switched linear systems. At last, the designed BT approach is applied to a bivariate turbofan engine and a micro-turbojet engine, wherein the hardware-in-the-loop tests and real bench tests are performed respectively, so as to demonstrate the significant improvement and potential for practical engineering.
... is inspired researchers to propose several more general methods; one of the commonly applied methods which exerts restrictions on the time period that subsystems being activated named average dwell-time (ADT) [25][26][27][28], and another effective method is the multiple Lyapunov functions (MLFs) [4,24,29,30]. For example, in [10,[31][32][33][34], ISS for switched systems exploited by ADT, and MLF-based results are presented in [24,29,[35][36][37]. ...
This paper investigates the Lyapunov characterizations of input-to-output stability (IOS) and input-to-state stability (ISS) for nonlinear switched systems. The restriction on the negativity of derivatives of Lyapunov functions is relaxed; additionally, subsystems are not restricted to be ISS/IOS throughout the state space. These problems are overcome by a newly proposed method, which extends multiple Lyapunov functions (MLFs) to an indefinite form. Indefinite multiple Lyapunov functions (iMLFs) are proposed on the basis of an inequality to estimate the upper boundary of the system state. Moreover, we apply the iMLFs method to verify the extended notions of IOS in a switched system. More relaxed sufficient conditions for ISS and IOS are proposed, a simulation experiment is carried out on a numerical example of the system, which demonstrates their effectiveness and superiority.
... The concept of dynamic dwell-time (DDT) is also used to analyze the stability of switched systems. 17,18 In the context of ADT switching, many control issues for asynchronously switched systems have been extensively discussed, such as the stabilization via state-feedback, [19][20][21] static output-feedback, 22,23 and dynamic output-feedback, 24,25 as well as the ∞ control via state-feedback 26,27 and dynamic output-feedback, 28,29 just to name a few. However, almost all the controllers in the aforementioned publications are configured as time-invariant forms, which are a bit more conservative to some extent. ...
Considering that actuator delay is inevitable and a discretized control design method is more preferable in practice, this article is concerned with the discretized quasi‐time‐dependent (QTD) control for continuous‐time asynchronously switched systems via dynamic output‐feedback. The persistent dwell‐time (PDT) is employed to analyze the stability and ‐gain for switched systems in the framework of multiple QTD Lyapunov function approach, and the concept of dynamic dwell‐time (DDT) is used to characterize a category of switching signals for asynchronously switched systems with measurable switching lag, which means that the controlled plant can adjust its dwell‐time on some mode dynamically according to the switching lag of the corresponding controller. First, the stability and ‐gain are analyzed for switched systems with PDT. Then, by constructing controller‐dependent multiple QTD Lyapunov functions, a QTD control synthesis condition is derived based on the obtained stability criterion. The developed controller can guarantee the closed‐loop stability and performance for the underlying system under a class of identified DDT switching signals. Subsequently, a discretized QTD control scheme is proposed by choosing the discretized quadratic version of the constructed Lyapunov functions. The synthesis condition is formulated as a set of finite‐dimensional LMIs, and hence it can be easily tractable. Finally, the proposed control scheme is applied to the tracking control of a mobile robot to illustrate its effectiveness and application potential.
This paper mainly investigates the finite-time rate anti-bump switching (RABS) control problem for switched systems. A novel multiple convex Lyapunov function is first proposed by constructing a convex combination of positive definite matrices for the RABS control problem of switched systems. By imposing a prespecified dwell time on state-dependent switching, a combined switching law is devised based on the new Lyapunov function to ensure that Zeno phenomenon is prevented. Then, a bumpless control scheme is proposed to reduce the big and undesired jump in the rate of system at switching instants while achieving the disturbance suppression in finite time. In the end, through an ingenious Simulink model to characterize the combined switching law and finite-time H ∞ rate anti-bump controller, an actual engine model is used to demonstrate the validity of the proposed approach.
In IMCS, the system degree of freedom (DOF) in the hardware increases from one to two. It first benefits the flexible configuration of individual flow paths, including regeneration and recuperation. In this chapter, multiple operating modes are established according to the achievable flow paths by the independent metering control. Considering the force and flow limitation of operating modes, the mode-switching logic is designed, such that the most efficient operating mode without losing controllability can be automatically selected. To solve an unstable and unsmooth switch, this chapter proposes a bumpless mode switch approach for the IMCS, which contains the dynamic dwell-time switch and a bidirectional latent tracking loop. The proposed strategy is verified by a mini-excavator. The comparison results show that the instability during mode mode-switching instant is eliminated, as well as the transitions of the cylinder velocity and pressure between the two modes are smoother. Meanwhile, the hysteresis of the mode switching caused by the dwell time is greatly decreased. In comparison to the conventional hydraulic control system, an IMCS could save energy without reducing the motion control performance using the proposed bumpless mode switch strategy.
In this paper, event-triggered L2 − L∞ filtering problem for a class of switched linear parameter varying systems with both mode-dependent average dwell time and asynchronous switching is concerned. A new sampled delay-dependent and parameterised filtering error switched system is constructed on the basis of event-triggered communication scheme. Results on exponential stability with a prescribed L2 − L∞ performance are proposed by exploiting the parameter-dependent Lyapunov–Krasovskii functional and the mode-dependent average dwell time method, which is less rigid with requirement of minimal average dwell time of the switching signal. Furthermore, the gains of the parameter-dependent filter with mismatched system mode are solved in terms of linear matrix inequalities, meanwhile, the parallel results with the matched case are also obtained. Finally, a numerical example is provided to show the effectiveness and potential of the proposed design scheme.
The paper discusses the issue of global asymptotic stabilization for non-smooth variable order nonlinear switched systems with partial unstable modes. The existence and uniqueness of solution for the considered system is firstly verified by utilizing Gronwall–Bellman inequality and the inductive method. Then, a slow switching strategy is performed for the stable modes and the unstable modes are handled by a fast switching mechanism. Under the framework of Filippov differential inclusion, sufficient stabilization criteria are derived by applying the mode-dependent average dwell time and dwell time schemes. Finally, some objects in real life are introduced and a numerical simulation is offered to show the validity of the obtained results.
In this paper we prove that a switched nonlinear system has several useful ISS-type properties under average dwell-time switching signals if each constituent dynamical system is ISS. This extends available results for switched linear systems. We apply our result to stabilization of uncertain nonlinear systems via switching supervisory control, and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide ISS properties with respect to the estimation errors. Illustrative examples are included.
We study the stability properties of linear switched systems
consisting of both Hurwitz stable and unstable subsystems using an
average dwell time approach. We show that if the average dwell time is
chosen sufficiently large and the total activation time of unstable
subsystems is relatively small compared with that of Hurwitz stable
subsystems, then exponential stability of a desired degree is
guaranteed. We also derive a piecewise Lyapunov function for the
switched system subjected to the switching law and the average dwell
time scheme under consideration, and we extend these results to the case
for which nonlinear norm-bounded perturbations exist in the subsystems.
We show that when the norms of the perturbations are small, we can
modify the switching law appropriately to guarantee that the solutions
of the switched system converge to the origin exponentially with large
average dwell time
A novel approach to the control of an automotive engine in the
cut-off region is presented. First, a hybrid model which describes the
torque generation mechanism and the power-train dynamics is developed.
Then, the cut-off control problem is formulated as a hybrid optimization
problem, whose solution is obtained by relaxing it to the continuous
domain and mapping its solution back into the hybrid domain. A formal
analysis as well as simulation results demonstrate the properties and
the quality of the control law
First, we review some previous work on networked control systems
(NCSs) and offer some improvements. Then, we summarize the fundamental
issues in NCSs and examine them with different underlying
network-scheduling protocols. We present NCS models with network-induced
delay and analyze their stability using stability regions and a hybrid
systems technique. Following that, we discuss methods to compensate
network-induced delay and present experimental results over a physical
network. Then, we model NCSs with packet dropout and multiple-packet
transmission as asynchronous dynamical systems and analyze their
stability. Finally, we present our conclusions
We first formulate a model for hybrid dynamical systems which
covers a very large class of systems and which is suitable for the
qualitative analysis of such systems. Next, we introduce the notion of
an invariant set for hybrid dynamical systems and we define several
types of (Lyapunov-like) stability concepts for an invariant set. We
then establish sufficient conditions for uniform stability, uniform
asymptotic stability, exponential stability, and instability of an
invariant set of hybrid dynamical systems. Under some mild additional
assumptions, we also establish necessary conditions for some of the
above stability types (converse theorems). In addition to the above, we
also establish sufficient conditions for the uniform boundedness of the
motions of hybrid dynamical systems (Lagrange stability). To demonstrate
the applicability of the developed theory, we present specific examples
of hybrid dynamical systems and we conduct a stability analysis of some
of these examples
During the last three decades, there has been an increasing interest on the modeling, analysis, synthesis, and control of switched systems. A survey on the current development of the research on switched systems is given. The main topics consist of i) common Lyapunov function; ii) stabilization of switched systems; iii) controllability of switched systems. The prospect of development is presented. Some auxiliary knowledge is illustrated in the appendix.
The stability of a multi-controller system is analyzed using Lyapunov theory. Stability is guaranteed by a switching strategy determined by a combination of separate Lyapunov functions for the different controllers. The behavior on sliding surfaces is analyzed using non-smooth analysis according to Filippov. Two examples illustrate the theory.
It is shown that switching among stable linear systems results in
a stable system provided that switching is
“slow-on-the-average”. In particular, it is proved that
exponential stability is achieved when the number of switches in any
finite interval grows linearly with the length of the interval, and the
growth rate is sufficiently small. Moreover, the exponential stability
is uniform over all switchings with the above property. For switched
systems with inputs this guarantees that several input-to-state induced
norms are bounded uniformly over all slow-on-the-average switchings.
These results extend to classes of nonlinear switched systems that
satisfy suitable uniformity assumptions. In this paper it is also shown
that, in a supervisory control context, scale-independent hysteresis can
produce switching that is slow-on-the-average and therefore the results
mentioned above can be used to study the stability of hysteresis-based
adaptive control systems
We study the control of redundant planar robotic manipulators
using a switched (or hybrid) control scheme, focusing on manipulators
with a degree of redundancy of one. We emphasize the effectiveness of
switched control systems with respect to stabilization and performance
enhancement for this class of manipulators. We present a simulation
study of logic-based switching control of a 3-DOF planar manipulator
under end-effector trajectory tracking and demonstrate the capabilities
of this scheme