Time-dependent Lyapunov Function Research Articles

Integral sliding mode control for uncertain impulsive stochastic systems

This paper studies the integral sliding mode control (ISMC) for uncertain impulsive stochastic systems. An ISMC law is proposed by designing a new integral sliding surface. Then, by constructing a piecewise time-dependent Lyapunov function, the uniformly almost surely exponentially stability conditions of the closed-loop system are established. Furthermore, an optimal design algorithm of solving control gains is formulated based on the established stable conditions and linear matrix inequalities (LMIs) theory. Finally, a simulation example and the simulation comparisons with the existing control methods are provided to demonstrate the effectiveness of the proposed ISMC scheme.

Analysis of the Relation Between L<sub>2</sub> Gain and Dwell Time in Discrete Time Switching System

This paper focuses on the design of <i>l</i>₂ gain controller for discrete time switching systems with dwell time (DT) switching. So far, the quantitative relationship between <i>l</i>₂ gain stability and dwell time is one of the unsolved problems in the field of switching systems. Different from previous studies, a novel dwell time dependent Lyapunov function (DTDLF) is proposed in this paper for the first time to obtain a less conservative stability criterion. The innovation of this Lyapunov function lies in that it is no longer built by the traditional two-step controller design method, but combines the two design tasks of controller and switching law into one framework, in which solving linear matrix inequalities (LMIs) involves only one step. Compared with the two-step design scheme, this method greatly simplifies the design process. Based on the resident time dependent Lyapunov function and the resident time switching strategy, a dwell time dependent (DTD) H_∞ controller is designed such that the system is globally uniformly asymptotically stable (GUAS) and has a guaranteed H_∞ performance index. Finally, a numerical example is given to verify the validity of the proposed design method. The simulation results show that the proposed design method can guarantee a shorter resident time limit and a smaller error result.

General kernel estimates of Schrödinger-type operators with unbounded diffusion terms

We first prove that the realization $A_{\mathrm {min}}$ of $A:={\operatorname {\mathrm {div}}}(Q\nabla )-V$ in $L^2({\mathbb {R}}^d)$ with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on $L^2({\mathbb {R}}^d)$ which coincides on $L^2({\mathbb {R}}^d)\cap C_b({\mathbb {R}}^d)$ with the minimal semigroup generated by a realization of $A$ on $C_b({\mathbb {R}}^d)$. Moreover, using time-dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of $A$ and deduce some spectral properties of $A_{\min }$ in the case of polynomially and exponentially growing diffusion and potential coefficients.

Membership-Mismatched Impulsive Exponential Stabilization for Fuzzy Unconstrained Multilayer Neural Networks With Node-Dependent Delays

This article focuses on the membership-mismatched impulsive exponential stabilization for fuzzy unconstrained multilayer neural networks (MNNs) with node-dependent time-varying delays (NDTVDs). To begin with, this work proposes a novel MNNs model with unconstrained interlayer and intralayer parameters, which may allow nodes in all layers to have inconsistent attributes and structures. Meanwhile, the novel model considers the NDTVDs and removes the strict constraints including node alignment and one-to-one interlayer connection to meet diverse modeling requirements in complex applications. Then, the proposed fuzzy impulsive controller does not need to share the same fuzzy parameters as the fuzzy MNNs model, reducing the implementation complexity of the fuzzy impulsive controller. To derive the main results using the augmented vector form of unconstrained MNNs, the sparse matrix method is proposed to convert the node-dependent delayed MNNs model into an equivalent model with multiple delays. Moreover, the time-dependent Lyapunov function (TDLF) technique is adopted to improve the reliability of the stabilization conditions by fully utilizing the state information of both the current and neighboring impulsive intervals. Finally, the main results are verified using numerical simulation.

Hybrid output regulation for linear impulsive systems with aperiodic jumps: A discrete-time feedback controller design approach

This paper is concerned with the problem of hybrid output regulation for a class of linear impulsive systems with aperiodic jumps. Firstly, by leveraging time-dependent Lyapunov function technique and impulsive control theory, sufficient conditions for achieving output regulation are obtained in state feedback case. Then, the results are extended to error feedback case by constructing an impulsive observer. In this framework, two novel hybrid controllers are designed. Such controllers only need the discrete-time system state or error signal for feedback. The complete procedures for controller designs are also presented. Finally, two illustrative examples, including a numerical example and an LC circuit, are given to show the validity and applicability of the proposed control laws.

Input-to-state stable synchronization for delayed Lurie systems via sampled-data control

&lt;p style='text-indent:20px;'&gt;This paper focuses on the input-to-state stable (ISS) synchronization issue for delayed Lurie systems via sampled-data control. Two time-dependent Lyapunov functionals are constructed for delay functions without and with differentiability restrictions, respectively. Sufficient conditions on the ISS synchronization are established via using the proposed Lyapunov functionals, the reciprocally convex-combination approach, and a few integral inequalities. The gain matrix of the needed sampled-data controller can be attained through solving a couple of linear matrix inequalities, which is numerically easy to handle. Furthermore, the present conditions are capable of ensuring the exponential synchronization of the considered Lurie systems in the absence of external disturbances. In the end, two examples are employed to demonstrate the effectiveness of the established ISS synchronization conditions.&lt;/p&gt;

New Explicit Criteria for Finite-time Stability of Singular Linear Systems Using Time-dependent Lyapunov Functions

New Explicit Criteria for Finite-time Stability of Singular Linear Systems Using Time-dependent Lyapunov Functions

Fixed-time stochastic synchronization of impulsive multi-weighted complex dynamical networks with non-chattering control

This paper studies the fixed-time stochastic synchronization problem of multi-weighted complex dynamical networks with impulsive effects. Different from most of previous networks, stochastic disturbances and impulsive effects are considered in this model. The non-chattering controller is utilized, and by devising a novel time-dependent Lyapunov function and dividing the impulsive intervals, a new sufficient criterion for the fixed-time stochastic synchronization (FDTSS) of multi-weighted complex dynamical networks (MWCDNs) is obtained. Secondly, compared with some existing results, this paper proposes a novel means of calculating the settling time by constructing comparison systems and exploiting the method of the classification discussion, and the settling time does not depend on the initial values of the system. In addition, compared with the existing fixed-time and finite-time controllers, the controller devised in this paper is continuous and do not involve sign functions, which non-chattering phenomenon in practical applications. Finally, the Lorenz system is submitted to certify the justifiability of the theoretical results.

Delay-dependent cluster synchronization of time-varying complex dynamical networks with noise via delayed pinning impulsive control

This paper investigates the problem of cluster synchronization of complex dynamical networks with noise and time-varying delays by using a delayed pinning impulsive control scheme. Different from the traditional impulsive control schemes without the effects of input delays, it designs a pinning impulsive control scheme to successfully address the aforementioned problem subject to impulsive input delays. By employing a time-dependent Lyapunov function and the mathematical induction, some novel criteria are established to guarantee the cluster synchronization of the noisy complex networks, revealing the closed relationship between the synchronization performance and the related factors, including the impulsive input delays, the number of the pinned nodes, the frequency and strength of the impulsive control, and the noisy perturbations. Some numerical examples and computer simulations are presented to illustrate the effectiveness of the theoretical results.

Global exponential stabilization of nonlinear systems via width time-dependent periodically intermittent smooth control

In this paper, the global exponential stability and stabilization problems for a class of nonlinear systems are investigated. Some sufficient conditions to guarantee global exponential stable and estimate the minimum admissible value of the control width are presented in virtue of time-dependent width Lyapunov functions. Furthermore, a periodically intermittent smooth controller with variant control width is introduced and theoretical analysis is provided. The smooth index function of periodically intermittent smooth control inputs is defined and the supremum (or least upper bound) of smooth index function set can be solved. On the basis of the analysis, the designed periodically intermittent smooth controller not only can globally exponentially stabilize the nonlinear systems, but also can control the exponential convergence rate of the nonlinear systems. Finally, numerical simulations are given to verify the obtained theoretical results.

Dynamic event-triggered [formula omitted] control for switched affine systems with sampled-data switching

This paper investigates the dynamic event-triggered L∞ control for switched affine systems under limited communication resources. For switched affine systems, affine terms bring the difficulty on the exclusion of triggering Zeno behavior. The proposed dynamic event-triggered mechanism can preclude the occurrence of triggering Zeno behavior even if affine terms and the exogenous disturbance coexist. Moreover, it further reduces the transmission frequency compared with the static version. The switching chattering phenomenon is avoided by the designed sampled-data switching rule. The co-design method of dynamic event-triggered scheme, switching controller and sampled-data switching rule is provided to achieve practical stability and the L∞ performance with respect to the external interference. Tools and techniques of time-dependent Lyapunov function are used to obtain sufficient conditions in terms of linear matrix inequalities. Finally, the effectiveness of the developed results is illustrated by an example.

Aperiodic Sampled-Data-Based Control for Interval Type-2 Fuzzy Systems via Refined Adaptive Event-Triggered Communication Scheme

This article is devoted to event-triggered stabilization for a class of interval type-2 (IT2) fuzzy systems with aperiodic sampling. First, the IT2 Takagi-Sugeno fuzzy model and sampled-data controllers are established subject to mismatched membership functions. Second, considering a nonuniform sampling case, a refined adaptive event-triggered communication scheme is proposed in a hierarchy form to dynamically adjust the direction and rate of the event-triggered threshold parameter by state changing trend and relative state error, respectively. Thus, a complete dual-directional regulating mechanism with sensitivity to state variation is reasonably created to give extra flexibility, which is beneficial for a preferable tradeoff between control performance and network resource. Third, considering the practical behaviors on the sampling interval, a novel integral type of time-dependent Lyapunov function is constructed. Then, the stability criterion and the controller design approach are derived. Finally, the numerical examples are provided to demonstrate the effectiveness and advantages of the proposed methods.

Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation

&lt;p style='text-indent:20px;'&gt;This paper investigates the control synthesis for discrete-time semi-Markov jump systems with nonlinear input. Observer-based controllers are designed in this paper to achieve a better performance and robustness. The nonlinear input caused by actuator saturation is considered as a group of linear controllers in the convex hull. Moreover, the elapse time and mode dependent Lyapunov functions are investigated and sufficient conditions are derived to guarantee the &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ H_\infty $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; performance index. The largest domain of attraction is estimated as the existing saturation in the system. Finally, a numerical example is utilized to verify the effectiveness and feasibility of the developed strategy.&lt;/p&gt;

Time-varying output-based Takagi–Sugeno fuzzy controller of uncertain nonlinear systems

This paper proves the ultimately boundedness analysis for the trajectory tracking problem between the states of an uncertain nonlinear system represented by a Takagi–Sugeno (T–S) system and a given set of desired trajectories. The design of an output feedback controller includes a weighted contribution of the models included in the T–S design. The proposed T–S fuzzy estimates the state variables based on the output information, exclusively. The output-based controller design uses a time-dependent Lyapunov function yielding the characterisation of ultimate boundedness for the trajectory tracking error. Sufficient conditions are obtained to ensure the existence of positive-definite solutions for two coupled time-varying matrix Riccati equations, which are needed to solve the tracking problem. A simplified scheme determines the gains for the feedback controller and observer. The proposed control law solves the trajectory tracking of an autonomous underwater vehicle. In this case, numerical solutions show that the controller forced the convergence of the tracking error after 2.0 seconds. An alternative control design approach based on linear matrix inequalities (LMI) is used for comparison purposes. The suggested controller forces a faster convergence of the tracking error than the LMI-based one and provides a smaller ultimate bound.

Effects of impulse delays on Lp-stability of a class of nonlinear time-delay systems

In this paper, a nonlinear time-delay system with delayed impulses and external disturbances is considered. The robustness problem of Lp-stability with respect to impulse-delays is addressed. In order to circumvent the technical difficulties caused by delayed impulses, the impulses-delayed system is modeled as an impulse-delay free system with switching impulses by means of an augmentation method. Based on the switching structure of the augmented system, a time-dependent Lyapunov function is constructed from the original Lyapunov function for the impulse-delay free case. Then by employing the new designed Lyapunov function combined with the use of Razumikhin-type arguments, sufficient conditions for exponential stability and Lp-stability are derived. The developed results characterize the relationship among the impulse intervals, the size of impulse-delays, and the level of Lp-gain. Two numerical examples are presented to illustrate the effectiveness of the developed methodology.

Mitigation of vortex-induced vibration lock-in using time-delay closed-loop control

We investigate the closed-loop control of a circular cylinder showing lock-in phenomena due to vortex-induced vibrations (VIV). The control action was implemented by a sampled-data proportional-integral-derivative (PID) controller to suppress the large amplitudes due to lock-in. The controller was first applied to a linearized system to observe its stability characteristics based on the eigenvalues of the system. Another method was also proposed, which employs a novel, time-dependent Lyapunov function that is positive definite at sampling times but not necessarily between the sampling times. A new set of sufficient conditions in terms of linear matrix inequalities is derived to obtain the sampled-data PID control gains for the VIV system. The PID controller tuned with these gains for various delays was applied to control the nonlinear responses of the circular cylinder during the lock-in. The results showed that the PID controller significantly reduced the rise in lock-in amplitude compared to only proportional control and for certain delays was able to completely mitigate the effects of lock-in. It was also observed that for delays ranging from 0.1 to 0.14 s, the nonlinear system was destabilized with increasing proportional gains as indicated by the eigenvalue analysis of the linearized system. Even under such situations, properly tuned integral and derivative gains could significantly reduce the amplitude rise otherwise observed due to lock-in of the uncontrolled system. Finally, an on-off control scheme was also proposed, which, if optimized properly, can restrict the lock-in amplitude to some prescribed limit by only using the control for some fraction of the total operational time. Thus, it can potentially save control power.

Pinning synchronization control of complex networks with partial couplings

In this article, the pinning synchronization problem of complex networks with a target node via sampled-data communications is considered. Due to partial couplings among the nodes in complex networks, a decoupling method is adopted to investigate each channel of complex networks independently. By constructing a time-dependent Lyapunov function, it is proved that the pinning synchronization of complex networks with a target node can be achieved if the control parameters are appropriately selected. Furthermore, further study is needed to investigate the pinning synchronization of complex networks in the presence of constant delay. A novel criterion is obtained using Jensen’s inequality and Wirtinger’s inequality. It is worth noting that the lower and upper bounds of the sampling intervals can be calculated by linear matrix inequality box of MATLAB. Theoretical results are well verified through a numerical simulation.

Finite-Time Stability of Switched Linear Time-Delay Systems Based on Time-Dependent Lyapunov Functions

In this paper, finite-time stability of switched linear time-delay systems has been addressed. By constructing a class of time-dependent common (multiple) Lyapunov functions, new explicit conditions for finite-time stability of the system under arbitrary switching and average dwell-time switching are established respectively. Compared with most of existing results in the literature, our results are easily verifiable by solving several linear matrix inequalities rather than complex matrix Riccati differential equations. The effectiveness of the proposed method is demonstrated by numerical examples.

Reliable Fuzzy Tracking Control of Near-Space Hypersonic Vehicle Using Aperiodic Measurement Information

This paper is concerned with reliable fuzzy tracking control for a near-space hypersonic vehicle (NSHV) subject to aperiodic measurement information and stochastic actuator failures. The NSHV dynamics is approximated by the Takagi–Sugeno fuzzy models and the stochastic failures are characterized by a Markov chain. Different with the existing tracking results on NSHV, only the aperiodic sampling measurements are available during system operation. To realize the tracking objective, a reliable fuzzy sampled-data tracking control strategy is presented. An appropriate time-dependent Lyapunov function is constructed to fully capture the real sampling pattern. The sampling-interval-dependent mean square exponential stability criterion with disturbance attenuation is then established. The solution of the tracking controller gains can be obtained by solving an optimization problem. Finally, the simulation studies on NSHV dynamics in the entry phase are performed to verify the validity of the developed fuzzy tracking control strategy.

Stability and $L_2$-Gain Analysis for Linear Time-Delay Systems With Delayed Impulses: An Augmentation-Based Switching Impulse Approach

In this paper, the stability and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain properties of linear impulsive delay systems with delayed impulses are studied. Commonly employed techniques, in which the delayed impulses are treated using Newton-Leibniz formula, may not be applicable to L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain analysis, since they make the disturbance input appear in the impulse part. In order to circumvent the difficulty, wefirstaugment the considered system to a time-delay system with switching nondelayed impulses. Due to the absence of delayed impulses, this new approach has advantages in constructing Lyapunov functions and handling the effects of impulse delays on the system performance. Switching-based time-dependent Lyapunov functions are introduced to deal with the resultant switching impulses of the augmented system. Sufficient conditions for exponential stability and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain properties are derived in terms of linear matrix inequalities. Numerical examples are provided to illustrate the efficiency of the new approach.