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Lyapunov stability analysis for nonlinear systems with state-dependent state delay
Abstract
This paper addresses the stability problem for systems with state-dependent state delay (delay which involves the state of the system). Different from the time-dependent delay, the state dependence of the delay makes the value of delay dependent on the state change, which indicates that it is impossible to exactly know a priori how far in the history the state-information is needed. We apply the Lyapunov stability theory to obtain sufficient conditions for exponential stability of the zero equilibrium. Then we apply those results to some specific examples to illustrate the effectiveness of the approach and our general results. A class of stabilizing memoryless controllers for a second-order system with state-dependent state delay is also proposed.
... Recently, there has been an influx of research on state-dependent delay (SDD), which implies that time delay may change depending on the current system state (see, e.g., previous research [5][6][7]). Nevertheless, once the time delay is related to system state, the bound of time delay is a priori unknown. ...
... Li et al. [8] studied the stability for nonlinear systems of fractional order with SDD, Li and Peng [9] investigated the uniform stability for nonlinear systems with SDD, and He et al. [10] analyzed finite-time stability for SDD systems, which means that the output or state of systems does not infinitely grow or diverge but fluctuates or tends to stabilize within a certain range in a finite amount of time. It is worth noting that the methods used in previous research [5][6][7][8][9][10] to handle SDD can only be employed for deterministic systems. ...
... Inspired by the preceding discussion, this paper aims to investigate the stochastic weak local exponential stability of SDSs with SDD subject to average-delay impulses by employing the Lyapunov-Krasovskii functional method and stochastic analytical skill. The primary contributions of this paper are as follows: (1) The SDD is introduced into the impulsive SDSs, where SDD is a stochastic variable relying on system state renders the approaches for handling SDD in earlier studies [5][6][7][8][9][10] no longer applicable. Based on this, we propose new sufficient conditions for the stochastic weak local exponential stability of SDSs with SDD. ...
In this paper, we study the stochastic weak local exponential stability of stochastic differential systems with state‐dependent delay (SDD) subject to average‐delay impulses by using Lyapunov–Krasovskii functional and stochastic analytical skill. Unlike time‐dependent delay discussed in the previous literature, we introduce SDD in impulsive stochastic differential systems, where SDD is a stochastic variable associated with system state, consequently leading to an indeterminate delay boundary. Our findings reveal that when destabilizing delayed impulses satisfying specific conditions are generated, the stability of stochastic differential systems with destabilizing delayed impulses and stable continuous stochastic dynamics can still be maintained. Additionally, if stable delayed impulses satisfy certain conditions, stability of the systems under consideration can also be achieved, irrespective of the stable status on the continuous stochastic dynamics. Finally, two examples are given to demonstrate the validity of theoretical results.
... In addition, state-dependent delay has been widely employed in practical applications, including drilling engineering [12], megacaryocyte modeling [13], age structured models [14], and virus infection model [15]. Moreover, many intriguing and significant results on systems with state-dependent delay have recently been published; see [16][17][18][19]. In [16], Zhang and Huang investigated the stability of stochastic delayed nonlinear systems based on impulses with state-dependent delay by using average impulsive interval (AII), the comparison principle, and differential inequalities. ...
... In [17], He et al. explored the finite-time stability of nonlinear systems with state-dependent delay through utilizing the Razumikhin technique. Using Lyapunov stability theory, Li and Yang established weak local exponential stability criteria for nonlinear systems with state-dependent delay in [18]. In [19], Zhang et al. conducted analysis of stochastic networks with state-dependent delay by combining the Lyapunov method and stochastic analysis techniques. ...
... There are plenty of intriguing conclusions in the literature concerning the control and analysis of nonlinear impulsive stochastic systems. However, the majority of recent studies have simply investigated impulsive stochastic systems without considering stochastic effects; see [15,17,18,20,21]. On the other hand, although [16,23] focused on stochastic impulsive systems, they all presented more conservative constraints on the rate coefficient for the Lyapunov function. ...
This article investigates the stability problem of impulsive stochastic switched systems with double state-dependent delays. In the designed system, unstable and stable impulses are taken into consideration, respectively, and they do not need to function simultaneously with switching behavior. Additionally, two new ideas, i.e., mode-dependent switching density and mode-dependent impulsive density, are developed. Based on the Lyapunov function method and comparison principle, the asymptotic stability criteria for an impulsive stochastic switched system with state-dependent delays are given. Moreover, the application of theoretical results to neural networks and the neural network-based lecture skills assessment of normal students is analyzed. Finally, two numerical examples are provided to illustrate the effectiveness and reliability of the theoretical criteria.
... Generally, impulsive systems are usually composed of three elements: A continuous dynamics governing the continuous evolution of the system between impulses, which is typically described by a Lyapunov-Razumikhin (L-R) approach, Lyapunov-Krasovskii (L-K) functional approach, Halanaytype inequalities, comparison principle, and so on. Owing to the important role in dynamical behavior estimation of engineering systems, time delay has been receiving increasing attention in various dynamical systems [32][33][34][35][36][37][38]. For instance, Zhu et al. considered the leakage delay existing in neural networks, and the delayed state-feedback control strategy was constructed for globally exponential stability, which showed the controller can be used in all states or in some state [32]. ...
... For instance, Zhu et al. considered the leakage delay existing in neural networks, and the delayed state-feedback control strategy was constructed for globally exponential stability, which showed the controller can be used in all states or in some state [32]. Yang et al. explored the state-dependent state delay making the value of delay dependent on the state change, which indicated that it was impossible to exactly know a priori how far the historical state information was needed, and the sufficient conditions for exponential stability of the zero equilibrium were derived by using the Lyapunov stability theory [33]. According to the different parts where the time delay exists, impulsive systems with time delay can be roughly divided into two classes: impulsive delayed systems (IDSs) [39][40][41][42] and delayed impulsive systems (DISs) [43][44][45][46]. ...
... Since time delay is very likely to be time-varying, Li and Cao addressed the impulsive systems with unbounded time-varying time delay and introduced a new impulsive delayed inequality that involved unbounded and non-differentiable time-varying time delay [142], where some sufficient conditions ensuring stability and stabilization of impulsive time-unvarying and time-varying systems were derived, respectively. Moreover, when the change of time lag is related to the system state, which is usually called the state-dependent time delay, some interesting works can be found in [33,143]. In [144], for the case that impulsive strengths were stochastic and impulsive intervals were confined by the AII and the case that both the impulsive intensity and density were stochastic, the ISS problems were considered for nonlinear impulsive systems, respectively. ...
The studies of impulsive dynamical systems have been thoroughly explored, and extensive publications have been made available. This study is mainly in the framework of continuous-time systems and aims to give an exhaustive review of several main kinds of impulsive strategies with different structures. Particularly, (i) two kinds of impulse-delay structures are discussed respectively according to the different parts where the time delay exists, and some potential effects of time delay in stability analysis are emphasized. (ii) The event-based impulsive control strategies are systematically introduced in the light of several novel event-triggered mechanisms determining the impulsive time sequences. (iii) The hybrid effects of impulses are emphatically stressed for nonlinear dynamical systems, and the constraint relationships between different impulses are revealed. (iv) The recent applications of impulses in the synchronization problem of dynamical networks are investigated. Based on the above several points, we make a detailed introduction for impulsive dynamical systems, and some significant stability results have been presented. Finally, several challenges are suggested for future works.
... Recently, it has shown an increasing research interest on the study of state-dependent delays (SDDs), see Krstic (2012, 2013), Hartung, Krisztin, Walther, and Wu (2006) and Li and Yang (2020). For instance, based on the usage of predictor-based compensators, the local stability for a class of nonlinear systems involving state-dependent input delays has been considered in Bekiaris-Liberis and Krstic (2012), and the robustness of such delayed systems has been addressed in Bekiaris-Liberis and Krstic (2013) later. ...
... For instance, based on the usage of predictor-based compensators, the local stability for a class of nonlinear systems involving state-dependent input delays has been considered in Bekiaris-Liberis and Krstic (2012), and the robustness of such delayed systems has been addressed in Bekiaris-Liberis and Krstic (2013) later. In contrast, systems with state-dependent state delays have been considered in Li and Yang (2020). After presenting some Lyapunov results on exponential stability, a class of memoryless controller design was proposed for second-order delayed systems in Li and Yang (2020), which was then applied to the position control of submarine systems. ...
... In contrast, systems with state-dependent state delays have been considered in Li and Yang (2020). After presenting some Lyapunov results on exponential stability, a class of memoryless controller design was proposed for second-order delayed systems in Li and Yang (2020), which was then applied to the position control of submarine systems. Note that SDD widely exists in many control problems and communication process, such as the turning process of mechanical models (Insperger, Steṕań, Hartung, & Turi, 2005), the automatic position control of vehicles (Li & Peet, 2013), and the signal transmission between neurons (Hartung et al., 2006). ...
Finite-time stability and stabilization problems of state-dependent delayed systems are studied in this paper. Different from discrete delays and time-dependent delays which can be well estimated over time, the information of state-dependent delays is usually hard to be estimated, especially when states are unknown or unmeasurable. To guarantee the stability of state-dependent delayed systems in the framework of finite time, a Razumikhin-type inequality is used, following which estimations on the settling time and the region of attraction are proposed. Moreover, the relationship between the variation speed of state-dependent delays and the size of the region of attraction is proposed. Then as an application of the theoretical result, finite-time stabilization is studied for a set of nonlinear coupled neural networks involving state-dependent transmission delay, where the design of memoryless finite-time controllers is addressed. Two numerical examples are given to show the effectiveness of the proposed results.
... Therefore, it is necessary and significant to investigate ISNSs with state-dependent delays. Based on Lyapunov functions methods, [21], [22] investigated the exponentially stability of nonlinear systems with state-dependent delays, respectively. By utilizing impulsive control theory, [23] derived stability criteria for impulsive nonlinear systems with state-dependent delay. ...
... c2) Compared with the existing results, based on removing some restrictive conditions, our works improve related existing results, and cover some existing research results as special cases. Specifically speaking, this brief relaxes some restrictions in [21], [22]. When the subsystems of ISNSs are the same one, that is, the switches do not exist, the results in [23] are the special cases of our works. ...
... Substituting (21) into (20), then ...
This brief addresses the problem of global asymptotic stability for impulsive switched nonlinear systems (ISNSs) with double state-dependent delays (DSDDs), where the state-dependent delays exist in both subsystems functions and impulsive functions. The innovation of our results is that: 1) Due to the ubiquitous asynchronous phenomenon between the switches and impulses in many practical systems, we investigate the asynchronous phenomenon for ISNSs with DSDDs. 2) Meanwhile, the stabilizing and destabilizing effects of state-dependent delays impulses are fully considered. 3) By utilizing multiple Lyapunov functions, the switching strategy and the impulsive strategy, based on the asynchronous phenomenon, some stability criteria on global asymptotic stability are presented for ISNSs with DSDDs. As a result, the existing results are improved and can be regarded as a special case of the stability criteria in this paper. A simulation example is provided to verify the effectiveness of the developed methods.
... It is mainly used in the control system to improve robustness in the presence of external forces. To clarify, consider the case that the LLAR is worn by the human object, hence the desired torque should be generated to minimize the steady-state-error e thus the auxiliary variable x converges to zero [31,32]. Theorem 1. ...
... where P is a positive definite matrix used to simplify algebra without any loss of generality; X T is the transpose of Equation (31); and Γ is the adaptation gain matrix, which is positive definite. Differentiating Equation (33) over time, it can be obtained, ...
... Substituting Equation (31) into Equation (34), we can get, ...
The idea of developing a multi-joint rehabilitation robot is to satisfy the demands for recovery of lower limb functionality in hemiplegic impairments and assist the physiotherapists with their therapy plans. This work aims at to implement the Lyapunov Adaptive and Swarm�Fuzzy Logic Control (LASFC) strategy of 4-degree of freedom (4 DoF) Lower Limb Assistive Robot (LLAR) application, in which the control law is an integration of swarm-fuzzy logic control (SFLC) and Lyapunov adaptive control (LAC) with particle swarm optimization (PSO). The controller is established based on the sliding filtered steady-state error for SFLC. Its parameters are tuned by using PSO for the mathematical model of LLAR. The fuzzy defuzzification membership is set based on the tuned parameters for the real-time control system. LAC strategy is determined using stability analysis of the system to choose the controller’s parameters by observation of the system’s output and reference. The control law implemented in LLAR is the integration of SFLC and LAC to adjust the input voltage of joints. The parameters tuned by PSO are compared with the genetic algorithm (GA) statistically. In addition, the real-time trajectory tracking of the proposed controller for each joint is compared with LAC and SFLC separately. The experiment revealed that the LASFC has superior performance to the other two methods in trajectory tracking. For example, the average error for left hip by LASFC is 53.57% and 68% lower than SFLC and LAC, respectively. By the statistical analysis, it can be ascertained that the LASFC strategy performed efficiently for real-time control of the joint trajectory tracking.
... The proposed techniques only considered actuator delays and have not investigated delays in state measurements or nonlinear systems with state delays. Compensation for known nonlinear systems with state delays is investigated in [34], [35], and compensation for uncertain nonlinear systems with state delays is explored in [36]- [39]. However, these techniques assume that the state is measurable, where such a strategy is not applicable when TDS attacks are applied to the state. ...
... where ω 4i ∈ R >0 is a parameter defined in the following section. With respect to (35) and (36), the function proj(·, ·) denotes a Lipschitz continuous projection operator defined in [50,Equation 4] that is used to boundŴ i andV i within userdefined compact sets. Remark 1. Recall that τ i is bounded by Assumption 6. ...
... Using (35), (36), (43), (46), (57), and Young's inequality, (56) can be upper bounded bẏ ...
A continuous controller is developed for a centralized network control system (NCS) that is composed of agents with nonlinear dynamics subject to a Time-Delay-Switch (TDS) attack and additive disturbances. Since the state tracking error is unmeasurable during TDS attacks, controllers cannot use the state tracking error to coordinate the NCS. Therefore, a novel error signal is designed to address this unique challenge and enable the NCS to achieve a formation control objective. Furthermore, a TDS attack mitigation strategy is developed, which uses both learning and model-based approaches to estimate an agent's state for TDS attack detection and compensation. Lyapunov-Krasovskii (LK) functionals are used in the stability analysis to prove that the tracking errors converge to a steady-state residual, which is a function of the system uncertainty and TDS attack properties. The leader-follower formation control problem for Unmanned Aerial Vehicles (UAVs) based on a pure pursuit guidance law is selected for a simulation study to validate the performance of the proposed method.
... In the submarine system, to keep the submarine at the equilibrium point, the deviation position information is obtained by the time between sending radio waves and receiving reflected radio waves, where the time is SD [30]. Recently, there has been a lot of work on the system with SD delay, in which [31] described the biomechanical models, [32], [33] solved the problem of existence, [34]- [36] studied the problem of system dynamics, [37], [38] explored the impulse control with SD delay, and [39] only gave the stability of nonlinear systems of IO. ...
... Besides, these conclusions in [30], [38] are valid only for partially linear systems. Although the stability of the IO linear and nonlinear system was proposed in [39], [41], they are only valid for IO systems and the differentiability of the SDDs needs to be satisfied. Thus, the stability of fractional systems with SDDs needs to be further explored. ...
... Thirdly, the conclusions obtained are easy to verify and can be applied to prototype models and more general SD systems. Especially, when applied to the FO submarine displacement system, compared with the IO case, the conditions of the conclusions obtained are weaker and the conclusions are more general, for example, the range of parametert is 1] times of that in [39]. ...
The stability of fractional-order (FO) nonlinear system with state-dependent delays (SDDs) is investigated. Unlike the usual time-dependent delays, the state-dependent (SD) delays make the size of the delays relative to the states, which makes the system uncertain when historical state information would be used. A Lemma on Riemann-Liouville derivative is first given to ensure the monotonicity of the considered function. Then, based on the Lyapunov method, several sufficient criteria are presented to guarantee the Mittag-Leffler stability of the discussed systems. In the end, three examples are applied to illustrate the correctness and applicability of our theoretical conclusions, including practical applications in submarine positioning models.
... For nonlinear and time-varying systems, such stability criteria do not apply [49]. Lyapunov stability analysis is the most common method for the stability determination of non-LTI systems [50]. ...
... Equations (34)-(49) of the building energy system are represented linearly in a matrix form as Equations (49) and (50). ...
... where D = Vector of disturbances. Small amplitude perturbation is studied by analyzing the dynamic behavior of the building energy system under study using the state-space model of Equation (49) and Equation (50). It is to be noted that these equations represent a steady-state equilibrium condition. ...
Building energy management system involves the development of control strategies for the heating, ventilation, and air-conditioning (HVAC), as well as lighting, systems. Building energy modeling is a significant part of designing such strategies. In order to analyze the feasibility of a building energy system model for any desired control strategy, a mathematical assessment tool is developed in this paper. A multi-input multi-output (MIMO) building energy system model, consisting of an outdoor wall, an external wall, two partition walls, one roof, and a ceiling, has been considered as the virtual test setup. A methodology for conducting stability and controllability assessment tests on the building energy model is proposed using inverse dynamics input theory (IDIT). IDIT enables the decoupling of control variables so as to enable the conversion of an MIMO system to a number of independent single-input single-output systems. The controllability is assessed based on the design properties for continuous systems: asymptotes and transmission zeros. The results show that the relative humidity and air temperature of the building space were controllable for all operating points; however, in unconditioned situations, where the humidity levels of the building space were greater than that of the outdoor levels, the models were unstable.
... Numerous control methods from classical controllers to Sliding Mode Control (SMC) [14][15][16] have been applied to QTP over the last few decades. Stability of SMC is improved using Lyapunov theory in [17][18][19][20][21][22]. Despite the fact that these controllers have been shown to work well in tracking the reference input, they have been observed to suffer from bigger settling times. ...
... The desired polynomial is represented by A * (s) [48,49]. Let us consider a SISO Linear Time Invariant (LTI) system described by the transfer function model in (19)(20)(21). ...
... R p = s n + a n s n−1 + · · · + a 0 (21) where y p is the output signal, G p (s) is a proper transfer function, and u p is the control input signal. The control law is considered to be ...
Many industrial processes have challenging issues such as complex nonlinear nature, high sensitivity to disturbances and existence of parametric uncertainties. A Quadruple Tank Process (QTP) is used in the laboratory as a scaled representation of many industrial processes involving liquid level control problems. Conventional controllers are relatively slow in tracking the reference inputs and in reacting to disturbances, and they tend to lack the ability in dealing with parametric changes in the QTP. The aim of this paper is to design and test an Adaptive Pole Placement Controller (APPC) and a robust Adaptive Sliding Mode Controller (ASMC) with the purpose of high-efficacy control of a minimum phase QTP. The controllers are tested via simulations and performance indices resulting from the simulation of these controllers are compared with a PID controller in terms of the three cases of robustness to set point variations, rejection of disturbance inputs, and robustness to parametric uncertainties. Simulation results reveal that the proposed adaptive control configurations perform better than the PID controller with lower performance indices and faster settling times due to the paramater estimation method used in the design processes. ASMC outperforms APPC in different variations of inputs and in regulation performances due to the invariant nature of the Sliding Mode Control (SMC). The results show that a decentralized ASMC can be used to improve the rapidity and robustness of many multivariable industrial process control systems.
... According to [21,24,27] and typical experimental data, the range of is generally 10 to 100, the range of is generally 0 to 0.01, and the range of is generally 0 to 0.01. Therefore, we can obtain that , and . ...
... Since and are constants, according to the limit calculation rule and the L'Hopital's rule, from (26), (27), (28), and (29) we can obtain (25) and (30), we find that for both the spring torque and the damping torque. Therefore, the first condition is satisfied. ...
This paper focuses on the influence of the disturbance rejection rate (DRR) and parasitic loop parameters on the stability domain of the roll-pitch seeker's guidance system. The DRR models of the roll-pitch seeker caused by different types of disturbance torques and the scale deviation of different sensors are established. The optimal DRR model of the roll-pitch seeker, which contains the scale deviation model, is proposed by formula derivation. The model of the roll-pitch seeker's guidance system is established and equivalently simplified by the dimensionless method. The Lyapunov stability criterion for stability analysis of the guidance system is given by means of the passivity theorem and related definitions and lemmas. A simplified model of the roll-pitch seeker's guidance system, which is suitable for the Lyapunov stability criterion, is established by formula derivation and equivalent transformation. Three conditions that satisfy the Lyapunov stability criterion are obtained. Mathematical simulation with Nyquist plots is used to analyze the influence of different DRR parameters on the stability domain of the roll-pitch seeker's guidance system. Simulation results of this paper can provide reference for the stability analysis of systems related to the roll-pitch seeker.
... The function ( ) is considered stable if there exists a positive-definite Lyapunov function ( ) (Li and Yang, 2020). For all in the domain of ( ), the Lyapunov function ( ) should be positive-definite, meaning that ( ) approaches values in the range [0, ∞). ...
The motivation for developing a rehabilitation lower-limb exoskeleton robot was to provide functional robot-assisted therapy for assisting physiotherapists in improving hemiplegic patients’ walking recovery. Rehabilitation tasks required robust and precise trajectory-tracking performance, mainly achieved with exoskeleton robots. This paper presents a study on the gait trajectory cycles of a rehabilitation lower-limb exoskeleton robot controlled by an Admittance Swarm Initialized Adaptive (ASIA). The aim of this paper was to develop a robust adaptive controller integrated with admittance model to overcome human–robot interaction forces generated by the wearer. The parameters of the ASIA controller were efficiently initialized using swarm beetle antenna searching. An experiment was conducted on a prototype lower limb exoskeleton with four degrees of freedom, involving a healthy human subject for gait trajectory analysis. The results demonstrated the effectiveness of the proposed method in terms of control performance, steady-state error reduction, and robustness. The statistical analysis revealed that the ASIA performed 63 %, 53 % and 48 % less in average error compared to adaptive conventional controllers used in the same exoskeleton platform. The findings ascertained the potential of the ASIA controller in improving human mobility through lower limb exoskeleton applications.
... If one did not manage finding one, it does not mean the equilibrium point being analyzed is unstable, it could be stable or it could be unstable. Meanwhile, this method involves with large calculations when we calculate the time derivative of the Lyapunov candidate function [83][84][85][86][87][88][89]. The graphical phase plane approach is a better choice to analyze the stability of an equilibrium point if quantitative result is not needed [90][91][92], and this approach is a visualization-based method, which can give the conclusion without being getting into calculation, and it is faster than the Lyapunov method when determining the stability of an equilibrium point. ...
The equilibrium state of a dynamical system can be divided into the equilibrium point and limit cycle. In this paper, the stability analysis of the equilibrium point and limit cycle of dynamical systems are presented through different and all possible approaches, and those approaches are compared as well. In particular, the author presented the stability analysis of the equilibrium point through phase plane approach, Lyapunov–LaSalle energy-based approach, and linearization approach, respectively, for two-dimensional nonlinear system, while the stability analysis of the limit cycle is analyzed by using the LaSalle local invariant set theorem and Poincaré–Bendixson theorem, which is only valid in two-dimensional systems. Different case studies are used to demonstrate the stability analysis of equilibrium point and limit cycle.
... The drone can be navigated autonomously or by a human pilot. There are several works about developing the navigation algorithm for the drone [8], [9], [10], [11]. Hodge et al. [12] presented a generic navigation algorithm that utilizes onboard sensors' data of the drone to navigate the drone to the target. ...
... Proof. The non-linear dynamic system can be shown to be asymptotically stable if such a Lyapunov function exists [30][31][32][33]. It is assumed that for a given positive-definite matrix Q, there is a positive-definite solution P such that PA + A T P = −Q, where P is the positive-definite matrix that is used to simplify the algebra without any loss of generality. ...
In this paper, we present a modelling, dynamic analysis, and controller tuning comparison for a five-degree-of-freedom (DoF) multi-joint robotic arm based on the Lyapunov-based Adaptive Controller (LAC). In most pick-and-place applications of robotic arms, it is essential to control the end-effector trajectory to reach a precise target position. The kinematic solution of the 5-DoF robotic arm has been determined by the Lagrangian technique, and the mathematical model of each joint has been obtained in the range of motion condition. The Proportional-Integral-Derivative (PID) control parameters of the LAC have been determined by the Lyapunov stability approach and are initialised by four observation methods based on the obtained transfer function. The effectiveness of the initialised controller’s parameters is compared by a unit step response as the desired input of the controller system. As a result, the average error (AE) for Ziegler–Nichols is 6.6%, 83%, and 53% lower than for Pettit & Carr, Chau, and Bucz. The performance of LAC for the robotic arm model is validated in a virtual 3D model under a robot operating system environment. The results of root mean square error by LAC are 0.021 (rad) and 0.025 (rad) for joint 1 and joint 2, respectively, which indicate the efficiency of the proposed LAC strategy in reaching the predetermined trajectory and the potential of minimizing the controller tuning complexity.
... Several articles have recently considered state-dependent delayed impulses from the point of view of stability [27,28,37,46] using Lyapunov functions/functionals. To compare, stability analysis of differential equations with state-dependent delay (without impulses) has been studied for several decades [11,20,18,29,31]. Stability of impulsive functional differential equations have been considered variously using Lyapunov functional-type methods [41,40,44,47] and a linearized stability result has been proven [9], but these require the continuous-time functional to be at least Lipschitz continuous with domain being a phase space of discontinuous functions. ...
We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for an impulsive differential equation with state-dependent delay to exhibit non-uniqueness of solutions. On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type condition on the state-dependent delay. Irrespective of uniqueness of solutions, we prove a result on linearized stability. As a specific application, we consider a scalar equation on the positive half-line with continuous-time negative feedback, non-negative state-dependent delayed nonlinearity and impulse effect functional satisfying affine bounds.
... The nonlinear dynamic system can be shown to be asymptotically stable if such a Lyapunov function exists [47][48][49]. In this paper, the closed-loop system of each joint is analyzed by Lyapunov stability theory. ...
... In [15] Singh and Shukla discussed the stability for fractional order (1, 2] stochastic differential equations in Banach's space. The authors discussed Lyapunov's stability for fractional order differential equations with state-dependent delay and time-varying delay in [16][17][18]. The authors used Lyapunov criteria and derived Lyapunov's function to obtain the result. ...
... In [15] Singh and Shukla discussed the stability for fractional order (1, 2] stochastic differential equations in Banach's space. The authors discussed Lyapunov's stability for fractional order differential equations with state-dependent delay and time-varying delay in [16][17][18]. The authors used Lyapunov criteria and derived Lyapunov's function to obtain the result. ...
In this article, our primary focus is on discussing the asymptotic stability of the semi-linear thermoelastic system. Results are obtained with the help of contraction mapping. We assume the Lipschitz condition on the nonlinear term to get the main result.
... In 2018, Li and Qin [28] focused on the periodic solutions of quaternion-valued delayed cellular neural networks. For more detailed works, we refer the readers to [10,[29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. ...
This paper mainly examines the stability and the existence of Hopf bifurcation of fractional-order BAM neural networks with multiple delays. Based on Laplace transform, stability criterion and Hopf bifurcation theory of fractional-order differential equations, a new sufficient criterion to guarantee the stability and the existence of Hopf bifurcation for the involved fractional-order BAM neural networks with multiple delays is established. The investigation manifests that time delay plays an important role in maintaining network stability and controlling the appearance of Hopf bifurcation of fractional-order BAM neural networks. Numerical simulations are performed to check the rationality of the analytical conclusions. The obtained theoretical predictions of this paper have extremely vital guiding significance in designing and controlling networks.
... In 2015, Rahmanian and Ghazavi [52] investigated a slider-crank mechanism with clearance and explained the sensitive dependence on clearance size. Moreover, authors in [53][54][55][56][57] considered the characteristics of the discontinuous systems with impul-sive, time-delay, state delay of state-dependence and so on. ...
This paper presents the nonlinear dynamics of a complex 2-DOF (two degree-of-freedom) system including nonlinear stiffness and damping elements, friction as well as impact, and the purpose of study is to give an original and deep investigation on the discontinuous dynamical behaviors for such a 2-DOF system through strict mathematical consideration. Firstly, the physical model of the system consisting of a ball and an object with curved track and viscoelastic limit devices is established by Coulomb friction and non-linear spring-damping model. And the eight motion states associated with free, sliding or stick motions are defined for the oscillator. Secondly, based on the non-smoothness/discontinuity resulted from impact/friction, the phase space is divided into different domains and boundaries in absolute and relative coordinates, respectively. Thirdly, some necessary and sufficient conditions for oscillator’s motion switching at separation boundaries are given by G-functions of the flow switchability theory in discontinuous dynamical systems. Finally, in order to better understand the switching criteria and the complexity of oscillator’s motion, some illustrative examples for several typical motions in system are studied by numerical simulation. The nonlinear spring-damping model is widely used as a shock absorber in machinery, aerospace, construction and other fields, which can accurately reflect the energy loss during impact process.
... Their control problems become sophisticated and challenging while multivariable systems are uncertain and nonlinear. Adaptive fuzzy control problems for unknown nonlinear MIMO systems were studied in [5][6][7]. ...
This paper investigates the switching mechanism-based event-trig-gered fuzzy adaptive control issue of multi-input and multi-output (MIMO) nonlinear systems with prescribed performance (PP). Utilizing fuzzy logic systems (FLSs) to approximate unknown nonlinear functions. By using the switching threshold strategy, the system has more flexibility in strategy selection. The proposed control scheme can better solve the communication resource limitation. On account of the Lyapunov stability theory, the stability of the controlled system is proved. And all signals of the controlled system are bounded. Moreover, the tracking errors are controlled in a diminutive realm of the origin within the PP bounded. Simultaneously, the Zeno behavior is avoided. Finally, illustrate the effectiveness of the control scheme that has been proposed by demonstrating some simulation consequences.
... The classical Lyapunov theorem focused on the system's stability over the infinite-time interval. This topic is studied widely by researchers (Binazadeh and Bahmani 2017;Bakefayat and Tabrizi 2016;Kumar et al. 2017;Li and Yang 2020;Liu et al. 2019). However, in some practical applications, the main concerns are on the behavior of the dynamical systems over a fixed or finite-time interval (Ning et al. 2017;Asadinia et al. 2019;Gholami and Shafiei 2020). ...
This paper presents robust fixed-time controllers for nonlinear discrete-time systems with time-varying delay and uncertain parameters. The nonlinear function has the Lipschitz condition, and the uncertain parameters are time-varying. Firstly, by selecting an appropriate Lyapunov–Krasovskii functional, sufficient conditions are extracted to guarantee the fixed-time boundedness in the presence of uncertain parameters and external disturbance. To obtain the state feedback controller gain, these sufficient conditions are presented in the form of linear matrix inequalities. Secondly, output feedback is employed to solve the problem in the presence of uncertain parameters. The effectiveness of the proposed method is also illustrated through two examples.
... Existence, uniqueness, and stability of many types nonlinear stochastic differential equations of integer order discussed in [1,7,12,14,15,17,[20][21][22]32,33,39] like as semi-linear evolution nonlocal Cauchy problem, second-order neutral stochastic differential equations, second-order neutral differential equation with statedependent delay, and second-order stochastic neutral partial differential equations. In [38] , the authors proved the Lyapunov stability analysis for nonlinear systems with state-dependent state delay. In [9] , the authors presented a survey work on fractional calculus and the application of fractional calculus. ...
MSC: 34A08 34D20 37C25 Keywords: Fractional differential equations Stochastic system Stability Mild solution Sine and cosine family of functions a b s t r a c t In this article, we discuss the asymptotic stability and mean square stability of stochastic differential equations of fractional-order 1 < α ≤ 2. We have considered the family of stochastic differential equations with variable delay in the state. For proving our main results, we apply the Banach fixed point theorem and imposed the Lipschitz condition on nonlinearity. Finally, we present an example to illustrate the obtained theory.
... Therefore, Lyapunov's direct method can be used to determine the stability of the method. An overview of Lyapunov's direct method and the definitions for Lyapunov stability can be found in (Narendra and Annaswamy 2012;Li and Yang 2020;Dong et al. 2020). For this algorithm, the Lyapunov candidate function can be chosen in the quadratic form (neglecting the time-varying notation ( ) for simplicity) as: ...
Systems are being developed for power distribution on future navy ships to effectively incorporate advanced electrical systems requiring DC power. These islanded DC microgrids provide new challenges to the management of power flow throughout the system, and thus require advanced controls to accurately regulate and stabilise the systems during changing operational conditions. Therefore, adaptive droop control is proposed as a power management layer for these maritime DC microgrids. This paper reports on a method that uses adaptation of the controller parameters to account for uncertainty and changes in the system in order to accurately regulate the power sharing among distributed generation sources and to stabilise the bus voltage of the system. The proposed method is applied to a virtual prototype of a medium voltage direct current (MVDC) ship power system and demonstrated through notional operational scenarios to test the effectiveness of the control algorithm utilising controller hardware in the loop (CHIL) experimentation.
This article focuses on the p th moment exponential stability of discrete‐time Markov switched stochastic systems with state‐dependent delay. By using Razumikhin technique, and stochastic analysis techniques, the state‐dependent delay with randomness due to the randomness of the system state can be addressed. With the help of stationary distribution of Markov switching and multiple Lyapunov–Krasovskii functionals approach, an appropriate switching rule is designed and a novel theorem on discrete‐time Markov switched stochastic systems with state‐dependent delay is established. Different from the previous results, the Markov switched with partially unstable subsystems and the state‐dependent delay with randomness are considered in this article, which brings many differences to the stability analysis of the system compared to traditional stability analysis. Furthermore, an illustrative example is performed to illustrate the effectiveness of the main theory.
This paper focuses on the asynchronous finite-time control of discrete-time impulsive switched positive time-delay systems (SPLTSs). The admissible edge-dependent average dwell time (AED-ADT) approach, in which the switching behavior is represented by a directed graph, is adopted in this paper to reduce the conservatism of the mode-dependent average dwell time (MDADT) method. The new sufficient conditions for the existence of a family of asynchronously switched controllers are derived by constructing the co-positive Lyapunov functional (CLF) for asynchronous intervals and the multiple piecewise co-positive Lyapunov functional (MPCLF) for synchronous intervals, ensuring that the resulting closed-loop system is finite-time stable. Then, the desired controller gains can be obtained by solving a standard linear programming (LP) problem. Finally, two numerical examples are presented to demonstrate the validity of the developed results.
Controlling an inverted pendulum towards an upright position is a difficult task. Backstepping control is an emerging tool for assisting this extremely nonlinear system to stabilize. Since several studies demonstrated fractional modern strategies with Oustaloup approximation, this current work proposes a novel fractional backstepping rule with improved biquadratic equiripple approximation method to stabilize the system with superior accuracy. On the basis of study in the frequency domain, a suitable fractional order is established. Closed-loop performances and control efforts between proposed fractional and conventional backstepping controllers are illustrated based on time domain analysis from a real-time perspective. By abruptly changing the system's parameters, the effectiveness of the proposed controller is also verified. A further fractional Lyapunov improved architecture is proposed to investigate control efficacy with proposed fractional backstepping strategy. The selection of tuning parameters of all control strategies is addressed analytically in depth. It is explored that the suggested fractional backstepping control scheme outperforms the conventional backstepping and fractional Lyapunov stability rules by effectively tracking desired position. This enhanced performance is achieved with relatively smooth control action. On the basis of error measurements, quantitative performance analysis is also subjected to all control strategies.
In this study, we consider the finite time stability (FTS) and finite time contractive stability (FTCS) of the stochastic functional systems. For the framework of Lyapunov-Ruzumikhin method, the sufficient conditions of FTS and FTCS are given first. Then the theory is applied to FTS and FTCS of linear time-varying stochastic functional systems. Finally, experiments on the numerical examples have demonstrated the superiority of the proposed method.
This paper is devoted to solving the problem of stochastically weakly locally exponential stability of switched stochastic systems with state-dependent delay. Different from the previous works, we introduce the state-dependent delay, which is a random variable dependent on the system state so that the bound of the delay is unknown. In addition, we also consider the influence of switching on the stability of the system. By using multiple Lyapunov-Krasovskii functionals and some novel stochastic analysis techniques, we obtain sufficient conditions for stochastically weakly locally exponential stability of the hybrid system. Two examples are provided to verify the correctness of the proposed results.
This paper is concerned with
–stabilization of switched positive systems with stochastic interval delay by using intermittent static output feedback control strategy. Different from previous results, this paper mainly focuses on the system state which is not completely measurable due to external disturbance and sensor fault, rather than the result of continuous and complete measurement of the system, which is more in line with the actual system situation, but it also yields much difficulty. To overcome the difficulty, a novel intermittent static output feedback control method is proposed in this paper, which is also the first time to be applied to the class modification system. More importantly, unlike the existing switched positive system which only considers deterministic time delay, the time delay considered in this paper occurs randomly in the interval, which has more research significance and value. To verify the correctness of the developed strategy, a practical model is provided.
This paper considers a wide class of smooth continuous dynamic nonlinear systems (control objects) with a measurable vector of state. The problem is to find a special function (Lyapunov function), which in the framework of the second Lyapunov method guarantees asymptotic stability for the above described class of nonlinear systems. It is well known that the search for a Lyapunov function is the "cornerstone" of mathematical stability theory. Methods for selecting or finding the Lyapunov function to analyze the stability of closed linear stationary systems, as well as for nonlinear objects with explicit linear dynamic and nonlinear static parts, have been well studied (see works by Lurie, Yakubovich, Popov, and many others). However, universal approaches to the search for the Lyapunov function for a more general class of nonlinear systems have not yet been identified. There is a large variety of methods for finding the Lyapunov function for nonlinear systems, but they all operate within the constraints imposed on the structure of the control object. In this paper we propose another approach, which allows to give specialists in the field of automatic control theory a new tool/mechanism of Lyapunov function search for stability analysis of smooth continuous dynamic nonlinear systems with measurable state vector. The essence of proposed approach consists in representation of some function through sum of nonlinear terms, which are elements of object's state vector, multiplied by unknown coefficients, raised to positive degrees. Then the unknown coefficients are selected using genetic algorithm, which should provide the function with all necessary conditions for Lyapunov function (in the framework of the second Lyapunov method).
In this paper, the composite anti-disturbances control problem is considered for a class of stochastic systems with multiple disturbances. The states of the system are assumed to be unavailable. A state observer and a disturbance observer are constructed to estimate the states and the matched disturbance respectively. Based on the estimated values of state observer and disturbance observer, a non-fragile composite controller is designed to achieve disturbance attenuation and rejection. By means of the technique of the disturbance compensation control and stochastic control theory, some sufficient conditions are obtained to guarantee that the closed-loop system is asymptotically bounded in mean square or asymptotically stable in probability. Finally, a numerical example is given to verify the validity of the obtained results.
This paper investigates the controllability of Markovian jump Boolean control networks (MJBCNs) by using graphical approach. To solve this problem, a deterministic directed graph is constructed to describe the stochastic evolutionary process of the considered MJBCN under the framework of semi-tensor product of matrices. Then, combining with breadth-first search, the special subgraph with breadth-first tree as the principal part is obtained. Based on this, the necessary and sufficient conditions to verify the controllability of MJBCNs are deduced, and the problem of determining the minimal controllable time is solved completely. Subsequently, two examples are given to illustrate the validity and correctness of the method.
Improving the stability and safety is of great significance for the in-wheel electric vehicle. There are many studies only concentrating on active front steering (AFS) control or direct yaw-moment control (DYC). However, When the in-wheel electric vehicle is under extreme conditions, AFS or DYC alone is not effective. In this paper, an integrated controller of AFS and DYC is proposed. Firstly, the ideal values of yaw rate and sideslip angle can be calculated based on the two-degree-of-freedom vehicle model. Secondly, the AFS controller is obtained according to the backstepping-based fast terminal sliding mode (FTSM). Then, the DYC controller which consists of the upper controller and the lower controller is constructed. The upper controller is developed via the integral-based second-order sliding mode (SOSM). The appropriate torque is outputted to each wheel by the lower controller. Finally, the simulation results show that the actual yaw rate and sideslip angle can approach the ideal ones as closely as possible under the proposed integrated controller.
This paper is concerned with the simultaneous exponential stabilization problem for a set of stochastic port-controlled Hamiltonian (PCH) systems. Due to the limited bandwidth of the channels, the phenomena of fading channels and transmission delays which are described by a time-varying stochastic model always occur in the communication channels from the controller to the actuator. Meanwhile, actuator saturation constraint is taken into account. On the basis of dissipative Hamiltonian structural and saturating actuator properties, those stochastic PCH systems are combined to generate an augmented system. By utilizing the stochastic analysis theory, sufficient criterions are given for the simultaneous stabilization controller design ensuring that the closed-loop system is simultaneously exponentially mean-square stable (SEMSS). For the case that there exist external disturbances in the systems, some results on stability analysis and controller design are given. The developed controller design scheme is proved by a three-helicopter model simulation example.
In this paper, we study the uniform stability (US) of nonlinear systems with state-dependent delay (SDD), where the SDD is not assumed to be a priori bounded since it is dependent on the state of the system. We use some Lyapunov functions coupled with differential inequality techniques tailed at SDD to obtain sufficient conditions for US of nonlinear systems. Two numerical examples and their simulations are given, where one of them is from the fluid-flow model of Transmission Control Protocol (TCP) behavior in active queue management (AQM) involving state-dependent queuing delay, to demonstrate the effectiveness and novelty of our proposed results.
In this article, impulsive control is firstly considered for the quaternion-based attitude stabilization of a rigid spacecraft. Inspired by the robustness of sliding mode control, a new concept of impulsive-sliding mode is proposed to design robust impulsive controller for attitude stabilization. It illustrates that the designed impulsive controller possesses simple structure and robustness with respect to external disturbances. In order to achieve the relaxation of continuous measurement, we utilize periodic state measurement that is more economical and practical than the continuous one. Namely, we can achieve attitude stabilization and resource economization by adopting only discrete state measurement and discrete control input. More interestingly, our results can overcome the difficulty that the upper bound of disturbances is unknown, which is often required to design adaptive controller to handle in previous results. Moreover, we prove that the phenomenon can be excluded under the proposed periodic event-triggering strategy and the interevent time has a uniform lower bound. An illustrative example is presented to show the feasibility of our results.
In this article, we consider the input-to-state stability (ISS) problem for a class of time-delay systems with intermittent large delays, which may cause the invalidation of traditional delay-dependent stability criteria. The topic of this article features that it proposes a novel kind of stability criterion for time-delay systems, which is delay dependent if the time delay is smaller than a prescribed allowable size. While if the time delay is larger than the allowable size, the ISS can be preserved as well provided that the large-delay periods satisfy the kind of duration condition. Different from existing results on similar topics, we present the main result based on a unified Lyapunov-Krasovskii function (LKF). In this way, the frequency restriction can be removed and the analysis complexity can be simplified. A numerical example is provided to verify the proposed results.
This paper investigates the problem of asynchronous control for nonlinear Markov jump systems (MJSs) on the basis of fuzzy quantized sampled-data controller. The modes of the designed fuzzy controller and the modes of the original system are asynchronously operational, which has been widely used in the real world and expressed by a hidden Markov model. Based on the mode-dependent Lyapunov-Krasovskii functional (LKF) approach and some improved inequalities, a novel criterion is established to ensure the stochastic stability of fuzzy nonlinear MJSs. As a particular case, with the help of novel LKF and new integral inequalities, new stability condition with less conservatism and low computational complexity for nonlinear quantized sampled-data control systems is obtained. Finally, the validity and merits of the developed method are shown by two numerical examples.
The exponential stability problem for impulsive systems subject to double statedependent delays is studied in this paper, where state-dependent delay (SDD) is involved in both continuous dynamics and discrete dynamics and the boundedness of it with respect to states is prior unknown. According to impulsive control theory, we present some Lyapunov-based sufficient conditions for the exponential stability of the concerned system. It is shown that the stabilizing effect of SDD impulses on an unstable SDD system changes the stability and achieves desired performance. In addition, the destabilizing effect of SDD impulses is also fully considered and the corresponding sufficient conditions are derived, which reveals the fact that a stable SDD system can maintain its performance when it is subject to SDD impulsive disturbance. As an application, the proposed result can be employed to the stability analysis of impulsive genetic regulatory networks (GRNs) with SDD and the corresponding sufficient conditions are proposed in terms of the model transformation technique and the linear matrix inequalities (LMIs) technique. In order to demonstrate the effectiveness and applicability of the derived results, we give two examples including impulsive GRNs with SDD and the impulsive controller design for the nonlinear system with SDD.
This article addresses the problems of fixed‐time stabilization for a class of quaternion fuzzy neural networks (QFNNs) with time‐varying delay. The QFNNs are developed by dividing our system into four real‐valued parts based on the Hamilton rule. Then, based on fixed‐time stability theory, some inequality techniques, and selecting the appropriate controllers and Lyapunov function, a novel criterion guaranteeing the fixed‐time stabilization and the finite‐time stabilization of the addressed system is derived. Finally, three numerical examples are presented to show the effectiveness of our theoretical results.
This paper characterizes the stability crossing curves of a class of linear systems with gamma-distributed delay with a gap. First, we describe the crossing set, i.e., the set of frequencies where the characteristic roots may cross the imaginary axis as the parameters change. Then, we describe the corresponding stability crossing curves, i.e., the set of parameters such that there is at least one pair of characteristic roots on the imaginary axis. Such stability crossing curves divide the parameter space ℝ + 2 defined by the mean delay and the gap into different regions. Within each such region, the number of characteristic roots on the right half complex plane is fixed. This naturally describes the regions of parameters where the system is stable. The classification of the stability crossing curves is also discussed. Some illustrative examples (Cushing equation in biology, traffic flow models in transportation systems, and control over networks of a simplified helicopter model) are also presented.
The exponential stability of a class of integral delay systems
with analytic kernels is investigated by using the
Lyapunov-Krasovskii functional approach. Sufficient
delay-dependent stability conditions and exponential estimates
for the solutions are derived. Special attention is paid to the
particular cases of polynomial and exponential kernels.
Networked control systems (NCSs) are spatially distributed systems for which the communication between sensors, actuators, and controllers is supported by a shared communication network. We review several recent results on estimation, analysis, and controller synthesis for NCSs. The results surveyed address channel limitations in terms of packet-rates, sampling, network delay, and packet dropouts. The results are presented in a tutorial fashion, comparing alternative methodologies
This paper investigates the global stabilization problem of discrete-time multiple integrators with bounded and delayed feedback. In the absence of input delay, two classes of nonlinear feedback laws are proposed. The first one consists of parallel connections of saturation functions and the other one consists of nested saturation functions. Some free parameters and the so-called state-dependent saturation functions are introduced into these two types of control laws which can help to improve the transient performance of the closed-loop system. In the presence of input delay, with the aid of a special canonical form, two types of nonlinear control laws are also proposed to achieve global stability of the closed-loop system. Two numerical examples are given to illustrate the effectiveness of the proposed methods.
This paper focuses on the fixed-time synchronization control methodology for a class of delayed memristor-based recurrent neural networks. Based on Lyapunov functionals, analytical techniques, and together with novel control algorithms, sufficient conditions are established to achieve fixed-time synchronization of the master and slave memristive systems. Moreover, the settling time of fixed-time synchronization is estimated, which can be adjusted to desired values regardless of the initial conditions. Finally, the corresponding simulation results are included to show the effectiveness of the proposed methodology derived in this paper.
In this paper, we present a computational approach to the problem of local stability of scalar systems with state-dependent delay. Our approach is to parameterize a class of positive quadratic Lyapunov-Krasovskii functionals using positive matrices. By constrain- ing the functional to be positive and its derivative to be negative, we can represent the problem of stability as a convex optimization problem and solve this problem using efficient algorithms for semidefinite programming. The accuracy of the approach is demonstrated using a set of numerical examples.
We consider nonlinear differential systems with state-dependent delayed impulses (impulses which involve the delayed state of the system for which the delay is state-dependent). Such systems arise naturally from a number of applications and the stability issue is complex due to the state-dependence of the delay. We establish general and applicable results for uniform stability, uniform asymptotic stability and exponential stability of the systems by using the impulsive control theory and some comparison arguments. We show how restrictions on the change rates of states and impulses should be imposed to achieve system’s stability, in comparison with general impulsive delay differential systems with state-dependent delay in the nonlinearity, or the differential systems with constant delays. In our approach, the boundedness of the state-dependent delay is not required but derives from the stability result obtained. Examples are given to demonstrate the sharpness and applicability of our general results and the proposed approach.
This paper is concerned with stabilization of (time-varying) linear systems with a single time-varying input delay by using the predictor based delay compensation approach. Differently from the traditional predictor feedback which uses the open-loop system dynamics to predict the future state and will result in an infinite dimensional controller, we propose in this paper a pseudo-predictor feedback (PPF) approach which uses the (artificial) closed-loop system dynamics to predict the future state and the resulting controller is finite dimensional and is thus easy to implement. Necessary and sufficient conditions guaranteeing the stability of the closed-loop system under the PPF are obtained in terms of the stability of a class of integral delay operators (systems). Moreover, it is shown that the PPF can compensate arbitrarily large yet bounded input delays provided the open-loop (time-varying linear) system is only polynomially unstable and the feedback gain is well designed. Comparison of the proposed PPF approach with the existing results is well explored. Numerical examples demonstrate the effectiveness of the proposed approaches.
In this paper we study exponential stability of the trivial solution of the state-dependent delay system ú x(t) = Pm i=1 Ai(t)x(ti(t, xt)). We show that under mild assumptions, the trivial solution of the state-dependent system is exponentially stable, if and only if the trivial solution of the corre- sponding linear time-dependent delay system ú y(t) = Pm i=1 Ai(t)y(ti(t, 0)) is exponentially stable. We also compare the order of the exponential stability of the nonlinear equation to that of its linearized equation. We show, that in some cases, the two orders are equal. As an application of our main result, we formulate a necessary and sufficient condition for the exponential stability of the trivial solution of a threshold-type delay system.
In this work, we consider the exponential stability of piecewise affine systems with time- and state-dependent delay, and delayed-state-dependent switching. The stability analysis is based on the use of Lyapunov-Krasovskii functionals, and is divided into two parts. First, global stability conditions are proposed in the case of systems with (state-independent) time-varying delay. Then, local stability conditions are derived in the case of systems with time- and state-dependent delay. In the latter case, estimations of the domain of attraction are also proposed. The theoretical results are applied to the congestion control problem, which can be modelled by such systems.
This chapter illustrates the recent work on equations with state dependent delays, with emphasis on particular models and on the emerging theory from the dynamical systems point of view. Several new results are presented. The chapter describes examples of differential equations with state dependent delays which arise in physics, automatic control, neural networks, infectious diseases, population growth, and cell production. Some of these models differ considerably from others, and most of them do not look simple. Typically the delay is not given explicitly as a function of what seems to be the natural state variable; the delay may be defined implicitly by a functional, integral or differential equation and should often be considered as part of the state variables.
In this paper we begin a study of the differential-delay equation
.
We prove the existence of periodic solutions for 0ɛɛ
0, where ɛ
0 is an optimal positive number. We investigate regularity and monotonicity properties of solutions x(t) which are defined for all t and of associated functions like η(t)=t−r(x(t)). We begin the development of a Poincar-Bendixson theory and phase-plane analysis for such equations. In a companion paper
these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions
as ɛ approaches zero.
We consider LTI finite-dimensional, completely controllable, but possibly open-loop unstable, plants, with arbitrarily long actuator delay, and the corresponding predictor-based feedback for delay compensation. We study the problem of inverse-optimal re-design of the predictor-based feedback law. We obtain a simple modification of the basic predictor-based controller, which employs a low-pass filter, and has been proposed previously by Mondie and Michiels for achieving robustness to discretization of the integral term in the predictor feedback law. The key element in our work is the employment of an infinite-dimensional “backstepping” transformation, and the resulting Lyapunov function, for the infinite dimensional systems consisting of the state of the ODE plant and the delay state. The Lyapunov function allows us to quantify the Lyapunov stability properties under the modified feedback, the inverse optimality of the feedback, and its disturbance attenuation properties. For the basic predictor feedback, the availability of the Lyapunov function also allows us to prove robustness to small delay mismatch (in both positive and negative directions).
In this paper we formulate a stability theorem by means of linearization around a trivial solution in the case of autonomous neutral functional differential equations with state-dependent delays. We prove that if the trivial solution of the linearized equation is exponentially stable, then the trivial solution of the nonlinear equation is exponentially stable as well. As an application of the main result, explicit stability conditions are given.
Compensation of state-dependent input delay for nonlinear systems
- Liberis