Sufficient Conditions For Exponential Stability Research Articles

Output Feedback Robust H∞ Control for Discrete 2D Switched Systems

The two-dimensional (2-D) system has a wide range of applications in different fields, including satellite meteorological maps, process control, and digital filtering. Therefore, the research on the stability of 2-D systems is of great significance. Considering that multiple systems exist in switching and alternating work in the actual production process, but the system itself often has external perturbation and interference. To solve the above problems, this paper investigates the output feedback robust H∞ stabilization for a class of discrete-time 2-D switched systems, which the Roesser model with uncertainties represents. First, sufficient conditions for exponential stability are derived via the average dwell time method, when the system’s interference and external input are zero. Furthermore, in the case of introducing the external interference, the weighted robust H∞ disturbance attenuation performance of the underlying system is further analyzed. An output feedback controller is then proposed to guarantee that the resulting closed-loop system is exponentially stable and has a prescribed disturbance attenuation level γ. All theorems mentioned in the article will also be given in the form of linear matrix inequalities (LMI). Finally, a numerical example is given, which takes two uncertain values respectively and solves the output feedback controller’s parameters by the theorem proposed in the paper. According to the required controller parameter values, the validity of the theorem proposed in the article is compared and verified by simulation.

ON SUFFICIENT CONDITIONS OF EXPONENTIAL STABILITY FOR SYSTEMS OF LINEAR AUTONOMOUS DIFFERENTIAL EQUATIONS WITH AFTEREFFECT

We consider systems of linear autonomous functional differential equa-tion with aftereffect and propose an approach to obtain effective sufficient conditions of exponential stability for these systems. In the approach we use the positiveness of the fundamental matrix of an auxiliary system (a comparison system) with concentrated and distributed delays.

Event-Triggered Exponential Stabilization for State-Based Switched Inertial Complex-Valued Neural Networks With Multiple Delays.

This article explores the exponential stabilization issue of a class of state-based switched inertial complex-valued neural networks with multiple delays via event-triggered control. First, the state-based switched inertial complex-valued neural networks with multiple delays are modeled. Second, by separating the real and imaginary parts of complex values, the state-based switched inertial complex-valued neural networks are transformed into two state-based switched inertial real-valued neural networks. Through the variable substitution method, the model of the second-order inertial neural networks is transformed into a model of the first-order neural networks. Third, an event-triggered controller with the transmission sequence is designed to study the exponential stabilization issue of neural networks constructed above. Then, by constructing the Lyapunov functions and based on some inequalities, we obtain sufficient conditions for exponential stabilization of the proposed neural networks. Furthermore, it is proved that the Zeno phenomenon cannot happen under the designed event-triggered controller. Finally, a simulation example is given to illustrate the correctness of the results.

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

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

Output-based event-triggered resilient control of uncertain NCSs under DoS attacks and quantisation

This paper studies output-based resilient control of uncertain networked control systems under denial-of-service (DoS) attacks, resilient event-triggered mechanism (ETM) and quantisation. Firstly, to save constrained system resources such as network bandwidth, using information of DoS attacks and sampled outputs of plant, an output-based resilient ETM is introduced. Secondly, a switched system model is built, which characterises effects of DoS attacks, the resilient ETM, plant uncertainty and quantisation in a unified framework. Thirdly, sufficient conditions for exponential stability are obtained with less conservativeness, which establish quantitative relationships among DoS attacks, the resilient ETM, plant uncertainty, quantisation, dynamic output feedback (DOF) controller and system stability. Further, co-design conditions are presented to compute parameters of the resilient ETM and DOF controller simultaneously. Finally, example illustrates effectiveness of the proposed method.

New Results on H∞ Control for Nonlinear Conformable Fractional Order Systems

This paper deals with H∞ control problem for nonlinear conformable fractional order systems. The authors first derive new sufficient condition for exponential stability of nonlinear conformable fractional order systems based on Lyapunov-like function method for conformable fractional order systems and linear matrix inequalities (LMIs) approach. Then, by introducing a new concepts of H∞ control problem for nonlinear conformable fractional order systems, the authors study H∞ performance analysis and H∞ state feedback controller design problems for the considered systems. In terms of LMIs, a sufficient condition is proposed to ensure the nonlinear conformable fractional order systems are not only exponentially stable, but also satisfy H∞ performance γ. An explicit expression for state feedback controllers is also designed to make the closed-loop system is exponentially stable with H∞ performance γ. Finally, numerical examples are given to illustrate the validity and effectiveness of the proposed results.

Late‐lumping fuzzy boundary geometric control of nonlinear partial differential systems

SummaryIn this article, a fuzzy boundary geometric controller that stabilizes a class of nonlinear distributed parameter systems (DPSs) is proposed. The design procedure relies on the use of Takagi‐Sugeno (T‐S) type fuzzy partial differential equation (PDE) model, which approximates the dynamical behavior of the nonlinear DPS. The T‐S fuzzy PDE model is constructed through “fuzzy blending” of local linear PDE models of infinite characteristic indexes. This is a challenging task in the design procedure of fuzzy PDE model‐based boundary controller in the framework of the well‐established geometric control theory. To overcome this constraint, it is proposed in this article to resort to the concept of extended operator in order to transform the T‐S fuzzy PDE model with boundary control to an equivalent fuzzy PDE model with punctual control and finite characteristic index. Based on the developed fuzzy model, a fuzzy boundary geometric controller is derived and sufficient conditions of exponential stability of the resulting closed‐loop system are established by employing the Lyapunov direct method. The stabilizing performance of the proposed fuzzy PDE model‐based boundary geometric controller is evaluated on benchmark control problems and compared with other existing control methods via numerical simulations.

Neural Contraction Metrics for Robust Estimation and Control: A Convex Optimization Approach

This letter presents a new deep learning-based framework for robust nonlinear estimation and control using the concept of a Neural Contraction Metric (NCM). The NCM uses a deep long short-term memory recurrent neural network for a global approximation of an optimal contraction metric, the existence of which is a necessary and sufficient condition for exponential stability of nonlinear systems. The optimality stems from the fact that the contraction metrics sampled offline are the solutions of a convex optimization problem to minimize an upper bound of the steady-state Euclidean distance between perturbed and unperturbed system trajectories. We demonstrate how to exploit NCMs to design an online optimal estimator and controller for nonlinear systems with bounded disturbances utilizing their duality. The performance of our framework is illustrated through Lorenz oscillator state estimation and spacecraft optimal motion planning problems.

Stability analysis of discrete-time coupled systems with delays

Addressed in this paper is the stability issue of discrete-time delayed coupled systems consisting of a difference equation and an algebraic equation in coupled states. First, a necessary and sufficient condition of exponential stability is proposed for coupled systems with constant delays. Second, periodicity and stability are investigated for periodic coupled systems. Third, based on the Lyapunov inequality, a necessary and sufficient condition is presented for exponential stability of general time-varying coupled systems. Finally, as an application of these results, stability of discrete-time singular systems is discussed.

Necessary and Sufficient Stability Condition by Finite Number of Mathematical Operations for Time-delay Systems of Neutral Type

We present a new necessary and sufficient condition of exponential stability of neutral-type linear time invariant (LTI) systems with one delay. The new stability criterion is given in terms of a block matrix, which exclusively depends on the delay Lyapunov matrix and is of finite dimension.

Sampled-data exponential synchronization of time-delay neural networks subject to random controller gain perturbations

• A non-fragile sampled-data control method is used to study the exponential synchronization of neural networks with mixed delays. • A sufficient exponential stability condition is developed by using a new two-sided looped Lyapunov functional. • A design scheme of the non-fragile sampled-data controller is proposed by using some decoupling techniques. In this paper, a non-fragile sampled-data control method is used to investigate the exponential synchronization of neural networks with discrete and distributed delays. The occurrence of controller gain perturbations is assumed to be random, which is described by a stochastic variable with the Bernoulli distribution. An extended two-sided looped Lyapunov functional is constructed, which efficiently utilizes available state information of the sampled instants. By using the two-sided looped Lyapunov functional and introducing suitable free weighting matrices, a sufficient condition is derived under which the resulting synchronization-error system is exponentially stable. Then, a design scheme of the non-fragile sampled-data controller is proposed with the aid of some decoupling techniques. At last, a numerical example is provided to illustrate the effectiveness and superiority of the proposed sampled-data control method.

Non-fragile observer-based adaptive control of uncertain nonlinear stochastic Markovian jump systems via sliding mode technique

This paper is focused on a novel observer-based sliding mode control strategy for a class of uncertain stochastic Markovian jump systems with nonlinear perturbation. A novel linear switching surface scheme is established for the system with the aid of a simplified non-fragile observer, from which the corresponding reaching motion adaptive control law is synthesized. Then, a new sufficient condition of exponential stability for the associated dynamics during the sliding mode is derived based upon the designed sliding function and linear matrix inequality approach. At last, two illustrative examples are conducted to justify the effectiveness and applicability of the proposed methodology.

Exponential Stabilization of Linear Time-Varying Differential Equations with Uncertain Coefficients by Linear Stationary Feedback

We consider a control system defined by a linear time-varying differential equation of n-th order with uncertain bounded coefficients. The problem of exponential stabilization of the system with an arbitrary given decay rate by linear static state or output feedback with constant gain coefficients is studied. We prove that every system is exponentially stabilizable with any pregiven decay rate by linear time-invariant static state feedback. The proof is based on the Levin’s theorem on sufficient conditions for absolute non-oscillatory stability of solutions to a linear differential equation. We obtain sufficient conditions of exponential stabilization with any pregiven decay rate for a linear differential equation with uncertain bounded coefficients by linear time-invariant static output feedback. Illustrative examples are considered.

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.

Stochastic master stability function for noisy complex networks.

In this paper, we broaden the master stability function approach to study the stability of the synchronization manifold in complex networks of stochastic dynamical systems. We provide necessary and sufficient conditions for exponential stability that allow us to discriminate the impact of noise. We observe that noise can be beneficial for synchronization when it diffuses evenly in the network. On the contrary, an excessively large amount of noise only acting on a subset of the node state variables might have disruptive effects on the network synchronizability. To demonstrate our findings, we complement our theoretical derivations with extensive simulations on paradigmatic examples of networks of noisy systems.

Positivity, exponential stability and disturbance attenuation performance for singular switched positive systems with time-varying distributed delays

This article analyzes the singular switched positive systems with time-varying distributed delays from the perspective of positivity, exponential stability and disturbance attenuation performance referring to both L1-gain and L∞-gain. For the system, a sufficient and necessary positivity condition is firstly developed by using the singular value decomposition technique. Then, a sufficient condition of exponential stability, which makes the considered system exponentially stable, is proposed on the basis of co-positive Lyapunov–Krasovskii functional and average dwell time techniques, and the obtained exponential decay rate can be adjusted in the light of various actual situations. Furthermore, the article analyzes the disturbance attenuation performance referring to both L1-gain and L∞-gain, and through the convex optimization approach, the optimal L1-gain and L∞-gain performance level could be established, respectively. Three examples are finally presented to show the feasibility and effectiveness of the obtained results.

A Network SIS Meta-Population Model with Transportation Flow

This paper considers a deterministic Susceptible-Infected-Susceptible (SIS) meta-population model for the spread of a disease in a strongly connected network, where each node represents a large population. Individuals can travel between the nodes (populations). We derive a necessary and sufficient condition for the healthy equilibrium to be the unique equilibrium of the system, and then in fact it is asymptotically stable for all initial conditions (a sufficient condition for exponential stability is also given). If the condition is not satisfied, then there additionally exists a unique endemic equilibrium which is exponentially stable for all nonzero initial conditions. We then consider time-delay in the travel between nodes, and further investigate the role of the mobility rate that governs the flow of individuals between nodes in determining the convergence properties. We find that sometimes, increasing mobility helps the system converge to the healthy equilibrium.

General condition for exponential stability of thermoelastic Bresse systems with Cattaneo's law

In this paper, we give a new and more general sufficient condition for exponential stability of thermoelastic Bresse systems with heat flux given by Cattaneo's law acting in shear and longitudinal motion equations. This condition, which we also prove to be necessary in some special cases, is given by a relation between the constants of the system and generalizes the well-known equal wave speed condition.

Periodicity and stability analysis of periodic switched positive systems

Addressed in this study are the minimal period and stability issues of periodic switched positive systems. With the assumption that all the system matrices, delays and switching signal are periodic, it is shown that the considered switched system is also periodic with the minimal period being a divisor of the least common multiple of the periods of subsystems and switching signal, and an algorithm is presented to determine the minimal period. Then some necessary and sufficient exponential stability conditions are proposed for periodic switched positive systems. These conditions are also extended to a more general class of periodic switched systems. Finally, a numerical example with three cases is provided to demonstrate the effectiveness of the theoretical results and reveals two interesting facts: The system matrices of periodic positive systems are unnecessarily non-negative and system delays influence system stability. These facts imply that there exist some remarkable differences between general switched positive systems and periodic switched positive systems.

Exponential Stability of Leakage Delay and Semi-Markovian Jump for Neutral-Type Neural Network

In this paper, a suitable Lyapunov-Krasovskii functional together with the inequality analysis technique for neutral-type system, and sufficient exponential stability conditions are proposed in the form of linear matrix inequalities (LMIs). Firstly, the system model of leakage delay and semi-Markovian process for neutral-type neural network is established, where the sojourn-time follows Weibull distribution, which is time-varying instead of being constant in the transition rate. Secondly, through combinations of stability analysis method, Lyapunov-Krasovskii functional and inequality analysis technique, some sufficient conditions have been achieved to undertake the exponential stabilization of given neural network system. Moreover, time delays of the system contain leakage and mixed delays, which are more general and appropriate. Finally, in order to show the many advantages of our model, the obtained results will be compared with previous works based on methods different from semi-Markovian jump or leakage delay.