Dependent Stability Conditions Research Articles

A further step to transient performance enhancement of discrete-time singular systems with time-varying delay

This paper is the first attempt to enhance the transient performance while simultaneously addressing the output regulation challenges for discrete singular time-delay systems subject to actuator saturation. The proposed controller consists of two linear and nonlinear parts, without any switching elements. The linear part is designed to solve the main regulation problem, while the nonlinear part is designed to enhance the performance of the closed-loop system in the face of input saturation. The delay-range dependent stability conditions of the closed-loop system are analysed using the Lyapunov-Krasovskii (LK) approach, which involves solving an LMI problem to determine the parameters of the proposed controller. Furthermore, the admissibility of the closed-loop system is assessed to ensure the system’s stability, regularity, and causality. Finally, the theoretical achievements are endorsed by simulations through numerical and practical examples.

Design of optimal PID controller for multivariable time-varying delay discrete-time systems using non-monotonic Lyapunov-Krasovskii approach

This paper is concerned with stability analysis and stabilization of time-varying delay discrete-time systems in Lyapunov-Krasovskii stability analysis framework. In this regard, a less conservative approach is introduced based on non-monotonic Lyapunov-Krasovskii (NMLK) technique. The proposed method derives time-varying delay dependent stability conditions based on Lyapunov-Krasovskii functional (LKF), which are in the form of linear matrix inequalities (LMI). Also, a PID controller designing algorithm is extracted based on obtained NMLK stability condition. The stability of the closed loop system is guaranteed using the designed controller. Another property that is important along with the stability, is the optimality of the controller. Thus, an optimal PID designing technique is introduced in this article. The proposed method can be used to design optimal PID controller for unstable multi-input multi-output time-varying delay discrete-time systems. The proposed stability and stabilization conditions are less conservative due to the use of non-monotonic decreasing technique. The novelty of the paper comes from the consideration of non-monotonic approach for stability analysis of time-varying delay discrete-time systems and using obtained stability conditions for designing PID controller. Numerical examples and simulations are given to evaluate the theoretical results and illustrate its effectiveness compared to the existing methods.

Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model

<p style='text-indent:20px;'>By extending some linear time delay systems stability techniques, this paper, focuses on continuous time delay nonlinear systems (TDNS) dependent delay stability conditions. First, by using the Takagi Sugeno Fuzzy Modeling, a novel relaxed dependent delay stability conditions involving uncommon free matrices, are addressed in Linear Matrix Inequalities (LMI). Then, as application a Nonlinear Carbon Dioxide Model is used and rewritten by a change of coordinate to the interior equilibrium point. Next, by using the non-linearity sector method the model is transformed to a corresponding Fuzzy Takagi Sugeno (TS) multi-model. Also, the maximum delay margin to which the model is stable, is identified. Finally, to prove the analytic results a numerical simulation is also performed and compared to other methods.</p>

Multiple delay-dependent noise-to-state stability for a class of uncertain switched random nonlinear systems with intermittent sensor and actuator faults

This paper focuses on multiple delay-dependent noise-to-state stability (NSS) for a class of switched random nonlinear systems against uncertainty terms and intermittent sensor and actuator faults. External disturbances, nonlinear functions, as well as measurement noise, are also taken into account. This is the first attempt to achieve dynamic output feedback controller design for uncertain switched random nonlinear systems subject to intermittent sensor and actuator faults. First, a controller is established to perform passive fault-tolerant control (FTC). Random systems are more common than Ito stochastic systems. Thus, compared with the previous works, the proposed controller has a wider application scope and is more feasible. Next, an augmented closed-loop system is exhibited to realize NSS. Moreover, a piecewise Lyapunov function is utilized with less conservatism than common Lyapunov function. The delay dependent stability conditions are gathered via linear matrix inequalities (LMIs) and controller matrices are earned. At last, the novelty and validity of the approach suggested in this paper are demonstrated through two simulation examples.

Regional stabilization of TCP/AQM system with polytopic uncertainties and homogeneous segmented time delay

In this paper, new local stabilization analysis for TCP/AQM process based on segmented delay is proposed. The new approach consists in describing the time delay as a sum of homogeneous loads delays plus the variation of the queuing delay resulting from the made actions inside the router buffer. By employing an appropriate new Lyapunov–Krasovskii functional that depends on the upper bounds of the loads, new segmented delays dependent local stabilization conditions are proposed. Furthermore, new results concerning the unsymmetrical saturation input are obtained. Finally, the theory is experimented on NS 3, which proves that the proposed Segment Controller (SegCont) mechanism is strong and robust.

Synchronization of Fractional Order Fuzzy BAM Neural Networks With Time Varying Delays and Reaction Diffusion Terms

In this article synchronization of fractional order fuzzy BAM neural networks with time varying delays and reaction diffusion terms is studied. The time varying delays consist of discrete delays and distributed delays are considered. Then, some sufficient conditions black are presented to guarantee the global asymptotic stability of the error system by using Lyapunov-Krasovskii functional having the double integral terms, we utilized Jensens inequality techniques and LMI approach. Accordingly, we accomplished synchronization of master-slave fuzzy BAMNNs. The delay dependent stability conditions are set up in terms of linear matrix inequalities(LMIs), which can be productively understood utilizing Matlab LMI control tool box. At last, illustrative numerical results have been provided to verify the correctness and effectiveness of the obtained results.

Delay range‐and‐rate dependent stability criteria for systems with interval time‐varying delay via a quasi‐quadratic convex framework

SummaryThis paper concerns the stability analysis of systems with interval time‐varying delay. A Lyapunov‐Krasovskii functional containing an augmented quadratic term and certain triple integral terms is constructed to integrate features of the truncated Bessel‐Legendre inequality less conservative than Wirtinger inequality that encompasses Jensen inequality, respectively, and to exploit merits of the newly developed double integral inequalities tighter than auxiliary function‐based, Wirtinger, and Jensen double integral inequalities. A new quadratic convex lemma is proposed to derive delay and its derivative dependent sufficient stability conditions in terms of linear matrix inequalities synthetically with reciprocal convex approach and affine convex combination. The efficiency of the presented method is illustrated on some classical numerical examples.

Stability Analysis and Memory Control Design of Polynomial Fuzzy Systems with Time Delay via Polynomial Lyapunov-Krasovskii Functional

This paper investigates the problems of delay-dependent stability analysis and memory control design of polynomial fuzzy systems with time delay. Using polynomial Lyapunov-Krasovskii functional and slack polynomial matrix variables, delay dependent sufficient stability and stabilizability conditions are derived in terms of sum of squares (SOS) which can be numerically (partially symbolically) solved via the recently developed SOSTOOLS. The main advantage of the proposed design is the reduction of conservatism for three great reasons. The first one is that polynomial matrices are not only dependent on the system state vector but also on the state vector with time delay. The second one is that the design conditions are formulated in delay dependent SOS. It is well known that the delay-dependent conditions are less conservative than those independent of time delay. The third one is that only correlated terms are used in the design of SOS. The simulation and comparison are given to illustrate the lesser conservativeness of the proposed result.

Delay Dependent Local Stabilization Conditions for Time-delay Nonlinear Discrete-time Systems Using Takagi-Sugeno Models

We propose convex conditions for stabilization of nonlinear discrete-time systems with time-varying delay in states through a fuzzy Takagi-Sugeno (T-S) modeling. These conditions are developed from a fuzzy Lyapunov-Krasovskii function and they are formulated in terms of linear matrix inequalities (LMIs). The results can be applied to a class of nonlinear systems that can be exactly represented by T-S fuzzy models inside a specific region called the region of validity. As a consequence, we need to provide an estimate of the set of safe initial conditions called the region of attraction such that the closed-loop trajectories starting in this set are assured to remain in the region of validity and to converge asymptotically to the origin. The estimate of the region of attraction is done with the aid of two sets: one dealing with the current state, and the other concerning the delayed states. Then, we can obtain the feedback fuzzy control law depending on the current state, x k , and the maximum delayed state vector, xk−d. It is shown that such a control law can locally stabilize the nonlinear discrete-time system at the origin. We also develop convex optimization procedures for the computation of the fuzzy control gains that maximize the estimates of the region of attraction. We present two examples to demonstrate the efficiency of the developed approach and to compare it with other approaches in the literature.

Membership‐Function‐Dependent Stability Analysis of Interval Type‐2 Polynomial Fuzzy‐Model‐Base Control Systems

In this paper, the stability analysis for interval type-2 (IT2) polynomial fuzzy-model-based (PFMB) control system using the information of membership functions is investigated. To tackle uncertainties, IT2 membership functions are used in the IT2 polynomial fuzzy model and IT2 polynomial fuzzy controller. The stability of IT2 PFMB control system is investigated based on the Lyapunov stability theory and both sets of membership function independent (MFI) and membership function dependent (MFD) stability conditions are derived on the basis of the sum-of-squares (SOS) approach. To make the stability conditions MFD, the boundary information of IT2 membership functions is used in the stability analysis. To extract richer information of IT2 membership functions, the operating domain is partitioned into sub-domains. In each sub-domain, the boundary information of IT2 membership functions and those of the upper and lower membership function are obtained. Furthermore, to further relax the conservativeness, a switching polynomial fuzzy controller, together with the informations obtained in each sub-domain, is employed in investigating the stability analysis. Numerical examples and simulation results are given to demonstrate the validity of MFD and MFD switching methods.

Evolutionary stability under limited population growth: Eco-evolutionary feedbacks and replicator dynamics

This paper further develops a new way of modelling evolutionary game models with an emphasis on ecological realism, concerned with how ecological factors determine payoffs in evolutionary games. Our paper is focused on the impact of strategically neutral growth limiting factors and background fitness components on game dynamics and the form of the stability conditions for the rest points constituted by the intersections of the frequency and density nullclines. It is shown that for the density dependent case, that at the stationary state, the turnover coefficients (numbers of newborns per single dead adult) are equal for all strategies. In addition, the paper contains a derivation of the EESS (eco-evolutionarily stable states) conditions, describing evolutionary stability under limited population growth. We show that evolutionary stability depends on the local geometry (slopes) of the intersecting nullclines. The paper contains examples showing that density dependence induces behaviour which is not compatible with purely frequency dependent static game theoretic ESS stability conditions. We show that with the addition of density dependence, stable states can become unstable and unstable states can be stabilised. The stability or instability of the rest points can be explained by a mechanism of eco-evolutionary feedback.

Stability analysis and feedback control of T-S fuzzy hyperbolic delay model for a class of nonlinear systems with time-varying delay

In this paper, a new T-S fuzzy hyperbolic delay model for a class of nonlinear systems with time-varying delay, is presented to address the problems of stability analysis and feedback control. Fuzzy controller is designed based on the parallel distributed compensation (PDC), and with a new Lyapunov function, delay dependent asymptotic stability conditions of the closed-loop system are derived via linear matrix inequalities (LMIs). Besides, considering the differences between the model and the real system, we extent the model to uncertain T-S fuzzy hyperbolic delay model. Based on the uncertain model, a robust $H_{infty}$ fuzzy controller is obtained and stability conditions are developed in terms of LMIs. The main advantage of the control based on T-S fuzzy hyperbolic delay model is that it can achieve small control amplitude via ``soft'' constraint approach. Finally, a numerical example and the Van de Vusse example are given to validate the advantages of the proposed method.

Robust Switched Controller Design for Nonlinear Continuous Systems

A novel approach is presented to robust switched controller design for nonlinear continuous-time systems under an arbitrary switching signal using the gain scheduling approach. The proposed design procedure is based on the robust multi parameter dependent quadratic stability condition. The obtained switched controller design procedure for nonlinear continuoustime systems is in at bilinear matrix form (BMI). The properties of the obtained design are illustrated on simulation examples.

Delay dependent stability conditions of static recurrent neural networks: a non‐linear convex combination method

A new method is developed for stability of static recurrent neural networks with time-varying delay in this study. Improved delay-dependent conditions in the form of a set of linear matrix inequalities are derived for this class of static nets through the newly proposed augmented Lyapunov–Krasovski functional. Our derivation employs a novel non-linear convex combination technique, that is, quadratic convex combination. Different from previous results, the property of quadratic convex function is fully taken advantage of without resort to the Jensen's inequality. A numerical example is provided to verify the effectiveness and superiority of the presented results.

On robustness of predictor feedback control of linear systems with input delays

This paper is concerned with the robustness of the predictor feedback control of linear systems with input delays. By applying certain equivalent transformations on the characteristic equation associated with the closed-loop system, we first transform the robustness problem of a predictor feedback control system into the stability problem of a neutral time-delay system containing an integral operator in the derivative. The range of the allowable input delay for this neutral time-delay system can be computed by exploring its delay dependent stability conditions. In particular, delay dependent stability conditions for the neutral time-delay system are established by partitioning the delay into segments. The conservatism of this method can be reduced when the number of segments in the partition is increased. Numerical examples are worked out to illustrate the effectiveness of the proposed method.

Observer based (Q,V,R)-α-dissipative control for TS fuzzy descriptor systems with time delay

This paper is concerned with the design of (Q,V,R)-α-dissipative control for continuous TS fuzzy descriptor systems with unmeasurable states and time-varying delay. Our attention is focused on the design of the fuzzy controller and the fuzzy observer such that the closed loop fuzzy descriptor system with time delay is not only regular, impulse-free and stable but also (Q,V,R)-α-dissipative. By using the matrix decoupling technique, new delay dependent stabilization conditions are presented in terms of Linear Matrix Inequalities (LMI). As a particular case, observer-based H∞ control is deduced from the proposed observer-based dissipative control result. Finally, a numerical example is given to demonstrate the effectiveness of our results.

Robustness of a discrete-time predictor-based controller for time-varying measurement delay

A predictor-based controller for time-varying delay systems is presented in this paper and its robustness properties for different uncertainties are analyzed. First, a time-varying delay dependent stability condition is expressed in terms of LMIs. Then, uncertainties in the knowledge of all plant-model parameters are considered and the resulting closed-loop system is shown to be robust with respect to these uncertainties. A significant improvement with respect to the same control strategy without predictor is achieved. The scheme is applicable to open-loop unstable plants and it has been tested in a real-time application to control the roll angle of a quad-rotor helicopter prototype. The experimental results show good performance and robustness of the proposed scheme even in the presence of long delay uncertainties.

New Delay Dependent Stability Analysis for Switched Neutral Systems with Mode-Dependent Delays under Arbitrary Switching Laws

In this paper, the stability analysis of switched uncertain neutral systems with mode-dependent delays under arbitrary switching rules is presented. Based on common Lyapunov functional, and combined with the analysis of matrix inequalities, the delay dependent stability conditions are obtained in the form of linear matrix inequalities(LMIs)which can be easily solved by LMI toolbox in Matlab. Finally, a numerical example illustrate that the proposed criteria are effective.

Robust Stability of Uncertain Switched Delay Systems with Nonlinear Perturbations

This paper focuses on the stability problem of a class of uncertain switched delay systems with nonlinear perturbations. Applying multiple functional technique, we establish a delay dependent stability condition via designing appropriate switching rule. It should emphasize that this result is a standard extension of linear delay differential equations. Meanwhile, this result is presented by LMIs and thus solved easily.

Stability Analysis of Switched Systems with Mixed Delayed and Nonlinear Perturbations

This note deals with a class of switched systems with nonlinear perturbations in mixed delayed state. Based on the multiple functional appoach, we establish a delay dependent stability condition via designing appropriate switching rule. This result indicates that if the norm of system matrices is small enough, the uniform asymptotic stability is always achieved.