Stability Switches of a Class of Fractional-Delay Systems With Delay-Dependent Coefficients

2018 ◽  
Vol 13 (11) ◽  
Author(s):  
Xinghu Teng ◽  
Zaihua Wang
Keyword(s):  
Time Delay ◽  
Delay Systems ◽  
Jacobian Determinant ◽  
Stability Switches ◽  
Characteristic Roots ◽  
Fractional Delay ◽  
Analysis Of Stability ◽  
The Stability ◽  
Key Steps ◽  
Delay Dependent

Stability of a dynamical system may change from stable to unstable or vice versa, with the change of some parameter of the system. This is the phenomenon of stability switches, and it has been investigated intensively in the literature for conventional time-delay systems. This paper studies the stability switches of a class of fractional-delay systems whose coefficients depend on the time delay. Two simple formulas in closed-form have been established for determining the crossing direction of the characteristic roots at a given critical point, which is one of the two key steps in the analysis of stability switches. The formulas are expressed in terms of the Jacobian determinant of two auxiliary real-valued functions that are derived directly from the characteristic function, and thus, can be easily implemented. Two examples are given to illustrate the main results and to show an important difference between the fractional-delay systems with delay-dependent coefficients and the ones with delay-free coefficients from the viewpoint of stability switches.

scholarly journals Improved Delay-Dependent Stability and Stabilization Conditions of T-S Fuzzy Time-Delay Systems via an Augmented Lyapunov-Krasovskii Functional Approach

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Zifan Gao ◽  
Jiaxiu Yang ◽  
Shuqian Zhu
Keyword(s):  
Time Delay ◽  
Delay Systems ◽  
Linear Matrix ◽  
Stability Conditions ◽  
State Variables ◽  
Time Varying Delay ◽  
Improved Stability ◽  
Stability And Stabilization ◽  
The Stability ◽  
Delay Dependent

This paper develops some improved stability and stabilization conditions of T-S fuzzy system with constant time-delay and interval time-varying delay with its derivative bounds available, respectively. These conditions are presented by linear matrix inequalities (LMIs) and derived by applying an augmented Lyapunov-Krasovskii functional (LKF) approach combined with a canonical Bessel-Legendre (B-L) inequality. Different from the existing LKFs, the proposed LKF involves more state variables in an augmented way resorting to the form of the B-L inequality. The B-L inequality is also applied in ensuring the positiveness of the constructed LKF and the negativeness of derivative of the LKF. By numerical examples, it is verified that the obtained stability conditions can ensure a larger upper bound of time-delay, the larger number of Legendre polynomials in the stability conditions can lead to less conservative results, and the stabilization condition is effective, respectively.

Stability of nonlinear time-delay systems satisfying a quadratic constraint

2016 ◽  
Vol 40 (3) ◽  
pp. 712-718 ◽  
Author(s):  
Mohsen Ekramian ◽  
Mohammad Ataei ◽  
Soroush Talebi
Keyword(s):  
Time Delay ◽  
Uncertain Systems ◽  
Delay Systems ◽  
Linear Matrix ◽  
Time Delay Systems ◽  
Matrix Inequalities ◽  
Quadratic Constraint ◽  
The Stability ◽  
Delay Dependent ◽  
Delay Dependent Stability

The stability problem of nonlinear time-delay systems is addressed. A quadratic constraint is employed to exploit the structure of nonlinearity in dynamical systems via a set of multiplier matrices. This yields less conservative results concerning stability analysis. By employing a Wirtinger-based inequality, a delay-dependent stability criterion is derived in terms of linear matrix inequalities for the nominal and uncertain systems. A numerical example is used to demonstrate the effectiveness of the proposed stability conditions in dealing with some larger class of nonlinearities.

scholarly journals A New Integral Inequality and Delay-Decomposition with Uncertain Parameter Approach to the Stability Analysis of Time-Delay Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Haiyang Zhang ◽  
Lianglin Xiong ◽  
Qing Miao ◽  
Yanmeng Wang ◽  
Chen Peng
Keyword(s):  
Time Delay ◽  
Integral Inequality ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Decomposition Approach ◽  
Delay Decomposition Approach ◽  
The Stability ◽  
Delay Dependent ◽  
Delay Dependent Stability ◽  
Delay Decomposition

This paper is concerned with the problem of delay-dependent stability of time-delay systems. Firstly, it introduces a new useful integral inequality which has been proved to be less conservative than the previous inequalities. Next, the inequality combines delay-decomposition approach with uncertain parameters applied to time-delay systems, based on the new Lyapunov-Krasovskii functionals and new stability criteria for system with time-delay have been derived and expressed in terms of LMIs. Finally, a numerical example is provided to show the effectiveness and the less conservative feature of the proposed method compared with some recent results.

scholarly journals Delay-Dependent Stability Analysis of TS Fuzzy Switched Time-Delay Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
Nawel Aoun ◽  
Marwen Kermani ◽  
Anis Sakly
Keyword(s):  
Time Delay ◽  
Stability Criterion ◽  
Delay Systems ◽  
Space Representation ◽  
Time Delay Systems ◽  
State Space Representation ◽  
Arbitrary Switching ◽  
The Stability ◽  
Delay Dependent ◽  
Arrow Form Matrix

This paper proposes a new approach to deal with the problem of stability under arbitrary switching of continuous-time switched time-delay systems represented by TS fuzzy models. The considered class of systems, initially described by delayed differential equations, is first put under a specific state space representation, called arrow form matrix. Then, by constructing a pseudo-overvaluing system, common to all fuzzy submodels and relative to a regular vector norm, we can obtain sufficient asymptotic stability conditions through the application of Borne and Gentina practical stability criterion. The stability criterion, hence obtained, is algebraic, is easy to use, and permits avoiding the problem of existence of a common Lyapunov-Krasovskii functional, considered as a difficult task even for some low-order linear switched systems. Finally, three numerical examples are given to show the effectiveness of the proposed method.

Delay Dependent Stability Analysis of Active Power Filter

2012 ◽  
Vol 220-223 ◽  
pp. 1592-1597
Author(s):  
Tao Li ◽  
Yong Qiang Liu
Keyword(s):  
Time Delay ◽  
Active Power Filter ◽  
Negative Influence ◽  
Active Power ◽  
Delay Systems ◽  
Linear Matrix ◽  
Power Filter ◽  
The Stability ◽  
Delay Dependent ◽  
Delay Dependent Stability

The active power filter (APF) is a dynamic harmonic filtering equipment. The time delay is unavoidable and it has a great negative influence on the stability of the APF system. Based on the introduction of its topology and control strategy, the mathematical model of APF with time delay is built. And the model is a time delay systems with bounded nonlinearity, so a new delay dependent stability criteria is derived and formulated in the form of linear matrix inequality (LMI). The maximum allowed time delay is solved, and the relationships between it and some other parameters are investigated and simulated. The result can be as a reference in the future work.

scholarly journals Fast Generation of Stability Charts for Time-Delay Systems Using Continuation of Characteristic Roots

2020 ◽  
Vol 15 (11) ◽  
Author(s):  
Surya Samukham ◽  
Thomas K. Uchida ◽  
C. P. Vyasarayani
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Grid Point ◽  
Galerkin Approximation ◽  
Computational Effort ◽  
Numerical Continuation ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Characteristic Roots ◽  
The Stability

Abstract Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method can significantly reduce the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method.

Measures of Robustness for Uncertain Time-Delay Linear Systems

1995 ◽  
Vol 117 (4) ◽  
pp. 633-635 ◽  
Author(s):  
Said Oucheriah
Keyword(s):  
Time Delay ◽  
Linear Systems ◽  
Robust Stability ◽  
Salient Feature ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Lyapunov Matrix ◽  
The Stability ◽  
Uncertain Time ◽  
Delay Dependent

Several delay-dependent criteria to test the stability of time-delay systems that were proposed require solving the Lyapunov matrix equation. This can be a troublesome task and often nontrivial. In this note, a delay-dependent sufficient condition that guarantees the robust stability of linear uncertain time-delay systems is presented. The stability test criterion derived in this paper is based on induced norms and matrix measures. The salient feature of the result obtained is its simplicity and ease in testing the robust stability of uncertain time-delay linear systems.

Stability and Performance Analysis of Time-Delayed Actuator Control Systems

2016 ◽  
Vol 138 (5) ◽  
Author(s):  
Bing Ai ◽  
Luis Sentis ◽  
Nicholas Paine ◽  
Song Han ◽  
Aloysius Mok ◽  
...  
Keyword(s):  
Time Delay ◽  
Time Windows ◽  
System Stability ◽  
Robotic Systems ◽  
Delay Systems ◽  
Physical Constraints ◽  
Characteristic Roots ◽  
And Performance ◽  
The Stability ◽  
Actuator Control

Time delay is a common phenomenon in robotic systems due to computational requirements and communication properties between or within high-level and low-level controllers as well as the physical constraints of the actuator and sensor. It is widely believed that delays are harmful for robotic systems in terms of stability and performance; however, we propose a different view that the time delay of the system may in some cases benefit system stability and performance. Therefore, in this paper, we discuss the influences of the displacement-feedback delay (single delay) and both displacement and velocity feedback delays (double delays) on robotic actuator systems by using the cluster treatment of characteristic roots (CTCR) methodology. Hence, we can ascertain the exact stability interval for single-delay systems and the rigorous stability region for double-delay systems. The influences of controller gains and the filtering frequency on the stability of the system are discussed. Based on the stability information coupled with the dominant root distribution, we propose one nonconventional rule which suggests increasing time delay to certain time windows to obtain the optimal system performance. The computation results are also verified on an actuator testbed.

On the stability of a class of nonlinear time delay systems

1999 ◽  
Author(s):  
Dan Ivancscu ◽  
Silviu-Iulian Niculcscu ◽  
Jcan-Michcl Dion ◽  
Luc Dugard
Keyword(s):  
Time Delay ◽  
Delay Systems ◽  
Time Delay Systems ◽  
The Stability
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