Conditions For Stability Of Linear Systems Research Articles

Enhancing Stability Criteria for Linear Systems with Interval Time-Varying Delays via an Augmented Lyapunov–Krasovskii Functional

This work investigates the stability conditions for linear systems with time-varying delays via an augmented Lyapunov–Krasovskii functional (LKF). Two types of augmented LKFs with cross terms in integrals are suggested to improve the stability conditions for interval time-varying linear systems. In this work, the compositions of the LKFs are considered to enhance the feasible region of the stability criterion for linear systems. Mathematical tools such as Wirtinger-based integral inequality (WBII), zero equalities, reciprocally convex approach, and Finsler’s lemma are utilized to solve the problem of stability criteria. Two sufficient conditions are derived to guarantee the asymptotic stability of the systems using linear matrix inequality (LMI). First, asymptotic stability criteria are induced by constructing the new augmented LKFs in Theorem 1. Then, simplified LKFs in Corollary 1 are proposed to show the effectiveness of Theorem 1. Second, asymmetric LKFs are shown to reduce the conservatism and the number of decision variables in Theorem 2. Finally, the advantages of the proposed criteria are verified by comparing maximum delay bounds in four examples. Four numerical examples show that the proposed Theorems 1 and 2 obtain less conservative results than existing outcomes. Particularly, Example 2 shows that the asymmetric LKF methods of Theorem 2 can provide larger delay bounds and fewer decision variables than Theorem 1 in some specific systems.

Further Results on the Input-to-State Stability of a Linear Disturbed System with Control Delay

In this paper, a theorem is obtained that gives sufficient input-to-state stability conditions for linear systems with control delay and additive disturbances. Stabilizing feedback is considered available in the absence of delay and disturbances. The mathematical tools are the Lyapunov–Krasovskii functional, the Jensen inequality and the double Hadamard inequality. The critical delay is highlighted.

Hierarchical stability conditions for linear systems with interval time-varying delay

This paper proposes the hierarchical stability conditions for linear systems with an interval time-varying delay. In the matter of the delay, the upper and lower bounds are restricted, but there is no constraint on its derivative. Firstly, based on the state vectors and multiple integral state vectors involved in the delay-related generalized free-matrix-based integral inequalities (GFIIs), the hierarchical Lyapunov-Krasovskii functional (LKF) candidates are constructed. Then, the GFIIs are applied for the approximation of the integral quadratic terms, which will introduce the delay related nonlinear terms in the LKF differential. Next, the novel adjusted matrix-valued high and odd degree polynomial negative definite conditions (NDCs) are provided to acquire the hierarchical linear matrix inequality (LMI) conditions and cope with the nonlinear terms introduced by the GFIIs. Eventually, the advantages of the obtained stability conditions are checked through some numeric examples.

New input‐to‐state stability conditions of linear systems with interval time‐varying delays and their application to power systems

AbstractIt is commonly believed that the Lyapunov‐Krasovskii functional method plays an important role in obtaining input‐to‐state stability conditions of systems with interval time‐varying delays. For a system with an interval time‐varying delay, an appropriate functional should be constructed during applying the method. Nevertheless, the delay has an interesting property that the delay in different delay intervals can have varying impacts on the system. In a way, the property is not considered adequately for the Lyapunov‐Krasovskii functional method, which may give rise to conservative conditions. This article proposes a new input‐to‐state stability analysis method by taking advantage of the ignored property. When the new method is adopted, multiple functionals are constructed for the delay in the different delay intervals. This means that the multiple functionals method is more applicable in practice than the Lyapunov‐Krasovskii functional method. With the help of the proposed method, new input‐to‐state stability conditions for linear delay systems are derived. A numerical example is presented to show that the obtained conditions are effective. At last, the newly derived conditions are applied to power systems with delayed load frequency control schemes.

Discretization of the Functionals With Prescribed Derivative for Linear Systems With Multiple Delays

This work aims to derive a sufficient stability condition for linear systems with multiple delays based on a discretization procedure of Lyapunov – Krasovskii functionals. This procedure relies on a piecewise constant approximation of matrix kernels of the functionals with prescribed derivatives. The approximation error is explicitly bounded, and the resulting stability test is expressed in terms of nonnegative definiteness of a block matrix that depends on the values of delay Lyapunov matrix at discretization points and the system matrices.

Advanced stability analysis for linear systems with time-varying delays via a generalized integral inequality

The improved stability conditions for linear systems with time-varying delays are investigated. The proposed integral inequality derived from the properties of simple matrices has been simply proved, and this is a generalized form of various integral inequalities. Thus, depending on N, various forms of integral inequality can be derived. Also, the stability results are obtained by extending the augmented terms of Lyapunov–Krasovskii functionals according to N. Finally, two well-known numerical examples show that the stability criteria derived through the proposed generalized integral inequality have less conservatism than those of the existing ones.

Stability analysis for systems with multiple/single time delays via a Cascade augmented L-K functional

This paper investigates stability of linear systems with multiple/single time-delays. Firstly, a three-level cascade augmented Lyapunov-Krasovskii (L-K) functional is introduced, in which interconnect information among delayed state vectors is fully taken into account. Based on a newly integral inequality and the cascade L-K functional, a novel stability criterion is derived for linear systems with multiple time-delays. Secondly, it is found that the proposed L-K functional is also suitable for linear systems with single time-delay if the delay-partitioning method is employed. This motivates us to obtain a less conservative stability condition for linear systems with single time-delay. Finally, Numerical examples are given to confirm the advantages of the proposed method.

Some augmented approaches to the improved stability criteria for linear systems with time-varying delays

This work investigates the improved stability conditions for linear systems with time-varying delays via various augmented approaches. Some augmented approaches are augmented Lyapunov-Krasovskii functionals, augmented zero equalities, and the augmented zero equality approach. At first, by constructing augmented Lyapunov-Krasovskii functionals including the state vectors which were not considered in the previous works and augmented zero equalities, a stability criterion is proposed in the forms of linear matrix inequalities. Through the proposed Lyapunov-Krasovskii functionals and an additional functional derived from the integral inequality, a slightly improved result is derived. The proposed results do not consider the increase in the computational complexity to achieve a larger delay bound. So, by applying the augmented zero equality approach, which is a method of grafting the proposed augmented zero equality proposed in Finsler Lemma, to the proposed result, an enhanced stability result was derived. Also, the computational complexity is reduced by appropriately adjusting any vector of the integral inequality utilized in the proposed criteria. By applying some numerical examples to the proposed conditions, the effectiveness and superiority of the results are confirmed.

Necessary stability conditions for linear systems with incommensurate delays

The Lyapunov matrix is a key element of the construction of Lyapunov–Krasovskii functionals with prescribed derivative for linear time-invariant time delay systems. Moreover, stability criteria which are based exclusively on the Lyapunov matrix have been developed recently. A weak point of the approach is a lack of effective techniques to compute the Lyapunov matrix when the delays are incommensurate, i.e. at least two of them are rationally independent. To overcome this difficulty, we present in this paper necessary stability conditions for systems with incommensurate delays based on the Lyapunov matrix of a “close” auxiliary system with commensurate delays, which can be computed by known techniques. An explicit condition for the choice of delays of the auxiliary system is provided. Illustrative examples are given.

Relaxed stability conditions for linear systems with time-varying delays via some novel approaches

<abstract> <p>In this paper, the stability problem with considering time-varying delays of linear systems is investigated. By constructing new augmented Lyapunov-Krasovskii (L-K) functionals based on auxiliary function-based integral inequality (AFBI) and considering Finsler's lemma, a stability criterion is derived. Based on the previous result, a less conservative result is proposed through the augmented zero equality approach. Finally, numerical examples are given to show the effect of the proposed criteria.</p> </abstract>

On improved delay-range-dependent stability condition for linear systems with time-varying delay via Wirtinger inequality

This paper studies the problem of delay-range-dependent stability analysis for the continuous-time linear systems with time-varying delay. A new and appropriate Lyapunov–Krasovskii (L–K) functional is constructed. To estimate the quadratic integral terms coming out from the derivative of L–K functional, utilize the well-known Wirtinger integral inequality together with the reciprocal convex lemma. Then, an improved delay-range-dependent stability condition is being established in terms of linear matrix inequalities (LMIs) in such a way that it can be effectively solved by using existing software (LMI toolbox in MATLAB). The delay upper bound results obtained by the developed stability condition are found to be less conservative than other recent results. Furthermore, the proposed stability criterion use the less number of decision variables and give the consistent delay bound results compared to some other methods. Two numerical examples are given to illustrate the effectiveness of the obtained stability condition compared to some recently published stability methods.

Necessary and sufficient stability conditions for linear systems with pointwise and distributed delays

A stability criterion for the exponential stability of systems with multiple pointwise and distributed delays is presented. Conditions in terms of the delay Lyapunov matrix are obtained by evaluating a Lyapunov–Krasovskii functional with prescribed derivative at a pertinent initial function that depends on the system fundamental matrix. The proof relies on properties connecting the delay Lyapunov matrix and the fundamental matrix, which are proven to be valid for both stable and unstable systems. The conditions are applied to the determination of the exact stability region for some examples.

Necessary Stability Conditions for Delay Systems With Multiple Pointwise and Distributed Delays

Necessary stability conditions for linear systems with multiple pointwise and distributed delays are presented. New properties that connect the Lyapunov delay matrix and the fundamental matrix of these systems are instrumental in proving that these conditions depend exclusively on the Lyapunov delay matrix. The result is illustrated by some examples.

Improved delay-dependent stability analysis for linear time-delay systems: Based on homogeneous polynomial Lyapunov–Krasovskii functional method

An improved delay-dependent stability condition for linear system with interval time-varying delay is provided in this paper. Different from previous methods, a homogeneous polynomial Lyapunov–Krasovskii functional is introduced, based on the new Lyapunov–Krasovskii functional method and wirtinger-based integral inequality technique, a novel delay-dependent stability criterion for mentioned system is obtained in terms of linear matrix inequalities, which is less conservative than most recent results. Numerical examples are also provided to show the advantage of the obtained result.

New Stability Condition for Linear Systems with Time-Varying Delay

Generally, the obtained results on delayed systems can be classified into two types: delay-independent ones and delay-dependent ones. Delay-dependent stabilization problem for a class linear system with interval time-varying delay is studied. Early first proposed stability analysis method for systems with time-varying delay in a range, but the method therein still leaves much room for improvement. A sufficient condition in terms of linear matrix inequalities (LMIs) is achieved by constructing a novel Lyapunov-Krasovskii functional with the idea of partitioning the time delay, and then using free-weighting matrix approach and adopting inequalities, the interval delay is dealt with successfully. Compared with former stability analysis approaches, this approach can overcome the defect of finding a common positive definite matrix, and reduce conservative greatly. Finally, one simulation example is given to illustrate the effectiveness of the methods.

Stabilizability of Solutions to Linear and Nonlinear Differential–Algebraic Equations

We consider a control system of first order ordinary differential equations that is unsolved with respect to the derivative of a sought vector-valued function and is identically degenerate in the domain. We obtain stabilizability conditions for linear systems with vector control and prove the stabilizability by linear approximation of a system with scalar input. Bibliography: 10 titles.

Necessary conditions for the exponential stability of time-delay systems via the Lyapunov delay matrix

SUMMARY Exponential necessary stability conditions for linear systems with multiple delays are presented. The originality of these conditions is that, in analogy with the case of delay free systems, they depend on the Lyapunov matrix function of the delay system. They are validated by examples for which the analytic characterization of the stability region is known. Copyright © 2013 John Wiley & Sons, Ltd.

Linear Matrix Inequality Stability Conditions from a Parameter-Dependent Lyapunov Functional for Linear Systems with a Time-Varying Delay

This paper proposes new stability conditions for linear systems with a time-varying state delay. The time-varying delay is treated as a polytopic uncertainty parameter, and a Lyapunov functional that is dependent on the delay parameter is constructed. Linear matrix inequality conditions of system stability and robust stability are derived. Numerical and practical examples are given to show the effectiveness and non-conservativeness of the proposed conditions.

Simplified approach to the asymptotical stability of linear systems with interval time-varying delay

The asymptotical stability conditions for linear systems with interval time-varying delay are presented in terms of linear matrix inequalities. The conditions are less conservative than some existing ones and turn out to be greatly simplified version of the stability result previously reported by He et al. (2007). A distinct feature of the present criteria are that they are obtained by only using a simply integral inequality and are not involved any free-weighting matrices. This feature makes the computational burden of the criteria largely reduce. Numerical example is given to verify the effectiveness of the proposed criteria.

Stability of Zero Dynamics of Sampled-Data Nonlinear Systems

One of the approaches to sampled-data controller design for nonlinear continuous-time systems consists of obtaining an appropriate model and then proceeding to design a controller for the model. Hence, it is important to derive a good approximate sampled-data model because the exact sampled-data model for nonlinear systems is often unavailable to the controller designers. Recently, Yuz and Goodwin have proposed an accurate approximate model which includes extra zero dynamics corresponding to the relative degree of the continuous-time nonlinear system. Such extra zero dynamics are called sampling zero dynamics. A more accurate sampled-data model is, however, required when the relative degree of a continuous-time nonlinear plant is two. The reason is that the closed-loop system becomes unstable when the more accurate sampled-data model has unstable sampling zero dynamics and a controller design method based on cancellation of the zero dynamics is applied. This paper derives the sampling zero dynamics of the more accurate sampled-data model and shows a condition which assures the stability of the sampling zero dynamics of the model. Further, it is shown that this extends a well-known result for the stability condition of linear systems to the nonlinear case.