Abstract
This paper addresses decentralized exponential stability problem for a class of linear largescale systems with timevarying delay in interconnection. The time delay is any continuous function belonging to a given interval, but not necessarily differentiable. By constructing a suitable augmented LyapunovKrasovskii functional combined with LeibnizNewton’s formula, new delaydependent sufficient conditions for the existence of decentralized exponential stability are established in terms of LMIs. Numerical examples are given to show the effectiveness of the obtained results.
Keywords:
decentralized exponential stability; largescale systems; uncertain systems; interval timevarying delay; Lyapunov function; linear matrix inequalities1 Introduction
The theory and applications of functional differential equations form an important part of modern nonlinear dynamics. Such equations are natural mathematical models for various real life phenomena where the aftereffects are intrinsic features of their functioning. In recent years, functional differential equations have been used to model processes in different areas such as population dynamics and ecology, physiology and medicine, economics, and other natural sciences. Stability analysis of linear systems with timevarying delays is fundamental to many practical problems and has received considerable attention [14]. Most of the known results on this problem are derived assuming only that the timevarying delay is a continuously differentiable function, satisfying some boundedness condition on its derivative: . In delaydependent stability criteria, the main concern is to enlarge the feasible region of stability criteria in a given timedelay interval. Interval timevarying delay means that a time delay varies in an interval in which the lower bound is not restricted to being zero. By constructing a suitable augmented Lyapunov functional and utilizing free weight matrices, some less conservative conditions for asymptotic stability are derived in [510] for systems with time delay varying in an interval. However, the shortcoming of the method used in these works is that the delay function is assumed to be differential and its derivative is still bounded: .
On the other hand, there has been a considerable research interest in largescale interconnected systems. A typical largescale interconnected system such as a power grid consists of many subsystems and individual elements connected together to form a large, complex network capable of generating, transmitting and distributing electrical energy over a large geographical area. In general, a largescale system can be characterized by a large number of variables representing the system, a strong interaction between subsystem variables, and a complex interaction between subsystems. The problem of decentralized control of largescale interconnected dynamical systems has been receiving considerable attention, because there is a large number of largescale interconnected dynamical systems in many practical control problems, e.g., transportation systems, power systems, communication systems, economic systems, social systems, and so on [1114]. The operation of largescale interconnected systems requires the ability to monitor and stabilize in the face of uncertainties, disturbances, failures and attacks through the utilization of internal system states. However, even with the assumption that all the state variables are available for feedback control, the task of effective controlling a largescale interconnected system using a global (centralized) state feedback controller is still not easy as there is a necessary requirement for information transfer between the subsystems [1518].
To the best of our knowledge, there has been no investigation on the exponential stability of largescale systems with timevarying delays interacted between subsystems. In fact, this problem is difficult to solve; particularly, when the timevarying delays are interval, nondifferentiable and the output is subjected to such timevarying delay functions. The time delay is assumed to be any continuous function belonging to a given interval, which means that the lower and upper bounds for the timevarying delay are available, but the delay function is bounded but not necessarily differentiable. This allows the timedelay to be a fast timevarying function and the lower bound is not restricted to being zero. It is clear that the application of any memoryless feedback controller to such timedelay systems would lead to closed loop systems with interval timevarying delays. The difficulties then arise when one attempts to derive exponential stability conditions. Indeed, the existing LyapunovKrasovskii functional and associated results in [11,14,15,1835] cannot be applied to solve the problem posed in this paper as they would either fail to cope with the nondifferentiability aspects of the delays, or lead to very complex matrix inequality conditions, and any technique such as matrix computation or transformation of variables fails to extract the parameters of the memoryless feedback controllers. This has motivated our research.
In this paper, we consider a class of largescale linear systems with interval timevarying delays in interconnections. Compared to the existing results, our result has its own advantages. (i) Stability analysis of previous papers reveals some restrictions: The time delay was proposed to be either timeinvariant interconnected or the lower delay bound is restricted to being zero, or the time delay function should be differential and its derivative is bounded. In our result, the above restricted conditions are removed for the largescale systems. In addition, the time delay is assumed to be any continuous function belonging to a given interval, which means that the lower and upper bounds for the timevarying delay are available, but the delay function is bounded but not necessarily differentiable. This allows the timedelay to be a fast timevarying function, and the lower bound is not restricted to being zero. (ii) The developed method using new inequalities for lower bounding cross terms eliminates the need for overbounding and provides larger values of the admissible delay bound. We propose a set of new LyapunovKrasovskii functionals, which are mainly based on the information of the lower and upper delay bounds. (iii) The conditions will be presented in terms of the solution of LMIs that can be solved numerically in an efficient manner by using standard computational algorithms [36].
The paper is organized as follows. Section 2 presents definitions and some wellknown technical propositions needed for the proof of the main results. Main result for decentralized exponential stability of largescale systems is presented in Section 3. Numerical examples showing the effectiveness of the obtained results are given in Section 4. The paper ends with conclusions and cited references.
2 Preliminaries
The following notations are used in this paper. denotes the set of all real nonnegative numbers; denotes the ndimensional space with the scalar product and the vector norm ; denotes the space of all matrices of dimensions; denotes the transpose of matrix A; A is symmetric if ; I denotes the identity matrix; denotes the set of all eigenvalues of A; ; denotes the set of all valued differentiable functions on ; stands for the set of all squareintegrable valued functions on . , ; denotes the set of all valued continuous functions on ; matrix A is called semipositive definite () if for all ; A is positive definite () if for all ; means . ∗ denotes the symmetric term in a matrix.
Consider a class of linear largescale systems with interval timevarying delays composed of N interconnected subsystems of the form
where , , is the state vector, the system matrices , are of appropriate dimensions.
The time delays are continuous and satisfy the following condition:
and the initial function , , with the norm
Definition 2.1 Given . The zero solution of system (2.1) is αexponentially stable if there exists a positive number such that every solution satisfies the following condition:
We end this section with the following technical wellknown propositions, which will be used in the proof of the main results.
Proposition 2.1For anyand positive definite matrix, we have
Proposition 2.2 (Schur complement lemma [37])
Given constant matricesX, Y, Zwith appropriate dimensions satisfying. Thenif and only if
Proposition 2.3[38]
For any constant matrixand scalarh, , such that the following integrations are well defined, then
3 Main results
In this section, we investigate the decentralized exponential stability of linear largescale system (2.1) with interval timevarying delays. It will be seen from the following theorem that neither freeweighting matrices nor any transformation are employed in our derivation. Before introducing the main result, the following notations of several matrix variables are defined for simplicity
The following is the main result of the paper, which gives sufficient conditions for the decentralized exponential stability of linear largescale system (2.1) with interval timevarying delays. Essentially, the proof is based on the construction of Lyapunov Krasovskii functions satisfying the Lyapunov stability theorem for a timedelay system [37].
Theorem 3.1Given. System (2.1) isαexponentially stable if there exist symmetric positive definite matrices, , , , , , and matrices, , , such that the following LMI holds:
Moreover, the solutionof the system satisfies
Proof We consider the following LyapunovKrasovskii functional for system (2.1):
where
It is easy to verify that
Taking the derivative of V in t along the solution of system (2.1), we have
Applying Proposition 2.3 and the LeibnizNewton formula
we have
Note that
Using Proposition 2.3 gives
then
Similarly, we have
respectively. Besides, using Proposition 2.3 again, we have
Hence,
Therefore, we have
By using the following identity relation:
we have
Adding all the zero items of (3.4) into (3.3), we obtain
Applying Proposition 2.1, we obtain
Therefore, applying inequalities (3.5) and noting that
we have
By condition (3.1), we obtain
Integrating both sides of (3.6) from 0 to t, we obtain
Furthermore, taking condition (3.2) into account, we have
then
This completes the proof of the theorem. □
Remark 3.1 Theorem 3.1 provides sufficient conditions for linear largescale system (2.1) in terms of the solutions of LMIs, which guarantees the closedloop system to be exponentially stable with a prescribed decay rate α. The developed method using new inequalities for lower bounding cross terms eliminates the need for overbounding and provides larger values of the admissible delay bound. Note that the timevarying delays are nondifferentiable; therefore, the methods proposed in [11,14,15,1835] are not applicable to system (2.1). LMI condition (3.2) depends on parameters of the system under consideration as well as the delay bounds. The feasibility of the LMIs can be tested by the reliable and efficient Matlab LMI Control Toolbox [36].
4 Numerical examples
In this section, we give a numerical example to show the effectiveness of the proposed result.
Example 4.1 This example is a largescale model composed of two machine subsystems as follows:
where the absolute rotor angle and angular velocity of the machine in each subsystem are denoted by and , respectively; the ith system coefficient ; the modulus of the transfer admittance ; the initial input ; the timevarying delays between the two machines in the subsystem:
It is worth nothing that the delay functions , are nondifferentiable; therefore, the methods in [11,14,15,1835] are not applicable to this system. By using LMI Toolbox in Matlab [36], LMIs (3.1) is feasible with , , , and
According to Theorem 3.1, the system is exponentially stable. Finally, the solution of the system satisfies
Figure 1 shows the trajectories of and of the closed loop system with the initial conditions , .
Figure 1. The trajectories of a solution of the linear largescale system.
The trajectories of a solution of the linear largescale system are shown in Figure 1, respectively.
5 Conclusion
In this paper, the problem of the decentralized exponential stability for largescale timevarying delay systems has been studied. The time delay is assumed to be a function belonging to a given interval, but not necessarily differentiable. By effectively combining an appropriate Lyapunov functional with the NewtonLeibniz formula and freeweighting parameter matrices, this paper has derived new delaydependent conditions for the exponential stability in terms of linear matrix inequalities, which allow simultaneous computation of two bounds that characterize the exponential stability rate of the solution. The developed method using new inequalities for lower bounding cross terms eliminates the need for overbounding and provides larger values of the delay bound. Numerical examples are given to show the effectiveness of the obtained result.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. The authors read and approved the final manuscript.
Acknowledgements
This work was supported by the Thailand Research Fund Grant, the Commission for Higher Education and Faculty of Science, Maejo University, Thailand. The second author is supported by the Center of Excellence in Mathematics, Thailand, and Commission for Higher Education, Thailand. The authors thank anonymous reviewers for valuable comments and suggestions, which allowed us to improve the paper.
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