<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5247pt" id="M1" height="16.9226pt" version="1.1" viewBox="-0.0657574 -12.3979 31.7288 16.9226" width="31.7288pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g198-9" d="M714 664L702 678C677 661 651 638 627 626C588 607 568 607 547 607C524 607 496 619 461 638C391 673 360 687 319 687C215 687 144 611 144 536S191 410 271 410C339 410 379 452 412 517L394 526C361 473 326 440 273 440C209 440 185 482 185 534S237 635 301 635C329 635 366 621 403 601C456 572 487 557 527 557C532 557 540 557 547 558C427 454 384 369 336 292C189 231 53 149 53 45C53 9 77 -15 110 -15C127 -15 146 -11 166 -4C260 31 369 152 443 298L550 339C495 254 446 155 446 77C446 21 481 -12 533 -12C594 -12 659 42 735 134L714 147C642 63 586 18 547 18C528 18 513 32 513 53C513 111 593 253 687 387C835 446 991 527 991 637C991 673 965 687 934 687C874 687 752 597 612 422L579 379C540 364 503 356 464 340C536 484 606 581 696 650L714 664ZM958 635C958 559 851 493 720 434C805 554 896 657 934 657C950 657 958 650 958 635ZM307 241C253 147 171 15 110 15C98 18 86 24 86 44C86 120 179 182 307 241Z"/></g><g transform="matrix(.013,0,0,-0.013,17.831,4.308)"><path id="g50-30" d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z"/></g></svg> Performance and Stability Analysis of Linear Systems with Interval Time-Varying Delays and Stochastic Parameter Uncertainties

Park, M. J.; Kwon, O. M.; Park, Ju H.; Lee, S. M.; Cha, E. J.

doi:https://doi.org/10.1155/2015/821086

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2015 | Article ID 821086 | https://doi.org/10.1155/2015/821086

Performance and Stability Analysis of Linear Systems with Interval Time-Varying Delays and Stochastic Parameter Uncertainties

M. J. Park,¹O. M. Kwon,¹Ju H. Park,²S. M. Lee,³and E. J. Cha⁴

Academic Editor: Quanxin Zhu

Received23 Apr 2014

Accepted30 May 2014

Published23 Feb 2015

Abstract

This paper deals with the problems of performance and stability analysis for linear systems with interval time-varying delays. It is assumed that the parameter uncertainties are of stochastic properties to represent random change of various environments. By constructing a newly augmented Lyapunov-Krasovskii functional, less conservative criteria of the concerned systems are introduced with the framework of linear matrix inequalities (LMIs). Four numerical examples are given to show the improvements over the existing ones and the effectiveness of the proposed methods.

1. Introduction

The mathematical models representing physical systems are generally not exact due to the various reasons such as noises and parameter changes in electrical elements. For this reason, in some cases, the stability of the mathematical model cannot guarantee the stability of the physical systems. In order to take into account such problem, the parameter uncertainties should be considered in the theoretical stability analysis for various systems. The aforementioned parameter uncertainties are the internal sources of the model, whereas the disturbances can be their external sources. Then, the objective of an performance analysis is to find a saddle point of objective functional calculus depending on the disturbance. In other words, we find its minimum for the worst-case disturbances. Moreover, from the point of view of stability, it is also needed to pay attention to a delay in the time. It is well known that time delays frequently occur in various systems due to the finite speed limit of information processing and transmission in the implementation of the systems. For this reason, the undesirable dynamic behaviors such as poor performance and instability can be caused by the wake of the delay.

In this regard, performance and/or stability of time-delay systems were dealt with in the literature [1–12]. Above all, in [5], the robust performance conditions for uncertain networked control systems with time-delay were derived by the use of some slack matrix variable. Jeong et al. [6] introduced the improved conditions of performance analysis and stability for systems with interval time-varying delays and uncertainties. In [10], the robust performance analysis and stability problems of linear systems with interval time-varying delays were investigated by constructing some new augmented Lyapunov-Krasovskii functional. Also, in order to obtain tighter lower bounds of integral terms of quadratic form, Wirtinger-based inequality in [11] is the recent remarkable tool in reducing the conservatism of delay-dependent stability criteria for dynamic systems with time delays. Therefore, there are scopes for further improved results in stability analysis with time-delay.

Returning to the concept of parameter uncertainties, in this work, it is assumed that the parameter uncertainties occur by stochastic property to represent random change of various environments. This exemplifies why considering the stochastic property includes the fact that when the earthquake happens, although the seismic intensity is the same, at all times, its wavy pattern and effects are different. However, the systems with stochastic parameter uncertainties have not been fully investigated yet. Specially, in this work, two stochastic indexes, the mean and the variance, are utilized. Thus, the concerned problems highlighting the difference between the effects of the mean and the variance on the systems will be dealt in this work.

With this motivation mentioned above, in this paper, the performance and stability problems to get improved sufficient conditions for uncertain systems with interval time-varying delays and stochastic parameter uncertainties are studied. Here, stability of system with interval time-varying delays has been a focused topic of theoretical and practical importance [13]. The interval time-varying delays mean that its lower bounds which guarantee the stability of system are not restricted to be zero and include networked control system as one of typical examples. To achieve this, by construction of a suitable augmented Lyapunov-Krasovskii functional and utilization of Wirtinger-based inequality [11], an performance condition is derived in Theorem 8 with the framework of LMIs which can be formulated as convex optimization algorithms which are amenable to computer solution [14]. Also, inspired by the works of [4, 12], the reciprocally convex and zero equality approaches are utilized with some decision variables to reduce the conservatism of the concerned condition. Based on the result of Theorem 8, performance condition with deterministic parameter uncertainties and an improved stability condition for the nominal form without parameter uncertainties and disturbances will be proposed, respectively, in Theorem 11 and Corollary 12. Finally, four numerical examples are included to show the effectiveness of the proposed methods.

Notation. The notations used throughout this paper are fairly standard. is the -dimensional Euclidean space, and denotes the set of all real matrices. is the space of square integrable vector on . For symmetric matrices and , means that the matrix is positive definite, whereas means that the matrix is nonnegative definite. , , and denote identity matrix, and zero matrices, respectively. refers to the Euclidean vector norm or the induced matrix norm. denotes the block diagonal matrix. For square matrix , means the sum of and its transposed matrix , that is, . means . means that the elements of matrix include the scalar value of , that is, . stands for the mathematical expectation operator. means the occurrence probability of the event .

2. Preliminaries and Problem Statement

Consider the uncertain systems with time-varying delays and disturbances given by where is the state vector, is the output vector, and is the disturbances; , , , , and , are the system matrices, and and are the parameter uncertainties of the form where , , and are real known constant matrices and is a real uncertain matrix function with Lebesgue measurable elements satisfying .

The delay is a time-varying continuous function satisfying where , , and are known constant values.

For simplicity of system representation, the system can be formulated as follows: Also, the following definition and lemmas will be used in main results.

Assumption 1. The parameter uncertainties are changed with the stochastic sequences , which are a family of time functions depending on the outcome of the set of experimental outcomes. Then, the uncertainty term, , is represented by where satisfies and . Here, and are mean and variance of , respectively.

Remark 2. After the introduction of the Bernoulli property to engineering, it has been applied in many situations such as random delays [15] and sensors fault [16]. In very recent times, various forms of randomly occurring concept, for example, randomly occurring uncertainties, randomly occurring nonlinearities, randomly occurring delays, and so on, are represented by the Bernoulli property [17, 18]. Besides, the Markov property, which is a favorite stochastic sequence, is used to describe the unexpected changes of parameters in hybrid systems [19–21]. It should be noted that the existing results utilizing Bernoulli and Markov property have not utilized the information about the variance. However, in this work, the system parameter uncertainties are described by the general stochastic property with its two indexes, mean and variance. By defining in (5), the variance value of will be considered in analyzing the robust performance of system (4). The necessity of these considerations will be explained in Example 1. Therefore, to analyze this problem mentioned above, in this work, the stochastic parameter uncertainties are dealt by adopting the property of the stochastic sequence, which contains two indexes, mean and variance, instead of studying the problem of previous stochastic analysis method considering Wiener process, that is, the form , where is Wiener process. Moreover, by utilizing the proposed model, the dynamic behavior of practical problem nearer to the random change of real environment will become accessible.

The aim of this paper is to investigate the performance and stability analysis of system (4) with interval time-varying delays and stochastic parameter uncertainties. Before deriving our main results, the following definition and lemmas are introduced.

Definition 3. -optimization seeks a state-feedback controller that minimizes the -norm of the system's closed-loop transfer function between the controlled output and the external disturbance , which belongs to ; that is, . Then, an equivalent definition of the -norm is where it is assumed that . Therefore, is the maximum possible gain in signal energy. This fact can be used to express constraints on the -norm in terms of LMIs. From the above it follows that is equivalent to

Lemma 4 (see [11]). The following inequality holds for a given matrix and all continuously differentiable functions in : where and .

Lemma 5 (see [12]). For any vectors , , constant matrices , , and real scalars satisfying that , the following inequality holds:

Lemma 6 (see [22]). Let , , and such that . Then, the following statements are equivalent:(i), , ,(ii), where is a right orthogonal complement of .

Lemma 7 (see [23]). For any matrices , , matrix , the following statements are equivalent:(i),(ii).

3. Main Results

In this section, some new performance and stability criteria for the system (4) will be derived. For convenience, the notations of several matrices are defined as follows: where are defined as block entry matrices, for example, .

Then, the following theorem is given by the main result.

Theorem 8. For given scalars and , the system (4) is stochastically stable with performance , stochastic parameters and , for and , if there exist positive definite matrices , , , and and a positive scalar , any symmetric matrices , and any matrices and satisfying the following LMIs: where are the two vertices of with the bounds of . That is, when and when .

Proof. Let us consider the following Lyapunov-Krasovskii functional candidate as follows: where with By infinitesimal operator in [24], the can be calculated as follows: Prior to obtaining the bound of , the is divided into the following two parts: Inspired by the work of [4], the following zero equalities with any symmetric matrices and are considered as a tool of reducing the conservatism of criterion: where By utilizing Lemma 4, calculating the and , and adding (19) into the , the following relations can be obtained as follows: where By Lemmas 4 and 5, the integral terms, , of the are bounded as follows: where , and .
Hence, Here, when hold, the bound of is valid.
In succession, with the relational expression between and , , from the system (4), there exists any scalar satisfying the following inequality where and its mathematical expectation is as follows: From (17) to (26), the has a new upper bound as follows: From Definition 3, with the zero initial condition, we rewrite as follows: Here, we get where .
From (29) with (30), considering and , the is bounded as follows: Thus, the following inequality is equivalent to the . Therefore, if the condition (32) then system (4) is asymptotically stable with performance .
Applying Lemmas 6 and 7 to (32) with leads to for any matrix , where .
The above condition is affinely dependent on . Therefore, if LMIs (11) hold, then system (4) is stochastically stable with performance and stochastic indexes and for and . It should be noted that the inequality (12) is satisfied if the inequalities (11) hold. This completes our proof.

Remark 9. To achieve the less conservatism of stability condition, Wirtinger-based inequality with the basic Lyapunov-Krasovskii functional was introduced in [11]. However, a newly Lyapunov-Krasovskii functional was not proposed. In view of this, the main contribution in this work is the use of included in a new Lyapunov-Krasovskii functional (14). As a result, some cross terms such as and are utilized in estimating the .

Remark 10. In deriving lower bounds of and obtained by calculating the time-derivative values of , Lemma 4 which is the remarkable result in reducing the conservatism of delay-dependent stability criteria is utilized. However, unlike the results in [11], the utilized vectors of the two quadratic integral terms and are . As a result, some new integral terms such as , , and are utilized as elements of the augmented vector , which is different from the works [11].

In case of the deterministic uncertainties, the following theorem can be obtained.

Theorem 11. For given scalars , , the system (4) is asymptotically stable with performance , for and , if there exist positive definite matrices , , , and , a positive scalar , any symmetric matrices , any matrices and satisfying the LMIs (13) and where all notations were defined in (10).

Proof. When the mean, , and the variance, , of are, respectively, 1 and 0, it means the uncertainties are deterministic. Therefore, by setting and in the result of Theorem 8, LMIs (35) can be easily obtained. So, it is omitted.

As a special case of Theorem 11, when the system (1) is the nominal form without parameter uncertainties and disturbances given by then, based on same Lyapunov-Krasovskii functional candidate in (14), the following corollary can be obtained.

Corollary 12. For given scalars , , the system (37) is asymptotically stable, for and , if there exist positive definite matrices , , , and , any symmetric matrices , and any matrices and satisfying the LMIs (13) and where is the two vertices of with and other notations were defined in (10).

Proof. Upper bound of time-derivative of (14) can be calculated as follows: where was defined in (10) and with replacing the block entry matrices to , which is very similar to the proofs of Theorems 8 and 11, so it is omitted.

Remark 13. When the information of is unknown, the corresponding results of Theorems 8, 11 and Corollary 12 can be obtained by choosing , respectively.

4. Illustrative Examples

Example 1. Consider the system (1) with For the above system, the maximum allowable delay bounds (MADBs) with various and , fixed and , and unknown are listed in Table 1. When the stochastic indexes (the mean and the variance ) increase, the MADBs become smaller, which means that Theorem 8 becomes more conservative. In order to verify this, the MADBs with ranges and are shown in Figure 1. This figure demonstrates that a larger or will lead to a smaller . Then, from Table 1 and Figure 1, it can be seen that the mean and the variance can be addressed in the parameter uncertainties since the MADBs for guaranteeing the performance are influenced by the stochastic indexes. Moreover, Figures 2 and 3 are drawn to show the state trajectories with and . At this time, the initial condition and the disturbance is 1 if and 0, otherwise, and the time-delays are and , respectively, in Figures 2 and 3. Also, in order to verify the stochastic indexes, is set as .

These figures give the relations between state trajectories and for the fixed and the relations between the state trajectories and for the fixed . Also, these figures show that a lager or will lead to the poor performance of system.

Here, one of significant points is that the effect of the mean and the variance on system performance is different. From Table 1, it can be seen that the growth of stochastic indexes leads to conservatism, whereas, from Figures 2 and 3, one can confirm the following two facts: (i) the mean deteriorates the dynamic behavior of systems (see Figure 2) and (ii) the variance influences the system performance (see Figure 3).

Example 2. Consider the system (1) with (41). For the above system, the results of MADBs with various , fixed unknown , and are listed in Table 2. By applying Theorem 11, it can be guaranteed that the MADBs under the same conditions are larger than the ones in the existing works which supports the fact that the proposed Lyapunov-Krasovskii functional and some utilized techniques effectively reduce the conservatism in performance.

Example 3. Consider the system (1) with For the above system, the minimized performance with various ordered pair and unknown are listed in Table 3. In this table, the recent results [5, 6, 10] are compared with ones in this works. From Table 3, it is clear that our results for this example give smaller than the ones in [5, 6, 10].

Example 4. Consider the system (1) with In Table 4, the results for different condition of various and for guaranteeing stability are compared with the results of the existing works. From Table 4, it can be shown that our result for this example gives larger MABD than the ones in [7–10].

5. Conclusions

The performance and stability analysis for linear systems with interval time-varying delays and disturbances were studied in this paper. In Theorem 8, the performance criterion for interval time-delayed systems with stochastic parameter uncertainties was proposed with the stochastic indexes, the mean and the variance . In Theorem 11, based on the result of Theorem 8, the interval time-delayed systems with deterministic parameter uncertainties were dealt. Afterward, in Corollary 12, the improved stability criterion for the nominal form of linear systems without parameter uncertainties and disturbances was derived. Four illustrative examples have been given to show the effectiveness and usefulness of the presented criteria. By utilizing the proposed criteria, future works will focus on solving various problems in [25–30].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (2008-0062611), and by a Grant of the Korea Healthcare Technology R&D Project, Ministry of Health&Welfare, Republic of Korea (A100054).

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Copyright © 2015 M. J. Park et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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