Research Article | Open Access

Guifang Cheng, Zhishuai Ding, Jianyin Fang, "Dissipativity Analysis of Linear State/Input Delay Systems", *Abstract and Applied Analysis*, vol. 2012, Article ID 458243, 14 pages, 2012. https://doi.org/10.1155/2012/458243

# Dissipativity Analysis of Linear State/Input Delay Systems

**Academic Editor:**Gaston Mandata N'Guerekata

#### Abstract

This paper discusses dissipativity problem for system of linear state/input delay equations. Motivated by dissipativity theory of control systems, we choose a new quadratic supply rate. Using the concept of dissipativity, necessary and sufficient conditions for the linear state/input delay systems to be dissipative and exponentially dissipative are derived. The connection of dissipativity with stability is also considered. Finally, passivity and finite gain are explored, correspondingly. The positive-real and bounded-real lemmas are derived.

#### 1. Introduction

Dissipativity theory is one of major issues in control system theory. It was developed by Willems [1] and further extended by Hill and Moylan [2–4]. And it has become one of the major approaches to study complex systems [1, 5, 6]. In some particular systems with certain physical meaning, for example, electrical network systems, physical system, and so forth, dissipativity, just as the name implies, shows that the energy stored inside the system is no more than energy supplied from outside the system. In the area of dynamical system theory, storage function and supply rate are introduced to express the generalized abstract energy, correspondingly.

For dissipative systems, the storage functions usually provide natural candidates for Lyapunov functions [6]. Therefore, in many cases, stability and stabilization problems can be solved once the dissipativity property is assured [7–14]. Dissipativity theory is applied to impulsive hybrid dynamical systems [8], discontinuous dynamical systems [9], and switched systems [10]. Also, dissipativity theory is an efficient tool for the analysis and design of composite systems. Especially, it is used to deal with some problems of stability and stabilization for interconnected systems [5, 6, 15]. Passivity and finite gain are two special cases of dissipativity which are important in the stability analysis of dynamical systems. The fact that system is passive is equivalent to that system satisfies positive-real or Kalman-Yakubovich-Popov (KYP) condition [12]. Many versions of positive-real lemmas are given for nonlinear [16, 17] and linear systems [18, 19]. The results about -gain are summarized in recent references [15, 20, 21]. In papers [22, 23], bounded real lemmas are presented and are applied to control.

On the other hand, time delay is often encountered in various practical systems [24]. The existence of delays is often one main reason of instability and poor performance of systems. A great number of results on the stability and stabilization problems of control systems with time delays have been reported in the literature (see [25, 26], and the references therein). Lyapunov-Krasovskii functional approach is usually used to cope with various problems [27]. Dissipative theory about time-delay systems is discussed in references [28–31]. Sufficient conditions that guarantee robustly stable and strictly passive for uncertain time-delay systems are derived [28]. Dissipativity of linear and nonlinear time-delay systems is, respectively, discussed in [29, 30]. Dissipative problems of singular time-delay systems are considered in [31]. Unlike the systems discussed in the literatures, we consider a special class of linear state/input delay systems. This paper considers the dissipative problem of linear state/input delay systems with a new supply rate and with a storage function substituted by Lyapunov-Krasovskii functional.

There is no systematic dissipativity theory about time-delay systems because of the existence of delays. We consider linear state/input delay systems in this paper. The cause is not only that it has a simply form but it contains state delay and input delay. Motivated by dissipative theory, the variants of supply rate contain not input and output but delay input. A new quadratic supply rate developed in our paper is a function of input, delay input, and output. Based on it, we give sufficient and necessary conditions of dissipativity and exponential dissipativity and show its relationship to dissipativity and stability of the system. Finally, we discuss passivity and -gain and derive position-real lemma and bounded-real lemma for the system.

*Notation.*

denotes the -dimensional Euclidean space, and stands for the Euclidean norm of any vector. is the set of real matrices, and , , denote, respectively, the eigenvalues, and the maximum eigenvalues, the minimum eigenvalues of any square matrix. We use to denote a symmetric positive-definite matrix. In a symmetric block matrix, the expression will be used to denote the submatrices that lie under the diagonal. Given a positive number , denotes the Banach space of continuous functions that maps the interval into with the topology of uniform convergence,and denotes the norm of continuous function, that is, for , . The space of square integrable functions is denoted by . And, for any , its norm is defined as

The rest of the paper is organized as follows. Problem formulation is presented in Section 2. We give some results about dissipativity analysis of linear state/input delay systems in Section 3. In Section 4, positive-real lemma and bounded-real lemma are derived. Finally, a brief conclusion is provided to summarize the paper.

#### 2. Problem Formulation

Consider the continuous-time systems with a state/input delay where is the state vector, is the delay state vector, is exogenous input, where is the set of admissible input, is the measured output vector. , , , , , and are constant system matrices with appropriate dimensions. is an unknown constant delay. which is a real-valued continuous function on with the topology of uniform convergence is the initial function. For any continuous initial function and any admissible input , there exists a unique solution to system (2.1) on which continuously depends on the initial data. If , we denote the segment of trajectory by

Assume that a real-valued function about input and output variants is the supply rate of system (2.1). We assume that for all admissible input and for any , the output of (2.1) is such that satisfies the follwing:

*Definition 2.1. *The linear time-delay system (2.1) with supply rate is said to be dissipative if there exists a nonnegative functional , called the storage functional, such that for all (), , and ,
where , and is a function about input and output, with .

*Remark 2.2. *The inequality (2.4) is called the dissipation inequality. The dissipation inequality implies that the increase in generalized system energy over a given time interval cannot exceed the generalized energy supply delivered to the system (2.1) during this time interval. Thus, some of the supplied generalized energy to the system is stored, and some is dissipative. The dissipated energy is nonnegative and equals the difference of the right-hand of dissipation inequality and the left-hand.

*Remark 2.3. *If using equality sign as a substitution for inequality sign in the dissipation inequality, then Definition 2.1 of dissipativity becomes the definition of losslessness. We call system (2.1) lossless with respect to supply rate .

The next definition shows the notion of available storage, which is the maximum amount of storage energy which may be extracted from a dynamical system.

*Definition 2.4. *The available storage, , of system (2.1) with supply rate is the functional defined by

Note that the available storage, whenever defined, is nonnegative, that is, for all , . The available storage plays an important role in determining whether a given system is dissipative or not. This is shown in the following proposition (see [1]).

Proposition 2.5. *The available storage, , is finite for all if and only if system (2.1) with respect to is dissipative. Moreover, any possible storage function satisfies and if is continuous, then itself is a possible storage function.*

Since supply rate is a function of input and output and system (2.1) contains delay input, the delay input should be considered as an argument of supply rate. The supply rate should be functional about input, delay input, and output, that is, . Motivated by Reference [1, 2], we are interested in studying dissipativity of system (2.1) with the following new quadratic supply rate: where , (), and () are known constant matrices with and being symmetric. Specially, two important special cases of dissipative system (2.1) are the following.(i)When , , dissipative system (2.1) is also called passive system.(ii)When , , dissipative system (2.1) is also called finite gain system.

*Remark 2.6. *Generally speaking, the storage functional is nonnegative and needs not to satisfy differentiability. Lyapunov-Krasovskii functional can be chosen instead of because it is an energy functional representing the abstract storage energy of system. If choose , then the dissipation inequality becomes
If functional is continuous differentiable, then the integral version of dissipation inequality (2.7) can be transformed into a differential version. Differentiating both sides of (2.7), we can get the following:
In this paper, we assume that has the following form of Lyapunov-Krasovskii functional:
where , are positive-definite matrices.

Before concluding this section, we introduce two definitions which will be used in the development of our results.

*Definition 2.7. *The linear time-delay system (2.1) is reachable from the origin, if for any given and , there exist and an admissible control such that the state can be driven from to .

*Remark 2.8. *We suppose that the system (2.1) is reachable from the origin.

*Definition 2.9. *The linear time-delay system (2.1) with and is zero-state detectable, if for any trajectory , there holds

#### 3. Dissipativity Analysis

Notice that dissipativity defined according to the Definition 2.1 represents an input-output property of the system. The following theorem, which is the central result of our paper, shows that dissipativity can also be characterized in terms of the coefficient matrices of system (2.1).

Theorem 3.1. *System (2.1) is said to be dissipative with respect to supply rate (2.6) if and only if there exist positive definite matrices , and constant matrices , () such that
*

*Proof. **Sufficiency. * Suppose that of form (2.9), and are given such that (3.1) is satisfied. Then, for any admissible control and any and , we consider the following performance index:

Using (3.1) and (2.9), we can get the following:
Obviously,
Integrating both sides of (3.4) from to , it yields
*Necessity. *Suppose that system (2.1) is dissipative with respect to the supply rate (2.6). We will show that the available storage, , is a solution of (3.1) for some appropriate matrices and . Due to (2.6) and Definition 2.4, it follows from Proposition 2.5 that the available storage of system (2.1) is finite for all , and satisfies that
for all and .

Differentiating both sides of the above inequality,
Defining a functional by
From (3.7), . In addition, it is known from (3.8) that is quadratic in , , and .

Thus there exist constant matrices , and () such that
Substituting (3.9) into (3.8), we have
for all and .

Now, let . Equating coefficients of equal powers, (3.1) is obtained.

Theorem 3.1 provides a dissipative criterion for system (2.1). Theorem 3.1 can be represented by linear matrix inequality (LMI). The Corollary 3.2 shows a LMI’s version of dissipative criterion.

Corollary 3.2. *System (2.1) is said to be dissipative with respect to supply rate (2.6) if and only if there exist positive definite matrices and such that
**
where .*

The theory of dissipative systems is used to investigate stability of system via Lyapunov methods. It is shown in the following theorem.

Theorem 3.3. *Suppose that system (2.1) is dissipative with smooth storage functional (2.9) with respect to the supply rate (2.6) and zero-state detectable, then the unforced system is Lyapunov stable if and asymptotically stable if . *

*Proof. *Suppose that system (2.1) is dissipative, according to the proof part of Theorem 3.1, and (3.10) is satisfied. Let , we have

If , then . The unforced system is Lyapunov stable (see [22, 23]).

If , then . By zero-state detectable and LaSalle’s invariance principle (see [22, 23]), the unforced system is asymptotically stable.

Next, we give some results of exponential dissipativity. First, the definition of exponential dissipativity is represented.

*Definition 3.4. *The linear time-delay system (2.1) with supply rate is said to be exponentially dissipative if there exists a nonnegative storage functional and a constant , such that for all (), , and ,
where and , with .

*Remark 3.5. *When we choose , the definition of exponential dissipativity becomes the definition of dissipativity.

Assume that has the following form of Lyapunov-Krasovskii functional: where , are positive definite matrices, and is a positive constant.

Theorem 3.6. *System (2.1) is said to be exponentially dissipative with respect to supply rate (2.6) if and only if there exist positive definite matrices , , constant matrices , () and a constant such that
*

*Proof. *Using (3.14), there is
Let performance index be
Note that . So if , we can directly obtain that
Substituting and into , it yields the following:
The sufficiency and necessity proof is similar to the corresponding part of Theorem 3.1. For the sake of brevity, we omit it.

We show the result of Theorem 3.6 by LMI.

Corollary 3.7. *System (2.1) is said to be exponentially dissipative with respect to supply rate (2.6) if and only if there exist positive definite matrices , and a constant such that
**
where .*

Similar to the result about the effect of dissipativity on the stability of unforced system, we will show that the exponentially dissipative system which satisfies that the property of zero-state detectability is exponentially stable.

Theorem 3.8. *Suppose that system (2.1) is exponentially dissipative with smooth storage functional (3.14) with respect to the supply rate (2.6) and zero-state detectable, then the unforced system is exponentially stable if .*

*Proof. *According to the conditions of Theorem 3.8 and the proof of Theorem 3.6, we can get the following:
If , then
Integrating both sides from to , it yields that
From (3.14), we get the following inequality:
where and .

By (3.23) and (3.24), we have
Any solution of unforced system satisfies that
Thus, the unforced system is exponentially stable.

#### 4. Positive-Real and Bounded-Real Conditions

##### 4.1. Positive-Real Lemma

When , we shall focus on studying dissipative systems with supply rate given by the inner product of the following form; For convenience, we characterize this choice by means of a separate definition.

*Definition 4.1. *The linear time-delay system (2.1) with supply rate (4.1) is said to be passive if it is dissipative with supply rate (4.1); that is, there exists a nonnegative storage functional , such that for all (), , and ,

The following theorem is positive-real lemma (KYP lemma) for linear time-delay system (2.1).

Theorem 4.2. *System (2.1) is said to be passive with respect to supply rate (4.1) if and only if there exist positive definite matrices , and constant matrices () such that
*

*Proof. *Let , and in Theorem 3.1. We can get . The result can be directly obtained.

We rewrite the positive-real condition of (4.3) in terms of LMI.

Corollary 4.3. *System (2.1) is said to be passive with respect to supply rate (4.1) if and only if there exists positive definite matrices and such that
**
where .*

##### 4.2. Bounded-Real Lemma

In this subsection, we shall be particularly interested in the quadratic supply rate where positive constant and .

*Definition 4.4. *The system (2.1) is said to have finite -gain from to less than or equal to , if it is dissipative with supply rate (4.5): that is, there exists a nonnegative storage functional , such that for all , , and any

When , obviously, . Inequality (4.6) can be rewritten as

The following theorem is bounded-real lemma for linear time-delay system (2.1).

Theorem 4.5. *System (2.1) is said to have finite -gain from to less than or equal to with respect to supply rate (4.5) if and only if there exist positive definite matrices , and constant matrices , () such that
*

*Proof. *The result can be directly obtained from the consequence of Theorem 3.1 with , , and .

By LMI, Theorem 4.5 can be rewritten as the following corollary.

Corollary 4.6. *System (2.1) is said to have finite -gain from to less than or equal to with respect to supply rate (4.5) if and only if there exist positive definite matrices and such that
**
where .*

*Remark 4.7. *The connections between passive property or finite gain property and unforced systems have not been shown in this section. They can be derived according to the results of Theorem 3.3 and Theorem 3.8.

#### 5. Conclusion

This paper considers dissipativity theory for linear state/input delay systems. Motivated by reference [1, 2], we consider a new type quadratic supply rate for linear state/input delay systems. The choice of the new supply rate also depends on the case of system itself. Based on new supply rate, the necessary and sufficient conditions of dissipativity and exponential dissipativity are represented. And, stability results of dissipative linear state/input delay systems are derived. Finally, we focus on passive and finite -gain systems and discuss positive-real and bounded-real condition, correspondingly. Positive-real lemma and bounded-real lemma based on new supply rate may help us to analyze the problem of stabilization using output or state feedback for the system with or without external disturbances, that is a future direction of research. Interconnected dissipative delay systems are also considered in future works.

#### Acknowledgments

This work is supported by the Natural Science Foundation of China (60874006), the Foundation of Henan Educational Committee (12A120008, 2008B120001), the Outstanding Youth Foundation of Henan Province (084100510010), and the 2012 Key Technologies R&D Program of Henan Scientific Committee (122102210061).

#### References

- J. C. Willems, “Dissipative dynamical systems—part I general theory—part II: linear systems with quadratic supply rates,”
*Archive for Rational Mechanics and Analysis*, vol. 45, pp. 321–351, 1972. View at: Publisher Site | Google Scholar | Zentralblatt MATH - D. Hill and P. Moylan, “The stability of nonlinear dissipative systems,”
*IEEE Transactions on Automatic Control*, vol. 21, no. 5, pp. 708–711, 1976. View at: Google Scholar | Zentralblatt MATH - D. Hill and P. Moylan, “Stability results for nonlinear feedback systems,”
*Automatica*, vol. 13, pp. 377–382, 1977. View at: Google Scholar - D. J. Hill and P. J. Moylan, “Dissipative dynamical systems: basic input-output and state properties,”
*Journal of the Franklin Institute*, vol. 309, no. 5, pp. 327–357, 1980. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Lozano, B. Brogliato, O. Egeland, and B. Maschke,
*Dissipative Systems Analysis and Control*, Springer, London, UK, 2000. - W. M. Haddad and V. Chellaboina,
*Nonlinear Dynamical Systems and Control*, Princeton University Press, Princeton, NJ, USA, 2008. - I. G. Polushin and H. J. Marquez, “On the existence of a continuous storage function for dissipative systems,”
*Systems and Control Letters*, vol. 46, no. 2, pp. 85–90, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Patra and A. Lanzon, “Stability analysis of interconnected systems with “mixed” negative-imaginary and small-gain properties,”
*IEEE Transactions on Automatic Control*, vol. 56, no. 6, pp. 1395–1400, 2011. View at: Publisher Site | Google Scholar - W. M. Haddad, V. Chellaboina, and S. G. Nersesov,
*Impulsive and Hybrid Dynamical Systems*, Princeton University Press, Princeton, NJ, USA, 2006. - W. M. Haddad and Q. Hui, “Dissipativity theory for discontinuous dynamical systems: basic input, state, and output properties, and finite-time stability of feedback interconnections,”
*Nonlinear Analysis*, vol. 3, no. 4, pp. 551–564, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Zhao and D. Hill, “Dissipativity theory for switched systems,”
*IEEE Transactions on Automatic Control*, vol. 53, no. 4, pp. 941–953, 2008. View at: Publisher Site | Google Scholar - C. I. Byrnes, A. Isidori, and J. C. Willems, “Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems,”
*IEEE Transactions on Automatic Control*, vol. 37, no. 6, pp. 770–784, 1992. View at: Google Scholar | Zentralblatt MATH - A. L. Fradkov and D. J. Hill, “Exponential feedback passivity and stabilizability of nonlinear systems,”
*Automatica*, vol. 34, no. 6, pp. 697–703, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH - I. Barkana, “Output feedback stabilizability and passivity in nonstationary and nonlinear systems,”
*International Journal of Adaptive Control and Signal Processing*, vol. 24, no. 7, pp. 568–591, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. J. van der Schaft,
*L*, Springer, London, UK, 2nd edition, 2000._{2}-Gain and Passivity Techniques in Nonlinear Control - A. Loria, E. Panteley, and H. Nijmeijer, “A remark on passivity-based and discontinuous control of uncertain nonlinear systems,”
*Automatica*, vol. 37, pp. 1481–1487, 2001. View at: Google Scholar - S. V. Gusev and A. L. Likhtarnikov, “Kalman-Popov-Yakubovich lemma and the S-procedure: a historical essay,”
*Automation and Remote Control*, vol. 67, no. 11, pp. 1768–1810, 2006. View at: Publisher Site | Google Scholar - A. Rantzer, “On the Kalman-Yakubovich-Popov lemma,”
*Systems and Control Letters*, vol. 28, no. 1, pp. 7–10, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH - D. Chu and R. C. E. Tan, “Algebraic characterizations for positive realness of descriptor systems,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 30, no. 1, pp. 197–222, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. I. Imura, T. Sugie, and T. Yoshikawa, “Characterization of the strict bounded real condition of nonlinear systems,”
*IEEE Transactions on Automatic Control*, vol. 42, no. 10, pp. 1459–1464, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. E. De Souza and L. Xie, “On the discrete-time bounded real lemma with application in the characterization of static state feedback ${H}_{\infty}$ controllers,”
*Systems and Control Letters*, vol. 18, no. 1, pp. 61–71, 1992. View at: Publisher Site | Google Scholar - E. Fridman and U. Shaked, “New bounded real lemma representations for time-delay systems and their applications,”
*IEEE Transactions on Automatic Control*, vol. 46, no. 12, pp. 1973–1979, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - U. Shaked, I. Yaesh, and C. E. De Souza, “Bounded real criteria for linear time-delay systems,”
*IEEE Transactions on Automatic Control*, vol. 43, no. 7, pp. 1016–1022, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. K. Hale,
*Theory of Functional Differential Equations*, Springer, New York, NY, USA, 2003. - K. Gu, V. L. Kharitonov, and J. Chen,
*Stability of Time-Delay Systems*, Birkhäuser, Boston, Mass, USA, 2003. - M. S. Mahmoud,
*Robust Control and Filtering for Time-Delay Systems*, Marcel Dekker, New York, NY, USA, 2000. - V. Kolmanovskii, S. I. Niculescu, and J. P. Richard, “On the Liapunov-Krasovskii functionals for stability analysis of linear delay systems,”
*International Journal of Control*, vol. 72, no. 4, pp. 374–384, 1999. View at: Publisher Site | Google Scholar - M. S. Mahmoud and L. Xie, “Passivity analysis and synthesis for uncertain time-delay systems,”
*Journal of Mathematical Problems in Engineering*, vol. 7, no. 5, pp. 455–484, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. S. Mahmoud and H. N. Nounou, “Dissipative analysis and synthesis of time-delay systems,”
*Mediterranean Journal of Measurements and Control*, vol. 1, pp. 97–108, 2005. View at: Google Scholar - M. S. Mahmoud, Y. Shi, and F. M. AL-Sunni, “Dissipativity analysis and synthesis of a class of nonlinear systems with time-varying delays,”
*Journal of the Franklin Institute*, vol. 346, no. 6, pp. 570–592, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. S. Mahmoud, F. M. Al-Sunni, and Y. Shi, “Dissipativity results for linear singular time-delay systems,”
*International Journal of Innovative Computing, Information and Control*, vol. 4, no. 11, pp. 2833–2845, 2008. View at: Google Scholar

#### Copyright

Copyright © 2012 Guifang Cheng 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.