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2010 | Buch

Stability Analysis and Robust Control of Time-Delay Systems

verfasst von: Min Wu, Yong He, Jin-Hua She

Verlag: Springer Berlin Heidelberg

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Über dieses Buch

"Stability Analysis and Robust Control of Time-Delay Systems" focuses on essential aspects of this field, including the stability analysis, stabilization, control design, and filtering of various time-delay systems. Primarily based on the most recent research, this monograph presents all the above areas using a free-weighting matrix approach first developed by the authors. The effectiveness of this method and its advantages over other existing ones are proven theoretically and illustrated by means of various examples. The book will give readers an overview of the latest advances in this active research area and equip them with a pioneering method for studying time-delay systems. It will be of significant interest to researchers and practitioners engaged in automatic control engineering.

Prof. Min Wu, senior member of the IEEE, works at the Central South University, China.

Inhaltsverzeichnis

Frontmatter
1.. Introduction
Abstract
In many physical and biological phenomena, the rate of variation in the system state depends on past states. This characteristic is called a delay or a time delay, and a system with a time delay is called a time-delay system. Timedelay phenomena were first discovered in biological systems and were later found in many engineering systems, such as mechanical transmissions, fluid transmissions, metallurgical processes, and networked control systems. They are often a source of instability and poor control performance. Time-delay systems have attracted the attention of many researchers [13] because of their importance and widespread occurrence. Basic theories describing such systems were established in the 1950s and 1960s; they covered topics such as the existence and uniqueness of solutions to dynamic equations, stability theory for trivial solutions, etc. That work laid the foundation for the later analysis and design of time-delay systems.
2.. Preliminaries
Abstract
This chapter provides basic knowledge and concepts on the stability of timedelay systems, including the concept of stability, H∞ norm, H∞ control, the LMI method, and some useful lemmas. They form the foundation for subsequent chapters.
3.. Stability of Systems with Time-Varying Delay
Abstract
Increasing attention is being paid to delay-dependent stability criteria for linear time-delay systems. Investigations of such criteria for constant delays [110] usually involve either some type of frequency-domain method or the Lyapunov-Krasovskii functional method in the time domain. For time-varying delays, studies of such criteria [1117] generally employ a fixed model transformation because frequency-domain methods and the discretized Lyapunov-Krasovskii functional method are too difficult to use in this case. Of the four types of fixed model transformations, three can handle time-varying delays: Model transformation I [1214]; Model transformation III [18]; and Model transformation IV, which uses both Park’s and Moon et al.’s inequalities. Model transformations III and IV are the most effective, and they can also be used to obtain delay-independent stability criteria. However, their use of fixed weighting matrices imposes certain limitations, as pointed out in Chapter 1.
4.. Stability of Systems with Multiple Delays
Abstract
If a linear system with a single delay, h, is not stable for a delay of some length, but is stable for h=0, then there must exist a positive number \( \overline h \) for which the system is stable for \( 0 \leqslant h \leqslant \overline h \). Many researchers have simply extended this idea to a system with multiple delays, but this simple extension may lead to conservativeness. For example, Fridman & Shaked [1, 2] investigated a linear system with two delays:
$$ \dot x\left( t \right) = A_0 x\left( t \right) + A_1 x\left( {t - h_1 } \right) + A_2 x\left( {t - h_2 } \right). $$
(4.1)
The upper bounds \( \overline h _1 \) and \( \overline h _2 \) on h 1 and h 2, respectively, are selected so that this system is stable for \( 0 \leqslant h_1 \leqslant \overline h _1 \) and \( 0 \leqslant h_2 \leqslant \overline h _2 \). However, the ranges of h 1 and h 2 that guarantee the stability of this system are conservative because they start from zero, even though that may not be necessary. One reason for this is that the relationship between h 1 and h 2 was not taken into account in the procedure for finding the upper bounds. Another point concerns a linear system with a single delay,
$$ \dot x\left( t \right) = A_0 x\left( t \right) + \left( {A_1 + A_2 } \right)x\left( {t - h_1 } \right), $$
(4.2)
which is a special case of system (4.1), namely, the case h 1=h 2. The stability criterion for system (4.2) should be equivalent to that for system (4.1) for h 1=h 2; but this equivalence cannot be demonstrated by the methods in [1,2].
5.. Stability of Neutral Systems
Abstract
A neutral system is a system with a delay in both the state and the derivative of the state, with the one in the derivative being called a neutral delay. That makes it more complicated than a system with a delay in only the state. Neutral delays occur not only in physical systems, but also in control systems, where they are sometimes artificially added to boost the performance. For example, repetitive control systems constitute an important class of neutral systems [1]. Stability criteria for neutral systems can be classified into two types: delay-independent [24] and delay-dependent [524]. Since the delayindependent type does not take the length of a delay into consideration, it is generally conservative. The basic methods for studying delay-dependent criteria for neutral systems are similar to those used to study linear systems, with the main ones being fixed model transformations. As mentioned in Chapter 1, the four types of fixed model transformations impose limitations on possible solutions to delay-dependent stability problems.
6.. Stabilization of Systems with Time-Varying Delay
Abstract
At present, there is no effective controller synthesis algorithm for solving delay-dependent stabilization problems, even for the simple situation of statefeedback; for output feedback, the problem is even more difficult. It is possible to use model transformations I and II to derive an LMI-based controller synthesis algorithm. However, as mentioned in [1,2], they add eigenvalues to the system, with the result that the transformed system is not equivalent to the original one. Thus, they have been abandoned in favor of model transformations III and IV, for which an NLMI is used to design a controller in synthesis problems.
7.. Stability and Stabilization of Discrete-Time Systems with Time-Varying Delay
Abstract
Increasing attention is being paid to the delay-dependent stability, stabilization, and H control of linear systems with delays [114]. The literature discusses discrete-time systems with two types of time-varying delays: small and non-small. For a small delay, [1517] presented methods of designing an H state-feedback controller. More recently, a time-varying interval delay, which is a kind of non-small delay, has become a focus of attention for both continuous-time systems [18] and discrete-time systems [9, 19, 20]. [9] solved the robust H control problem using an output-feedback controller; but the limitations of that approach are that matrix inequalities must be solved to obtain the decision matrix variables, and only a range of delays can be dealt with. [21] handled the problem of designing an H filter by using a finite-sum inequality. [20] used Moon et al.’s inequality and criteria containing both the range and upper bound of the time-varying delay for the delay-dependent output-feedback stabilization of discrete-time systems with a time-varying state delay. And [19] derived H control criteria using a descriptor model transformation in combination with Moon et al.’s inequality for uncertain linear discrete-time systems with a time-varying interval delay; in that approach, the delay was decomposed into a nominal part and an uncertain part.
8.. H ∞ Control Design for Systems with Time-Varying Delay
Abstract
During the last decade, considerable attention has been devoted to the problems of delay-dependent stability, stabilization, and H∞ controller design for time-delay systems [16]. However, as pointed out in [7, 8], most studies to date have ignored some useful terms in the derivative of the Lyapunov-Krasovskii functional [1, 5, 911]. Although [7, 8] retained these terms and established an improved delay-dependent stability criterion for systems with a time-varying delay, there is room for further investigation. For instance, in [1, 5, 712], the delay, d(t), where 0 ≦ d(t)h, was often increased to h. And in [7, 8], another term, h-d(t), was also taken to be equal to h; that is, h=d(t)+(h-d(t)) was increased to 2h, which may lead to conservativeness. Moreover, these methods are not applicable to systems with a time-varying interval delay.
9.. H ∞ Filter Design for Systems with Time-Varying Delay
Abstract
One problem with estimating the state from corrupted measurements is that, if we assume that the noise source is an arbitrary signal with bounded energy, then the well-known Kalman filtering scheme is no longer applicable. To handle that case, H∞ filtering was proposed in [1]. It provides a guaranteed noise attenuation level [25]. H∞ filtering for time-delay systems has been a hot topic in recent years, and design methods for delay-independent H∞ filters have been presented in [610].
10.. Stability of Neural Networks with Time-Varying Delay
Abstract
Neural networks are useful in signal processing, pattern recognition, static image processing, associative memory, combinatorial optimization, and other areas [1]. Although considerable effort has been expended on analyzing the stability of neural networks without a delay, in the real world such networks often have a delay due, for example, to the finite switching speed of amplifiers in electronic networks and to the finite signal propagation speed in biological networks. So, the stability of different classes of neural networks with a delay has become an important topic [220]. The criteria in these papers are based on various types of stability (asymptotic, complete, absolute, exponential, and so on); and they can be classified into two categories according to their dependence on information about the length of a delay: delay-independent [310, 13] and delay-dependent [2, 1120]. Since delay-independent criteria tend to be conservative, especially when the delay is small or varies within an interval, the delay-dependent type receives greater attention.
11.. Stability of T-S Fuzzy Systems with Time-Varying Delay
Abstract
Takagi-Sugeno (T-S) fuzzy systems [1] combine the flexibility of fuzzy logic and the rigorous mathematics of a nonlinear system into a unified framework. A variety of analytical methods have been used to express asymptotic stability criteria for them in terms of LMIs [25]. All of these methods are for systems with no delay. In the real world, however, delays often occur in chemical, metallurgical, biological, mechanical, and other types of dynamic systems. Furthermore, a delay usually causes instability and degrades performance. Thus, the analysis of the stability of T-S fuzzy systems is not only of theoretical interest, but also of practical value [624].
12.. Stability and Stabilization of NCSs
Abstract
Closed control loops in communication networks are becoming more and more common as network hardware becomes cheaper and use of the Internet expands. Feedback control systems in which control loops are closed through a real-time network are called NCSs. In an NCS, network-induced delays of variable length occur during data exchange between devices (sensor, controller, actuator) connected to the network. This can degrade the performance of the control system and can even destabilize it [111]. It is important to make these delays bounded and as small as possible. On the other hand, it is also necessary to design a controller that guarantees the stability of an NCS for delays less than the maximum allowable delay bound (MADB) [12], which is also called the maximum allowable transfer interval (MATI) [5, 6].
13.. Stability of Stochastic Systems with Time-Varying Delay
Abstract
Stochastic phenomena are common in many branches of science and engineering, and stochastic perturbations can be a source of instability in systems. This has made stochastic systems an interesting topic of research; and stochastic modeling has become an important tool in science and engineering. Increasing attention is now being paid to the stability, stabilization, and H∞ control of stochastic time-delay systems [16].
14.. Stability of Nonlinear Time-Delay Systems
Abstract
Since Lur’e brought up the subject of absolute stability in [1], three common ways of dealing with the problem of the absolute stability of Lur’e control systems have emerged. One is to use the Popov frequency-domain criterion [25]. The problem with it is that it is not suitable for dealing with multiple nonlinearities because it is not geometrically intuitive and cannot be examined by illustration. The second is to use the extended Popov frequency-domain criterion [6,7]. The problem here is that the condition is only sufficient. The third is a method based on a Lyapunov function in the Lur’e form [8,9]. It produces necessary and sufficient conditions for the existence of a Lyapunov function in the Lur’e form that ensures the absolute stability of a Lur’e control system with multiple nonlinearities in a bounded sector; but the conditions simply exist and are unsolvable.
Backmatter
Metadaten
Titel
Stability Analysis and Robust Control of Time-Delay Systems
verfasst von
Min Wu
Yong He
Jin-Hua She
Copyright-Jahr
2010
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-03037-6
Print ISBN
978-3-642-03036-9
DOI
https://doi.org/10.1007/978-3-642-03037-6

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