select different controllers according to different component failures, and therefore better performance of the closed-loop system can be expected. If an AFTC is designed properly, it will be able to deal with unforeseen faults and maintain the system stability and acceptable level of performance in the presence of fault. Some preliminary results on AFTC can be found in [15]-[17] and references therein. Compared with the fruitful FTC results for various dynamic **systems**, relatively few efforts were made to investigate FTC issue for **switched** **systems**. [18] and [20] considered the passive FTC issue for discrete-**time** **switched** **systems**. In [19], passive FTC for **switched** nonlinear **systems** in lower triangular was studied. **Switched** system belongs to hybrid system, which consists of several subsystems and a switching signal that specifies which subsystem will be activated along the system trajectory at each instant of **time**. Many real-world process and **systems** can be modeled as **switched** **systems**, including chemical processes, computer **controlled** **systems**, **switched** circuits, and so on. During the past three decades, fruitful theoretic results have been reported for **switched** **systems**, for examples [21]-[25] and references therein. On the other hand, **time** delays are the inherent features of many physical process and the big sources of instability and poor performances. Meanwhile, **switched** **systems** with **time** delay have strong engineering background, such as in network control **systems** [29] and power **systems** [30]. More recently, many theoretical studies were conducted for **switched** **systems** with **time** delays [31]-[33].

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condition expressed in terms of the Lyapunov-Metzler inequalities is proposed in [27]. This condition does not require each subsystem to be stable and contains as a special case the quadratic stability condition. The underlying minimum switching strategy [47] is a special case that satisfies the hard-to-check non-increasing condition of multiple Lyapunov functions and it is easy to utilize. Sliding mode (chattering) motion is shown to be stable based on this min- switching approach, but is undesirable in practice as there is always a limit to the rate at which the physical system can perform switching. On the other hand, by assuming stability of each subsystem, slow switching can be achieved based on the minimum dwell **time** or average dwell **time** logics [39, 46]. These results have been extended to the case where not all subsystems are stable, and its associated disturbance attenuation properties are discussed in [85]. The discrete- **time** version is discussed in [87] and is extended to the so-called asynchronous switching in [86]. Nevertheless, it could not handle the case where all subsystems are unstable. A state- feedback path-wise switching approach has also been developed for stabilizing switching design of discrete-**time** **switched** linear **systems** in [69]. The proposed switching law generates universal switching signals for the stabilization problem. However, this approach is difficult to incorporate H ∞ performance criterion and address **controlled** performance issue.

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An extreme case of intermittent control is impulsive control which has been gained increasing interest and intensively researched 33, 34. The prominent characteristic of impulsive control is that the states of **controlled** system will “jump” at certain discrete **time** moments, namely, the control is with zero duration of **time**. Because the states of **controlled** **systems** are changed directly, impulsive control is an eﬀective approach when the states are observable, but it seems to be invalid when the states of **controlled** **systems** are unobservable. Our interest focuses on the class of intermittent control with **time** duration, namely, the control is activated in certain nonzero **time** intervals, and oﬀ in other **time** intervals. Specifically, the control law is of the form

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Security lights are an additional safety measure homeowners can use to protect their property. Crime simply cannot occur as easily if there are no dark places for intruders to hide. Darkness **controlled** light switch can be used for security and eliminate wasted power consumption, waste energy, and waste money. These new lights do a better job of aiming light only where it is needed. The light will turn on during darkness and off during brightness. Darkness **controlled** light switch for security lighting can be installed at residential areas, walkways, shopping centers, malls, access roads, parking lots and other outdoor areas. [6]

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Diffusion: Upon contact with aqueous fluid of the gastrointestinal tract (GIT), water gets diffused into the core of the particle. Drug dissolution get occurs and the drug solutions disperse across the release coat to the exterior. Osmosis: Under the right circumstances when water is allowed to enter, an osmotic pressure can be created inside the interior of the particle. Due to this the drug is expelled out of the particle into the outside through the coating Erosion: In some cases coatings can be designed to wear away gradually with **time**, thus delivering the drug contained within the particle.

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Different release patterns, as well as short and reproducible gastric residence **time** (Dashevsk &Mohamad, 2006). Multiparticulate **systems** consists pellets of different release profile which can be of any type like **time** dependent, pH dependent, microflora activated system. Site and **time** specific oral drug delivery have recently been of great interest in pharmaceutical field to achieve improved therapeutic efficacy. Gastro retentive drug delivery system is an approach to prolong gastric residence **time**, thereby targeting site specific drug release in upper gastrointestinal (GI) tract.

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Cirrcadian rhythm are an adaptive phenomena relating to predictabe changes in environmental factors that regulate many body functions like metabolism, sleep pattern, hormone production and physiology. Synchronizing drug delivery in a consistent manner with body’s circadian rhythm is the basic concept for chronotherapeutic drug delivery system. The safety and efficacy of a drug can be improved by conducting the peak plasma concentration with the circadian rhythm of the body. The **time** **controlled** or pulsatile drug delivery **systems** are the best approach for chemotherapy. It offers rapid and transient release of certain molecules within short period which is a **time** and site specific drug delivery system. These **systems** delivers the drug at the right **time** in the right amount for patient suffering from circadian phase dependent disorders like asthma, myocardial infarction, angina pectoris, hypertension, arthritic, epilepsy etc. So various **systems** like osmotic and coated system are being made.

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This paper addresses the stability problem of a class of **switched** positive nonlinear **systems** (SPNSs). Both continuous-**time** **systems** and discrete-**time** **systems** are studied. By applying the minimum dwell-**time** (MDT) approach, we design **time**-dependent switching rules under which the continuous-**time** SPNSs is

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On the other hand, The static output feedback problem is one of the most important open questions in control engineering. A survey of static output-feedback control is given by [16]. Conditions often involve two Riccati equations coupled by a spectral radius condition[17]. [18] presented an algorithm for computing the optimal H 2 static output- feedback gain that is in standard use, along with a convergence proof. Up to now, the exist results related to positive **switched** **systems** main focus on the state feedback controller design, the robust output feedback control of positive **switched** **systems** with **time**-varying delays was not fully investigated, which motivates our present study.

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In practical engineering **systems**, it is unavoidable that the **systems** are affected by unexpected external surroundings or sudden changes in signals. If the fault can not be deleted immediately, it may cause a big impact on the whole system and even cause potentially catastrophic damage. For the past decades, many theories and techniques have been developed in dealing with fault detection problem for various dynamic **systems**[5, 6, 7]. Presently, several popular methods have been developed in dealing with this issue such as model-based fault detection approach, generalized likelihood method etc. In reviewing of the development of these theories and techniques for different fault detection system designs, one of the commonly adopted ways in fault detection is to introduce a performance index and formulate the fault detection as an optimization problem [8]. The H norm of transfer-function matrix from unknown input to residual is accepted as a suitable and effective measure to estimate the influence of the unknown inputs; the H norm of transfer function from fault to residual has been proposed to evaluate the system sensitivity to the faults.

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Positive **switched** **systems**, which consist of a number of positive subsystems and a switching signal governing the switching among this subsystems, have been paid much attention in recent years, see [1-5] and references therein. The problems of stability analysis and controller synthesis of the **systems** have been investigated by many researchers due to their broad applications in communication networks [6], network employing TCP[7], **systems** theory[8-11], and so on. So far, many existing results related to stability analysis focus on the asymptotic stability or exponential stability in the area of positive **switched** **systems**, which reflects the behavior of the system in an infinite **time** interval. But in many practical conditions, one is more interested in what happens over a finite-**time** interval. The concept of finite-**time** stability(FTS) was firstly defined in [12]. Then F. Amato et.al [13] extended this definition to finite **time** boundedness (FTB) when they dealt with the behavior of the state in the presence of external disturbances. Some related results can be found in [14-19]. Recently, [20] firstly extends the concept of FTS to positive **switched** **systems** and gives some FTS conditions of positive **switched** **systems**. In [21], the problem of finite-**time** stability and stabilization of fractional-order positive **switched** **systems** is considered via mode-dependent average dwell **time** approach. The problem of finite-**time** L 1 control for a class of positive **switched**

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Abstract: In this paper, the index-1 notion for arbitrarily **switched** discrete-**time** linear singular **systems** (SDLS) has been introduced. Based on the Bohl exponents of SDLS as well as properties of associated positive **switched** **systems**, some necessary and sufficient conditions have been established for exponential stability.

On the other hand, in many practical applications, **time** de- lays and uncertainties are regularly encountered in dynamic **systems**, which lead to poor performance and even instability in some control **systems**, even make the machine unable to work properly. Therefore, the subject of stability analysis of **switched** **systems** with **time**-varying delays has attracted considerable attention due to strong engineering background in the past few years. For instance, process control system- s[17], networked control [18] and power **systems** [19]. Free weighting matrix method and matrix deformation technique are adopted to reduce the conservatism of delay-dependent criteria of **switched** linear **systems**. Obviously, those methods are more realistic and of great significance to study. To men- tion a few, robustly exponential stability for uncertain neutral **systems** with **time**-varying delays and nonlinear perturbations are investigated by matrix deformation technique in [20]. [16] considered the mean-square exponential stability of **switched** stochastic neutral **systems** with **time**-varying delay under asynchronous switching by the free weight matrix method. Moreover, the issues of robustly exponential stability H ∞ control for uncertain discrete **switched** **systems** with interval **time**-varying delay and the new sufficient stability condition with delay dependence are presented in [21]. However, based on the above discussion, the problem of exponential stabilization for a class of **switched** nonlinear uncertain **systems** with **time**-varying delay has not been well reported, which motivates the present study. Specially, in order to overcome some difficulties caused by the delay and nonlinearity behavior, this paper focuses on the methods of the bounded **time** delays and matrix deformation technique such that the resulting system is exponentially stable.

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The uncertainty is unavoidable in practical engineering due to the parameter drafting, modeling error, and component aging. The controllers or filtering obtained based on nominal **systems** cannot be employed to get the desired performance. Therefore, more and more researchers are devoted to robust control or robust filtering problems; see, for instance, [1–4]. On the other hand, **time**-delay often exists in the practical engineering **systems** and is the main reason of the instability and poor performance of the **systems**. **Time**-delay **systems** have been widely studied during the past two decades [5–7]. In order to get less conservative results, more and more approaches have been proposed to develop delay-dependent conditions for discrete-**time** system with **time**-varying delay. For examples, Jensen’s inequality is proposed in [8]; delay- partitioning method is utilized in [9]; improved results are obtained by using convex combination approach in [10].

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circuit switched and packet switched communication in cellular systems, and support data transmission (not only voice). • The technology which has first supported this evolution[r]

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Remark By the properties of discrete-**time** **systems** associated with the Schrödinger operator and (ii) and (iii) of Switching rule , it is not hard to ﬁnd that the dwell **time** in each **time** interval is at least greater than . So there exist ﬁnite switchings in any ﬁnite **time** interval. Thus, Switching rule is well deﬁned.

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Most control schemes are focused on the faultless system, such as the results of [13–30]. As is well known, actuator faults [31–36] occur frequently in techniques. When the sys- tem encounters actuator failure, it can damage the stability of the system and even crash the system. Therefore, this emphasizes the importance of designing fault-tolerant control in practical applications. Furthermore, by using the approximation ability of FLS in [37], an adaptive controller has been created to deal with the system uncertainty and unknown actuator failure [31]. Thus, as an eﬀective solution, designing appropriate fault-tolerant controllers has been widely used in fault handling. An adaptive fault-tolerant controller was designed for large **systems** with actuator failures [38]. The developed control scheme can be used to avoid the problem of “explosion of complexity” [35]. Then several output feedback control strategies have been developed for nonlinear **switched** **systems** with ac- tuator faults, by using ﬁlter observers, adaptive methods and a backstepping technique [39–43]. For instance, an adaptive compensation controller was constructed by utilizing the backstepping technique [39]. An adaptive distributed controller was proposed to en- sure the realization of tracking error [41]. However, the fault-tolerant control strategies in [32–43] can guarantee system performances in inﬁnite **time**.

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