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 - and references therein. Compared with the fruitful FTC results for various dynamic systems, relatively few efforts were made to investigate FTC issue for switchedsystems.  and  considered the passive FTC issue for discrete-timeswitchedsystems. In , 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 switchedsystems, including chemical processes, computer controlledsystems, switched circuits, and so on. During the past three decades, fruitful theoretic results have been reported for switchedsystems, for examples - 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, switchedsystems with time delay have strong engineering background, such as in network control systems  and power systems . More recently, many theoretical studies were conducted for switchedsystems with time delays -.
condition expressed in terms of the Lyapunov-Metzler inequalities is proposed in . 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  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 . The discrete- time version is discussed in  and is extended to the so-called asynchronous switching in . 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-timeswitched linear systems in . 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.
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 controlledsystems are changed directly, impulsive control is an eﬀective approach when the states are observable, but it seems to be invalid when the states of controlledsystems 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
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. 
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.
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.
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 timecontrolled 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.
This paper addresses the stability problem of a class of switched positive nonlinear systems (SPNSs). Both continuous-timesystems and discrete-timesystems are studied. By applying the minimum dwell-time (MDT) approach, we design time-dependent switching rules under which the continuous-time SPNSs is
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 . Conditions often involve two Riccati equations coupled by a spectral radius condition.  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 switchedsystems main focus on the state feedback controller design, the robust output feedback control of positive switchedsystems with time-varying delays was not fully investigated, which motivates our present study.
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 . 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.
Positive switchedsystems, 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 , network employing TCP, 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 switchedsystems, 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 . Then F. Amato et.al  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,  firstly extends the concept of FTS to positive switchedsystems and gives some FTS conditions of positive switchedsystems. In , the problem of finite-time stability and stabilization of fractional-order positive switchedsystems is considered via mode-dependent average dwell time approach. The problem of finite-time L 1 control for a class of positive switched
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 switchedsystems, 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 switchedsystems with time-varying delays has attracted considerable attention due to strong engineering background in the past few years. For instance, process control system- s, networked control  and power systems . 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 .  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 switchedsystems with interval time-varying delay and the new sufficient stability condition with delay dependence are presented in . 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.
Switchedsystems are a kind of hybrid systems composed of a family of subsystems and a logical rule that orchestrates switching between these subsystems. Due to the physical properties or various environmental factors, many real-world systems can be modeled as switchedsystems, such as networked control systems [, ], robot control systems , and so on. Switchedsystems have drawn increasing attention during the past decades due to their wide applications. Common Lyapunov function method , multiple Lyapunov function method  and average dwell time approach [, ] have been proposed to study the stability of such systems.
Switched dynamical system (SDS for short) exhibits a wide variety of complex phenom- ena which cannot be dealt with by the classical theory, but are typically observed in many models of real systems; for instance, stick-slip, chattering, grazing-sliding, and jump phe- nomena were observed in an automotive brake system, impact contact model of a church bell, electronic switches, and genetic networks, respectively. For historical overviews and references, see [3, 5, 6, 10, 17, 18, 22].
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 ; delay- partitioning method is utilized in ; improved results are obtained by using convex combination approach in .
Remark By the properties of discrete-timesystems 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.
Switchedsystems, which consist of two or more subsystems and a switching signal or- chestrating switching between these subsystems, have attracted a growing interest in re- cent years. The viability kernel for switchedsystems has been largely studied, and several results have been reported in [20–28]. Aubin and Lygeros introduced a notion of hybrid strategy and proved convergence of the iterative algorithm by using non-smooth analysis tools in . Margellos and Lygeros proposed a method of computing the viable set for hy- brid systems  in the context of optimal control, and a complete characterization of the computation of the viable set was provided based on dynamic programming. Haimovich et al. in  also developed a method of computing the invariant set for continuous-timeswitched linear systems with disturbances and arbitrary switching. Abate and Prandini in  studied probabilistic reachability over a ﬁnite horizon for a class of discrete time stochastic hybrid systems. Lygeros et al. in  presented some simple properties of the invariant sets for hybrid automata, and the invariance principle was extended to hybrid automata. By establishing a link between reachability, viability, and invariance problems and viscosity solutions of a special form of the Hamilton–Jacobi equation, the numeri- cal algorithm developed for the approximation of viscosity solutions to partial diﬀerential equations was extended to viability and invariance computation in .
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 , an adaptive controller has been created to deal with the system uncertainty and unknown actuator failure . 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 . The developed control scheme can be used to avoid the problem of “explosion of complexity” . Then several output feedback control strategies have been developed for nonlinear switchedsystems 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 . An adaptive distributed controller was proposed to en- sure the realization of tracking error . However, the fault-tolerant control strategies in [32–43] can guarantee system performances in inﬁnite time.