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A typical teleoperation system consists of a human operator, master robot, communication networks, slave robot, and environment. Teleoperation can be classified as unilateral and bilateral.

The control of bilateral teleoperators is a highly active research due to the complexity of their nonlinear dynamics, the time delays in the communications as well as the wide range of practical real applications. A bilateral networked teleoperation is a networked dual-robot system exchanging sensed feedback and command signals between master and slave robots through network channels where a human operator controls the master robot, and senses the motion and/or force information communicated back from the slave to the master.

Mobile manipulators have been extensively studied because of their reconfigurability and adaptability [265, 306] while multiple mobile manipulators in cooperation can perform a task which is impossible by a single robot.

The teleoperation system usually consists of three components, i.e., the master robot, the slave robot, and the communication network. In the chapter, we present the modeling of three components. First, we describe the mathematical formulation of kinematics and dynamics for the master and slave robot; for simplification, we take the robotic manipulators as a concrete example for master and slave robots. Modeling master and slave robots help us understand the physical properties and are the basis to develop relevant control algorithms. To derive kinematics and dynamics models, we assume that all the physical parameters of the robot are known, e.g., dimensions of links, relative positions and orientations of connected parts, mass distributions of the links, joints, and motors. While in practice, it is not straightforward to obtain accurate values for all these parameters. In addition, in this chapter we assume an ideal system with rigid bodies, joints without backlash, and accurately modeled friction. Second, the presence of a communication network in a control loop induces many imperfections such as varying transmission delays, varying sampling/transmission intervals, packet losses, communication constraints, and quantization effects, which can degrade the control performance significantly and can even lead to instability. In this chapter, we give the mathematical description for the modeling of time delays existing in the communication network.

Recently, cooperation of multiple manipulators have attracted the attention of many researchers [266–307].

This paper discusses the flight control strategy based on intelligent adaptive control and dynamic inversion. The primary use of the Optimal Control Modification (OCM) adaptive control is to add damping to the neural network controller weight update law so as to reduce high-frequency oscillations in the weights and to prevent parameter drift in the absence of persistent excitation. The OCM is applied to the inner loop control of an UAV during flight conditions to investigate its flight control augmentation capability to a dynamic inversion (DI) controller subject to off-nominal flight conditions.

In the recent years, a networked control system (NCS) has been broadly employed in control systems due to its cost-effective and flexible structure. The major challenge in networked control systems is the warranty system’s performance in case of network-induced time delays or data losses. This paper presents a comparison between predictive and predictive fuzzy controller for the DC motor via network. The networked predictive control (NPC) and networked predictive fuzzy control (NPFC) strategy are proposed to compensate the random time delay and data packet dropout in forward channel which can effectively achieve desired control performance of networked control systems. Simulation results are presented to demonstrate the proposed performance.

This paper is concerned with the optimal state estimation problem under linear dynamic systems when the sampling rates of different sensors are different. The noises of different sensors are cross-correlated and coupled with the system noise of the previous step. By use of the projection theory and induction hypothesis repeatedly, a sequential fusion estimation algorithm is derived. The algorithm is proven to be optimal in the sense of Linear Minimum Mean Square Error(LMMSE). Finally, a numerical example is presented to illustrate the effectiveness of the proposed algorithm.

SMC is a system whose structure is intentionally changed with a discontinuous control which drives the phase trajectory to a stable hyperplane or manifold. The high-speed switching forces the state to slide along the surface until it converges.

Attitude control problem for rigid spacecraft with highly nonlinear characteristics has attracted a great deal of interest for its important application [130],[196]. The attitude motion of a rigid body is represented by a set of two vector equations, namely, the kinematic equation which relates to the time derivatives of the orientation angles to the angular velocity vector and the dynamic equation which describes the time evolution of the angular velocity vector [262]. In [247], passivity-based control is proposed to ensure the asymptotical convergence of the attitude tracking without angular velocity measurements.

The attitude control problem for a rigid spacecraft with highly nonlinear characteristics in three dimensions has already been extensively studied (see, e.g., [298, 130, 239, 290, 283, 196, 140] and references therein). From a practical point of view, the design of efficient and low-cost attitude controllers is an important issue for aerospace industry. Indeed, a large variety of nonlinear control schemes, using the quaternion and angular velocity in the feedback, have been proposed recently. For instance, in [261], the attitude tracking control problem for an uncertain rigid spacecraft without angular velocity measurements is addressed.

The PID controller is the most widely used controller and has occupied a dominant position in industrial process control system for over 90 years. However, with the development of science and technology, the requirements of control system focus on control accuracy and rapid respond speed, and the increasingly high demands ability to adapt to the changes in the environment, then the classical PID can not obtain satisfied control performance with some shortcomings.

Proportional navigation (PN) has attracted a amount of interest in the missile interception and design of guidance laws due to the high effectiveness and simple implementation [299, 332, 221, 7]. Specially, PN guidance laws can be easily applied to many-to-one engagement scenarios and achieve the cooperative simultaneous attack of multiple missiles against their common target [123]. However, the PN guidance law is more applicable for the task of intercepting a non-maneuvering target or a weakly maneuvering target. In practice, target acceleration can change rapidly. For intercepting a target with powerful maneuvering capability, the performance of PN is degraded and it may be ineffective for some orientations between missile and target.

As is well known, the accurate guidance of a missile is a highly-concerned problem. Over these decades, persistent efforts have been made to meet the increasingly need for the accuracy of missile guidance. While there has been rapid improvement in realizing the physical conditions for high-speed missile guidance and intercepting missiles, various theories in this field are also created. Dated back to 30 years ago, Guelman [89] made a comprehensive study of PN, which is widely considered as a very significant and fundamental theory of missile guidance. Since then, PN guidance law has been complemented and extended. For example, Becker [19] obtained a closed form solution of PN. Ghawghawe et al.

Proportional navigation (PN) has attracted a amount of interest in the missile interception and continues to be a benchmark for new missile guidance laws due to the high effectiveness and simple implementation [87, 285, 97, 248]. It is well known that PN is the optimal guidance law with a navigation ratio N = 3, and it can result in successful interceptions under a wide range of engagement conditions. However, the PN guidance law is more applicable for the task of intercepting a non-maneuvering target or a weakly maneuvering target. In practice, target acceleration can change rapidly. For intercepting a target with powerful maneuvering capability, the performance of PN is degraded and it may be ineffective for some orientations between missile and target.

Attitude control and stabilization problem for rigid spacecraft systems with highly nonlinear characteristics has attracted a great deal of interest for its important application [216]-[196]. From a practical point of view, the design of efficient and low-cost attitude controllers is an important issue for aerospace industry. In general, the spacecraft motion is governed by the so-called kinematic equations and dynamic equations [262]. Actually, these mathematical descriptions are highly nonlinear and the conventional linear control techniques are not suitable for the control design.

Control of the attitude dynamics of a rigid body is an important and practical problem. The interest is motivated by its key role in many space missions such as, satellite surveillance, space station docking and installation, spacecraft formation flying, etc. SMC is known as one of the most powerful techniques to handle nonlinear systems with uncertainties and bounded external disturbances. SMC has been used to address for the attitude control of spacecraft [168, 44, 196, 274, 263]. [44] investigates the attitude tracking and disturbance rejection problems of spacecraft and converts attitude control into a global stabilization problem. In [196], some spacecraft-attitude tracking problems are resolved by HOSMC laws.

Flight Vehicle includes aircraft, spacecraft, rocket and missile. Attitude tracking and stabilization are key technologies of modern flight vehicle systems which have been served as an interesting benchmark in the past decades. In the attitude equations, two types of uncertainties are paid attention widely, one is the external disturbance which arises from the unexpected environmental torques, and another is the model uncertainty existing in the inertia matrix of the spacecraft. The presence of the external disturbance and the inertia uncertainty makes the attitude control problem more complicated. From a practical point of view, the design of efficient and low-cost attitude controllers is an important issue for aerospace industry.

For the reason of that high frequency sampling will increase the possibility of network congestion, high frequency constricts of NCSs is more affected by time delays and packets dropout than low frequency constricts. Therefore, designing a high frequency system to describe NCSs and satisfying stability performance by using delta operator approach is a key task in this topic. To deal with finite frequency requirements, both weighting functions and frequency gridding has been utilized in [150] and [286], respectively. KYP lemma establishes equivalence between such frequency domain inequality for a transfer function and a LMI for its state space realization [198]. It originates from Popov’s criterion [191]. The Popov’s frequency condition is equivalent to existence of a certain simple Lyapunov function form [133, 282]. Based on the famous Shannon sampling theory, sampling frequency is much larger than system frequency. Goodwin introduced delta operator method to deal with high frequency sampling in [82]. A number of results on delta operator method have been reported in literatures, such as [215, 285, 284]. The generalized KYP lemma in Chapter 9 has given a direct treatment of the finite frequency performance, thus it is can avoid computational burden and guarantee gain property performances simultaneously as in [115, 118, 119].

NCSs is defined as a feedback control system that the control loops are closed through a real-time network [40, 60]. The general theory for NCSs has received much attention in recent years, see for examples [73, 97, 108, 167, 244], and the references therein. A switched linear model has also been proposed to describe NCS in [304]. Predictive NCSs have also been investigated in [155, 156]. A sampling period scheduling of NCSs with multiple control loops has been addressed in [187]. The values of packets dropout and network-induced delays in NCSs are derived from the values of the input delay by the viewpoint of a logic ZOH [272]. The logic ZOH is assumed to be both time-driven and event-driven and has the logical capability of comparing the time steps of the arrived control input packets and choosing the latest one to control the process.

Delta operator based implementations have gained interest due to their excellent finite word length performance under fast sampling [82, 170]. Delta operator models have been studied extensively, for example [137, 142, 176, 177, 194]. Some control problems have also been obtained in recent years, such as [33, 123, 215, 222, 307]. Recently, a robust SMC method has been considered for a class of time-delay DOSs [254]. Furthermore, Markovian jump systems have attracted a lot of researchers from control and operations research communities, for example [95, 208, 211, 245, 247], The stability and stabilization problems for both continuous-time and discrete-time Markovian jump systems were investigated in [298]. Some results on Markovian jump systems with time delays have also been given in [299, 300].

Recently, the problems of the filtering has been studied by a number of authors, see for example [9, 71, 72, 178] and the references therein. Both a Kalman-filter-based iterative algorithm and a recursive frequency estimator using linear prediction have been considered in [305]. In [122], the synthesis of two observer-based nonlinear control algorithms was performed within the globally linearizing control framework. A problem of optimal linear Kalman filtering with packet losses has been considered in [109]. When parameter uncertainty arises in a system model, the robust H ∞ filtering problem has been studied in [75]. Because H ∞ filtering provides a bound for the worstcase estimation error without the need for knowledge of noise statistics, it has been demonstrated that H ∞ filtering has the advantage of being less sensitive to uncertainties of the underlying system. The class of Markovian jump systems is an important family of systems subject to abrupt variations. The Markovian jump systems have finite modes, which may jump from one to another at different times and between different modes. A system with such a “jumping” character may be modeled as a hybrid system, and the parameter jumps among different modes can be seen as discrete events. Note that the concept of Markovian jump filter has already been used in some papers, for example [6, 20, 209, 275]. Robust Kalman filtering for continuous systems with Markovian jumping parameters was given in [160, 208].

In the last two decades, fault detection has attracted considerable attention from many researchers due to the increasing demand for reliability and safety in industrial processes, for example [135, 188, 288]. Some results on fault detection in frequency domain have been reported in literatures, such as [1, 56, 302]. The effects of modeling disturbances and faults usually occupy in different frequency ranges. Therefore, designing an filter to detect the fault by satisfying rigorous performances in finite frequency domain is a key task in this topic. Though weighting functions [150] and frequency gridding [286] can deal with the finite frequency requirements. However, both of the two methods are hard to avoid computational burden and guarantee gain property performances simultaneously. With the aid of the generalized KYP Lemma, new LMI conditions were given for the finite frequency performance index [115, 119]. There are several results in the literature along this line, please refer to [116, 117, 118].

Goodwin constructed delta operator instead of traditional shift operator for sampling continuous systems in [82]. The proposed method can unify some previous related results of the continuous and discrete systems into the framework of the DOSs framework. A class of systems in delta domain has been studied in [32, 140, 266]. The problem of system instability when the sampling time is fast can be solved by using delta operator model [49]. The delta operator model also has the advantage of better numerical properties at high sampling rates [219]. In contrast to the discrete shift operator approximates, the delta operator approach means that the Euler derivative can lead to a quasi-continuous time s-domain model for high sampling frequencies, see for example, [54] and the reference therein. The robustness problem for some DOSs with parametric uncertainties also has been investigated. There are a few papers consider the stability of DOSs such as [214] and [215], and H ∞ control problem has been reported in [52], which proved that the discrete-time controller approach to the continuous-time systems using delta operator.

Fuzzy technique has been widely and successfully used in nonlinear system modeling and control for more than two decades. T-S fuzzy model was introduced in [223]. The T-S fuzzy system has emerged as one of active and fruitful areas of fuzzy control in recent years, see for example [44, 47, 148, 210, 216] and the references therein. Fuzzy automata and syntactic analysis approach have been used for fault diagnosis [199]. Two different methods on simulate fuzzy random variables have been presented in [81]. Fuzzy H ∞ control has been investigated in [76, 152, 179]. By using a fuzzy linear model, an MIMO adaptive sliding mode controller has been proposed in systems [37]. Fuzzy filter has been established by using Lyapuonv function approach in [181, 297]. A stabilization problem for T-S fuzzy systems with nonuniform uncertain sampling was investigated in [65]. By a fuzzy output feedback controller, a stabilizing problem of nonlinear systems has been addressed in [180]. Stabilization analysis for a T-S discrete fuzzy model has been presented based on a non-PDC control law [85]. Using a non-PDC control law, an extensive result has been presented for T-S discrete fuzzy systems in [45].

Over the past decades, much attention has been focused on the problems of robust control for uncertain standard state-space systems, refer to [10, 11, 68, 73, 278] and so on. Furthermore, the state observer design problem has been also paid considerable attention at the same time, for example [163, 164, 165, 280]. A robust non-fragile H ∞ observer-based control has been investigated for continuous-time delayed systems [31]. Based on input-output linearization method, a kind of state observers has been presented for nonlinear systems in [128]. Some T-S fuzzy observers have been proposed based on some relaxed stability conditions [158]. Variable structure control strategy by using sliding-mode concept has been widely studied and developed for control and state estimation problems since the works of Utkin in [232]. A robust predictor based SMC for discrete systems has been designed in [260]. The problem of SMC for continuous systems has been investigated in [259] and [211]. The robust adaptive SMC for discrete systems has been given in [265]. Sensorless SMC of induction motors has been considered in [195]. Moreover, some problems of SMC and identification for uncertain stochastic systems have been investigated in [13].

It is well known that each design specification is often given not for the entire frequency range but rather for a low frequency range of relevance, for example a closed-loop shaping control design typically requiring small sensitivity. Weighting method has been proven useful to deal with the finite frequency requirements [150]. However, the additional weights tend to increase the system complexity and the process of selecting appropriate weights is tedious and time-consuming. Another alternative approach has also been used to grid the frequency axis [286], but it suffers from the lack of a rigorous performance guarantee in the design process. Different from weighting function and frequency gridding methods, a generalized KYP lemma has been proposed in [119]. It can avoid computational burden and guarantee gain property performances simultaneously [115]. There are several results in the literature along this line, see for example [114, 116, 117]. Since the notion of positive realness plays a central role in systems theory [3], many researchers have considered positive real control for linear systems such as in [161, 162, 270], and references therein. The finite frequency positive realness approach for dynamical system design from a control perspective has been given in [118].

Control systems in which information from the sensors and controllers is sent over an electronic communication network are called NCSs. The general theory for NCSs have attracted much research interest in recent years, such as [73, 108, 167] for the network-induced delay issue, [97, 244, 304] for packets dropout in digital communication networks, [153, 200, 205] for the H ∞ control performance of NCSs, [157, 166] for Markovian characterization of NCSs, [155, 156, 306] for predictive NCSs, [94, 187] for sampling strategy of NCSs. For limited bandwidth networks, a nonuniform sampling strategy may be achieved better performance than a uniform sampling strategy [220]. The nonuniform sampled-data NCSs have been extensively studied such as in [201, 241, 287]. A better idea is to shorten network-induced delay and reduce packet dropout by taking into account the system QoS criterion, i.e., the network throughput. As a method of remote control, a QoS-based remote control scheme for a NCS has been presented in [39, 138]. Fortunately, Kalman filter firstly presented by Kalman in [132] is an efficient recursive filter that estimates the state of a linear dynamic system from a series of noisy measurements. In [281], a continuous Kalman filter connected with sampler has been used in NCSs. Delta operator method is a reasonable approach than traditional shift operator in sampling continuous-time systems for short sampling period [53, 82, 170, 137]. A delta operator Riccati difference equation was gained for discrete-time Kalman filter in [202], in which the detailed DOKF algorithm had not been given.

Kalman filter is a very powerful tool in control system design since 1960 when Kalman offered his original paper [132]. Kalman filter design problems for control systems have been given in [130, 183, 225, 256] and the references therein. The convergence analysis of an extended Kalman filter for nonlinear deterministic systems has been presented in [21]. Kalman filter was also used in GPS/SINS integrated navigation fusion algorithm in [36]. Kalman filter was also used for networked data fusion [262]. Kalman filter has also been widely used in many engineering and scientific areas, such as [7, 8, 92, 131, 229]. The control of an inverted pendulum has been studied and demonstrated in control laboratories in connection with the control of launching the rocket for decades. There is a long list of references related to inverted pendulum control for example [80, 184, 217, 224]. In this chapter, we use the inverted pendulum model in an experiment to illustrate the effectiveness of the designed Kalman filter and its implementation.

Robustness of a control system is very important since there exist many various uncertainties in practical application. In the theory of robust control, it has been frequently encountered in several class of systems which include NCSs [66, 68], quantized control systems [67], discrete singular hybrid systems [264], input delay systems [289, 291], state delay systems [12, 74, 75, 88] and so on. Goodwin constructed delta operator instead of traditional shift operator for sampling continuous systems [82]. The model parameters in DOS will approach to the ones in continuous system as sampling period approach to zero, so as control results [54]. It has been shown better numerical properties by using delta operator than using shift operator at high sampling rates in [219]. The robustness problem for DOSs with parametric uncertainties was first presented in [215]. Traditionally, SMC as a general design tool for robust control systems has been gained much attention. The problems of SMC for linear systems with time delays have been investigated in [211, 254, 259, 260] and so on.

In recent years, networked control and data fusion technology have become popular control problems which have been extensively studied under various assumptions and scenarios [197, 235. 98, 145, 91, 2, 206]. There has been a growing interest in the design of controllers based on the network systems such as traffic, communication, aviation and spaceflight [236]. Particularly, the rapid rising of Internet makes Internet based control systems accomplish remote monitoring and adjustment over a long distance. This makes the control systems benefit from the ways of retrieving data and reacting to plant fluctuations from anywhere around the world at any time [34, 179, 129, 127]. In NCSs, the plant, controller, sensor, actuator and reference command are connected through a network. As the structure of NCSs is different from that of traditional control systems, there exist various specific problems in NCSs, for example, network delay, loss of data packets, network security and safety [213].

As we know, two quantizers are used frequently. The uniform quantizer is introduced in above chapter, as for logarithmic quantizer, wonderful results have been obtained. In [40], a logarithmic quantizer is firstly presented for stabilization of a linear discrete-time system. Ref. [51] shows an alternative proof for the optimal design and extends the results to quantized output feedback and quantized quadratic performance control using the sector bound method. Output feedback control of discrete-time linear systems using a finite-level quantizer is studied in [52]. A new approach based on sector bound method is used to analyze the stability of quantized feedback control systems in [237]. Remote control system affected by quantized signal is considered in [86]. Based on a quantization dependent Lyapunov function, the study on stability analysis of quantized feedback control system is given in [55]. The problems of discontinuous stabilization and robust stabilization of nonlinear systems are discussed in [21] and [139], respectively. Ref. [57] considers the problem of dynamic out-feedback stabilization of NCSs. In [66] and [67], the adaptive quantized control is considered and an adaptive feedback control law is given to ensure Lyapunov stable and x(k)→0 as k→ ∞. In [185], quantization and packet dropout are considered simultaneously, packet losses rate and unstable poles of the plant are considered to ensure different stability of the system, such as stochastically quadratically stable and mean square practically stable.

Recently there have been many of interests in NCSs, that is, control systems closed via possibly shared communication links with delay/bandwidth constraints. Particularly, Internet based control systems allow remote monitoring and adjustment of plants over the Internet. This enables the control system to benefit from the way it retrieves data and reacts to plant fluctuations from anywhere around the world at any time, see for example [56, 78], and references therein. The main advantages of NCSs are low cost, simple installation and maintenance, and potentially high reliability. As the structure of NCSs is different from that of traditional control systems, there exist various specific problems in NCSs, for example, quantization, network delay, loss of data packets, network security and safety [148]. In recent years, more and more network techniques have been applied to control systems [133], much attention has been paid to the study of control design and stability analysis of NCSs ([29, 131, 229, 221, 106, 122]). A state estimation problem involving bit-rate communication capacity constraints is studied in [119]. A current survey of the emerging field of NCSs is provided in [7] and Hespanha, et. al review several recent results on estimation, analysis, and controller synthesis for NCSs in [70]. In [206], a practical architecture and some algorithms for the networked data fusion system with packet losses and variable delays are given, in which optimal state estimates through network are presented. However, one important issue is not considered in [206], that is, the measurement or estimated states have to be quantized before they are sent through network.

It is clear that fault diagnosis (including fault detection and isolation, FDI) has been becoming an important subject in modern control theory and practice. Since the beginning of 1970s, research in fault diagnosis has been gaining increasing consideration world-wide in both theory and application [134, 47]. This development was mainly stimulated by the trend of automation towards more complexity and the growing demand for higher availability and security of control systems. However, a strong impetus also comes from the side of modern control theory estimation and parameter identification that have been made feasible by the spectacular progresses of computer technology [26].

The research of fault detection (FD) stems from its practical application to a variety of industries such as aerospace, energy systems, and process control to name a few. The main function of a FD scheme is to detect a fault when it happens, which may then be acted on by sending alarm signals, taking protection measures, or reconfiguring a running control scheme [135, 176]. The observer based FD scheme is currently receiving much attention [227], Generally speaking, an observer based FD system consists of an observer based residual generator and a residual evaluator. It is the state of the art that problems related to the observer based FD system design are mainly addressed in the context of improving system robustness against unknown disturbances and simultaneously enhancing system sensitivity to faults. Study on solving such problems builds the recent research which focus on designing observer based FD systems [69, 82].

Most of the earlier works are based on the measurements observed by sensors with synchronous samples at the same sampling rate [10, 20, 23, 28, 39]. Only a few pieces of work deal with asynchronous multirate multisensor data fusion. Based on continuous time systems, Alouani with his group [4] and Bar-Shalom et al. [10, 8] present some effective algorithms for asynchronous multisensor systems. As far as the discrete time systems are concerned, the related researches include the approaches based on multiscale system theory [72, 11, 12, 195, 211, 226], the batch process methods [104], and the algorithms based on the designing of multirate filter banks [42], etc. In the literature listed above, the missing of observations is rarely concerned, which is inclined to encounter in many application fields including communication, navigation, etc. For filtering of incomplete measurements, there are some interesting results. Among these, the algorithms presented by the team of Wang are promising that it has proper computation complexity and can generate nearly optimal state estimate [194]. Based on a discrete-time linear dynamic system, Kalman filtering with intermittent observations is studied in [169], where the arrival of the observations is modeled as a random process, and the statistical convergence property of Kalman filter is given. The modified Riccati equation is studied by Boers and his group [15]. Some useful results are presented as far as a single sensor observing a single target which is described by a linear state space model is concerned. Kalman filtering with faded measurements is studied in [173]. By use of peak covariance as an estimate of filtering deterioration caused by packet losses, the stability of Kalman filtering with Markovian packet losses is studied in [81] based on a linear time-invariant system. Bar-Shalom studies the state estimation with out of sequence measurements based on a time-invariant dynamic system [8]. Xia, Shang, Chen and Liu study the networked data fusion with packet losses and variable delays, and an optimal state estimate is generated [206]. However, in all these interesting papers, multirate systems are not concerned. The multi-rate linear minimum mean squared error state estimation problem is solved by use of the lifting technique [99]. While, asynchronous sampling and data losses are not concerned in this reference. To sum up, there are few results on the fusion of multirate sensors that sample asynchronously with measurements randomly missing. This motivates us for the present study.

In this chapter, we consider how to design a sliding surface and reaching motion controller for a class of time-delay systems with mismatched uncertainties and exogenous disturbance, where the parameters of mismatched uncertainties are known to lie in a priori specified intervals and the matched uncertainties and exogenous disturbance are assumed to be bounded with unknown bound. The aim of this chapter is to combine adaptive control and variable structure control to realize their individual advantages. The use of variable structure control improves the transient response of the overall system significantly. As a result, the adaptive law proposed is able to reduce the chattering due to the implementation of variable structure controller, and possesses the desired characteristics of robustness and good performance. Sufficient conditions for the existence of linear sliding surfaces are derived. The solution to the condition can be used to characterize linear sliding surfaces, and by using smooth projection method for adaptive control law the reaching motion controller is designed.

Many papers have considered the problem of SMC for classes of uncertain systems with time-delay [161, 147, 189, 160, 10, 8, 9]. Although much attention has been given to the control problem of discrete systems with time-delay [96, 127, 168, 154, 51], few results have been reported on robust control of systems with time-delay in discrete time with discrete SMC.

Mainly due to these advantages variable structure control theory has found applications in various kinds of plants (see, e.g. [86], [175], [33], [85]. Other SMC schemes are proposed for linear systems with state delay [161], [104], [100], and system with input delay [147]. Their methods have the same benefits, but cannot be applied to systems with both state and input delay. In a recent paper [63], the authors propose the design methods of a SMC for systems with both input and state delay. However, the results of [63] involve the tuning and factorizing of a symmetric positive-definite matrix. Because no tuning procedure for such matrix is available, this makes the use of these methods somehow difficult, especially when one wants to find the largest possible bound for the time delay which ensures stabilization.

Recently, much attention has been given to the delay-dependent stability and stabilization of continuous systems with time-delay, see for example,[146, 139, 99, 44, 188, 53, 109, 185, 78, 198, 203]. In those papers, many kinds of Lyapunov-Krasovskii functional are proposed in order to derive less conservative stability conditions. Correspondingly, some of the techniques adopted in the above papers have been extended to the stability and stabilization problem for discrete systems with time-delay. Less conservative stability conditions for discrete systems with time-delay are also derived, see, [53, 201] for example. However, discrete system with time-delay has an important feature, that is, it can be transformed into augmented delay-free system [3, 170]. Then, the stability and stabilization of such a system can be solved with a simple Lyapunov function. Although the dimensions of augmented systems could be larger if the time-delay is large, the stability conditions are simple and convex, which can be checked easily by today’s fast developing computing techniques.