Showing papers in "International Journal of Robust and Nonlinear Control in 2013"

Distributed containment control of multi‐agent systems with general linear dynamics in the presence of multiple leaders

Abstract: SUMMARY This paper considers the containment control problems for both continuous-time and discrete-time multi-agent systems with general linear dynamics under directed communication topologies. Distributed dynamic containment controllers based on the relative outputs of neighboring agents are constructed for both continuous-time and discrete-time cases, under which the states of the followers will asymptotically converge to the convex hull formed by those of the leaders if, for each follower, there exists at least one leader that has a directed path to that follower. Sufficient conditions on the existence of these dynamic controllers are given. Static containment controllers relying on the relative states of neighboring agents are also discussed as special cases. Copyright © 2011 John Wiley & Sons, Ltd.

Robust stability and stabilization of uncertain linear positive systems via integral linear constraints : L-1-gain and L-infinity-gain characterization

Abstract: Copositive linear Lyapunov functions are used along with dissipativity theory for stability analysis and control of uncertain linear positive systems. Unlike usual results on linear systems, linear ...

Stochastic stability and robust stabilization of semi-Markov jump linear systems

Abstract: SUMMARY The semi-Markov jump linear system (S-MJLS) is more general than the Markov jump linear system (MJLS) in modeling some practical systems. Unlike the constant transition rates in the MJLS, the transition rates of the S-MJLS are time varying. This paper focuses on the robust stochastic stability condition and the robust control design problem for the S-MJLS with norm-bounded uncertainties. The infinitesimal generator for the constructed Lyapunov function is first derived. Numerically solvable sufficient conditions for the stochastic stability of S-MJLSs are then established in terms of linear matrix inequalities. To reduce the conservativeness of the stability conditions, we propose to incorporate the upper and lower bounds of the transition rate and meanwhile apply a new partition scheme. The robust state feedback controller is accordingly developed. Simulation studies and comparisons demonstrate the effectiveness and advantages of the proposed methods. Copyright © 2012 John Wiley & Sons, Ltd.

Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach

Abstract: SUMMARY This paper is concerned with the problem of consensus in directed networks of multiple agents with intrinsic nonlinear dynamics and sampled-data information. A new protocol is induced from a class of continuous-time linear consensus protocols by implementing data-sampling technique and a zero-order hold circuit. On the basis of a delayed-input approach, the sampled-data multi-agent system is converted to an equivalent nonlinear system with a time-varying delay. Theoretical analysis on this time-delayed system shows that consensus with asymptotic time-varying velocities in a strongly connected network can be achieved over some suitable sampled-data intervals. A multi-step procedure is further presented to estimate the upper bound of the maximal allowable sampling intervals. The results are then extended to a network topology with a directed spanning tree. For the case of the topology without a directed spanning tree, it is shown that the new protocol can still guarantee the system to achieve consensus by appropriately informing a fraction of agents. Finally, some numerical simulations are presented to demonstrate the effectiveness of the theoretical results and the dependence of the upper bound of maximal allowable sampling interval on the coupling strength. Copyright © 2012 John Wiley & Sons, Ltd.

Distributed output regulation of leader–follower multi‐agent systems

Abstract: SUMMARY In this paper, distributed leader–follower control algorithms are presented for linear multi-agent systems based on output regulation theory and internal model principle. By treating a leader to be followed as an exosystem, the proposed framework can be used to generalize existing multi-agent coordination solutions to allow the identical agents to track an active leader with different dynamics and unmeasurable variables. Moreover, the obtained results for multi-agent coordination control are an extension of previous work on centralized and decentralized output regulation to a distributed control context. Necessary and sufficient conditions for the distributed output regulation problem are given. Finally, distributed output regulation of some classes of multi-agent systems with switching interconnection topologies are discussed via both static and dynamic feedback. Copyright © 2011 John Wiley & Sons, Ltd.

Stability of a class of switched positive linear time-delay systems

Abstract: SUMMARY This paper addresses the problem of stability for a class of switched positive linear time-delay systems. As first attempt, the Lyapunov–Krasovskii functional is extended to the multiple co-positive type Lyapunov–Krasovskii functional for the stability analysis of the switched positive linear systems with constant time delay. A sufficient stability criterion is proposed for the underlying system under average dwell time switching. Subsequently, the stability result for system under arbitrary switching is presented by reducing multiple co-positive type Lyapunov–Krasovskii functional to the common co-positive type Lyapunov–Krasovskii functional. A numerical example is given to show the potential of the proposed techniques. Copyright © 2012 John Wiley & Sons, Ltd.

Stability and robustness of large platoons of vehicles with double-integrator models and nearest neighbor interaction

Abstract: SUMMARY We study the stability and robustness of a large platoon of vehicles, where each vehicle is modeled as a double integrator, for two decentralized control architectures: predecessor following and symmetric bidirectional. In the predecessor-following architecture, the control action on each agent only depends on the information from its immediate front neighbor, whereas in the symmetric bidirectional architecture, it depends equally on the information from both its immediate front neighbor and back neighbor. We prove asymptotic stability of the formation for a class of nonlinear controllers with sector nonlinearity, with the linear controller as a special case. We show that the convergence rate of the predecessor-following architecture is much faster than that of the symmetric bidirectional architecture. However, the predecessor-following architecture suffers high algebraic growth of initial errors. We also establish scaling laws (with N) of certain H ∞ norms of the formation that measure its robustness to external disturbances for the linear case. It is shown that the robustness performance grows geometrically in N for predecessor-following architecture but only polynomially in N for symmetric-bidirectional architecture. Extensive numerical simulations are conducted to verify the predictions for the linear case and empirically estimate the corresponding performance metrics for a saturation-type nonlinear controller. On the basis of the analytical and numerical results, it is seen that the symmetric bidirectional architecture outperforms the predecessor-following architecture in all measures of performance. Within the predecessor-following architecture, the nonlinear controller is seen to perform better in general than the linear one. A number of design guidelines are provided on the basis of these conclusions. Copyright © 2012 John Wiley & Sons, Ltd.

Novel delay‐derivative‐dependent stability criteria using new bounding techniques

Abstract: SUMMARY This paper studies the stability of linear systems with interval time-varying delays. By constructing a new Lyapunov–Krasovskii functional, two delay-derivative-dependent stability criteria are formulated by incorporating with two different bounding techniques to estimate some integral terms appearing in the derivative of the Lyapunov–Krasovskii functional. The first stability criterion is derived by using a generalized integral inequality, and the second stability criterion is obtained by employing a reciprocally convex approach. When applying these two stability criteria to check the stability of a linear system with an interval time-varying delay, it is shown through some numerical examples that the first stability criterion can provide a larger upper bound of the time-varying delay than the second stability criterion. Copyright © 2012 John Wiley & Sons, Ltd.

Decentralized sliding-mode control for attitude synchronization in spacecraft formation

Abstract: SUMMARY This paper addresses attitude synchronization and tracking problems in spacecraft formation in the presence of model uncertainties and external disturbances. A decentralized adaptive sliding mode control law is proposed using undirected interspacecraft communication topology and analyzed based on algebraic graph theory. A multispacecraft sliding manifold is derived, on which each spacecraft approaches desired time-varying attitude and angular velocity while maintaining attitude synchronization with the other spacecraft in the formation. A control law is then developed to ensure convergence to the sliding manifold. The stability of the resulting closed-loop system is proved by application of Barbalat's Lemma. Simulation results demonstrate the effectiveness of the proposed attitude synchronization and tracking methodology. Copyright © 2012 John Wiley & Sons, Ltd.

Adaptive consensus with a virtual leader of multiple agents governed by locally Lipschitz nonlinearity

Abstract: This paper investigates adaptive flocking of multi-agent systems (MASs) with a virtual leader. All agents and the virtual leader share the same intrinsic nonlinear dynamics, which satisfies a locally Lipschitz condition and depends on both position and velocity information for the agent itself. Under the assumption that the initial network is connected, an approach to preserving the connectivity of the network is proposed. On the basis of the Lyapunov stability theory, an adaptive flocking control law is derived, for making the MASs track the virtual leader without collision. Finally, a numerical example is presented to illustrate the effectiveness of the theoretical results.

Robust control of robot manipulators based on uncertainty and disturbance estimation

Abstract: SUMMARY In this work, uncertainty and disturbance estimation (UDE) based robust trajectory tracking controller for rigid link manipulators was proposed. The UDE was employed to estimate the composite uncertainty that comprises the effects of system nonlinearities, external disturbances, and parametric uncertainties. A feedback linearization based controller was designed for trajectory tracking, and the same was augmented by the UDE-estimated uncertainties to achieve robustness. The resulting controller however required measurement of joint velocities apart from the joint positions. To address the issue, an observer that employed the UDE-estimated uncertainties for robustness was proposed, giving rise to the UDE-based controller–observer structure. Closed-loop stability of the overall system was established. The notable feature of the proposed design was that it neither required accurate plant model nor any information about the uncertainty. Also, the design needed only joint position measurements for its implementation. To demonstrate the effectiveness, simulation results of the proposed approach as applied to the trajectory tracking control of two-link robotic manipulator and comparison of its performance with some of the well-known existing controllers were presented. Lastly, hardware implementation of the proposed design for trajectory control of Quanser's single-link flexible joint module was carried out, and it was shown that the proposed strategy offered a viable approach for designing implementable robust controllers for robots. Copyright © 2011 John Wiley & Sons, Ltd.

Reliable H ∞ filter design for T–S fuzzy model-based networked control systems with random sensor failure

Abstract: SUMMARY This paper investigates robust and reliable H ∞ filter design for a class of nonlinear networked control systems: (i) a T-S fuzzy model with its own uncertainties is used to approximate the nonlinear dynamics of the plant, (ii) a new sensor failure model with uncertainties is proposed, and (iii) the signal transfer of the closed-loop system is under a networked communication scheme and therefore is subject to time delay, packet loss, and/or packet out of order. Four new theorems are proved to cover the conditions for the robust mean square stability of the systems under study in terms of LMIs, and a decoupling method for the filter design is developed. Two examples, one of them is based on a model of an inverted pendulum, are provided to demonstrate the design method. Copyright © 2011 John Wiley & Sons, Ltd.

Finite-Time Attitude Stabilization for Rigid Spacecraft

Abstract: This chapter investigates the finite-time attitude stabilization problem for rigid spacecraft in the presence of inertia uncertainties and external disturbances. Three nonsingular terminal sliding mode (NTSM) controllers are designed to make the spacecraft system converge to its equilibrium point or a region around its equilibrium point in finite time. In addition, these novel controllers are singularity-free, and the presented adaptive NTSM control (ANTSMC) laws are chattering-free. A rigorous proof of finite-time convergence is developed. The proposed ANTSMC algorithms combine NTSM, adaptation, and a constant plus power rate reaching law. Because the algorithms require no information about inertia uncertainties and external disturbances, they can be used in practical systems, where such knowledge is typically unavailable. Simulation results support the theoretical analysis.

On the control of Markov jump linear systems with no mode observation: application to a DC Motor device

Abstract: SUMMARY This paper deals with the control problem of discrete-time Markov jump linear systems for the case in which the controller does not have access to the state of the Markov chain. A necessary optimal condition, which is nonlinear with respect to the optimizing variables, is introduced, and the corresponding solution is obtained through a variational convergent method. We illustrate the practical usefulness of the derived approach by applying it in the control speed of a real DC Motor device subject to abrupt power failures.Copyright © 2012 John Wiley & Sons, Ltd.

Finite‐time formation control of multiagent systems via dynamic output feedback

Abstract: SUMMARY Finite-time formation control of multiple second-order agents via dynamic output feedback is investigated in this paper Under the assumption that the velocities of all agents cannot be measured, a continuous consensus algorithm is first proposed such that the states of all agents will reach an agreement in finite time Then, the consensus algorithm is applied to the finite-time formation control, including stationary formation and moving formation, respectively Rigorous proof shows that all agents will converge to the desired formation pattern in finite time Finally, an example is given to verify the efficiency of the proposed method Copyright © 2012 John Wiley & Sons, Ltd

Distributed consensus control for multi-agent systems using terminal sliding mode and Chebyshev neural networks

Abstract: This paper investigates the problem of consensus tracking control for second-order multi-agent systems in the presence of uncertain dynamics and bounded external disturbances. The communication ?ow among neighbor agents is described by an undirected connected graph. A fast terminal sliding manifold based on lumped state errors that include absolute and relative state errors is proposed, and then a distributed finite-time consensus tracking controller is developed by using terminal sliding mode and Chebyshev neural networks. In the proposed control scheme, Chebyshev neural networks are used as universal approximators to learn unknown nonlinear functions in the agent dynamics online, and a robust control term using the hyperbolic tangent function is applied to counteract neural-network approximation errors and external disturbances, which makes the proposed controller be continuous and hence chattering-free. Meanwhile, a smooth projection algorithm is employed to guarantee that estimated parameters remain within some known bounded sets. Furthermore, the proposed control scheme for each agent only employs the information of its neighbor agents and guarantees a group of agents to track a time-varying reference trajectory even when the reference signals are available to only a subset of the group members. Most importantly, finite-time stability in both the reaching phase and the sliding phase is guaranteed by a Lyapunov-based approach. Finally, numerical simulations are presented to demonstrate the performance of the proposed controller and show that the proposed controller exceeds to a linear hyperplane-based sliding mode controller. Copyright (C) 2011 John Wiley & Sons, Ltd.

Razumikhin–Nussbaum‐lemma‐based adaptive neural control for uncertain stochastic pure‐feedback nonlinear systems with time‐varying delays

Abstract: SUMMARY This paper addresses the problem of adaptive neural control for a class of uncertain stochastic pure-feedback nonlinear systems with time-varying delays. Major technical difficulties for this class of systems lie in: (1) the unknown control direction embedded in the unknown control gain function; and (2) the unknown system functions with unknown time-varying delays. Based on a novel combination of the Razumikhin–Nussbaum lemma, the backstepping technique and the NN parameterization, an adaptive neural control scheme, which contains only one adaptive parameter is presented for this class of systems. All closed-loop signals are shown to be 4-Moment semi-globally uniformly ultimately bounded in a compact set, and the tracking error converges to a small neighborhood of the origin. Finally, two simulation examples are given to demonstrate the effectiveness of the proposed control schemes. Copyright © 2012 John Wiley & Sons, Ltd.

Adaptive high-order dynamic sliding mode control for a flexible air-breathing hypersonic vehicle

Abstract: SUMMARY The problem of tracking control for a longitudinal air-breathing hypersonic vehicle model with flexible effects and intricate couplings between the engine dynamics and flight dynamics is investigated in this paper. In order to overcome the analytical intractability of this model, a simplified model is constructed during the feedback controller design. High-order dynamic sliding mode control is proposed to force the velocity and altitude of the flexible air-breathing hypersonic vehicle (FAHV) to the desired reference commands in finite time. In addition, the adaptive control method is employed to identify bounded uncertainties for eliminating the requirement of boundaries needed in the robust controller design and to guarantee the property of finite time stability without information of upper bound of uncertainties. Finally, simulations are presented to illustrate the effectiveness of the control strategies.Copyright © 2013 John Wiley & Sons, Ltd.

L2-Gain analysis of event-triggered networked control systems: a discontinuous Lyapunov functional approach

Abstract: SUMMARY In this paper, the problems of exponential stability and -gain analysis of event-triggered networked control systems (NCSs) with network-induced delays are studied. We first propose event-triggering conditions in the sensor side and controller side, respectively. Because the implementation of our event-triggering scheme only needs periodic supervision of the system state at the constant sampling instants, instead of being monitored continuously, it is expected that the scheme will improve the resource utilization. Taking the network-induced delays into account and using delay system approach, we constructed a unified model of NCSs with hybrid event-triggering schemes. On the basis of this model, sufficient conditions for the exponential stability and -gain analysis are developed in the form of LMIs by using a discontinuous Lyapunov–Krasovskii functional approach. Moreover, the corresponding results can be further extended to more general cases, where the system matrices of the considered plant contain parameter uncertainties, represented in either polytopic or norm-bounded frameworks. In addition, as a special case, we also present the exponential stability, -gain analysis, and the control feedback gain design of event-triggered NCSs without considering the effects of network-induced delays and event-triggering condition in the controller side. Finally, a simulation example is provided to illustrate the usefulness and effectiveness of the proposed hybrid event-triggering schemes.Copyright © 2012 John Wiley & Sons, Ltd.

Reaction wheel fault tolerant control for spacecraft attitude stabilization with finite‐time convergence

Abstract: SUMMARY The problem of fault tolerant attitude stabilization with finite-time convergence is investigated for spacecraft with redundant actuators. On the basis of the sliding mode control technique, a robust controller is derived with uncertain inertia parameters, actuator faults, and external disturbances explicitly addressed. It is shown that finite-time reachability into the small neighborhood of sliding surface, and faster time convergence of attitude orientation are achieved. To address actuator input constraints, an adaptive fault tolerant controller is further proposed. One feature of the proposed strategy is that the design of the fault tolerant control does not require any fault detection and isolation mechanism to detect, separate, and identify actuator faults. The attitude stabilization performance using the controller is evaluated through a numerical example. Copyright © 2012 John Wiley & Sons, Ltd.

Adaptive fuzzy decentralized output feedback control for stochastic nonlinear large‐scale systems using DSC technique

Abstract: SUMMARY In this paper, an adaptive fuzzy decentralized backstepping output feedback control approach is proposed for a class of uncertain large-scale stochastic nonlinear systems without the measurements of the states. The fuzzy logic systems are used to approximate the unknown nonlinear functions, and a fuzzy state observer is designed for estimating the unmeasured states. Using the designed fuzzy state observer, and by combining the adaptive backstepping technique with dynamic surface control technique, an adaptive fuzzy decentralized output feedback control approach is developed. It is shown that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are semi-globally uniformly ultimately bounded in probability, and the observer errors and the output of the system converge to a small neighborhood of the origin by choosing appropriate design parameters. A simulation example is provided to show the effectiveness of the proposed approaches. Copyright © 2011 John Wiley & Sons, Ltd.

Adaptive fault‐tolerant backstepping control against actuator gain faults and its applications to an aircraft longitudinal motion dynamics

Abstract: SUMMARY The problem of fault-tolerant control (FTC) for a class of uncertain nonlinear systems with actuator faults is discussed, and an observer-based FTC scheme is proposed Adaptive fuzzy observers are designed to provide a bank of residuals for fault detection and isolation Using a backstepping approach, we proposed a novel fault diagnosis algorithm, which removes the classical assumption that the time derivative of the output error should be known Further, an accommodation scheme is proposed to compensate for the effect of the fault, where it is not needed to know the bounds of the time derivative of the fault The proposed controller guarantees that all signals of the closed-loop system are semi-globally uniformly ultimately bounded and converge to a small neighborhood of the origin by appropriately choosing designed parameters In addition, a sufficient condition for the existence of an fault detection and isolation observer is derived using Lyapunov stability theory Finally, a numerical example and a practical aircraft longitudinal motion dynamics are used to demonstrate the effectiveness of the proposed FTC approach Copyright © 2013 John Wiley & Sons, Ltd

Robust adaptive neural network control for environmental boundary tracking by mobile robots

Abstract: SUMMARY The paper addresses the problem of environmental boundary tracking for the nonholonomic mobile robot with uncertain dynamics and external disturbances. To do environmental boundary tracking, a reference velocity is designed for the nonholonomic mobile robot. In this paper, a radial basis function neural network (NN) is used to approximate a nonlinear function containing the uncertain model terms and the elements of the Hessian matrix of the environmental concentration function. Then, the NN approximator is combined with a robust control to construct a robust adaptive NN control for the mobile robot to track the desired environment boundary. It is proved that the tracking error can be guaranteed to converge to zero in the ultimate. Simulation results are presented to illustrate the stability of the robust adaptive control. Copyright © 2011 John Wiley & Sons, Ltd.

Synchronization of coupled neural networks with random coupling strengths and mixed probabilistic time-varying delays

Abstract: SUMMARY This paper investigates the global asymptotic synchronization in an array of coupled neural networks with random coupling strengths, probabilistic interval time-varying coupling delays as well as unbounded distributed delays (mixed delays). Two important integral inequalities that include the Jensen's inequality as a special case are developed. On the basis of the developed inequalities, the properties of random variables and Lyapunov functional method, several delay-dependent sufficient synchronization criteria are derived for the considered model. The derived synchronization criteria are formulated by linear matrix inequalities (LMIs) and can be easily verified by using MATLAB LMI Toolbox. Some existing results are improved and extended by taking different values of parameters of the obtained results. Numerical simulations are finally given to demonstrate the effectiveness of the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.

Static output feedback control for positive linear continuous-time systems

Abstract: SUMMARY This paper studies the problem of designing the static output feedback controller for the positive linear continuous-time systems. On the basis of a system augmentation approach, a novel characterization on the stable condition of the closed-loop system is firstly established. Then, a necessary and sufficient condition is given to ensure the existence of the desired static output feedback controller, and an iterative linear matrix inequality algorithm is presented to compute the feedback gain matrix. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.

Finite‐time consensus tracking for harmonic oscillators using both state feedback control and output feedback control

Abstract: SUMMARY This paper investigates the distributed finite-time consensus-tracking problem for coupled harmonic oscillators The objective is to guarantee a team of followers modeled by harmonic oscillators to track a dynamic virtual leader in finite time Only a subset of followers can access the information of the virtual leader, and the interactions between followers are assumed to be local We consider two cases: (i) The followers can obtain the relative states between their neighbors and their own; and (ii) Only relative outputs between neighboring agents are available In the former case, a distributed consensus protocol is adopted to achieve the finite-time consensus tracking In the latter case, we propose a novel observer-based dynamic protocol to guarantee the consensus tracking in finite time Simulation examples are finally presented to verify the theoretical analysis Copyright © 2012 John Wiley & Sons, Ltd

Probability-dependent gain-scheduled control for discrete stochastic delayed systems with randomly occurring nonlinearities

Abstract: SUMMARY In this paper, the gain-scheduled control problem is addressed by using probability-dependent Lyapunov functions for a class of discrete-time stochastic delayed systems with randomly occurring sector nonlinearities. The sector nonlinearities are assumed to occur according to a time-varying Bernoulli distribution with measurable probability in real time. The multiplicative noises are given by means of a scalar Gaussian white noise sequence with known variances. The aim of the addressed gain-scheduled control problem is to design a controller with scheduled gains such that, for the admissible randomly occurring nonlinearities, time delays and external noise disturbances, the closed-loop system is exponentially mean-square stable. Note that the designed gain-scheduled controller is based on the measured time-varying probability and is therefore less conservative than the conventional controller with constant gains. It is shown that the time-varying controller gains can be derived in terms of the measurable probability by solving a convex optimization problem via the semi-definite programme method. A simulation example is exploited to illustrate the effectiveness of the proposed design procedures. Copyright © 2012 John Wiley & Sons, Ltd.

Exponential H ∞ filtering for singular systems with Markovian jump parameters

Abstract: SUMMARY This paper is concerned with the H ∞ filter design for continuous-time singular systems with Markovian jump parameters, whose system mode is transmitted through an unreliable network. In contrast to the traditionally mode-dependent or mode-independent filtering method, a new partially mode-dependent filter is established via using a mode-dependent Lyapunov function, where the stochastic property of mode available to a filter is considered. Sufficient conditions for the existence of H ∞ filter are obtained as strict linear matrix inequalities. Finally, numerical examples are used to show the effectiveness of the given theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.

Stabilization and regulator design for a one‐dimensional unstable wave equation with input harmonic disturbance

Abstract: SUMMARY This paper considers the parameter estimation and stabilization of a one-dimensional wave equation with instability suffered at one end and uncertainty of harmonic disturbance at the controlled end. The backstepping method for infinite-dimensional system is adopted in the design of the adaptive regulator. It is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the estimated parameter is shown to be convergent to the unknown parameter as time goes to infinity. Copyright © 2011 John Wiley & Sons, Ltd.

Global stabilization of a class of uncertain upper‐triangular systems under sampled‐data control

Abstract: SUMMARY This paper considers the problem of using a sampled-data controller to globally stabilize a class of uncertain upper-triangular systems First, we design a continuous-time controller by integrating the nested saturation and Lyapunov design methods together Then, the explicit formula for the maximum allowable sampling period is computed such that the discretized controller will guarantee global stability and robustness against uncertainties of the closed-loop system The bound of a proposed sampled-data controller can be adjusted to any small level to accommodate the actuation bound in practical implementation Copyright © 2012 John Wiley & Sons, Ltd