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The wave equation is studied under the effect of a persistent external perturbation. A dynamical distributed controller is suggested, based on an infinite-dimensional generalization of second-order sliding-mode control techniques, that provides the exponential attainment of a sufficiently smooth arbitrary reference of the state trajectory. The control system comprises of both feedforward and feedback parts, the latter being a discontinuous term directly connected to the plant input through a dynamical filter that augments the system state smoothing out the discontinuity of the feedback control loop. As a result, a continuous input is applied to the plant. A constructive Lyapunov-based proof of convergence of the proposed control algorithm is carried out and supporting numerical results are presented.

This paper considers a design method of nonlinear time-variant variable structure control system with uncertainty and parameter perturbation. First, a state-dependant differential Ricatti equation (SDDRE) is given, then a sliding sector is designed with the solution of this SDDRE. Using the sliding sector, a control law is designed such that the system state enters the sliding sector in finite time, and inside the sliding sector a Lyapunov function decreases respect to the system state, system can be stabilized globally and quadratically with the control low. An example is also given to show the validity of theoretical results.

Resonance phenomena of a harmonically excited system with mul-tiple potential well play an important role in nonlinear dynamics research.In this paper, we investigate the resonant behaviours of a discontinuous dynamical system with double well potential derived from the SD oscillator to gain better understanding of the transition of resonance mechanism. Firstly,the time dependent Hamiltonian is obtained for a Duffing type discontinuous system modelling snap-through buckling. This system comprises two subsystems connected at x = 0, for which the system is discontinuous. We constructa series of generating functions and canonical transformations to obtain the canonical form of the system to investigate the complex resonant behavioursof the system. Furthermore, we introduce a composed winding number to explore complex resonant phenomena. The formulation for resonant phenomena given in this paper generalizes the formulation of n Omega0 = m Omega used in the regular perturbation theory, where n and m are relative prime integers, Omega 0 and Omega are the natural frequency and external frequencies respectively. Understanding the resonant behaviour of the SD oscillator at the discontinuousphase enables us to further reveal the vibrational energy transfer mechanism between smooth and discontinuous nonlinear dynamical systems

We propose a variable structure control design method for a class of multivariable uncertain state-delayed systems which can be represented by polytopic models. In terms of LMIs, we derive a sufficient condition for the existence of a linear sliding surface guaranteeing the asymptotic stability of the sliding mode dynamics. We parameterize the sliding surface by using the solution of the LMI existence condition. We also give a switching feedback control strategy guaranteeing stable sliding mode. By using a numerical example, we show that our method supplements the existing results and it can be better than the existing results.

This technical note investigates the design of sliding mode control subject to packet losses. It is assumed that there exists a communication network in the feedback loop, and the dropout of data packet may occur. First, an estimation method is proposed to compensate the packet dropout. Subsequently, a discrete-time integral sliding surface involving dropout probability is introduced and a sliding mode controller is designed. By using the stochastic Lyapunov method, the state trajectories are shown to enter into (in mean square) a neighborhood of the specified sliding surface. Meanwhile, the stability of sliding mode dynamics is also ensured. Finally, numerical simulation example is provided.

This study presents robust control architecture in the sense of variable structure control via a backstepping design. By using systematic backstepping design techniques, closed-loop behavior of an n-order nonlinear system can be transformed into a stability and convergence problem of a fast switched 2nd order system. There are two main parts contained within the proposed control algorithm; one is a nominal control effort generated according to the Lyapunov stability criterion during recursive backstepping processes, and the other belongs to a smooth robust control law designed to eliminate the effects of unknown lumped perturbations. Finally, a Genesio system is used as an illustrated example to demonstrate the robustness of the control algorithm. The feasibility and properties of the proposed method are given by numerical simulations.

S PIN is an autorotational state of aircraft at high angles of attack characterized by large rotation rate in yaw when compared with rates in roll and pitch. In a typical spin, an aircraft rotates about its center of gravity and an axis perpendicular to Earth descending vertically at high speed following a downward corkscrew path [1]. It is one of themost dangerous phenomena encountered bymany of the modern fighter aircraft required to fly in high-angle-of-attack flight regimes. Improper functioning of aerodynamic control surfaces during spin makes it difficult for the pilot to control the aircraft, leading to many a problem, such as spatial disorientation and uncontrolled motion, resulting in fatal accidents and subsequent loss of aircraft. Spin states for many high-angle-of-attack aircraft models have been computed by using bifurcation analysis and continuation technique methodology. Bifurcation analysis results provide the onset points of bifurcations or the critical values of control surface deflections that may inadvertently or voluntarily land an aircraft into spin. Standard control inputs using proper deflection of rudder to control yaw rate with simultaneous application of elevator to reduce the angle of attack have been recommended for spin recovery [1]. Design of new-generation fighter aircraft with new configurations, however, cannot rely on standard piloting strategies. Instead, a uniform approach to design aircraft-model-based recovery strategies is called for. An introduction to spin problem and needs to design nonlinear controllers to recover aircraft from spin has been reported recently in [2]. Identifying spin and level-trim states from a bifurcation analysis of a nonlinear aircraft model, Raghavendra et al. [2] developed a nonlinear dynamic inversion (NDI)-techniquebased spin-recovery controller. NDI-technique-based methods have emerged as popular control design techniques in aircraft flight dynamics. Controllers based on the NDI techniques, however, come with certain associated disadvantages, such as separation of dynamics based on timescales, resulting in complicated control architecture, lack of robustness due to external uncertainties and unmodeled plant dynamics, and inability to achieve control saturation limits [3]. In this Note, we present a sliding-mode (SM) controller based on variable-structure control technique for spin recovery of aircraft. Variable-structure-technique-based controllers have been found to be robust in the presence of system uncertainties and external disturbances, and, usually result in simpler control algorithms [4]. Controller presented in this Note uses results from a bifurcation analysis of the high-angle-of-attack research vehicle (HARV) model of F-18 available in literature [2]. This Note is organized as follows. In Sec. II, a brief description of the aircraft model with reference states for SM controller design is presented. In Sec. III, SM control design technique is explained. Results and discussions are presented in Sec. IV and conclusions follow thereafter in Sec. V.

The major task for any monitoring system is to detect upcoming faults as early as possible. Rotor failures are responsible for a large percentage of total induction motor failures. Thus, a new nonintrusive and in-service approach has been proposed in this paper to detect one broken rotor bar in induction motor using only input quantities information. The method is not affected by the type of load and other asymmetries and it is capable of providing reliable fault diagnose without maintenance specialist intervention. The proposed sensorless technique is based on two real time state space discrete models that are used to estimate the flux of both the stator and the rotor and analyzes the differences obtained in torque when the two models are employed. One of the observers was designed for rotor resistance disturbance rejection. Firstly, simulation results have been conducted and sensitivity and robustness also have been checked for the proposed method. Secondly, an experimental setup has been constructed to implement the new technique in on-line model. The results obtained validate the proposed model and show the reliability of the new method.

This paper proposes an actuator fault-tolerant control scheme designed for a class of nonlinear systems, and then applied to an underwater Remotely Operated Vehicle (ROV) used for inspection purposes. Fault detection is achieved using a Variable Structure System (VSS) observer, while a sliding-mode based approach has been used both for fault isolation and ROV control, after the application of an input decoupling nonlinear state transformation to the ROV model. Finally, control reconfiguration is performed exploiting the inherent redundancy of actuators. An extensive simulation study has been also performed, supporting the effectiveness of the proposed approach.

The design of variable structure system inputs which are constrained by saturation is studied. For a LTI system which satisfies some conditions, it is shown that appropriate bounded controllers guarantee the system’s global stability and maximize the sliding mode domain on the switching surfaces. Stability conditions of variable structure systems with constrained inputs are relaxed, and the stability of the closed-loop system is guaranteed by using passivity theory of linear passive systems. Moreover, nonlinear sliding surfaces are discussed for variable structure controller design, and a novel nonlinear switching surface is proposed. Finally, the proposed methods are applied to a 2nd order LTI system to show their usefulness.

There exists a dynamic mapping relationship between the sliding variable and the output variable in a linear, time-invariant (LTI), single-input-single-output (SISO) system. This relationship is invariant of the matched disturbances, which lies in the control space. In this note, a continuous-time dynamic inverse compensator is proposed to reconstruct the sliding variable from the system output. This inverse compensator is stable and causal provided that the original SISO system is minimum phase and that the input-output relative degree is less than or equal to one. In case the system relative degree is greater than one, a digital approximation for the inverse compensator is introduced to track the sliding variable in discrete-time to the accuracy of O(T), where T is the sampling period. Thus the discrete-time variable structure control (VSC) law can be implemented without full state accessibility.

In this article the control of mechanical systems with elastic joints is addressed. This kind of system is characterised by a high relative degree. The mechanical control input must be designed to be continuous. Different reduced order observers are introduced and conditions are found to guarantee the exponential convergence of the observation errors. The error stability is ensured provided a certain matrix inequality has a solution. The proposed second-order sliding mode point-to-point controller based on the observed state is proven to stabilise the mechanical system with elastic joints about the desired reference position.

This paper presents a robust output feedback controller based on sliding modes and indirect adaptive control applied to the angular shaft speed control of a three-phase induction motor. This approach has remarkable stability properties and offers a straightforward and intuitive tuning of its parameters. Moreover, a simplified version of the algorithm, named compact indirect variable structure model reference adaptive control (CIVS-MRAC), provides not only an easy design procedure, but also a computational implementation of low complexity, thus motivating its application in industrial environments.

This study presents an adaptive sliding mode controller for a micro-electro-mechanical systems (MEMS) z-axis vibratory gyroscope. The adaptive model reference state tracking control with sliding mode controller is developed to estimate the angular velocity, the damping and stiffness coefficients of MEMS vibratory gyroscope in real time. An adaptive controller that can estimate the unknown upper bound of the parameter uncertainties and external disturbance is derived and the stability of the closed-loop system is established. The adaptive sliding mode controller is designed so that the trajectory of the driving axes can track the state of the reference vibration model. The effectiveness of the proposed scheme is demonstrated through numerical simulations on an MEMS vibratory gyroscope.

This paper is concerned with the state estimation and sliding-mode control problems for continuous-time Markovian jump singular systems with unmeasured states. Firstly, a new necessary and sufficient condition is proposed in terms of strict linear matrix inequality (LMI), which guarantees the stochastic admissibility of the unforced Markovian jump singular system. Then, the sliding-mode control problem is considered by designing an integral sliding surface function. An observer is designed to estimate the system states, and a sliding-mode control scheme is synthesized for the reaching motion based on the state estimates. It is shown that the sliding mode in the estimation space can be attained in a finite time. Some conditions for the stochastic admissibility of the overall closed-loop system are derived. Finally, a numerical example is provided to illustrate the effectiveness of the proposed theory.

This paper investigates the robust H ∞ control problem for a class of uncertain switched delay systems with parameter uncertainties, unknown nonlinear perturbations, and external disturbance. Based on the multiple Lyapunov functions method, a sufficient condition for the solvability of the robust H ∞ control problem is derived by employing a hysteresis switching law and variable structure controllers. When the upper bounds of the nonlinear perturbations are unknown, an adaptive variable structure control strategy is developed. The use of the adaptive technique is to adapt the unknown upper bounds of the nonlinear disturbances so that the objective of asymptotic stabilization with an H ∞-norm bound is achieved under the hysteresis switching law. A numerical example illustrates the effectiveness of the proposed design methods.

A new state-of-health (SOH) estimation method for lithium batteries that uses a dual-sliding-mode observer is presented. The capacity fade and resistance deterioration were estimated by the dual-sliding-mode observer. The dual-sliding-mode observer consists of a fast-paced time-varying observer and a slow-paced time-varying observer. The fast-paced time-varying observer estimates parameters, such as the state of charge and terminal voltage, and polarization effects. The slow-paced time-varying observer estimates the SOH in terms of the capacity fade and resistance deterioration. The convergence of the proposed observer was proved by the Lyapunov equation. The structure of the proposed system is simple and easy to implement and shows robust control properties against modeling errors and temperature variations. The test results show that the proposed observer system has superior tracking performance than previous ampere-counting method.

The chaotic librational motion of a tether system in elliptic orbit has been studied and controlled by delayed feedback control up to date. Contrary to the past literatures that addressed the periodic stability of tether systems, this paper studies the local stability of in-plane/out-of-plane motion of electrodynamic tether (EDT) systems in elliptic orbits with respect to the true anomaly. From the viewpoint of the local stability of tether systems, in this paper, provided that the electric current is used only for controlling the librational motion, a new simple delayed feedback control, named “switching delayed-feedback control (SDFC)” is proposed, which combines a delayed-feedback control (DFC) and variable structure system (VSS) concept. The effectiveness of the proposed method to stabilize the librational in-plane/out-of-plane periodic motion of an electrodynamic tether system in an inclined elliptic orbit with high eccentricity was demonstrated, and compared with that of the traditional delayed-feedback controllers by numerical simulations.

Actual endoscopes and boroscopes, widely used in industry and in minimal invasive surgery, have considerable limitations, mainly due to their low number of degrees of freedom and their manual operation. Two different solutions for the electrical actuation of articulated endoscopes are presented in this paper. The technical constraints for this kind of application are very limited space for the actuators and high performance in terms of torque and angular reach. The first solution classically consists in a 2 d.o.f. structure steered by two pairs of antagonist shape memory alloy (SMA) wires. The sizing and preload determination for those actuators follow an original analytical approach. The second solution consists in a multi-d.o.f. structure actuated by thin NiTi springs mounted in an antagonist configuration and directly integrated in the structure of the endoscope. The geometry of the springs is obtained by optimization through genetic algorithms and finite elements method. Experiments show good adequacy between real behaviour and numerical model and also validate the approach. This study is also enhanced by a control scheme specifically developed for SMA actuators in an antagonist configuration. It is based on a first order sliding mode scheme, which has the advantage of a great structural simplicity. The experimental results show that this solution can reach a good compromise between the dynamic behaviour of the actuator, its energy consumption and the structural lifetime of the endoscope.

It is shown that the higher the order r of the homogeneous sliding mode is, the less sensitive it is to the presence of unaccounted-for fast stable actuators. With ? being an infinitesimal actuator time constant, the sliding variable magnitude is proved to be proportional to ?r . The system chattering is shown to be not amplified in the presence of actuators.

A higher order sliding mode observer is proposed for asymptotic identification of the full state vector and the vector of unknown inputs for MIMO nonlinear causal systems with unstable internal dynamics. The problem is addressed via consistent application of exact higher order sliding mode differentiators in conjunction with the method of stable system center. A numerical example illustrates the performance of the proposed algorithm.

The main objective in this study is the vibrational control of a one-link flexible arm system. A variable structure system (VSS) nonlinear observer has been proposed in order to reduce the oscillation when controlling the angle-of the flexible arm. The parameters of the nonlinear observer are optimized using a modified version of the simultaneous perturbation stochastic approximation (SPSA) algorithm. The SPSA algorithm is especially useful when the number of parameters to be adjusted is large, and makes it possible to estimate them simultaneously. For the vibration and position control, a model reference sliding-mode control (MR-SMC) has been proposed. The MR-SMC parameters are also optimized using a modified version of the SPSA algorithm. The simulations show that vibrational control of a one-link flexible arm system can be achieved more efficiently using our method. Therefore, by applying the MR-SMC method to a nonlinear observer, we can improve the performance in this kind of model using our proposed SPSA algorithm, and we can determine the control parameters very easy and efficiently.

Based on the variable structure system theory, an output feedback fuzzy control design method is developed for a class of uncertain nonlinear systems that can be represented by a T-S (Takagi-Sugeno) fuzzy model. The uncertain fuzzy systems under consideration may have mismatched uncertainties in the state matrix. As the local controller a compensator-based sliding mode output feedback controller with a nonlinear discontinuous term is used. In terms of linear matrix inequalities (LMIs), a sufficient condition is derived for the existence of the fuzzy controller guaranteeing a stable sliding motion. It is shown that the compensator gain and the switching surface parameter matrix can be characterized in terms of the solution of the LMI existence condition. Finally, an LMI-based design algorithm and numerical design examples are given.

This paper considers an optimization problem in the setting of switched-mode hybrid dynamical systems, where the control variable (independent variable) consists of the mode-switching times, and the performance criterion is a cost functional defined on the system's state trajectory. The system is deterministic, nonlinear, and autonomous, and its state variable cannot be measured and hence it has to be estimated. We propose an on-line, Newton-like optimization algorithm that recomputes the control variable by attempting to optimize the cost-to-go at equally-spaced epochs. The main result concerns the algorithm's convergence rate, which can vary from sublinear to quadratic depending on its computing rate and the state estimation error.

State estimation for uncertain systems affected by external noises is an important problem in control theory. This paper deals with a state observation problem when the dynamic model of a plant contains uncertainties or it is completely unknown. Differential neural network (NN) approach is applied in this uninformative situation but with activation functions described by wavelets. A new learning law, containing an adaptive adjustment rate, is suggested to imply the stability condition for the free parameters of the observer. Nominal weights are adjusted during the preliminary training process using the least mean square (LMS) method. Lyapunov theory is used to obtain the upper bounds for the weights dynamics as well as for the mean squared estimation error. Two numeric examples illustrate this approach: first, a nonlinear electric system, governed by the Chua's equation and second the Lorentz oscillator. Both systems are assumed to be affected by external perturbations and their parameters are unknown.

This study presents a dynamic post-filtering approach to a single-input-single-output sampled-data variable structure output feedback control problem. Given a sliding surface, Σ(x)=0 (x being the system state vector), designed through standard procedures with the assumption of full state accessibility, a dynamic filter, denoted post-filter, is employed to synthesise Σ(x) by means of output variable alone. In contrast with a state observer, the post-filter is invariant of the matched disturbances; and it requires only partial information of the system parameters. Thus, the robustness property of the variable structure system is retained. Let T be the sampling period. A sampled-data realisation of the post-filter is done by zero-order hold. Depending on the system relative degree, either Σk or Σk−1 can be obtained for implementing the variable structure control law. A discrete-time switching-type control law, which is originally designed to satisfy the continuous-time reaching condition, will lead to a quasi-sliding motion, forming a boundary layer of thickness O(T) about the sliding surface. The stability of the post-filter is ensured if the system output is in minimum phase and that the sliding mode dynamics are chosen properly.

A variable structure control (VSC) algorithm is proposed with a nonlinear time-varying (NTV) sliding sector for a class of nonlinear time-variant systems represented in a state-dependent linear time-variant form. The NTV sliding sector, a subset of the state space is designed by the state-dependent differential Riccati equation (SDDRE). A VSC law is designed such that the system enters the sliding sector in finite time and a Lyapunov function candidate decreases inside the NTV sliding sector. An NTV control and a sliding mode control algorithms are proposed based on the solution of the SDDRE as well.

In this paper, we discuss the classical problem of attitude stabilization for a rigid spacecraft with external disturbances. Almost all the control laws for attitude stabilization problem in the literature can only guarantee at best exponential convergence of the attitude with infinite settling time. We employ finite-time control techniques including terminal sliding mode approach and continuous finite-time control approach to develop feedback control laws for stabilizing spacecraft attitude in terms of Rodrigues parameters. The attitude of the rigid spacecraft will be stabilized to the origin in finite time in the absence of disturbances and a small region of the origin in the presence of disturbances. Rigorous proof is given. Numerical simulation results show the effectiveness of the method.

The multi-linear variable separation approach is reviewed in this article. The method has been recently established and successfully solved a large number of nonlinear systems. One of the most exciting findings is that the basic multi-linear variable separation solution can be expressed by a universal formula including two (1+1)-dimensional functions, and at least one is arbitrary for integrable systems. Furthermore, the method has been extended in two different ways so as to enroll more low dimensional functions in the solution.

The problem of designing robust tracking controls for a large class of robots actuated by brushed direct current motors is addressed. This class of electrically driven robots can be perturbed by plant uncertainties, unmodelled time-varying perturbations and external disturbances. Adaptive neural network (or fuzzy) systems are employed to approximate the behaviours of uncertain dynamics, and the variable structure system control algorithm is used to efficiently eliminate the effect of both the approximation error and the time-varying perturbation. Consequently, a hybrid robust adaptive dynamic feedback tracking controller is developed such that all the states and signals of the closed-loop system are bounded and the asymptotic bound on the trajectory tracking error can be made arbitrarily small. Finally, simulation examples are presented to demonstrate the effectiveness and the tracking performance of the proposed control algorithm.

In this paper, a novel “composite reaching law” was explained in details: 1) the equation of discrete motion for a control system; 2) the design of discrete-time variable structure control. In addition, the model of a three-storey shear-type building structure was used to verify the effectiveness of the discrete variable structure control method. The results of numerical example analysis of the model show that the control law can effectively reduce the peak value of seismic response of the building structure and the chattering effect of the control system.

This paper presents an NMPC of a supermarket refrigeration system. This is a hybrid process involving switched nonlinear dynamics and discrete events, on/off manipulated variables (valves and compressors), continuous controlled variables (goods temperatures) and, finally, several operation constraints. The hybrid controller is based on a parameterization of the on/off control signals in terms of time of occurrence of events instead of using directly binary values; in this way, the optimization problem can be re-formulated as an NLP problem. A rigorous model of a real supermarket refrigeration system is presented, as well as results of the hybrid controller operating on it.

The component mode synthesis (CMS) method is widely used to establish reduced dynamic models of complex structures so that iterative problems such as flutter analyses can be efficiently analyzed with reasonable cost and time. In the present study, a structural coupling method is developed for the dynamic analysis of a nonlinear structure consisting of substructures connected by nonlinear interfaces such as nonlinear hinge joints or sliding mode conditions. In order to verify the coupling method extended to consider the hinge joints, a numerical plate model consisting of two substructures and torsional springs is synthesized, and its modal parameters are compared with analysis data. The extended coupling method is further improved to consider the sliding mode condition. The improved coupling method is applied to a three-substructure-model with nonlinearity of sliding lines between the substructures. Finally, using the proposed coupling method, a dynamic model of a tilting structure consisting of two substructures with sliding line conditions is synthesized, and its dynamic characteristics are investigated. The analysis results show that the improved coupling method is effectively applicable to the dynamic analysis of a nonlinear structure with the sliding mode condition.

A new slip model derived by molecular dynamics has been used to investigate the ultra-thin gas-lubricated slider bearings beneath the three bushings of an electrostatic micromotor in micro-electro-mechanical systems (MEMS). Modified Reynolds equation is proposed based on the modified slip model. Analytical solutions for flow rate, pressure distribution, load carrying capacity and streamwise location using the modified Reynolds equation are obtained and compared with the results gained from those in the literature. It demonstrates that the new second-order slip model is of greater accuracy than that predicted by the first-order, second-order slip models and MMGL model and produces a good approximation to variable hard sphere (VHS) and variable soft sphere (VSS) models, which agree well with the solution obtained from the linearized Boltzmann equation. It is indicated that the slip effect reduces the pressure distribution and load carrying capacity, and shifts the streamwise location of the load carrying capacity, which should not be ignored to study the step-shaped slider bearings in micromotors for MEMS devices.

This paper treats the question of attitude maneuver control and elastic mode stabilization of a flexible spacecraft based on adaptive sliding mode theory and active vibration control technique using piezoelectric materials. More precisely, a modified positive position feedback (PPF) scheme is developed to design the PPF compensator gains in a more systematical way to stabilize the vibration modes in the inner loop, in which a cost function is introduced to be minimized by the feedback gains subject to the stability criterion at the same time. Based on adaptive sliding mode control theory, a discontinuous attitude control law is derived to achieve the desired position of the spacecraft, taking explicitly into account the mismatched perturbation and actuator constraints. In the attitude control law, an adaptive mechanism is also embedded such that the unknown upper bound of perturbation is automatically adapted. Once the controlled attitude control system reaches the switching hyperplane, the state variables can be driven into a small bounded region. An additional attractive feature of the attitude control method is that the structure of the controller is independent of the elastic mode dynamics of the spacecraft, since in practice the measurement of flexible modes is not easy or feasible. The proposed control strategy has been implemented on a flexible spacecraft. Both analytical and numerical results are presented to show the theoretical and practical merit of this approach.

A set of novel nonlinear variable structure excitation and steam-valving controllers are proposed in this paper. On the basis of the classical dynamic equations of a generator, excitation control and steam valving control are simultaneously considered. Design of these controllers combines the differential geometry theory with the variable structure controlling theory. The mathematical model in the form of “an affine nonlinear system” is set up for the control design of a large-scale power plant. The dynamic performance of the nonlinear variable structure controllers proposed for a single machine connected to an infinite bus power system is simulated. Simulation results show that the nonlinear variable structure excitation and steam-valving controllers give satisfactory dynamic performance and good robustness.

This paper formulates and solves a variable structure control problem for the state control of quantum systems. Following the concept of wavefunction controllability, we first present a specific quantum system which is wavefunction uncontrollable under the action of any one of two controllers with different fixed structures. However, we prove that the system is wavefunction controllable if the control input is allowed to switch between the two controllers. An extension and the potential applications of the proposed control strategy are briefly discussed and the general steps of variable structure control design for this class of quantum systems are also presented. The variable structure control scheme provides an alternative quantum engineering approach for the manipulation of quantum systems.

Although discontinuous, non-smooth or switched dynamical systems have been broadly studied, there still exist dynamical behaviours which are not completely understood. In particular, those related to transitions between state-space regions and to crossings through discontinuity surfaces. The problem of specifying system transitions becomes crucial when multiple discontinuity surfaces or switching elements are present. In this case, a great deal of care has to be taken in order to simulate the system and to verify its properties. Obtaining a computational model appears to be an elegant way for the specification of the transitions and event-triggered phenomena involved in the dynamical behaviour of discontinuous systems. This is the goal of this paper. A class of discontinuous dynamical systems including control inputs and outputs is reinterpreted within the hybrid-automata framework, and what is referred as to discontinuous-dynamical-systemhybrid automaton is proposed. An example is used, which corresponds to a 2-degrees-of-freedom electromechanical system including discontinuous friction and sliding-mode control. This system exhibits several discontinuity surfaces which give rise to different types of transitions and dynamical behaviours. New computational-kind phenomena may arise when the system is reinterpreted as a hybrid dynamical system. Nevertheless, this paper only deals with the modelling and specification of the system.

This paper treats the question of simultaneous robust attitude control and vibration suppression of orbiting spacecraft with flexible appendages. The spacecraft consists of a rigid body and two flexible appendages and the finite dimensional representation of the flexible spacecraft is assumed to be of arbitrary order. Robust nonlinear variable structure control (VSC) strategy integrated with input shaping technique is concerned for the pitch angle control and elastic vibration suppression under actuator saturation limit. More specially, the input shaper is implemented outside of the feedback loop, which is designed for the reference model and achieves the exact elimination of residual vibration; while for the feedback loop, the variable structure controller is designed to make the closed-loop system behave like the reference model with the input shaper in the presence of parametric uncertainty, external disturbances and actuator saturation. To prevent the presence of input saturation from destroying the system performance, a saturation compensator is designed as well for the variable structure attitude control system. For the synthesis of the attitude controller, only the pitch angel and its derivative are used. Simulation results are presented which show that in the closed loop, pitch angel control and elastic mode stabilization are accomplished in spite of uncertainty and external disturbance.

Non-smooth characteristics are, in general, the source of difficulties for the modeling and simulation of natural systems. These characteristics are usually related to either the friction phenomenon or the discontinuous behavior as intermittent contacts. This article develops an experimental investigation concerning non-smooth systems with discontinuous support. An experimental apparatus is developed in order to analyze the nonlinear dynamics of a single-degree of freedom system with discontinuous support. The apparatus is composed by an oscillator constructed by a car, free to move over a rail, connected to an excitation system. The discontinuous support is constructed considering mass–spring systems separated by a gap to the car position. This apparatus is instrumented to obtain all the system state variables. System dynamical behavior shows a rich response, presenting dynamical jumps, bifurcations and chaos. Different configurations of the experimental set up are treated in order to evaluate the influence of the internal impact within the car and also support characteristics in the system dynamics.

We consider the sliding-mode output feedback controller (SMOFC) design problem for a class of uncertain multivariable systems. We first design a stabilizing SMOFC for matched uncertain systems. Using linear matrix inequalities (LMIs), we derive a necessary and sufficient condition for the existence of a linear sliding surface depending on outputs and compensator states. Using the solution of the LMI existence condition, we characterize the gain matrices. We give the nonlinear switching feedback gain guaranteeing the reachability condition. Second, we give an LMI-based design method to combine various useful performance criteria which can be used to guarantee a desired robust performance in spite of mismatched uncertainties. The performance criteria include alpha-stability, LQ performance, H 2/H infin performance, and peak-to-peak gain bound. In particular, we show that by including H infin performance constraints, we can easily solve the SMOFC design problem for challenging system models to which the previous methods are not easily applicable. Finally, we give a numerical design example showing that our method can be successfully applied to the problem of designing reduced-order SMOFCs for uncertain time-delay systems or mismatched uncertain systems.

In this paper we present a stability theory for discontinuous dynamical systems (DDS): continuous-time systems whose motions are not necessarily continuous with respect to time. We show that this t...

The aim of this paper is to spread the sliding-mode control design technique on the new class of the ac drives: an induction motor fed by a three-level voltage-source inverter (3LVSI). A comprehensive investigation of possible drive ldquovariable structuresrdquo was carried out. Based on this analysis, an original two-step design procedure allows the use of the ldquoclassicalrdquo result of sliding-mode theory for the real variable structure system with more than 2m variable structures ( m is the control space order). Based on this design procedure, the original control algorithm that includes a choice condition for 3LVSI input dc voltage and a switch table for the 3LVSI semiconductor switches are designed. The performance of the considered control structure has been examined by simulation.

A variable structure control system for output tracking in the presence of a class of mismatched uncertainty is studied in this paper. For the mismatched uncertainty, we use an unknown input observer to estimate the states and then apply the VSS control methodology to attain asymptotic output tracking for an arbitrary desired trajectory. The robustness of the VSS control system in the presence of mismatched uncertainty is analyzed using the quadratic stability approach.

A robust speed controller for an induction motor drive that achieves quantitative control and considers the effects of nonlinearities is presented in this paper. First, a two-degree-of-freedom controller (2DOFC) is designed according to the estimated model for the nominal case and given speed specifications. Then, the control characteristics and limitations of the conventional robust controller are investigated. Accordingly, a modified robust speed controller based on disturbance cancellation is developed, wherein tuning of disturbance cancellation is performed via a variable structure system (VSS) controller. The estimated model mismatched error and its derivative are used as the driven inputs of the VSS controller. Greater robustness and flexibility in achieving a compromise between control effort and control performance can be achieved using the proposed controller. The effectiveness of the proposed controller is demonstrated by simulation and measured results.

SUMMARYSliding mode observer is a variable structure system where the dynamics of a nonlinear system is altered via application of a high-frequency switching control. This paper presents a non-linear sliding mode observer for wheel linear slip and slip angle estimation of a single wheel based on its kinematic model and velocity measurements with added noise to simulate actual on-board sensor measurements. Lyapunov stability theory is used to establish the stability conditions for the observer. It is shown that the observer will converge in a finite time, provided the observer gains satisfy constraints based on a stability analysis. To validate the observer, linear and two-dimensional (2D) test rigs are specially designed. The sliding mode observer is tested under a variety of conditions and it is shown that the sliding mode observer can estimate wheel slip and slip angle to a high accuracy. It is also shown that the sliding mode observer can accurately predict wheel slip and slip angle in the presence of noise, by testing the performance of the sliding mode observer after adding white noise to the measurements. An extended Kalman filter is also developed for comparison purposes. The sliding mode observer is better in terms of prediction accuracy.

In this paper, the variable structure control problem is presented for Takagi-Sugeno fuzzy systems with uncertainties and external disturbances. The sliding surfaces for the T-S fuzzy system are proposed by using a Lyapunov function and a fuzzy Lyapunov function, respectively. And we design the variable structure controllers such that the global T-S fuzzy system confined on the sliding surfaces is asymptotically stable. Two examples are given to illustrate the effectiveness of our proposed methods.

This paper describes a variable structure control for a class of uncertain fractional-order systems with state-delay. A sliding surface based on a fractional integration is proposed. Then a robust control law is derived to ensure the existence of sliding mode and to overcome the uncertainty of the closed loop system. A condition on the uncertainties to ensure the finite time stability of the system on sliding mode is derived.