Dissertations / Theses on the topic 'Optimal discrete-time sliding mode'

Research into observer design for non-linear discrete-time systems has produced many design methods. There is no general design method however and that provides the motivation to search for a new simple and realizable design method. In this thesis, an observer for non-linear discrete-time systems is designed using the sliding mode technique. The equation of the observer error is split into two parts; the first part being a linearized model of the system and the second part an uncertain vector. The sliding mode technique is introduced to eliminate the uncertainty caused by the uncertain vector in the observer error equation. By choosing the sliding surface and the boundary layer, the observer error is attracted to the sliding surface and stays within the sliding manifold. Therefore, the observer error converges to zero. The proposed technique is applied to two cases of observers for nonlinear discrete-time systems. The second case is chosen to be a particular practical system, namely the non-linear discrete-time ball and beam system. The simulations show that the sliding mode technique guarantees the convergence of the observer error for both systems.

Sliding mode control is a well-known approach to the problem of the control of uncertain systems, since it is invariant to a class of parameter variations. Well-established investigations have shown that the sliding mode controller/ observer is a good approach from the point of view of robustness, implementation, numerical stability, applicability, ease of design tuning and overall evaluation. In the sliding mode control approach, the controller and/ or observer is designed so that the state trajectory converges to a surface named the sliding surface. It is desired to design the sliding surface so that the system stability is achieved. Many new methods and design techniques for the sliding controller/ observer are presented in this thesis. LQ frequency shaping sliding mode is a way of designing the sliding surface and control. By using this method, corresponding to the weighting functions in conventional quadratic performance, a compensator can be designed. The intention of observer design is to find an estimate for the state and, if the input is unknown, estimate a suitable input. Using the sliding control input form, a suitable estimated input can be obtained. The significance of the observer design method in this thesis is that a discontinuous observer for full order systems, including disturbance inputs, is designed. The system may not be ideally in the sliding mode and the uncertainty may not satisfy the matching condition. In discrete-time systems instead of having a hyperplane as in the continuous case, there is a countable set of points comprising a so-called lattice; and the surface on which these sliding points lie is named the latticewise hyperplane. Control and observer design using the discrete-time sliding mode, the robust stability of the sliding mode dynamics and the problem of stabilization of discrete-time systems are also studied. The sliding mode control of time-delay systems is also considered. Time-delay sliding system stability is studied for the cases of full information about the delay and also lack of information. The sliding surface is delay-independent as for the traditional sliding surface, and the reaching condition is achieved by applying conventional discontinuous control. A well-known method of control design is to specify eigenvalues in a region in the left-hand half-plane, and to design the gain feedback matrix to yield these eigenvalues. This method can also be used to design the sliding gain matrix. The regions considered in this thesis are, a sector, an infinite vertical strip, a disc, a hyperbola and the intersection ii of two sectors. Previous erroneous results are rectified and new theory developed. The complex Riccati equation, positivity of a complex matrix and the control of complex systems are significant problems which arise in many control theory problems and are discussed in this thesis.

This thesis concerns the development of output-based sliding mode control schemes for discrete time, linear time invariant systems. Unlike most of the work given in the literature in this area, the work is concemed with the development of static output feedback based discrete time sliding mode control schemes for non-minimum phase, non-square systems with arbitrary relative degree and which include unmatched uncertainties. The key concept of extended outputs in discrete time will be introduced. It will be shown that by identifying a minimal set of present and past outputs an augmented system can be obtained which permits the design of a sliding manifold based upon output information only, which renders the sliding manifold stable. Any transmission zeros of the augmented plant will also be shown to be among the transmission zeros of the original plant. It will also be shown that if the extended outputs chosen span the state zero directions of an invariant zero of the system, then the invariant zero disappears from the augmented system. Linear matrix inequalities are then used for sliding surface design. For non-minimum phase, non-square systems with unmatched uncertainties, it will be shown that in some cases the extended outputs can be chosen such that the effect of the disturbance on the sliding surface can be nullified. If this is possible, a procedure to obtain a Lyapunov matrix, which simultaneously satisfies a Riccati inequality and a structural constraint and which is used to formulate the control law that satisfies the reachability condition has been given. For the general case, where the sliding surface is a function of the disturbance, a control law will be chosen such that the effect of the disturbance on the augmented outputs and the sliding manifold will be minimized. Another key contribution of this work is the use of extended outputs for reconfigurable control under sensor loss. The reconfigurable control methodology presented in this work is in discrete time and is a static output feedback based control scheme, unlike most of the reconfigurable control schemes given in the literature which require an estimator and which are continuous time based schemes. Suitable examples, which include multiple sensor failures and a benchmark problem taken from the literature which represents the lateral dynamics of the F-14 aircraft, have been chosen to show the effectiveness of the proposed control design methodologies. - L Abstract T his thesis concerns the development of out put-based sliding mode control schemes for discrete time, linear time invariant systems. Unlike most of the work given in the literature in this area, the work is concerned with the development of static output feedback based discrete time sliding mode control schemes for non-minimum phase, non-square systems with arbitrary relative degree and which include unmatched uncertainties. The' key concept of extended outputs in discrete time will be introduced. It will be shown that by identifying a minimal set of present and past outputs an augmented system can be obtained which permits the design of a sliding manifold based upon output information only, which renders the sliding manifold stable. Any transmission zeros of the augmented plant will also be sho,wn to be among the transmission zeros of the original plant. It will also be shown that- if the extended outputs chosen span the state zero directions of an invariant zero of the system, then the invariant zero disappears from the augment.ed system. Linear matrix inequalities are then used for sliding surface design. For nonminimurn phase, non-square systems with unmatched uncertainties, it will be shown that in some cases the extended outputs can be chosen such ,that the effect of the disturbance on the sliding surface can be nullified. If this is possible, a procedure to obtain a Lyapunov matrix, which simultaneously satisfies a Riccati inequality and a structural constraint and which is used to formulate the control law t hat satisfies the reachability condit ion has been given. For the general case, where the sliding surface is a function of the disturbance, a control law will be chosen such that the effect of the disturbance on the augmented outputs and the sliding manifold will be minimized. Another key contribution of t his work is the use of extended outputs for reconfigurable control under sensor loss. The reconfigura~le control methodology presented in this work is in discrete time and is a static output feedback based control scheme, unlike most of t he reconfigurable control schemes given in the literature which require an estimator and which are continuous time based schemes. Suitable examples, which include multiple sensor failures and a benchmark problem taken from the literature which represents the lateral dynamics of the F-14 aircraft have been chosen to show the effectiveness of the proposed control design methodologies.

Este trabalho aborda o problema de controle ótimo por modos deslizantes via função penalidade para sistemas de tempo discreto. Para resolver este problema será desenvolvido uma estrutura matricial alternativa baseada no problema de mínimos quadrados ponderados e funções penalidade. A partir desta nova formulação é possível obter a lei de controle ótimo por modos deslizantes, as equações de Riccati e a matriz do ganho de realimentação através desta estrutura matricial alternativa. A motivação para propormos essa nova abordagem é mostrar que é possível obter uma solução alternativa para o problema clássico de controle ótimo por modos deslizantes.
This work introduces a penalty function approach to deal with the optimal sliding mode control problem for discrete-time systems. To solve this problem an alternative array structure based on the problem of weighted least squares penalty function will be developed. Using this alternative matrix structure, the optimal sliding mode control law of, the matrix Riccati equations and feedback gain were obtained. The motivation of this new approach is to show that it is possible to obtain an alternative solution to the classic problem of optimal sliding mode control.

Sliding mode control (SMC) has been successfully applied to many practical control problems due to its attractive features such as invariance to matched uncertainties. The characteristic feature of a continuous-time SMC system is that sliding mode occurs on a prescribed manifold, where switching control is employed to maintain the state on the surface. When a sliding mode is realized, the system exhibits some superior robustness properties with respect to external matched uncertainties. However, the realization of the ideal sliding mode requires switching with an infinite frequency. Control algorithms are now commonly implemented in digital electronics due to the increasingly affordable microprocessor hardware although the essential conceptual framework of the feedback design still remains to be in the continuous-time domain. Discrete sliding mode control has been extensively studied to address some basic questions associated with the sliding mode control of discrete-time systems with relatively low switching frequencies. However, the complex dynamical behaviours due to discretization in continuous-time SMC systems have not yet been fully explored. In this thesis, the discretization behaviours of SMC systems are investigated. In particular, one of the most frequently used discretization schemes for digital controller implementation, the zero-order-holder discretization, is studied. First, single-input SMC systems are discretized, stability and boundary conditions of the digitized SMC systems are derived. Furthermore, some inherent dynamical properties such as periodic phenomenon, of the discretized SMC systems are studied. We also explored the discretization behaviours of the disturbed SMC systems. Their steady-state behaviours are discussed using a symbolic dynamics approach under the constant and periodic matched uncertainties. Next, discretized high-order SMC systems and sliding mode based observers are explored using the same analysis method. At last, the thesis investigates discretization effects on the SMC systems with multiple inputs. Some conditions are first derived for ensuring the

This thesis describes the development of robust discrete time sliding mode controllers where only output information is available. A connection between discrete time sliding mode controllers and so-called min-max controllers is described. New conditions for the existence of stabilizing output feedback discrete time sliding mode controllers are given for non-square systems with bounded matched uncertainties. A novel sliding surface is described; this in itself is not realizable through outputs alone, but it gives rise to a control law which depends only on outputs. An explicit procedure is also described which shows how a Lyapunov matrix, which satisfies both a discrete Riccati inequality and a structural constraint, can be obtained using LMI optimization. This Lyapunov matrix is used to calculate the robustness bounds associated with the closed-loop system.;For systems which are not static output feedback stabilisable, a compensation scheme is proposed and a dynamic output feedback discrete time sliding mode controller is described with a simple parameterisation of the available design freedom.;Initially, a regulation problem, to drive all plant states to zero, is considered. Then a new scheme which incorporates tracking control using integral action is proposed for both the static and dynamic output feedback discrete time sliding mode controller. The scheme requires only that the plant has no poles or zeros at the origin and therefore with an appropriate choice of surface, the controller can be applied to non-minimum phase systems.;The theory described is demonstrated for various engineering systems including implementation on a DC-motor rig in real-time and simulations on a nonlinear, non-minimum phase model of a Planar Vertical Take-Off and Landing aircraft. The effectiveness of the controller is further proven by its application for control of the longitudinal dynamics of a detailed combat aircraft model call the high Incidence Research Model, a benchmark problem used by the Group for Aeronautical Research and Technology in Europe. Simulations with real-time pilot input commands have been carried out on a Real Time All Vehicle Simulator and good results obtained.

Orientador: José Paulo Fernandes Garcia
Banca: Jean Marcos de Souza Ribeiro
Banca: Cristiano Quevedo Andrea
Resumo: Usando dois controladores digitais com modos deslizantes, é proposto neste trabalho dois esquemas que minimizam os efeitos degenerativos causados pelo atraso no tempo de compu- tação do sinal de controle, que aqui é tratado como falha. Um observador robusto com modos deslizantes é utilizado neste trabalho, uma vez que nem sempre é possível ter acesso a todos os estados do sistema. Neste trabalho o observador tem um papel fundamental na detecção e acomodação da falha, pois através de um banco de observadores é gerado um resíduo que pos- sibilita a detecção da falha e determina qual controlador deve estar atuando sobre o sistema a ser controlado. Para validar os métodos propostos, são realizadas simulações e experimentos nos modelos do pêndulo invertido e no helicóptero 3DOF; ambos equipamentos da Quanser
Abstract: Using two digital controllers with sliding mode schemes that minimizes the degenerative effects caused by the delay in the computation time of the control signal are proposed in this work, which is here treated as failure. A robust observer with sliding mode is shown in this work, since it is not always possible to have access to all system states, but in this work the observer has a key role in the failure detection and accommodation, as observers are generated through a residue that directs the performance of the controller on the system being controlled. To test the proposed methods, simulations and experiments are performed on models of the inverted pendulum and the helicopter 3DOF, both Quanser equipment
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Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
Usando dois controladores digitais com modos deslizantes, é proposto neste trabalho dois esquemas que minimizam os efeitos degenerativos causados pelo atraso no tempo de compu- tação do sinal de controle, que aqui é tratado como falha. Um observador robusto com modos deslizantes é utilizado neste trabalho, uma vez que nem sempre é possível ter acesso a todos os estados do sistema. Neste trabalho o observador tem um papel fundamental na detecção e acomodação da falha, pois através de um banco de observadores é gerado um resíduo que pos- sibilita a detecção da falha e determina qual controlador deve estar atuando sobre o sistema a ser controlado. Para validar os métodos propostos, são realizadas simulações e experimentos nos modelos do pêndulo invertido e no helicóptero 3DOF; ambos equipamentos da Quanser
Using two digital controllers with sliding mode schemes that minimizes the degenerative effects caused by the delay in the computation time of the control signal are proposed in this work, which is here treated as failure. A robust observer with sliding mode is shown in this work, since it is not always possible to have access to all system states, but in this work the observer has a key role in the failure detection and accommodation, as observers are generated through a residue that directs the performance of the controller on the system being controlled. To test the proposed methods, simulations and experiments are performed on models of the inverted pendulum and the helicopter 3DOF, both Quanser equipment

Le contrôle par mode glissant est une technique d'automatique qui possède une longue histoire, la littérature remontant jusqu'au année 50. Son essence est la suivante : le contrôle est définit comme étant l'image d'une fonction discontinue de la variable de glissement, contraignant le système à évolué sur une variété, le système glisse alors dessus, d'où le nom. Cette variable de glissement est elle définie à partir de l'état du système. Les développements ont mené à la constitution d'une théorie bien établie à propos de cette technique, avec de nombreuses propriétés théoriques fort intéressante. Toutefois ceci ne porte que sur la version continue, c'est à dire quand le contrôle peut changer de valeur à chaque instant. En comparaison la version discrète du ce contrôleur est définie par le fait que la valeur du contrôle ne peut changer qu'à des instants isolés discrets. On a alors une fonction en escalier, constante sur la période d'échantillonnage. Cette situation est rencontrée par exemple lorsque le contrôleur est implémenté à l'aide d'un micro-contrôleur, ce qui est le cas dans nombre d'applications industrielles. Le principal problème avec le mode glissant est l'apparition d'un phénomène largement indésirable, le chattering (ou broutement) avec la version discrète du contrôleur, où même déjà en simulation. Dans ce dernier cas, nous appelons ceci du chattering numérique que nous attribuons à une mauvaise discrétisation du contrôle. L'approche développée ici se focalise sur ce point et est largement inspirée par les travaux effectués en mécanique non régulière, où ce type de comportement a aussi été observé lors de la simulation de système avec frottements et/où impacts. L'idée principale est de discretisé le contrôle de manière implicite et non explicite. Ceci permet d'éliminer le chattering numérique dans les cas simples (systèmes linéaires par exemple) où bien de le réduire grandement. Pour mener à bien l'analyse, des outils provenant de l'analyse convexe ainsi que des inégalités variationnelles en dimension finie sont utilisés. Le contrôleur proposé possède des propriétés intéressantes et proches de celles du temps continu. Ainsi on peut montrer que la variable de glissement est régie par une dynamique stable en temps finie, avec une fonction de Lyapunov. Le contrôle discret convergence vers celui du cas continu quand la période d'échantillonnage tends vers 0. Une atténuation d'éventuelles perturbations de type "matching" peut être établie. Ces travaux ont essentiellement portés sur le contrôle par mode glissant classique. L'algorithme dit twisting a pu être discrétisé avec la même technique et sa stabilité en temps finie grâce à une fonction de Lyapunov a pu être montrée. Ces propriétés ont été vérifiée en simulation, mais aussi de manière expérimentale. Ainsi des essais ont pu être menés sur deux banc d'essai: le premier est basé sur un système electropneumatique où à la fois le contrôle par mode glissant classique ainsi que le twisting ont pu être implémentés. L'objectif étant de suivre une trajectoire de référence. Le second système est un pendule inverse où le système doit être stabilisé à la position d'équilibre instable. Ici seul le contrôleur classique a été testé. L'analyse des données expérimentales a permis de mettre en lumière les performances supérieures des contrôleurs proposés par rapport à ceux classiquement usités. Les objectifs de contrôle sont mieux atteint et le chattering est grandement diminué
Sliding Mode Control is a control technique with a long history, with research efforts dating back to the 50's. The basic idea is to define the control input as a discontinuous function of the sliding variable, which solely depends on the state, and to constraint the system to evolve on a manifold, hence the term sliding. Over the years a strong theory was build around this technique, but only in continuous time. In our context, this means that control input value can change value at any time. The discrete-time case is when the control input can only change at isolated time instants and the dynamical system on which the control is still a continuous-time process. The control input is therefore a step function. This case appears when the controller is digitally implemented, for instance with the help of a microcontroller. This kind of setup is nowadays ubiquitous in benchmarks and industrial applications. One of the main limitation of the applicability of sliding mode control is the chattering phenomenon that is witnessed when this control technique is applied in practice, but already in simulations. In contrast to previous approaches, we single out the chattering that is already witnessed in simulation, even with no disturbance and with perfect knowledge of the dynamics. This is called the numerical chattering and one of its distinct feature is the constant chattering, or high-frequency bang-bang behavior, of the control input. This naturally induces a chattering of the sliding variable. The claim that this type of chattering is usually predominant and that it is due to a bad discretization of the signum multifunction. The approach developed in this work was inspired by the research effort in the nonsmooth mechanical to properly simulate some systems like those with dry friction and/or unilateral constraints. The main point is to discretize the signum in an implicit fashion, that is its argument is the value of the sliding variable at the end of the next sampling period. With this change, the numerical chattering can be removed in the simplest cases, largely attenuated. The research effort was focused on classical sliding mode controller, rather than the higher order ones. The frameworks used to perform the analysis are convex analysis and variational inequalities. This discrete-time controller enjoys several interesting theoretical properties. First it is finite-time Lyapunov stable: the sliding variable goes to 0 in finite-time. The discrete-time control input converges to the continuous-time one as the sampling period goes to 0. The control action also attenuates the effect of matched perturbations. Also the increase of the gain of the controller does not affect the performances when the system is sliding. The twisting controller can be discretized in the same way and is also finite-time Lyapunov stable. This good theoretical properties have been verified in simulations, but also on experimental setups. Two tests were conducted: the first one on an electropneumatic system, where both the classical first-order sliding mode controller and the twisting algorithm were tested. The objective was to track a reference trajectory. The second one was an inverted pendulum on a cart with only the classical SMC. The goal was to stabilize the system at the unstable equilibrium. The analysis from the data collected during those experiments shows that the proposed controllers perform better than the their explicitly discretized versions. The performances are better and the chattering is effectively reduced

L’objectiu d’aquest treball és el disseny d’un controlador de velocitat per a una cinta de córrer de manera que la freqüència cardíaca d’una persona que corre sobre ella segueixi un perfil determinat, potencialment variable en el temps, preespecificado pels metges per a la recuperació cardíaca de la persona. Inicialment es considera el model matemàtic que relaciona la velocitat de la cinta de córrer amb la freqüència cardíaca de la persona. Un dels aspectes importants del model és la determinació dels seus paràmetres. En aquest sentit, en primer lloc es tracta el problema de l’estimació paramètrica, que es formula com un problema d’optimització que és resolt amb una tècnica heurística coneguda com Particle Swarm Optimization (PSO). Aquesta és la primera vegada que aquesta tècnica s’utilitza en l’estimació de models cardíacs i és una de les contribucions de la tesi. A continuació, es dissenya un controlador en mode lliscant de tipus super-Twisting per dur a terme el control robust de la cinta en presència d’incertesa paramètrica i dinàmica no modelada. Els experiments numèrics duts a terme mostren que el algoritmes estimació emprat és capaç d’obtenir valors molt precisos per als paràmetres del sistema i que l’enfocament de control utilitzat obté un error de seguiment nul asimptòticament, aconseguint els objectius del control. En tots dos casos, la velocitat de la cinta es troba en el rang dels 2-4 km \/ h, rang que no s’ha utilitzat anteriorment en els treballs previs. Finalment, en l’última part d’aquest treball, es dissenya un controlador robust en temps discret. Inicialment es dissenya un controlador de tipus linealització per realimentació, però es mostra que la seva robustesa és pobre. Per resoldre aquest problema es proposa un altre mètode basat en l’estimació simultània estat-paràmetres. Tot i que els resultats obtinguts són prometedors, l’enfocament presenta algunes problemes com la no identificació dels paràmetres i la generació d’algunes oscil·lacions no a la sortida. Atès que l’aproximació del control lliscant amb super-Twisting ha proporcionat un gran resultat en temps continu, es proposa un controlador lliscant amb super-Twisting en el cas discret. Per a això, inicialment es dissenya un control lliscant per al sistema linealitzat i discretitzat. Es dissenya un control lliscant amb observador i el senyal de control es reconstrueix per mitjà d’un retenidor d’ordre zero (ZOH). No obstant això, el sistema és intrínsecament nolineal, de manera que s’estén el disseny al sistema nolineal discretitzat, per al qual s’aplica el controlador amb super-Twisting. El controlador proposat és capaç d’aconseguir un seguiment de la trajectòria excel·lent sense la presència de chattering i sense la necessitat d’un observador d’estat, el que és una de les contribucions de la tesi. En aquest cas també, la velocitat de la cinta es troba en el rang dels 2-4 km \/ h, rang que no s’ha utilitzat anteriorment en els treballs previs.
El objetivo de este trabajo es el diseño de un controlador de velocidad para una cinta de correr de tal forma que la frecuencia cardíaca de una persona que corre sobre ella siga un perfil determinado, potencialmente variable en el tiempo, preespecificado por los médicos para la recuperación cardíaca de la persona. Inicialmente se considera el modelo matemático que relaciona la velocidad de la cinta de correr con la frecuencia cardíaca de la persona. Uno de los aspectos importantes del modelo es la determinación de sus parámetros. En este sentido, en primer lugar se trata el problema de la estimación paramétrica, que se formula como un problema de optimización que es resuelto con una técnica heurística conocida como Particle Swarm Optimization (PSO). Esta es la primera vez que esta técnica se utiliza en la estimación de modelos cardíacos y es una de las contribuciones de la tesis. A continuación, se diseña un controlador en modo deslizante de tipo super-twisting para llevar a cabo el control robusto de la cinta en presencia de incertidumbre paramétrica y dinámica no modelada. Los experimentos numéricos llevados a cabo muestran que el algoritmos estimación empleado es capaz de obtener valores muy precisos para los parámetros del sistema y que el enfoque de control utilizado obtiene un error de seguimiento nulo asintóticamente, consiguiendo los objetivos del control. En ambos casos, la velocidad de la cinta se encuentra en el rango de los 2-4 km\/h, rango que no se ha utilizado anteriormente en los trabajos previos. Finalmente, en la última parte de este trabajo, se diseña un controlador robusto en tiempo discreto. Inicialmente se diseña un controlador de tipo linealización por realimentación, pero se muestra que su robustez es pobre. Para resolver este problema se propone otro método basado en la estimación simultánea estado-parámetros. Aunque los resultados obtenidos son prometedores, el enfoque presenta algunas problemas como la no identificación de los parámetros y la generación de algunas oscilaciones no en la salida. Dado que la aproximación del control deslizante con super-twisting ha proporcionado un gran resultado en tiempo continuo, se propone un controlador deslizante con super-twisting en el caso discreto. Para ello, inicialmente se diseña un control deslizante para el sistema linealizado y discretizado. Se diseña un control deslizante con observador y la señal de control se reconstruye por medio de un retenedor de orden cero (ZOH). No obstante, el sistema es intrínsecamente nolineal, por lo que se extiende el diseño al sistema nolineal discretizado, para el que se aplica el controlador con super-twisting. El controlador propuesto es capaz de lograr un seguimiento de la trayectoria excelente sin la presencia de chattering y sin la necesidad de un observador de estado, lo que es una de las contribuciones de la tesis. En este caso también, la velocidad de la cinta se encuentra en el rango de los 2-4 km\/h, rango que no se ha utilizado anteriormente en los trabajos previos.
The objective of this work is to design a heart rate (HR) controller for a treadmill so that the HR of an individual running on it tracks a pre-specied, potentially time-varying profile specified by doctors for the cardiac recovery of the person. Initially, we consider a mathematical model relating the relationship between the speed of the treadmill and HR of the person running on it. An important issue in this model is the determination of its parameters. Thus, we first tackle the parameter estimation problem in this model which is formulated as an optimization one, that is solved through a heuristic technique known as Particle Swarm Optimization. This is the first time that this technique is used for the estimation of cardiac models and is a contribution of the thesis. Afterward, a super- twisting sliding mode controller is designed to perform the robust control of treadmill’s speed in the presence of potential unmodelled dynamics and parametric uncertainties. Numerical examples show that the estimation procedure is able to obtain accurate values for the system’s parameters while the proposed control approach is able to obtain zero tracking error without chattering, definitely achieving the control objectives. In both cases, the range of treadmill’s speed goesfrom 2 to 14 km\/h, range that is not usually employed in previous studies. Finally, in the last part of this work, the objective is to design a discrete-time robust controller. Initially, a feedback linearization-based controller is designed, but it has poor robustness properties. In order to solve this problem, we propose another method consisting in the Joint parameter-state estimation based-control. However, this approach does not identify the parameters and it offers some oscillations. To solve all of these problems and regarding the previous Chapter, we used the discrete-time sliding mode controller method to complete our study. In the first part of this Section, as designing a nonlinear model directly is hard, we decided to linearize the model and then discretize it. Furthermore, the continuous control is generated by a zero-order hold (ZOH). On the other hand, since the nonlinear relationship describes a better relation between HR and speed, a nonlinear model is used in the last part of this thesis. The final and best controller is a discrete-time super-twisting system that avoids chattering and achieves very good robustness and tracking in the system. The great systematic procedure to design of the controller, the perfect tracking and the avoidance of using an observer for this system are other advantages of this approach. The simulation results in this work that presented in the speed range of 2-14 km, a range that is not usually employed in previous studies to the control of the heart rate during treadmill exercise.

碩士
國立臺灣師範大學
工業教育學系
84
Abstract Sliding mode controller within the domain of variable structure control has been developed over thirty years. It applied suitable sliding mode by means of switching the structure of control systems. The system states might be pulled above sliding line (plane). Variable structure controller has very good insensitivity for disturbance or uncertainty system. But when the study uses digital computer to achieve sliding model control , the sliding condition of sliding mode controller is different from that of traditional continuous system. Therefore, it is necessary to design a controller which can correspond to discrete sliding condition. Furthermore, when the system has exogenous disturbances , and we only know the statistic characteristic of these disturbances, how to satisfy the sliding condition and design optimal hitting control law turn into main points of the study. In this thesis, we propose a new method to design discrete sliding mode controller for nonlinear systems. Discrete sliding mode controller is designed based on Lyapunov function. The approach is guaranteed to satisfy the stable discrete time sliding mode condition. It also manipulate the theory of optimal expectation value of statistics. An optimal hitting control law is designed based on the consideration of stochastic disturbance. Besides, a switching algorithm is presented to attenuate the chattering phenomenon. Base on the above research results, It can get effective verification by means of computer software simulation, when acts on linear and nonlinear system.

碩士
大同大學
機械工程研究所
90
Abstract --- In general, a multilayer LVPZT bender actuator (MLBA) is in the presence of the unmodeled dynamics, e.g., hysteresis, bending modes, measurement noise and external disturbance. It is difficult to obtain a precise linear model of the MLBA that is finite-dimensional. Spillover associated with a reduced-order model has been known to result in the instability of the closed-loop system. In this situation, a low-pass filter is employed to attenuate the effect of the unmodeled dynamics. A parameter estimation of recursive least-squares method is applied to model the data of the filtered output and the corresponding input that is a narrow-band signal. The so-called “high-displacement piezoelectric actuator system (HDPAS)” contains a MLBA and a low-pass filter. Then the identified model to represent the HDPAS is used for the controller design of a dead-beat to its filtered switching surface. Because the HDPAS is in the face of the output disturbance caused by hysteresis, flexible structure, measurement noise, external disturbance and modeling error, the system performance is poor and the system instability probably occurs. Hence, the norm of the sensitivity function between the filtered switching surface and the output disturbance is simultaneously minimized to reduce its effect. For further improving the tracking accuracy, a switching control based on Lyapunov redesign is constructed. Finally, the experiments of the HDPAS by the PID control and the proposed control are arranged to verify the usefulness of the proposed control.

碩士
國立成功大學
航空太空工程學系碩博士班
91
To achieve the required system performance, the control system should robust to system uncertainties and disturbances. The sliding mode controller can be designed systematically with robustness to matched uncertainties. Therefore, it has been widely applied to many theorical studies and industrial applications. The high order sliding mode controller can be considered as an extension of conversional sliding mode controller with less state information required and can suppress or reduce chattering due to the controller output. This make the high-order sliding mode controller more realizable then the conversional sliding mode controller. In this thesis, discrete systems with disturbances are considered. An approach to design 2-order discrete sliding mode controller is proposed. The result can be extended to the high order cases. Using the Lyapunov stability criterion, the resulting system is guaranteed to stay around the desired sliding surface with a specific boundary. The performance of the proposed control system can be achieve, the design objective which indicate its feasibility of the proposed discrete high-order sliding mode controller.

碩士
國立成功大學
航空太空工程學系
86
In control system design, it is often required that the system behavior is insensitive to the parameter variations and external disturbances. To this problem, variable structure system control is developed and widely applied. Traditional sliding mode control possesses the characteristics that the controlled system is invariant to model uncertainty and external disturbance when the state enters the sliding surface, and the system response follows the dynamics of the sliding surface. However, when the sliding mode controller is realized by digital computer, it is impossible for the control input to switch in a very high frequency and the sliding mode motion will not occurred. The respective robustness vanishes, and the system may become unstable when the sampling period is too long. In this dissertation, a discrete-time sliding control law, which is applied immediately after sensing the system states, is developed to guarantee the existence of the weak-pseudo-sliding mode along the prescribed hyperplane. Due to the effect of the computational delay, an one-sample-delay discrete-time sliding mode control law is developed. Concept of the "modified weak- pseudo-sliding mode" is proposed. The upper bounds of the sampling periods are also determined to ensure the stability.

碩士
大同大學
電機工程學系(所)
96
In this thesis, we adopt two kinds of reaching law based on discrete sliding-mode control for discrete-time bilinear systems. The reaching law is a differential equation which specifies the dynamics of a sliding surface function . The system states are unknown and we have not need to construct the observers or dynamic compensators to estimate system states. This method use the reaching law for designing the controller in bilinear system. We will design a reaching control law to ensure the bilinear system trajectories can satisfy with the reaching law. Simulation results are presented to demonstrate the feasibility of the proposed control scheme.

碩士
國立中山大學
電機工程研究所
84
A methodology of designing discrete-time sliding mode controllers is proposed in this thesis for a class of multi- input multi-output (MIMO) discrete system. This sliding mode controller is composed of two types of controllers. One is the equivalent control law. which can drive the state trajectories into a bounded region. The other is a switching controller, which can drive the state trajectories into a band region. The combina- tion of these two controller are robust sliding mode controllers which can drive all the state trajectories toward zero if there is no model uncertainties. However if the model uncertainties and/or disturbances exist, the state trajectories will be driven into a bounded region. Several numerical examples are also given for demonstrate the feasibility of the proposed control scheme.

碩士
大同大學
電機工程學系(所)
98
In this thesis, the multivariable discrete time bilinear systems are considered. We adopt two kinds of reaching law that used by quasi-sliding mode control theory to obtain two reaching control laws. Then we apply two reaching control laws which drive the state of the controlled system to a band around the sliding surface. And the system states will remain in a band in finite time. Finally the design techniques are illustrated through two numerical examples. They will demonstrate the feasibility of the proposed controller.

碩士
國立臺灣大學
土木工程學系
85
The sliding mode control ( SMC ) is known to be robust against variations in certain system parameters or external excitations . It is often used to solvethe control problems of system parametric uncertainties . The purpose of this paper is to study the differences between the linear control strategy and sliding mode control strategy . We also consider the influence of time delay effect and use modal condensation method to reduce the degree of freedoms of the structure . In this paper , we utilize two control types of buildings ,ATMD system and base isolation system , to prove the control efficient in numerical analysis .

碩士
中興大學
電機工程學系所
99
In this thesis, the single chip (AT89S51) is the core of the temperature control platform. By using the method of discrete-time on-off sliding mode control, the system temperature output can be maintained within a small variance about the command value. The controlled process is a hair dryer heater actuated by a solid state relay (SSR). The mathematical model of the plant is obtained by using the MATLAB System Identification Toolbox to process the open-loop experimental data, yielding a transfer function for the system. Since the SSR actuator can only issue on-off commands, a direct implementation of on-off control law simply with the temperature error information will result in a zigzag type of temperature output behavior called the chattering phenomenon. We propose the discrete-time on-off sliding mode control method for the system to attenuate the amount of chattering as well as to follow the temperature command. The sliding surface variable is a linear combination of the current and the past temperature errors. Thus the on-off command is determined by the sign of the sliding surface variable, rather than the error variable. Mathematically, this sliding surface serves as a digital low-pass filter, which rejects the high frequency chattering of the temperature error. By tuning the free parameter in the sliding surface, we are able to adjust the bandwidth of the low-pass filter, as a means of attenuating the error magnitude. The experimental results show that the temperature error can be reduced substantially by regulating the parameter of the proposed control scheme. They confirm the stability and accuracy accomplished by the proposed discrete-time on-off sliding mode control method.

碩士
國立中山大學
電機工程學系研究所
96
Based on the Lyapunov stability theorem, a methodology of designing robust discrete-time model reference variable structure state tracking controller is proposed in this thesis for a class of multi-input multi-output (MIMO) discrete-time systems. This variable structure controller is composed of three types of controllers. The first one is the feedback control law, which can eliminate the nominal term in the derivative of a Lyapunov function. The second one is the switching control law, which can determine the decreasing rate of the Lyapunov function. The third one is the adaptive control law, which is used to overcome the perturbations. The resultant robust variable structure controllers are capable of driving all the trajectories of tracking errors toward a small bounded region. The information of upper bound of the perturbation, which is not a constant and is dependent on the norm of state variable, is not required beforehand due to some adaptive mechanisms are embedded in the proposed control scheme, and the stability of the overall controlled system is guaranteed. A numerical example and a practical example are given to demonstrate the feasibility of the proposed control scheme.

碩士
國立臺灣師範大學
工業教育研究所
89
The control law derived by traditional sliding mode control is based on a continuous-time system, which is well known to be robust to the uncertainties of the system and insensitive to parameter variations and to external disturbances. However, if the control law designed on the basis of a continuous sliding mode is directly implemented for discrete-time systems using a digital computer, the performance of control is often unsatisfying on the prescribed specifications, even systems may become unstable. The discrete-time sliding mode controller (DSMC) was derived to improve these problems; however, the suitable control laws are difficult to obtain due to the rigorous limited conditions in the design of DSMC. Therefore, this thesis presents a novel design of discrete-time sliding mode using differentiable cerebellar model articulation controller (DCMAC) so that the above drawbacks can be overcome. Where DCMAC is used to real-time assist the original DSMC. Taking advantage of the excellent capability of DCMAC in nonlinear function learning and patterns generalization promotes the control performance and simplifies the formular deriving process in original DSMC. Compared with the conventional DSMC, this new control design provides a more simple method to design the control law of DSMC and reduces its difficulty of design. According to simulated results, this controller can significantly reduce the tracking error, and effectively elevate the accuracy in control process. Finally one experiment for Ball-Beam Balancing System using proposed controller has performed to demonstrate the feasibility and the control performance in practical control application.

博士
國立清華大學
動力機械工程學系
103
A novel concept of global quasi-sliding mode control scheme which is extended from a global sliding mode control to establish a quasi-sliding motion that ensures a zigzag motion, which will directly move forward to the original phase portrait throughout the entire sliding dynamic response in discrete-time. The new design of an augmented forcing function is followed by three conditions extended from global sliding mode control. In order to reduce chattering phenomenon from zigzag motion, we also propose a redesign of continuous approximation with smooth saturation and utilize an equivalent D2 wavelet filter to decide smooth interval according to sampling time in system. Finally, in discrete time, we propose a global quasi-sliding mode approach for the different hardware platforms. The experimental results show that it matches global quasi-sliding mode theory and maintain zigzag motion throughout the entire response. And then generalized smooth saturation is adopted to ensure that switching control term will output more force and state response will be converge faster. Therefore, the switching function adopted generalized smooth function gsat(.) would be a balance between control force response and state response.

碩士
大同大學
電機工程學系(所)
98
In this thesis, a quasi-sliding mode controller for discrete-time linear system with delay input and disturbance is proposed. By using the method of state transformation, a system with the input delay and disturbance delay is transformed to a delay free system. Based on Lyapunov stability theorem , a quasi-sliding mode controller is designed to ensure the system state trajectories were stable to remain in a small band around the desired sliding surface. Finally, two illustrative examples are provided to demonstrate the effectiveness of the proposed controller.

碩士
國立交通大學
電機與控制工程系所
96
In this thesis, we extend Bartoszewicz’s design of quasi-sliding mode control for discrete-time linear systems (1998) to a class of discrete-time nonlinear systems. The selected sliding surface in this extension is allowed to be nonlinear rather than only linear one. Three schemes for the selection of sliding surfaces, and two quasi-sliding mode controllers according to the variation rate of disturbances are presented in this study. The obtained analytic results are also employed to study the control of a trailer-truck system. Simulation results with comparisons to those of fuzzy controllers demonstrate the benefits of the proposed schemes.

碩士
大同大學
電機工程學系(所)
100
For a class of uncertain nonlinear discrete-time multi-input-multi-output (MIMO) systems, this thesis presents an integral-type sliding mode control (ISMC) method to deal with the control problem. First, a robust observer is used to estimate unknown states of the controlled system. Next, the reaching condition and the sliding condition of the system behavior can be satisfied by the proposed control law based on the integral-type sliding surface. Moreover, some adaptive laws are introduced to estimate the upper bounds of the unknown uncertainties. By the Lyapunov theory, the presented integral sliding mode control not only guarantees the robust stability of the overall closed-loop system, but also achieves the precision estimation. The feasibility of the proposed method will be confirmed by computer simulation in this thesis.

碩士
國立臺北科技大學
自動化科技研究所
106
In this thesis, the problems of implementation and the control for the quadrotor is investigated based on the discrete-time sliding mode control approach. Firstly, the embedded system with gyroscope and the global positioning system module are utilized to establish the hardware of the quadrotor. In addition, the Kalman filter is adopted to eliminate the measurement errors of the accelerometer, the gyroscope and the magnetometer. Moreover, the desired position can be obtained by global positioning system. Based on these information, the discrete time sliding mode controller is adopted to track the desired position. The simulations and the experiments shows the validity and the feasibility of the developed quadrotor.

碩士
國立交通大學
機械工程系
91
Near-field optical disk drives can overcome the diffraction limit of conventional techniques, thereby substantially increasing storage capacity and density. Since the precision of track pitches and the flying height lie within range of nanometers, it is required to enhance control performance to improve focusing speed and accuracy. In this study, a PZT bender is used as a fine actuator of the pickup head. Due to hysteresis of the PZT material, a discrete sliding mode controller inherited with robust properties is developed to deal with model uncertainty and external disturbances, so as to maintain a stable flying height. In the experiment, a gap capacitance sensor utilizes an FM system to extract capacitance change corresponding to flying height variation. Furthermore, to investigate the transient response at the head/disk interface, wavelet transform with flexible time-frequency resolution is applied to analyze transient behavior, and identify the dynamic characteristics as the system subject to impulsive shock loading.

博士
國立清華大學
動力機械工程學系
90
A discrete-time sliding-mode control scheme for a certain class of systems is proposed to guarantee the existence of sliding-mode as well as to alleviate the chattering. Firstly, a process using auxiliary compensator is introduced and the explicit condition to ensure the one-sided behavior, deadbeat sliding-mode and quasi-sliding mode is derived, which is merely an inequality constraint on controller parameters. This condition also guarantees that the state trajectory starting from any initial condition, will reach the boundary layer of the switching surface in a finite number of steps. Moreover, the controller parameters can be determined via the correlation between the pole location in the z-plane and the time-domain response characteristics. Next, a robust discrete-time sliding-mode approach for a class of perturbed systems is proposed. To overcome the difficulty of conservative design due to over-estimated upper bounds on system perturbations; the discrete-time sliding-mode control law employs a smooth function to alleviate the chattering phenomenon. Conditions for stability are analyzed and given. Also, an estimated reaching time can be pre-calculated. Finally, we propose a discrete-time hierarchical sliding-mode approach for dual-stage systems. Here, we combine the dual-stage positioning servo design phase into only one stage. Therefore, the performance of the coarse tracking and fine tracking stages can be ensured simultaneously. The settling time of the controlled positioning system can be assigned through the choice of a pre-specified sliding hyperplanes. Hence, the proposed methodology can avoid the drawbacks of mode switching control design, such as discontinuity in control input, incompatible dynamic characteristics, and difficulty in estimating the overall settling time and is easy to be implemented for practical applications. Simulation and experimental studies of an uninterruptible power supply control systems, seek control of an optical pick-up head and the dual-stage positioning servo systems are performed to validate the feasibility and the effectiveness of these approaches.