Event-triggered control for piecewise affine discrete-time systems

In the present work, we study the problems of stability analysis of piecewise-affine (PWA) discrete-time systems, and trigger-function design for discrete-time event-triggered control systems. We propose a representation for piecewise-affine systems in terms of ramp functions, and we rely on Lyapunov theory for the stability analysis. The proposed implicit piecewise-affine representation prevents the shortcomings of the existing stability analysis approaches of PWA systems. Namely, the need to enumerate regions and allowed transitions of the explicit representations. In this context, we can emphasize two benefits of the proposed approach: first, it makes possible the analysis of uncertainty in the partition and, thus, the transitions. Secondly, it enables the analysis of event-triggered control systems for the class of PWA systems since, for ETC, the transitions cannot be determined as a function of the state variables. The proposed representation, on the other hand, implicitly encodes the partition and the transitions. The stability analysis is performed with Lyapunov theory techniques. We then present conditions for exponential stability. Thanks to the implicit representation, the use of piecewise quadratic Lyapunov functions candidates becomes simple. These conditions can be solved numerically using a linear matrix inequality formulation. The numerical analysis exploits quadratic expressions that describe ramp functions to verify the positiveness of extended quadratic forms. For ETC, a piecewise quadratic trigger function defines the event generator. We find suitable parameters for the trigger function with an optimization procedure. As a result, this function uses the information on the partition to reduce the number of events, achieving better results than the standard quadratic trigger functions found in the literature. We provide numerical examples to illustrate the application of the proposed representation and methods.Ce manuscrit présente des résultats sur l’analyse de stabilité des systèmes affines par morceaux en temps discret et sur le projet de fonctions de déclenchement pour des stratégies de commande par événements. Nous proposons une représentation pour des systèmes affines par morceaux et l’on utilise la théorie de stabilité de Lyapunov pour effectuer l’analyse de stabilité globale de l’origine. La nouvelle représentation implicite que nous proposons rend plus simple l’analyse de stabilité car elle évite l’énumération des régions et des transitions entre régions tel que c’est fait dans le cas des représentations explicites. Dans ce contexte nous pouvons souligner deux avantages principaux, à savoir I) la possibilité de traiter des incertitudes dans la partition qui définit le système et, par conséquent des incertitudes dans les transitions, II) l’analyse des stratégies de commande par événements pour des systèmes affines par morceaux. En effet, dans ces stratégies les transitions ne peuvent pas être définies comme des fonctions des variables d’état. La théorie de stabilité de Lyapunov est utilisée pour établir des conditions pour la stabilité exponentielle de l’origine. Grâce à la représentation implicite des partitions nous utilisons des fonctions de Lyapunov quadratique par morceaux. Ces conditions sont données par des inégalités dont la solution numérique est possible avec une formulation par des inégalités matricielles linéaires. Ces formulations numériques se basent sur des expressions quadratiques décrivant des fonctions rampe. Pour des stratégies par événement, une fonctions quadratique par morceaux est utilisée pour le générateur d’événements. Nous calculons les paramètres de ces fonctions de déclenchement a partir de solutions de problèmes d’optimisation. Cette fonction de déclenchement quadratique par morceaux permet de réduire le nombre de d’événementsen comparaison avec les fonctions quadratiques utilisées dans la littérature. Nous utilisons des exemples numériques pour illustrer les méthodes proposées.No presente trabalho, são estudados os problemas de análise de estabilidade de sistemas afins por partes e o projeto da função de disparo para sistemas de controle baseado em eventos em tempo discreto. É proposta uma representação para sistemas afins por partes em termos de funções rampa, e é utilizada a teoria de Lyapunov para a análise de estabilidade. A representação afim por partes implícita proposta evita algumas das deficiências das abordagens de análise de estabilidade de sistemas afins por partes existentes. Em particular, a necessidade de anumerar regiões e transições admissíveis das representações explícitas. Neste contexto, dois benefícios da abordagem proposta podem ser enfatizados: primeiro, ela torna possível a análise de incertezas na partição, e, assim, nas transições. Segundo, ela permite a análise de sistemas de controle baseado em eventos para a classe de sistemas afins por partes, já que, para o controle baseado em eventos, as transições não podem ser determinadas como uma função das variáveis de estado. A representação proposta, por outro lado, codifica implicitamente a partição e as transições. A análise de estabilidade é realizada com técnicas da teoria de Lyapunov. Condi- ções para a estabilidade exponencial são então apresentadas. Graças à representação implícita, o uso de funções candidatas de Lyapunov se torna simples. Essas condições podem ser resolvidas numéricamente usando uma formulação de desigualdades matriciais lineares. A análise numérica explora expressões quadráticas que descrevem funções de rampa para verificar a postivividade de formas quadráticas extendidas. Para o controle baseado em eventos, uma função de disparo quadrática por partes define o gerador de eventos. Parâmetros adequados para a função de disparo sãoencontrados com um procedimento de otimização. Como resultado, esta função usa informação da partição para reduzir o número de eventos, obtendo resultados melhores do que as funções de disparo quadráticas encontradas na literatura. Exemplos numéricos são fornecidos para ilustrar a aplicação da representação e mé- todos propostos

A Switching Controller for a class of MIMO Bilinear Systems with Time-Delay

International audienceIn this paper we propose a state-dependent switching controller for MIMO bilinear systems with constant delays in both the state and the input. The control input is assumed to be restricted to take only a finite number of values. The stability analysis of the closed-loop is based on a Lyapunov-Krasovskii functional, and the design is reduced to solve a system of linear matrix inequalities. The controller can be designed by considering (state) delay-dependent or delay-independent conditions

A CENTER MANIFOLD THEORY-BASED APPROACH TO THE STABILITY ANALYSIS OF STATE FEEDBACK TAKAGI-SUGENO-KANG FUZZY CONTROL SYSTEMS

The aim of this paper is to propose a stability analysis approach based on the application of the center manifold theory and applied to state feedback Takagi-Sugeno-Kang fuzzy control systems. The approach is built upon a similar approach developed for Mamdani fuzzy controllers. It starts with a linearized mathematical model of the process that is accepted to belong to the family of single input second-order nonlinear systems which are linear with respect to the control signal. In addition, smooth right-hand terms of the state-space equations that model the processes are assumed. The paper includes the validation of the approach by application to stable state feedback Takagi-Sugeno-Kang fuzzy control system for the position control of an electro-hydraulic servo-system

Естимација крутости и адаптивно управљање код попустљивих робота

Although there has been an astonishing increase in the development of nature- inspired robots equipped with compliant features,i.e.soft robots, their full potential has not been exploited yet. One aspect is that the soft robotics research has mainly focused on their position control only, whilest iffness is managed in open loop. Moreover, due to the difficulties of achieving consistent production of the actuation systems for soft articulated robots and the time-varyingnatureoftheirinternalflexibleelements,whicharesubjecttoplasticdeformation overtime,itiscurrentlyachallengetopreciselydeterminethejointstiffness. . In this regard, the thesis puts an emphasis on stiffness estimation and adaptive control for soft articulated robots driven by antagonistic Variable Stiffness Actuators (VSAs) with the aim to impose the desired dynamics of both position and stiffness, which would finally contribute to the overall safety and improved performance of a soft robot. By building upon Unknown Input Observer (UIO) theory, invasive and non-invasive solutions for estimation of stiffness in pneumatic and electro-mechanical actuators are proposed and in the latter case also experimentally validated. Beyond the linearity and scalability advantage, the approaches have an appealing feature that torque and velocity sensors are not needed. Once the stiffness is determined, innovative control approaches are introduced for soft articulated robots comprising an adaptive compensator and a dynamic decoupler. The solutions are able to cope with uncertainties of the robot dynamic model and, when the desired stiffness is constant or slowly-varying, also of the pneumatic actuator. Their verification is performed via simulations and then the pneumatic one is successfully tested on an experimental setup. Finally, the thesis shows via extensive simulations the effectiveness of adaptive technique ap- plied to soft-bodied robots, previously deriving the sufficient and necessary conditions for the controller convergence.Iako se danas izuzetno intenzivno radi na razvoju robota inspirisanih prirodom koje odlikuje elastična struktura, njihov puni potencijal jox uvek nije iskorišćen. Sa jedne strane, istraživanja u oblasti popustljivih robota su uglavnom fokusirana samo na upravljanje njihovom pozicijom, dok se krutost reguliše u otvorenoj sprezi. Pored toga, zbog poteškoća u postiznju konzistentne proizvodnje aktuatora i promenljive prirode njihovih elastičnih elemenata, koji su vremenom podlo_ni plastičnoj deformaciji, trenutno je izazov precizno odrediti krutost zglobova robota. U cilju doprinosa poboljšanja_u performansi i bezbednosti rada popustivih robota, teza prikazuje doprinos proceni krutosti i adaptivnog simultanog upravljanja pozicijom i krutosti antagonističkih aktuatora promenljive krutosti (VSA). Oslanjajući se na teoriju opservera nepoznatih ulaza (UIO), predložena su invazivna i neinvazivna rešenja za procenu krutosti u pneumatskim i elektromehaničkim aktuatorima i eksperimentalno verifikovana u slučaju druge grupe aktuatora. Pored linearnosti i skalabilnosti, ovi pristupi imaju privlaqnu osobinu da senzori momenta i brzine nisu potrebni. Teza predla_e inovativne sisteme upravljanja koji poseduju adaptivni kompenzator i dinamički dekupler. Predložene metode upravljanja demonstriraju mogućnost da kompenzuju nesigurnosti dinamičkog modela robota bez obzira da li je on pogođen električnim ili pneumatskim aktuatorima. Nakon simulacija, razvijeno upravljanje je verifikovano i na pneumatskom robotu. Na kraju teze, obimne simulacije pokazuju efikasnost adaptivne tehnike kada se primeni na robote sa fleksibilnim linkovima, prethodno izvodeći dovoljne i potrebne uslove za konvergenciju kontrolera

Hybrid modeling and control of mechatronic systems using a piecewise affine dynamics approach

This thesis investigates the topic of modeling and control of PWA systems based on two experimental cases of an electrical and hydraulic nature with varying complexity that were also built, instrumented and evaluated. A full-order model has been created for both systems, including all dominant system dynamics and non-linearities. The unknown parameters and characteristics have been identi ed via an extensive parameter identi cation. In the following, the non-linear characteristics are linearized at several points, resulting in PWA models for each respective setup. Regarding the closed loop control of the generated models and corresponding experimental setups, a linear control structure comprised of integral error, feed-forward and state-feedback control has been used. Additionally, the hydraulic setup has been controlled in an autonomous hybrid position/force control mode, resulting in a switched system with each mode's dynamics being de ned by the previously derived PWA-based model in combination with the control structure and respective mode-dependent controller gains. The autonomous switch between control modes has been de ned by a switching event capable of consistently switching between modes in a deterministic manner despite the noise-a icted measurements. Several methods were used to obtain suitable controller gains, including optimization routines and pole placement. Validation of the system's fast and accurate response was obtained through simulations and experimental evaluation. The controlled system's local stability was proven for regions in state-space associated with operational points by using pole-zero analysis. The stability of the hybrid control approach was proven by using multiple Lyapunov functions for the investigated test scenarios.publishedVersio

Robust Iterative Learning Control for Pneumatic Muscle with State Constraint and Model Uncertainty

In this paper, we propose a novel iterative learning control (ILC) scheme for precise state tracking of pneumatic muscle (PM) actuators. Two critical issues are considered in our scheme: 1) state constraints on PM position and velocity; 2) uncertainties of the PM model. Based on the three-element form, a PM model is constructed that takes both parametric and nonparametric uncertainties into consideration. By introducing the composite energy function (CEF) approach incorporated with a barrier Lyapunov function (BLF), full state constraints of PM will not be violated and uncertainties are effectively compensated. Through rigorous analysis, we show that under proposed ILC scheme, uniform convergence of PM state tracking errors are guaranteed. Simulation results validate the performance of the proposed scheme

Pseudo Euler-Lagrange and Piecewise Affine Control Applied to Surge and Stall in Axial Compressors

This thesis addresses the control of the axial compressor surge and stall phenomena using Pseudo Euler-Lagrange and Piecewise Affine (PWA) controller synthesis techniques. These phenomena are considered as major gas turbine compressor instabilities that may result in failures such as the engine flame-out or severe mechanical damages caused by high blade vibration. The common approach towards the detection of the rotating stall and surge is to install various types of pressure sensors, hot wires and velocity probes. The inception of the rotating stall and surge is recognized by the presence of pressure fluctuation and velocity disturbances in the gas stream that are obtained through sensors. The necessary measure is then taken by applying proper stall and surge stabilizing control actions. The Lyapunov stability of pseudo Euler-Lagrange systems in the literature is extended to include additional nonlinear terms. Although Lyapunov stability theory is considered as the cornerstone of analysis of nonlinear systems, the generalization of this energy-based method poses a drawback that makes obtaining a Lyapunov function a difficult task. Therefore, proposing a method for generating a Lyapunov function for the control synthesis problem of a class of nonlinear systems is of potential importance. A systematic Lyapunov-based controller synthesis technique for a class of second order systems is addressed in this thesis. It is shown, in terms of stability characteristics, that the proposed technique provides a more robust solution to the compressor surge suppression problem as compared to the feedback linearization and the backstepping methods. The second contribution is a proposed new PWA approximation algorithm. Such an approximation is very important in reducing the complexity of nonlinear systems models while keeping the global validity of the models. The proposed method builds upon previous work on piecewise affine (PWA) approximation methods, which can be used to approximate continuous functions of n-variables by a PWA function. Having computed the PWA model of the stall and surge equations, the suppression problem is then solved by using PWA synthesis techniques. The proposed solution is shown to have higher damping characteristics as compared to the backstepping nonlinear method