209 research outputs found
Stability analysis and stabilization of linear aperiodic sampled-data systems subject to input constraints
Motivados pelo crescente uso de controladores embarcados em diferentes aplicações, onde um protocolo de comunicação Ă© responsável pela transmissĂŁo de dados entre algoritmos computacionais, atuadores e sensores, a análise e o controle de sistemas amostrados foram abordados em muitos trabalhos. Nesse contexto, a amostragem aperiĂłdica pode ser vista como uma abstração matemática empregada para representar, na teoria, o efeito de imperfeições no canal de comunicação, como instabilidades, flutuações e, em alguns casos, perda de pacotes de dados. AlĂ©m disso, devido a limitações fĂsicas dos atuadores, restrições de entrada e, em particular, a saturação sĂŁo onipresentes em problemas reais de controle. Essas restrições sĂŁo fonte de comportamentos nĂŁo-lineares e de degradação do desempenho. Em muitos casos, apenas a estabilidade local (ou regional) do sistema em malha fechada pode ser garantida na presença de restrições e nĂŁo-linearidades de entrada, mesmo para plantas lineares. Este trabalho lida com sistemas lineares amostrados aperiodicamente em que a entrada de controle, sujeita a restrições (por exemplo, saturação), Ă© calculada com base em uma realimentação de estados do sistema. Dois problemas principais sĂŁo abordados. O primeiro consiste na análise de estabilidade da origem de tais sistemas com a determinação de estimativas da regiĂŁo de atração da origem (RAO). O segundo, por sua vez, corresponde ao projeto de controle, onde uma lei de controle de realimentação de estados Ă© calculada para otimizar o tamanho de uma estimativa da RAO do sistema em malha fechada resultante. Os mĂ©todos propostos sĂŁo baseados no uso de programação semidefinida ou linear e, portanto, podem ser facilmente aplicados na prática. Um dos mĂ©todos propostos considera uma realimentação de estados linear sujeita a saturação e funções de Lyapunov quadráticas, resultando em estimativas elipsoidais da RAO do sistema. Dois outros mĂ©todos lidam com a análise de estabilidade do sistema amostrado sujeito a saturação fornecendo estimativas poliedrais da RAO. Devido Ă sua flexibilidade, a adoção de poliedros em vez de elipsĂłides permite uma redução de conservadorismo, mas Ă© muito exigente em termos de complexidade computacional. Motivada por esse fato, esta tese tambĂ©m propõe um mĂ©todo de projeto de controle baseado em uma estratĂ©gia alternativa, onde a complexidade dos poliedros Ă© fixada a priori. Essa ideia resulta em um problema de otimização com restrições bilineares, onde uma lei de controle linear por partes estabilizadora de complexidade relativamente baixa Ă© encontrada para o sistema amostrado. Os mĂ©todos mencionados acima consideram uma abordagem nĂŁo-estocástica, onde limites inferior e superior sĂŁo impostos para o intervalo de amostragem do sistema, o qual Ă© desconhecido e variante no tempo. Como contribuição adicional, esta tese tambĂ©m considera uma abordagem estocástica. Um mĂ©todo de projeto de controle Ă© proposto para a estabilização global no sentido quadrático mĂ©dio do sistema amostrado, onde a lei de realimentação de estados linear Ă© sujeita a nĂŁo-linearidades que satisfazem a uma condição de setor e os intervalos de amostragem correspondem a variáveis aleatĂłrias com a distribuição de Erlang. A possibilidade de perda de pacotes de dados tambĂ©m Ă© explicitamente levada em consideração atravĂ©s da distribuição de Bernoulli. AlĂ©m disso, o mĂ©todo proposto, que se baseia na teoria de processos de Markov determinĂsticos por partes, resulta em condições de estabilização nĂŁo-conservadoras no caso linear sem restrições de entrada.Motivated by the growing use of embedded controllers in different applications, where a communication protocol is responsible for the transmission of data between computer algorithms, actuators and sensors, the analysis and control design for sampled-data control systems have been addressed in many works. In this context, aperiodic sampling can be seen as a modeling abstraction employed to represent, in a theoretical framework, the effect of imperfections on the communication channel such as sampling jitters, fluctuations and, in some cases, packet dropouts. Moreover, due to physical limitations of actuators, input constraints and, in particular, input saturation are ubiquitous in real control problems. These constraints are source of nonlinear behaviors and performance degradation. In many cases, only local (or regional) stability of the closed-loop system can be ensured in the presence of actuators constraints and nonlinearities, even for linear plants. This work deals with linear aperiodic sampled-data systems where the control input, subject to constraints (e.g. saturation), is computed based on a feedback of the system state. It focuses on two main problems. The first one regards the stability analysis of the origin of such systems, with the determination of estimates of the region of attraction of the origin (RAO). The second one, in turn, corresponds to the control design, where a state-feedback control law is computed in order to enlarge an estimate of the RAO of the resulting closed-loop system. The proposed methods are based on the use of semidefinite or linear programming and can therefore be easily applied in practice. One of the proposed methods considers a linear saturating feedback of the system state and quadratic Lyapunov functions, leading to ellipsoidal estimates of the RAO of the system. Two other methods deal with the stability analysis of the sampled-data system subject to input saturation providing polyhedral estimates of the RAO. Because of their flexibility, adopting polyhedrons instead of ellipsoids allows a reduction of conservatism, but is very demanding in terms of computational complexity. Motivated by this fact, this thesis also proposes a control design method based on an alternative strategy, where the complexity of the polytopes is fixed a priori. This idea results in an optimization problem with bilinear constraints, where a stabilizing piecewise linear control law of relatively low complexity is found for the sampled-data system. The aforementioned methods consider a non-stochastic framework, where lower and upper bounds are imposed for the unknown, time-varying sampling interval of the system. As an additional contribution, this thesis also considers a stochastic setting. A control design method is proposed for the global stabilization in the mean square sense of the sampled-data system, where the linear feedback control law is subject to sector bounded nonlinearities and the sampling intervals are assumed to be random variables with the Erlang distribution. The possibility of packet dropouts is also explicitly taken into account through the Bernoulli distribution. Moreover, the proposed approach, which is based onthe framework of Piecewise Deterministic Markov Processes, leads to non-conservative stabilization conditions in the unconstrained linear case.MotivĂ© par l’utilisation croissante de contrĂ´leurs embarquĂ©s dans diffĂ©rentes applications, oĂą un protocole de communication est responsable par la transmission de donnĂ©es entre les algorithmes numĂ©riques, les actionneurs et les capteurs, l’analyse et la conception de contrĂ´le pour les systèmes de contrĂ´le Ă©chantillonnĂ©es ont Ă©tĂ© abordĂ©es dans de nombreux travaux. Dans ce contexte, l’échantillonnage apĂ©riodique peut ĂŞtre considĂ©rĂ© comme une abstraction mathĂ©matique employĂ©e pour reprĂ©senter, dans un cadre thĂ©orique, l’effet des imperfections sur le canal de communication telles que la gigue d’échantillonnage, les fluctuations et, dans certains cas, les pertes de paquets. De plus, en raison des limitations physiques des actionneurs, les contraintes d’entrĂ©e et, en particulier, la saturation des entrĂ©es sont omniprĂ©sentes dans les problèmes de contrĂ´le rĂ©els. Ces contraintes sont une source de comportements non-linĂ©aires et de dĂ©gradation de la performance. Dans de nombreux cas, seule la stabilitĂ© locale (ou rĂ©gionale) du système en boucle fermĂ©e peut ĂŞtre assurĂ©e en prĂ©sence de contraintes et de non-linĂ©aritĂ©s des actionneurs, mĂŞme pour les systèmes linĂ©aires. Ce travail traite des systèmes linĂ©aires Ă©chantillonnĂ©es apĂ©riodiquement oĂą l’entrĂ©e de commande, soumise Ă des contraintes (par exemple la saturation), est calculĂ©e sur la base d’un retour d’état du système. Il se concentre sur deux problèmes principaux. Le premier consiste en l’analyse de stabilitĂ© de l’origine de tels systèmes avec la dĂ©termination d’estimations de la rĂ©gion d’attraction de l’origine (RAO). Le deuxième, Ă son tour, correspond Ă la conception de la commande, oĂą une loi de commande Ă retour d’état est calculĂ©e afin d’agrandir une estimation de la RAO du système en boucle fermĂ©e rĂ©sultant. Les mĂ©thodes proposĂ©es sont basĂ©es sur la programmation semi-dĂ©finie ou linĂ©aire et peuvent donc ĂŞtre facilement appliquĂ©es dans la pratique. L’une des mĂ©thodes proposĂ©es considère un retour d’état linĂ©aire soumis Ă la saturation et des fonctions de Lyapunov quadratiques, conduisant Ă des estimations ellipsoĂŻdales de la RAO du système. Deux autres mĂ©thodes traitent de l’analyse de stabilitĂ© du système Ă©chantillonnĂ© soumis Ă la saturation des entrĂ©es fournissant des estimations polyĂ©driques de la RAO. En raison de leur flexibilitĂ©, l’adoption de polyèdres au lieu d’ellipsoĂŻdes permet une rĂ©duction du conservatisme mais est très exigeante en termes de complexitĂ© de calcul. MotivĂ©e par ce fait, cette thèse propose Ă©galement une mĂ©thode de conception de contrĂ´le basĂ©e sur une stratĂ©gie alternative, oĂą la complexitĂ© des polyèdres est fixĂ©e a priori. Cette idĂ©e se traduit par un problème d’optimisation avec des contraintes bilinĂ©aires, oĂą une loi de commande linĂ©aire par morceaux stabilisante de complexitĂ© relativement faible est trouvĂ©e pour le système Ă©chantillonnĂ©. Les mĂ©thodes mentionnĂ©es ci-dessus considèrent un cadre non stochastique, oĂą des limites infĂ©rieure et supĂ©rieure sont imposĂ©es pour l’intervalle d’échantillonnage inconnu et variable dans le temps du système. Comme contribution supplĂ©mentaire, cette thèseconsidère Ă©galement un cadre stochastique. Une mĂ©thode de conception de contrĂ´le est proposĂ©e pour la stabilisation globale dans le sens quadratique moyen du système Ă©chantillonnĂ©, oĂą la loi de contrĂ´le linĂ©aire de retour d’état est soumise Ă des non-linĂ©aritĂ©s dĂ©limitĂ©es par secteur et les intervalles d’échantillonnage sont supposĂ©s ĂŞtre des variables alĂ©atoires avec la distribution d’Erlang. La possibilitĂ© de pertes de paquets est aussi explicitement prise en compte via la distribution de Bernoulli. De plus, l’approche proposĂ©e, qui est basĂ©e sur le cadre des processus de Markov dĂ©terministes par morceaux, conduit Ă des conditions de stabilisation non conservatrices dans le cas linĂ©aire sans contraintes
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For an unknown linear system, starting from noisy open-loop input-state data
collected during a finite-length experiment, we directly design a linear
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Correct-By-Construction Control Synthesis for Systems with Disturbance and Uncertainty
This dissertation focuses on correct-by-construction control synthesis for Cyber-Physical Systems (CPS) under model uncertainty and disturbance. CPSs are systems that interact with the physical world and perform complicated dynamic tasks where safety is often the overriding factor. Correct-by-construction control synthesis is a concept that provides formal performance guarantees to closed-loop systems by rigorous mathematic reasoning. Since CPSs interact with the environment, disturbance and modeling uncertainty are critical to the success of the control synthesis. Disturbance and uncertainty may come from a variety of sources, such as exogenous disturbance, the disturbance caused by co-existing controllers and modeling uncertainty. To better accommodate the different types of disturbance and uncertainty, the verification and control synthesis methods must be chosen accordingly. Four approaches are included in this dissertation. First, to deal with exogenous disturbance, a polar algorithm is developed to compute an avoidable set for obstacle avoidance. Second, a supervised learning based method is proposed to design a good student controller that has safety built-in and rarely triggers the intervention of the supervisory controller, thus targeting the design of the student controller. Third, to deal with the disturbance caused by co-existing controllers, a Lyapunov verification method is proposed to formally verify the safety of coexisting controllers while respecting the confidentiality requirement. Finally, a data-driven approach is proposed to deal with model uncertainty. A minimal robust control invariant set is computed for an uncertain dynamic system without a given model by first identifying the set of admissible models and then simultaneously computing the invariant set while selecting the optimal model. The proposed methods are applicable to many real-world applications and reflect the notion of using the structure of the system to achieve performance guarantees without being overly conservative.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145933/1/chenyx_1.pd
Actuation-Aware Simplified Dynamic Models for Robotic Legged Locomotion
In recent years, we witnessed an ever increasing number of successful hardware implementations of motion planners for legged robots. If one common property is to be identified among these real-world applications, that is the ability of online planning.
Online planning is forgiving, in the sense that it allows to relentlessly compensate for external disturbances of whatever form they might be, ranging from unmodeled dynamics to external pushes or unexpected obstacles and, at the same time, follow user commands. Initially replanning was restricted only to heuristic-based planners that exploit the low computational effort of simplified dynamic models. Such models deliberately only capture the main dynamics of the system, thus leaving to the controllers the issue of anchoring the desired trajectory to the whole body model of the robot. In recent years, however, we have seen a number of new approaches attempting to increase the accuracy of the dynamic formulation without trading-off the computational efficiency of simplified models.
In this dissertation, as an example of successful hardware implementation of heuristics and simplified model-based locomotion, I describe the framework that I developed for the generation of an omni-directional bounding gait for the HyQ quadruped robot. By analyzing the stable limit cycles for the sagittal dynamics and the Center of Pressure (CoP) for the lateral stabilization, the described locomotion framework is able to achieve a stable bounding while adapting to terrains of mild roughness and to sudden changes of the user desired linear and angular velocities.
The next topic reported and second contribution of this dissertation is my effort to formulate more descriptive simplified dynamic models, without trading off their computational efficiency, in order to extend the navigation capabilities of legged robots to complex geometry environments. With this in mind, I investigated the possibility of incorporating feasibility constraints in these template models and, in particular, I focused on the joint torques limits which are usually neglected at the planning stage.
In this direction, the third contribution discussed in this thesis is the formulation of the so called actuation wrench polytope (AWP), defined as the set of feasible wrenches that an articulated robot can perform given its actuation limits. Interesected with the contact wrench cone (CWC), this yields a new 6D polytope that we name feasible wrench polytope (FWP), defined as the set of all wrenches that a legged robot can realize given its actuation capabilities and the friction constraints. Results are reported where, thanks to efficient computational geometry algorithms and to appropriate approximations, the FWP is employed for a one-step receding horizon optimization of center of mass trajectory and phase durations given a predefined step sequence on rough terrains.
For the sake of reachable workspace augmentation, I then decided to trade off the generality of the FWP formulation for a suboptimal scenario in which a quasi-static motion is assumed.
This led to the definition of the, so called, local/instantaneous actuation region and of the global actuation/feasible region. They both can be seen as different variants of 2D linear subspaces orthogonal to gravity where the robot is guaranteed to place its own center of mass while being able to carry its own body weight given its actuation capabilities. These areas can be intersected with the well known frictional support region, resulting in a 2D linear feasible region, thus providing an intuitive tool that enables the concurrent online optimization of actuation consistent CoM trajectories and target foothold locations on rough terrains
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In this paper, we focus on non-conservative obstacle avoidance between robots
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non-smooth, such that distance constraints cannot be enforced directly in the
optimization problem. To handle this challenge, we employ non-smooth control
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using a quadratic program. Our approach is proven to guarantee system safety.
We theoretically analyze the smoothness properties of the minimum distance
quadratic program and its KKT conditions. We validate our approach by
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robotic systems with strictly convex and polytopic shapes. To our best
knowledge, this is the first time a real-time QP problem can be formulated for
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