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 é responsável pela transmissão 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 aperiódica 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. Além disso, devido a limitações físicas dos atuadores, restrições de entrada e, em particular, a saturação são onipresentes em problemas reais de controle. Essas restrições são fonte de comportamentos não-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 não-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), é calculada com base em uma realimentação de estados do sistema. Dois problemas principais são abordados. O primeiro consiste na análise de estabilidade da origem de tais sistemas com a determinação de estimativas da região de atração da origem (RAO). O segundo, por sua vez, corresponde ao projeto de controle, onde uma lei de controle de realimentação de estados é calculada para otimizar o tamanho de uma estimativa da RAO do sistema em malha fechada resultante. Os métodos propostos são baseados no uso de programação semidefinida ou linear e, portanto, podem ser facilmente aplicados na prática. Um dos métodos 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 métodos lidam com a análise de estabilidade do sistema amostrado sujeito a saturação fornecendo estimativas poliedrais da RAO. Devido à sua flexibilidade, a adoção de poliedros em vez de elipsóides permite uma redução de conservadorismo, mas é muito exigente em termos de complexidade computacional. Motivada por esse fato, esta tese também propõe um método de projeto de controle baseado em uma estratégia alternativa, onde a complexidade dos poliedros é 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 é encontrada para o sistema amostrado. Os métodos mencionados acima consideram uma abordagem não-estocástica, onde limites inferior e superior são impostos para o intervalo de amostragem do sistema, o qual é desconhecido e variante no tempo. Como contribuição adicional, esta tese também considera uma abordagem estocástica. Um método de projeto de controle é proposto para a estabilização global no sentido quadrático médio do sistema amostrado, onde a lei de realimentação de estados linear é sujeita a não-linearidades que satisfazem a uma condição de setor e os intervalos de amostragem correspondem a variáveis aleatórias com a distribuição de Erlang. A possibilidade de perda de pacotes de dados também é explicitamente levada em consideração através da distribuição de Bernoulli. Além disso, o método proposto, que se baseia na teoria de processos de Markov determinísticos por partes, resulta em condições de estabilização não-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.Motivé par l’utilisation croissante de contrôleurs embarqués dans différentes applications, où un protocole de communication est responsable par la transmission de données entre les algorithmes numériques, les actionneurs et les capteurs, l’analyse et la conception de contrôle pour les systèmes de contrôle échantillonnées ont été abordées dans de nombreux travaux. Dans ce contexte, l’échantillonnage apériodique peut être considéré comme une abstraction mathématique employée pour représenter, dans un cadre théorique, 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’entrée et, en particulier, la saturation des entrées sont omniprésentes dans les problèmes de contrôle réels. Ces contraintes sont une source de comportements non-linéaires et de dégradation de la performance. Dans de nombreux cas, seule la stabilité locale (ou régionale) du système en boucle fermée peut être assurée en présence de contraintes et de non-linéarités des actionneurs, même pour les systèmes linéaires. Ce travail traite des systèmes linéaires échantillonnées apériodiquement où l’entrée de commande, soumise à des contraintes (par exemple la saturation), est calculée sur la base d’un retour d’état du système. Il se concentre sur deux problèmes principaux. Le premier consiste en l’analyse de stabilité de l’origine de tels systèmes avec la détermination d’estimations de la région d’attraction de l’origine (RAO). Le deuxième, à son tour, correspond à la conception de la commande, où une loi de commande à retour d’état est calculée afin d’agrandir une estimation de la RAO du système en boucle fermée résultant. Les méthodes proposées sont basées sur la programmation semi-définie ou linéaire et peuvent donc être facilement appliquées dans la pratique. L’une des méthodes proposées considère un retour d’état linéaire soumis à la saturation et des fonctions de Lyapunov quadratiques, conduisant à des estimations ellipsoïdales de la RAO du système. Deux autres méthodes traitent de l’analyse de stabilité du système échantillonné soumis à la saturation des entrées fournissant des estimations polyédriques de la RAO. En raison de leur flexibilité, l’adoption de polyèdres au lieu d’ellipsoïdes permet une réduction du conservatisme mais est très exigeante en termes de complexité de calcul. Motivée par ce fait, cette thèse propose également une méthode de conception de contrôle basée sur une stratégie alternative, où la complexité des polyèdres est fixée a priori. Cette idée se traduit par un problème d’optimisation avec des contraintes bilinéaires, où une loi de commande linéaire par morceaux stabilisante de complexité relativement faible est trouvée pour le système échantillonné. Les méthodes mentionnées ci-dessus considèrent un cadre non stochastique, où des limites inférieure et supérieure sont imposées pour l’intervalle d’échantillonnage inconnu et variable dans le temps du système. Comme contribution supplémentaire, cette thèseconsidère également un cadre stochastique. Une méthode de conception de contrôle est proposée pour la stabilisation globale dans le sens quadratique moyen du système échantillonné, où la loi de contrôle linéaire de retour d’état est soumise à des non-linéarités délimitées par secteur et les intervalles d’échantillonnage sont supposés être des variables aléatoires avec la distribution d’Erlang. La possibilité de pertes de paquets est aussi explicitement prise en compte via la distribution de Bernoulli. De plus, l’approche proposée, qui est basée sur le cadre des processus de Markov déterministes par morceaux, conduit à des conditions de stabilisation non conservatrices dans le cas linéaire sans contraintes

Controller design for robust invariance from noisy data

For an unknown linear system, starting from noisy open-loop input-state data collected during a finite-length experiment, we directly design a linear feedback controller that guarantees robust invariance of a given polyhedral set of the state in the presence of disturbances. The main result is a necessary and sufficient condition for the existence of such a controller, and amounts to the solution of a linear program. The benefits of large and rich data sets for the solution of the problem are discussed. A numerical example about a simplified platoon of two vehicles illustrates the method

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

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

Regelungstheorie

The workshop “Regelungstheorie” (control theory) covered a broad variety of topics that were either concerned with fundamental mathematical aspects of control or with its strong impact in various fields of engineering

Invariant Set-based Methods for the Computation of Input and Disturbance Sets

This dissertation presents new methods to synthesize disturbance sets and input constraints set for constrained linear time-invariant systems. Broadly, we formulate and solve optimization problems that (a) compute disturbance sets such that the reachable set of outputs approximates an assigned set, and (b) compute input constraint sets guaranteeing the stabilizability of a given set of initial conditions. The proposed methods find application in the synthesis and analysis of several control schemes such as decentralized control, reduced-order control, etc., as well as in practical system design problems such as actuator selection, etc. The key tools supporting the develpment of the aforementioned methods are Robust Positive Invariant (RPI) sets. In particular, the problems that we formulate are such that they co-synthesize disturbance/input constraint sets along with the associated RPI sets. This requires embedding existing techniques to compute RPI sets within an optimization problem framework, that we facilitate by developing new results related to properties of RPI sets, polytope representations, inclusion encoding techniques, etc. In order to solve the resulting optimization problems, we develop specialized structure-exploiting solvers that we numerically demonstrate to outperform conventional solution methods. We also demonstrate several applications of the methods we propose for control design. Finally, we extend the methods to tackle data-driven control synthesis problems in an identification-for-control framework