Energy Shaping of Underactuated Systems via Interconnection and Damping Assignment Passivity-Based Control with Applications to Planar Biped Robots

The sought goal of this thesis is to show that total energy shaping is an effective and versatile tool to control underactuated mechanical systems. The performance of several approaches, rooted in the port-Hamiltonian formalism, are analyzed while tackling distinct control problems: i) equilibrium stabilization; ii) gait generation; iii) gait robustication. Firstly, a constructive solution to deal with interconnection and damping assignment passivity-based control (IDA-PBC) for underactuated two-degree-of-freedom mechanical systems is proposed. This strategy does not involve the resolution of any partial differential equation, since explicit solutions are given, while no singularities depending on generalized momenta are introduced by the controller. The methodology is applied to the stabilization of a translational oscillator with a rotational actuator system, as well as, to the gait generation for an underactuated compass-like biped robot (CBR). Then, the problem of gait generation is addressed using dissipative forces in the controller. In this sense, three distinct controllers are presented, namely simultaneous interconnection and damping assignment passivity-based control with dissipative forces, energy pumping-and-damping passivity-based control (EPD-PBC), and energy pumping-or-damping control. Finally, EPD-PBC is used to increase the robustness of the gait exhibited by the CBR over uncertainties on the initial conditions. The passivity of the system is exploited, as well as, its hybrid nature (using the hybrid zero dynamics method) to carry out the stability analysis. Besides, such an approach is applied to new gaits that are generated using IDA-PBC. Numerical case studies, comparisons, and critical discussions evaluate the performance of the proposed approaches

Contributions to ida-pbc with adaptive control for underactuated mechanical systems

This master thesis is devoted to developing an adaptive control scheme for the well- known Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) technique. The main objective of this adaptive scheme is to asymptotically stabilize a class of Underactuated Mechanical Systems (UMSs) in the presence of uncertainties (not necessarily matched). This class of UMSs is characterized by the solvability of the Partial Differential Equation (PDE) resulting from the IDA-PBC technique. Two propositions are stated in this work to design the adaptive IDA-PBC. One of the main properties of these propositions is that even though the parameter estimation conver- gence is not guaranteed, the adaptive IDA-PBC achieves asymptotic stabilization. To illustrate the effectiveness of these propositions, this work performs simulations of the Inertia Wheel Inverted Pendulum (IWIP) system, considering a time-dependent input disturbance, a type of physical damping, i.e., friction (not considered in the standard IDA-PBC methodology), and parameter uncertainties in the system (e.g., inertia).Tesi

Modeling, Simulation and Control of Doubly-Fed Induction Machine Controlled by Back-to-Back converter

Aquesta Tesi estudia el control d'un sistema complex, un sistema d'emmagatzement d'energia cinètica, incloent les seves especificacions de control, modelat, disseny de controladors, simulacions, muntatge i validació experimental.Primerament, s'estudia l'interconnexió i control dels sistemes electromecànics. Es presenta el formalisme Hamiltonià (PCHS) en general, i després particularitzant en els sistemes electromecànics, inclòs els sistemes d'estructura variable (VSS).L'IDA-PBC (Interconnection and damping assignment-passivity based control) és una tècnica de control basat en els PCHS. En aquesta Tesi s'estudien el problemes que apareixen en controlar, per IDA-PBC, sortides de grau relatiu u quan el paràmetres nominals del controlador són incerts. Per evitar-los es proposa introduir una acció integral que pot ésser interpretada dins l'estructura Hamiltoniana.En aquesta Tesi també es presenten dues modificacions que permeten millorar el rang d'aplicacions de la tècnica IDA-PBC. Primer, es demostra que el fet de descomposar la tècnica de l'IDA-PBC en deformar la funció d'energia i una injecció de fregament, redueix el conjunt de sistemes que es poden estabilitzar mitjançant aquest mètode. Per evitar aquest problema, es proposa fer simultàniament els dos passos donant lloc a l'anomenat SIDA-PBC. Per altre costat, el mètode IDA-PBC requereix el coneixement de la funció energia (o Hamiltonià). Això representa un problema perquè, en general, el punt d'equilibri depèn de paràmetres incerts. En aquest treball es desenvolupa una metodologia per seleccionar l'estructura Hamiltoniana que redueix aquesta dependència dels paràmetres. Aquesta tècnica permet millorar la robustesa dels les sortides d'ordre relatiu superior a u.El sistema d'emmagatzement d'energia cinètica consisteix en una màquina d'inducció doblament alimentada (DFIM) amb un volant d'inèrcia, controlada pel rotor per un convertidor de potència back-to-back (B2B). L'objectiu és gestionar el flux d'energia entre la DFIM i una càrrega local connectada a la xarxa, commutant entre diferents punts de funcionament. Per això es planteja una gestió de l'energia, basada en la velocitat òptima de la DFIM.Pel què fa al control de la DFIM, es proposa un nou esquema de control que ofereix importants avantatges, i que és considerablement més senzill que el mètode clàssic, el vector control. Aquest nou controlador permet una fàcil descomposició de les potències activa i reactiva de l'estator, i el seu control a través de les tensions de rotor. Aquest disseny s'obté aplicant el procediment que millora la robustesa de l'IDA-PBC.S'han estudiat d'altres controladors, com el vector control clàssic. També a partir de la tècnica IDA-PBC, on l'equació en derivades parcials que apareix en aplicar el mètode es pot resoldre fixant l'energia en llaç tancat, i afegint nous termes a la matriu d'interconnexió. Per obtenir un controlador definit globalment s'afegeix un terme de fregament depenent dels estats, que desacobla la part elèctrica i mecànica del sistema. Finalment, també es prova que mitjançant el SIDA-PBC es pot modelar l'energia total (elèctrica i mecànica) de la DFIM. Tots aquest controladors han estat simulats i comparats. El controlador robust IDA-PBC s'ha validat experimentalment amb uns resultats satisfactoris. A la Tesi també es presenta un controlador que permet el flux bidireccional de potència pel B2B. L'estudi de la dinàmica zero adverteix que les tècniques de control estàndard no garanties en l'estabilitat en ambdós direccions, i per això s'utilitza un controlador IDA-PBC. Pel disseny s'utilitza un model basat en GSSA (generalized state space averaging), on es descomposa i es trunca el sistema per determinades freqüències, i que permet expresar els objectius de control (tensió constant al bus de contínua i factor de potència unitari) com un problema de regulació. Les simulacions i els resultats experimentals validen, tant la llei de control, com les simplificacions efectuades.Els controladors proposats i validats experimentalment són usats, finalment, per implementar la gestió de potència del sistema d'emmegatzement d'energia cinètica. Els resultats confirmen el bon comportament del sistema i dels controladors IDA-PBC proposats.This Thesis studies a complex multidomain system, the Flywheel Energy Storage System, including the control objectives specification, modeling, control design, simulation, experimental setup assembling and experimental validation stages.The port interconnection and control of electromechanical systems is studied. The port Hamiltonian formalism is presented in general, and particularized for generalized electromechanical systems, including variable structure systems (VSS).Interconnection and damping assignment-passivity based control (IDA-PBC) is a well known technique for port Hamiltonian systems (PCHS). In this Thesis we point out the kind of problems that can appear in the closed-loop structure obtained by IDA-PBC methodsfor relative degree one outputs, when nominal values are used in a system with uncertain parameters. To correct this, we introduce an integral control, which can be cast into the Hamiltonian framework.This Thesis also presents two new approaches which improve the range of applicability of the IDA-PBC technique. First, we show that the standard two-stage procedure used in IDA-PBC consisting of splitting the control action into the sum of energy-shaping and damping injection terms is not without loss of generality, and effectively reduces the set of systems that can be stabilized with IDA-PBC. To overcome this problem we suggest to carry out simultaneously both stages and refer to this variation of the method as SIDA-PBC.Secondly, we present an improvement of the IDA-PBC technique. The IDA-PBC method requires the knowledge of the full energy (or Hamiltonian) function. This is a problem because, in general, the equilibrium point which is to be regulated depends on uncertain parameters. We show how select the target port-Hamiltonian structure so that this dependence is reduced. This new approach allows to improve the robustness for higher relative degree outputs.The Flywheel Energy Storage System consists of a doubly-fed induction machine (DFIM), controlled through the rotor voltage by a power electronics subsystem (a back-to-back AC/AC converter (B2B)), and coupled to flywheel. The control objective is to optimally regulate the power flow between the DFIM and a local load connected to the grid, and this is achieved by commuting between different steady-state regimes. A police management based on the optimal speed for the DFIM is proposed.In this Thesis we propose a new control scheme for the DFIM that offers significant advantages, and is considerably simpler, than the classical vector control method. This controller allows an easy decomposition of the active and reactive powers on the stator side and their regulation, acting on the rotor voltage, via stator current control. This design was obtained applying the new robust IDA-PBC procedure.Other controllers are also designed along the dissertation. The classical vector control is studied. We also apply the classic IDA-PBC technique. It is shown that the partial differential equation that appears in this method can be circumvented by fixing the desired closed-loop total energy and adding new terms to the interconnection structure. Furthermore, to obtain a globally defined control law we introduce a state--dependent damping term that has the nice interpretation of effectively decoupling the electrical and mechanical parts of the system. This results in a globally convergent controller parameterized by two degrees of freedom. Finally, we also prove that with SIDA-PBC we can shape the total energy of the full (electrical and mechanical) dynamics of the DFIM. These different controllers (vector control, IDA-PBC, SIDA-PBC and robust IDA-PBC) are simulated and compared. The IDA-PBC robust controller is also experimentally tested and shown to work satisfactorily.A controller able to achieve bidirectional power flow for the B2B converter is presented. Standard techniques cannot be used since it is shown that no single output yields a stable zero dynamics for power flowing both ways. The controller is computed using standard IDA-PBC techniques for a suitable generalized state space averaging truncation of the system, which transforms the control objectives, namely constant output voltage dc-bus and unity input power factor, into a regulation problem. Simulation and experimental results for the full system confirm the correctness of the simplifications introduced to obtain the controller.The proposed and tested controllers for the DFIM and the B2B are used to implement the power management policy. These results show a good performance of the flywheel energy storage system and also validate the IDA-PBC technique, with the proposed improvements

Solution to IDA-PBC PDEs by Pfaffian Differential Equations

Finding the general solution of partial differential equations (PDEs) is essential for controller design in some methods. Interconnection and damping assignment passivity based control (IDA-PBC) is one of such methods in which the solution to corresponding PDEs is needed to apply it in practice. In this paper, such PDEs are transformed to corresponding Pfaffian differential equations. Furthermore, it is shown that upon satisfaction of the integrability condition, the solution to the corresponding third order Pfaffian differential equation may be obtained quite easily. The method is applied to the PDEs of IDA-PBC in some benchmark problems such as Magnetic levitation system, Pendubot and underactuated cable driven robot to verify its applicability

Energy-Based Control for the Cart-Pole System in Implicit Port-Hamiltonian Representation

This master thesis is devoted to the design, analysis, and experimental validation of an energy-based control strategy for the well-known benchmark cart-pole system in implicit Port-Hamiltonian (PH) representation. The control scheme performs two tasks: swingup and (local) stabilization. The swing-up controller is carried out on the basis of a generalized energy function and consists of bringing the pendulum trajectories from the lower (stable) position to a limit cycle (homoclinic orbit), which passes by the upright (unstable) position, as well as the cart trajectories to the desired point. The (local) stabilizing controller is designed under a novel algebraic Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) technique and ensures the upright (asymptotic) stabilization of the pendulum as well as the cart at a desired position. To illustrate the effectiveness of the proposed control scheme, this work presents simulations and real-time experiments considering physical damping, i.e., viscous friction. The results are additionally contrasted with another energy-based control strategy for the cart-pole system in explicit Euler-Lagrange (EL) representation.Diese Masterarbeit widmet sich dem Entwurf, der Analyse und der experimentellen Validierung einer energiebasierten Regelstrategie für das bekannte Benchmarksystem des inversen Pendels auf einem Wagen in impliziter Port-Hamiltonscher (PH) Darstellung. Das Regelungssystem erfüllt zwei Aufgaben: das Aufschwingen und (lokale) Stabilisierung. Das Aufschwingen erfolgt auf Grundlage der generalisierten Energiefunktion und besteht darin, sowohl die Trajektorien des Pendels von der unteren (stabilen) Position in einen Grenzzyklus (homokliner Orbit) zu bringen, wobei die (instabile) aufrechte Lage passiert wird, als auch den Wagen in einer gewünschten Position einzustellen. Die (lokale) Regelung zur Stabilisierung ist nach einer neuartigen algebraischen Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) Methode konzipiert und gewährleistet die aufrechte (asymptotische) Stabilisierung des Pendels sowie die Positionierung des Wagens an einem gewünschten Referenzpunkt. Um die Funktionalität des entworfenen Regelungssystems zu veranschaulichen, werden in dieser Masterarbeit Simulationen und Echtzeit-Experimente unter Berücksichtigung der physikalischen Dämpfung, d.h. der viskosen Reibung, vorgestellt. Die Ergebnisse werden zusätzlich mit einem weiteren energiebasierten Regelungsansatz für das System des inversen Pendels auf einem Wagen in expliziter Euler-Lagrange (EL) Darstellung verglichen.Tesi