ISSUES IN THE CONTROL OF HALFSPACE SYSTEMS

By the name HALFSPACE SYSTEMS, this dissertation refers to systems whose dynamics are modeled by linear constraints of the form Exk+1 \u3c= Fxk + Buk (where E, F 2 andlt;mn, B 2 andlt;mp). This dissertation explores the concepts of BOUNDEDNESS, STABILITY, IRREDUNDANCY, and MAINTAINABILITY (which is the same as REACHABILITY OF A TARGET TUBE) that are related to the control of halfspace systems. Given that a halfspace system is bounded, and that a given static target tube is reachable for this system, this dissertation presents algorithms to MAINTAIN the system in this target tube. A DIFFERENCE INCLUSION has the form xk+1 = Axk + Buuk, where xk, xk+1 2 andlt;n, uk 2 andlt;p, A 2 andlt;nn, Bu 2 andlt;np, Ai 2 andlt;nn, Bj 2 andlt;np, and A and Bu belong to the convex hulls of (A1,A2, . . . ,Aq) and (B1, B2, . . . , Br) respectively. This dissertation investigates the possibility that halfspace systems have equivalent difference inclusion representation for the case of uk = 0. An affirmitive result in this direction may make it possible to apply to halfspace systems the control theory that exists for difference inclusions

Primal-dual interior-point algorithms for linear programs with many inequality constraints

Linear programs (LPs) are one of the most basic and important classes of constrained optimization problems, involving the optimization of linear objective functions over sets defined by linear equality and inequality constraints. LPs have applications to a broad range of problems in engineering and operations research, and often arise as subproblems for algorithms that solve more complex optimization problems. ``Unbalanced'' inequality-constrained LPs with many more inequality constraints than variables are an important subclass of LPs. Under a basic non-degeneracy assumption, only a small number of the constraints can be active at the solution--it is only this active set that is critical to the problem description. On the other hand, the additional constraints make the problem harder to solve. While modern ``interior-point'' algorithms have become recognized as some of the best methods for solving large-scale LPs, they may not be recommended for unbalanced problems, because their per-iteration work does not scale well with the number of constraints. In this dissertation, we investigate "constraint-reduced'' interior-point algorithms designed to efficiently solve unbalanced LPs. At each iteration, these methods construct search directions based only on a small working set of constraints, while ignoring the rest. In this way, they significantly reduce their per-iteration work and, hopefully, their overall running time. In particular, we focus on constraint-reduction methods for the highly efficient primal-dual interior-point (PDIP) algorithms. We propose and analyze a convergent constraint-reduced variant of Mehrotra's predictor-corrector PDIP algorithm, the algorithm implemented in virtually every interior-point software package for linear (and convex-conic) programming. We prove global and local quadratic convergence of this algorithm under a very general class of constraint selection rules and under minimal assumptions. We also propose and analyze two regularized constraint-reduced PDIP algorithms (with similar convergence properties) designed to deal directly with a type of degeneracy that constraint-reduced interior-point algorithms are often subject to. Prior schemes for dealing with this degeneracy could end up negating the benefit of constraint-reduction. Finally, we investigate the performance of our algorithms by applying them to several test and application problems, and show that our algorithms often outperform alternative approaches

Robust Spline Path Following for Redundant Mechanical Systems

Path following controllers make the output of a control system approach and traverse a pre-specified path with no a priori time-parametrization. The first part of the thesis implements a path following controller for a simple class of paths, based on transverse feedback linearization (TFL), which guarantees invariance of the path to be followed. The coordinate and feedback transformation employed allows one to easily design control laws to generate arbitrary desired motions on the path for the closed-loop system. The approach is applied to an uncertain and simplified model of a fully actuated robot manipulator for which none of the dynamic parameters are measured. The controller is made robust to modelling uncertainties using Lyapunov redesign. The experimental results show a substantial improvement when using the robust controller for path following versus standard state feedback. In the second part of the thesis, the class of paths and systems considered are extended. We present a method for path following control design applicable to framed curves generated by spline interpolating waypoints in the workspace of kinematically redundant mechanical systems. The class of admissible paths include self-intersecting curves. Kinematic redundancies of the system are resolved by designing controllers that solve a suitably defined constrained quadratic optimization problem that can be easily tuned by the designer to achieve various desired poses. The class of redundant systems considered include mobile manipulators for a large class of wheeled ground vehicles. The result is a path following controller that simultaneously controls the manipulator and mobile base, without any trajectory planning performed on the mobile base. The approach is experimentally verified using the robust controller developed in the first part of the thesis on a 4-degree-of-freedom (4DOF) redundant manipulator and a mobile manipulator system with a differential drive base

Motion Control of the Hybrid Wheeled-Legged Quadruped Robot Centauro

Emerging applications will demand robots to deal with a complex environment, which lacks the structure and predictability of the industrial workspace. Complex scenarios will require robot complexity to increase as well, as compared to classical topologies such as fixed-base manipulators, wheeled mobile platforms, tracked vehicles, and their combinations. Legged robots, such as humanoids and quadrupeds, promise to provide platforms which are flexible enough to handle real world scenarios; however, the improved flexibility comes at the cost of way higher control complexity. As a trade-off, hybrid wheeled-legged robots have been proposed, resulting in the mitigation of control complexity whenever the ground surface is suitable for driving. Following this idea, a new hybrid robot called Centauro has been developed inside the Humanoid and Human Centered Mechatronics lab at Istituto Italiano di Tecnologia (IIT). Centauro is a wheeled-legged quadruped with a humanoid bi-manual upper-body. Differently from other platform of similar concept, Centauro employs customized actuation units, which provide high torque outputs, moderately fast motions, and the possibility to control the exerted torque. Moreover, with more than forty motors moving its limbs, Centauro is a very redundant platform, with the potential to execute many different tasks at the same time. This thesis deals with the design and development of a software architecture, and a control system, tailored to such a robot; both wheeled and legged locomotion strategies have been studied, as well as prioritized, whole-body and interaction controllers exploiting the robot torque control capabilities, and capable to handle the system redundancy. A novel software architecture, made of (i) a real-time robotic middleware, and (ii) a framework for online, prioritized Cartesian controller, forms the basis of the entire work

A Copositive Framework for Analysis of Hybrid Ising-Classical Algorithms

Recent years have seen significant advances in quantum/quantum-inspired technologies capable of approximately searching for the ground state of Ising spin Hamiltonians. The promise of leveraging such technologies to accelerate the solution of difficult optimization problems has spurred an increased interest in exploring methods to integrate Ising problems as part of their solution process, with existing approaches ranging from direct transcription to hybrid quantum-classical approaches rooted in existing optimization algorithms. While it is widely acknowledged that quantum computers should augment classical computers, rather than replace them entirely, comparatively little attention has been directed toward deriving analytical characterizations of their interactions. In this paper, we present a formal analysis of hybrid algorithms in the context of solving mixed-binary quadratic programs (MBQP) via Ising solvers. We show the exactness of a convex copositive reformulation of MBQPs, allowing the resulting reformulation to inherit the straightforward analysis of convex optimization. We propose to solve this reformulation with a hybrid quantum-classical cutting-plane algorithm. Using existing complexity results for convex cutting-plane algorithms, we deduce that the classical portion of this hybrid framework is guaranteed to be polynomial time. This suggests that when applied to NP-hard problems, the complexity of the solution is shifted onto the subroutine handled by the Ising solver

Advances in gain-scheduling and fault tolerant control techniques

This thesis presents some contributions to the state-of-the-art of the fields of gain-scheduling and fault tolerant control (FTC). In the area of gain-scheduling, the connections between the linear parameter varying (LPV) and Takagi-Sugeno (TS) paradigms are analyzed, showing that the methods for the automated generation of models by nonlinear embedding and by sector nonlinearity, developed for one class of systems, can be easily extended to deal with the other class. Then, two measures, based on the notions of overboundedness and region of attraction estimates, are proposed in order to compare different models and choose which one can be considered the best one. Later, the problem of designing state-feedback controllers for LPV systems has been considered, providing two main contributions. First, robust LPV controllers that can guarantee some desired performances when applied to uncertain LPV systems are designed, by using a double-layer polytopic description that takes into account both the variability due to the varying parameter vector and the uncertainty. Then, the idea of designing the controller in such a way that the required performances are scheduled by the varying parameters is explored, which provides an elegant way to vary online the behavior of the closed-loop system. In both cases, the problem reduces to finding a solution to a finite number of linear matrix inequalities (LMIs), which can be done efficiently using the available solvers. In the area of fault tolerant control, the thesis first shows that the aforementioned double-layer polytopic framework can be used for FTC, in such a way that different strategies (passive, active and hybrid) are obtained depending on the amount of available information. Later, an FTC strategy for LPV systems that involves a reconfigured reference model and virtual actuators is developed. It is shown that by including the saturations in the reference model equations, it is possible to design a model reference FTC system that automatically retunes the reference states whenever the system is affected by saturation nonlinearities. In this way, a graceful performance degradation in presence of actuator saturations is incorporated in an elegant way. Finally, the problem of FTC of unstable LPV systems subject to actuator saturations is considered. In this case, the design of the virtual actuator is performed in such a way that the convergence of the state trajectory to zero is assured despite the saturations and the appearance of faults. Also, it is shown that it is possible to obtain some guarantees about the tolerated delay between the fault occurrence and its isolation, and that the nominal controller can be designed so as to maximize the tolerated delay.Aquesta tesi presenta diverses contribucions a l'estat de l'art del control per planificació del guany i del control tolerant a fallades (FTC). Pel que fa al control per planificació del guany, s'analitzen les connexions entre els paradigmes dels sistemes lineals a paràmetres variants en el temps (LPV) i de Takagi-Sugeno (TS). Es demostra que els mètodes per a la generació automàtica de models mitjançant encastament no lineal i mitjançant no linealitat sectorial, desenvolupats per una classe de sistemes, es poden estendre fàcilment per fer-los servir amb l'altra classe. Es proposen dues mesures basades en les nocions de sobrefitació i d'estimació de la regió d'atracció, per tal de comparar diferents models i triar quin d'ells pot ser considerat el millor. Després, es considera el problema de dissenyar controladors per realimentació d'estat per a sistemes LPV, proporcionant dues contribucions principals. En primer lloc, fent servir una descripció amb doble capa politòpica que té en compte tant la variabilitat deguda al vector de paràmetres variants i la deguda a la incertesa, es dissenyen controladors LPV robustos que puguin garantir unes especificacions desitjades quan s'apliquen a sistemes LPV incerts. En segon lloc, s'explora la idea de dissenyar el controlador de tal manera que les especificacions requerides siguin programades pels paràmetres variants. Això proporciona una manera elegant de variar en línia el comportament del sistema en llaç tancat. En tots dos casos, el problema es redueix a trobar una solució d'un nombre finit de desigualtats matricials lineals (LMIs), que es poden resoldre fent servir algorismes numèrics disponibles i molt eficients. En l'àrea del control tolerant a fallades, primerament la tesi mostra que la descripció amb doble capa politòpica abans esmentada es pot utilitzar per fer FTC, de tal manera que, en funció de la quantitat d'informació disponible, s'obtenen diferents estratègies (passiva, activa i híbrida). Després, es desenvolupa una estratègia de FTC per a sistemes LPV que fa servir un model de referència reconfigurat combinat amb la tècnica d'actuadors virtuals. Es mostra que mitjançant la inclusió de les saturacions en les equacions del model de referència, és possible dissenyar un sistema de control tolerant a fallades que resintonitza automàticament els estats de referència cada vegada que el sistema es veu afectat per les no linealitats de la saturació en els actuadors. D'aquesta manera s'incorpora una degradació elegant de les especificacions en presència de saturacions d'actuadors. Finalment, es considera el problema de FTC per sistemes LPV inestables afectats per saturacions d'actuadors. En aquest cas, es porta a terme el disseny de l'actuador virtual de tal manera que la convergència a zero de la trajectòria d'estat està assegurada tot i les saturacions i l'aparició de fallades. A més, es mostra que és possible obtenir garanties sobre el retard tolerat entre l'aparició d'una fallada i el seu aïllament, i que el controlador nominal es pot dissenyar maximitzant el retard tolerat

Predictive Control of Linear Uncertain Systems

Predictive control is a very useful tool in controlling constrained systems, since the constraints can be satisfied explicitly by the optimisations. Sets, namely, reachable sets, controllable sets, invariant sets, etc, play fundamental roles in designing predictive control strategies for uncertain systems. Meanwhile, in addition to the commonly assumed boundedness of the uncertainty, the explicit use of its stochastic properties can lead to imprq\fement in system response. This thesis is concerned with robust set theories, mainly for reachable sets, with applications to time-optimal control; and the use of stochastic properties of the uncertainty to achieve less conservative controls. In the first part of this thesis, we focus on LTI systems subject to, additional to the usual constraints, a constraint on the control change between sample times. One key ingredient in controlling such constrained systems is the initial control value, which, via analyses and simulations, is shown to be a useful extra degree of freedom. Reachable sets that incorporate this influential initial control value are derived and analyzed, with theoretical as well as computational algorithms developed for both nominal and uncertain systems under different types of feedback policy. Following this, the reachable set is discussed in connection with time-optimal control to obtain desired control laws. In addition, controllable sets, stabilisable sets and invariant sets for such constrained uncertain systems are studied. In the second part, the uncertainties are assumed to have stochastic properties. They are exploited in three different ways: the expected worst-case is used instead of the worst-case to achieve less conservative control even when the uncertainty is relatively large; the stochastic invariant set is proposed to provide alternative methods for approximating disturbance invariant sets; the relaxed set difference is developed to obtain less restrictive controls and/or replacing probabilistic constraint or slack variables.Imperial Users onl