Region of Attraction Estimation Using Invariant Sets and Rational Lyapunov Functions

This work addresses the problem of estimating the region of attraction (RA) of equilibrium points of nonlinear dynamical systems. The estimates we provide are given by positively invariant sets which are not necessarily defined by level sets of a Lyapunov function. Moreover, we present conditions for the existence of Lyapunov functions linked to the positively invariant set formulation we propose. Connections to fundamental results on estimates of the RA are presented and support the search of Lyapunov functions of a rational nature. We then restrict our attention to systems governed by polynomial vector fields and provide an algorithm that is guaranteed to enlarge the estimate of the RA at each iteration

A Semi-Definite Programming Approach to Stability Analysis of Linear Partial Differential Equations

We consider the stability analysis of a large class of linear 1-D PDEs with polynomial data. This class of PDEs contains, as examples, parabolic and hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary coupled PDEs. Our approach is Lyapunov based which allows us to reduce the stability problem to the verification of integral inequalities on the subspaces of Hilbert spaces. Then, using fundamental theorem of calculus and Green's theorem, we construct a polynomial problem to verify the integral inequalities. Constraining the solution of the polynomial problem to belong to the set of sum-of-squares polynomials subject to affine constraints allows us to use semi-definite programming to algorithmically construct Lyapunov certificates of stability for the systems under consideration. We also provide numerical results of the application of the proposed method on different types of PDEs

A Convex Approach to Hydrodynamic Analysis

We study stability and input-state analysis of three dimensional (3D) incompressible, viscous flows with invariance in one direction. By taking advantage of this invariance property, we propose a class of Lyapunov and storage functionals. We then consider exponential stability, induced L2-norms, and input-to-state stability (ISS). For streamwise constant flows, we formulate conditions based on matrix inequalities. We show that in the case of polynomial laminar flow profiles the matrix inequalities can be checked via convex optimization. The proposed method is illustrated by an example of rotating Couette flow.Comment: Preliminary version submitted to 54rd IEEE Conference on Decision and Control, Dec. 15-18, 2015, Osaka, Japa

Semi-definite programming and functional inequalities for Distributed Parameter Systems

We study one-dimensional integral inequalities, with quadratic integrands, on bounded domains. Conditions for these inequalities to hold are formulated in terms of function matrix inequalities which must hold in the domain of integration. For the case of polynomial function matrices, sufficient conditions for positivity of the matrix inequality and, therefore, for the integral inequalities are cast as semi-definite programs. The inequalities are used to study stability of linear partial differential equations.Comment: 8 pages, 5 figure

VEÍCULO AUTÔNOMO POR VISÃO COMPUTACIONAL

Veículos autônomos são veículos que dispensam a intervenção de um motorista. O presente projeto tem o objetivo de desenvolver um veículo autônomo, seguidor de linha, utilizando visão computacional. No veículo é acoplado um Raspberry Pi3, onde é processado o algoritmo para o reconhecimento da linha através da biblioteca OpenCV da linguagem de programação Python, uma câmera Pi, que captura as imagens que são processadas no algoritmo, e também os motores de acionamento das rodas e da direção do veículo

Barrier Functionals for Output Functional Estimation of PDEs

We propose a method for computing bounds on output functionals of a class of time-dependent PDEs. To this end, we introduce barrier functionals for PDE systems. By defining appropriate unsafe sets and optimization problems, we formulate an output functional bound estimation approach based on barrier functionals. In the case of polynomial data, sum of squares (SOS) programming is used to construct the barrier functionals and thus to compute bounds on the output functionals via semidefinite programs (SDPs). An example is given to illustrate the results.Comment: 8 pages, 1 figure, preprint submitted to 2015 American Control Conferenc

Development and testing of model predictive control strategies for spacecraft formation flying

Satellite Formation Flying (SFF) is a key technology for several future missions, since, with respect to a single spacecraft, it allows better performances, new capabilities, more flexibility and robustness to failure and cost reduction. Despite these benefits, however, this new concept poses several signicant design challenges and requires new technologies. The Guidance, Navigation and Control (GNC) system is a key element in the SFF concept since it must be reliable in coordinating all the satellites fying in formation during each mission phase, guaranteeing formation integrity and preventing from formation evaporation, and, at the same time, efficient in using the limited on board resources. Model Predictive Control (MPC), also referred to as Receding Horizon Control, is a modern optimal control technique that seems to be suitable for these purposes because of its three main features: model-based control scheme, constraints handling ability and replanning nature. The final aim of my Ph.D. activities was to develop and test MPC strategies for SFF applications. This task was accomplished by means of both computer simulations and experimental tests conducted on both the MIT Synchronized Position Hold Engage & Reorient Experimental Satellites (SPHERES) testbed and the SFF Hardware Simulator under development at the Center of Studies and Activities for Space "Giuseppe Colombo" (CISAS), University of Padova. MPC capabilities were first tested in computer simulations in carrying out a formation acquisition maneuver for two space vehicles, taking into account two scenarios: a Leader-Follower (LF) formation and Projected Circular Orbit (PCO) formation. The performances of the MPC-based controller were compared with those of a Linear Quadratic Regulator (LQR) based controller in the presence of active constraints on the maximum control acceleration, evaluating also the effects of the gravitational harmonics J2 and J3 and atmospheric drag perturbations on the proposed maneuvers. Simulation results of both scenarios showed that, with similar performances in tracking the same reference state trajectory in terms of settling time, the MPC controller is more efficient (less delta-v requirement) than the LQR controller also in the perturbed cases, allowing a delta-v requirement reduction by 40% in the LF formation scenario and by 30% in the PCO formation scenario. The next activity concerned the development of some guidance and control strategies for a Collision-Avoidance scenario in which a free-flying chief spacecraft follows temporary off-nominal conditions and a controlled deputy spacecraft performs a collision avoidance maneuver. The proposed strategy consists on a first Separation Guidance that, using a computationally simple, deterministic and closed-form algorithm, takes charge of avoiding a predicted collision. When some safe conditions on the relative state vector (position and velocity) are met, a subsequent Nominal Guidance takes over. Genetic Algorithms are used to compute a pair of reference state trajectories in order to place the deputy spacecraft in a bounded safe or "parking" trajectory, while minimizing the propellant consumption and avoiding the formation evaporation. The performances of a LQR and a MPC in tracking these reference trajectories were compared, showing how a MPC controller can reduces the total delta-v requirement by 5 - 10% with respect to a LQR controller. MPC capabilities were then evaluated on the MIT SPHERES testbed in simulating the close-proximity phase of the rendez-vous and capture maneuver for the Mars Orbital Sample Return (MOSR) scenario. Better performances of MPC with respect to PD in executing this maneuver were conrmed both in a Matlab simulator and in the MIT SPHERES software simulator, with a total delta-v requirement reduction by 10-15 %. The proposed MPC control strategy was then tested using the SPHERES Flat Floor facility at the MIT Space System Laboratory. The last part of my research activities was devoted to the SFF Hardware Simulator of the University of Padova. My contributions to this project dealt with: (a) conclusion of the designing, building and testing of the five main subsystems of the hardware simulator; (b) software development for the hardware simulator and its Matlab software simulator; (c) preparatory experimental activities aimed at characterizing the thrust force performed by the on board thrusters and estimating the hardware simulator inertia properties; and (d) test of attitude control maneuvers with the use of predictive controllers. In particular, three main tests were carried out with the hardware simulator moving at one degree of freedom about the yaw axis. The first one aimed at tuning a Kalman Filter to properly estimate the yaw axis angular velocity using a double-integrator as dynamic model and angular position measurements provided by the yaw quadrature encoder. With the use of a simple Kalman Filter, the yaw angular position and velocity could be estimated with an error less than 0.1 ° and 0.1°/s, respectively. In the second test, an explicit MPC was used to perform a 170° slew maneuver of the hardware simulator attitude module about the yaw axis. The final target angular position was reached with an error less than 0.5° in 20 s. In the third test, a 3 degrees of freedom attitude reference trajectory was first computed using pseudospectral optimization methods for a repointing maneuver with active constraints on the attitude trajectory. The state trajectory was then projected along the satellite z-Body axis and tracked in the hardware simulator using an explicit MPC. Experimental results showed that with an explicit MPC the reference trajectories can be tracked with an error less that 1.5° for the angular position and less than 1°/s for the angular velocity, both in dynamic conditions. The final target state was reached with an error less than the estimation accuracy. The SFF Hardware Simulator is a ground-based testbed for the development and verification of GNC algorithms that in the present configuration allows the development and testing of advanced controls for attitude motion and in its final form will enable the derivation of control strategies for Formation Flight and Automated Rendezvous and Docking

Metrological characterization of a vision-based system for relative pose measurements with fiducial marker mapping for spacecrafts

An improved approach for the measurement of the relative pose between a target and a chaser spacecraft is presented. The selected method is based on a single camera, which can be mounted on the chaser, and a plurality of fiducial markers, which can be mounted on the external surface of the target. The measurement procedure comprises of a closed-form solution of the Perspective from n Points (PnP) problem, a RANdom SAmple Consensus (RANSAC) procedure, a non-linear local optimization and a global Bundle Adjustment refinement of the marker map and relative poses. A metrological characterization of the measurement system is performed using an experimental set-up that can impose rotations combined with a linear translation and can measure them. The rotation and position measurement errors are calculated with reference instrumentations and their uncertainties are evaluated by the Monte Carlo method. The experimental laboratory tests highlight the significant improvements provided by the Bundle Adjustment refinement. Moreover, a set of possible influencing physical parameters are defined and their correlations with the rotation and position errors and uncertainties are analyzed. Using both numerical quantitative correlation coefficients and qualitative graphical representations, the most significant parameters for the final measurement errors and uncertainties are determined. The obtained results give clear indications and advice for the design of future measurement systems and for the selection of the marker positioning on a satellite surface