arXiv subjects

Rafael Vazquez

Publications and source records attributed to Rafael Vazquez.

At least 37 records · Page 2Linked to original sources

arXiv2024

Single track orbit determination analysis for low Earth orbit with approximated J2 dynamics

In the domain of Space Situational Awareness (SSA), the challenges pertaining to orbit determination and catalog correlation are notably pronounced, partly attributable to the escalating presence of non-cooperative satellites engaging in unspecified maneuvers at irregular intervals. This study introduces an initial orbit determination methodology reliant upon data obtained from a single surveillance radar, such as the Spanish Space Surveillance and Tracking Surveillance Radar (S3TSR). The need for fast algorithms within an operational context is considered here as the main design driver. The result is a least-squares fitting procedure that incorporates an analytically formulated approximation of the dynamics under the J2 perturbation, valid for short-term propagation. The algorithm makes use of all available observables, including range-rate, which makes it distinct from other similar methods. The method is compared in a battery of synthetic tests against a classical range and angles fitting method (GTDS) to study the effect of the track length and density of measurements on the full state estimation. The presented methodology is quite versatile, and it is leveraged to improve the estimation quality by adding information of the object's orbital plane obtained from predictions. The resulting method has been named OPOD.

physics.space-ph↗

arXiv2024

Towards a MATLAB Toolbox to compute backstepping kernels using the power series method

In this paper, we extend our previous work on the power series method for computing backstepping kernels. Our first contribution is the development of initial steps towards a MATLAB toolbox dedicated to backstepping kernel computation. This toolbox would exploit MATLAB's linear algebra and sparse matrix manipulation features for enhanced efficiency; our initial findings show considerable improvements in computational speed with respect to the use of symbolical software without loss of precision at high orders. Additionally, we tackle limitations observed in our earlier work, such as slow convergence (due to oscillatory behaviors) and non-converging series (due to loss of analiticity at some singular points). To overcome these challenges, we introduce a technique that mitigates this behaviour by computing the expansion at different points, denoted as localized power series. This approach effectively navigates around singularities, and can also accelerates convergence by using more local approximations. Basic examples are provided to demonstrate these enhancements. Although this research is still ongoing, the significant potential and simplicity of the method already establish the power series approach as a viable and versatile solution for solving backstepping kernel equations, benefiting both novel and experienced practitioners in the field. We anticipate that these developments will be particularly beneficial in training the recently introduced neural operators that approximate backstepping kernels and gains.

eess.SY↗

arXiv2024

MPC for Tracking applied to rendezvous with non-cooperative tumbling targets ensuring stability and feasibility

A Model Predictive Controller for Tracking is introduced for rendezvous with non-cooperative tumbling targets in active debris removal applications. The target's three-dimensional non-periodic rotational dynamics as well as other state and control constraints are considered. The approach is based on applying an intermediate coordinate transformation that eliminates the time-dependency due to rotations in the constraints. The control law is then found as the solution to a QP problem with linear constraints and dynamics, as derived from the HCW equations, that provides feasibility and stability guarantees by means of a terminal LQR and dead-beat region. The proposed control algorithm performs well in a realistic simulation scenario, namely a near rendezvous with the Envisat spacecraft.

eess.SY↗

arXiv2024

A hybrid dynamical system approach to the impulsive control of spacecraft rendezvous (extended version)

This paper introduces a hybrid dynamical system methodology for managing impulsive control in spacecraft rendezvous and proximity operations under the Hill-Clohessy-Wiltshire model. We address the control design problem by isolating the out-of-plane from the in-plane dynamics and present a feedback control law for each of them. This law is based on a Lyapunov function tailored to each of the dynamics, capable of addressing thruster saturation and also a minimum impulse bit. These Lyapunov functions were found by reformulating the system's dynamics into coordinates that more intuitively represent their physical behavior. The effectiveness of our control laws is then shown through numerical simulation. This is an extended version of an ECC24 article of the same name, which includes the proofs omitted for lack of space.

eess.SY↗

arXiv2024

Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation

To stabilize PDE models, control laws require space-dependent functional gains mapped by nonlinear operators from the PDE functional coefficients. When a PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent, a gain-scheduling (GS) nonlinear design is the simplest approach to the design of nonlinear feedback. The GS version of PDE backstepping employs gains obtained by solving a PDE at each value of the state. Performing such PDE computations in real time may be prohibitive. The recently introduced neural operators (NO) can be trained to produce the gain functions, rapidly in real time, for each state value, without requiring a PDE solution. In this paper we introduce NOs for GS-PDE backstepping. GS controllers act on the premise that the state change is slow and, as a result, guarantee only local stability, even for ODEs. We establish local stabilization of hyperbolic PDEs with nonlinear recirculation using both a "full-kernel" approach and the "gain-only" approach to gain operator approximation. Numerical simulations illustrate stabilization and demonstrate speedup by three orders of magnitude over traditional PDE gain-scheduling. Code (Github) for the numerical implementation is published to enable exploration.

eess.SY↗

arXiv2022

Backstepping Control of a Hyperbolic PDE System with Zero Characteristic Speed States

While for coupled hyperbolic PDEs of first order there now exist numerous PDE backstepping designs, systems with zero speed, i.e., without convection but involving infinite-dimensional ODEs, which arise in many applications, from environmental engineering to lasers to manufacturing, have received virtually no attention. In this paper, we introduce single-input boundary feedback designs for a linear 1-D hyperbolic system with two counterconvecting PDEs and $n$ equations (infinite-dimensional ODEs) with zero characteristic speed. The inclusion of zero-speed states, which we refer to as {\em atachic}, may result in non-stabilizability of the plant. We give a verifiable condition for the model to be stabilizable and design a full-state backstepping controller which exponentially stabilizes the origin in the $\mathcal{L}^{2}$ sense. In particular, to employ the backstepping method in the presence of atachic states, we use an invertible Volterra transformation only for the PDEs with nonzero speeds, leaving the zero-speed equations unaltered in the target system input-to-state stable with respect to the decoupled and stable counterconvecting nonzero-speed equations. Simulation results are presented to illustrate the effectiveness of the proposed control design.

math.OC↗

arXiv2022

Output Feedback Control of Radially-Dependent Reaction-Diffusion PDEs on Balls of Arbitrary Dimensions

Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, well-posedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying reaction (namely, analyticity and evenness) and showing the existence and convergence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems.

math.OC↗

arXiv2022

Rapid Stabilization of Timoshenko Beam by PDE Backstepping

In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) x (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H^1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate.

math.OC↗

arXiv2022

Manoeuvre detection in Low Earth Orbit with Radar Data

This work outlines and assesses several methods for the detection of manoeuvres in Low Earth Orbit (LEO) from surveillance radar data. To be able to detect manoeuvres, the main starting assumption is that the object under analysis has an orbit known with a sufficient degree of precision. Based on the precise (a posteriori) orbit and radar data, several manoeuvre detection methods are presented; one is based on unscented Kalman filtering, whereas two others algorithms are based on reachability analysis of the state, which correlates its prediction set with the next track from the radar. The filtering algorithm can be extended for several radar tracks, whereas the reachability-based methods are more precise in detecting manoeuvres. Then, to inherit the best properties of both classes of algorithms, a manoeuvre detection filter that combines both concepts is finally presented. Manoeuvre detection results are presented first for simulated scenarios -- for validation and calibration purposes -- and later for real data. Radar information comes from the Spanish Space Surveillance Radar (S3TSR), with real manoeuvre information and high-quality ephemerides. The results show promise, taking into account that a single surveillance radar is the only source of data, obtaining manoeuvre detection rates of more than 50% and false positive rates of less than 10%.

math.OC↗

arXiv2019

BPX Preconditioners for Isogeometric Analysis Using (Truncated) Hierarchical B-splines

We present the construction of additive multilevel preconditioners, also known as BPX preconditioners, for the solution of the linear system arising in isogeometric adaptive schemes with (truncated) hierarchical B-splines. We show that the locality of hierarchical spline functions, naturally defined on a multilevel structure, can be suitably exploited to design and analyze efficient multilevel decompositions. By obtaining smaller subspaces with respect to standard tensor-product B-splines, the computational effort on each level is reduced. We prove that, for suitably graded hierarchical meshes, the condition number of the preconditioned system is bounded independently of the number of levels. A selection of numerical examples validates the theoretical results and the performance of the preconditioner.

math.NA↗

arXiv2019

Folding Bilateral Backstepping Output-Feedback Control Design For an Unstable Parabolic PDE

We present a novel methodology for designing output-feedback backstepping boundary controllers for an unstable 1-D diffusion-reaction partial differential equation with spatially-varying reaction. Using "folding" transforms the parabolic PDE into a 2X2 coupled parabolic PDE system with coupling via folding boundary conditions. The folding approach is novel in the sense that the design of bilateral controllers can be generalized to centering around arbitrary points, which admit additional design parameters for both the state-feedback controller and the state observer. The design can be selectively biased to achieve different performance indicies (e.g. energy, boundedness, etc). A first backstepping transformation is designed to map the unstable system into a strict-feedback intermediate target system. A second backstepping transformation is designed to stabilize the intermediate target system. The invertibility of the two transformations guarantees that the derived state-feedback controllers exponentially stabilize the trivial solution of the parabolic PDE system in the L^2 norm sense. A complementary state observer is likewise designed for the dual problem, where two collocated measurements are considered at an arbitrary point in the interior of the domain. The observer generates state estimates which converge to the true state exponentially fast in the L^2 sense. Finally, the output feedback control law is formulated by composing the state-feedback controller with the state estimates from the observer, and the resulting dynamic feedback is shown to stabilize the trivial solution of the interconnected system in the L^2 norm sense. Some analysis on how the selection of these points affect the responses of the controller and observer are discussed, with simulations illustrating various choices of folding points and their effect on the stabilization.

math.OC↗

arXiv2019

Bilateral Boundary Control Design for a Cascaded Diffusion-ODE System Coupled at an Arbitrary Interior Point

We present a methodology for designing bilateral boundary controllers for a class of systems consisting of a coupled diffusion equation with an unstable ODE at an arbitrary interior point. A folding transformation is applied about the coupling point, transforming the system into an ODE with an input channel consisting of two coupled diffusive actuation paths. A target system with an exponentially stable trivial solution in the sense of L^2 X R^n is proposed, and the stability property is shown via the Lyapunov method. The stabilizing control laws are formulated via tiered Volterra transformations of the second kind, establishing an equivalence relation between the stable target system and the original plant under boundary feedback. Stability properties of the plant under feedback is inferred from the equivalence relation. The well-posedness of the backstepping transformations involved are studied, and the existence of bounded Volterra kernels is shown, constituting a sufficient condition for the invertibility of the Volterra transformations.

math.OC↗

arXiv2018

Nonlinear Bilateral Output-Feedback Control for a Class of Viscous Hamilton-Jacobi PDEs

We tackle the boundary control and estimation problems for a class of viscous Hamilton-Jacobi PDEs, considering bilateral actuation and sensing, i.e., at the two boundaries of a 1-D spatial domain. First, we solve the nonlinear trajectory generation problem for this type of PDEs, providing the necessary feedforward actions at both boundaries. Second, in order to guarantee trajectory tracking with an arbitrary decay rate, we construct nonlinear, full-state feedback laws employed at the two boundary ends. Third, a nonlinear observer is constructed, using measurements from both boundaries, which is combined with the full-state feedback designs into an observer-based output-feedback law. All of our designs are explicit since they are constructed interlacing a feedback linearizing transformation (which we introduce) with backstepping. Due to the fact that the linearizing transformation is locally invertible, only regional stability results are established, which are, nevertheless, accompanied with region of attraction estimates. Our stability proofs are based on the utilization of the linearizing transformation together with the employment of backstepping transformations, suitably formulated to handle the case of bilateral actuation and sensing. We illustrate the developed methodologies via application to traffic flow control and we present consistent simulation results.

math.OC↗

arXiv2017

Hierarchical B-spline complexes of discrete differential forms

In this paper, we introduce the hierarchical B-spline complex of discrete differential forms for arbitrary spatial dimension. This complex may be applied to the adaptive isogeometric solution of problems arising in electromagnetics and fluid mechanics. We derive a sufficient and necessary condition guaranteeing exactness of the hierarchical B-spline complex for arbitrary spatial dimension, and we derive a set of local, easy-to-compute, and sufficient exactness conditions for the two-dimensional setting. We examine the stability properties of the hierarchical B-spline complex, and we find that it yields stable approximations of both the Maxwell eigenproblem and Stokes problem provided that the local exactness conditions are satisfied. We conclude by providing numerical results showing the promise of the hierarchical B-spline complex in an adaptive isogeometric solution framework.

math.NA↗

arXiv2016

Bilateral Boundary Control of One-Dimensional First- and Second-Order PDEs using Infinite-Dimensional Backstepping

This paper develops an extension of infinite-dimensional backstepping method for parabolic and hyperbolic systems in one spatial dimension with two actuators. Typically, PDE backstepping is applied in 1-D domains with an actuator at one end. Here, we consider the use of two actuators, one at each end of the domain, which we refer to as bilateral control (as opposed to unilateral control). Bilateral control laws are derived for linear reaction-diffusion, wave and 2X2 hyperbolic 1-D systems (with same speed of transport in both directions). The extension is nontrivial but straightforward if the backstepping transformation is adequately posed. The resulting bilateral controllers are compared with their unilateral counterparts in the reaction-diffusion case for constant coefficients, by making use of explicit solutions, showing a reduction in control effort as a tradeoff for the presence of two actuators when the system coefficients are large. These results open the door for more sophisticated designs such as bilateral sensor/actuator output feedback and fault-tolerant designs.

math.OC↗

arXiv2016

Boundary Control of Coupled Reaction-Advection-Diffusion Systems with Spatially-Varying Coefficients

Recently, the problem of boundary stabilization for unstable linear constant-coefficient coupled reaction-diffusion systems was solved by means of the backstepping method. The extension of this result to systems with advection terms and spatially-varying coefficients is challenging due to complex boundary conditions that appear in the equations verified by the control kernels. In this paper we address this issue by showing that these equations are essentially equivalent to those verified by the control kernels for first-order hyperbolic coupled systems, which were recently found to be well-posed. The result therefore applies in this case, allowing us to prove H^1 stability for the closed-loop system. It also shows an interesting connection between backstepping kernels for coupled parabolic and hyperbolic problems.

math.OC↗

arXiv2016

Boundary control of a singular reaction-diffusion equation on a disk

Recently, the problem of boundary stabilization for unstable linear constant-coefficient reaction-diffusion equation on N-balls has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. This work deals with radially-varying reaction coefficients under revolution symmetry conditions on a disk (the 2-D case). Under these conditions, the equations become singular in the radius. When applying the backstepping method, the same type of singularity appears in the backstepping kernel equations. Traditionally, well-posedness of the kernel equations is proved by transforming them into integral equations and then applying the method of successive approximations. In this case, the resulting integral equation is singular. A successive approximation series can still be formulated, however its convergence is challenging to show due to the singularities. The problem is solved by a rather non-standard proof that uses the properties of the Catalan numbers, a well-known sequence frequently used in combinatorial mathematics.

math.OC↗

arXiv2015

Boundary exponential stabilization of 1-D inhomogeneous quasilinear hyperbolic systems

This paper deals with the problem of boundary stabilization of first-order n\times n inhomogeneous quasilinear hyperbolic systems. A backstepping method is developed. The main result supplements the previous works on how to design multi-boundary feedback controllers to realize exponential stability of the original nonlinear system in the spatial H^2 sense.

math.OC↗