Model Predictive Control for Spacecraft Rendezvous in Elliptical Orbits with On/Off Thrusters

IFAC Workshop on Advanced Control and Navigation for Autonomous Aerospace Vehicles. 08/06/2015. SevillaIn previous works, the authors have developed a trajectory planning algorithm for spacecraft rendezvous which computed optimal Pulse-Width Modulated (PWM) control signals, for circular and eccentric Keplerian orbits. The algorithm is initialized by solving the impulsive problem first and then, using explicit linearization and linear programming, the solution is refined until a (possibly local) optimal value is reached. However, trajectory planning cannot take into account orbital perturbations, disturbances or model errors. To overcome these issues, in this paper we develop a Model Predictive Control (MPC) algorithm based on the open-loop PWM planner and test it for elliptical target orbits with arbitrary eccentricity (using the linear time-varying Tschauner-Hempel model). The MPC is initialized by first solving the open-loop problem with the PWM trajectory planning algorithm. After that, at each time step, our MPC saves time recomputing the trajectory by applying the iterative linearization scheme of the trajectory planning algorithm to the solution obtained in the previous time step. The efficacy of the method is shown in a simulation study where it is compared to MPC computed used an impulsive-only approach

Robust Model Predictive Control for Spacecraft Rendezvous with Online Prediction of Disturbance Bounds

IFAC Workshop Aerospace Guidance, Navigation and Flight Control Systems (AGNFCS' 09) Samara, RUSSIA June 30 - July 2, 2009A Model Predictive Controller is introduced to solve the problem of rendezvous of spacecraft, using the HCW model and including additive disturbances and line-of-sight constraints. It is shown that a standard MPC is not able to cope with disturbances. Then a robust Model Predictive Control that introduces the concepts of robust satisfaction of constraints is proposed. The formulation also includes a predictor of the disturbance properties which are needed in the robust algorithm. In simulations it is shown that the robust MPC scheme is able to handle not only additive disturbances (which are the ones used in the formulation) but also large multiplicative disturbances and unmodelled dynamics (due to eccentricity of the orbit of the target spacecraft)

Trajectory Planning for Spacecraft Rendezvous in Elliptical Orbits with On / Off Thrusters

The 19th World Congress of the International Federation of Automatic Control 2014 Cape Town, SudáfricaIn a previous work, the authors developed a trajectory planning algorithm for spacecraft rendezvous which computed optimal Pulse-Width Modulated (PWM) control signals, assuming that the target was moving in a circular Keplerian orbit. In this paper we extend the algorithm to the case of an elliptical target orbit with arbitrary eccentricity. Since the orbit is elliptical, the linear time-varying Tschauner-Hempel model is used, whose exact solution is possible by using true (or eccentric) anomaly instead of time (which is directly related to both via Kepler's equation). Unlike in the circular case, computing the PWM solution itself requires numerical integration. However, explicit linearization around the computed solution turns out to be possible and is exploited for rapidly improving the solution using linear programming (LP) techniques. The algorithm is initialized by solving the impulsive problem first; the impulses are converted to PWM signals, which are used as an initial guess. Using the explicit linearization and LP, the solution is refined until a (possibly local) optimal value is reached. The efficacy of the method is shown in a simulation study where it is compared to the impulsive-only approach

Trajectory Planning for Spacecraft Rendezvous with On / Off Thrusters

18th World CongressThe International Federation of Automatic ControlMilano (Italy) August 28 - September 2The objective of this work is to present a trajectory planning algorithm for spacecraft rendezvous that is able to incorporate Pulse-Width Modulated (PWM) control signals. The algorithm is based on linearization around a previously computed solution. To initialize the algorithm, a first solution needs to be obtained. To do so, the trajectory planning problem is solved using Pulse-Amplitude Modulated (PAM) control signals; these are then converted to PWM signals, which are used as an initial guess. Iterating, the solution is refined until an optimal value is reached. Simulations show that this method converges after a few iterations. The algorithm is simple and fast, hence it could be implemented online or used together with a Model Predictive Controller

Explicit output-feedback boundary control of reaction-diffusion PDEs on arbitrary-dimensional balls

This paper introduces an explicit output-feedback boundary feedback law that stabilizes an unstable linear constant-coefficient reaction-diffusion equation on an n-ball (which in 2-D reduces to a disk and in 3-D reduces to a sphere) using only measurements from the boundary. The backstepping method is used to design both the control law and a boundary observer. To apply backstepping the system is reduced to an infinite sequence of 1-D systems using spherical harmonics. Well-posedness and stability are proved in the L2 and H1 spaces. The resulting control and output injection gain kernels are the product of the backstepping kernel used in control of one-dimensional reaction-diffusion equations and a function closely related to the Poisson kernel in the n-ball.Ministerio de Economía y Competitividad MTM2015-65608-

Explicit boundary control of a reaction-diffusion equation on a disk

This paper introduces an explicit full-state boundary feedback law that stabilizes an unstable linear constant-coefficient reaction-diffusion equation on a disk. The backstepping method is used to design the control law. To apply backstepping the system is reduced to an infinite sequence of 1-D systems using Fourier series. H2 well-posedness and stability are proved, which implies that the solution is at least continuous on the disk

Pulse-width predictive control for LTV systems with application to spacecraft rendezvous

This work presents a Model Predictive Controller (MPC) that is able to handle Linear Time-Varying (LTV) plants with Pulse-Width Modulated (PWM) control. The MPC is based on a planner that employs a Pulse-Amplitude Modulated (PAM) or impulsive approximation as a hot-start and then uses explicit linearization around successive PWM solutions for rapidly improving the solution by means of quadratic programming. As an example, the problem of rendezvous of spacecraft for eccentric target orbits is considered. The problem is modeled by the LTV Tschauner–Hempel equations, whose state transition matrix is explicit; this is exploited by the algorithm for rapid convergence. The efficacy of the method is shown in a simulation study.Ministerio de Economía y Competitividad DPI2008–05818Ministerio de Economía y Competitividad MTM2015-65608-

Implementation in MATLAB of a Multiplicative Extended Kalman Filter for live estimation of a smart device's attitude

Volume 54, Issue 12This work introduces the tools used to teach the Kalman Filter (KF) to Aerospace Engineering students in the University of Seville. In particular, an easy-to-set-up application is introduced; based on the Matlab framework on its 2020b (or newer versions), it is able to display the attitude of a smart device in real time through a wireless connection to a computer. This tool is a simple yet powerful educative resource when teaching about the KF, since it showcases its performance while allowing the student to understand how a complex real-world problem can be solved using a relatively simple implementation of the KF; in particular, the Multiplicative Extended KF (MEKF) is chosen, but the framework can be easily adapted to other versions such as the Extended or Unscented KF. In addition, the tool allows the student to be aware of the inner workings of the filter itself, learning about its advantages and limitations compared with other attitude estimation algorithms. The student is also able to understand how the filter needs to be tuned, and to observe the results of the experiments in a visual and straightforward manner.Ministerio de Ciencia e Innovación (MICIN). España PGC2018-100680-B-C2

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 (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. As a first step, 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 appearing in combinatorial mathematics.Ministerio de Economía y Competitividad MTM2015-65608-

Propagation of Initial Mass Uncertainty in Aircraft Cruise Flight

The propagation of initial mass uncertainty in cruise flight is studied. Two cruise conditions are analyzed: one with given cruise fuel load and the other with given cruise range. Two different distributions of initial mass are considered: uniform and gamma type. The generalized polynomial chaos method is used to study the evolution of mean and variance of the aircraft mass. To compute the mass distribution function as a function of time, two approximate methods are developed. These methods are also applied to study the distribution functions of the flight time (in the case of given fuel load) and of the fuel consumption (in the case of given range). The dynamics of mass evolution in cruise flight is defined by a nonlinear equation, which can be solved analytically; this exact solution is used to assess the accuracy of the proposed methods. Comparison of the numerical results with the exact analytical solutions shows an excellent agreement in all cases, hence verifying the methods developed in this work