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Control theory is a field of great practical importance with well-established theoretical fundations for systems described by ordinary differential equations (ODEs). However, many systems cannot be adequately modelled by using ODEs and one needs to use instead partial differential equations (PDEs). Systems that are described by PDEs include fluid mechanics model, channel flows, convection problems, magnetohydrodynamic problems, etc... These systems belong to the class of Distributed Parameter Systems (i.e., systems whose state is infinite-dimensional, also including delay systems). For distributed parameter systems, control theory is still under development and there are many challenges (both theoretical and practical) that need to be tackled.

Backstepping is a powerful tool coming from nonlinear control theory that is proving to be of great utility for designing boundary controllers and observers that solve some of these open problems, and remarkably so for many types of evolution equations (both parabolic and hyperbolic). However there are still a number of unsolved problems for which the backstepping technique represent a potential resolution tool. In particular, applications often pose problems that would require the extension of the method to coupled systems, to two or more dimensions, and/or to complicated geometries.

Some selected publications:

- R. Vazquez and M. Krstic,
*Control of Turbulent and Magnetohydrodynamic Channel Flows*, Birkhauser, 2007. - R. Vazquez and M. Krstic, "Control of 1-D parabolic PDEs with Volterra nonlinearities - Part I: Design," Automatica, vol. 44, 2778-2790 , 2008.
- R. Vazquez, E. Schuster and M. Krstic, "A closed-form feedback controller for stabilization of 3D magnetohydrodynamic channel flow," ASME Journal of Dynamic Systems, Measurement and Control, vol. 131, 041001, 2009.
- J.-M. Coron, R. Vazquez, M. Krstic, and G. Bastin, “Local exponential H2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping,” SIAM Journal on Control and Optimization, vol. 51(3), 2005-2035, 2013.
- R. Vazquez and M. Krstic, "Explicit boundary control of a reaction-diffusion equation on a disk," IFAC World Congress, 2014.
- R. Vazquez and M. Krstic, “Marcum Q-functions and Explicit Kernels for Stabilization of 2x2 Linear Hyperbolic Systems with Constant Coefficients,” System and Control Letters, vol. 68, 33-42, 2014.
- J. Qi, R. Vazquez, M. Krstic, "Multi-agent Deployment in 3-D via PDE Control," IEEE Transactions on Automatic Control, 2015.
- Long Hu, Florent Di Meglio, Rafael Vazquez, Miroslav Krstic, "Control of Homodirectional and General Heterodirectional Linear Coupled Hyperbolic PDEs," to appear, IEEE Transactions on Automatic Control, 2016.

On-orbit spacecraft rendezvous, this is, the controlled encounter of two or more spacecraft in space, is an extremely important practical problem that has already been put to practice in numerous space missions. For instance the Apollo program required the rendezvous between the command module and the lunar module, and the development of techniques for solving the rendezvous problem was an important milestone for the American space program.

Today, the main challenge of spacecraft rendezvous consists on developing an autonomous system with low weight and consumption able to obtain a safe encounter without wasting any fuel. Numerous optimization and control techniques can be applied to achieve a successful rendezvous. However modern automatic rendezvous systems are rather heavy and expensive and it is still a challenge to develop light, low-cost systems for small satellites that would allow them, for instance, to rendezvous with large spacecraft such as space stations.

Some selected publications:

- F. Gavilan, R. Vazquez, E. F. Camacho"Robust Model Predictive Control for Spacecraft Rendezvous with Online Prediction of Disturbance Bounds," AGNFCS‘09, Samara, Russia, 2009.
- R. Vazquez, F. Gavilan, E. F. Camacho, "Trajectory Planning for Spacecraft Rendezvous withOn/Off Thrusters," IFAC World Congress, 2011.
- F. Gavilan, R. Vazquez and E. F. Camacho, "Chance-constrained Model Predictive Control for Spacecraft Rendezvous with Disturbance Estimation", Control Engineering Practice, vol 20 (2), 111-122, 2012.
- R. Vazquez, F. Gavilan, and E.F. Camacho, "Trajectory Planning for Spacecraft Rendezvous in Elliptical Orbits with On/Off Thrusters," IFAC World Congress, 2014.

The age of Unmmaned Aerial Vehicles (UAVs), popularly known as drones, is already here. With it, a huge number of applications are arising both in the civil and military fields. UAVs require well-designed guidance, control and navigation (GNC) systems, able to carry out successfully any mission within the vehicle capabilities.

In particular, flight controllers are of paramount importance to maintain a stable flight with good levels of performance. While there are already commercial generic systems with decent performance that can be installed on board to carry out the GNC functions, they are usually rather expensive and oftentimes require extensive knowledge of aircraft aerodynamics, which requires wind tunnel experiments or even flight testing. However, nowadays it is possible (even with off-the-shelf hardware and at a low cost) to develop algorithms that do not require any a priori knowledge of aircraft aerodynamics and are even able to cope with sudden system changes due to faults or reconfiguration. One example of such algorithms is adapative control, in particular adaptive backstepping.

Guidance algorithms (the high level systems that issue references for the flight controllers) pose a different set of challenges. For instance, a shortcoming of typical commercial systems is that they tend to degrade considerably in the presence of wind and/or uncertainties, or are unable to provide time synchronization capabilities. It is however possible to develop predictive guidance algorithms with more functionalities than conventional systems, that are able to overcome uncertainties and include time synchronization in a cost-optimal way, using for instance Model Predictive Control techniques.

Some selected publications:

- F. Gavilan, R. Vazquez, J. A. Acosta, "Output-Feedback Control of the Longitudinal Flight Dynamics Using Adaptative Backstepping," 2011 CDC/ECC, Orlando, Florida, 2011.
- F. Gavilan, J. A. Acosta, R. Vazquez, "Control of the longitudinal flight dynamics of an UAV using adaptive backstepping," IFAC World Congress, 2011.
- F. Gavilan, R. Vazquez and J. A. Acosta, "Adaptive Backstepping Control for UAV Longitudinal Flight Dynamics with Thrust Saturation," Journal of Guidance, Control and Dynamics, 2014.
- F. Gavilan, R. Vazquez, E. F. Camacho, "An Iterative Model Predictive Control Algorithm for UAV Guidance," IEEE Transactions on Aerospace and Electronic Systems, vol. 51, No. 3, pp. 2406 - 2419, 2015.

Real systems are characterized, among other things, by the ubiquitous presence of uncertainty. Be it in the form of measurement errors, unmodelled dynamics, unknown system parameter, or errors in the initial conditions (or several of these at the same time), in applications it is just unavoidable to have a certain degree of uncertainty present in the system.

However, oftentimes unknown parts of the systems or parameters can be modelled by making use of the theory of statistics, in particular by using distribution functions. The inclusion of probabilistic modelling makes the plant dynamics stochatic (be it continuous or discrete in time).

There is a theoretical body that allows, from these stochastic models, to determine the system evolution. The state of the system would, however, be known not in an exact way, but rather described in a probabilistic way (as a distribution function that evolves in time). This theory of stochastic differential equations also allows us to quantify uncertainty propagation, i.e., how uncertainty rises (or diminishes) with time.

Applications are very diverse, ranging from evolution of droplet populations to the dynamics of aircraft when subject to uncertainty in the performance models or the initial conditions.

Some selected publications:

- R. Vazquez and A. M. Gañan-Calvo, "An operational calculus framework to characterize droplet size populations from turbulent breakup by a small number of parameters," Journal of Physics A, vol. 43, 185501, 2010.
- R. Vazquez, D. Rivas, "Propagation of Initial Mass Uncertainty in Aircraft Cruise Flight," Journal of Guidance, Control and Dynamics, vol. 36 (2), 415-429, 2013

The number of Earth Observation Satellites (EOSs) is on the rise, mainly due to the ever-increasing demand of images of our planet. EOSs are required to capture these pictures with as much resolution as possible and, oftentimes, as soon as possible or even on the fly (for example when catastrophic events happen).

Managing these observational resources has been done in the past mostly in a manual fashion, and on a satellite-by-satellite basis. However, there is nowadays a true market for satellite images that allows the combination of observations from many different satellites to cover any region of interest. Selecting a set of satellites in an optimal cost-effective way, taking into account observational constraints, and in a short time-frame, require the development of optimal planners able to tackle challenging computational geometry problems together with orbital mechanics and the user preferences.

There are other instances of optimal planning problems in space missions such as the satellite-antenna assignation problem (for a large number of both antennas and satellites) or the management of observational modes for agile satellites, which requires the coordination of the satellite attitude with the relative position of the different targets as the satellite moves along its orbit.

All these applications require the development of heuristic algorithms which are able to rapidly provide feasible solutions (approaching as much as possible the global optimum). These algorithms pose a challenge as they have to be integrated with tools from computational geometry and orbital mechanics and verified and simplemented in realistic simulation software (for instance, SaVoir).

Some selected publications:

- J. Galan, R. Vazquez, E. Carrizosa, C. V. Flores, F. Perea, F. M. Crespo, "Towards a visual tool for swath acquisition planning in multiple-mission EOSs," 7th International Workshop on Planning and Scheduling for Space IWPSS-11, 2011.
- R. Vazquez, F. Perea, J. Galan-Vioque, "Resolution of an Antenna-Satellite assignment problem by means of Integer Linear Programming," Aerospace Science and Technology, 2014.
- F. Perea, R. Vazquez, J. Galan-Vioque, "Swath acquisition planning in multiple-mission EOSs: exact and heuristic approaches," IEEE Transactions on Aerospace and Electronic Systems, vol. 51, No. 3, pp. 1717-1725, 2015.

© 2007-2014 Rafael Vazquez Valenzuela