Finite-Dimensional Observation and Control of Ensembles of Linear Systems (FOCENS)
Final Report Abstract
Ensemble control is an emerging field of mathematical systems and control theory that is concerned with the task of simultaneously controlling a large, potentially infinite, number of systems, using open-loop inputs and/or feedback controllers which have to be chosen independently of the individual subsystems. Thus ensemble control is very much at the core of robust control theory and motivated by a wide range of engineering applications. Moreover, in robotics and systems engineering there has recently been much interest in studying motion control problems for spatio-temporal systems such as infinite platoons of vehicles. Such shift-invariant systems can often be transformed to a linear system over a suitably chosen functions space resulting in a linear ensemble. A slightly different aspect of ensemble control refers to the effect that in large scale systems it is often impossible to control individual states due to noise and system size. Therefore, a more realistic goal is to steer finite dimensional output quantities such as the mean value and/or the covariance of the entire ensemble. An instance where this arises is e.g. in classical thermodynamics, where the conversion of heat into work and vice versa effects the control of averages like temperature and pressure. The involved researchers provided contributions to a comprehensive structure theory for parameter-dependent linear systems (linear ensembles) and derived new results on controlling density distributions. In detail, the following findings were obtained: We investigated thoroughly the ensemble control problem for linear parameter-dependent systems. As indicated above, the challenge of this class of problems results from the fact that the control task has to be achieved simultaneously for all parameters using open-loop inputs which are independent of the parameter. As a consequence of this requirement most results in the literature on infinite dimensional linear systems are either not applicable or hardly verifiable. Here, our structural results on controllability allow to decompose ensembles into smaller subensembles, to analyze and “solve” them independently and to recompose everything. Following this strategy, we provided generalizations of existing results on uniform ensemble controllability as well as new findings on Lq-ensemble controllability. Moreover, we obtained a characterization of uniform ensemble controllability for time-variant linear systems and new sufficient conditions for output ensemble controllability. Concerning the control of probability density functions we derived: two sufficient controllability criteria for moment control problems generalizing a former result by R. Brockett and a new controllability result for parallel connected moment control problems which is of particular interest due to its bilinear nature. Taking into account the limited funding period, we think the goals of the project have been reached in a most satisfying way.
Publications
- Uniform and Lq-ensemble reachability of parametric linear systems
G. Dirr and M. Schönlein
- Controlling mean and variance in ensembles of linear systems. Proc. 10th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2016), Monterey, USA, pp. 1036–1041, 2016
G. Dirr, U. Helmke and M. Schönlein
(See online at https://doi.org/10.1016/j.ifacol.2016.10.301) - Uniform ensemble controllability of parametric systems. Proc. 22nd International Symposium on Mathematical Theory of Networks and Systems (MTNS 2016), Minneapolis, USA, pp. 677–684, 2016
G. Dirr, U. Helmke and M. Schönlein
