Finite-Dimensional Observation and Control of Ensembles of Linear Systems
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. In applications, such as classical thermodynamics, where the conversion of heat into work and vice versa effects the control of averages like temperature and pressure, it is often impossible to control individual states due to noise and system size. Therefore, ensemble control also refers to the more realistic task of steering finite dimensional output quantities such as the mean value and/or the covariance of an entire ensemble. The project FOCENS considers parameter-dependent linear systems and investigates the problem of controlling entire families of systems using open-loop inputs which are independent of the parameter so that given control tasks are achieved simultaneously. A consequence of this restriction is that most results in the literature on infinite dimensional linear systems are either not applicable or hardly verifiable. In recent years much effort has been spent on the derivation of pointwise testable necessary and sufficient conditions for the existence of suitable open-loop inputs. This project addresses the subsequent question how such inputs can be computed. Moreover, the project also explores whether the incorporation of feedback techniques enlarges the range of ensemble control problems that can be solved. In this project we thoroughly investigated the problem of constructing suitable open-loop inputs. We distinguish between continuous-time and discrete-time systems. As for discrete-time systems the solution formula can be represented by polynomials, we explored constructive methods from (complex) polynomial approximation theory. Based on derived sufficient conditions for the existence of such inputs, the project provides two procedures to compute suitable inputs for discrete-time single-input systems if the corresponding polynomial approximation takes place on the real line or on the unit circle. Moreover, using conformal mappings we provide techniques to extend this approach to Jordan arcs. Moreover, if the matrices A(θ) are diagonalizable, one of the methods can also applied to continuous-time single input systems. For continuous-time systems the solution formula is given by a linear integral operator. The project considers the moment collocation technique from inverse problems, as it inherits well-known methods from control theory such as the controllability Gramian. The approach is to sample the linear ensemble at finitely many parameters. This yields a classical finite-dimensional parallel connection for which the minimal L2 -norm input function is given in terms of the controllability Gramian. To derive a lower bound on the number of sampling parameters so that the input for the parallel conncetion yields a suitable input for the entire linear ensemble, we refined and extended a standard convergence result to vectorvalued integral equations. Moreover, we also developed two consensus methods that do not require the computation of the inverse of the controllability Gramian of the parallel connection. In either case, a suitable input is obtained by computing for each parameter an appropiate input and run a consesus scheme. The FOCENS project also explores how the usage of mixtures of open-loop and feedback controllers enlarges the range of ensemble control tasks on the space of continuous functions. We distinguish between the single-input and the multi-input case. Assuming the necessary condition that a single-input pair (A, b) is pointwise reachable, we showed that this implies the existence of a feedback f such that the pair (A + bf, b) is uniformly ensemble reachable. The analysis of multi-input systems is more involved. We showed that a pair (A, B) is restricted feedback equivalent to an uniformly ensemble reachable pair if it is pointwise reachable and the parameter-dependent linear system has constant Kronecker indices. Furthermore, the project presents a first attempt to investigate the possibility to apply open-loop and feedback controllers with a parameter-independent feedback matrix. Examplarily, we tackled the controllled harmonic oscillator. For this case we provide sufficient conditions for the existence of suitable parameterindependent feedback gains. If the desired terminal states also satisfy a Lipschitz condition, we derived an estimate on the approximation accuracy illustrating the infuence of the feedback gain.
Publications
- Ensemble reachability of parametric harmonic oscillators via mixed open-loop and feedback control. Proc. Appl. Math. Mech. 18(1):e201800467, 2018
M. Schönlein
(See online at https://doi.org/10.1002/pamm.201800467) - Ensemble reachability of homogenous parameter-depedent systems. Proc. Appl. Math. Mech. 20(1):e202000342
M. Schönlein
(See online at https://doi.org/10.1002/pamm.202000342) - Computation of open-loop inputs for uniformly ensemble controllable systems
M. Schönlein
(See online at https://doi.org/10.48550/arXiv.2105.14812) - Feedback equivalence and uniform ensemble reachability
M. Schönlein
(See online at https://doi.org/10.48550/arXiv.2106.00658) - Polynomial methods to construct inputs for uniformly ensemble reachable systems
M. Schönlein
(See online at https://doi.org/10.48550/arXiv.2105.14963) - Uniform and Lq -ensemble reachability of parametric linear systems. Journal of Differential Equations 283:216–262, 2021
G. Dirr and M. Schönlein
(See online at https://doi.org/10.1016/j.jde.2021.02.032)
