Uniform ensemble controllability for one-parameter families of time-invariant linear systems

Abstract In this paper we derive necessary as well as sufficient conditions for approximate controllability of parameter-dependent linear systems in the supremum norm. Using tools from complex approximation theory, we prove the existence of parameter-independent open-loop controls that steer the zero initial state of an ensemble of linear systems uniformly to a prescribed family of terminal states. New necessary conditions for uniform ensemble controllability of single-input systems are derived. Our results extend earlier ones of Li for ensemble controllability of linear systems.

[1]  P. Revesz.  Interpolation and Approximation , 2010 .

[2]  Fernando Paganini,et al.  Distributed control of spatially invariant systems , 2002, IEEE Trans. Autom. Control..

[3]  George Finlay Simmons,et al.  Introduction to Topology and Modern Analysis , 1963 .

[4]  Jonathan R. Partington,et al.  On controllability of diagonal systems with one-dimensional input space , 2006, Syst. Control. Lett..

[5]  S. Liberty,et al.  Linear Systems , 2010, Scientific Parallel Computing.

[6]  Timothy Bretl,et al.  Approximate Steering of a Unicycle Under Bounded Model Perturbation Using Ensemble Control , 2012, IEEE Transactions on Robotics.

[7]  Paul A. Fuhrmann,et al.  On controllability and observability of systems connected in parallel , 1975 .

[8]  Navin Khaneja,et al.  On the Stochastic Control of Quantum Ensembles , 2000 .

[9]  Jr-Shin Li,et al.  Ensemble Control of Finite-Dimensional Time-Varying Linear Systems , 2008, IEEE Transactions on Automatic Control.

[10]  Ulrich Eggers,et al.  Introduction To Infinite Dimensional Linear Systems Theory , 2016 .

[11]  Jr-Shin Li,et al.  Ensemble Control of Bloch Equations , 2009, IEEE Transactions on Automatic Control.

[12]  D. Gaier,et al.  Lectures on complex approximation , 1987 .

[13]  Dirk Aeyels,et al.  Vehicle Platoons Through Ring Coupling , 2008, IEEE Transactions on Automatic Control.

[14]  Henryk Gzyl,et al.  THE WEIERSTRASS APPROXIMATION THEOREM AND LARGE DEVIATIONS , 1997 .

[15]  Michel Verhaegen,et al.  Distributed Control for Identical Dynamically Coupled Systems: A Decomposition Approach , 2009, IEEE Transactions on Automatic Control.

[16]  Uwe Helmke,et al.  Control of Ensembles of Single-Input Continuous-Time Linear Systems , 2013 .

[17]  M. Hazewinkel.  A PARTIAL SURVEY OF THE USES OF ALGEBRAIC GEOMETRY IN SYSTEMS AND CONTROL THEORY , 1979 .

[18]  N. Khaneja,et al.  Control of inhomogeneous quantum ensembles , 2006 .

[19]  Roberto Triggiani.  On the lack of exact controllability for mild solutions in Banach spaces , 1975 .

[20]  P. Fuhrmann.  A Polynomial Approach to Linear Algebra , 1996 .

[21]  M. Newman,et al.  Interpolation and approximation , 1965 .

[22]  Eduardo Sontag.  Linear Systems over Commutative Rings. A Survey , 1976 .

[23]  Uwe Helmke,et al.  Reachability, observability and strict equivalence of networks of linear systems , 2013, Math. Control. Signals Syst..

[24]  Paul A. Fuhrmann,et al.  On weak and strong reachability and controllability of infinite-dimensional linear systems , 1972 .

[25]  R. Brockett.  On the control of a flock by a leader , 2010 .

[26]  S. N. Mergelyan,et al.  On the representation of functions by series of polynomials on closed sets , 1953 .