Recent years have witnessed a wave of research activities in systems science toward the study of population systems. The driving force behind this shift was geared by numerous emerging and ever-changing technologies in life and physical sciences and engineering, from neuroscience, biology, and quantum physics to robotics, where many control-enabled applications involve manipulating a large ensemble of structurally identical dynamic units, or agents. Analyzing fundamental properties of ensemble control systems in turn plays a foundational and critical role in enabling and, further, advancing these applications, and the analysis is largely beyond the capability of classical control techniques. In this paper, we consider an ensemble of time-invariant linear systems evolving on an infinite-dimensional space of continuous functions. We exploit the notion of separating points and techniques of polynomial approximation to develop necessary and sufficient ensemble controllability conditions. In particular, we introduce an extended notion of controllability matrix, called Ensemble Controllability Gramian. This means enables the characterization of ensemble controllability through evaluating controllability of each individual system in the ensemble. As a result, the work provides a unified framework with a systematic procedure for analyzing control systems defined on an infinite-dimensional space by a finite-dimensional approach.
We study possibilities to control an ensemble (a parameterized family) of nonlinear control systems by a single parameter-independent control. Proceeding by Lie algebraic methods we establish genericity of exact controllability property for finite ensembles, prove sufficient approximate controllability condition for a model problem in $\mathbb{R}^3$, and provide a variant of Rashevsky-Chow theorem for approximate controllability of control-linear ensembles.
We investigate the task of controlling ensembles of initial and terminal state vectors of parameter-dependent linear systems by applying parameter-independent open loop controls. Necessary, as well as sufficient, conditions for ensemble controllability are established, using tools from complex approximation theory. For real analytic families of linear systems it is shown that ensemble controllability holds only for systems with at most two independent parameters. We apply the results to networks of linear systems and address the question of open-loop robust synchronization.
In this paper, we study uniform ensemble controllability (UEC) of linear ensemble systems defined in an infinite-dimensional space through finite-dimensional settings. Specifically, with the help of the Stone-Weierstrass theorem for modules, we provide an algebraic framework for examining UEC of linear ensemble systems with diagonalizable drift vector fields through checking the controllability of finite-dimensional subsystems in the ensemble. The new framework renders a novel concept of ensemble controllability matrix, which rank-condition serves as a sufficient and necessary condition for UE…
In the paper, we introduce and address the problem of structural averaged controllability for linear ensemble systems. We provide examples highlighting the differences between this problem and others. In particular, we show that structural averaged controllability is strictly weaker than structural controllability for single (or ensembles of) linear systems. We establish a set of necessary or sufficient conditions for sparsity patterns to be structurally averaged controllable.
This note presents a sufficient condition for partial approximate ensemble controllability of a set of bilinear conservative quantum systems in an infinite dimensional Hilbert space. The proof relies on classical geometric and averaging control techniques applied on finite dimensional approximation of the infinite dimensional system. The results are illustrated with the planar rotation of a linear molecule.
Problems involving control of large ensmebles of structurally identical dynamical systems, called \emph{ensemble control}, arise in numerous scientific areas from quantum control and robotics to brain medicine. In many of such applications, control can only be implemented at the population level, i.e., through broadcasting an input signal to all the systems in the population, and this new control paradigm challenges the classical systems theory. In recent years, considerable efforts have been made to investigate controllability properties of ensemble systems, and most works emphasized on linea…
We study one-parametric perturbations of finite dimensional real Hamiltonians depending on two controls, and we show that generically in the space of Hamiltonians, conical intersections of eigenvalues can degenerate into semi-conical intersections of eigenvalues. Then, through the use of normal forms, we study the problem of ensemble controllability between the eigenstates of a generic Hamiltonian.
We address an open problem in ensemble control: Whether there exist controllable linear ensemble systems over multi-dimensional parameterization spaces? We provide a negative result: Any real-analytic linear ensemble system is not $\mathrm{L}^p$-controllable, for $2\leq p \leq \infty$, if its parameterization space contains an open set in $\mathbb{R}^d$ for $d \geq 2$.
In this paper, we study the control of a class of time-invariant linear ensemble systems whose natural dynamics are linear in the system parameter. This class of ensemble control systems arises from practical engineering and physical applications, such as transport of quantum particles and control of uncertain harmonic systems. We establish explicit algebraic criterions to examine controllability of such ensemble systems. Our derivation is based on the notion of polynomial approximation, where the elements of the reachable set of the ensemble system are represented in polynomials of the system…
Ensemble control, an emerging research field focusing on the study of large populations of dynamical systems, has demonstrated great potential in numerous scientific and practical applications. Striking examples include pulse design for exciting spin ensembles in quantum physics, neurostimulation for relieving neurological disorder symptoms, and path planning for steering robot swarms. However, the control targets in such applications are generally large-scale complex and severely underactuated ensemble systems, research into which stretches the capability of techniques in classical control an…
We study the controlled dynamics of the {\it ensembles of points} of a Riemannian manifold $M$. Parameterized ensemble of points of $M$ is the image of a continuous map $γ:Θ\to M$, where $Θ$ is a compact set of parameters. The dynamics of ensembles is defined by the action $γ(θ) \mapsto P_t(γ(θ))$ of the semigroup of diffeomorphisms $P_t:M \to M, \ t \in \mathbb{R}$, generated by the controlled equation $\dot{x}=f(x,u(t))$ on $M$. Therefore any control system on $M$ defines a control system on (generally infinite-dimensional) space $\mathcal{E}_Θ(M)$ of the ensembles of points. We wish to esta…
Symmetric spaces arise in wide variety of problems in Mathematics and Physics. They are mostly studied in Representation theory, Harmonic analysis and Differential geometry. As many physical systems have symmetric spaces as their configuration spaces, the study of controllability on symmetric space is quite interesting. In this paper, a driftless control system of type \dot{x}= \sum_{i=1}^m u_if_i(x) is considered on a symmetric space. For this we have established global controllability condition which is illustrated by few examples of exponential submanifolds of SE(3) and random matrix ensemb…
In this paper, we tackle the long-standing challenges of ensemble control analysis and design using a convex-geometric approach in a Hilbert space setting. Specifically, we formulate the control of linear ensemble systems as a convex feasibility problem in a Hilbert space, which can be solved by iterative weighted projections. Such a non-trivial geometric interpretation not only enables a systematic design principle for constructing feasible, optimal, and constrained ensemble control signals, but also makes it possible for numerical examination of ensemble reachability and controllability. Fur…
Controlling a large population, in the limit, a continuum, of structurally identical dynamical systems with parametric variations is a pervasive task in diverse applications in science and engineering. However, the severely underactuated nature and the inability to avail comprehensive state feedback information of such ensemble systems raise significant challenges in analysis and design of ensemble systems. In this paper, we propose a moment-based ensemble control framework, which incorporates and expands the method of moments in probability theory to control theory. In particular, we establis…
Legendre-Moment Transform for Linear Ensemble Control and Computation
Washington University in St. Louis（华盛顿大学圣路易斯分校）；Tongji University（同济大学）；Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education（教育部智能计算与应用重点实验室（同济大学））
Synchronization of oscillations is a phenomenon prevalent in natural, social, and engineering systems. Controlling synchronization of oscillating systems is motivated by a wide range of applications from neurological treatment of Parkinson's disease to the design of neurocomputers. In this article, we study the control of an ensemble of uncoupled neuron oscillators described by phase models. We examine controllability of such a neuron ensemble for various phase models and, furthermore, study the related optimal control problems. In particular, by employing Pontryagin's maximum principle, we an…
Control of complex networks requires both structure and dynamics
School of Informatics and Computing, Indiana University（印第安纳大学信息学与计算学院）；Program in Cognitive Science, Indiana University（印第安纳大学认知科学项目）；Instituto Gulbenkian de Ciencia（古尔本基安科学研究所）
