Shen Zeng

Ensemble Observability of Linear Systems

We address the observability problem for ensembles that are described by probability distributions. The problem is to reconstruct a probability distribution of the initial state from the time-evolution of the probability distribution of the output under a classical finite-dimensional linear system. We present two solutions to this problem, one based on formulating the problem as an inverse problem and the other one based on reconstructing all the moments of the distribution. The first approach leads us to a connection between the reconstruction problem and mathematical tomography problems. In the second approach we use the framework of tensor systems to describe the dynamics of the moments, which leads to a more systems theoretic treatment of the reconstruction problem. Furthermore we show that both frameworks are inherently related. The appeal of having two dual viewpoints, the first being more geometric and the second one being more systems theoretic, is illuminated in several examples of theoretical or practical importance.

An inverse problem of tomographic type in population dynamics

In this paper we address an inverse problem on populations described by probability distributions. From a theoretical point of view, this problem can be seen as a natural extension to the classical observability problem. We consider a population that is described by a classical linear finite-dimensional system in which the initial state is a random vector subject to a non-parametric probability distribution. The problem is to reconstruct this initial state distribution from the time-evolution of the probability distribution of the output. We reveal as a novel viewpoint, that, at its core, this problem is a tomography problem which is a well-known subject in the field of inverse problems. Furthermore we show how this tomography problem is inherently linked with the observability properties of the finite-dimensional system thereby establishing a beautiful link between a control theoretic question and tomography problems.

Sampled Observability and State Estimation of Linear Discrete Ensembles

We consider the problem of reconstructing the initial states of a finite group of structurally identical linear systems in the situation that output measurements of the individual systems are received at discrete time steps and in an anonymized manner: While we do know all output measurements of the individual systems in the group, we do not know which output measurement corresponds to which system. This state estimation problem addresses the essence of state estimation problems for populations, in which the output measurements of the individual systems are given only as statistics. We adopt a measure theoretical approach in which the group is modelled by an LTI system describing the structure of the individual systems and an initial state which is expressed by a discrete measure. In this framework we derive a geometric characterization for the state estimation to admit a unique solution, which combined with a result on the observability of linear systems under irregular sampling, yields a sufficient condition for the sampled observability of discrete ensembles. As a supplement to our theoretical findings, we provide illustrations by means of simulation examples. Furthermore we consider the practical state estimation problem under noisy output measurements.

On the state estimation problem for discrete ensembles from discrete-time output snapshots

We consider the problem of reconstructing the initial states of a group of structurally identical LTI systems in the situation that output measurements of the individual systems in the group are received at discrete time steps and in an anonymized manner: While we do know all output measurements of the individual systems in the group, we do not know which output measurement corresponds to which system. This specific state estimation problem is motivated by emerging problems in applied fields that are dealing with populations of structurally identical systems for which it is only possible to measure the population as a whole, e.g. due to economic or technological reasons. We adopt a measure theoretical approach in which the group is modelled by an LTI system describing the structure of the individual systems and an initial state which is expressed by a discrete measure. We derive characterizations for the state estimation to admit a unique solution.

Identifiability of population models via a measure theoretical approach

Abstract Heterogeneity in cell populations is a major factor in the dynamics of cellular systems in living tissue or microbial colonies. This heterogeneity needs to be taken into account for the interpretation of experimental observations as well as in the construction of predictive models for cellular systems. A common modelling framework for heterogeneous cell population is by an infinite ensemble of single cell models. The state of a cell population is in this framework modelled by the distribution of the single cell states. In this paper we study under which conditions the population model is identifiable, i.e., we can determine the initial distribution of cell states and parameters from a dynamic output distribution. We derive a necessary condition on the single cell model based on the classical observability results from linear and nonlinear control theory. Our results are illustrated via examples.

Sampled observability of discrete heterogeneous ensembles from anonymized output measurements

We consider the problem of reconstructing the initial states of a group of heterogeneous LTI systems in the situation that output measurements of the subsystems are received at discrete time steps and in an anonymized manner. By the latter we refer to the fact that in each time step, the association of the output measurements of the subsystems and the subsystem that produced it is lost. Therefore, the output measurements at a given time step are called output snapshot, as they capture only the groups outputs as a whole. This setup is motivated by state estimation problems for populations of dynamical systems in which the output measurements of the subsystems are only given as statistics. We give a theoretical result concerning the number of measurement times needed in order to be able to uniquely reconstruct the initial states, and furthermore illustrate a practical method for state estimation under noisy output snapshots.

On the ensemble observability problem for nonlinear systems

We consider the ensemble observability problem for nonlinear systems and illuminate it from both theoretical and practical perspectives. In the ensemble observability problem we would like to reconstruct a density of initial states from the evolution of the density of outputs under a nonlinear system. For the theoretical question of ensemble observability, i.e. the uniqueness of the reconstruction of the initial state density, we discuss possible approaches. One approach is based on the natural connection between the ensemble observability problem with nonlinear tomography problems. Through this formulation, the problem also becomes amenable to computational solutions. We present an implementation of Algebraic Reconstruction Techniques from computerized tomography which is well-suited for reconstructing the initial state density. We illustrate this method on a simulation example.

Linear systems with quadratic outputs

We study linear systems with quadratic outputs. Our goal is to analyze and control the evolution of such quadratic outputs under a given linear control system. In this spirit, we first attempt to find a differential equation governing the evolution of the outputs in order to consequently be able to control this evolution. We derive tangible algebraic conditions for being able to find such a differential equation and explicitly determine controls which bilinearize that differential equation.

Structured optimal feedback in multi-agent systems: A static output feedback perspective ☆

In this paper we demonstrate how certain structured feedback gains necessarily emerge as the optimal controller gains in two linear optimal control formulations for multi-agent systems. We consider the cases of linear optimal synchronization and linear optimal centroid stabilization. In the former problem, the considered cost functional integrates squared synchronization error and input, and in the latter, the considered cost functional integrates squared sum of the states and input. Our approach is to view the structures in the feedback gains in terms of a static output feedback with suitable output matrices and to relate this fact with the optimal control formulations. We show that the two considered problems are special cases of a more general case in which the optimal feedback to a linear quadratic regulator problem with cost functionals integrating squared outputs and inputs is a static output feedback. A treatment in this light leads to a very simple and general solution which significantly generalizes a recent result for the linear optimal synchronization problem. We illustrate the general problem in a geometric light.

Collinear dynamical systems

We characterize coupled linear dynamical systems whose solutions remain collinear for all times whenever they are initialized collinearly. With this characterization at hand, we further ask whether linear control systems can be forced to exhibit such collinear solutions via appropriate choice of state feedback. Last, we extend our analysis to multiple collinear systems, therein also encountering coplanar dynamical systems and, in general, linear dynamical systems living on certain Grassmannians.