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1. Controllability of Sobolev-Type Linear Ensemble Systems

   https://doi.org/10.1109/CDC45484.2021.9683659  

   Zhang, W; Tie, L; Li, J.-S. (December 2021, 60th IEEE Conference on Decision and Control)

   Systems composed of large ensembles of isolated or interacted dynamic units are prevalent in nature and engineered infrastructures. Linear ensemble systems are inarguably the simplest class of ensemble systems and have attracted intensive attention to control theorists and practionars in the past years. Comprehensive understanding of dynamic properties of such systems yet remains far-fetched and requires considerable knowledge and techniques beyond the reach of modern control theory. In this paper, we explore the classes of linear ensemble systems with system matrices that are not globally diagonalizable. In particular, we focus on analyzing their controllability properties under a Sobolev space setting and develop conditions under which uniform controllability of such ensemble systems is equivalent to that of their diagonalizable counterparts. This development significantly facilitates controllability analysis for linear ensemble systems through examining diagonalized linear systems. 

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2. Finite Sample Identification of Bilinear Dynamical Systems

   Sattar, Yahya; Oymak, Samet; Ozay, Necmiye (January 2022, Proceedings of the IEEE Conference on Decision Control)

   Bilinear dynamical systems are ubiquitous in many different domains and they can also be used to approximate more general control-affine systems. This motivates the problem of learning bilinear systems from a single trajectory of the system’s states and inputs. Under a mild marginal meansquare stability assumption, we identify how much data is needed to estimate the unknown bilinear system up to a desired accuracy with high probability. Our sample complexity and statistical error rates are optimal in terms of the trajectory length, the dimensionality of the system and the input size. Our proof technique relies on an application of martingale small-ball condition. This enables us to correctly capture the properties of the problem, specifically our error rates do not deteriorate with increasing instability. Finally, we show that numerical experiments are well-aligned with our theoretical results. 

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3. Sparse Linear Ensemble Systems and Structural Controllability

   https://doi.org/10.1109/TAC.2021.3097289  

   Chen, Xudong (July 2021, IEEE Transactions on Automatic Control)

   The paper introduces and solves a structural controllability problem for continuum ensembles of linear time-invariant systems. All the individual linear systems of an ensemble are sparse, governed by the same sparsity pattern. Controllability of an ensemble system is, by convention, the capability of using a common control input to simultaneously steer every individual systems in it. A sparsity pattern is structurally controllable if it admits a controllable linear ensemble system. A main contribution of the paper is to provide a graphical condition that is necessary and sufficient for a sparsity pattern to be structurally controllable. Like other structural problems, the property of being structural controllable is monotone. We provide a complete characterization of minimal sparsity patterns as well. 

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4. Scalable computation of input-normal/output-diagonal balanced realization for control-affine polynomial systems

   https://doi.org/10.1016/j.sysconle.2025.106178  

   Corbin, Nicholas A; Sarkar, Arijit; Scherpen, Jacquelien MA; Kramer, Boris (October 2025, Systems & Control Letters)

   We present a scalable tensor-based approach to computing input-normal/output-diagonal nonlinear balancing transformations for control-affine systems with polynomial nonlinearities. This transformation is necessary to determine the states that can be truncated when forming a reduced-order model. Given a polynomial representation for the controllability and observability energy functions, we derive the explicit equations to compute a polynomial transformation to induce input-normal/output-diagonal structure in the energy functions in the transformed coordinates. The transformation is computed degree-by-degree, similar to previous Taylor-series approaches in the literature. However, unlike previous works, we provide a detailed analysis of the transformation equations in Kronecker product form to enable a more scalable implementation. We derive the explicit algebraic structure for the equations, present rigorous analyses for the solvability and algorithmic complexity of those equations, and provide general purpose open-source software implementations for the proposed algorithms to stimulate broader use of nonlinear balanced truncation model reduction. We demonstrate that with our efficient implementation, computing the nonlinear transformation is approximately as expensive as computing the energy functions using state-of-the-art methods. 

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5. A Randomized Distributed Kaczmarz Algorithm and Anomaly Detection

   https://doi.org/10.3390/axioms11030106  

   Keinert, Fritz; Weber, Eric S. (March 2022, Axioms)

   The Kaczmarz algorithm is an iterative method for solving systems of linear equations. We introduce a randomized Kaczmarz algorithm for solving systems of linear equations in a distributed environment, i.e., the equations within the system are distributed over multiple nodes within a network. The modification we introduce is designed for a network with a tree structure that allows for passage of solution estimates between the nodes in the network. We demonstrate that the algorithm converges to the solution, or the solution of minimal norm, when the system is consistent. We also prove convergence rates of the randomized algorithm that depend on the spectral data of the coefficient matrix and the random control probability distribution. In addition, we demonstrate that the randomized algorithm can be used to identify anomalies in the system of equations when the measurements are perturbed by large, sparse noise. 

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