arXiv subjects

Jr-Shin Li

Publications and source records attributed to Jr-Shin Li.

At least 19 recordsLinked to original sources

arXiv2025

EBIF: Exact Bilinearization Iterative Form for Control-Affine Nonlinear Systems

In this paper, we develop a novel framework, Exact Bilinearization Iterative Form (EBIF), for transforming a nonlinear control-affine system into an exact finite-dimensional bilinear representation. In contrast to most existing approaches which generally lead to an infinite-dimensional representation, the proposed EBIF approach yields an iterative procedure for constructing a finite set of smooth coordinate functions that define an embedding, enabling an exact bilinear representation of the original nonlinear dynamics. Leveraging tools from algebra and differential geometry, we establish both necessary and sufficient conditions for a nonlinear system to be exactly bilinearizable. We further illustrate how the EBIF-induced bilinear systems facilitate reachability analysis and control design. Through theoretical analysis and numerical simulations, we demonstrate the effectiveness of the EBIF framework and highlight its potential in simplifying control synthesis for nonlinear systems.

math.OC↗

arXiv2025

One Ring to Rule Them All: Constrained Distributional Control for Massive-Scale Heterogeneous Robotic Ensemble Systems

Ensemble control aims to steer a population of dynamical systems using a shared control input. This paper introduces a constrained ensemble control framework for parameterized, heterogeneous robotic systems operating under state and environmental constraints, such as obstacle avoidance. We develop a moment kernel transform that maps the parameterized ensemble dynamics to the moment system in a kernel space, enabling the characterization of population-level behavior. The state-space constraints, such as polyhedral waypoints to be visited and obstacles to be avoided, are also transformed into the moment space, leading to a unified formulation for safe, large-scale ensemble control. Expressive signal temporal logic specifications are employed to encode complex visit-avoid tasks, which are achieved through a single shared controller synthesized from our constrained ensemble control formulation. Simulation and hardware experiments demonstrate the effectiveness of the proposed approach in safely and efficiently controlling robotic ensembles within constrained environments.

cs.RO↗

arXiv2025

Robust Quantum State Generation in Symmetric Spin Networks

In this work, we consider a parameterized Ising model with long-range symmetric pairwise interactions on a network of spin $\frac{1}{2}$ particles. The system is designed with symmetric dynamics, allowing for the reduction of the state space to a subspace defined by the set of Dicke states. We propose a method for designing robust electromagnetic amplitude pulses based on a moment quantization approach. The introduced parameter accounts for uncertainties in the electromagnetic field, resulting in a family of distinct Hamiltonians. By employing a discretized moment-based quantization technique, we design a control pulse capable of simultaneously steering an infinite collection of dynamical systems to compensate for parameter variations. This approach benefits from the duality between the infinite-dimensional parameterized system and its finite-dimensional trucnated moment dynamics. Simulation results demonstrate the efficacy of this method in achieving states of significant interest in quantum sensing, including the GHZ and W states.

quant-ph↗

arXiv2025

Distributional Control of Ensemble Systems

Ensemble control offers rich and diverse opportunities in mathematical systems theory. In this paper, we present a new paradigm of ensemble control, referred to as distributional control, for ensemble systems. We shift the focus from controlling the states of ensemble systems to controlling the output measures induced by their aggregated measurements. To facilitate systems-theoretic analysis of these newly formulated distributional control challenges, we establish a dynamic moment kernelization approach, through which we derive the distributional system and its corresponding moment system for an ensemble system. We further explore optimal distributional control by integrating optimal transport concepts and techniques with the moment representations, creating a systematic computational distributional control framework.

math.OC↗

arXiv2025

Robust Quantum Control for Bragg Pulse Design in Atom Interferometry

We formulate a robust optimal control algorithm to synthesize minimum energy pulses that can transfer a cold atom system into various momentum states. The algorithm uses adaptive linearization of the evolution operator and sequential quadratic programming to iterate the control towards a minimum energy pulse that achieves optimal target state fidelity. Robustness to parameter variation is achieved using Legendre polynomial approximation over the domain of variation. The method is applied to optimize the Bragg beamsplitting operation in ultra-cold atom interferometry. Even in the presence of 10-40% variability in the initial momentum dispersion of the atomic cloud and the intensity of the optical pulse, the algorithm reliably converges to a control protocol that robustly achieves unprecedented momentum levels with high fidelity for a single-frequency multi-photon Bragg diffraction scheme (e.g. $|\pm 40\hbar k\rangle$). We show the advantages of our method by comparison to stochastic optimization using sampled parameter values, provide detailed sensitivity analyses, and performance of the designed pulses is verified in laboratory experiments.

quant-ph↗

arXiv2024

Reinforcement Learning for Infinite-Dimensional Systems

Interest in reinforcement learning (RL) for large-scale systems, comprising extensive populations of intelligent agents interacting with heterogeneous environments, has surged significantly across diverse scientific domains in recent years. However, the large-scale nature of these systems often leads to high computational costs or reduced performance for most state-of-the-art RL techniques. To address these challenges, we propose a novel RL architecture and derive effective algorithms to learn optimal policies for arbitrarily large systems of agents. In our formulation, we model such systems as parameterized control systems defined on an infinite-dimensional function space. We then develop a moment kernel transform that maps the parameterized system and the value function into a reproducing kernel Hilbert space. This transformation generates a sequence of finite-dimensional moment representations for the RL problem, organized into a filtrated structure. Leveraging this RL filtration, we develop a hierarchical algorithm for learning optimal policies for the infinite-dimensional parameterized system. To enhance the algorithm's efficiency, we exploit early stopping at each hierarchy, demonstrating the fast convergence property of the algorithm through the construction of a convergent spectral sequence. The performance and efficiency of the proposed algorithm are validated using practical examples in engineering and quantum systems.

eess.SY↗

arXiv2024

Convergence of Iterative Quadratic Programming for Robust Fixed-Endpoint Transfer of Bilinear Systems

We present a computational method for open-loop minimum-norm control synthesis for fixed-endpoint transfer of bilinear ensemble systems that are indexed by two continuously varying parameters. We suppose that one ensemble parameter scales the homogeneous, linear part of the dynamics, and the second parameter scales the effect of the applied control inputs on the inhomogeneous, bilinear dynamics. This class of dynamical systems is motivated by robust quantum control pulse synthesis, where the ensemble parameters correspond to uncertainty in the free Hamiltonian and inhomogeneity in the control Hamiltonian, respectively. Our computational method is based on polynomial approximation of the ensemble state in parameter space and discretization of the evolution equations in the time domain using a product of matrix exponentials corresponding to zero-order hold controls over the time intervals. The dynamics are successively linearized about control and trajectory iterates to formulate a sequence of quadratic programs for computing perturbations to the control that successively improve the objective until the iteration converges. We use a two-stage computation to first ensure transfer to the desired terminal state, and then minimize the norm of the control function. The method is demonstrated for the canonical uniform transfer problem for the Bloch system that appears in nuclear magnetic resonance, as well as the matter-wave splitting problem for the Raman-Nath system that appears in ultra-cold atom interferometry.

math.OC↗

arXiv2024

Legendre-Moment Transform for Linear Ensemble Control and Computation

Ensemble systems, pervasive in diverse scientific and engineering domains, pose challenges to existing control methods due to their massive scale and underactuated nature. This paper presents a dynamic moment approach to addressing theoretical and computational challenges in systems-theoretic analysis and control design for linear ensemble systems. We introduce the Legendre-moments and Legendre-moment transform, which maps an ensemble system defined on the $L^2$-space to a Legendre-moment system defined on the $\ell^2$-space. We show that this pair of systems is of one-to-one correspondence and shares the same controllability property. This equivalence admits the control of an ensemble system through the control of the corresponding Legendre-moment system and inspires a unified control design scheme for linear ensemble systems using structured truncated moment systems. In particular, we develop a sampling-free ensemble control design algorithm, then conduct error analysis for control design using truncated moment systems and derive error bounds with respect to the truncation orders, which are illustrated with numerical examples.

math.OC↗

arXiv2023

An Iterative Approach to Data-Driven Inference for Decoding Oscillator Network Structures

In complex networks, interactions between multiple agents give rise to an array of intricate global dynamics, ranging from synchronization to cluster formations. Decoding the connectivity structure as well as the types of interactions from measurement data is the first step toward understanding these intriguing behaviors. In this paper, we present a bilinear optimization framework to infer both the connectivity and interaction functions of oscillator networks with the identical class of coupling functions. We then propose an iterative algorithm to solve the resulting bilinear problem and illustrate its convergence. We validate our approach on both simulated and noisy experimental datasets, where we demonstrate its effectiveness compared with existing approaches.

eess.SY↗

arXiv2022

Koopman Bilinearization of Nonlinear Control Systems

Koopman operators, since introduced by the French-born American mathematician Bernard Koopman in 1931, have been employed as a powerful tool for research in various scientific domains, such as ergodic theory, probability theory, geometry, and topology. The current use of Koopman operators mainly focuses on the characterization of spectral properties of ergodic dynamical systems. In this paper, we step forward from unforced dynamical systems to control systems and establish a systematic Koopman control framework. Specifically, we rigorously derive a differential equation system governing the dynamics of the Koopman operator associated with a control system, and show that the resulting system is a bilinear system evolving on an infinite-dimensional Lie group, which directly leads to a global bilinearization of control-affine systems. Then, by integrating techniques in geometric control theory with infinite-dimensional differential geometry, this further offers a two-fold benefit to controllability analysis: the characterization of controllability for control-affine systems in terms of de Rham differential operators and the extension of the Lie algebra rank condition to infinite-dimensional systems. To demonstrate the applicability, we further adopt the established framework to develop the Koopman feedback linearization technique, which, as one of the advantages, waives the controllability requirement for the systems to be linearized by using the classical feedback linearization technique. In addition, it is worth mentioning that a distinctive feature of this work is the maximum utilization of the intrinsic algebraic and geometric properties of Koopman operators, instead of spectral methods and the ergodicity assumption of systems, which further demonstrate a significant advantage of our work and a substantial difference between the presented and existing research into Koopman operators.

math.OC↗

arXiv2022

Controllability Canonical Forms of Linear Ensemble Systems

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 and dynamical systems theory to the very limit. This paper then devotes to advancing our knowledge about controllability of linear ensemble systems by integrating tools in modern algebra into the technique of separating points developed in our recent work. In particular, we give an algebraic interpretation of the dynamics of linear systems in terms of actions of polynomials on vector spaces, and this leads to the development of the functional canonical form of matrix-valued functions, which can also be viewed as the generalization of the rational canonical form of matrices in linear algebra. Then, leveraging the technique of separating points, we achieve a necessary and sufficient characterization of uniform ensemble controllability for time-invariant linear ensemble systems as the ensemble controllability canonical form, in which the system and control matrices are in the functional canonical and block diagonal form, respectively. This work successfully launches a new research scheme by adopting and tailoring finite-dimensional methods to tackle control problems involving infinite-dimensional ensemble systems, and lays a solid foundation for a more inclusive ensemble control theory targeting a much broader spectrum of control and learning problems in both scientific research and practice.

math.OC↗

arXiv2022

Bilinear Systems Induced by Proper Lie Group Actions

In the study of induced bilinear systems, the classical Lie algebra rank condition (LARC) is known to be impractical since it requires computing the rank everywhere. On the other hand, the transitive Lie algebra condition, while more commonly used, relies on the classification of transitive Lie algebras, which is elusive except for few simple geometric objects such as spheres. We prove in this note that for bilinear systems induced by proper Lie group actions, the underlying Lie algebra is closely related to the orbits of the group action. Knowing the pattern of the Lie algebra rank over the manifold, we show that the LARC can be relaxed so that it suffices to check the rank at an arbitrary single point. Moreover, it removes the necessity for classifying transitive Lie algebras. Finally, this relaxed rank condition also leads to a characterization of controllable submanifolds by orbits.

eess.SY↗

arXiv2021

On Uniform Ensemble Controllability of Diagonalizable Linear Ensemble Systems

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 UEC of linear ensembles. We provide several examples demonstrating that the proposed approach well-encompasses existing results and analyzes UEC of linear ensembles not addressed by literature.

math.OC↗

arXiv2021

Ensemble Recognition in Reproducing Kernel Hilbert Spaces through Aggregated Measurements

In this paper, we study the problem of learning dynamical properties of ensemble systems from their collective behaviors using statistical approaches in reproducing kernel Hilbert space (RKHS). Specifically, we provide a framework to identify and cluster multiple ensemble systems through computing the maximum mean discrepancy (MMD) between their aggregated measurements in an RKHS, without any prior knowledge of the system dynamics of ensembles. Then, leveraging the gradient flow of the newly proposed notion of aggregated Markov parameters, we present a systematic framework to recognize and identify an ensemble systems using their linear approximations. Finally, we demonstrate that the proposed approaches can be extended to cluster multiple unknown ensembles in RKHS using their aggregated measurements. Numerical experiments show that our approach is reliable and robust to ensembles with different types of system dynamics.

cs.LG↗

arXiv2021

Interpretable Design of Reservoir Computing Networks using Realization Theory

The reservoir computing networks (RCNs) have been successfully employed as a tool in learning and complex decision-making tasks. Despite their efficiency and low training cost, practical applications of RCNs rely heavily on empirical design. In this paper, we develop an algorithm to design RCNs using the realization theory of linear dynamical systems. In particular, we introduce the notion of $\alpha$-stable realization, and provide an efficient approach to prune the size of a linear RCN without deteriorating the training accuracy. Furthermore, we derive a necessary and sufficient condition on the irreducibility of number of hidden nodes in linear RCNs based on the concepts of controllability and observability matrices. Leveraging the linear RCN design, we provide a tractable procedure to realize RCNs with nonlinear activation functions. Finally, we present numerical experiments on forecasting time-delay systems and chaotic systems to validate the proposed RCN design methods and demonstrate their efficacy.

cs.LG↗

arXiv2021

Structural Controllability on Graphs for Drifted Bilinear Systems over Lie Groups

In this paper, we study graphical conditions for structural controllability and accessibility of drifted bilinear systems over Lie groups. We consider a bilinear control system with drift and controlled terms that evolves over the special orthogonal group, the general linear group, and the special unitary group. Zero patterns are prescribed for the drift and controlled dynamics with respect to a set of base elements in the corresponding Lie algebra. The drift dynamics must respect a rigid zero-pattern in the sense that the drift takes values as a linear combination of base elements with strictly non-zero coefficients; the controlled dynamics are allowed to follow a free zero pattern with potentially zero coefficients in the configuration of the controlled term by linear combination of the controlled base elements. First of all, for such bilinear systems over the special orthogonal group or the special unitary group, the zero patterns are shown to be associated with two undirected or directed graphs whose connectivity and connected components ensure structural controllability/accessibility. Next, for bilinear systems over the special unitary group, we introduce two edge-colored graphs associated with the drift and controlled zero patterns, and prove structural controllability conditions related to connectivity and the number of edges of a particular color.

math.OC↗

arXiv2021

Optimization of periodic input waveforms for global entrainment of weakly forced limit-cycle oscillators

We propose a general method for optimizing periodic input waveforms for global entrainment of weakly forced limit-cycle oscillators based on phase reduction and nonlinear programming. We derive averaged phase dynamics from the mathematical model of a limit-cycle oscillator driven by a weak periodic input and optimize the Fourier coefficients of the input waveform to maximize prescribed objective functions. In contrast to the optimization methods that rely on the calculus of variations, the proposed method can be applied to a wider class of optimization problems including global entrainment objectives. As an illustration, we consider two optimization problems, one for achieving fast global convergence of the oscillator to the entrained state and the other for realizing prescribed global phase distributions in a population of identical uncoupled noisy oscillators. We show that the proposed method can successfully yield optimal input waveforms to realize the desired states in both cases.

nlin.AO↗

arXiv2020

Combinatorics-Based Approaches to Controllability Characterization for Bilinear Systems

The control of bilinear systems has attracted considerable attention in the field of systems and control for decades, owing to their prevalence in diverse applications across science and engineering disciplines. Although much work has been conducted on analyzing controllability properties, the mostly used tool remains the Lie algebra rank condition. In this paper, we develop alternative approaches based on theory and techniques in combinatorics to study controllability of bilinear systems. The core idea of our methodology is to represent vector fields of a bilinear system by permutations or graphs, so that Lie brackets are represented by permutation multiplications or graph operations, respectively. Following these representations, we derive combinatorial characterization of controllability for bilinear systems, which consequently provides novel applications of symmetric group and graph theory to control theory. Moreover, the developed combinatorial approaches are compatible with Lie algebra decompositions, including the Cartan and non-intertwining decomposition. This compatibility enables the exploitation of representation theory for analyzing controllability, which allows us to characterize controllability properties of bilinear systems governed by semisimple and reductive Lie algebras.

math.OC↗