Infinite-dimensional Linear Systems Research Articles

Closed-loop Koopman operator approximation

This paper proposes a method to identify a Koopman model of a feedback-controlled system given a known controller. The Koopman operator allows a nonlinear system to be rewritten as an infinite-dimensional linear system by viewing it in terms of an infinite set of lifting functions. A finite-dimensional approximation of the Koopman operator can be identified from data by choosing a finite subset of lifting functions and solving a regression problem in the lifted space. Existing methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems, such as unstable systems, in an open-loop fashion. The proposed method leverages the linearity of the Koopman operator, along with knowledge of the controller and the structure of the closed-loop (CL) system, to simultaneously identify the CL and plant systems. The advantages of the proposed CL Koopman operator approximation method are demonstrated in simulation using a Duffing oscillator and experimentally using a rotary inverted pendulum system. An open-source software implementation of the proposed method is publicly available, along with the experimental dataset generated for this paper.

On BIBO Stability of Infinite-Dimensional Linear State-Space Systems

On BIBO Stability of Infinite-Dimensional Linear State-Space Systems

Solvability of Time-varying Infinite-dimensional Linear Port-Hamiltonian Systems

Solvability of Time-varying Infinite-dimensional Linear Port-Hamiltonian Systems

Hybrid quantum algorithms for flow problems.

For quantum computing (QC) to emerge as a practically indispensable computational tool, there is a need for quantum protocols with end-to-end practical applications-in this instance, fluid dynamics. We debut here a high-performance quantum simulator which we term QFlowS (Quantum Flow Simulator), designed for fluid flow simulations using QC. Solving nonlinear flows by QC generally proceeds by solving an equivalent infinite dimensional linear system as a result of linear embedding. Thus, we first choose to simulate two well-known flows using QFlowS and demonstrate a previously unseen, full gate-level implementation of a hybrid and high precision Quantum Linear Systems Algorithms (QLSA) for simulating such flows at low Reynolds numbers. The utility of this simulator is demonstrated by extracting error estimates and power law scaling that relates [Formula: see text] (a parameter crucial to Hamiltonian simulations) to the condition number [Formula: see text] of the simulation matrix and allows the prediction of an optimal scaling parameter for accurate eigenvalue estimation. Further, we include two speedup preserving algorithms for a) the functional form or sparse quantum state preparation and b) in situ quantum postprocessing tool for computing nonlinear functions of the velocity field. We choose the viscous dissipation rate as an example, for which the end-to-end complexity is shown to be [Formula: see text], where [Formula: see text] is the size of the linear system of equations, [Formula: see text] is the solution error, and [Formula: see text] is the error in postprocessing. This work suggests a path toward quantum simulation of fluid flows and highlights the special considerations needed at the gate-level implementation of QC.

Approximate moving horizon estimation for switching conservative linear infinite-dimensional systems

This work addresses moving horizon estimation for switching conservative linear infinite-dimensional systems described by partial differential equations (PDE), where the plant and measurement equations are corrupted with bounded disturbances, and the system mode is regarded as an unknown and unpredictable discrete state to be estimated. To address the issues associated with unbounded operators (induced by boundary or point observation and disturbance) and facilitate discrete-time moving horizon estimator design, the Cayley–Tustin transformation is deployed for model time-discretization without any spatial discretization or model reduction while preserving model essential properties. A series of observability concepts along with corresponding properties are proposed and analyzed for the switching linear infinite-dimensional discrete-time systems. A moving horizon estimation algorithm that accounts for state/output and mode estimation and constraint handling is proposed. Based on the proposed observability properties, we prove the stability of the proposed moving horizon estimator. The derived results are demonstrated by an undamped Schrödinger equation with switching group-velocity dispersion.

A dynamic-mode-decomposition-based acceleration method for unsteady adjoint equations at low Reynolds numbers

The computational cost of unsteady adjoint equations remains high in adjoint-based unsteady aerodynamic optimization. In this letter, the solution of unsteady adjoint equations is accelerated by dynamic mode decomposition (DMD). The pseudo-time marching of every real-time step is approximated as an infinite-dimensional linear dynamical system. Thereafter, DMD is utilized to analyze the adjoint vectors sampled from these pseudo-time marching. First-order zero frequency mode is selected to accelerate the pseudo-time marching of unsteady adjoint equations in every real-time step. Through flow past a stationary circular cylinder and an unsteady aerodynamic shape optimization example, the efficiency of solving unsteady adjoint equations is significantly improved. Results show that one hundred adjoint vectors contains enough information about the pseudo-time dynamics, and the adjoint dominant mode can be precisely predicted only by five snapshots produced from the adjoint vectors, which indicates DMD analysis for pseudo-time marching of unsteady adjoint equations is efficient.

DC operating points of nonlinear circuits and generalized Carleman linearization

PurposeThe purpose of this paper is to present a procedure for approximating DC operating points of nonlinear circuits. The presented approach can also be applied in case of multiple DC operating points.Design/methodology/approachA generalized Carleman linearization is used, which transforms an algebraic nonlinear equation into an equivalent infinite-dimensional linear system. In general, no close-form solution can be given for the infinite-dimensional linear system. Hence, the infinite-dimensional linear system is approximated by a finite one over a predefined interval using a self-consistent technique. The presented procedure allows to approximate all possible DC operating points within a predefined interval. To isolate all DC operating points, the initial interval is gradually divided into subintervals.FindingsIt is shown that the presented approach is not restricted to the polynomial case and allows to approximate all DC operating points. The presented approach can be applied in case of multiple DC operating points and does not depend on the domain of attraction of the DC operating points.Originality/valueA new procedure for the approximation of DC operating points of nonlinear circuits based on a generalized Carleman linearization is presented. This approach can be applied in case of multiple DC operating points and is independent of the domain of attraction. Further, this generalized approach is not restricted to the polynomial case and can be applied to a variety of circuits.

Saturated Output Regulation of Distributed Parameter Systems with Collocated Actuators and Sensors⋆

This paper addresses the problem of output regulation of infinite-dimensional linear systems subject to input saturation. We focus on strongly stabilizable linear dissipative systems with collocated actuators and sensors. We generalize the output regulation theory for finite-dimensional linear systems subject to input saturation to the class of considered infinite-dimensional linear systems. The theoretic results are illustrated with an example where we consider the output regulation of a flexible satellite model that is composed of two identical flexible solar panels and a center rigid body.

Approximation of infinite-dimensional observer-based state feedback for systems with boundary control and observation

Infinite-dimensional linear systems with unbounded input and output operators are considered. For the purpose of finite-dimensional observer-based state feedback, an observer approximation scheme will be developed which can be directly combined with existing late-lumping controllers and observer output injection gains. It relies on a decomposition of the feedback gain, resp. observer output injection gain, into a bounded and an unbounded part. Based on a perturbation result, the spectrum-determined growth condition is established, for the closed loop.

Koopman Representation for Boolean Networks

Boolean networks, as logical dynamical systems whose system states are Boolean variables, arise from applications in biology, computer networks, and social networks etc. The representation and control of Boolean networks have attracted a lot of attention in recent years. On a parallel line of research, Koopman developed an operator view of nonlinear dynamical systems in early 1930s, which shows that using observable functions all nonlinear dynamics can be represented as (infinite dimensional) linear systems. In this paper, we present a framework for representing Boolean networks via Koopman approach. First of all, we construct addition and scalar multiplication operations over the set of logical functions over the binary field, defining a linear Boolean function space. Then associated with any Boolean mapping, the induced Koopman operator is defined as an operator over such Boolean function space. Next, we show that if there exists a set of logical functions that are invariant in the Koopman sense, a Boolean network can be represented by a finite-dimensional linear system. This Koopman perspective for Boolean networks is also extended to controlled Boolean networks.

Robustness of polynomial stability with respect to sampling

We provide a partially affirmative answer to the following question on robustness of polynomial stability with respect to sampling: “Suppose that a continuous-time state-feedback controller achieves the polynomial stability of the infinite-dimensional linear system. We apply an idealized sampler and a zero-order hold to a feedback loop around the controller. Then, is the sampled-data system strongly stable for all sufficiently small sampling periods? Furthermore, is the polynomial decay of the continuous-time system transferred to the sampled-data system under sufficiently fast sampling?” The generator of the open-loop system is assumed to be a Riesz-spectral operator whose eigenvalues are not on the imaginary axis but may approach it asymptotically. We provide conditions for strong stability to be preserved under fast sampling. Moreover, we estimate the decay rate of the state of the sampled-data system with a smooth initial state and a sufficiently small sampling period.

Approximate and Exact Controllability of Switched Infinite-Dimensional Linear Systems

This article addresses the approximate and exact controllability issues of switched infinite-dimensional linear systems. First, a controllability map that is related to switching sequence and a controllability Gramian operator are proposed, based on which two necessary and sufficient conditions are established for approximate and exact controllability, respectively. Second, it shows that the obtained conditions can degenerate into the existing controllability criteria of infinite-dimensional linear systems, switched finite-dimensional linear systems, and a single finite-dimensional linear system. Finally, the new results are applied to analyze the approximate controllability of switched linear parabolic systems and the exact controllability of switched linear wave systems. Two examples are given to illustrate the effectiveness of the proposed approaches.

Fault tolerant time optimization for a class of time-varying switched infinite dimensional linear systems

This paper addresses the issue of finding the optimal switching times for the time-varying switched infinite dimensional linear system with discrete faults that lead to the deviation of the actual switching signal from the normal one. A robust fault tolerant time optimization scheme based on an equivalent time transformation and a min–max mechanism is proposed to obtain the upper bound of optimal values of cost functional of all possible discrete faults. For any detectable discrete fault, an active fault tolerant time optimization scheme equipped with a switching detection technique and a gradient descent algorithm is presented to minimize every post-fault cost functional, which reduces the conservation of the robust one.

Team MMSE Precoding With Applications to Cell-Free Massive MIMO

This article studies a novel distributed precoding design, coined <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">team minimum mean-square error</i> (TMMSE) precoding, which rigorously generalizes classical centralized MMSE precoding to distributed operations based on transmitter-specific channel state information (CSIT). Building on the so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">theory of teams</i> , we derive a set of necessary and sufficient conditions for optimal TMMSE precoding, in the form of an infinite dimensional linear system of equations. These optimality conditions are further specialized to cell-free massive MIMO networks, and explicitly solved for two important examples, i.e., the classical case of local CSIT and the case of unidirectional CSIT sharing along a serial fronthaul. The latter case is relevant, e.g., for the recently proposed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">radio stripe</i> concept and the related advances on sequential processing exploiting serial connections. In both cases, our optimal design outperforms the heuristic methods that are known from the previous literature. Duality arguments and numerical simulations validate the effectiveness of the proposed team theoretical approach in terms of ergodic achievable rates under a sum-power constraint.

Cost Evaluation of Approximate Controllability and Fault Recoverability for Switched Infinite-Dimensional Linear Systems

This article finds the minimum cost that is needed to achieve the approximate controllability of switched infinite-dimensional linear systems (IDLSs) by a newly defined Gramian operator. Such a new cost evaluation method helps to establish a fault recoverability condition, which reveals the ability of faulty switched IDLSs to maintain the approximate controllability under given energetic constraints. An example of the swarming system is taken to illustrate the theoretical results.

Feedback theory approach to positivity and stability of evolution equations

In this paper, we study the positivity and (uniform) exponential stability of a large class of perturbed semigroups. Our approach is essentially based on feedback theory of infinite-dimensional linear systems. The obtained results are applied to the stability of hyperbolic systems including those with a delay at the boundary conditions. The efficiency of our theoretical results is illustrated by some numerical simulations.

Learning Causal Estimates of Linear Operators from Noisy Data

This paper studies the identification problem of linear systems from a set of noisy input-output trajectories. The problem is formulated and solved as a least-square regularized estimate on a suitable function space of finite-bandwidth operators. This abstract setting is well suited to represent a broad class of finite- and infinite-dimensional linear systems. We determine the value of the regularization parameter as a function of the amount of noise on the learning trajectories and we show how to obtain recursive and causal estimates for the case of linear dynamical systems.

Stabilization of 2 × 2 linear hyperbolic systems with delayed feedback boundary

This paper is concerned with the stabilization of linear hyperbolic systems with time lags in the boundary feedback. The well-posedness of such system is established. Moreover, we derived necessary and sufficient conditions on stabilization of hyperbolic systems with delayed feedback boundary. Our approach is mainly based on the feedback theory of infinite dimensional linear systems and the theory of positive semigroups.

Koopman Operator Based Modeling for Quadrotor Control on SE(3)

In this letter, we propose a Koopman operator based approach to describe the nonlinear dynamics of a quadrotor on SE(3) in terms of an infinite-dimensional linear system which evolves in the space of observables (lifted space) and which is more appropriate for control design purposes. The major challenge when using the Koopman operator is the characterization of a set of observables that can span the lifted space. Most of the existing methods either start from a set of dictionary functions and then search for a subset that best fits the underlying nonlinear dynamics or they rely on machine learning algorithms to learn these observables. Instead of guessing or learning the observables, in this work we derive them in a systematic way for the quadrotor dynamics on SE(3). In addition, we prove that the proposed sequence of observables converges pointwise to the zero function, which allows us to select only a finite set of observables to form (an approximation of) the lifted space. Our theoretical analysis is also confirmed by numerical simulations which demonstrate that by increasing the dimension of the lifted space, the derived linear state space model can approximate the nonlinear quadrotor dynamics more accurately.

Locally positive stabilization of infinite-dimensional linear systems by state feedback

An overview is given of the locally positive stabilization problem for positive infinite-dimensional linear systems with a bounded control operator. The impossibility of solving this problem by using a nonnegative input is established. Two methods for solving the problem by means of state feedback, namely spectral decomposition and control invariance, are described. The results are illustrated by means of a perturbed diffusion equation with Dirichlet boundary conditions and a diffusion equation with Neumann boundary conditions and pointwise control.