Reduced-order System Research Articles

Order Reduction of Real Time Electromechanical Systems by Using a New Model Order Reduction Method and Controller Design

Order Reduction of Real Time Electromechanical Systems by Using a New Model Order Reduction Method and Controller Design

Finite-frequency model order reduction of linear and bilinear systems via low-rank approximation

In this paper, we first investigate the finite-frequency model order reduction for linear systems based on low-rank Gramian approximations. An efficient algorithm for computing low-rank approximations of the finite-frequency and frequency-dependent Gramians based on Laguerre functions is proposed. The approach constructs the low-rank decomposition factors of the finite-frequency Gramians or frequency-dependent Gramians through a recursive formula of Laguerre functions expansion coefficient vectors and then combines the low-rank square root method and frequency-dependent balanced truncation method to obtain the reduced-order models. In this process, it avoids dealing with the matrix-valued functions and solving the related (generalized) Lyapunov matrix equations directly, making them computationally efficient. Furthermore, the above method is successfully extended to bilinear systems, and a corresponding efficient computation method for low-rank approximations of the finite-frequency Gramians of bilinear systems is derived. Finally, some numerical simulations are provided to illustrate the effectiveness of our proposed algorithms.

Model order reduction for the input–output behavior of a geothermal energy storage

In this article, we consider a geothermal energy storage system in which the spatio-temporal temperature distribution is modeled by a heat equation with a time-dependent convection term. Such storage systems are often embedded in residential heating systems. The control and management of such systems requires knowledge of aggregated characteristics of the temperature distribution in the storage. These describe the input–output behavior of the storage, the associated energy flows, and their response to charging and discharging processes. Our aim is to derive an efficient, approximate description of these characteristics by using low-dimensional systems of ordinary differential equations (ODEs). This leads to a model order reduction problem for a large-scale linear system of ODEs resulting from the semidiscretization of the heat equation combined with a linear algebraic output equation. In a first step, we approximated the nonautonomous system of ODEs by a linear time-invariant system. Then, we applied Lyapunov balanced truncation model order reduction to approximate the output by a reduced-order system that has only a few state equations but almost the same input–output behavior. The results of our extensive numerical experiments show the efficiency of the applied model order reduction method. We found that only a few suitably chosen ODEs are sufficient to achieve good approximations of the input–output behavior of the storage.

Improved a posteriori error bounds for reduced port-Hamiltonian systems

Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical error bound and (b) an error bound based on an auxiliary linear problem, to the case of port-Hamiltonian systems. The approaches rely on a secondary approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are adapted to port-Hamiltonian systems. The mathematical relationship between the two methods is discussed both theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid–structure interaction.

A statistical high-order reduced model for nonlinear random heterogeneous materials with three-scale micro-configurations

An effective statistical higher-order three-scale reduced homogenization (SHTRH) method is established to analyze the nonlinear random heterogeneous materials with multiple micro-configurations. Firstly, the various unit cell functions based on the microscale and mescoscale regions are given, and two expected homogenization coefficients are computed through Kolmogorov's strong laws of large number. Further, the nonlinear homogenized equations are formulated, and the corresponding reduced-order multiscale systems for displacement and stress solutions are derived by using the high-order unit cell solutions and homogenized solutions. The key features of the new statistical multiscale methods are (i) the novel reduced models established to solve the inelastic problems of random composites at a fraction of cost, (ii) the high-order homogenized solutions which do not need high-order continuity for the macro solutions of the random problems and (iii) the statistical high-order multiscale algorithms developed for analyzing the nonlinear random composites with three-scale structures. Finally, the effectiveness and correctness of the algorithm are confirmed according to several hyperelastic, plasticity and damage periodic/random composites with multiple-scale configurations. The computation shows that the proposed SHTRH methods are useful for analyzing the macroscopic nonlinear performance, and can efficiently catch the microscopic and mesoscopic information for the random heterogeneous composites.

Moth Flame Optimization for Model Order Reduction of Complex High Order Linear Time-Invariant Systems

Moth Flame Optimization for Model Order Reduction of Complex High Order Linear Time-Invariant Systems

Structure-preserving model reduction based on multiorder Arnoldi for discrete-time second-order systems

This paper investigates the new structure-preserving model reduction methods based on multiorder Arnoldi for discrete-time second-order systems by using discrete Laguerre polynomials. The expansion coefficients of discrete-time second-order system and its equivalent first-order system satisfy the implicit recursive expressions in space spanned by discrete Laguerre polynomials, and then multiorder Arnoldi algorithm is applied to the implicit recursive expressions to produce the column-orthogonal matrices, where the orthogonal projection matrix for the equivalent first-order system is constructed by partitioning the resulting column-orthogonal matrix appropriately. The reduced-order systems can preserve the special structures of the original systems as well as match a desired number of expansion coefficients of the original outputs. The error estimate of output variables is also given for discrete-time second-order system. Two benchmark examples are simulated to illustrate the effectiveness of the proposed methods.

Order Reduction of z-Domain Interval Systems by Advanced Routh Approximation Method

Order Reduction of z-Domain Interval Systems by Advanced Routh Approximation Method

Balanced truncation for quadratic-bilinear control systems

We discuss model order reduction (MOR) for large-scale quadratic-bilinear (QB) systems based on balanced truncation. The method for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (Systems Control Lett. 21, 143-153 1993). These formulations of Gramians are not only challenging to compute for large-scale systems but hard to utilize also in the MOR framework. This work proposes algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems. We then show their relation to a certain type of generalized quadratic Lyapunov equation. Furthermore, we quantify the reachability and observability subspaces based on the proposed Gramians. Consequently, we propose a balancing algorithm, allowing us to find those states that are simultaneously hard to reach and hard to observe. Truncating such states yields reduced-order systems. We also study sufficient conditions for the existence of Gramians, and a local stability of reduced-order models obtained using the proposed balanced truncation scheme. Finally, we demonstrate the proposed balancing-type MOR for QB systems using various numerical examples.

Krylov subspace model order reduction of linear dynamical systems with quadratic output

This paper presents Krylov subspace model order reduction of linear dynamical systems with quadratic output. To reach this goal, the quadratic transfer function with two variables is taken into consideration. The Krylov subspaces are established at the finite and the infinite frequency points where both the one-sided and the two-sided projection cases are discussed. The moment matching is accordingly studied and the quadratic transfer function of the reduced system resulting from the two-sided projection case can match more moments. Finally, numerical results illustrate the performance of the proposed Krylov model order reduction methodology.

Distributed formation containment control for multi-agent systems via dynamic event-triggering communication mechanism

This paper focuses on resolving the distributed formation containment control issue in general multi-agent systems (MASs) possessing multiple leader agents subject to limited communication resources. The unnecessary data transmissions among agents in MASs are reduced by utilizing a unique dynamic event-triggering communication mechanism (DETCM), which can build communication time sequences for each agent automatically. Based on the sampled states of agents at event-triggering instants, a novel distributed formation containment control protocol is proposed. By mathematical analysis, the formation containment control problem is turned into the problem of stability in a reduced-order system. The global formation containment of the MASs is finally achieved through the application of Lyapunov-Krasovskii functionals and general inequalities. Moreover, a valid example is also given to support the validity and effectiveness of the findings.

Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.

Gaussian Process-Based Learning Control of Underactuated Balance Robots with an External and Internal Convertible Modeling Structure

Abstract External and internal convertible (EIC) form-based motion control is one of the effective designs of simultaneous trajectory tracking and balance for underactuated balance robots. Under certain conditions, the EIC-based control design is shown to lead to uncontrolled robot motion. To overcome this issue, we present a Gaussian process (GP)-based data-driven learning control for underactuated balance robots with the EIC modeling structure. Two GP-based learning controllers are presented by using the EIC structure property. The partial EIC (PEIC)-based control design partitions the robotic dynamics into a fully actuated subsystem and a reduced-order underactuated system. The null-space EIC (NEIC)-based control compensates for the uncontrolled motion in a subspace, while the other closed-loop dynamics are not affected. Under the PEIC- and NEIC-based, the tracking and balance tasks are guaranteed and convergence rate and bounded errors are achieved without causing any uncontrolled motion by the original EIC-based control. We validate the results and demonstrate the GP-based learning control design using two inverted pendulum platforms.

Mixed Riccati–Lyapunov Balanced Truncation for Order Reduction of Electrical Circuit Systems

Mixed Riccati–Lyapunov Balanced Truncation for Order Reduction of Electrical Circuit Systems

Tensorial Parametric Model Order Reduction of Nonlinear Dynamical Systems

Tensorial Parametric Model Order Reduction of Nonlinear Dynamical Systems

Period-doubling cascade route to chaos in an initially curved microbeam resonator exposed to fringing-field electrostatic actuation

This paper explores the chaotic dynamics of a piezoelectrically laminated initially curved microbeam resonator subjected to fringing-field electrostatic actuation, for the first time. The resonator is fully clamped at both ends and is coated with two piezoelectric layers, encompassing both the top and bottom surfaces. The nonlinear motion equation which is obtained by considering the nonlinear fringing-field electrostatic force, includes geometric nonlinearities due to the mid-plane stretching and initial curvature. The motion equation is discretized using Galerkin method and the reduced order system is numerically integrated over the time for the time response. The variation of the first three natural frequencies with respect to the applied electrostatic voltage is determined and the frequency response curve is obtained using the combination of shooting and continuation methods. The bifurcation points have been examined and their types have been clarified based on the loci of the Floquet exponents on the complex plane. The period-doubled branches of the frequency response curves originating from the period doubling (PD) bifurcation points are stablished. It's demonstrated that the succession PD cascades leads to chaotic behavior. The chaotic behavior is identified qualitatively by constructing the corresponding Poincaré section and analyzing the response's associated frequency components. The bifurcation diagram is obtained for a wide range of excitation frequency and thus the exact range in which chaotic behavior occurs for the system is determined. The chaotic response of the system is regularized and controlled by applying an appropriate piezoelectric voltage which shifts the frequency response curve along the frequency axis.

Current Sensorless Pole-Zero Cancellation Output Voltage Control for Uninterruptible Power Supply Systems with a Three-Phase Inverter

This article presents a proportional–derivative (PD) type output voltage regulator without the current feedback, taking into account system parameter and load variations. The main advantages are given as follows: First, the first-order output voltage derivative observer is developed without the requirement of system parameter information, which makes it possible to stabilize the system without current sensing. Second, a simple self-tuner implements the feedback-loop adaptation by updating the desired dynamics accordingly. Third, the observer-based active damping injection for the PD-type controller results in the closed-loop system order reduction to 1 by the pole-zero cancellation, including the disturbance observer as a feed-forward term. The prototype uninterruptible power supply system comprised of a 3 kW three-phase inverter, inductors, and capacitors verifies the practical merits of the proposed technique for linear and nonlinear loads.

Cascade NMPC-PID control strategy of active heave compensation system for ship-mounted offshore crane

In this paper, the problem associated with active heave motion compensation during heavy-lift operations performed by ship-mounted offshore cranes beneath rough marine environment is investigated. To effectively decouple the correlated motion of the suspended payload from the support vessel via the secondary regulated active heave compensation (SRAHC) system, a novel hierarchical control strategy is proposed by integrating a cascade control structure with an adaptive robust control scheme that incorporates a vertical motion forecast algorithm based on autoregressive integrated moving average (ARIMA) model, a conventional proportional-integral-derivative (PID) framework, and a nonlinear model predictive control (NMPC) method. The introduction of the cascade structure is motivated by its prompt error response against control lag and efficient system order reduction for computational burden alleviation, which forms a crucial foundation for the NMPC-based real-time regulation of PID gains, ensuring optimal evolution of the SRAHC system while improving its practical feasibility. Furthermore, the enhancements in heave compensation and trajectory tracking performance, noise resistance, constraint satisfaction, and engineering application potential of the proposal are demonstrated through a thorough comparative analysis conducted in a practical co-simulation research system.

Identifying dominant flow structures in a bubbling gas-particle fluidized bed using the spectral proper orthogonal decomposition

This study applies the spectral proper orthogonal decomposition (SPOD) to analyze the spatio-temporal characteristics associated with the flow fields of a bubbling fluidized bed. The results suggest there is no singular dominant frequency linked to the highest energy levels; rather, there is a spectrum of frequencies. These frequencies are consistent with the natural frequency from published correlations. SPOD analysis allows for the capture of quasi-periodicity associated with the flow that arises from the semi-periodic evolution of bubbles in the bed and reveals the presence of multiple co-existing spatio-temporal patterns in the particle volume fraction and velocity fields. The dominant SPOD modes can be used for prediction, control, design, optimization, and reduced-order system representation. The capability of a few dominant modes to yield a low-rank reconstruction of the flow fields is demonstrated. The intermittent behavior of the bed is examined through frequency-time analysis of the gas pressure field.

Model order reduction, a novel method using krylov sub-spaces and genetic algorithm

Model Order Reduction (MOR) of complex and large systems in Electrical engineering, continuous to be an attractive field for Engineers and Scientists over the last few decades, this complexity of models makes the control designs and simulation using Computer Aided Design (CAD) more and more difficult and consuming a lot of time. There for, accurate, robust and fast algorithms for simulation are needed. The goal of MOR is to replace the original system by an appropriate reduced system which preserves the main properties of the original one such that stability and passivity. Several analytical MOR techniques have been proposed in the literature over the past few decades, to approximate high order linear dynamic systems like Krylov sub-space techniques and SVD (Singular Value Decomposition) techniques. However, most of these techniques lead to computationally demanding, time consuming, iterative procedures that usually result in non-robustly stable models with poor frequency response resemblance to the original high order model in some frequency ranges. Recently a set of new techniques based on Artificial Intelligence (AI) were proposed in [1] for MOR. This article considers the problem of model order reduction of Linear Time In varying (LTI) systems. It is described by first and second order ordinary differential equations model. A tow steps method for model order reduction of LTI systems is proposed here, which combined features of an analytic technique (Krylov approach) and an AI technique (Genetic Algorithm). In the first step, the size of the original model is reduced to an intermediate order, using an analytical technique based on Krylov sub-spaces. In the final step of the reduction process, an AI approach based on Genetic Algorithm (GA) is applied to obtain an optimized nominal model.