Balanced Model Reduction Research Articles

Confined Frequency-Interval Gramian Framework-Based Balanced Model Reduction

Some of the existent frequency-limited model reduction techniques result in unstable reduced-order models (ROMs). Further, the ROMs of a few methods differ significantly from the actual model, resulting in a significant approximation error. Therefore, this work presents an innovative structure of confined frequency-intervals Gramians for continuous-time systems. The developed ROM ensures stability and minimal inaccuracy in approximation error over the desired frequency intervals. Also, the developed technique provides an easy-to-calculate error-bound. Compared to conventional approaches, the suggested approach offers consistent results, illustrating its usefulness. Further, the numerical cases support the effectiveness of the proposed scheme.

Nearly Linear-Phase 2-D Recursive Digital Filters Design Using Balanced Realization Model Reduction

This paper presents a new method for the design of separable-denominator 2-D IIR filters with nearly linear phase in the passband. The design method is based on a balanced realization model reduction technique. The nearly linear-phase 2-D IIR filter is designed using 2-D model reduction from a linear-phase 2-D FIR filter, which serves as the initial filter. The structured controllability and observability Gramians Ps and Qs serve as the foundation for this technique. onal positive-definite matrices that satisfy 2-D Lyapunov equations. An efficient method is used to compute these Gramians by minimizing the traces of Ps and Qs under linear matrix inequality (LMI) constraints. The use of these Gramians ensures that the resulting 2-D IIR filter preserves stability and can be implemented using a separable-denominator 2-D filter with fewer coefficients than the original 2-D FIR filter. Numerical examples show that the proposed method compares favorably with existing techniques.

Tracking the trajectory of an object in a noisy environment with unknown statistics: A novel robust Kalman filter residue-based approach

A novel robust Kalman filter (KF)-based controller design approach is proposed to accurately track a specified trajectory under unknown stochastic disturbance, known deterministic disturbance, and measurement noise. The system is nonlinear and is approximated by a piecewise-linear dynamic model. The Box–Jenkins model is an augmented model of the signal and the disturbance, is non-controllable and observable, while the signal model is controllable and observable. An emulator-based two-stage identification is employed to obtain an accurate system model needed to design the robust controller. The system and its KF are identified, and the signal and output errors are estimated. From the identified models, the signal, its KF, the disturbance model, and the whitening filter are all obtained using balanced model reduction techniques. It is shown that the signal model is a transfer matrix description relating the system output and KF residual, and the residual is the whitened output error. The disturbance model is identified by inverse filtering. A new combined feedforward–feedback controller is designed and implemented using an internal model of the reference driven by both the error between the reference and the signal estimate, and by the feedforwarded reference signal. The proposed scheme was successfully evaluated on a simulated autonomously guided drone.

Balanced truncation model reduction with a priori error bounds for LTI systems with nonzero initial value

In standard balanced truncation model order reduction, the initial condition is typically ignored in the reduction procedure and is assumed to be zero instead. However, such a reduced-order model may be a bad approximation to the full-order system, if the initial condition is not zero. In the literature there are several attempts for modified reduction methods at the price of having no error bound or only a posteriori error bounds which are often too expensive to evaluate. In this work we propose a new balancing procedure that is based on a shift transformation on the state. We first derive a joint projection reduced-order model in which the part of the system depending only on the input and the one depending only on the initial value are reduced at once and we prove an a priori error bound. With this result at hand, we derive a separate projection procedure in which the two parts are reduced separately. This gives the freedom to choose different reduction orders for the different subsystems. Moreover, we discuss how the reduced-order models can be constructed in practice. Since the error bounds are parameter-dependent we show how they can be optimized efficiently. We conclude this paper by comparing our results with the ones from the literature by a series of numerical experiments.

Optimal Sensor and Actuator Selection Using Balanced Model Reduction

Optimal sensor and actuator selection is a central challenge in high-dimensional estimation and control. Nearly all subsequent control decisions are affected by these sensor and actuator locations. In this article, we exploit balanced model reduction and greedy optimization to efficiently determine sensor and actuator selections that optimize observability and controllability. In particular, we determine locations that optimize scalar measures of observability and controllability using greedy matrix QR pivoting on the dominant modes of the direct and adjoint balancing transformations. Pivoting runtime scales linearly with the state dimension, making this method tractable for high-dimensional systems. The results are demonstrated on the linearized Ginzburg–Landau system, for which our algorithm approximates known optimal placements computed using costly gradient descent methods.

Control for stability improvement of high-speed train bogie with a balanced truncation reduced order model

The high-speed train bogie is a typical underactuated mechanical system. It has been difficult to design controls to increase the stability against hunting motion with limited sensor measurements and actuation hardware. This paper develops a control design for the stability improvement of the bogie by making use of the balanced truncation reduced order model of the bogie. When the train travels at speed higher than the critical speed, the bogie has unstable modes. The balanced truncation model reduction (BTMR) is usually applied to stable systems. In this paper, we first apply the real Schur decomposition to separate the stable and unstable modes of the bogie. The BTMR is then carried out for the stable subsystem. An augmented -balanced truncation algorithm is considered to deal with the dependence of the BTMR on initial conditions. The reduced order subsystem is then combined with the unstable subsystem. A linear quadratic regulator control together with a Luenberger observer for the state estimation is designed for the bogie with non-zero initial conditions. Simulation results show that the control designed with the balanced truncation reduced order model performs very well in terms of the stability improvement by significantly increasing the critical train speed.

Error bounds for port-Hamiltonian model and controller reduction based on system balancing

We study linear quadratic Gaussian (LQG) control design for linear port-Hamiltonian systems. To this end, we exploit the freedom in choosing the weighting matrices and propose a specific choice which leads to an LQG controller which is port-Hamiltonian and, thus, in particular stable and passive. Furthermore, we construct a reduced-order controller via balancing and subsequent truncation. This approach is closely related to classical LQG balanced truncation and shares a similar a priori error bound with respect to the gap metric. By exploiting the non-uniqueness of the Hamiltonian, we are able to determine an optimal pH representation of the full-order system in the sense that the error bound is minimized. In addition, we discuss consequences for pH-preserving balanced truncation model reduction which results in two different classical H∞-error bounds. Finally, we illustrate the theoretical findings by means of two numerical examples.

A dual reduction strategy for reduce-order modeling of periodic control system

Model order reduction (MOR) of periodic systems using the Krylov subspace methods received lots of interest in last few decades. In this paper, a structured Krylov subspace based model reduction for linear discrete-time periodic (LDTP) control system has been proposed using the corresponding lifted form. The proposed model reduction strategy first finds a reduced order Lyapunov equation via a projection where the projection subspace is computed by using the periodic Arnoldi method. Then, by applying the balanced truncation model reduction approach, it forms a reduced order model of the original system. In both cases, the periodic structure is ensured in the solutions and systems’ dynamics. This paper also presents numerical results to confirm the efficiency and accuracy of the proposed algorithm.

Robust Output Feedback Control Design for Inertia Emulation by Wind Turbine Generators

Wind generation has gained widespread use as a renewable energy source. Most wind turbines and other renewables connected to the grid through converters result in a reduction in the natural inertial response to grid frequency changes. The doubly-fed induction generator (DFIG) can be controlled to compensate for this reduction and, in fact, provide faster response than traditional synchronous machines. This paper proposes to design observer based output feedback linear quadratic regulator (LQR) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> control laws to realize the inertia emulation function and deliver fast frequency support. The aim is to track the reference speed by a diesel synchronous generator (DSG) in order to reach the desired inertia. The control signal is computed based on a reduced order model using the balanced truncation technique. A comparison with selective modal analysis (SMA) and balanced truncation model reduction techniques is presented. Comprehensive results show the effective emulation of synthetic inertia by implementing the control laws on a nonlinear three-phase diesel-wind system. The proposed technique is analyzed for different short circuit ratio (SCR) scenarios.

Balanced Truncation Model Reduction for Symmetric Second Order Systems---A Passivity-Based Approach

We introduce a model reduction approach for linear time-invariant second order systems based on positive real balanced truncation. Our method guarantees asymptotic stability and passivity of the reduced order model as well as the positive definiteness of the mass and stiffness matrices. Moreover, we receive an a priori gap metric error bound. Finally, we show that our method based on positive real balanced truncation preserves the structure of overdamped second order systems.

Complex Frequency Domain Model Reduction: A Comparison of Techniques

A mathematical model of the plant to be controlled must first be identified in order to design a controller to improve the process’ performance and stability. This mathematical model usually is in the form of complex frequency domain state-space realization or continuous/ discrete time domain regression algorithm. This paper works on the complex frequency domain state-space realization. Reducing the order of the state-space model of the physical plant is usually desirable since it reduces computational time and hardware implementation for the resulting controller designed based on the model. For a frequency domain state space realization of a linear time-invariant and non-causal plant, there are numerous methods available to reduce its order. These include Balancing Model Reduction, Frequency-Weighted Balancing, Low-Frequency Approximation and Balanced Truncation with Reduced Error Bound. Besides background information and a summary, the model-reduction methodology for each of the above-said techniques is also presented. Using a two-input two-output six-state model for a large turbo-generator and a large industrial boiler with 8 states, 2 inputs and 2 outputs model, this paper provides a comparison between three of the four methodologies mentioned above in terms of the spectral norms of the model reduction error.

Decentralized vibration control of smart constrained layer damping plate

The traditional centralized control strategy to vibration suppression of large-scale thin plate structures may increase the design difficulty of the controller. In this article, a decentralized vibration active control method is proposed to suppress the vibration of the thin plate structure with smart constrained layer damping treatment. First, the dynamics model of the smart constrained layer damping plate is established based on the finite element method, and the characteristics of viscoelastic materials with temperature and frequency are described by Golla-Hughes-McTavish damping model. Subsequently, a decentralized subsystem control model is obtained from the balanced model reduction method and complex mode truncation method. The modal test proves that the theoretical model is accurate. Then, the particular emphasis is placed on the stability and vibration attenuation of a decentralized system, which is composed of multiple subsystems. The local state feedback stabilization, using interaction of local state feedback and output feedback, is introduced to achieve system stability. To solve the practical problem of local state feedback, a decentralized controller with an observer is developed by adopting the pole placement method. Finally, the numerical simulation and hardware-in-the-loop experiment under different excitation are performed to investigate the effectiveness of decentralized vibration active control. The results demonstrate that the decentralized controller can effectively suppress the vibration, especially under mixed periodic signal and Gauss white noise signal.

Modeling and Control of Inductive Power Transfer System Supplied by Multiphase Phase-Controlled Inverter

A multiphase inverter with phase-shifted control is proposed for the inductive power transfer (IPT) of electric vehicles where the charging voltage can be regulated by adjusting the angle among the inverter phases. An equivalent circuit model is developed and linearized to analyze the system dynamic characteristics at its operating point. In terms of the high-order model, a balanced model reduction method is used to remove the fast modes that are outside the desired system bandwidth and insignificant to controller design. In addition, the model is discretized and the communication delay of the feedback loop is considered in the discretized model. Based on this model, the parameters of a PI controller are designed to achieve the expected performance indexes. A 6.5 kW S-S compensation IPT system prototype with constant voltage control is built and tested to verify the control performance. The experimental results show that the charging voltage can be maintained constant within 6 ms and has no overshoot under disturbances, which verifies that the closed-loop system with PI controller operates properly and efficiently for IPT. Besides, the measured transmission efficiency of the system at a receiving power of 6.5 kW is 95.1%.

Balanced Model Reduction for Linear Time-Varying Symmetric Systems

The goal of this paper is to develop balancing theory for the linear time-varying (LTV) symmetric systems. To this end, first, we extend the concept of symmetry in terms of the dual system. Then, we define the cross Gramian for the LTV systems. For LTV symmetric systems, we establish a connection among the controllability, observability, and cross Gramians. In particular, if one of these three Gramians is obtained, the other two Gramians can be constructed. Based on this fact, we show that the symmetry structure is preserved under balanced truncation if the Hankel singular values are pointwise distinct.

Balanced model reduction of linear systems with nonzero initial conditions: Singular perturbation approximation

In this article we study balanced model reduction of linear control systems using the singular perturbation approximation. Balanced model reduction techniques have been successfully applied to systems with homogeneous initial conditions, with one of their most important features being a priori L2 and H∞ bounds for the approximation error. The main focus of this work is to derive an L2 error bound for the singular perturbation approximation for system with inhomogeneous initial conditions, extending related work on balanced truncation. This L2 error bound measures the difference between the input-output maps of the original and of the reduced initial value systems. The advantages and flexibility of this approach are demonstrated with a variety of numerical examples.

H∞ Suboptimal controller design for adaptive optic systems

In this paper, an alternative approach to conventional [Formula: see text] control is proposed for adaptive optic (AO) systems. In order to account for the dynamics of the deformable mirror (DM), the Kirchhoff plate equation is considered. Phase wavefront aberrations, which are the effect of atmospheric turbulence, are modelled using the orthogonal set of Zernike polynomials. The first part of this study concerns the derivation of mathematical models for the DM and the atmospheric turbulence. The AO systems used for disturbance rejection concentrate on a specific frequency band, as the disturbance occurs in that region. However, the conventional weighted [Formula: see text] controller is not applicable due to its order. The proposed controller handles this problem by using frequency weighted balanced model reduction. The efficiency of the proposed approach is examined on Zernike’s tip and tilt, focus, astigmatism, coma, trefoil, spherical, secondary astigmatism and quadrafoil modes.

Error Bound for Non-Zero Initial Condition Using the Singular Perturbation Approximation Method

In this article, we present up to date results on the balanced model reduction techniques for linear control systems, in particular the singular perturbation approximation. One of the most important features of this method is it allows for an a priori L 2 and H ∞ bounds for the approximation error. This method has been successfully applied for systems with homogeneous initial conditions, however, the main focus in this work is to derive an L 2 error bound for singular perturbation approximation for system with inhomogeneous initial conditions, extending the work by Antoulas et al. The theoretical results are validated numerically.

Optimal control of linear systems with balanced reduced-order models: Perturbation approximations

In this article we study balanced model reduction of linear systems for feedback control problems. Specifically, we focus on linear quadratic regulators with collocated inputs and outputs, and we consider perturbative approximations of the dynamics in the case that the Hankel singular values corresponding to the hardly controllable and observable states go to zero. To this end, we consider different perturbative scenarios that depend on how the negligible states scale with the small Hankel singular values, and derive the corresponding limit systems as well as approximate expressions for the optimal feedback controls. Our approach that is based on a formal asymptotic expansion of an algebraic Riccati equations associated with the Pontryagin maximum principle and that is validated numerically shows that model reduction based on open-loop balancing can also give good closed-loop performance.

Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping

<p style='text-indent:20px;'>We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

Robust Kalman filter‐based least squares identification of a multivariable system

A novel direct identification using the residual model of Kalman filter (KF) is proposed for multiple-input and multiple-output Box–Jenkins model of the system formed of the signal and disturbance models using the residual model relating the input and output of the system without any a priori knowledge of the statistics of the disturbance and measurement noise corrupting the output. To avoid a non-linear optimisation, the auto-regressive and moving average (MA) residual model is approximated by a high-order MA model, so that the unknown parameters of the KF enter the residual model linearly. A key property of the KF is established, namely that the transfer matrix of the signal model is the matrix fraction description (MFD) model relating the residual and the system input and output. A two-stage identification method is developed here. In stage 1, a high-order KF of the system is identified using the robust, and computationally efficient least-squares method to capture completely both the signal and disturbance models. In stage 2, the KF for the signal is derived using the balanced model reduction technique. The signal model is derived using the key MFD property. The performance of the proposed scheme is successfully evaluated on both simulated and physical systems.