Balanced Truncation Algorithm Research Articles

Application of Zhou’s Balanced Truncation Algorithm for Controlling the Balance of Bicyrobot

Application of Zhou’s Balanced Truncation Algorithm for Controlling the Balance of Bicyrobot

Order Reduction of LTI Systems Using Balanced Truncation and Particle Swarm Optimization Algorithm

Order Reduction of LTI Systems Using Balanced Truncation and Particle Swarm Optimization Algorithm

Control Design for the Ward–Leonard System in Wind Turbines

A robust optimal controller for the Ward-Leonard system in a wind turbine is used to meet the performance and stability requirements when the system parameters change. However, designing according to the robust optimal method often leads to a high-order controller. This study investigated the application of order reduction algorithms to simplify the controller and help it better meet the real control problem. Comparative evaluation of the order reduction controller methods showed that the second-order reduction controller according to Moore's balanced truncation algorithm was the most suitable to replace the higher-order controller. The step response quality of the system was better when using a second-order reduction controller than a higher-order controller.

The Application of LQG Balanced Truncation Algorithm to the Digital Filter Design Problem

This paper presents a method for using a model reduction algorithm to design low-order digital filters. Designing an IIR digital filter that meets the specifications often leads to a high-order digital filter. To reduce the computation time and increase the response rate of the filter, we need to reduce the order of the high-order digital filter. Applying the LQG balanced truncation algorithm to reduce the demand for high-order digital filters shows that low-order filters can completely replace high-order digital filters. The simulation results show that the use of the LQG balanced truncation algorithm in order to reduce the filter order is correct and efficient.

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.

Decentralized Robust Frequency Regulation of Multi-terminal HVDC-linked Grids

This paper proposes a new strategy for optimal grid frequency regulation (FR) in an interconnected power system where regional ac grids and an offshore wind farm are linked via a multi-terminal high voltage direct-current (MTDC) network. In the proposed strategy, decentralized <i>H</i>_&#x221E; controllers are developed to coordinate the operations of synchronous generators and MTDC converters, thus achieving optimal power sharing of interconnected ac grids and minimizing frequency deviations in each grid. To develop the controllers, robust optimization problems are formulated and solved using a dynamic model of the hybrid MTDC-linked grids with model parameter uncertainty and decentralized control inputs and outputs. The model orders of the controllers are then reduced using a balanced truncation algorithm to eliminate unobservable and uncontrollable state variables while preserving their dominant response characteris- tics. Sensitivity and eigenvalue analyses are conducted focusing on the effects of grid measurements, parameter uncertainty levels, and communication time delays. Comparative case studies are also carried out to verify that the proposed strategy improves the effectiveness, stability, and robustness of real-time FR in MTDC-linked grids under various conditions.

Model reduction of unstable systems based on balanced truncation algorithm

Model reduction of a system is an approximation of a higher-order system to a lower-order system while the dynamic behavior of the system is almost unchanged. In this paper, we will discuss model order reduction (MOR) strategies for unstable systems, in which the method based on the balanced truncation algorithm will be focused on. Since each MOR algorithm has its strengths and weakness, practical applications should be suitable for each specific requirement. Simulation results will demonstrate the correctness of the algorithms.

Application of model reduction for robust control of self-balancing two-wheeled bicycle

In recent years, balance control of two-wheeled bicycle has received more attention of scientists. One difficulty of this problem is the control object is unstable and constantly impacted by noise. To solve this problem, the authors often use robust control algorithms. However, robust controller of self-balancing two-wheeled bicycle are often complex and higher order so affect to quality during real controlling. The article introduces the stochastic balanced truncation algorithm based on Schur analysis and applies this algorithm to reduce order higher order robust controller in control balancing two-wheeled bicycle problem. The simulation results show that the reduced 4th and 5th order controller arcoording to the stochastic balanced truncation algorithm based on Schur analysis can control the two-wheeled bicycle model. The reduced 3rd order controller cannot control the balance of the two-wheeled bicycle model. The reduced 4th and 5th order controller can replace the original controller while the performance of the control system is ensured. Using reduced 5th, 4th order controller will make the program code simpler, reducing the calculation time of the self-balancing two-wheel control system. The simulation results show the correctness of the model reduction algorithm and the robust control algorithm of two-wheeled self-balancing two-wheeled bicycle.

Efficient IC hotspot thermal analysis via low-rank Model Order Reduction

Efficient full-chip thermal simulation is among the most challenging problems facing the EDA industry today, due to the need for solution of very large systems of equations that require unreasonably long computational times. However, in most cases, temperature is not required to be computed at every point of the IC but only at certain hotspots, in order to assess the circuit's compliance with thermal specifications. This makes the thermal analysis problem amenable to Model Order Reduction techniques. System-theoretic techniques like Balanced Truncation offer very reliable bounds for the approximation error, which can be used to control the order and accuracy of the reduced models during creation, at the expense of greater computational complexity to create them. In this paper, we propose a computationally efficient low-rank Balanced Truncation algorithm based on extended Krylov subspace method, which retains all the system-theoretic advantages in the reduction of model order for fast hotspot thermal simulation. Experimental results demonstrate around 97% order reduction and very tight accuracy bounds.

Mxpfit: A library for finding optimal multi-exponential approximations

mxpfit is a library implemented in C++ to find optimal approximations of functions by multi-exponential sums with complex-valued parameters. The library provides an interface for evaluating exponents and coefficients from sampling data on a uniform grid using the fast Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm originally proposed by Potts and Tasche (2015). The parameters can be estimated efficiently from a sampling data even including noise. A modified balanced truncation algorithm to find the multi-exponential sum with a smaller order is also provided. These features are useful for finding optimal exponential sum approximations of analytic functions or large-scale numerically sampled data set. Program summaryProgram Title: mxpfitProgram Files doi:http://dx.doi.org/10.17632/vjygkwpss4.1Licensing provisions: MITProgramming language: C++Nature of problem: Nonlinear curve fitting by multi-exponential function, frequency estimation of signals, and reduction of multi-exponential sumSolution method: The modified fast Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm originally proposed in [1] is utilized for the parameter estimation of multi-exponential sums from sampled data on a uniform grid. In this method, partial Lanczos bidiagonalization is used to find a low-rank approximation of a Hankel matrix constructed from sampled data. An overdetermined Vandermonde system is solved by the conjugate gradient least square (CGLS) method with preconditioner to obtain the coefficients of the multi-exponential sum. The truncation of the multi-exponential sum is performed by a modified balanced truncation method in [2]. The projection matrix to obtain the truncated system is determined by computing con-eigenvalues of a Gramian matrix, which can be computed efficiently with highly relative accuracy using the algorithm in [3].Additional comments: A C++ compiler supporting the C++11 standard is required to compile programs using the mxpfit library. mxpfit depends on the Eigen (http://eigen.tuxfamily.org/) and FFTW3 (http://www.fftw.org/) libraries. CMake (https://cmake.org/) is used for building example programs.

The extended balanced truncation algorithm

The extended balanced truncation algorithm

Fixed-order controller for reduced-order model for damping of power oscillation in wide area network

The interconnected power systems are complex and stabilizing control design still remains challenging task. The use of wide area monitoring system (WAMS) offers an integrated measurement-based and model-based control, which suits to the operation of large electric power system (EPS), along with online analysis. This paper presents a study on fixed-order controller design for equivalent network of coherent generator in order to stabilize inter-area electromechanical oscillations in the system. Firstly, the coherent generators in each area of large EPS are determined by mutual information theory, which represents the dynamic equivalence. Then network of each area with input–output variables of the selected generator that participates dominantly is reduced to lower size by square-root variant of balanced truncation algorithm. The dynamics and important oscillation modes are verified in equivalent representation of each area. Finally a local controller (decentralized) in each coherent area and a centralized controller between two coherent areas for selected generator are designed by reducing the H∞ norm of its closed loop transfer function as much as possible. These controllers feed supplementary control signal in addition to one fed by local conventionally tuned PSS. The decentralized controller for selected generator is fed by local bus power or generator’s speed signal. On other hand, the centralized controller uses difference of power flow/speed of generators as input signal to dampen the oscillations between equivalent networks of two areas. The simulation results reveal effective damping of power/speed oscillations achieved by designed controller with respect to conventional PSS implemented. The robustness of controller is verified for heavy and light load operating conditions.

Balanced Truncation Approach to Power System Model Order Reduction

This article demonstrates the application of balanced truncation based model order reduction to the task of dynamic reduction of power systems. The entire power system is separated into an external area and a study area; dynamic reduction of the external area is conducted. The benefit of applying the balanced truncation technique is that key input–output relationships between these areas are retained during the reduction process. For perturbations originating in the study area, patterns in the dynamic behavior of the external area, such as generator coherency, are well captured by the reduced equivalent. The efficiency of the balanced truncation algorithm is also explored in the context of changing system conditions. Illustrative applications using a test system representative of the northern grid of India have led to some key findings on the coherency of the generators in this grid.

Balanced Truncation for Discrete Time Markov Jump Linear Systems

This technical note investigates the model reduction problem for mean square stable discrete time Markov jump linear systems. For this class of systems a balanced truncation algorithm is developed. The reduced order model is suboptimal, however the approximation error, which is captured by means of the stochastic gain, is bounded from above by twice the sum of singular numbers associated to the truncated states of each mode. Such a result allows rigorous simplification of the dynamics of each mode in an independent manner with respect to a metric which is relevant from a robust control point of view.

Balanced Truncation for a Class of Stochastic Jump Linear Systems and Model Reduction for Hidden Markov Models

This paper develops a generalization of the balanced truncation algorithm applicable to a class of discrete-time stochastic jump linear systems. The approximation error, which is captured by means of the stochastic L 2 gain, is bounded from above by twice the sum of singular numbers associated to the truncated states, similar to the case of linear time-invariant systems. A two step model reduction algorithm for hidden Markov models is also developed. The first step relies on the aforementioned balanced truncation algorithm due to a topological equivalence established between hidden Markov models and a subclass of stochastic jump linear systems. In a second step the positivity constraints, which reflect the hidden Markov model structure, are enforced by solving a low dimensional optimization problem.

Multiplicative approximation of transfer functions with frequency weighting

An algorithm for transfer function order reduction is presented, which generalizes the balanced stochastic truncation algorithm to allow for input weighting. An example illustrates use of the algorithm to secure smaller dB error in selected frequency bands through the introduction of the weighting.