Balanced Truncation Method Research Articles

Low-Rank Alternating Direction Implicit Iteration in pyMOR

The low-rank alternating direction implicit (LR-ADI) iteration is an effective method for solving large-scale Lyapunov equations. In the software library pyMOR, solutions to Lyapunov equations play an important role when reducing a model using the balanced truncation method. In this article we introduce the LR-ADI iteration as well as pyMOR, while focusing on its features which are relevant for integrating the iteration into the library. We compare the run time of the iteration's pure pyMOR implementation with those achieved by external libraries available within the pyMOR framework.

Structure‐preserving interval‐limited balanced truncation reduced models for port‐Hamiltonian systems

In this study, the authors propose structure-preserving balanced truncation methods of port-Hamiltonian systems over the frequency and time intervals. First, the port-Hamiltonian system is reduced based on the frequency-interval controllability and observability Gramians. In order to reduce the computational cost, the frequency-interval cross Gramian is utilised to yield the frequency-interval balanced truncation method. For the symmetric port-Hamiltonian systems, the authors prove that these two methods can generate equivalent reduced systems. Additionally, they are also devoted in exploring two structure-preserving balanced truncation methods over time intervals, where one is based on the time interval controllability and observability Gramians and the other is based on the time interval cross Gramian. All the resulting reduced systems have port-Hamiltonian structure, and as a consequence, they are passive. Two numerical examples are simulated to demonstrate the efficiency of the proposed methods.

Towards robust controller design using $$\mu $$-synthesis approach for speed regulation of an uncertain wind turbine

Wind turbines are subjected to factors like fatigue, aerodynamics, structural flexibility and wind turbulence which lead to uncertain behaviour. This affects stability and deteriorates the performance of the large structured wind turbine. In order to reduce the effects of uncertainties on the system performance and its structure, a robust controller design is necessary. In this paper, it is proposed to design a $$\mu $$-synthesis-based robust controller to overcome the effects due to uncertainties. A $$\pm \,25\%$$ variation in the values of the system matrix elements is considered for analysis in this paper. A 109th order of the proposed $$\mu $$-controller is obtained, wherein it is reduced to a 7th order controller by using the balanced truncation method. The robust stability and robust performances are satisfactorily achieved with both these controllers. Furthermore, the worst-case performance of the uncertain wind turbine is also analysed.

PSO Based reduced order modelling of autonomous AC microgrid considering state perturbation

Reduced order modelling of complex autonomous microgrid system is crucial to its small signal modelling and stability concerns. To reduce the storage requirements and computational time, the order of such microgrids can be reduced by Model Order Reduction (MOR) techniques. This paper presents an optimal reduction technique, which retains dominant poles of the original system and achieves subsequent error minimization through the Particle Swarm Optimization algorithm (PSO). The 36th order complex microgrid system is reduced to 9th order approximant, which retains the significant dynamics of the original system. The simulation results reflect the superiority of the proposed method as compared to the balanced truncation method in terms of the time and frequency domain analysis of the reduced order equivalents. State perturbation in the state space model has also been considered in full as well as reduced order system dynamics and eigenvalue analysis for system stability.

Two‐dimensional FIR filters model order reduction and frequency transformation: Maximum error invariance

AbstractThis paper presents a numerical study of the maximum error in model order reduction (MOR) for finite impulse response (FIR) 2‐D digital filter under frequency transformation, which is based on the Roesser state‐space formulation. This study discuss the maximum error obtained from the 2‐D balanced truncation method after and before using the frequency transformation. Several numerical examples are realized to show the invariant of the maximum error in all frequency transformation cases (single variable, two‐variable, and hybrid variable transformations).

Reduced-order modelling of parametric systems via interpolation of heterogeneous surrogates

This paper studies parametric reduced-order modeling via the interpolation of linear multiple-input multiple-output reduced-order, or, more general, surrogate models in the frequency domain. It shows that realization plays a central role and two methods based on different realizations are proposed. Interpolation of reduced-order models in the Loewner representation is equivalent to interpolating the corresponding frequency response functions. Interpolation of reduced-order models in the (real) pole-residue representation is equivalent to interpolating the positions and residues of the poles. The latter pole-matching approach proves to be more natural in approximating system dynamics. Numerical results demonstrate the high efficiency and wide applicability of the pole-matching method. It is shown to be efficient in interpolating surrogate models built by several different methods, including the balanced truncation method, the Krylov method, the Loewner framework, and a system identification method. It is even capable of interpolating a reduced-order model built by a projection-based method and a reduced-order model built by a data-driven method. Its other merits include low computational cost, small size of the parametric reduced-order model, relative insensitivity to the dimension of the parameter space, and capability of dealing with complicated parameter dependence.

Design of inter-stations H∞ decoupling controller in VSC-HVDC system

The purpose of this work is to reduce the mutual interference between converter stations when the AC power or DC voltage fluctuates in VSC-HVDC system. The global small signal mathematical model of VSC-HVDC system is established firstly. Then, according to the principle of mixed sensitivity H∞ robust control, the appropriate weight function is selected and the decoupling controller between stations is deduced. In addition, the high-order controller deduced is reduced to an appropriate order which is beneficial to engineering application by using the balanced truncation method of coprime factors. Finally, the electromagnetic transient simulation is carried out to verify that the H∞ decoupling controller between stations can be effectively weakened the interaction between two converter stations and improves the dynamic performance and robust stability of the system.

Balanced Truncation for a Special Class of Bilinear Descriptor Systems

In this letter, we investigate balanced truncation for a class of continuous-time bilinear descriptor systems, appearing, e.g., in constraint circuit analysis or resulting from semi-discretization of the Navier–Stokes equations. For this, as widely done in the literature, we first aim at transforming the bilinear descriptor system into an equivalent ODE system by means of projectors for which the standard balanced truncation method for bilinear systems can be applied. Subsequently, we discuss how to solve the arising generalized Lyapunov equations corresponding to the equivalent ODE, without requiring the explicit computation of the projectors. The efficiency of the proposed algorithm is illustrated through a numerical example, which also shows its competitiveness with an iterative ${\mathcal {H}}_{2}$ -optimal model order reduction scheme.

Model Order Reduction of Wind Farms: Linear Approach

This paper presents three types of linear model order reduction (MOR) technique, namely singular value decomposition (SVD) based, Krylov-based, and modal truncation based type applied to large-scale wind farm models. The first type includes a Balanced Truncation (BT) and Alternating Direction Implicit (ADI)-based BT method, while the second type encompasses a Rational Krylov (RK), and Iterative Rational Krylov Algorithm (IRKA) method. In the third type, a Subspace Accelerated MIMO Dominant Pole Algorithm (SAMDP) method is used. The effectiveness of these methods are tested on practical-sized wind farms with 90, 120, and 210 doubly fed induction generators (DFIGs). Merits and demerits of each method are discussed in detail. The reduced-order model (ROM) of wind farm is validated against the full-order model (FOM) in terms of frequency domain indices and waveform agreement at the point of common coupling (PCC).

Infinite-dimensional bilinear and stochastic balanced truncation with explicit error bounds

Along the ideas of Curtain and Glover (in: Bart, Gohberg, Kaashoek (eds) Operator theory and systems, Birkhäuser, Boston, 1986), we extend the balanced truncation method for (infinite-dimensional) linear systems to arbitrary-dimensional bilinear and stochastic systems. In particular, we apply Hilbert space techniques used in many-body quantum mechanics to establish new fully explicit error bounds for the truncated system and prove convergence results. The functional analytic setting allows us to obtain mixed Hardy space error bounds for both finite-and infinite-dimensional systems, and it is then applied to the model reduction of stochastic evolution equations driven by Wiener noise.

Model Order Reduction of Linear Systems via the Cross Gramian and SVD

On the basis of the cross Gramian and the singular value decomposition, this brief investigates model order reduction of linear time-invariant systems in the view of the symmetric case and the nonsymmetric case. It guarantees the reduced system for the nonsymmetric system to be data-sparse. The asymptotical stability of the reduced system is discussed and an output error bound is provided. The correlation of the Hankel singular values in the proposed method and the balanced truncation method is presented. Finally, numerical results indicate that the reduced system of the symmetric system is equivalent to that resulting from balanced truncation, while the reduced system of the nonsymmetric system has a good performance.

Balanced Truncation for Model Order Reduction of Linear Dynamical Systems with Quadratic Outputs

We investigate model order reduction (MOR) for linear dynamical systems, where a quadratic output is defined as a quantity of interest. The system can be transformed into a linear dynamical system with many linear outputs. MOR is feasible by the method of balanced truncation, but suffers from the large number of outputs in approximate methods. To ameliorate this shortcoming we derive an equivalent quadratic-bilinear system with a single linear output and analyze the properties of this system. We examine MOR for this system via the technique of balanced truncation, which requires a stabilization of the system. Therein, the solution of two quadratic Lyapunov equations is traced back to the solution of just two linear Lyapunov equations. We present numerical results for several test examples comparing the two MOR approaches.

Riemannian Optimal Model Reduction of Stable Linear Systems

In this paper, we develop a method for solving the problem of minimizing the $H^2$ error norm between the transfer functions of original and reduced systems on the set of stable matrices and two Euclidean spaces. That is, we develop a method for identifying the optimal reduced system from all stable linear systems. However, it is difficult to develop an algorithm for solving this problem, because the set of stable matrices is highly non-convex. To overcome this issue, we show that the problem can be transformed into a tractable Riemannian optimization on the product manifold of the set of skew-symmetric matrices, the manifold of the symmetric positive-definite matrices, and two Euclidean spaces. The stability of the reduced systems constructed using the optimal solutions to our problem is preserved. To solve the reduced problem, the Riemannian gradient and Hessian are derived and a Riemannian trust-region method is developed. The initial point in the proposed approach is selected using the output from the balanced truncation (BT) method. Numerical experiments demonstrate that our method considerably improves the results given by BT in the sense of the $H^2$ norm, and also provides reduced systems that are globally near-optimal solutions to the problem of minimizing the $H^{\infty}$ error norm. Moreover, we show that our method provides a better reduced model than BT from the viewpoint of the frequency response.

Balancing Related Model Reduction with the MORLAB Toolbox

AbstractBalancing related model reduction methods are based on the idea of the balanced truncation method and can be used to preserve certain system properties like passivity or contractivity. The MORLAB toolbox provides efficient implementations for a big set of such methods. We will give an overview about the implemented methods and test some of these methods on a benchmark example.

Structure Preserving Truncation of Nonlinear Port Hamiltonian Systems

In this paper, we present a novel balancing method for nonlinear port Hamiltonian systems based on the Hamiltonian and the controllability function. This corresponding balanced truncation method results in a reduced-order model that is still in port Hamiltonian form in contrast to the traditional balanced truncation method based on the controllability and observability functions.

Gramian‐based model‐order reduction of constrained structural dynamic systems

This study discusses model reduction techniques for second-order index 3 descriptor systems using the balanced truncation methods; in particular, linearised equations of motion with holonomic constraints are considered which arise in mechanics and multibody dynamics. It is shown that the index 3 system can be converted into an equivalent form of index 0 system by projecting it onto the hidden manifold. When model reduction is applied to the projected system, explicit formulation of the projected system is not required. The low-rank alternating direction implicit iteration is also discussed for solving the projected Lyapunov equations of the underlying descriptor system efficiently in an implicit way. The theoretical results are illustrated by numerical experiments.

Embedded order reduction of hierarchical systems within an mCCM framework

PurposeThe purpose of this paper is to present the embedding of linear and nonlinear order reduction methods in a theoretical framework for handling hierarchically interconnected dynamical systems.Design/methodology/approachBased on the component connection modeling (CCM), a modified framework called mCCM for describing interconnected dynamic systems especially with hierarchical structures is introduced and used for order reduction purposes. The balanced truncation method for linear systems and the trajectory piecewise linear reduction scheme are used for the order reduction of systems described within the mCCM framework.FindingsIt is shown that order reduction methods can be embedded into the context of interconnected dynamical systems with the benefit of having a further degree of freedom in form of the hierarchical level, on which the order reduction is performed.Originality/valueThe aspect of hierarchy within system descriptions is exploited for order reduction purposes. This distinguishes the presented approach from common methods, which already start with single large-scale systems without explicitly considering hierarchical structures.

Model Order Reduction by Using the Balanced Truncation and Factor Division Methods

ABSTRACTThe aim of this paper is to construct a new model order reduction method for linear dynamic systems. In this technique, the denominator polynomial of the reduced order model (ROM) is obtained by the balanced truncation method and the numerator polynomial is calculated by the factor division technique. This method is proposed for overcoming the existing problems of the balanced truncation method such as mismatch of steady-state values of the reduced model and the original model. The proposed method guarantees the preservation of stability and steady-state value of the original system in the ROM; hence, it preserves the advantages of balanced truncation and factor division methods in the reduced model. A special advantage of the proposed technique is that it guarantees the preservation of the first few time moments of the original system in the ROM. The mathematical modeling of the dc–dc converter is also developed and by using the proposed method its fourth-order original system is reduced to a second-order model in such a way that the reduced system preserves the fundamental features of the original system. In order to show the accuracy and effectiveness of the proposed technique, three standard numerical examples are taken from the literature. The proposed method is compared with the recent and popular techniques of model order reduction with the help of MATLAB.

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.

Coprime factors reduction of distributed nonstationary LPV systems

ABSTRACTThis paper is on the coprime factors reduction of distributed systems formed by discrete-time, heterogeneous, nonstationary linear parameter-varying subsystems. The subsystems are represented in a linear fractional transformation framework and interconnected over arbitrary directed graphs, and the communication between the subsystems is subjected to a delay of one time-step. Two methods for constructing a contractive coprime factorisation for the full-order system are proposed. This factorisation forms an augmented system which is reducible by the structure-preserving balanced truncation method. A reduced-order contractive coprime factorisation is obtained from which the reduced-order system can be formed. A robustness theorem is also provided to interpret the error bound from coprime factors reduction in terms of robust stability of the closed-loop system. A numerical example is considered at the end of the paper.