Balanced Model Reduction Research Articles

Parametric Models of Thermal Transfer Impedances within a Successive Node Reduction Based Thermal Simulation Environment

In this paper we present a new, direct computational method for calculating thermal transfer impedances between two separate locations of a given physical structure, aimed at the implementation into a field-solver based on the SUNRED (SUccessive Node REDuction) algorithm. We tested the method symbolically with a simple 2D example with multiple combinations of Dirichlet and Neumann type boundary conditions. Also, for time domain transient analysis different types of thermal loads such as prescribed unit-step change in dissipation or temperature were assumed. A model of a typical MCPCB assembled LED was also created. With that model we studied the inverse problem of predicting the thermal conditions at the junction (the "driving point") from the transient signal measured at the thermal test point on the MCPCB (the "monitoring point") which is a typical task in simple LED thermal management designs. Results obtained by the proposed new calculation method and results obtained by conventional numerical simulations differ less than the uncertainty of the traditional solution method itself. The drawback of the accuracy is the high computational cost. This increased computational need can be mitigated by introducing the combination of the balanced model reduction and the SUNRED algorithms

Numerical study of polynomial feedback laws for a bilinear control problem

An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium.

Model reduction of constrained mechanical systems in M-M.E.S.S.

We discuss balanced truncation model reduction of constrained mechanical systems. The incorporation of the algebraic constraints leads to linear differential-algebraic control systems of index 2 or 3. Then for the square-root method in balanced truncation, the solution of projected Lyapunov equations is required. In this paper we discuss how to avoid the explicit construction of the projectors and the formulation of a projection-avoiding balanced truncation method which is suitable for efficient numerical computations. We have implemented this method in the Matlab package M-M.E.S.S. and present some numerical results for illustration.

Balanced truncation model reduction of nonstationary systems interconnected over arbitrary graphs

This paper deals with the balanced truncation of discrete-time, linear time-varying, heterogeneous subsystems interconnected over finite arbitrary directed graphs. The information transfer between the subsystems is subject to a communication latency of one time-step. The method guarantees the preservation of the interconnection structure and further allows for its simplification. In addition to truncating temporal states associated with the subsystems, the method allows for the order reduction of spatial states associated with the interconnections between the subsystems and even the removal of whole interconnections. Upper bounds on the ℓ2-induced norm of the resulting error system are derived. The method is illustrated through an example.

The design of delay-dependent wide-area DOFC with prescribed degree of stability α for damping inter-area low-frequency oscillations in power system

In this paper, the delay-dependent wide-area dynamic output feedback controller (DOFC) with prescribed degree of stability is proposed for interconnected power system to damp inter-area low-frequency oscillations. Here, the prescribed degree of stability α is used to maintain all the poles on the left of s=−α in the s-plane. Firstly, residue approach is adopted to select input-output control signals and the schur balanced truncation model reduction method is utilized to obtain the reduced power system model. Secondly, based on Lyapunov stability theory and transformation operation in complex plane, the sufficient condition of asymptotic stability for closed-loop power system with prescribed degree of stability α is derived. Then, a novel method based on linear matrix inequalities (LMIs) is presented to obtain the parameters of DOFC and calculate delay margin of the closed-loop system considering the prescribed degree of stability α. Finally, case studies are carried out on the two-area four-machine system, which is controlled by classical wide-area power system stabilizer (WAPSS) in reported reference and our proposed DOFC respectively. The effectiveness and advantages of the proposed method are verified by the simulation results under different operating conditions.

Structure preserving iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems

In this paper, we develop structure preserving iterative schemes to solve the periodic discrete-time projected Lyapunov equations associated to analysis and design of discrete-time descriptor systems exploiting the reflexive generalized inverses of the periodic matrices associated with these systems. In particular, we extend the Smith method to solve the large-scale projected periodic discrete-time algebraic Lyapunov equations in lifted form. A low-rank version of this method is also presented, avoiding the explicit lifted formulation and working directly with the periodic matrix coefficients. Moreover, we consider an application of the Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

Solving Parameter-Dependent Lyapunov Equations Using the Reduced Basis Method with Application to Parametric Model Order Reduction

Our aim is to numerically solve parameter-dependent Lyapunov equations using the reduced basis method. Such equations arise in parametric model order reduction. We restrict ourselves to the systems that affinely depend on the parameter, as our main strategy is the min-$\theta$ approach. In those cases, we derive various a posteriori error estimates. Based on these estimates, a greedy algorithm for constructing reduced bases is formulated. Thanks to the derived results, a novel so-called parametric balanced truncation model reduction method is developed. Numerical examples are presented.

Efficient Techniques for Solving the Periodic Projected Lyapunov Equations and Model Reduction of Periodic Systems

We have presented the efficient techniques for the solutions of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form. These types of problems arise in model reduction and state feedback problems of periodic descriptor systems. Two most popular techniques to solve such Lyapunov equations iteratively are the low-rank alternating direction implicit (LR-ADI) method and the low-rank Smith method. The main contribution of this paper is to update the LR-ADI method by exploiting the ideas of the adaptive shift parameters computation and the efficient handling of complex shift parameters. These approaches efficiently reduce the computational cost with respect to time and memory. We also apply these iterative Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. We illustrate numerical results to show the performance and accuracy of the proposed methods.

Balanced truncation model reduction for linear time-varying systems

ABSTRACTA practical procedure based on implicit time integration methods applied to the differential Lyapunov equations arising in the square root balanced truncation method is presented. The application of high-order time integrators results in indefinite right-hand sides of the algebraic Lyapunov equations that have to be solved within every time step. Therefore, classical methods exploiting the inherent low-rank structure often observed for practical applications end up in complex data and arithmetic. Avoiding the additional effort in treating complex quantities, a symmetric indefinite factorization of both the right-hand side and the solution of the differential Lyapunov equations is applied.

Partial Fraction Expansion Based Frequency Weighted Balanced Singular Perturbation Approximation Model Reduction Technique with Error Bounds

In this paper, a new frequency weighted partial fraction expansion based model reduction technique is developed based on the partial fraction expansion approach. In order to further reduce the frequency weighted approximation error, singular perturbation approximation is incorporated into the algorithm. This technique results in stable reduced order models regardless if single sided or double sided weights are used. Error bounds are also derived for the proposed method. For minimization of the frequency weighted approximation error, free parameters are introduced into the algorithm. A numerical example is provided in order to validate the proposed algorithm.

Digital and analog integer delayed modeling and control for multivariable systems with multiple time delays in states, inputs and outputs

This paper presents a new approximated integer delayed modeling and control technique for Continuous-time Fractional Delay Differential Equations (CFDDEs). Using the impulse response sequences of the CFDDE and the balanced model-reduction method, a delay-free discrete-time state-space model is constructed. Then, an equivalent Discrete-time Integer Delay Difference Equation (DIDDE) is obtained by transforming the obtained discrete-time state-space model into a controller-type block companion form. Furthermore, based on the obtained DIDDE, an equivalent Continuous-time Integer Delay Differential Equation (CIDDE) is determined by means of the newly developed Chebyshev's bilinear approximation method. For digital control of the CFDDE, an optimal Discrete-time Integer Delayed Control Law (DIDCL) is designed using the conventional discrete-time LQR approach together with the obtained delay-free discrete-time state-space model. On the other hand, for continuous-time control of the CFDDE, a Continuous-time Integer Delayed Control Law (CIDCL) is determined from the designed DIDCL by means of the inverse Chebyshev's bilinear approximation method. Finally, digital and analog integer delayed observers are constructed for the implementations of the developed DIDCL and CIDCL, respectively. An illustrative example is given to demonstrate the effectiveness of the proposed method.

A Frequency Limited Interval Gramians-Based Model Reduction Technique with Error Bounds

A balanced model reduction technique that is based on frequency limited interval Gramians is proposed. The technique provides stable models and also yields frequency response error bounds. Numerical examples are also presented. The results are comparable with other frequency limited interval Gramians-based model reduction techniques.

A frequency weighted model order reduction technique and error bounds

A new frequency weighted technique for balanced model reduction is proposed. The proposed technique not only provides stable reduced order models for the case when both input and output weightings are included but also yields frequency response error bounds. The method is illustrated using numerical examples and the results are compared with other frequency weighted model reduction techniques.

Stability Preserving Model Reduction Technique and Error Bounds Using Frequency-Limited Gramians for Discrete-Time Systems

A frequency-limited interval Gramians-based balanced model reduction technique for discrete-time systems is proposed. The technique provides stable models and also yields frequency-response error bounds. Numerical examples are also presented. The results are comparable with other existing frequency-limited interval Gramians-based model reduction techniques.

Optimality of balanced proper orthogonal decomposition for data reconstruction II: Further approximation results

In our earlier paper Singler (2010) [7], we showed two separate data sets can be optimally approximated using balanced proper orthogonal decomposition (POD) modes derived from the data. In this work, we prove new results concerning the approximation capability of the balanced POD modes. We give exact computable expressions for the errors between the individual data sets and the low order balanced POD data reconstructions. We also consider approximating elements of the Hilbert space using various projections onto the balanced POD modes. We discuss the relevance of these results to balanced POD model reduction of nonlinear partial differential equations.

State-space representation of Li-ion battery porous electrode impedance model with balanced model reduction

This paper presents an approximate time-domain solution for physics-based electrochemical lithium-ion cell battery models. The time-domain solution is represented in state-space form and can be easily used for the design of a state estimator or controller. It uses an interconnection-of-system approach to derive a state-space representation of a battery impedance model and provides a reduced order model based via the balanced truncation method. Simulation results are also provided to show the performance of the proposed model in the frequency domain.

Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems

We discuss the numerical solution of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

Port Hamiltonian System in Descriptor Form for Balanced Reduction: Application to a Nanotweezer

This paper proposes a method of balanced model reduction for constrained linear port Hamiltonian systems. Constrained linear port Hamiltonian systems are first written in a canonical descriptor form such that the Hamiltonian structure is preserved. The computations of the controllability and observability Gramians are then used to derive the balanced port Hamiltonian representation of the system. The method of flow constraint is applied to reduce the system. Finally, numerical simulations for the reduction of a micro mechanical actuator model is given to illustrate the effectiveness of the proposed method.

Frequency Weighted Model Order Reduction Technique and Error Bounds for Discrete Time Systems

Model reduction is a process of approximating higher order original models by comparatively lower order models with reasonable accuracy in order to provide ease in design, modeling and simulation for large complex systems. Generally, model reduction techniques approximate the higher order systems for whole frequency range. However, certain applications (like controller reduction) require frequency weighted approximation, which introduce the concept of using frequency weights in model reduction techniques. Limitations of some existing frequency weighted model reduction techniques include lack of stability of reduced order models (for two sided weighting case) and frequency response error bounds. A new frequency weighted technique for balanced model reduction for discrete time systems is proposed. The proposed technique guarantees stable reduced order models even for the case when two sided weightings are present. Efficient technique for frequency weighted Gramians is also proposed. Results are compared with other existing frequency weighted model reduction techniques for discrete time systems. Moreover, the proposed technique yields frequency response error bounds.

Design of decoupling and tracking controllers for continuous-time transfer function matrices with multiple time delays

This paper presents an extended adjoint decoupling method together with a reference model-based sliding mode tracking method, to design a decoupling and tracking controller for continuous-time transfer function matrices with multiple (integer/fractional) time delays in both denominators and numerators. First, for obtaining the diagonally decoupled subsystems, a decoupler is designed by utilizing the extended adjoint decoupling method. Then, by using sampled data from the unit-step response of the decoupled subsystems, the conventional balanced model-reduction method is carried out to obtain the approximated delay-free/single-delay continuous-time models for the decoupled subsystems with multiple time delays. For the integral of time multiplied by absolute error (ITAE) reference model tracking, a chain observer is designed to establish the virtual estimated states for the decoupled subsystems by utilizing the obtained approximated continuous-time models. At last, we develop a sliding mode tracking controller together with a disturbance observer (DOB), to achieve reference model tracking and disturbance rejection. Illustrative examples are given to demonstrate the effectiveness of the proposed method.