Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition

Circuits with delay elements are very popular and important in the simulation of very-large-scale integration (VLSI) systems. Neutral systems (NSs) with multiple constant delays (MCDs), for example, can be used to model the partial element equivalent circuits (PEECs), which are widely used in high-frequency electromagnetic (EM) analysis. In this paper, the model order reduction (MOR) problem for the NS with MCDs is addressed by moment matching method. The nonlinear exponential terms coming from the delayed states and the derivative of the delayed states in the transfer function of the original NS are first approximated by a Padé approximation or a Taylor series expansion. This has the consequence that the transfer function of the original NS is exponential-free and the standard moment matching method for reduction is readily applied. The Padé approximation of exponential terms gives an expanded delay-free system, which is further reduced to a delay-free reduced-order model (ROM). A Taylor series expansion of exponential terms lets the inverse in the original transfer function have only powers-of-s terms, whose coefficient matrices are of the same size as the original NS, which results in a ROM modeled by a lower-order NS. Numerical examples are included to show the effectiveness of the proposed algorithms and the comparison with existing MOR methods, such as the linear matrix inequality (LMI)-based method.

During the past three decades, evolutionary computing techniques have grown manifold in tackling all sorts of optimization problems. Genetic algorithm (GA) is one of the most popular EAs because it is easy to implement and is conducive for noisy environment. Similarly, amongst several swarm intelligence techniques, bacterial foraging optimization (BFO) is the recent popular algorithm being used for many practical applications. Depending on the complexity of the problem concerned, there is need for hybridized techniques which help in balancing exploration and exploitation capability over the search space. Many hybridized techniques have been developed recently to tackle such problems. This paper proposes a hybridization of GA and BFO to solve a real-life unconstrained electrical engineering problem. This unconstrained optimization problem is a model order reduction (MOR) problem of linear time invariant continuous single input and single output (SISO) system.

Krylov subspace methods for model order reduction of bilinear control systems

Riemannian optimization approach to structure-preserving model order reduction of integral-differential systems on the product of two Stiefel manifolds

In this paper, we present an adaptive framework for constructing a pseudo‐optimal reduced model for the frequency‐limited‐optimal model order reduction problem. We show that the frequency‐limited pseudo‐optimal reduced‐order model has an inherent property of monotonic decay in error if the interpolation points and tangential directions are selected appropriately. We also show that this property can be used to make an automatic selection of the order of the reduced model for an allowable tolerance in error. The proposed algorithm adaptively increases the order of the reduced model such that the frequency‐limited‐norm error decays monotonically irrespective of the choice of interpolation points and tangential directions. The stability of the reduced‐order model is also guaranteed. Additionally, it also generates the approximations of the frequency‐limited system Gramians that monotonically approach the original solution. Further, we show that the low‐rank alternating direction implicit iteration method for solving large‐scale frequency‐limited Lyapunov equations implicitly performs frequency‐limited pseudo‐optimal model order reduction. We consider two numerical examples to validate the theory presented in the paper.

Balanced truncation (BT) model order reduction (MOR) is known for its superior accuracy and computable error bounds. Balanced stochastic truncation (BST) is a particular BT procedure that provides a general, structure-independent MOR framework to preserve both passivity and stability of original models. Its application toward large scale systems, however, has been limited by the complexity of solving large size continuous time algebraic Riccati equations (CAREs). This paper introduces a novel quadratic extension of the alternating direction implicit (ADI) iteration, called QADI, that efficiently solves a CARE. A Cholesky factor variant of QADI, called CFQADI, further exploits low rank matrices and and produces solution in factor form that greatly accelerates BST. Remarkable efficiency of the proposed BST/(CF)QADI integration is demonstrated with numerical examples.

We propose a new approach to computing a parametric solution (the so-called p-solution) to parametric interval linear systems. Solving such system is an important part of many scientific and engineering problems involving uncertainties. The parametric solution has many useful properties. It permits to compute an outer solution, an inner estimate of the interval hull solution, and intervals containing the lower and upper bounds of the interval hull solution. It can also be employed for solving various constrained optimisation problems related to the parametric interval linear system. The proposed approach improves both outer and inner bounds for the parametric solution set. In this respect, the new approach is competitive to most of the existing methods for solving parametric interval linear systems. Improved bounds on the parametric solution set guarantees improved bounds for the solutions of related optimisation problems.

Sample selection based on sensitivity analysis in parameterized model order reduction

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.

An accelerated subspaces recycling strategy for the deflation of parametric linear systems based on model order reduction

This letter proposes to automate generation of reduced-order models used for accelerated S-parameter computation by applying a posteriori model error estimators. So far, a posteriori error estimators were used in Reduced Basis Method (RBM) and Proper Orthogonal Decomposition (POD) to select frequency points at which basis vectors are generated. This letter shows how a posteriori error estimators can be applied to automatically select the order of the reduced model in second-order Model Order Reduction (MOR) methods. Three different error estimators are investigated and compared in order to arrive at a new MOR scheme that is fast, reliable, and fully automated. The effectiveness of the proposed approach is verified by very high accuracy of the computed scattering parameters ( S-parameters) for an example of a waveguide filter over a prescribed frequency band.

Various versions are compared of the multidimensional method of orthogonal regression in the problem of estimating the parameters of linear systems by short sections of transient processes with additive measuring perturbations. The methods are classified by the degree of the use of information on linear links between counts of trajectories of a dynamic object.

In this paper, we will discuss the problem of optimal model order reduction of bilinear control systems with respect to the generalization of the well-known ${\cal H}_2$-norm for linear systems. We revisit existing first order necessary conditions for ${\cal H}_2$-optimality based on the solutions of generalized Lyapunov equations arising in bilinear system theory and present an iterative algorithm which, upon convergence, will yield a reduced system fulfilling these conditions. While this approach relies on the solution of certain generalized Sylvester equations, we will establish a connection to another method based on generalized rational interpolation. This will lead to another way of computing the ${\cal H}_2$-norm of a bilinear system and will extend the pole-residue optimality conditions for linear systems, also allowing for an adaption of the successful iterative rational Krylov algorithm to bilinear systems. By means of several numerical examples, we will then demonstrate that the new techniques outperform the method of balanced truncation for bilinear systems with regard to the relative ${\cal H}_2$-error.

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

In this paper, we generalize existing frameworks for $\mathcal{H}_2\otimes\mathcal{L}_2$-optimal model order reduction to a broad class of parametric linear time-invariant systems. To this end, we derive first-order necessary ptimality conditions for a class of structured reduced-order models, and then building on those, propose a stability-preserving optimization-based method for computing locally $\mathcal{H}_2\otimes\mathcal{L}_2$-optimal reduced-order models. We also make a theoretical comparison to existing approaches in the literature, and in numerical experiments, show how our new method, with reasonable computational effort, produces stable optimized reduced-order models with significantly lower approximation errors.