Model order reduction of fully parameterized systems by recursive least square optimization

Abstract

Abstract-This paper presents a method for the model order reduction of fully parameterized linear dynamic systems. In a fully parameterized system, not only the state matrices, but also can the input/output matrices be parameterized. This algorithm is based on neither conventional moment-matching nor balancedtruncation ideas; instead, it uses "optimal (block) vectors" to construct the projection matrix such that the system errors in the whole parameter space could be minimized. This minimization problem is formulated as a recursive least square (RLS) optimization and then solved at a low cost. Our algorithm is tested by a set of multi-port multi-parameter cases with both intermediate and large parameter variations. The numerical results show that high accuracy is guaranteed, and that very compact models can be obtained for multi-parameter models due to the fact that the ROM size is independent of the parameter number in our flow. I. INTRODUCTIONDesign and process parameter variations need very careful consideration in submicron digital, mixed-signal and RF analog integrated circuit design. Parameterized model order reduction (PMOR) techniques can generate compact models reflecting the impact induced by design or process variations. Such techniques have become highly desirable in the EDA community in order to accelerate the time-consuming design space exploration, sensitivity analysis and automatic synthesis.Several PMOR techniques have been developed for both linear and nonlinear circuits. Most are based on moment matching techniques [1]-[11] due to the numerical efficiency. These algorithms normally assume that the closed forms of the parameterized state-space models are given, or that the parameters' statistical distributions are known. With similar assumptions, the positive-real balanced truncation method [12] has been modified for parameterized interconnect model reduction [13], [14]. In many cases the designers do not know the exact symbolic forms of the parameterized circuit equations. As a result, neither moment matching nor positivereal balanced truncation can be used in the PMOR flow. A numerically efficient and flexible method is the variational PMTBR scheme [15], which starts from the state-space model and uses a cheap sampling scheme to approximate the Gramian matrix. This approach is capable of preserving passivity for symmetric systems (such as RC circuits), but not for general RLC interconnect models. When the system equations are not available, one can treat the original model as a black box, and then use system identification techniques to construct macromodels from simulated or measured data by, for example, the quasi-convex optimization method [16], [17].

Stochastic Testing Method for Transistor-Level Uncertainty Quantification Based on Generalized Polynomial Chaos

“…• SG and ST are intrusive solvers as they both directly compute the gPC coefficients by simulating a larger-size coupled DAE only once. With gPC approximations, they both start from the residual function (10). SG sets up a larger-size coupled deterministic model by Galerkin testing, whereas ST uses a collocation testing technique.…”

Gramian‐based model order reduction of parameterized time‐delay systems

“…In fact, the exponential growth of the ROM size with respect to the parameter number and expansion order is an open question in parameterized MOR community. Although a recent optimization-based work [45] whose resulting ROM size is independent of the number of parameter has been proposed, no solution has been proposed for the time-delay cases. For the non-smooth parameter dependence, the optimization-based idea [45] can be used for systems without delay effects, and its ROM size is also independent of the number of parameters.…”

A Comprehensive Comparison of Contemporary Parameterized Model Order Reduction Techniques

“…Taylor expansion is not used in approximating the parameterized system matrices. This implies that passivity is preserved for the parameterized RLC interconnect model [54].…”

“…Consider a multi-parameter linear time invariant (LTI) fully parameterized dynamic system as given by the following relations [54] ( The objective of this method is to find a fully parameterized reduced order model of order q where q << n, which approximates the characteristics of the original system with a high degree of accuracy. The reduced order model can be given by the following set of equations [54] ( )( ) = ( ) ( ) + ( ) ( )…”

“…As mentioned in the earlier sections, this can be achieved if an appropriate projection matrix Q can be computed such that the reduced matrices are related to the original matrices as [54] ( ) = ( ) , ( ) = ( ) , ( ) = ( ), ( ) = ( ) (3.54)…”