H∞ model reduction of switched LPV systems via semi-time-varying reduced-order model

2016 ◽  
Vol 98 ◽  
pp. 25-32 ◽  
Author(s):  
Lixian Zhang ◽  
Wei Xing Zheng ◽  
Huijun Gao
Keyword(s):  
Model Reduction ◽  
Reduced Order Model ◽  
Time Varying ◽  
Order Model ◽  
Reduced Order ◽  
Lpv Systems

scholarly journals H∞ Model Reduction of 2D Markovian Jump System with Roesser Model

2012 ◽  
Vol 6-7 ◽  
pp. 135-142
Author(s):  
Xue Song Han ◽  
Yu Bo Duan
Keyword(s):  
Model Reduction ◽  
Reduced Order Model ◽  
Markovian Jump Systems ◽  
Linear Matrix ◽  
Roesser Model ◽  
Markovian Jump ◽  
Order Model ◽  
Existence Of A Solution ◽  
Reduced Order ◽  
Jump Systems

This paper extends the results obtained for one-dimensional Markovian jump systems to investigate the problem of H∞model reduction for a class of linear discrete time 2D Markovian jump systems with state delays in Roesser model which is time-varying and mode-independent. The reduced-order model with the same randomly jumping parameters is proposed which can make the error systems stochastically stable with a prescribed H∞ performance. A sufficient condition in terms of linear matrix inequalitiesSubscript text(LMIs) plus matrix inverse constraints are derived for the existence of a solution to the reduced-order model problems. The cone complimentarity linearization (CCL) method is exploited to cast them into nonlinear minimization problems subject to LMI constraints. A numerical example is given to illustrate the design procedures.

scholarly journals Model Reduction for Kinetic Models of Biological Systems

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 863
Author(s):  
Neveen Ali Eshtewy ◽  
Lena Scholz
Keyword(s):  
Nonlinear Systems ◽  
Model Reduction ◽  
Kinetic Models ◽  
Initial Conditions ◽  
Reduced Order Model ◽  
Order System ◽  
High Dimensional ◽  
Order Model ◽  
Model Order ◽  
Reduced Order

High dimensionality continues to be a challenge in computational systems biology. The kinetic models of many phenomena of interest are high-dimensional and complex, resulting in large computational effort in the simulation. Model order reduction (MOR) is a mathematical technique that is used to reduce the computational complexity of high-dimensional systems by approximation with lower dimensional systems, while retaining the important information and properties of the full order system. Proper orthogonal decomposition (POD) is a method based on Galerkin projection that can be used for reducing the model order. POD is considered an optimal linear approach since it obtains the minimum squared distance between the original model and its reduced representation. However, POD may represent a restriction for nonlinear systems. By applying the POD method for nonlinear systems, the complexity to solve the nonlinear term still remains that of the full order model. To overcome the complexity for nonlinear terms in the dynamical system, an approach called the discrete empirical interpolation method (DEIM) can be used. In this paper, we discuss model reduction by POD and DEIM to reduce the order of kinetic models of biological systems and illustrate the approaches on some examples. Additional computational costs for setting up the reduced order system pay off for large-scale systems. In general, a reduced model should not be expected to yield good approximations if different initial conditions are used from that used to produce the reduced order model. We used the POD method of a kinetic model with different initial conditions to compute the reduced model. This reduced order model is able to predict the full order model for a variety of different initial conditions.

scholarly journals Sampling-free parametric model reduction of systems with structured parameter variation

2018 ◽  
Author(s):  
Christopher Beattie ◽  
Serkan Gugercin ◽  
Zoran Tomljanović
Keyword(s):  
Model Reduction ◽  
Transfer Functions ◽  
Linear Time ◽  
Mass Matrix ◽  
Reduction Process ◽  
Parametric Model ◽  
Reduced Order Model ◽  
Computationally Efficient ◽  
Order Model ◽  
Reduced Order

We consider a parametric linear time invariant dynamical systems represented in state-space form as $$E \dot x(t) = A(p) x(t) + Bu(t), \\ y(t) = Cx(t),$$ where $E, A(p) \in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m} $ and $C\in \mathbb{R}^{l\times n}$. Here $x(t)\in \mathbb{R}^{n} $ denotes the state variable, while $u(t)\in \mathbb{R}^{m}$ and $y(t)\in \mathbb{R}^{l}$ represent, respectively, the inputs and outputs of the system. We assume that $A(p)$ depends on $k\ll n$ parameters $p=(p_1, p_2, \ldots, p_k)$ such that we may write $$A(p)=A_0+U\,\diag (p_1, p_2, \ldots, p_k)V^T,$$ where $U, V \in \mathbb{R}^{n\times k}$ are given fixed matrices.We propose an approach for approximating the full-order transfer function $H(s;p)=C(s E -A(p))^{-1}B$ with a reduced-order model that retains the structure of parametric dependence and (typically) offers uniformly high fidelity across the full parameter range. Remarkably, the proposed reduction process removes the need for parameter sampling and thus does not depend on identifying particular parameter values of interest. Our approach is based on the classic Sherman-Morrison-Woodbury formula and allows us to construct a parameterized reduced order model from transfer functions of four subsystems that do not depend on parameters, allowing one to apply well-established model reduction techniques for non-parametric systems. The overall process is well suited for computationally efficient parameter optimization and the study of important system properties. One of the main applications of our approach is for damping optimization: we consider a vibrational system described by $$ \begin{equation}\label{MDK} \begin{array}{rl} M\ddot q(t)+(C_{int} + C_{ext})\dot q(t)+Kq(t)&=E w(t),\\ z(t)&=Hq(t), \end{array} \end{equation} $$ where the mass matrix, $M$, and stiffness matrix, $K$, are real, symmetric positive-definite matrices of order $n$. Here, $q(t)$ is a vector of displacements and rotations, while $ w(t) $ and $z(t) $ represent, respectively, the inputs (typically viewed as potentially disruptive) and outputs of the system. Damping in the structure is modeled as viscous damping determined by $C_{int} + C_{ext}$ where $C_{int}$ and $C_{ext}$ represent contributions from internal and external damping, respectively. Information regarding damper geometry and positioning as well as the corresponding damping viscosities are encoded in $C_{ext}= U\diag{(p_1, p_2, \ldots, p_k)} U^T$ where $U \in \mathbb{R}^{n\times k}$ determines the placement and geometry of the external dampers. The main problem is to determine the best damping matrix that is able to minimize influence of the disturbances, $w$, on the output of the system $z$. We use a minimization criteria based on the $\mathcal{H}_2$ system norm. In realistic settings, damping optimization is a very demanding problem. We find that the parametric model reduction approach described here offers a new tool with significant advantages for the efficient optimization of damping in such problems.

scholarly journals Adaptive POD-DEIM model reduction based on an improved error estimator

2018 ◽  
Author(s):  
Sridhar Chellappa ◽  
Lihong Feng ◽  
Peter Benner
Keyword(s):  
Dynamical Systems ◽  
Model Reduction ◽  
Nonlinear Model ◽  
Reduced Order Model ◽  
Error Estimator ◽  
Order Model ◽  
Reduced Order ◽  
Nonlinear Model Reduction ◽  
Nonlinear Approximations ◽  
Output Error

For reliable, efficient, rapid simulations of dynamical systems, a reduced order model (ROM) with certified accuracy is highly desirable. The ROM is derived from a full order model (FOM) through model reduction. In this work, we propose an adaptive approach for nonlinear model reduction by making use of a suitable output error estimator, thus enable the generation of a compact ROM, with appropriate balance between the state and nonlinear approximations.

scholarly journals A Reduction Methodology Using Free-Free Component Eigenmodes and Arnoldi Enrichment

2015 ◽  
Author(s):  
Hadrien Tournaire ◽  
Franck Renaud ◽  
Jean-Luc Dion
Keyword(s):  
Modal Analysis ◽  
Model Reduction ◽  
Reduction Method ◽  
Reduced Order Model ◽  
Reduced Model ◽  
Order Model ◽  
Reduced Order ◽  
Complex Problems

In order to perform faster simulations, the model reduction is nowadays used in industrial contexts to solve large and complex problems. However, the efficiency of such an approach is sometimes cut by the interface size of the reduced model and its reusability. In this article, we focus on the development of a reduction methodology for the build of modal analysis oriented and updatable reduced order model whose size is not linked to their contacting interface. In order to allow latter model readjusting, we impose the use of eigenmodes in the reduction basis. Eventually, the method introduced is coupled to an Arnoldi based enrichment algorithm in order to improve the accuracy of the reduced model produced. In the last section the proposed methodology is discussed and compared to the Craig and Bampton reduction method. During this comparison we observed that even when not enriched, our work enables us to recover the Craig and Bampton accuracy with partially updatable and smaller reduced order model.

Control relevant model reduction: a reduced order model for ‘model IV’ fluid catalytic cracking units

1994 ◽  
Vol 4 (1) ◽  
pp. 3-14 ◽  
Author(s):  
Frank Z. Tatrai ◽  
Paul A. Lant ◽  
Peter L. Lee ◽  
Cameron Ian T ◽  
Robert B. Newell
Keyword(s):  
Model Reduction ◽  
Catalytic Cracking ◽  
Reduced Order Model ◽  
Fluid Catalytic Cracking ◽  
Order Model ◽  
Relevant Model ◽  
Reduced Order

Delay-Dependent γ-Suboptimal H∞ Model Reduction for Neutral Systems With Time-Varying Delays

2005 ◽  
Vol 128 (2) ◽  
pp. 394-399 ◽  
Author(s):  
Qing Wang ◽  
James Lam ◽  
Shengyuan Xu ◽  
Liqian Zhang
Keyword(s):  
Model Reduction ◽  
Linear Matrix ◽  
Optimization Approach ◽  
Time Varying ◽  
Neutral Systems ◽  
Order Model ◽  
Reduced Order ◽  
Reduction Problem ◽  
Bounded Realness ◽  
Delay Dependent

In this paper, the model reduction problem of neutral systems with time-varying delays is studied with γ suboptimality under the H∞ measure. A delay-dependent bounded realness condition of the H∞ norm is given via linear matrix inequalities (LMIs). Based on such a condition, a sufficient condition to characterize the existence of the reduced-order models is given in terms of LMIs with inverse constraints. By employing a sequential convex optimization approach, a reduced-order model can be computed with H∞ error less than some prescribed scalar γ.

scholarly journals Projection-based model reduction with dynamically transformed modes

2020 ◽  
Vol 54 (6) ◽  
pp. 2011-2043
Author(s):  
Felix Black ◽  
Philipp Schulze ◽  
Benjamin Unger
Keyword(s):  
Model Reduction ◽  
Classical Model ◽  
Order Reduction ◽  
Basis Functions ◽  
Reduced Order Model ◽  
Order Model ◽  
Optimal Basis ◽  
Reduced Order ◽  
Dimensional Approximation ◽  
Special Case

We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions for a similar approximation quality. Analogously to the MFEM framework, the reduced model is designed to minimize the residual, which is also the basis for an a posteriori error bound. Moreover, since the dependence of the transformation operators on the reduced state is nonlinear, the resulting reduced order model is obtained by projecting the original evolution equation onto a nonlinear manifold. Furthermore, for a special case, we show a connection between our approach and the method of freezing, which is also known as symmetry reduction. Besides the construction of the reduced order model, we also analyze the problem of finding optimal basis functions based on given data of the full order solution. Especially, we show that the corresponding minimization problem has a solution and reduces to the proper orthogonal decomposition of transformed data in a special case. Finally, we demonstrate the effectiveness of our method with several analytical and numerical examples.

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