Formula omitted] model reduction for uncertain switched linear discrete-time systems

In this paper, the problem of robust H∞ filtering for switched linear discrete-time systems with polytopic uncertainties is investigated. Based on the mode-switching idea and parameter-dependent stability result, a robust switched linear filter is designed such that the corresponding filtering error system achieves robust asymptotic stability and guarantees a prescribed H∞ performance index for all admissible uncertainties. The existence condition of such filter is derived and formulated in terms of a set of linear matrix inequalities (LMIs) by the introduction of slack variables to eliminate the cross coupling of system matrices and Lyapunov matrices among different subsystems. The desired filter can be constructed by solving the corresponding convex optimization problem, which also provides an optimal H∞ noise-attenuation level bound for the resultant filtering error system. A numerical example is given to show the effectiveness and the potential of the proposed techniques. Copyright © 2006 John Wiley & Sons, Ltd.

The reduction of the large order models of complex systems into a sufficiently low dimensional order in such a way that important dynamic characteristics of the real system are preserved, is a nontrivial task. The motivations for such a model order reduction are either to reduce computations for analysis and practical control design or to simplify the control system structure. In the present article, major techniques in model order reduction are briefly discussed. The characteristic similarities and differences are highlighted, their advantages and disadvantages are underlined, and further research requirements and trends are commented upon. Introduction and Perspective One of the central issues in the active control of complex systems such as large flexible space structures (LFSS) is the derivation of a correct mathematical model for both the controlled and the uncontrolled dynamical systems. Theoretically, there are infinitely many elastic modes or degrees of freedom (DOF) in the distributed parameter (DP) models of LFSS, usually with very low natural damping. Moreover, the flexible modes contribute to the actual deformation of the structure.' The truly infinite dimensional character of LFSS models has to be approximated by some fidelity finite (but usually very large) dimensional model. The normal approach taken by engineers to achieve this end is via modal models with a large number of modes that provide a reasonable representation of the spacecraft dynamic characteristics. However, a difficult problem still remains to be the development of a model of lowenough dimensional order that it can be utilized by the onboard controller, yet high enough dimensional order that it preserves the dynamic characteristics of the real system represented and controlled. The motivations for such a reduction are either to reduce computations for analysis and practical control design or to simplify the control system structure. Discretization procedures will not be discussed herein and it will be assumed that a large finite dimensional model has been generated somehow and for implementation and other practical considerations, the model needs to be further reduced. *This work was-supported, in part, by the Air Force Wright Aeronautical, Air Force Systems Command, Flight Dynamics Lab, Structures and Dynamics Division, while the author was employed with HR-Textron, Valencia, California. There are various approaches to model order reduction; some are or pseudo-optimal and others are ad-hoc methods based on practical considerations and engineering experience and judgment. These techniques are known by different names such as reduction, condensation, economization, aggregation, optimal projection, and other combinations thereof. It has often been pointed out that such techniques generally constitute application of Rayleigh-Ritz/Galerkin optimization, and matrix transformation methods to the eigenvalue/eigenvector formulation for structural dynamic problems. The reduction or the condensation methods are based on transformations of the coordinates in the equations of motion that essentially maintain the invariance of the quadratic forms of the potential and kinetic energies. An important feature of these methods is that the reduced order model often loses the basic characteristics of the original system. Considerable progress was made by Likins, Ohkami, and wong5 in this respect. However, they failed to develop a general enough criterion that could reduce the system model in an sense without significantly affecting the eigenvalues of the original model. The optimization or the mathematical reduction procedure, on the other hand, is based on the reduction of the eigenvalue problem to a smaller size based on some optimality criterion (usually quadratic). In these techniques, the reductions are carried out without actually resorting to approximations, i .e. , without truncatiag any coordinates or states of the original system. Balanced model reduction of linear timeinvariant dynamical systems is essentially based on the controllability and observability relations of the states of the system. Subsystem models are obtained by deleting those states that contribute the least to the controllability and observability (or the impulse response) of the original system7 and thus are only in this sense. A more recent approach to model reduction is proposed in Skelton's work,* where each state of the system model is assigned a relative to a given basis, via a quadratic criterion, and the states with the least cost are deleted in a systematic manner. The resulting reduced model is a function of the state-space basis, and thus there is no guarantee for optimality for all choices. The latest development in model order reduction techniques is the work by Hyland and erns stein.^ Herein first order necessary conditions for reduced order modeling of linear time-invariant systems are derived via a pair of modified Lyapunov equations coupled by a nonorthogonal projection. This approach reveals the possibility of multiple extrema forsome of the abovementioned methods. Copyright @ 1988 by the American Institute of Aeronautics and Astronautics. Inc. All Rights Reserved.

In this paper, an algorithm for order reduction of linear multivariable systems is proposed using the combined advantages of the dominant pole retention method and the error minimisation by Particle Swarm Optimisation (PSO) technique. The denominator of the reduced order transfer function matrix is obtained by retaining the dominant poles of the original system while the numerator terms of the lower order transfer matrix are determined by minimising the Integral Square Error (ISE) in between the transient responses of original and reduced order models using PSO technique. The proposed algorithm guarantees stability of the reduced order transfer function matrix if the original High Order System (HOS) is stable and is having superior features, including easy implementation, stable convergence characteristic and good computational efficiency. The proposed algorithm has been applied successfully to the transfer function matrix of a 10th order two-input two-output linear time invariant model of a practical power system. The performance of the algorithm is tested by comparing the relevant computer simulation results.

In this paper, we present four projection-based model order reduction algorithms grounded in Laguerre functions, specifically tailored for linear systems with quadratic outputs. The first algorithm utilises a Galerkin projection strategy, where the projection matrix is construced from the Laguerre expansion coefficient vectors of the system's state impulse response. The resulting reduced-order model accurately matches selected Laguerre expansion coefficients of the original system's impulse response output. The remaining three algorithms implement Petrov-Galerkin projections combined with the low-rank square root balancing method. While these methods share an identical low-rank factor for the controllability Gramian, they differ in their selection of the observability Gramian, which is chosen from either the linearised system, the quadratic-bilinearized system, or the original linear system with quadratic outputs. A notable advantage of these algorithms is their ability to avoid solving large-scale Lyapunov equations. Instead, all low-rank factors are directly constructed from the Laguerre expansion coefficient vectors via recurrence formulas, substantially improving computational efficiency. The effectiveness and performance of the proposed algorithms are demonstrated through a series of numerical experiments.

In this paper differentiation method is proposed for driving reduced order model (ROM) of Multi-input multi - output (MIMO) linear dynamic system. The proposed method is based on differentiation of polynomials numerator and denominators. The reduction procedure is very simple given better response when compared with other reduction techniques and original high order system. Also, the proposed method guarantees given stability of the reduced models of MIMO system if the original high order MIMO system is stable. The proposed approach of MIMO system order reduction model is illustrated with the help of an Two-input and Two-output (2×2) transfer function matrix and the results of MIMO transfer function matrix are compared with the recently published reduction modal order techniques.

The paper extends the Routh approximation method for one-dimensional (1-D) discrete systems to two-dimensional (2-D) discrete systems for finding stable reduced-order models from a stable high-order 2-D linear discrete separable-denominator system (SDS). The extension is achieved by exploring new properties of the 1-D Routh canonical model and establishing new 2-D bilinear Routh canonical models. Without explicitly performing bilinear transformations, a computationally-efficient procedure is presented for finding the bilinear Routh reduced-order models. The properties of the obtained 2-D bilinear Routh approximants are discussed in detail. In addition, a new 2-D bilinear Routh canonical state-space realisation is presented from which the low-dimensional state-space models corresponding to the bilinear Routh approximants can be obtained by a direct truncation procedure. Furthermore, the relationships among the states of the bilinear Routh reduced-dimension model, the aggregated model, and the original system are explored. Numerical examples are given to demonstrate the effectiveness of the proposed method.

This paper is concerned with the problem of ℋ ∞ model reduction for linear distributed-delay systems. For a given stable system containing both discrete and distributed delays in the state, our attention is focused on the construction of reduced-order model of the same structure, which guarantees the corresponding error system to be asymptotically stable and has a prescribed H∞ norm error performance. This problem is solved through a delay-dependent approach. By defining a new Lyapunov functional and making use of the Newton-Leibniz formula, a delay-dependent bounded real lemma is first established in terms of linear matrix inequalities (LMIs). Based on this performance criterion and by employing the projection lemma, sufficient conditions are formulated for the existence of admissible reduced-order models. Owing to the presence of matrix inverse constraints, these conditions are intrinsically nonconvex. An algorithm based on the sequential linear programming matrix method is proposed for verifying these conditions, which can be readily implemented by using standard numerical software. Desired reduced-order models can be constructed if these conditions are satisfied. The effectiveness of the proposed model reduction method is illustrated via a numerical example.

On Padé-type model order reduction of J-Hermitian linear dynamical systems

A structure-preserving machine learning framework for accurate prediction of structural dynamics for systems with isolated nonlinearities

Optimal model reduction via delay approximation

A numerical comparison of simplified Galerkin and machine learning reduced order models for vaginal deformations

A Linear Quadratic Regulator (LQR)-based least-squares output feedback control procedure using a complex mode procedure is developed for the optimal vibration control of high-order asymmetric discrete system. An LQ Regulator is designed for a reduced-order model obtained by neglecting high-frequency complex modes of the original system. The matrix transformations between physical coordinates and complex mode coordinates are derived. The complex mode approach appears to provide more accurate reduced-order models than the normal mode approach for asymmetric discrete systems. The proposed least-squares output feedback control procedure takes advantage of the fact that a full-state feedback control is possible without using an observer. In addition, the lateral vibration of a high-order rotor system can be effectively controlled by monitoring one single location along the rotor shaft, i.e., the number of measured states can be much less than the number of eigenvectors retained in producing the reduced-order model while acceptable performance of the controller is maintained. The procedure is illustrated by means of a 52 degree-of-freedom finite element based rotordynamic system. Simulation results show that LQ regulators based on a reduced-order model with 12 retained eigenvalues can be accurately approximated by using feedback of four measured states from one location along the rotor shaft. The controlled and uncontrolled transient responses, using various numbers of measured states, of the original high-order system are shown. Comparisons of reduced-order model results using normal modes and complex modes are presented. The spillover problem is discussed for both collocated and noncollocated cases based on this same example.

The ability to conduct fast and reliable simulations of dynamic systems is of special interest to many fields of operations. Such simulations can be very complex and, to be thorough, involve millions of variables, making it prohibitive in CPU time to run repeatedly for many different configurations. Reduced-Order Modeling (ROM) provides a concrete way to handle such complex simulations using a realistic amount of resources. However, when the original dynamical system is very large, the resulting reduced-order model, although much “thinner”, is still as tall as the original system, i.e., it has the same number of equations. In some extreme cases, the number of equations is prohibitive and cannot be loaded in memory. In this work, we combine traditional interval constraint solving techniques with a strategy to reduce the number of equations to consider. We describe our approach and report preliminary promising results.

SUMMARY This paper deals with the problem of finite frequency H∞ full-order filter design for discrete-time and continuous-time linear systems, with polytopic uncertainties. Based on the generalized Kalman–Yakubovich–Popov lemma and a parameter-dependent Lyapunov function, a set of sufficient conditions are established in terms of matrix inequalities, ensuring that the filtering error system is stable and the H∞ attenuation level, from disturbance to the estimation error, is smaller than a given value over a prescribed finite frequency domain of the external disturbances. Then, in order to linearize and relax the obtained matrix inequalities, we introduce a large number of slack variables by applying Finsler's lemma twice, which provides extra degrees of freedom in optimizing the guaranteed H∞ performance. This leads to performance improvement and reduction of conservatism in the solution. It is shown later that the robust filter gains can be obtained by solving a set of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness and the less conservatism of the proposed approach in comparison with the existing methods. Copyright © 2016 John Wiley & Sons, Ltd.

— In this paper, an effective procedure to determine the reduced order model of higher order linear time invariant discrete systems is discussed. A new procedure has been proposed for evaluating Time moments of the original high order system. Numerator and denominator polynomials of reduced order model are obtained by considering first few redefined time moments of the original high order system. The proposed method has been verified using numerical examples. Keywords- order reduction; eigen values; large scale discrete systems; modeling. I.