ℋ ∞ model reduction of linear systems with distributed delay

The problem of model reduction of linear systems with certain interconnection structure is considered in this paper. To preserve the interconnection structure between subsystems in the reduction, special care needs to be taken. This problem is important and timely because of the recent focus on complex networked systems in control engineering. Two different model reduction methods are introduced and compared in this paper. Both methods are extensions to the well‐known balanced truncation method. Compared with earlier work in the area these methods use a more general linear fractional transformation framework, and utilize linear matrix inequalities. Furthermore, new approximation error bounds that reduce to classical bounds in special cases are derived. The so‐called structured Hankel singular values are used in the methods, and indicate how important states in the subsystems are with respect to a chosen input–output map for the entire interconnected system. It is shown how these structured Hankel singular values can be used to select an approximation order. Finally, the two methods are applied to a model of a mechanical device. Copyright © 2008 John Wiley & Sons, Ltd.

This paper investigates the problem of Hankel norm model reduction for linear systems with time-varying delay in the state. For a given stable system, our attention is focused on the construction of reduced-order model, which guarantees the corresponding error system to be asymptotically stable and has a specified Hankel norm error performance. Two different approaches are proposed to solve this problem. One casts the model reduction into a convex optimization problem by using a linearization procedure, and the other is based on the cone complementarity linearization idea, which casts the model reduction into a sequential minimization problem subject to linear matrix inequality constraints. Both continuous and discrete time cases are considered. A numerical example is provided to show the effectiveness of the proposed theory.

For model reduction, the approximation performance sometimes needs to be enhanced over a specific frequency range. Motivated by this fact, the paper investigates generalized $$H_{\infty }$$H? model reduction for stable two-dimensional (2-D) discrete systems represented by the Roesser model and the Fornasini---Machesini local state-space model, respectively. The generalized $$H_{\infty }$$H? norm of 2-D systems is introduced to evaluate the approximation error over a specific finite frequency (FF) domain. In light of the 2-D generalized Kalman---Yakubovich---Popov lemmas, sufficient conditions in terms of linear matrix inequalities are derived for the existence of a stable reduced-order model satisfying a specified generalized $$H_{\infty }$$H? level. Several examples are provided to illustrate the effectiveness and advantages of the proposed method. Compared with most of the existing 2-D model reduction results, the proposed method has the following merits: (1) The generalized $$H_\infty $$H? model reduction problems are considered for both important types of 2-D models, and no structural assumption is made for the plant model, so that our method has a broader applicable scope. (2) The reduced-order model is guaranteed to be stable, and an upper bound on the generalized $$H_{\infty }$$H? error is provided. Moreover, no frequency weighting function is needed. (3) The proposed method is applicable for 2-D model reduction with multiple FF specifications.

We discuss algorithms for balanced truncation (BT) based model reduction of linear systems. BT is known to have good global approximation properties and to preserve important system properties. A computable error bound allows to choose the order of the reduced-order model adaptively. We will emphasize those aspects that makes the application of BT to models arising in circuit simulation a non-straightforward task. In recent years, these issues have been addressed by several authors. We will survey some of these developments and demonstrate that BT is now suitable for linear descriptor systems encountered in circuit simulation.

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.

H∞ Control of Distributed and Discrete Delay Systems via Discretized Lyapunov Functional

This paper considered the state estimation for stochastic neural networks of neutral type with discrete and distributed delays. By using available output measurements, the state estimator can approximate the neuron states, and the asymptotic property of the state error is mean square exponential stable and also almost surely exponential stable in the presence of discrete and distributed delays. Under the Lipschitz assumptions for the activation functions and the measurement nonlinearity, a delay-dependent linear matrix inequality (LMI) criterion is proposed to guarantee the existence of the desired estimators by constructing an appropriate Lyapunov-Krasovskii function. It is shown that the existence conditions and the explicit expression of the state estimator can be parameterised in terms of the solution to a LMI. Finally, two numerical examples are presented to demonstrate the validity of the theoretical results and show that the theorem can provide less conservative conditions.

Model Order Reduction for Magneto-Quasistatic Equations

Discrete bilinear stochastic systems with time-varying delay: Stability analysis and control synthesis

This paper is concerned with the problem of reliable \(\mathrm H_2\) control for discrete-time T-S fuzzy bilinear systems with actuator faults and infinite-distributed delays. A discrete-time homogeneous Markov chain is used to represent the stochastic behavior of actuator faults. Based on a stochastic fuzzy Lyapunov function, the fuzzy observer-based controllers are designed for T-S fuzzy bilinear delay systems, two improved stability conditions are proposed to ensure that the resultant closed-loop system is exponentially stable in the mean-square sense with an \(\mathrm H_2\) performance index. It is shown the fuzzy observer-based controller can be obtained by solving a set of nonlinear minimization problem involving linear matrix inequalities (LMIs) constraints. An iterative algorithm making use of sequential linear programming matrix method (SLPMM) to derive a single-step LMI condition for fuzzy observer-based control design. Finally, an illustrative example is provided to demonstrate the effectiveness of the results proposed in this paper.

Improved delay-dependent robust passivity criteria for uncertain neural networks with discrete and distributed delays

<p style='text-indent:20px;'>We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

A note on control of a class of discrete-time stochastic systems with distributed delays and nonlinear disturbances

Proper orthogonal decomposition (POD) finds an orthonormal basis yielding an optimal reconstruction of a given dataset. We consider an optimal data reconstruction problem for two general datasets related to balanced POD, which is an algorithm for balanced truncation model reduction for linear systems. We consider balanced POD outside of the linear systems framework, and prove that it solves the optimal data reconstruction problem. The theoretical result is illustrated with an example.

We address the problem of model order reduction (MOR) of parametrized dynamical systems. Motivated by reduced basis (RB) methods for partial differential equations, we show that some characteristic components can be transferred to model reduction of parametrized linear dynamical systems. We assume an affine parameter dependence of the system components, which allows an offline/online decomposition and is the basis for efficient reduced simulation. Additionally, error control is possible by a posteriori error estimators for the state vector and output vector, based on residual analysis and primal-dual techniques. Experiments demonstrate the applicability of the reduced parametrized systems, the reliability of the error estimators and the runtime gain by the reduction technique. The a posteriori error estimation technique can straightforwardly be applied to all traditional projection-based reduction techniques of non-parametric and parametric linear systems, such as model reduction, balanced truncation, moment matching, proper orthogonal decomposition (POD) and so on.