The optimal $\mathcal{H}_2$ model reduction problem is of great importance in the area of dynamical systems and simulation. In the literature, two independent frameworks have evolved focusing either on solution of Lyapunov equations on the one hand or interpolation of transfer functions on the other, without any apparent connection between the two approaches. In this paper, we develop a new unifying framework for the optimal $\mathcal{H}_2$ approximation problem using best approximation properties in the underlying Hilbert space. This new framework leads to a new set of local optimality conditions taking the form of a structured orthogonality condition. We show that the existing Lyapunov- and interpolation-based conditions are each equivalent to our conditions and so are equivalent to each other. Also, we provide a new elementary proof of the interpolation-based condition that clarifies the importance of the mirror images of the reduced system poles. Based on the interpolation framework, we describe an iteratively corrected rational Krylov algorithm for $\mathcal{H}_2$ model reduction. The formulation is based on finding a reduced order model that satisfies interpolation-based first-order necessary conditions for $\cHtwo$ optimality and results in a method that is numerically effective and suited for large-scale problems. We illustrate the performance of the method with a variety of numerical experiments and comparisons with existing methods.

/pdf/mathcal-h-2-model-reduction-for-large-scale-linear-dynamical-xoow2ovn5b.pdf

$\mathcal{H}_2$ Model Reduction for Large-Scale Linear Dynamical Systems

The problem of passing from a linear time-invariant high-order controller designed for a linear time-invariant plant (of presumably high order) to a low-order approximation of the controller is discussed. The approximation problem is often best posed as a frequency-weighted L/sub infinity / approximation problem. Many different controller representations are possible, giving different performances of the various reduction algorithms. >

Controller reduction: concepts and approaches

There is much theory for the use of covariance matrices in both identification and in state estimation. However, there exists no theory for the control of covariances. The need for a theory of covariance control may be argued from two points: 1) Many engineering systems have performance requirements which are naturally stated in terms of root-mean-square (RMS) values of the system states or outputs and 2) the various theories of identification, estimation, and model reduction use covariances as a measure of performance. Hence a theory on covariance control may help unify the modeling and control problem. This paper introduces a theory for designing linear feedback controllers so that the closed loop system achieves a specified state covariance.

A covariance control theory

First-order necessary conditions for quadratically optimal reduced-order modeling of linear time-invariant systems are derived in the form of a pair of modified Lyapunov equations coupled by an oblique projection which determines the optimal reduced-order model. This form of the necessary conditions considerably simplifies previous results of Wilson [1] and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of Moore [2]. The possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation-type algorithm is proposed for computing these local extrema. A component-cost analysis of the model-error criterion similar to the approach of Skelton [3] is utilized at each iteration to direct the algorithm to the global minimum.

/pdf/the-optimal-projection-equations-for-model-reduction-and-the-l64ohcw7xl.pdf

The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore

A new and direct approach to stochastic model reduction is developed. The order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic Riccati equations. Also the concept of balanced stochastic realization (BSR) plays a fundamental role. Asymptotic stability of the reduced-order realization is established, and spectral domain interpretations for the BSR are given.

A transformation approach to stochastic model reduction

A brief summary of component cost analysis (CCA) is given. CCA provides a means to determine the contribution of each dynamical element of the system in the overall performance metric. Using CCA a theory of cost-equivalent realizations is developed. Cost-equivalent realizations are realizations that have the same value of the performance metric as the original system, and they can be of lower order than Kalman's minimal realization.

/pdf/component-cost-analysis-of-large-scale-systems-3g0d7xbago.pdf

Component cost analysis of large scale systems

Linear system approximation via covariance equivalent realizations

The standard linear quadratic Gaussian (LQG) optimal controller is not minimal in general, even if the plant is minimal. Hence, the order of the optimal controller can be less than that of the plant. This note shows that the original balancing method of Moore [1] can be used to generate these reduced order optimal controllers, but the newer balancing methods recently proposed for controller reduction might not generate the reduced order optimal controllers, even when they exist.

A note on balanced controller reduction

Covariance Equivalent Realizations with Application to Model Reduction of Large-Scale Systems

The ideas of cost decomposition is summarized to aid in the determination of the relative cost (or 'price') of each component of a linear dynamic system using quadratic performance criteria. In addition to the insights into system behavior that are afforded by such a component cost analysis CCA, these CCA ideas naturally lead to a theory for cost-equivalent realizations.

/pdf/component-cost-analysis-of-large-scale-systems-39pzb9g64e.pdf

A. Yousuff

Papers

Component cost analysis of large scale systems

Linear system approximation via covariance equivalent realizations

A note on balanced controller reduction

Covariance Equivalent Realizations with Application to Model Reduction of Large-Scale Systems

Component Cost Analysis of Large-Scale Systems