Near-optimal frequency-weighted interpolatory model reduction

Abstract

This paper develops an interpolatory framework for weighted-${\mathcal{H}}_2$ model reduction of MIMO dynamical systems. A new representation of the weighted-${\mathcal{H}}_2$ inner products in MIMO settings is introduced and used to derive associated first-order necessary conditions satisfied by optimal weighted-${\mathcal{H}}_2$ reduced-order models. Equivalence of these new interpolatory conditions with earlier Riccati-based conditions given by Halevi is also shown. An examination of realizations for equivalent weighted-${\mathcal{H}}_2$ systems leads then to an algorithm that remains tractable for large state-space dimension. Several numerical examples illustrate the effectiveness of this approach and its competitiveness with Frequency Weighted Balanced Truncation and an earlier interpolatory approach, the Weighted Iterative Rational Krylov Algorithm.

Introduction

Consider a multiple input/multiple output (MIMO) linear dynamical system having a state-space realization (which will be presumed minimal) given by

$$\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$$$$y\left(t\right)=Cx\left(t\right)+Du\left(t\right)$$

where $A\in {\mathbb{R}}^{n\times n},B\in {\mathbb{R}}^{n\times m},C\in {\mathbb{R}}^{p\times n}$, and $D\in {\mathbb{R}}^{p\times m}$ are constant matrices. The variables $x\left(t\right)\in {\mathbb{R}}^n$, $u\left(t\right)\in {\mathbb{R}}^m$ and $y\left(t\right)\in {\mathbb{R}}^p$ are, respectively, the state, the input, and the output of the system. The transfer function of this system is $G\left(s\right)=C{(sI-A)}^{-1}B+D\in {\mathbb{C}}^{p\times m}$. Following common usage, the underlying system will also be denoted by $G$. The circumstances of interest for us presume very large state-space dimensions relative to the input/output dimensions, $n\gg m,p$. This leads to fundamental difficulties for any task that involves optimization or control of this system. This in turn motivates model reduction: finding a reduced order model (ROM),

$${\dot{x}}_r\left(t\right)=A_r{x}_r\left(t\right)+B_{r}u\left(t\right)\text{,}$$$$y_r\left(t\right)=C_r{x}_r\left(t\right)+D_{r}u\left(t\right)$$

with an associated transfer function $G_r\left(s\right)=C_r{(sI-A_r)}^{-1}{B}_r+D_r$ where $A_r\in {\mathbb{R}}^{n_r\times n_r},B_r\in {\mathbb{R}}^{n_r\times m},C_r\in {\mathbb{R}}^{p\times n_r}$, and $D_r\in {\mathbb{R}}^{p\times m}$. The goal is to produce a greatly reduced state-space dimension, $n_r\ll n$, yet still assure that $y_r\left(t\right)\approx y\left(t\right)$ over a large class of inputs $u\left(t\right)$. This is accomplished by requiring $G_r\left(s\right)$ to approximate $G\left(s\right)$ very well, in an appropriate sense, which we interpret as making $G_r\left(s\right)-G\left(s\right)$ small with respect to an appropriate system norm.

For example, one may consider approximations that attempt to minimize either the ${\mathcal{H}}_2$-error:

$${\Vert G-G_r\Vert }_{{\mathcal{H}}_2}\stackrel{\text{def}}{=}{\left(\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }{\Vert G\left(ı\omega \right)-G_r\left(ı\omega \right)\Vert }_F^{2}d\omega \right)}^{1/2}\text{,}$$

or the ${\mathcal{H}}_{\infty }$-error:

$${\Vert G-G_r\Vert }_{{\mathcal{H}}_{\infty }}\stackrel{\text{def}}{=}\underset{\omega \in \mathbb{R}}{sup}{\Vert G\left(ı\omega \right)-G_r\left(ı\omega \right)\Vert }_2\text{.}$$

Here ${\Vert M\Vert }_F^2={\sum }_{i,j}{\left|m_{ij}\right|}^2$ denotes the Frobenius norm and ${\Vert M\Vert }_2$ denotes the spectral norm of the matrix $M$. Notice that to ensure that the first error measure is even finite, it is necessary that $D_r=D$.

“Typical” inputs, $u\left(t\right)$, often will have their power concentrated in known frequency ranges, and so, some frequency ranges will naturally be more important than others with regard to ROM fidelity. This leads in a natural way to consideration of weighted system errors designed in such a way so as to enhance accuracy in certain frequency ranges while permitting larger errors at other frequencies, and towards that end we consider, weighted measures of system error such as

$${\Vert G_r-G\Vert }_{{\mathcal{H}}_2\left(W\right)}\stackrel{\text{def}}{=}{\Vert (G_r\left(s\right)-G\left(s\right))W\left(s\right)\Vert }_{{\mathcal{H}}_2}$$

and

$${\Vert G_r-G\Vert }_{{\mathcal{H}}_{\infty }\left(W\right)}\stackrel{\text{def}}{=}{\Vert (G_r\left(s\right)-G\left(s\right))W\left(s\right)\Vert }_{{\mathcal{H}}_{\infty }}$$

where $W\left(s\right)\in {\mathbb{R}}^{m\times m_w}$ is a given input weighting (a “shaping filter”). One may specify an output weighting as well, however in the interest of clarity and brevity, we do not do this here. We focus on weighted-${\mathcal{H}}_2$ measures of error so that for a given system, $G\in {\mathcal{H}}_2\left(W\right)$, one seeks a reduced system $G_r\in {\mathcal{H}}_2\left(W\right)$ solving:

$$G_r=\underset{\mathrm{ord}\left(\tilde{G}\right)\le n_r}{argmin}{\Vert G-\tilde{G}\Vert }_{{\mathcal{H}}_2\left(W\right)}\text{.}$$

A variety of shaping filters can be considered. For example, if $W\left(s\right)$ were to be chosen to be a transfer function associated with a band-pass filter then approximation errors at frequencies within the passband would be penalized, while approximation error at frequencies lying outside the passband would be discounted.

Another choice of shaping filter arises from controller reduction: Suppose we wish to stabilize a linear dynamical system, $\mathbb{G}$, having order $n_0$ with a controller, $G$, having order $n$, that connects to $\mathbb{G}$ via a feedback loop. Many control design methodologies will produce stabilizing controllers with order that will be no smaller than the order of the system being stabilized: $n\approx n_0$, (see e.g., [1], [2]). High-order systems generally lead to high-order controllers, and high-order controllers are generally undesirable because high order typically translates into complex hardware implementations that may be costly and exhibit degraded performance. Replacing $G$ with a reduced order controller, $G_r$, with order $n_r\ll n$ can address these shortcomings, however it often is not enough to simply require $G_r$ to be a good approximation to $G$. In order to accurately recover closed-loop performance, plant dynamics need to be taken into account during the reduction process. This may be achieved through frequency weighting: Given a stabilizing controller $G$, if a reduced model, $G_r$, has the same number of unstable poles as $G$ and

$${\Vert [G-G_r]\cdot \mathbb{G}{[I+\mathbb{G}G]}^{-1}\Vert }_{{\mathcal{H}}_{\infty }}<1\text{,}$$

then, if $G_r$ is used to replace $G$, $G_r$ will also be a stabilizing controller  [3], [2]. Seeking $G_r$ to minimize a weighted measure of ${\mathcal{H}}_2$ error as in (3) is an effective proxy, using the weight $W\left(s\right)=\mathbb{G}\left(s\right){[I+\mathbb{G}\left(s\right)G\left(s\right)]}^{-1}$. This approach has been considered in [1], [3], [4], [5], [6], [7], [8], [9], [10] and references therein, leading then to variants of frequency-weighted balanced truncation. Related methods in [11], [12], [13] are tailored instead towards minimizing a similarly weighted ${\mathcal{H}}_2$ error, as we do here.

The main contributions of this paper are threefold. First, we develop a new analysis framework through the introduction of a linear mapping from ${\mathcal{H}}_2\left(W\right)$ to ${\mathcal{H}}_2$ that gives a new representation of the weighted-${\mathcal{H}}_2$ inner product for MIMO systems. This representation allows us to rewrite the weighted-${\mathcal{H}}_2$ inner product as a regular (unweighted) ${\mathcal{H}}_2$ inner product and leads to interpolatory first-order necessary conditions for optimal weighted-${\mathcal{H}}_2$ approximation. This analysis framework allows us to extend the interpolatory conditions of  [14] for the SISO weighted-${\mathcal{H}}_2$ problem to the MIMO case, and more generally allows us greater flexibility in treating more general settings that involve non-trivial feedthrough terms, which play a crucial role in the weighted-${\mathcal{H}}_2$ problem. Second, we show that this new interpolation framework is equivalent to the Riccati-based formulation of Halevi  [11], thus assuring the accuracy of the Riccati-based optimality formulation at a much lower cost. Finally, via a detailed examination and a new state-space realization for equivalent weighted-${\mathcal{H}}_2$ systems, we propose a numerical algorithm for weighted-${\mathcal{H}}_2$ approximation that remains tractable for large state-space dimension. Unlike the heuristic algorithm introduced in  [14], which is inspired by optimality conditions but does not attempt to satisfy them, the algorithm proposed here is “near optimal” in the sense that it directly approximates the weighted optimality conditions and approaches true optimality as reduction order grows.

The rest of the paper is organized as follows: In Section  2, we introduce the new formulation for the weighted-${\mathcal{H}}_2$ inner product for MIMO systems based on a bounded linear transformation from ${\mathcal{H}}_2\left(W\right)$ to ${\mathcal{H}}_2$ with which we derive interpolatory optimality conditions. The equivalence of these conditions to those of Halevi  [11] is proved in Section  3 followed in Section  4 by a description of a numerical algorithm for optimal weighted-${\mathcal{H}}_2$ approximation based on these conditions. Several numerical examples are given in Section  5; a summary and conclusions are offered in Section  6.