3.3 Gramian-based methods
3.4.5 H 2 -optimal interpolation-based model reduction for SISO
We now come to the model reduction technique which was alluded to in the introduction of the Subsection 3.3.4. In the case of SISO systems, this method constructs the reduced-order model that satisfies the interpolation-based first order necessary H2-optimality conditions. These conditions, established in 1967 by Meier and Luenberger [78] are stated in the next theorem.
Theorem 3.4.4 (Meier-Luenberger conditions for SISO continuous systems).
Given an asymptotically stable SISO continuous-time LTI dynamical system with transfer functionH(s) =C(sIN−A)−1B.LetH(s)ˆ be a local minimizer of dimension n for the optimal H2 model reduction problem (Problem 3.3.2) and suppose thatH(s)ˆ has simple poles at λˆk, k = 1, . . . , n. ThenH(s)ˆ interpolates both H(s) and its first derivative at −ˆλ∗k, k= 1, . . . , n:
H−ˆλ∗k= ˆH−ˆλ∗k, H−ˆλ∗k= ˆH−λˆ∗k,
9=
; for k = 1, . . . , n. (3.4.13)
Recently, Gugercin et al. [57, 58] presented a new proof for Theorem 3.4.4 and also established the first order necessaryh2-optimality conditions for discrete-time systems.
Theorem 3.4.5 (Meier-Luenberger conditions for SISO discrete systems).
Given an asymptotically stable SISO discrete-time LTI dynamical system with transfer function H(s) = C(sIN −A)−1B. Let H(s)ˆ be a local minimizer of dimension n for the optimal h2 model reduction problem (Problem 3.3.3) and suppose thatH(s)ˆ has simple poles at λˆk, k = 1, . . . , n. ThenH(s)ˆ interpolates both H(s) and its first derivative at ˆ1
λ∗k, k= 1, . . . , n: H
ˆ1 λ∗k
= ˆH
ˆ1 λ∗k
, H
ˆ1 λ∗k
= ˆH
ˆ1 λ∗k
,
9>
=
>; for k = 1, . . . , n. (3.4.14)
An efficient algorithm that, if it converges, satisfies the first order necessary conditions is given in Figure 3.3. We stress that the interpolation-based first order necessary conditions presented in this subsection as well as the IRKA algorithm are applicable only for asymptotically stable SISO systems with simple poles. We established analogous results for SISO and MIMO systems with multiple poles [24, 103]. Similar results were simultaneously and inde-pendently obtained by Gugercin et al. [58] and Van Dooren et al. [36]. In this thesis, we establish a further generalization of these results to stable and unstable MIMO systems with simple poles.
3.4. INTERPOLATION-BASED METHODS 57 An Iterative Rational Krylov Algorithm (IRKA):
1. Choose initial shifts σk, fork = 1, . . . , n.
2. Choose Vn and Wn such that Wn∗Vn =In and
columnspan(Vn) = span{(σ1IN −A)−1B, . . . ,(σnIN −A)−1B} columnspan(Wn) = span{(σ1IN −A∗)−1C∗, . . . ,(σnIN −A∗)−1C∗} 3. While (not converged)
a) Compute ˆA=Wn∗AVn
b) Compute eigenvalues ˆλ1, . . . ,ˆλn of matrix ˆA c) Assign
•for continuous-time systems: σk =−λˆ∗k, k = 1, . . . , n
•for discrete-time systems: σk= ˆ1
λ∗k, k = 1, . . . , n d) Update Vn and Wn so that Wn∗Vn=In and
columnspan(Vn) = span{(σ1IN −A)−1B, . . . ,(σnIN −A)−1B} columnspan(Wn) = span{(σ1IN −A∗)−1C∗, . . . ,(σnIN −A∗)−1C∗} 4. ˆA=Wn∗AVn,Bˆ =Wn∗B,Cˆ =CVn
5. ˆH(s) = ˆC(sIn−A) ˆˆ B.
Figure 3.3: An iterative rational Krylov-based model reduction method (IRKA) [57, 58].
References
Several books [4, 30, 18, 82, 100], dozens of dissertations, e.g. [13, 15, 22, 54, 56, 61, 71, 83, 101, 113], and surveys, e.g. [5, 8, 46, 52, 112], hundreds of technical reports, e.g. [24, 27, 34, 38, 39, 57, 103, 105], and articles, e.g.
[9, 10, 17, 21, 34, 86], has been written in the area of model order reduction.
It is thus understandable that the list of model reduction techniques discussed in this thesis can by no way be exhaustive. We presented either the best known and the most commonly used methods or the ones that are of interest in next chapters. Some techniques, like e.g. Guyan reduction [88], Laguerre methods [29], proper orthogonal decomposition [7, 75] or procedures suitable for second and higher order systems [28, 72, 73, 94], have not been discussed.
For the recent account of theory and rather complete study of model reduction techniques we refer the reader to [4, 100] and references therein.
Chapter 4
H 2 ,α -optimal model order reduction
4.1 Introduction
Minimizing theH2-norm of the error between the original asymptotically stable LTI models and its asymptotically stable reduced-order approximants is con-sidered to be one of the oldest approaches to model order reduction. It has been extensively investigated in the last forty years, see e.g. [11, 12, 47, 60, 95, 98, 109] and references therein. The problem of finding the global optimum is a very difficult task. Furthermore, in the case of MIMO LTI systems, up to authors knowledge, the existence of such a global minimum has not yet been clarified. Therefore, most of current methods try to find a reduced-order model satisfying the first order necessary conditions. Among many papers de-voted to this problem are the notable contributions of Meier and Luenberger [78], Wilson [79, 106, 107] and Hyland and Bernstein [64], which are of special interest in this thesis (see also Section 3.3.4 and 3.4.5).
Since theH2-norm is not defined for unstable systems, all theH2-norm optimal model reduction techniques are only designed for approximating stable sys-tems. But some processes, such as weather development, are in their nature unstable and the unstable part plays a crucial role. On the other hand, some asymptotically stable systems have several poles lying very close to imaginary axis. For such systems some transformations, e.g. balancing, can lead to a system which due to rounding errors is not stable any more. A common approach here is to divide the system into stable and unstable subsystems.
Then the unstable part is retained and reduction techniques are applied only to the stable part. There are two main disadvantages of this approach. On the one hand the process of dividing the system into stable and unstable parts
59
is computationally expensive for large-scale dynamical systems. On the other hand, reducing only the stable part is not very beneficial when the dimension of the unstable part of the system is large. In the latter case, if we choose the size of the reduced model to be smaller than the size of unstable part, then the stable part is entirely removed. Additionally, the unstable subsystem is reduced in such a way that it is difficult to quantify how much information is being lost.
The main contribution of this dissertation is the development of a new tan-gential Hermite interpolation-based model reduction technique which aims to minimize theH2,α-norm of the error between the original and the reduced-order systems. One major advantage of this technique is that it can be applied to both, stable and unstable systems. In the latter case, the aforementioned diffi-culties are avoided. Moreover, the method can be used for reducing not only SISO systems, which for a long time was a restriction for interpolation-based H2-optimal methods, but also MIMO LTI systems.
We introduce here the concept ofα-boundedness which will play a crucial role in the model reduction technique, established in this thesis.
Definition 4.1.1(α-boundedness). A continuous-time LTI dynamical system of the form (3.1.1) is calledα-bounded, if all eigenvalues λ1, . . . , λN of the state matrix A are elements of the set C−α := {s ∈ C : (s) < α}, i.e. (λk) < α for all k= 1, . . . , N.
Throughout the thesis, and denote the real and the imaginary part of a complex number, respectively.
It is clear that the poles of any LTI model considered in this thesis are all fi-nite. Therefore, there exists a finiteα ∈R such that the system isα-bounded.
Consequently, the H2,α-norm of the continuous-time LTI system exists and is defined via (2.3.11). In Section 4.2, we will give a simple proof for a use-ful alternative expression for this norm. Next, the H2,α-norm optimization problem is formally stated. In Subsection 4.3.2, based on the new formula of the norm, the tangential Hermite interpolation-based first order necessary conditions for this problem are formulated. Moreover, a MIMO iterative ra-tional interpolation algorithm, MIRIAm, aiming to satisfy the aforementioned conditions, is presented in Section 4.3.3. The potential of the new method is illustrated by several experiments with a model of the compact disc player and compared to some other existing methods. In view of the fact that the H2-optimal model reduction problem can be considered as a special case of the H2,α-optimal model reduction, the two sets of gramian-based first order neces-sary conditions of Wilson [79, 106, 107] and Hyland-Bernstein [64] are extended to the case ofα-bounded systems. We conclude this chapter proving that these generalized gramian-based conditions are equivalent to theH2,α-optimal con-ditions developed in this thesis.