Model Order Reduction: Techniques and Tools
Abstract
Model order reduction (MOR) is here understood as a computational technique to reduce the order of a dynamical system described by a set of ordinary or differentialalgebraic equations (ODEs or DAEs) to facilitate or enable its simulation, the design of a controller, or optimization and design of the physical system modeled. It focuses on representing the map from inputs into the system to its outputs, while its dynamics are treated as a black box so that the largescale set of describing ODEs/DAEs can be replaced by a much smaller set of ODEs/DAEs without sacrificing the accuracy of the inputtooutput behavior.
Problem Description
To simplify the description, only continuoustime systems are considered here. The discretetime case can be treated mostly analogously; see, e.g., Antoulas (2005).
 \(\G(\cdot ) \widetilde{ G}(\cdot )\_{\mathcal{H}_{\infty }}\), whereHere, σ _{max} is the largest singular value of the matrix F(s). This minimizes the maximal magnitude of the frequency response of the error system and by the PaleyWiener theorem bounds the \(\mathcal{L}_{2}\)norm of the output error.$$\F(.)\_{\mathcal{H}_{\infty }} =\mathrm{ sup}_{s\in \mathbb{C}_{+}}\sigma _{\max }(F(s)).$$
 \(\G(\cdot ) \widetilde{ G}(\cdot )\_{\mathcal{H}_{2}}\), where (with \(\imath = \sqrt{1}\))This ensures a small error \(\y(\cdot ) \tilde{ y}(\cdot )\_{\mathcal{L}_{\infty }(0,\infty )} =\mathrm{ sup}_{t>0}\y(t) \tilde{ y}(t)\_{\infty }\) (with ∥ . ∥ _{ ∞ } denoting the maximum norm of a vector) uniformly over all inputs u(t) having bounded \(\mathcal{L}_{2}\)energy, that is, \(\int _{0}^{\infty }u{(t)}^{T}u(t)dt \leq 1\); see Gugercin et al. (2008).$$\F(\cdot )\_{\mathcal{H}_{2}}^{2} = \frac{1} {2\pi }\displaystyle\int _{\infty }^{+\infty }\mathrm{tr}\left (F{(\imath \omega )}^{{\ast}}F(\imath \omega )\right )d\omega .$$
Besides a small approximation error, one may impose additional constraints for the ROM. One might require certain properties (such as stability and passivity) of the original systems to be preserved. Rather than considering the full nonnegative real line in time domain or the full imaginary axis in frequency domain, one can also consider bounded intervals in both domains. For these variants, see, e.g., Antoulas (2005) and Obinata and Anderson (2001).
Methods
In the following, we will briefly discuss the main classes of methods to construct suitable matrices V and W: truncationbased methods and interpolationbased methods. Other methods, in particular combinations of the two classes discussed here, can be found in the literature. In case the original LTI system is real, it is often desirable to construct a real reducedorder model. All of the methods discussed in the following either do construct a real reducedorder system or there is a variant of the method which does. In order to keep this exposition at a reasonable length, the reader is referred to the cited literature.
Truncation Based Methods
More suitable reducedorder systems can be obtained by balanced truncation. To introduce this concept, we no longer need to assume A to be diagonalizable, but we require the stability of A in the sense of (6). For simplicity, we assume E = I. For treatment of the DAE case (E≠I), see Benner et al. (2005, Chap. 3). Loosely speaking, a balanced representation of an LTI system is obtained by a change of coordinates such that the states which are hard to reach are at the same time those which are difficult to observe. This change of coordinates amounts to an equivalence transformation of the realization (A, B, C, D) of (1) called statespace transformation as in (7), where T now is the matrix representing the change of coordinates. The new system matrices (T ^{− 1} AT, T ^{− 1} B, CT, D) form a balanced realization of (1). Truncating in this balanced realization the states that are hard to reach and difficult to observe results in a ROM.
In balanced coordinates the Gramians P and Q of a stable minimal LTI system satisfy \(P = Q =\mathrm{ diag}(\sigma _{1},\ldots ,\sigma _{n})\) with the Hankel singular values \(\sigma _{1} \geq \sigma _{2} \geq \ldots \geq \sigma _{n} > 0.\) The Hankel singular values are the positive square roots of the eigenvalues of the product of the Gramians PQ, \(\sigma _{k} = \sqrt{\lambda _{k } (PQ)}.\) They are system invariants, i.e., they are independent of the chosen realization of (1) as they are preserved under statespace transformations.

Compute the Cholesky factors S and R of the Gramians such that P = S ^{ T } S, Q = R ^{ T } R.

Compute the singular value decomposition of \(S{R}^{T} = \Phi \Sigma {\Gamma }^{T}\), where Φ and Γ are orthogonal matrices and Σ is a diagonal matrix with the Hankel singular values on its diagonal. \(T = {S}^{T}\Phi {\Sigma }^{\frac{1} {2} }\) yields the balancing transformation (note that \({T}^{1} = {\Sigma }^{\frac{1} {2} }{\Phi }^{T}{S}^{T} = {\Sigma }^{\frac{1} {2} }{\Gamma }^{T}R\)).
 Partition \(\Phi ,\Sigma ,\Gamma \) into blocks of corresponding sizes,with \(\Sigma _{1} =\mathrm{ diag}(\sigma _{1},\ldots ,\sigma _{r})\) and apply T to (1) to obtain (7) with$$\Sigma = \left [\begin{array}{@{}c@{\quad }c@{}} \Sigma _{1}\quad & \\ \quad &\Sigma _{2}\end{array} \right ],\ \ \Phi = \left [\begin{array}{c} \Phi _{1} \\ \Phi _{2}\end{array} \right ],\ \ {\Gamma }^{T} = \left [\begin{array}{c} \Gamma _{1}^{T} \\ \Gamma _{2}^{T}\end{array} \right ],$$and \(CT = \left [CV \ C_{2}\right ]\) for \(W = {R}^{T}\Gamma _{1}\Sigma _{1}^{\frac{1} {2} }\) and \(V = {S}^{T}\Phi _{1}\Sigma _{1}^{\frac{1} {2} }.\) Preserving the r dominant Hankel singular values by truncating the rest yields the reducedorder model as in (5).$${ T}^{1}AT = \left [\begin{array}{@{}c@{\quad }c@{}} {W}^{T}AV \quad &A_{12} \\ A_{21} \quad &A_{22}\end{array} \right ], {T}^{1}B = \left [\begin{array}{c} {W}^{T}B \\ B_{2}\end{array} \right ],$$(10)
As the explicit computation of the balancing transformation T is numerically hazardous, one usually uses the equivalent balancingfree square root algorithm (Varga 1991) in which orthogonal bases for the column spaces of V and W are computed. The so obtained ROM is no longer balanced, but preserves all other properties (error bound, stability). Furthermore, it is shown in Benner et al. (2000) how to implement the balancingfree square root algorithm using lowrank approximations to S and R without ever having to resort to the square solution matrices P and Q of the Lyapunov equations (9). This yields an efficient algorithm for balanced truncation for LTI systems with large dense matrices. For systems with largescale sparse A efficient algorithms based on sparse solvers for (9) exist; see Benner (2006).
By replacing the solution matrices P and Q of (9) by other pairs of positive (semi)definite matrices characterizing alternative controllability and observability related system information, one obtains a family of model reduction methods including stochastic/boundedreal/positivereal balanced truncation. These can be used if further properties like minimum phase, passivity, etc. are to be preserved in the reducedorder model; for further details, see Antoulas (2005) and Obinata and Anderson (2001).
Besides SPA, another related truncation method that is not based on projection is optimal Hankel norm approximation (HNA). The description of HNA is technically quite involved; for details, see Zhou et al. (1996) and Glover (1984). It should be noted that the so obtained ROM usually has similar stability and accuracy properties as for balanced truncation.
InterpolationBased Methods
Another family of methods for MOR is based on (rational) interpolation. The unifying feature of the methods in this family is that the original TFM (2) is approximated by a rational matrix function of lower degree satisfying some interpolation conditions (i.e., the original and the reducedorder TFM coincide, e.g., \(G(s_{0}) =\widetilde{ G}(s_{0})\) at some predefined value s _{0} for which A − s _{0} E is nonsingular). Computationally this is usually realized by certain Krylov subspace methods.
The classical approach is known under the name of momentmatching or Padé(type) approximation. In these methods, the transfer functions of the original and the reduced order systems are expanded into power series, and the reducedorder system is then determined so that the first coefficients in the series expansions match. In this context, the coefficients of the power series are called moments, which explains the term moment matching.
Theoretically, the matrix V (and W) can be computed by explicitly forming the columns which span the corresponding Krylov subspace \(\mathcal{K}_{k}(F,H)\) and using the GramSchmidt algorithm to generate unitary basis vectors for \(\mathcal{K}_{k}(F,H).\) The forming of the moments (the Krylov subspace blocks F ^{ j } H) is numerically precarious and has to be avoided under all circumstances. Using Krylov subspace methods to achieve an interpolationbased ROM as described above is recommended. The unitary basis of a (block) Krylov subspace can be computed by employing a (block) Arnoldi or (block) Lanczos method; see, e.g., Antoulas (2005), Golub and Van Loan (2013), and Freund (2003).
In the case when an oblique projection is to be used, it is not necessary to compute two unitary bases as above. An alternative is then to use the nonsymmetric Lanczos process (Golub and Van Loan 2013). It computes biunitary (i.e., W ^{∗} V = I _{ r }) bases for the above mentioned Krylov subspaces and the reducedorder model as a byproduct of the Lanczos process. An overview of the computational techniques for momentmatching and Padé approximation summarizing the work of a decade is given in Freund (2003) and the references therein.
In general, the discussed MOR approaches are instances of rational interpolation. When the expansion point is chosen to be s _{0} = ∞, the moments are called Markov parameters and the approximation problem is known as partial realization. If s _{0} = 0, the approximation problem is known as Padé approximation.
As the use of one single expansion point s _{0} leads to good approximation only close to s _{0}, it might be desirable to use more than one expansion point. This leads to multipoint momentmatching methods, also called rational Krylov methods; see, e.g., Ruhe and Skoogh (1998), Antoulas (2005), and Freund (2003).
In contrast to balanced truncation, these (rational) interpolation methods do not necessarily preserve stability. Remedies have been suggested; see, e.g., Freund (2003).
The use of complexvalued expansion points will lead to a complexvalued reducedorder system (3). In some applications (in particular, in case the original system is real valued), this is undesired. In that case one can always use complexconjugate pairs of expansion points as then the entire computations can be done in real arithmetic.
The methods just described provide good approximation quality locally around the expansion points. They do not aim at a global approximation as measured by the \(\mathcal{H}_{2}\) or \(\mathcal{H}_{\infty }\)norms. In Gugercin et al. (2008), an iterative procedure is presented which determines locally optimal expansion points w.r.t. the \(\mathcal{H}_{2}\)norm approximation under the assumption that the order r of the reduced model is prescribed and only 0th and 1storder derivatives are matched. Also, for multiinput multioutput systems (i.e., m and p in (1) are both larger than one), no full moment matching is achieved, but only tangential interpolation: \(G(s_{j})b_{j} =\widetilde{ G}(s_{j})b_{j},\) \(c_{j}^{{\ast}}G(s_{j}) = c_{j}^{{\ast}}\widetilde{G}(s_{j}),\) \(c_{j}^{{\ast}}G'(s_{j})b_{j} = c_{j}^{{\ast}}\widetilde{G}'(s_{j})b_{j},\) for certain vectors b _{ j }, c _{ j } determined together with the optimal s _{ j } by the iterative procedure.
Tools
Almost all commercial software packages for structural dynamics include modal analysis/truncation as a means to compute a ROM. Modal truncation and balanced truncation are available in the MATLAB^{®;} Control System Toolbox and the MATLAB^{®;} Robust Control Toolbox.
Numerically reliable, welltested, and efficient implementations of many variants of balancingbased MOR methods as well as Hankel norm approximation and singular perturbation approximation can be found in the Subroutine Library In Control Theory (SLICOT, http://www.slicot.org) (Varga 2001). Easytouse MATLAB interfaces to the Fortran 77 subroutines from SLICOT are available in the SLICOT Model and Controller Reduction Toolbox (http://slicot.org/matlabtoolboxes/basiccontrol); see Benner et al. (2010). An implementation of moment matching via the (block) Arnoldi method is available in MOR for ANSYS^{®;}(http://modelreduction.com/Software.html).

Oberwolfach Model Reduction Benchmark Collectionhttp://simulation.unifreiburg.de/downloads/benchmark/

NICONET Benchmark Exampleshttp://www.icm.tubs.de/NICONET/benchmodred.html
The MOR WiKi http://morwiki.mpimagdeburg.mpg.de/morwiki/ is a platform for MOR research and provides discussions of a number of methods, links to further software packages (e.g., MOREMBS and MORPACK), as well as additional benchmark examples.
Summary and Future Directions
MOR of LTI systems can now be considered as an established computational technique. Some open issues still remain and are currently investigated. These include methods yielding good approximation in finite frequency or time intervals. Though numerous approaches for these tasks exist, methods with sharp local error bounds are still desirable. A related problem is the reduction of closedloop systems and controller reduction. Also, the generalization of the methods discussed in this essay to descriptor systems (i.e., systems with DAE dynamics), secondorder systems, or unstable LTI systems has only been partially achieved. An important problem class getting a lot of current attention consists of (uncertain) parametric systems. Here it is important to preserve parameters as symbolic quantities in the ROM. Most of the current approaches are based in one way or another on interpolation. MOR for nonlinear systems has also been a research topic for decades. Still, the development of satisfactory methods in the context of control design having computable error bounds and preserving interesting system properties remains a challenging task.
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