Non-projection methods

From Model order reduction site at MIT

Contents 1 Introduction 2 Hankel optimal model reduction. 3 State residualization, as opposed to state truncation 4 Transfer function fitting methods 4.1 Vector fitting and rational fitting algorithms 5 References

Introduction

This category of model reduction methods doesn't involve any projections. Therefore, state-space of the resultant approximations has no connection to the unreduced system. The most widely used methods from the category of linear non-projection type are the following:

1. Hankel optimal model reduction
2. State-residualization (or singular-perturbation approximation)
3. Transfer function fitting methods.

Hankel optimal model reduction.

This kind of reduction is well described in the original article ^[1], and is applicable only to LTI systems in form (A, B, C, D). This method is quite involved, both from the theoretical and computational perspectives. As the starting point, such reduction requires obtaining a balanced realization. This reduction has twice less theoretical a-priori error bound (measured in the Hankel norm) than the balanced-truncation:

where σ_i(H(s)) denotes i-th Hankel singular value, and H_r(s) denotes a Hankel-optimal reduced model of order q.

State residualization, as opposed to state truncation

As we have mentioned before, all projection-based methods effectively used some state-space transformation, after which they partitioned the state-space in the following way:

where the states in the vector x2 are considered "unimportant", and explicitly set to zero:

x₂ = 0

The alternative to this is a state residualization, which, instead, sets the derivatives of x₂ to zero.

Let's assume that we have partitioned the initial system (A, B, C, D) conformally with the states x₁ and x₂. By setting , after elimination of x₂, we obtain the following reduced system:

One can see that any projection-based reduced system (i.e. obtained by state truncation, preserving matrix D) preserves the transfer function at the infinite frequency, whereas state residualization preserves the value of transfer function at the zero frequency.

If the state residualization is performed on the balanced system, then the reduced model posesses the same error bound, as for the balanced truncation ^[2]

Transfer function fitting methods

These methods are aiming at approximating the transfer function of the original system directly. Usually, the input to such methods are samples of the original transfer function .

Some of the benefits of such approach:

1. The original system may not necessarily be in the state-space form; one can extract the frequency points by using, for example, a field solver. Even if the original system is in the state-space form, usually initial model is sparse, therefore obtaining a frequency response is a fast, and embarassingly parallel problem.
2. The reduced orders (for small number of input and output signals) are usually very small, outperforming projection-based models.
3. Since such methods operate directly with rational approximations, the output state-space model usually has a sparse form (for example, tri-diagonal in the vector-fitting algorithm), which is easy to simulate.
4. The output may not necessarily be a state-space model! For example, one can seek an approximation in the form of a pure delay in series with the state-space.
5. The algorithm complexity does not directly depend on the complexity of the original system.
6. One can easily control the frequency bandwidth of interest and introduce frequency weighting.

Drawbacks of such approach:

1. Such methods do not have a-priori error bounds, without any assumptions on the nature of the underlying system.
2. Choice of the reduced order is unknown, and it's difficult to guess which order is sufficient for the approximation, so usually, the error is being estimated a-posteriori.
3. Obtaining a higher-order model cannot make use of any results from the lower-order model.

Methods of this class include:

1. Vector fitting algorithm, and it's equivalent rational fitting algorithm^[3],^[4]
2. Convex-optimization based approach of K.C.Sou et al. ^[5]

Vector fitting and rational fitting algorithms

Vector-fitting algorithm, first introduced in ^[3], tries to find a rational approximation of the single-input single-output sampled transfer function in the pole-residue form:

It can be readily generalized to the system having a single input and multiple outputs (SIMO). The method requires starting pole locations.

Rational-fitting algorithm ^[4] tries to find the rational approximation in the alternative form:

where P_i(s) denotes some polynomials of order i, constructed specifically to avoid ill-conditioning in the least-squares fit. This method has the same properties as the vector-fitting algorithm, however it does not need the starting poles.

References

1. ↑ K. Glover, "All Optimal Hankel Norm Approximation of Linear Multivariable Systems, and Their L-infinity error Bounds," Int. J. Control, vol. 39, no. 6, pp. 1145-1193, 1984.
2. ↑ Y. Liu, B. Anderson "Singular Perturbation Approximation of Balanced systems", Proceedings of the 28^th conference on Decision and Control (1989).
3. ↑ ^3.1 3.2 B. Gustavsen and A. Semlyen, "Rational approximation of frequency domain responses by vector fitting", IEEE Trans. Power Delivery, vol. 14, no. 3, pp. 1052-1061, July 1999
4. ↑ ^4.1 4.2 C. P. Coelho, J. R. Phillips, and L. M. Silveira, “Robust rational function approximation algorithm for model generation,” in Proc. 36th Design Automation Conf. New Orleans, LA, June 1999, pp. 207-21
5. ↑ Kin Cheong Sou; Megretski, A.; Daniel, L., "A quasi-convex optimization approach to parameterized model order reduction", Proceedings of the 42^nd Design Automation Conference, pp.933 - 938, 2005