HNA is an absolute error method which tries to minimize $$ || G - G_r ||_H = \min $$$$ || V \ (G - G_r) \ W ||_H = \min $$ where V and W denote output and input weightings.

Inputs

G

LTI model to be reduced.

nr

The desired order of the resulting reduced order system Gr. If not specified, nr is chosen automatically according to the description of key "order".

…

Optional pairs of keys and values. "key1", value1, "key2", value2.

opt

Optional struct with keys as field names. Struct opt can be created directly or by function options. opt.key1 = value1, opt.key2 = value2.

Outputs

Gr

Reduced order state-space model.

info

Struct containing additional information.

info.n

The order of the original system G.

info.ns

The order of the alpha-stable subsystem of the original system G.

info.hsv

The Hankel singular values corresponding to the projection op(V)*G1*op(W), where G1 denotes the alpha-stable part of the original system G. The ns Hankel singular values are ordered decreasingly.

info.nu

The order of the alpha-unstable subsystem of both the original system G and the reduced-order system Gr.

info.nr

The order of the obtained reduced order system Gr.

Option Keys and Values

’order’, ’nr’

The desired order of the resulting reduced order system Gr. If not specified, nr is the sum of info.nu and the number of Hankel singular values greater than max(tol1, ns*eps*info.hsv(1);

’method’

Specifies the computational approach to be used. Valid values corresponding to this key are:

’descriptor’

Use the inverse free descriptor system approach.

’standard’

Use the inversion based standard approach.

’auto’

Switch automatically to the inverse free descriptor approach in case of badly conditioned feedthrough matrices in V or W. Default method.

’left’, ’v’

LTI model of the left/output frequency weighting. The weighting must be antistable. \(|| V \ (G-G_r) \dots ||_H = \min\)

’right’, ’w’

LTI model of the right/input frequency weighting. The weighting must be antistable. \(|| \dots (G-G_r) \ W ||_H = \min\)

’left-inv’, ’inv-v’

LTI model of the left/output frequency weighting. The weighting must have only antistable zeros. \(|| V^{-1}(G-G_r) \dots ||_H = \min\)

’right-inv’, ’inv-w’

LTI model of the right/input frequency weighting. The weighting must have only antistable zeros. \(|| \dots (G-G_r)\,W^{-1} ||_H = \min\)

’left-conj’, ’conj-v’

LTI model of the left/output frequency weighting. The weighting must be stable. \(|| conj(V)\,(G-G_r) \dots ||_H = \min\)

’right-conj’, ’conj-w’

LTI model of the right/input frequency weighting. The weighting must be stable. \(|| \dots (G-G_r) \ conj(W) ||_H = \min\)

’left-conj-inv’, ’conj-inv-v’

LTI model of the left/output frequency weighting. The weighting must be minimum-phase. \(|| conj(V^{-1}) \, (G-G_r) \dots ||_H = \min\)

’right-conj-inv’, ’conj-inv-w’

LTI model of the right/input frequency weighting. The weighting must be minimum-phase. \(|| \dots (G-G_r) \ conj(W^{-1}) ||_H = \min\)

’alpha’

Specifies the ALPHA-stability boundary for the eigenvalues of the state dynamics matrix G.A. For a continuous-time system, ALPHA <= 0 is the boundary value for the real parts of eigenvalues, while for a discrete-time system, 0 <= ALPHA <= 1 represents the boundary value for the moduli of eigenvalues. The ALPHA-stability domain does not include the boundary. Default value is 0 for continuous-time systems and 1 for discrete-time systems.

’tol1’

If ’order’ is not specified, tol1 contains the tolerance for determining the order of the reduced model. For model reduction, the recommended value of tol1 is c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. tol1 < 1. If ’order’ is specified, the value of tol1 is ignored.

’tol2’

The tolerance for determining the order of a minimal realization of the ALPHA-stable part of the given model. tol2 <= tol1 < 1. If not specified, ns*eps*info.hsv(1) is chosen.

’equil’, ’scale’

Boolean indicating whether equilibration (scaling) should be performed on system G prior to order reduction. Default value is true if G.scaled == false and false if G.scaled == true. Note that for MIMO models, proper scaling of both inputs and outputs is of utmost importance. The input and output scaling can not be done by the equilibration option or the prescale function because these functions perform state transformations only. Furthermore, signals should not be scaled simply to a certain range. For all inputs (or outputs), a certain change should be of the same importance for the model.

Approximation Properties:

• Guaranteed stability of reduced models
• Lower guaranteed error bound
• Guaranteed a priori error bound $$ \sigma_{r+1} \leq || (G-G_r) ||_{\infty} \leq 2 \sum_{j=r+1}^{n} \sigma_j $$

Algorithm
Uses SLICOT AB09JD, Copyright (c) 1996-2025, SLICOT, available under the BSD 3-Clause (License and Disclaimer).

Source Code: hnamodred