SPA is an absolute error method which tries to minimize 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 of the alpha-stable part of the original system G, 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 chosen automatically such that states with Hankel singular values info.hsv > tol1 are retained.

’left’, ’output’

LTI model of the left/output frequency weighting V. Default value is an identity matrix.

’right’, ’input’

LTI model of the right/input frequency weighting W. Default value is an identity matrix.

’method’

Approximation method for the L-infinity norm to be used as follows:

’sr’, ’s’

Use the square-root Singular Perturbation Approximation method.

’bfsr’, ’p’

Use the balancing-free square-root Singular Perturbation Approximation method. Default method.

’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]. Default value is info.ns*eps*info.hsv(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. If not specified, ns*eps*info.hsv(1) is chosen.

’gram-ctrb’

Specifies the choice of frequency-weighted controllability Grammian as follows:

’standard’

Choice corresponding to a combination method [4] of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method.

’enhanced’

Choice corresponding to the stability enhanced modified combination method of [4].

’gram-obsv’

Specifies the choice of frequency-weighted observability Grammian as follows:

’standard’

Choice corresponding to a combination method [4] of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method.

’enhanced’

Choice corresponding to the stability enhanced modified combination method of [4].

’alpha-ctrb’

Combination method parameter for defining the frequency-weighted controllability Grammian. abs(alphac) <= 1. If alphac = 0, the choice of Grammian corresponds to the method of Enns [1], while if alphac = 1, the choice of Grammian corresponds to the method of Lin and Chiu [2,3]. Default value is 0.

’alpha-obsv’

Combination method parameter for defining the frequency-weighted observability Grammian. abs(alphao) <= 1. If alphao = 0, the choice of Grammian corresponds to the method of Enns [1], while if alphao = 1, the choice of Grammian corresponds to the method of Lin and Chiu [2,3]. Default value is 0.

’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.

References
[1] Enns, D. Model reduction with balanced realizations: An error bound and a frequency weighted generalization. Proc. 23-th CDC, Las Vegas, pp. 127-132, 1984.

[2] Lin, C.-A. and Chiu, T.-Y. Model reduction via frequency-weighted balanced realization. Control Theory and Advanced Technology, vol. 8, pp. 341-351, 1992.

[3] Sreeram, V., Anderson, B.D.O and Madievski, A.G. New results on frequency weighted balanced reduction technique. Proc. ACC, Seattle, Washington, pp. 4004-4009, 1995.

[4] Varga, A. and Anderson, B.D.O. Square-root balancing-free methods for the frequency-weighted balancing related model reduction. (report in preparation)

Algorithm
Uses SLICOT AB09ID, Copyright (c) 2020, SLICOT, available under the BSD 3-Clause (License and Disclaimer).

Source Code: spamodred