Function File: [Gr, info] = spamodred (G, …)
Function File: [Gr, info] = spamodred (G, nr, …)
Function File: [Gr, info] = spamodred (G, opt, …)
Function File: [Gr, info] = spamodred (G, nr, opt, …)
Model order reduction by frequency weighted Singular Perturbation Approximation (SPA).
The aim of model reduction is to find an LTI system Gr of order
nr (nr < n) such that the input-output behaviour of Gr
approximates the one from original system G.
SPA is an absolute error method which tries to minimize
$$ || G - G_r ||_{\infty} = \min $$$$ || V \ (G - G_r) \ W ||_{\infty} = \min $$
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 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
-
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.
-
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.
-
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.
-
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) 1996-2025, SLICOT, available under the BSD 3-Clause
(License and Disclaimer).
Source Code:
spamodred