ssequiv
Description
ssequiv
performs an equivalence transformation on a
state-space model.
Input Model | Transformed Model |
---|---|
|
|
Here:
The left-transformation matrix is defined as .
The right-transformation matrix is defined as .
e1
ande2
are either –1 or 1. That is, TL isT1
or its inverse and TR isT2
or its inverse.
Examples
Equivalence Transformation for State-Space Model
Perform an equivalence transform for a state space model.
Generate a random state-space model and a transformation matrix.
rng(0) sys = rss(5); T1 = randn(5); rng(2) T2 = randn(5);
Perform the following transformation on sys
.
sysT = ssequiv(sys,T1,-1,T2,1);
Plot the frequency response of both models.
bode(sys,'b',sysT,'r--') legend
ans = Legend (sys, sysT) with properties: String: {'sys' 'sysT'} Location: 'northeast' Orientation: 'vertical' FontSize: 8.1000 Position: [0.8059 0.8814 0.1486 0.0884] Units: 'normalized' Use GET to show all properties
The responses of both models match closely.
Equivalence Transformation for Generalized State-Space Models
ssequiv
applies state transformation only to the state vectors of the numeric portion of the generalized model.
Create a genss
model.
sys = rss(2,2,2) * tunableSS('a',2,2,3) + tunableGain('b',2,3)
Generalized continuous-time state-space model with 2 outputs, 3 inputs, 4 states, and the following blocks: a: Tunable 2x3 state-space model, 2 states, 1 occurrences. b: Tunable 2x3 gain, 1 occurrences. Type "ss(sys)" to see the current value and "sys.Blocks" to interact with the blocks.
Specify transformation matrices.
T1 = [1 -2;3 5]; T2 = T1;
Apply the transformation
.
tsys = ssequiv(sys,T1,1,T2,-1);
Compare this transformed model with the model from decomposed tsys
.
sigma(sys,tsys,"r--")
The responses of both models match closely.
Input Arguments
sys
— Dynamic system
dynamic system model
Dynamic system, specified as a SISO, or MIMO dynamic system model. Dynamic systems that you can use include:
Continuous-time or discrete-time numeric LTI models, such as
ss
ordss
models.Generalized or uncertain LTI models, such as
genss
oruss
(Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)For such models, the state transformation is applied only to the state vectors of the numeric portion of the model. For more information about decomposition of these models, see
getLFTModel
and Internal Structure of Generalized Models.Identified state-space
idss
(System Identification Toolbox) models. (Using identified models requires System Identification Toolbox™ software.)
If sys
is an array of state-space models,
ssequiv
applies the transformation T
to each
individual model in the array.
T1
— Left-transformation matrix
matrix | []
Left-transformation matrix, specified as an
n-by-n square invertible matrix, where
n is the number of states. The left-transformation matrix is
defined as , that is, T1
or its inverse.
To omit the left transformation, specify T1
and
e1
as []
.
For example, the following code transforms A,
B, C, E of
sys
to ATT,
B, CTT,
ETT.
sys = ssequiv(sys,[],[],T',1);
T2
— Right-transformation matrix
matrix | []
Right-transformation matrix, specified as an
n-by-n square invertible matrix, where
n is the number of states. The right-transformation matrix is
defined as , that is, T1
or its inverse.
To omit the right transformation, specify T2
and
e2
as []
.
For example, the following code transforms A,
B, C, E of
sys
to T-1A,
T-1B, C,
T-1E.
sys = ssequiv(sys,T,-1,[],[]);
Output Arguments
sysT
— Transformed model
dynamic system model
Transformed state-space model, returned as a dynamic system model of the same type
as sys
.
Version History
Introduced in R2023b
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