Structured parameterization lets you exclude specific parameters from estimation by setting these parameters to specific values. This approach is useful when you can derive state-space matrices from physical principles and provide initial parameter values based on physical insight. You can use this approach to discover what happens if you fix specific parameter values or if you free certain parameters.
There are two stages to the structured estimation procedure:
Specify the state-space model structure, as described in Specify the State-Space Model Structure
Estimate the free model parameters, as described in Estimate State-Space Models at the Command Line
This approach differs from estimating models with free and canonical parameterizations, where it is not necessary to specify initial parameter values before the estimation. For free parameterization, there is no structure to specify because it is assumed to be unknown. For canonical parameterization, the structure is fixed to a specific form.
Note: To estimate structured state-space models in the System Identification app, define the corresponding model structures at the command line and import them into the System Identification app. |
To specify the state-space model structure:
Use idss
to
create a state-space model. For example:
A = [0 1; 0 -1]; B = [0; 0.28]; C = eye(2); D = zeros(2,1); m = idss(A,B,C,D,K,'Ts',T)
creates a discrete-time state-space structure, where A
, B
, C
, D
,
and K
specify the initial values for the free parameters. T
is
the sample time.
Use the Structure
property of the
model to specify which parameters to estimate and which to set to
specific values.
For example, if you want to fix A(1,2)=A(2,1)=0
,
use:
m.Structure.a.Value(1,2) = 0; m.Structure.a.Value(2,1) = 0; m.Structure.a.Free(1,2) = false; m.Structure.a.Free(2,1) = false;
The estimation algorithm only estimates the parameters in A
for
which m.Structure.a.Free
is true
.
Use physical insight, whenever possible, to initialize the parameters
for the iterative search algorithm. Because it is possible that the
numerical minimization gets stuck in a local minimum, try several
different initialization values for the parameters. For random initialization,
use init
. When the model structure
contains parameters with different orders of magnitude, try to scale
the variables so that the parameters are all roughly the same magnitude.
Alternatively, to quickly configure the parameterization and
whether to estimate feedthrough and disturbance dynamics, use ssform
.
Use ssest
to
estimate the model, as described in Estimate State-Space Models at the Command Line.
The iterative search computes gradients of the prediction errors
with respect to the parameters using numerical differentiation. The
step size is specified by the nuderst
command.
The default step size is equal to 10^{–4} times
the absolute value of a parameter or equal to 10^{–7},
whichever is larger. To specify a different step size, edit the nuderst
MATLAB^{®} file.
You estimate state-space models with
structured parameterization when you know some parameters of
a linear system and need to estimate the others. These models are
therefore similar to grey-box models. However, in this toolbox, the
"grey box modeling" terminology is used only when referring to idgrey
and idnlgrey
models.
In these models, you can specify complete linear or nonlinear models
with complicated relationships between the unknown parameters.
If you have independent unknown matrix elements in a linear
state-space model structure, then it is easier and quicker to use
state-space models with structured parameterizations. For imposing
dependencies, or to use more complex forms of parameterization, use
the idgrey
model and the associated greyest
estimator. For more information,
see Grey-Box Model Estimation.
This example shows how to estimate the unknown parameters of a discrete-time model.
In this example, you estimate in the following discrete-time model:
Suppose that the nominal values of the unknown parameters (
) are -1
, 2
, 3
, 4
,and 5
, respectively.
The discrete-time state-space model structure is defined by the following equation:
Construct the parameter matrices and initialize the parameter values using the nominal parameter values.
A = [1,-1;0,1]; B = [2;3]; C = [1,0]; D = 0; K = [4;5];
Construct the state-space model object.
m = idss(A,B,C,D,K);
Specify the parameter values in the structure matrices that you do not want to estimate.
S = m.Structure; S.a.Free(1,1) = false; S.a.Free(2,:) = false; S.c.Free = false; m.Structure = S;
D
is initialized, by default, as a fixed value, and K
and B
are initialized as free values. Suppose you want to fix the initial states to known zero values. To enforce this, configure the InitialState
estimation option.
opt = ssestOptions;
opt.InitialState = 'zero';
Load estimation data.
load iddata1 z1;
Estimate the model structure.
m = ssest(z1,m,opt);
where z1
is name of the iddata
object. The data can be time-domain or frequency-domain data. The iterative search starts with the nominal values in the A
, B
, C,
D
, and K
matrices.
This example shows how to estimate the unknown parameters of a continuous-time model.
In this example, you estimate in the following continuous-time model:
This equation represents an electrical motor, where is the angular position of the motor shaft, and is the angular velocity. The parameter is the inverse time constant of the motor, and is the static gain from the input to the angular velocity.
The motor is at rest at t=0
, but its angular position
is unknown. Suppose that the approximate nominal values of the unknown parameters are
and
.
The variance of the errors in the position measurement is 0.01
, and the variance in the angular velocity measurements is 0.1
. For more information about this example, see the section on state-space models in System Identification: Theory for the User, Second Edition, by Lennart Ljung, Prentice Hall PTR, 1999.
The continuous-time state-space model structure is defined by the following equation:
Construct the parameter matrices and initialize the parameter values using the nominal parameter values.
A = [0 1;0 -1]; B = [0;0.25]; C = eye(2); D = [0;0]; K = zeros(2,2); x0 = [0;0];
The matrices correspond to continuous-time representation. However, to be consistent with the idss
object property name, this example uses A
, B
, and C
instead of F
, G
, and H
.
Construct the continuous-time state-space model object.
m = idss(A,B,C,D,K,'Ts',0);
Specify the parameter values in the structure matrices that you do not want to estimate.
S = m.Structure; S.a.Free(1,:) = false; S.a.Free(2,1) = false; S.b.Free(1) = false; S.c.Free = false; S.d.Free = false; S.k.Free = false; m.Structure = S; m.NoiseVariance = [0.01 0; 0 0.1];
The initial state is partially unknown. Use the InitialState
option of the ssestOptions
option set to configure the estimation behavior of X0
.
opt = ssestOptions; opt.InitialState = idpar(x0); opt.InitialState.Free(2) = false;
Estimate the model structure.
load(fullfile(matlabroot, 'toolbox', 'ident', 'iddemos', 'data', 'dcmotordata')); z = iddata(y, u, 0.1); m = ssest(z, m, opt);
The iterative search for a minimum is initialized by the parameters in the nominal model m
. The continuous-time model is sampled using the same sample time as the data during estimation.
Simulate this system using the sample time T=0.1
for input u
and the noise realization e
.
e = randn(300,2); u1 = idinput(300); simdat = iddata([],u1,'Ts',0.1); simopt = simOptions('AddNoise', true, 'NoiseData', e); y1 = sim(m,simdat,simopt);
The continuous system is sampled using Ts=0.1
for simulation purposes. The noise sequence is scaled according to the matrix m.NoiseVariance
.
If you discover that the motor was not initially at rest, you can estimate
by setting the second element of the InitialState
parameter to be free.
opt.InitialState.Free(2) = true; m_new = ssest(z, m, opt);