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8.2 Frequency Domain System Identification
The frequency domain state-space system identification problem is to identify the state-space model from a given set of frequency responses. Thus, given a set of frequency responses  ,  , ... at  , where  ,  is the period of the discretization, and  are the frequencies, and a set of the associated frequency weights  , the problem is to find matrices  ,  ,  , and  such that the estimated frequency responses
are close to  . Mathematically, the problem is to minimize the error index
where  is a suitable weight.
The method implemented in FrequencyResponseIdentify is based on the singular value decomposition of a Hankel matrix of Markov parameters that are constructed from the coefficient matrices of the left matrix fraction (see, e.g., Juang (1994)) as
Here  is taken to be the smallest order of the approximation that is necessary to identify the system with the number of states limited by the option MaxOrder. Once the coefficient matrices are determined by solving the least-squares problem, the Markov parameters  are determined as
 for  and
 for 
Frequency domain system identification.
Special emphasis can be given to certain frequencies by using the option FrequencyWeights, which specifies the list of weights to be used for the least-squares fitting. FrequencyWeights Automatic specifies the frequencies with the same weight. All the options of the function ImpulseResponseIdentify can be used and they have the same meaning as in the time-domain case.
Options for the frequency domain system identification.
Make sure the application is loaded.
In[1]:=
Load the collection of test examples.
In[2]:=
This is a discrete-time state-space system of a simple process control of a paper machine.
In[3]:=
Out[3]=
Here is the transfer function of the defined system.
In[4]:=
Out[4]=
This is an a priori estimate of the order of the state-space system to be identified.
In[5]:=
Here is the number of measurements that is enough to identify the system.
In[6]:=
This is the sampling rate of the system.
In[7]:=
Out[7]=
Here are some selected frequencies.
In[8]:=
These points in the  -plane correspond to the selected frequencies.
In[9]:=
The frequency points are coupled with the corresponding transfer matrices.
In[10]:=
This identifies the state-space system from its sampled transfer function.
In[11]:=
Out[11]=
Here is the transfer function of the identified state-space system.
In[12]:=
Out[12]=
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