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8.1.2 Identification Using Input-Output Data (Subspace System Identification Method)
The method implemented here is due to Moonen, De Moor, Vandenberghe, and Vandewalle (1989). The method is based on the singular value decomposition of the Hankel matrix  , where  and  are defined directly from the input and output data as follows:
 , 
If the input is persistently exciting of order  , then the systems of order  are identifiable. The best results are obtained when the input is white noise. When the number of available measurements is large, so that  , system identification becomes robust with respect to the measurement errors.
Time-domain system identification using the output response.
All the options of the function ImpulseResponseIdentify can be used and they have the same meaning.
Make sure the application is loaded.
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Load the collection of test examples.
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This is a discrete-time state-space model of a steam power system.
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The ratios of each Hankel singular value to the largest one identify the strong and weak modes.
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Here is a number of measurements that is enough to identify the system.
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Make sure NormalDistribution is available.
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This constructs the white noise input.
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This simulates the output response with random initial conditions.
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The multiplicative white noise of amplitude  is introduced into the impulse response.
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The two weaker modes of this state-space system are obliterated by the added noise.
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This Bode plot denotes the original and identified models by solid and dashed lines, respectively.
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