Frequency Domain System Identification
System identification deals with the problem of identifying a model to accurately describe the response of a physical system to some input. This worksheet uses a spring-mass-damper system to illustrate the problem where the structure of the model is known and the parameters of the model are to be identified.
Identifying the model parameters that best describe the physical system is accomplished by exciting the system with a realistic input signal. The resulting output is then converted to the frequency domain and the parameters are estimated using a least-squares approximation approach.
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System Definition
Signal Generation and Simulation
Discrete Fourier Transform Calculations
Model-Based Parameter Estimation
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System Definition
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The parameters, variables, and equations that define the spring-mass-damper model are found in the following sections:
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Parameter Definition
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Name
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Value
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Units
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Parameters
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Mass of the object
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Damping coefficient ()
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Spring constant
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Simulation time
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Sampling time
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Number of samples
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Noise standard deviation
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Retrieve parameters
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Variable Definition
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Name
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Description
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Input Variables
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Input force on the mass
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Output Variables
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Output position of the mass
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Model Definition
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Signal Generation and Simulation
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Excitation Input
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System Response
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To excite the system, we apply a discrete chirp signal that sweeps the frequency spectrum from 0.01 Hz to 1 Hz over 50 seconds.
noise is added to reflect a realistic application.
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The system response to can be obtained using the DynamicSystems[Simulate] command. The response can be seen in the following plot.
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Discrete Fourier Transform Calculations
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The results of converting the input, output and model signals to the frequency domain can be seen in the plots below. The Maple commands used to generated the Discrete Fourier Transform are found in the code edit region.
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Model-Based Parameter Estimation
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The Maple commands used to obtain the model-based parameter values are found in the following code edit region.
Parameter Estimation Proc
Using Maple's optimization routines, the parameter values that best describe the physical model were found to be:
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The difference in parameter values between those measured and those obtained through the estimation process are shown below.
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The plot below shows the frequency response of the measured and the estimated model.
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