Frequency Domain System Identification

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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.

System Definition

•	Parameter Definition

•	Variable Definition

•	Model Definition

Signal Generation and Simulation

Discrete Fourier Transform Calculations

Model-Based Parameter Estimation

System Definition

The parameters, variables, and equations that define the spring-mass-damper model are found in the following sections:

Parameter Definition

Name	Value	Units
Parameters
Mass of the object
Damping coefficient ()
Spring constant
Simulation time
Sampling time
Number of samples
Noise standard deviation

Retrieve parameters

Variable Definition

Name	Description
Input Variables
	Input force on the mass
Output Variables
	Output position of the mass

Model Definition

M*(diff(diff(y(t), t), t))+b*(diff(y(t), t))+k*y(t) = u(t)

(1)

(2)

(3)

[[[`Diff. Equation`],[`discrete; sampletime = .5e-1`],[`1 output(s); 1 input(s)`],[`inputvariable = `[u(q)]],[`outputvariable = `[y(q)]],[de` = `[400. M y(q)-800. M y(q+1)+400. M y(q+2)-20. b y(q)+20. b y(q+1)+k y(q)=u(q)]]]

(4)

Signal Generation and Simulation

Excitation Input

System Response

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.

InputSignal := Chirp(1, .99*(1/50), 0.1e-1, hertz = true, discrete = true, samplecount = Ns, sampletime = Ts):

noise is added to reflect a realistic application.

NoiseVector := convert([seq(Statistics[Sample](NoiseRV, i)[1], i = 1 .. Ns)], Vector[column]):

The system response to can be obtained using the DynamicSystems[Simulate] command. The response can be seen in the following plot.

OutputResponse := Simulate(MassSpringDamperSys, NoisyInput, parameters = ParameterList, initialconditions = InitialConditions)[]:

Discrete Fourier Transform Calculations

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.