Estimating a Partially Known Continuous Transfer Function Model (System Identification Toolkit)
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Filename: continuous_tf_model.zip
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This case study illustrates how to use prior knowledge about an RLC circuit to define and estimate a partially known continuous transfer function model for the RLC circuit. You can use the LabVIEW System Identification VIs to complete this task.
The following figure shows an RLC circuit, where R is the resistor, L is the inductor, C is the capacitor, iL is the inductor current, and uC is the capacitor voltage. This RLC circuit uses u as the input voltage and uC as the output voltage, y.
In this RLC circuit, R is 1.5 W. L and C are unknown but have approximate values of 0.1 H and 0.1 F, respectively. Complete the following steps to identify the values of L and C.
- Apply a wide-band voltage to u and measure the output y simultaneously.
- Define a model for this circuit. Because you have information about the approximate values of L and C, you can build a partially known state-space or partially known transfer function model.
- Estimate the model you defined in step 2 and then calculate L and C from the estimated model.
The Continuous Transfer Function Model of an RLC Circuit example VI guides you through defining and estimating a transfer function model for the RLC circuit. This example VI uses a chirp signal from 0.5 Hz to 6 Hz as the stimulus signal. The response to the chirp signal is the response signal.
Refer to the Continuous State-Space Model of an RLC Circuit example VI for information about defining and estimating a state-space model for the RLC circuit. You can access this example VI by selecting Help»Find Examples to display the NI Example Finder and then navigating to the Toolkits and Modules»System Identification folder.
This case study first guides you through defining a continuous transfer function model for the RLC circuit shown in the above figure. Defining an equation to represent the RLC circuit involves the following three steps:
1. Use the following Laplace equation to represent the relationship between the capacitor voltage uC and the inductor current iL of this RLC circuit.
(1)
where s represents the s domain, or the complex frequency domain.
Use the following Laplace equation to represent the voltage relationship in this RLC circuit.
(2)
Because the capacitor voltage uC is the output voltage y in this RLC circuit, you can deduce the continuous transfer function model between u and y by manipulating Equations 1 and 2, as shown by the following equation:
(3)
2. Use the following equation to represent a second-order continuous transfer function model.
(4)
where Kp is the transfer function gain, w is the natural frequency, and r is the damping ratio.
3. Obtain the following equation from Equations 3 and 4 to define the RLC circuit using a second-order continuous transfer function model.
(5)
You therefore can deduce w and r from Equation 5 as follows:
(6)
(7)
You can use the SI Create Partially Known Continuous Transfer Function Model VI to define a second-order continuous transfer function model for this RLC circuit. You also can use this VI to incorporate prior knowledge about this circuit when defining the model.
The following figure shows the block diagram for building the partially known continuous transfer function model of the RLC circuit.
You can deduce from Equation 5 that the static gain is 1. Using prior knowledge, you know that C is approximately 0.1 F and L is a positive value of approximately 0.1 H. From Equations 6 and 7, you therefore can deduce an initial guess of 10 for the natural frequency and an initial guess of 0.75 for the damping ratio. The following figure shows the initial guesses and upper and lower limits for the Static Gain, Natural Freq, and Damping Ratio inputs you then can specify in this example VI.
Next, you can estimate the transfer function model with the SI Estimate Partially Known Continuous Transfer Function Model VI, as shown in the following figure.
The SI Estimate Partially Known Continuous Transfer Function Model VI estimates each parameter of the model. Using the SI Draw Model VI, you can display the model for this RLC circuit in a picture indicator, as shown in the following figure.
You also can obtain the information of the natural frequency w and damping ratio r, as shown in the following figure.
Using Equation 3, you can build the block diagram to deduce the values of the inductance L and the capacitance C. The following figure shows the identification results.
Therefore, you can identify this partially known RLC circuit by obtaining the values of 0.20 H for L and 0.02 F for C, respectively.
Requirements
Filename: continuous_tf_model.zip
Software Requirements
Application Software: LabVIEW Professional Development System 8.2, LabVIEW Full Development System 8.2
Toolkits and Add-Ons: LabVIEW System Identification Toolkit 3.0
Language(s): LabVIEW
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