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LabVIEW System Identification Toolkit User Manual. Circuit System Control Design Toolkit User Manual 4-8 ni.com Chapter 4 Connecting Models The input of this system is the voltage v. The output of this system is the total current i, which is the sum of currents i1 and i2. R1 and R2 are resistors, and L1 and L2 are inductors.
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When C(z), D(z), and F(z) equal 1, the general-linear polynomial model reduces to an autoregressive with exogenous terms (ARX) model. This model is the simplest model that incorporates the stimulus signal. However, the ARX model captures some of the stochastic dynamics as part of the system dynamics. In this model, the transfer function of the deterministic part G(z–1, θ) of the system and the transfer function of the stochastic part H(z–1, θ) of the system have the same set of poles. This coupling can be unrealistic. The system dynamics and stochastic dynamics of a system do not share the same set of poles all the time. You can reduce this disadvantage if the signal-to-noise ratio is high.
When the disturbance e(k) of a system is not white noise, the coupling between the deterministic and stochastic dynamics can bias the estimation of the ARX model. You can set the model order higher than the actual model order to minimize the estimation error, especially when the signal-to-noise ratio is low. However, increasing the model order can change some dynamic characteristics of the model, such as the stability of the model.
Use the SI Estimate ARX Model VI to estimate ARX models. The identification method for the ARX model is the least squares method, which is a special case of the prediction error method. The least squares method is the most efficient polynomial estimation method because this method solves linear regression equations in analytic form. Moreover, the solution is unique. Refer to the LabVIEW System Identification Toolkit Algorithm References manual for more information about the least squares and prediction error methods.
The following equation shows the form of the ARX model.
A(z)y(k) = B(z)u(
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k - n) + e(k)where | u(k) is the system inputs |
y(k) is the system outputs | |
n is the system delay | |
e(k) is the system disturbance |
A(z) and B(z) are polynomial with respect to the backward shift operator z–1 and defined by the following equations.
Note The backward shift operator makes z–nu(k) = u(k - n). |
The following figure depicts the signal flow of an ARX model.
where | u is the system inputs |
e is the system disturbance | |
y is the system outputs |
SISO
The following is the time domain equation for the ARX SISO model.
where | kA order |
kb is the B order | |
n is the system delay | |
e(k) is the system disturbance |
Refer to the Estimate Polynomial Models VI in the labviewexamplesSystem IdentificationGetting StartedParametric Estimation.llb for an example that demonstrates how to estimate ARX models for an unknown system.
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System identification, the first step in the model-based control design process, involves building mathematical models of a dynamic system based on a set of measured stimulus and response data samples. You can use system identification in a wide range of applications, including mechanical engineering, biology, physiology, meteorology, economics, and model-based control design. For example, engineers use a system model of the relationship between the fuel flow and the shaft speed of a turbojet engine to optimize the efficiency and operational stability of the engine. Biologists and physiologists use system identification techniques in areas such as eye pupil response and heart rate control. Meteorologists and economists build mathematical models based on historical data for use in forecasting.
The LabVIEW System Identification Toolkit provides the following tools.
System Identification VIs
The System Identification Toolkit provides VIs that you can use to preprocess raw data from a dynamic system and develop a model that reflects the behavior of that system. The Data Preprocessing VIs enable you to analyze the response of a plant or dynamic system to a certain stimulus. After analyzing the data, you can use the Parametric Model Estimation, Nonparametric Model Estimation, Partially Known Model Estimation, Recursive Model Estimation, and/or Frequency-Domain Model Estimation VIs to estimate a model for the plant or dynamic system. Finally, you can use the Model Validation or Model Analysis VIs to determine whether the model accurately describes the dynamics of the identified system.
The System Identification VIs enable you to customize a LabVIEW block diagram to achieve specific goals. You also can use other LabVIEW VIs and functions to enhance the functionality of the application. Creating a LabVIEW application using the System Identification VIs requires basic knowledge about programming in LabVIEW. Refer to the LabVIEW Fundamentals and Getting Started with LabVIEW manuals for more information about the LabVIEW programming environment.
The following case studies demonstrate how to use the System Identification VIs to estimate different model representations by using time-domain or frequency domain data.
Toolkit User Guide
- System Identification Case Study—Guides you through the entire system identification process. This case study demonstrates how to preprocess time-domain data from a dynamic system, estimate an ARX and state-space model by using the time-domain data, and validate the models to ensure they accurately reflect the dynamic system.
- Partially Known Model Estimation Case Study—Demonstrates how to estimate a state-space model by using prior knowledge about the system you want to define.
- Frequency-Domain Model Estimation Case Study—Demonstrates how to estimate and validate a state-space and transfer function model by using frequency-domain data from a dynamic system.
System Identification Assistant
If you do not have prior knowledge about programming in LabVIEW, you can use the NI System Identification Assistant to develop a model that reflects the behavior of a certain dynamic system. You access the System Identification Assistant by launching LabVIEW and selecting Tools»Control Design and Simulation»Launch System Identification Assistant.
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Using the System Identification Assistant, you can create a project that encompasses the whole system identification process. In a single project, you can load or acquire raw data into the System Identification Assistant, preprocess the data, estimate a model that describes the system, and then validate the accuracy of the model. SignalExpress provides windows in which you can see the raw data, the response data, the estimated model, the validation results, and the mathematical equations that describe the model.
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After creating a project in SignalExpress, you can convert the project to a LabVIEW block diagram and customize the block diagram in LabVIEW. This conversion enables you to enhance the capabilities of the application. Refer to the LabVIEW SignalExpress Help, available in the LabVIEW SignalExpress environment by selecting Help»LabVIEW SignalExpress Help, for more information about using the assistant to develop models.