How to measure IAE ISE ITAE ITSE in simulink

In control systems, performance indexes like IAE (Integral of Absolute Error), ISE (Integral of Squared Error), ITAE (Integral of Time-weighted Absolute Error), and ITSE (Integral of Time-weighted Squared Error) are crucial for evaluating the efficiency and effectiveness of a controller. These metrics help engineers compare different controllers and determine which one gives the best system response. In this blog post, we’ll guide you through the process of measuring these performance indexes in Simulink, using a basic control system setup.

Introduction to Performance Indexes

Performance indexes like IAE, ISE, ITAE, and ITSE are integral tools for assessing control system behavior. These indexes measure the error between the system's desired output and its actual response. The values derived from these calculations help engineers understand how well a controller is performing and identify areas of improvement.

In this tutorial, we’ll use a second-order system with a Proportional (P) controller, a feedback loop, and a Simulink model to demonstrate how to measure these performance metrics.

Setting Up the Simulink Model

We begin by creating a simple control system model using a second-order transfer function, a P-controller, and a negative feedback loop. The goal is to simulate the system's response and compare its performance by measuring the key performance indexes. The setup includes:

• A second-order plant transfer function.

• A P-controller with no tuning (used for demonstration purposes).

• Negative feedback to close the loop.

This setup serves as a baseline for measuring the system’s performance indexes.

Measuring System Response

Once the Simulink model is created, you can simulate the system’s response. During simulation, you'll observe key parameters like overshoot, undershoot, and settling time. These values give you an initial idea of how the system is behaving.

However, IAE, ISE, ITAE, and ITSE cannot be directly measured from the simulation scope. To measure these values, we need to calculate the integral of the error terms (absolute error, squared error, etc.).

Calculating Performance Indexes

1. IAE (Integral of Absolute Error)IAE is calculated by integrating the absolute error over time. In Simulink, you can measure IAE by:

   • Using an integrator block to accumulate error over time.

   • Applying an absolute value block to the error signal.

   • Saving the output to the workspace as a variable for analysis.

2. ISE (Integral of Squared Error)ISE is calculated by integrating the squared error over time. To compute ISE in Simulink:

   • Use an integrator block to accumulate the squared error.

   • Apply a square block to the error signal.

   • Save the result to the workspace as "ISE".

3. ITAE (Integral of Time-weighted Absolute Error)ITAE is calculated by integrating the time-weighted absolute error. Here’s how to compute it in Simulink:

   • Multiply the absolute error by the clock block (which represents time).

   • Integrate this product to get the ITAE value.

   • Save the result to the workspace.

4. ITSE (Integral of Time-weighted Squared Error)ITSE is similar to ITAE but with squared error. To measure ITSE:

   • Square the error signal before multiplying it by time.

   • Use an integrator block to accumulate the time-weighted squared error.

   • Save the result to the workspace as "ITSE".

Visualizing the Results

After running the simulation, you can visualize the performance indexes using a scope in Simulink. To observe all four performance indexes simultaneously:

• Configure the scope to display four inputs.

• Rename each input for clarity (IAE, ISE, ITAE, ITSE).

• Each performance index will be displayed in a different color for easy identification.

This allows you to compare the performance of the control system over time and assess how well the controller is managing the system's behavior.

Extracting Average Values for Performance Indexes

While visualizing the performance indexes over time is useful, often you need a single value to summarize the controller’s effectiveness. In this case, you can calculate the average value for each performance index.

For example:

• For IAE, calculate the average of the IAE values over time.

• Similarly, calculate the average for ISE, ITAE, and ITSE.

To extract these values, you can use formulas in the Simulink workspace, such as:

• mean(out.IAE) to get the average IAE value.

• Repeat this for ISE, ITAE, and ITSE.

These average values provide a clearer picture of the overall controller performance.

Comparing Different Controllers

Once you have the performance indexes for a single controller, you can swap the P-controller for other types of controllers (like a PI, PID, or fuzzy controller) to compare their performance. By calculating and comparing the IAE, ISE, ITAE, and ITSE values for each controller, you can identify which controller provides the best response for your system.

Conclusion

In this post, we’ve shown you how to measure key performance indexes—IAE, ISE, ITAE, and ITSE—in Simulink. These metrics are essential for evaluating the performance of controllers in control systems. By following this guide, you can analyze your system's performance and compare different controllers effectively.

For engineers and control system designers, being able to calculate and interpret these performance indexes is critical to ensuring that systems operate optimally. Experimenting with different controllers and observing their performance using these indexes helps you choose the best control strategy for your application.