Optimizing a Controller with HyperStudy

チュートリアルレベル：上級Learn how to set up a model of a controlled system, prepare it for interaction with HyperStudy, and perform an optimization in Altair HyperStudy.

重要: Available only with Twin Activate commercial edition.

Files for This Tutorial

PID.scm, PID Optimization.hstx

A finished version of the model you build in this tutorial along with any files required to complete the tutorials are available from the Demo Browser: Tutorial Models > Integration and Collaboration > Optimizing a Controller with HyperStudy or at this location: <installation_directory>/Tutorial Models/Integration and Collaboration/Optimizing a Controller with HyperStudy.

Overview

This overview discusses the objective, system description, performance metric, model components, key observations, steps, expected output, and challenges of the Optimizing a Controller with HyperStudy tutorial.

Objective

The goal of this problem is to optimize the tuning of a Proportional-integral-derivative (PID) controller (K_p, K_i, K_d) for a second-order system to minimize the Integral of Time-weighted Absolute Error (ITAE). The ITAE is a widely used performance metric that evaluates the control system's response. It emphasizes early error correction and penalizes errors over time.

Adjustment (optimization) of the three parameters can be a complex task that requires automation and mathematical optimization. This tutorial demonstrates how to couple Twin Activate and HyperStudy to perform parameter optimization.

System Description

The system being controlled is a second-order plant represented by the transfer function:

G (s) = \frac{1}{s^{2} + s + 1}

This transfer function models a generic second-order system with a damping ratio (ζ = 0.5) and natural frequency (ω_n = 1).

A PID controller is used to control the plant and minimize the error between the reference input (step signal) and the plant output. The controller has three tuning parameters:

K_p = Proportional gain
K_i = Integral gain
K_d = Derivative gain

These parameters influence the system's rise time, overshoot, settling time, and steady-state error.

A step input is used as the reference signal. The step input starts at 0 and transitions to a value of 1 at t = 0. The system's task is to track this step input.

Performance Metric: ITAE

The ITAE (Integral of Time-weighted Absolute Error) is used as the performance metric. It is defined as:

I T A E = \int_{0}^{T} t \cdot |e (t)| d t

Where:

e(t) is the error between the reference input and the plant output ( $e (t) = r (t) - y (t)$ ).
t is time, which weights the error more heavily as time progresses.
T is the total simulation time.

The goal is to minimize ITAE, which means:

Reduce large errors early in the response (where t is small).
Prevent sustained errors later in the response (where t is large and errors are penalized more).

Model Components

PID Controller:
- Inputs: Error signal (e(t)).
- Outputs: Control signal to the plant.
- Role: Adjust the plant's output to minimize the error and stabilize the system.
Second-Order Plant:
- Inputs: Control signal from the PID controller.
- Outputs: System response (y(t)).
- Role: Represents the dynamics of the system being controlled.
Summation Block:
- Inputs: Reference signal (r(t)) and plant output (y(t)).
- Outputs: Error signal ( $e (t) = r (t) - y (t)$ )
- Role: Computes the error, which is used by the PID controller.
Step Input:
- Initial value: 0.
- Final value: 1.
- Role: Provides the reference signal for the system to track.
ITAE Calculation:
- Computes $|e (t)|$ : The absolute value of the error.
- Multiplies $|e (t)|$ by time (t): Penalizes errors over time.
- Integrates over time (T): Accumulates the weighted error.

Key Observations

Importance of Proper Gains:
- K_p: Affects the system's responsiveness.
- K_i: Eliminates steady-state error.
- K_d: Reduces overshoot and oscillations. Improper gains can lead to instability, overshoot, or sluggish behavior.
Why ITAE is Used:
- ITAE prioritizes correcting early errors.
- Penalizes systems that take a long time to settle or exhibit sustained oscillations.
Simulation Time: The simulation must run long enough (T=10 to 30 seconds) to capture the system's transient and steady-state behavior.

Steps in the Model Setup

Define the Plant: Use a transfer function block to model:

$G (s) = \frac{1}{s^{2} + s + 1}$
Add the PID Controller: Use a PID block to process the error signal and generate a control signal.
Compute Error (e(t)): Use a summation block to compute $e (t) = r (t) - y (t)$
Reference Signal: Use a step input block with:
- Initial value: 0
- Final value: 1
- Step time: 0
Calculate ITAE: Use blocks to:
- Compute the absolute error ( $|e (t)|$ ).
- Multiply $|e (t)|$ by the time signal (t).
- Integrate the result to obtain ITAE.
Display Results: Use scopes to display:
- The system response (y(t))
- The accumulated ITAE value

Expected Output

System Response: The plant output (y(t)) should track the step input (r(t)) with minimal overshoot, fast rise time, and minimal steady-state error.
ITAE Value: Lower ITAE values indicate better performance, reflecting small errors corrected quickly.

Key Challenges

Tune the PID Gains:
- Properly adjust K_p, K_i, and K_d to balance responsiveness and stability.
- Avoid overshoot, instability, or sluggish response.
ITAE Calculation: Ensure accurate computation of the absolute error and its time-weighted integral.
System Dynamics:
- Understand how the second-order plant interacts with the PID controller.
- The figure below shows the reference and response signals, along with the nature of ITAE of this model for the base model.

図 2. Reference and Response Signals and ITAE Before Optimization

Prepare the Model in Twin Activate for Optimization

Prepare the model by adding values to the initialization script and adding a ToCSV block.

When the model is ready and running well, make the following changes.

Add the following values of K_p, K_i, K_d to the initialization script:
- Kp=10
- Ki=5
- Kd=1
図 3. Initialization Script Where Design Variables are Declared

Use these variables in the PID block.
図 4. Variables Used to Define the Gains in the PID Block
Add a ToCSV block with the following values:
- CSV Filename: 'ITAE.csv'
- Number of exported variables: 1
- Variable name: 'ITAE', Datatype: 'double'
- Sub-sampling rate: 1
- Name of the first column: 'time'
- Select Variable names should be quoted
When HyperStudy calls Twin Activate during any run, it expects that a CSV file is created. This CSV file is used as an output source. It should contain all the response vectors that are required in HyperStudy.

In this case, you need to dump the ITAE values, into a CSV file. The final value of this vector is used in HyperStudy to define a response.
図 5. The PID model with ITAE

The model is now ready to be linked with HyperStudy.

Optimize using HyperStudy

These steps show you how to perform an optimization in HyperStudy.

HyperStudy is a multi-disciplinary design exploration, study, and optimization software. HyperStudy lets you explore, understand, and improve your system's designs using methods such as design-of-experiments and optimization. HyperStudy generates intelligent variations of the parameters of any system model and reveals relationships between these parameters and the system responses.

HyperStudy provides engineers and designers a user-friendly environment to:

Improve Design Performance and Quality: It includes state-of-the-art, innovative optimization, design of experiments, and stochastic methods for rapid assessment and improvement of design performance and quality.
Reduce Development Time and Costs: It helps engineers reduce trial-and-error iterations and hence helps to reduce both the design development and testing time.
Increase Productivity through Easy-to-use Environment: HyperStudy's step-by-step process guides you in setting up and carrying out design studies. Its open architecture allows easy integration with 3rd party solvers.
Perform Powerful Dataset Analysis: A comprehensive set of post processing and data mining methods simplify and aid an engineer’s job of analyzing and understanding large simulation datasets.
Improve Simulation Correlation: HyperStudy's optimization capabilities can be applied to improve correlation of analysis models with test results or with other models.

For more information, see HyperStudy Overview.

Open HyperStudy and create a new HyperStudy file.
Under Setup > Define Models, click Add Model.
Add the SCM file prepared earlier. For Type, choose Twin Activate.
Set Solver execution script is set to Twin Activate and ensure the path to the script is defined correctly.

The path can be defined in Edit > Register Solver Script.
図 6. Twin Activate Model Added in HyperStudy
Click Import or Import Variables to import all the design variables defined in the initialization script of the Twin Activate file.

注: All independent variables are imported as design variables in HyperStudy. The Define Input Variables tab looks like the image below.

図 7. Design Variables in HyperStudy
Change the lower and upper bounds to 0.1 and 20, respectively, for all three design variables.

図 8. Upper and Lower Bounds
In the Test Models tab, click Run Definition.
If the model runs without errors, three green check marks are shown under Test.

図 9. Indication that the Model has Run Successfully

The stdout.txt file listed in the message log should not show any errors. This means that the model has successfully run from HyperStudy.
Click Define Output Responses.
Click Data Sources.
You should see that two output data sources have been added: ToCSV_time and ToCSV_ITAE. These represent the two columns of data from the ITAE.csv file. You only need the last value of the ToCSV_ITAE vector.
Click Define Output Responses, and then click Add Output Response.
Under expression type, enter ITAE[indexofmax(time)], and then click Evaluate.
The last value is now shown in the Value column.
図 10. Using the Last Value of ITAE as a Response

The model setup in HyperStudy is now complete.
To add an optimization study, right-click in the left browser window or click Next > Add in the Define Output Responses tab.
For Definition from, select Setup as the value.
The model, design variables, and output responses are created in the new optimization study.
From the Test Models tab in the browser, click Run Definition.
Ensure the model runs without any errors.

図 11. Running the Model in the Optimization Study
From the Define Output Responses tab in the browser on the left, select Minimize > Goals.
The optimization algorithm tries to minimize the value of last value of ITAE.
注: You might have to click Evaluate to populate the value of the response.
図 12. Defining the Goal in HyperStudy
Click Specifications in the browser.
Select the Adaptive Response Surface Method as the optimization mode on the left side of the tab. Set the following parameters on the right side:
- Number of Evaluations: 23
- Absolute Convergence: 0.001
- Relative Convergence (%): 1.000
- Input Variable Convergence: 0.001
- On Failed Evaluation: Ignore failed evaluations
Click Apply to use the modified settings.
図 13. Optimizer Settings
Open the Evaluate tab in the browser, and then click Evaluate Tasks.
When the optimization is complete, look for the row highlighted in green. This lists the optimized value of ITAE and the corresponding design variable values (Initial: 1.038; Optimized: 0.0402).

図 14. Optimum Values of Design Variables

The CSV files from run folder 20 (optimized solution) can be compared with the initial value in run folder 1.
図 15. Reduction of ITAE using Optimized Values of the Design Variables

図 16. Comparison of Response from Base Values of PID and Optimized Values of PID

As can be seen in the above figure, the response from the optimized values of K_p, K_i, K_d happen to produce a much smoother and better response.