Welcome back! In this article, we’re going to delve into the world of data-driven control, specifically focusing on balanced truncation and balanced proper orthogonal decomposition (BPOD). These techniques are incredibly useful when dealing with large systems and can help us reduce the complexity of our models while maintaining accuracy. Let’s dive in and explore these concepts further.
Contents
What is Balanced Truncation and BPOD?
Balanced truncation and BPOD are both model reduction techniques that allow us to reduce the order of a system while preserving its essential dynamics. These methods are particularly helpful when dealing with systems that have multiple inputs and outputs (MIMO).
To better understand these techniques, let’s take a look at an example. Suppose we have a system with two inputs, two outputs, and a state dimension of 100. We can use MATLAB’s built-in command, dss, to generate a random state-space system with these specifications. For the sake of this demonstration, we will use a pre-generated example.
Understanding the Hankel Singular Values
Before we dive into the implementation of balanced truncation and BPOD, it’s essential to visualize the Hankel singular values of our system. These values provide insights into the energy distribution of the system’s dynamics.
In MATLAB, we can easily obtain the Hankel singular values by performing the Hankel singular value decomposition (hsvd) on our system. By plotting these values on a logarithmic scale, we can see how aggressively we can truncate our system.
In our example, we observe that the first ten singular values capture over 90% of the energy. This indicates that we can achieve a reasonably accurate model by reducing the system’s order to 10.
Implementing Balanced Truncation
Implementing balanced truncation in MATLAB is incredibly straightforward. We can accomplish this reduction with a single line of code: balred(sys, R). This command provides a reduced-order model with a rank of R. The resulting model consists of ABCD matrices, which we can use for further analysis and control implementation.
It’s important to note that for smaller systems like ours, balanced truncation is sufficient. However, for larger systems, we may need to employ balanced proper orthogonal decomposition (BPOD).
Understanding Balanced Proper Orthogonal Decomposition (BPOD)
Balanced proper orthogonal decomposition (BPOD) is a more involved technique that extends the capabilities of balanced truncation to larger systems. While balanced truncation can be accomplished with a single line of code, BPOD requires multiple steps.
In BPOD, we start by computing the impulse response of our full system. Additionally, we calculate the impulse response of the adjoint system. These responses yield large data matrices that we can use to construct the Hankel matrix, perform the singular value decomposition, and compute the direct and adjoint modes.
Finally, we project the dynamics of the original system onto the reduced dynamics using the computed modes. This process yields a balanced model with a reduced rank.
Comparing Balanced Truncation and BPOD
Now that we have implemented both balanced truncation and BPOD, let’s compare their performance by analyzing the impulse responses of the reduced models.
In our example, we observe that the reduced models obtained through balanced truncation and BPOD closely approximate the full order model. However, it’s important to note that more aggressive reduction yields less accurate models. For our system, a rank of 10 provides good results, capturing most of the input-output energy while significantly reducing computation time.
FAQs
Q: Can I use balanced truncation and BPOD for system identification based on experimental data?
A: Yes! These techniques can be adapted for system identification based on experimental data. By leveraging the mathematical framework of balanced truncation and BPOD, you can build accurate models even when you only have measurement data.
Conclusion
Balanced truncation and balanced proper orthogonal decomposition (BPOD) are powerful techniques that allow us to reduce the order of large systems while preserving their essential dynamics. By employing these methods, we can create more tractable models for control implementation, significantly reducing computation time without sacrificing accuracy.
For more in-depth exploration of system identification based on experimental data, stay tuned for our next article. In the meantime, feel free to experiment with balanced truncation and BPOD in MATLAB and discover how these techniques can enhance your control systems.
