Welcome back! Today, we are going to explore the fascinating world of data-driven control. Specifically, we will delve into the realm of the eigensystem realization algorithm, a method that allows us to create a linear dynamical system that accurately represents our data. This algorithm is designed for experimental systems, where we have access to measurement data and are able to describe it using linear dynamics.
Imagine you have a system and you want to understand its behavior. You start by applying an impulse to the system and observing how its measurements change over time. If the system follows linear dynamics, meaning its response can be represented by an impulse response, we can collect a time series of measurements.
Now, let’s visualize this process. You measure the system’s response, represented by the data matrix Y at different time points. In a single-input, single-output system, Y at time 0 corresponds to the D matrix, while Y at time 1, Y at time 2, and so on, correspond to different combinations of the C, B, and A matrices. These matrices capture the underlying linear dynamics that govern the system.
The first step in the eigensystem realization algorithm is to extract the D matrix by isolating the first measurement of Y, Y0. With the D matrix in hand, we turn our attention to the remaining measurements, Y1 through YK, where K represents the number of measurements collected.
To unravel the best-fit model for the B and C matrices, we stack the measurement data into a Hankel matrix. This matrix, called H, allows us to capture the temporal evolution of the measurements. By constructing H using the available data, we can uncover the hidden patterns that shape the system’s dynamics.
Next, we perform a singular value decomposition (SVD) on the Hankel matrix H. Through this decomposition, we can identify the dominant left and right singular vectors, represented by U and V, respectively. These vectors capture the eigen time series that effectively describe the system’s behavior.
Once we have these dominant modes, we can truncate the SVD to reduce the dimensionality of the problem. We select a truncation rank, R, based on the singular values of H. The larger the singular values, the more important the corresponding modes are in capturing the system’s behavior.
Now, we are ready to build our reduced-order model. Utilizing the truncated U, V, and Sigma matrices, we can construct the matrices A tilde, B tilde, and C tilde. These matrices form our discovered model that accurately represents the system’s impulse response data.
By collecting measurement data and applying the eigensystem realization algorithm, we can uncover the hidden dynamics of the system. The discovered model, represented by A tilde, B tilde, C tilde, and D, provides us with valuable insights into the system’s behavior. Although the tilde states may seem abstract, they represent the eigen time-delay coordinates that capture the essential dynamics of the system.
In summary, the eigensystem realization algorithm is a powerful tool for discovering the underlying dynamics of a system purely from measurement data. By leveraging the concepts of eigen time series and truncation, we can create an accurate model that captures the system’s behavior. So, unleash the power of data-driven control and unlock the secrets hidden within your measurements.
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