Welcome back, my dear tech enthusiasts! In our last lecture, we delved into the captivating world of Eigen System Realization Algorithm (ERA). Today, we unveil a groundbreaking technique that involves fitting a linear dynamical system model to measured impulse response data. Picture this: a cutting-edge aerospace system where we harness the power of data to construct a best-fit linear model of rank R. Intrigued? Let’s dive in and explore this remarkable journey together.
Contents
The Essence of Impulse Response
Before we embark on our quest, let’s pause for a tantalizing moment. What exactly is an impulse response and what kind of response does it elicit? Brace yourself for a fascinating revelation. The impulse response represents the system’s output when it encounters a unit impulse input. In other words, it reveals the system’s inherent behavior when subjected to a sudden burst of energy. Imagine the possibilities!
The Discrete-Time Impulse Response Unveiled
Now, let’s embark on an exciting thought experiment to uncover the discreet-time impulse response. To simplify matters, let’s focus on single-input and single-output systems. Visualize a scenario where B and C are vectors, and D is a matrix. In this context, the input U is represented as a vector of ones. Intriguing, isn’t it?
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Let’s dive into the conceptual realm and unlock the secrets hidden within. At time k=0, as we assume zero initial conditions, U takes the form of the identity matrix, with X and Y each being zero. In this enchanting realm, Y is equal to the matrix D. Simple, yet profoundly effective.
As we fast forward to time k=1, an enchanting transformation occurs. The state X takes the form of B, as the impulse response springs into action. Meanwhile, Y becomes the product of matrix C and vector B, with D gracefully stepping aside. The system comes alive!
Curiosity compels us to venture further into the uncharted territories of time. Time k=2 beckons us with its mysteries. U retreats into the shadows, forever becoming zero. The mystical A matrix comes into play, multiplying its power with vector B. The ethereal composition of Y emerges, delicately woven from the enchanting combination of matrix C and the mighty A squared B.
Allow yourself a moment to marvel at the enchanting pattern that unfolds before our eyes. A pattern that resembles a symphony of K minus 1, A, B, C, and A to the power of K minus 1, B. It is in this mesmerizing dance that the expression of our impulse response finds its rhythm.
The Era of Eigen System Realization
As we journey deeper into the realms of complexity, we encounter the magnificent Eigen System Realization Algorithm. Brace yourself, for this algorithm assumes the existence of an impulse response. However, we recognize that in the realm of reality, acquiring such a response can be a formidable challenge. Fear not, my friends, for we have a cunning solution.
In our quest for knowledge, we shall feed our system a white noise input or a cunningly crafted pseudo-random input. Through this ingenious trickery, we can extract the elusive impulse response and unravel the enigma concealed within the matrices. Be warned, though, for the challenge intensifies if our impulse response fails to drop to zero. The mathematical expressions become untamed, wild creatures. But fret not, for we shall tackle those complexities in due time.
For now, let us bask in the glory of the Eigen System Realization Algorithm. Armed with a treasure trove of measured data, we embark on a voyage of system identification. With these sacred measurements, we weave a reduced-order model, a masterpiece of controlled perfection.
And so, my dear friends, we conclude today’s chapter on the hypnotic wonders of data-driven control. We have explored the captivating world of discrete-time impulse response and its potential for transformative power. Join me in our next adventure, where we shall unravel the intricacies of system identification. Until then, keep your curiosity aflame and your thirst for knowledge unquenchable.
Thank you, as always, for joining me on this exhilarating journey.
