Welcome back! We’ve stumbled upon a fascinating concept in control systems that can truly unleash the potential of data-driven control. But hold on tight, because we’re about to correct a little typo that slipped through in our previous discussion.
You see, there’s a hat-complex-conjugate-transpose lurking within this intricate world. And it turns out, our earlier description missed a crucial detail. Brace yourself, because this discovery will shed a whole new light on the subject.
Let’s break it down step by step. We have this matrix, cunningly symbolized as A hat. Here’s where things get interesting. When we take the star (or transpose) of A hat, it transforms into T star times A star times T inverse star.
Wait, but how does one transpose a product of matrices? Well, it’s as simple as flipping the order and transposing each one individually. So, the transpose of T star times A star times T inverse star becomes T star times A star times T inverse star.
Ah, now we see the mistake we made. Instead of T inverse star, it should actually be T star. Similarly, instead of T inverse star, it should be T star. These subtle changes make a world of difference.
But here’s the amazing part. The magic of mathematics comes into play, and the T inverse star times T star cancels out, leaving us with the elegant equation: T inverse times WC times T inverse complex conjugate transpose.
Phew! That may sound incredibly complex, but fear not. You can easily compute these mind-bending calculations in MATLAB with just a single line of code. Indeed, it’s a fancy mathematical notation, but at its core, it’s simply the inverse of T transposed.
And guess what? This mind-boggling revelation isn’t just limited to the WC hat equation. It also holds true for the observability cramming method. So, armed with this newfound knowledge, go ahead and embark on a meticulous journey, working through the intricacies of this transformative process on a pristine sheet of paper. Once you immerse yourself in the details, you’ll be convinced that this is the key to unlocking the power of Gramian transforms.
But wait, there’s more! This extraordinary property of variable transformation will guide us in designing a remarkable T that strikes the perfect balance between controllability and observability.
So there you have it, my dear friends! We’ve delved into the depths of data-driven control, surfacing with a fresh understanding of the pivotal role that variable transformation plays in control systems. Prepare yourself for a thrilling adventure as we explore the intricate dance between controllability and observability.
Stay tuned for more exciting secrets and discoveries in the world of technology!
Techal
