07.1 Integral Control (LQR) - Part 1
We have come a long way from open-loop plant 01.1 Basics of Open Loop to closed-loop plant 02.2 Closed Loop System with Feedback Control.
But we typically do not have access to the states. By states, I mean the $x$ vector itself, like the numerical value of the $x$ vector. I am not referring to the $A,B,C,D$ matrices. Those are the model, and they are obtained either from first principles or from system ID.
Anyway, since we do not have direct information about the states (again, the $x$ vector), we use an observer to estimate the states: 06.1 Luenberger Observer (State Estimation) - Full State Feedback - 2.
And with the closed loop, the control law 02.4 Controller or Compensator Design, and the observer, we have built an augmented system that can give us the response of a plant/system.
07.3 Draft - Summary Equations for Closed Loop, Observer, Controller
But how did we end up with an integral controller? And how does reference tracking come into play?
\[e(t) = r(t)-y(t)\]Integral control comes from the reference tracking problem. For instance, say you want your output $\mathbf{y}$ to be a certain way. So then you calculate the error, right?
\[x_I(t) = \int e(\tau)d\tau\]So you have an error vector. You take it and sum it up to get the integrated error, right? Something like this:
We can understand why we integrate the error, but why do we say that it is equal to $\mathbf{x_I}$?
Because we wanted to create a controller state. Why?
Because output (y) is determined by the plant states (x), and tracking error depends on (y), we add a controller state ($\mathbf{x_I}$) to remember accumulated tracking error and help drive (y) to the reference.
Also, when you say integration, that essentially means a $\frac{1}{s}$ term in the frequency/Laplace domain. Therefore, the integral part has its own dynamics.
Anyway, once we have our controller state, $\mathbf{x_I}$, we define its derivative so that it can be augmented into a state-space system:
\[\mathbf{\dot{x_I}} =\mathbf{e(t)} = \mathbf{r(t)-\mathbf{y(t)}}\]\[\mathbf{u(t)=-\mathbf{G_0 x(t)-G_I x_I(t)}}\]But why would you want that $\mathbf{x_I}$ thing anyway?
Because we are planning to use a control law like this: