In the last post, the process of linearization was covered in some detail. At the end of the analysis the equation of motion that resulted was
\[ \frac{d}{dt} \delta \bar S_{\bar f} = A \delta \bar S_{\bar f} + \bar \eta \; .\]
Generally, there is no reason to actually carry all that notational machinery around and typically the dynamical variables are often, generically, called $$\bar x$$.
Two main ingredients remain to be introduced. The first one is the control applied by a man-made actuator that introduces a general force into the equations of motion. Typically, the control, usually denoted by $$\bar u$$, is not an array of the same size as the state. The second ingredient, an output, denoted as $$\bar y$$. The output takes a few words to explain.
Typically, the dynamics of the system are not completely observable. For example, the motion of a projectile may be measure strictly by a radar gun, revealing the time history of the speed along the line of sight between the bore site of the gun and the projectile.
The combined system containing both the controls and the outputs is given by
\[ \dot{\bar x} (t) = \bar f (\bar x, \bar u, t) \; , \]
for the state evolution, and
\[ \bar y(t) = \bar g(\bar x, \bar u, t) \; .\]
Linearization, allows these equations to be written in state-matrix form as
\[ \dot{\bar x}(t) = \mathbf{A}(t) \bar x(t) + \mathbf{B}(t) \bar u(t) + \bar \eta \]
and
\[ \bar y(t) = \mathbf{C}(t) \bar x(t) + \mathbf{D}(t) \bar u(t) + \bar \rho\; .\]
The dynamical noise $$\bar \eta$$ and the measurement noise $$\bar \rho$$ are usually dropped or combined into the control term $$\bar u$$.
The above equations constitute the equations of modern control theory. Ogata, when describing these equations makes a distinction that is a bit difficult to reconcile with his emphasis on the Laplace Transform.
Most of his points are straightforward: the presence, at least initially, of nonlinear equations; the use of multiple inputs $$\bar u$$ and multiple outputs $$\bar y$$; and the presence of either time-varying or time independent terms. What is hard to understand is this distinction between modern control theory being essentially a time-domain approach, while the conventional approach uses frequency methods.
The idea of time- and frequency-domain methods standing side-by-side is a fruitful one in quantum mechanics. Why this distinction is so sharply drawn in the world of the controls engineer will, I suppose, reveal itself, in time.