In the last post, the process of linearization was covered in some detail.\u00a0 At the end of the analysis the equation of motion that resulted was<\/p>\n
\\[ \\frac{d}{dt} \\delta \\bar S_{\\bar f} = A \\delta \\bar S_{\\bar f} + \\bar \\eta \\; .\\]<\/p>\n
Generally, there is no reason to actually carry all that notational machinery around and typically the dynamical variables are often, generically, called $$\\bar x$$.<\/p>\n
Two main ingredients remain to be introduced.\u00a0 The first one is the control applied by a man-made actuator that introduces a general force into the equations of motion.\u00a0 Typically, the control, usually denoted by $$\\bar u$$, is not an array of the same size as the state.\u00a0 The second ingredient, an output, denoted as $$\\bar y$$.\u00a0 The output takes a few words to explain.<\/p>\n
Typically, the dynamics of the system are not completely observable.\u00a0 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.<\/p>\n
The combined system containing both the controls and the outputs is given by<\/p>\n
\\[ \\dot{\\bar x} (t) = \\bar f (\\bar x, \\bar u, t) \\; , \\]<\/p>\n
for the state evolution, and<\/p>\n
\\[ \\bar y(t) = \\bar g(\\bar x, \\bar u, t) \\; .\\]<\/p>\n
Linearization, allows these equations to be written in state-matrix form as<\/p>\n
\\[ \\dot{\\bar x}(t) = \\mathbf{A}(t) \\bar x(t) + \\mathbf{B}(t) \\bar u(t) + \\bar \\eta \\]<\/p>\n
and<\/p>\n
\\[ \\bar y(t) = \\mathbf{C}(t) \\bar x(t) + \\mathbf{D}(t) \\bar u(t) + \\bar \\rho\\; .\\]<\/p>\n
The dynamical noise $$\\bar \\eta$$ and the measurement noise $$\\bar \\rho$$ are usually dropped or combined into the control term $$\\bar u$$.<\/p>\n
The above equations constitute the equations of modern control theory.\u00a0 Ogata, when describing these equations makes a distinction that is a bit difficult to reconcile with his emphasis on the Laplace Transform.<\/p>\n
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.\u00a0 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.<\/p>\n
The idea of time- and frequency-domain methods standing side-by-side is a fruitful one in quantum mechanics.\u00a0 Why this distinction is so sharply drawn in the world of the controls engineer will, I suppose, reveal itself, in time.<\/p>\n","protected":false},"excerpt":{"rendered":"