Admissibility

Recall that the system state \(\def\xvec{\boldsymbol x} \xvec(t) \) and control \(\def\uvec{\boldsymbol u} \uvec(t) \) are the instantaneous values of state and control at any given time. We also speak of the state trajectory, or control history, to refer to the function as a whole, over the entire time period \( t ∈ [t_0, t_f] \).

Not all conceivable values for the state and control vectors are allowable. For example, in a control system intending to drive a car, speed limits must be obeyed, the car must not run out of gasoline, and—even more fundamentally—the car must stay on the road!

A simple way to indicate such constraints is with predicates on system and/or control values, for example the inequalities: \[ |x_1(t)| ≤ 2 \quad\text{or}\quad {-M} ≤ u(t) ≤ 0. \] For the soft-landing problem, one important constraint is listed right there in the title: the vertical velocity at touch-down must be very close to zero. (The landing surface will ensure that the vertical velocity after touchdown is zero, a process known as lithobraking.)

Optimality

Now we begin to touch on the name of this blog.

Once we have determined the set of “admissible” states and controls, there must be some way of choosing one (control curve–state curve)-pair out of the admissible region. In classical control theory, the measure is often the system response to step or ramp input, or a frequency response characteristic.

For our purposes, we define a very general performance measure: \[ J = h(\xvec(t_f), t_f) + \int_{t_0}^{t_f} g(\xvec(t),\uvec(t),t)\,dt \] for some scalar functions \( h \) and \( g \). An optimal system is one where \( J \) is minimized; this minimal value for the performance measure we denote \( J^* \).

Feedback

When I wrote \( \uvec(t) \) above, that is not to imply that the applied control at any given time has been pre-programmed into the system, though such systems do exist. Most systems have a control law, mapping the system’s current state to a control input that will get the system to do as we wish it to. Using the “—*” notation, this means \[ \uvec^*(t) = \uvec^*(\xvec(t),t). \]

Ah, but how to determine that law? More on that next time.