Stated loosely, the Maximum Principle of Pontryagin says
that to take a system from one state to another in the shortest possible time,
subject to constraints on the magnitude of certain variables, one should
operate continuously at the extremes, shifting from one extreme condition to
another in a systematic way dependent on the initial and final states. For
example, to take your car from rest at one place to rest at another in the
shortest possible time you should (obviously) apply maximum acceleration up to
the last possible moment such that maximum braking will just bring you to a
screeching halt at the desired spot. Control systems of this sort are often
rather picturesquely called “bang-bang” systems. As an example,
consider a second-order system characterized by state variables A1
(t) and A2 (t) satisfying the state equations
where x(t) and y(t) are the input and output, respectively.
a) Using L-transforms, solve for Y(s) = A1 (S) = L [A1 (t)]
and A2 (S) = L [A2(t)] in terms of X(s) = L [x(t)) , A1(0), and A2(0).
b) Consider the particular initial state )..1 (0) = 1, )..2
(0) = 2. Solve for yet) if xCt) = 0, t > O. Under these conditions, how long
would it take the system to come essentially to rest? For example, how long
would it take for both the state variables to become less than 10% of their
c) Show that the system can be brought to rest at time T2
(that is, yet) == 0, all t > T2) by the input shown in the figure
if the values of T1 and T2 are properly chosen. Find Tl
and T2, and compare the resulting value of T2 with the
time to come essentially to rest when xCt) = 0, t > O.