# Stated loosely, the Maximum Principle of Pontryagin says that to take a system from one state to…

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 A_{1}

(t) and A_{2} (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

initial values?

c) Show that the system can be brought to rest at time T_{2}

(that is, yet) == 0, all t > T_{2}) by the input shown in the figure

if the values of T_{1} and T_{2} are properly chosen. Find T_{l}

and T_{2}, and compare the resulting value of T_{2} with the

time to come essentially to rest when xCt) = 0, t > O.