Everything Is Under Control
Can control theory save the economy from going down the tubes?
On first acquaintance, the idea of feedback control seems straightforward enough. Consider the design of a cruise-control system for an automobile. A minimal version measures the current speed of the car, compares it with the desired speed, then adjusts the throttle by an amount proportional to the difference. If the car slows somewhat—perhaps on an upgrade—the controller senses the discrepancy and opens the throttle wider, so that the car regains some of the lost speed.
But there is more to control theory than this simple proportional-feedback mechanism. A drawback of pure proportional control is that the car never quite attains the requested speed; as the error diminishes, so does the feedback signal, and the system settles into a state with some nonzero offset from the correct velocity. The offset can be eliminated by another form of feedback, based not on the error itself but on the integral of the error with respect to time. In effect, the integral measures the cumulative error, which keeps growing if the speed differs even slightly from the set point. Thus integral control ensures that over the long term the net error approaches zero and the average speed converges on the set-point speed.
Yet integral control has drawbacks of its own. Suppose the car cannot maintain a commanded speed of 60 on an upgrade; an integral controller might compensate by going 80 on the other side of the hill, which could get you a speeding ticket. More generally, integral control has a tendency to overshoot and oscillate around the set point. A remedy is to add still another form of feedback, based on the time derivative of the error signal. Derivative feedback opposes rapid changes in speed and thus tends to damp out oscillations.