Use of The Complex Plane in Control Theory
In control theory, one use of the complex plane is known as the 's-plane'. It is used to visualise the roots of the equation describing a system's behaviour (the characteristic equation) graphically. The equation is normally expressed as a polynomial in the parameter 's' of the Laplace transform, hence the name 's' plane.
Another related use of the complex plane is with the Nyquist stability criterion. This is a geometric principle which allows the stability of a closed-loop feedback system to be determined by inspecting a Nyquist plot of its open-loop magnitude and phase response as a function of frequency (or loop transfer function) in the complex plane.
The 'z-plane' is a discrete-time version of the s-plane, where z-transforms are used instead of the Laplace transformation.
Read more about this topic: Complex Plane
Famous quotes containing the words complex, plane, control and/or theory:
“What we do is as American as lynch mobs. America has always been a complex place.”
—Jerry Garcia (19421995)
“At the moment when a man openly makes known his difference of opinion from a well-known party leader, the whole world thinks that he must be angry with the latter. Sometimes, however, he is just on the point of ceasing to be angry with him. He ventures to put himself on the same plane as his opponent, and is free from the tortures of suppressed envy.”
—Friedrich Nietzsche (18441900)
“There has been something crude and heartless and unfeeling in our haste to succeed and be great. Our thought has been Let every man look out for himself, let every generation look out for itself, while we reared giant machinery which made it impossible that any but those who stood at the levers of control should have any chance to look out for themselves.”
—Woodrow Wilson (18561924)
“We commonly say that the rich man can speak the truth, can afford honesty, can afford independence of opinion and action;and that is the theory of nobility. But it is the rich man in a true sense, that is to say, not the man of large income and large expenditure, but solely the man whose outlay is less than his income and is steadily kept so.”
—Ralph Waldo Emerson (18031882)