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Control theory, simply, is the process of designing a system such that, when given a specified objective, the system will achieve that objective. It is the process by which an input signal undergoes a transformation to result in an output signal; in an ideal (zero error) case, the input is the same as the output. This isn't quite as simple as, say, just multiplying the input by 1: control is meant to model real situations, such as a driver's slight increased pressure on the gas pedal making the car go 1 mph faster, where there are many electrical and mechanical (or even organic) connections between the input and the output. This is all mathematically modeled by the Laplace transforms of differential equations. In a closed-loop (feedback) system, the output is continuously compared to the input and adjusted accordingly; an open-loop system has no feedback. See control theory for a somewhat better explanation.

The transformation of input to output is mathematically modeled by a transfer function, which is usually a ratio of polynomials in the Laplace variable 's'. The roots of the numerator are called "zeros" and the roots of the denominator are "poles". The value of the poles is a most important consideration, because at those values, the transfer function heads for infinity. The system is unstable if the poles are greater than zero.

If the open-loop transfer function is denoted L(s), the closed-loop transfer funtion is L(s)/(1+L(s)). It is the job of the controls engineer to adjust this transfer function so as to acheive the desired output. A common way of achieving this is to multiply L(s) by a whole number - the "gain," denoted by 'K'. The poles of the closed-loop transfer function, therefore, occur where 1+KL(s) = 0. The engineer wants to ensure that values of s that satisfy this equation 1) are less than zero and 2) satisfy specific performance specifications, which we won't go into here. A root-locus diagram, roughly speaking, is a plot in the complex plane of all possible closed-loop poles for every gain, and its analysis sheds light on the stability and accuracy of the control system.

To draw a root-locus diagram:

  1. Start by plotting the poles and zeros of the open-loop transfer function on the complex plane.
  2. Each branch will start (K=0) at a pole and end (K -> ∞) at a zero, or if there are more poles than zeros, at infinity. The number of branches is equal to the number of poles.
  3. For any point on the real axis, if the total number of real open-loop roots lying to the right of that point is odd, the root-locus curve passes through that point. If that number is even, the curve will pass above and below that point, or will approach infinity in an entirely different direction.
  4. The branches which approach infinity (there will be (#poles-#zeros = ρ) of them) do so at the angle γl = (1+2l)π / ρ with respect to the real axis, where l = 0, 1, 2, ..... These asymptotes will all intersect the real axis at σa = (1/ρ) * ((sum of the real part of all poles) - (sum of the real part of all zeros)).
  5. The points at which the curve departs the real axis (to become a complex conjugate pair) are known as "break-out" points; their opposite are "break-in" points. They almost always occur perpendicular to the real axis, and their locations can be found by setting the derivative of the open-loop transfer function equal to zero, and solving for s.
For example, the root-locus diagram for the open-loop transfer function L(s) = K(s+3)/(s*(s+2)) would look something like this:

         ____<___     |
        /        \    |
       |          |   |
       |  -3  -2  |   0     
        \____<___/    |
The root-locus curve here consists of the equal signs and the curved portion above and below the real axis. < and > denote increasing K.

The result should be symmetric about the real axis. Each point on the curve corresponds to a closed-loop pole, and also the the gain K necessary to achieve that pole. If the diagram crosses to the right hand side of the imaginary axis, the system is unstable for values of K that place the locus there. In this way, by finding the desired poles on a root-locus diagram, the proper gain value for a stable, accurate system can be found. Usually, this is done with the help of a computer program like Matlab, though for simple systems it's not too bad by hand.

The root-locus method was invented in 1948 by Walter R. Evans, a noted electrical and aerospace engineer.

To all the control freaks out there: sorry if I've totally mangled this topic in my attempt to bring it to the masses; please let me know if there are any glaring errors, and I'll fix 'em.

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