An elementary treatment of the robust stabilization theorem can be found in Chapter 3 of

J.C. Doyle, B. Francis, A. Tannenbaum, Feedback Control Theory, Macmillan Publishing Company, 1992.

Mathematical developments that support the Robust Stabilization Applet are described below.

Consider the multiplicative uncertainty model for the plant,

where P(jw ) is the nominal plant, W(jw ) is the frequency weighting function, and D (jw ) is the uncertainty parameter. We want to derive maximum and minimum values for the magnitude and angle of as the magnitude of D (jw ) ranges from 0 to 1 (- ¥ db to 0 db) and the angle of D (jw ) varies over a 2p range in radians.

Since the max and min values are developed for each frequency, we often drop the argument jw for simplicity, and write

Thus, taking into account the properties of the uncertainty parameter, we want to compute max and min values of and for 0 £ a £ |W| and 0 £ f < 2p .

Using Euler’s formula to write

it is easy to see that for any nonnegative a this quantity has a maximum at j = 0 and a minimum at j = p . The corresponding maximum value is | 1 + a | and taking into account the range of a gives the result

The minimum value of with respect to j occurs at j = p , and is |1 - a | . The range of a then gives

and

To consider , first note that if |W| ³ 1, so that a can be unity, then is arbitrary in the range 0 to 2p . That is, can take any value if |W(jw )| ³ 1. If |W| < 1, so that 0 £ a < 1, then Euler’s formula gives

Differentiating this expression with respect to f , and setting the result to 0 gives the necessary condition cos j = - a for a maximum or minimum. This condition implies that

Substituting into the expression for and using properties of the inverse tangent gives that the maximum and minimum values with respect to j are given by

This expression is monotone with respect to a , for a between 0 and 1, and therefore

and

Given a unity feedback system with uncertain plant and a compensator C(jw ), the uncertain closed-loop system is described by

In order to derive maximum and minimum values for , as ranges over the uncertain plant family, we simplify the notation by setting

where

and the various bounds depend on C(jw ) and on the maximum and minimum values for the magnitude and phase of derived above, which in turn depend on W(jw ). In this notation,

To compute the maximum value, first choose q to minimize the denominator. Because of the bounds on q this is slightly more complicated than the similar calculation above. But by sketching a vector diagram it is easy to verify that the solution divides into three cases:

It remains to choose b to maximize

A straightforward calculation gives

The maximization thus devolves into three cases:

(i) If for , then is nondecreasing in b , and

(iii) If for some , then further analysis gives that is increasing for and decreasing for . Therefore

To compute the minimum value of

first choose q to maximize the denominator. Again a vector diagram gives the result

Now choosing b to minimize

is based on the derivative

and again involves three cases:

(i) If for , then is nondecreasing in b , and

(iii) If for some , then further analysis along the lines of the previous case (iii) gives