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Reference

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.

Technical Details

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

Derivations for the Uncertain Plant Description

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

Derivations for the Uncertain Closed-Loop Magnitude

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

*(ii) *If for , then is nonincreasing 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

*(ii)* * *If for , then is nonincreasing in b
, and

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