42 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL
by the superscripts of the lower and upper bounds of the intervals.
Example 2.4Take polynomialsP 2 (x)=a 0 +a 1 x+a 2 x^2 wherea 0 ∈[0. 9 , 1 .1],
a 1 ∈[− 0. 9 , 0 .1],anda 2 ∈[0. 9 , 1 .1]. The four Kharitonov canonical polynomials
are
K 1 (x)=0. 9 − 0. 9 x+1. 1 x^2
K 2 (x)=1.1+0. 1 x+0. 9 x^2
K 3 (x)=1. 1 − 0. 9 x+0. 9 x^2
K 4 (x)=0.9+0. 1 x+1. 1 x^2Two of these polynomials,K 1 (x)andK 3 (x), have roots with positive real parts,
0. 409 and 0. 5 , so the interval-coefficient polynomial with these intervals does not
represent a stable system. On the other hand, ifa 0 ∈[1. 0 , 1 .1],a 1 ∈[0. 9 , 1 .0],
anda 2 ∈[0. 9 , 1 .0], the four Kharitonov canonical polynomials are
K 1 (x)=1+0. 99 x+x^2
K 2 (x)=1.1+x+0. 99 x^2
K 3 (x)=1.1+0. 99 x+0. 99 x^2
K 4 (x)=1+x+x^2all of whose roots have negative real parts, and we know that a system producing
these polynomials is stable.
2.5 Controllerdesign...........................
There is a two-fold objective in the design of controllers. First, the overall
control loop must be stable. Second, the input to the plant must be such that
the desired set-point is achieved in minimum time within some specified criteria.
Both of these objectives require full knowledge of the plant.^1
The fundamental objective in feedback control is to make the output of a
plant track the desired input. Generally, this is referred to asset-point control.
The controller can be placed either in the forward path (in series) with the plant
or in the feedback path. Figures 2.14 (a) and (b) illustrate such configurations.
Most of the conventional design methods in control systems rely on the so-
calledfixed-configurationdesign in that the designer at the outset decides
the basic configuration of the overall designed system and the location where
the controller is to be positioned relative to the controlled process. The prob-
lem then involves the design of the elements of the controller. Because most
control efforts involve the modification or compensation of the system perfor-
mance characteristics, the general design usingfixed configuration is also called
compensation.
(^1) We are restricting our discussion of classical methods to only those that have a direct
bearing on the fuzzy controllers which will be discussed later. The reader should be aware
that classical methods exist for many other types of controller designs that are very powerful
and have been used extensively in modern control systems.