2.4. STABILITY 41
2.4.4 Robuststability
The problem of robust stability in control of linear systems is to ensure system
stability in the presence of parameter variations. We know that the origin is
asymptotically stable if all the eigenvalues of the matrixAhave negative real
parts. WhenAis ann◊nmatrix, there areneigenvalues that are roots of
the associated characteristic polynomialPn(x)=
Pn
k=0akxk.Thus,whenthecoefficientsakare known (given in terms of operating parameters of the plant),
it is possible to check stability since there are onlyfinitely many roots to check.
These parameters might change over time due to various factors such as
wearing out or aging; and hence, the coefficientsakshould be put in tolerance
intervals[a−k,a+k], allowing each of them to vary within the interval. Thus, we
have an infinite family of polynomialsPn, indexed by coefficients in the intervals
[a−k,a+k],k =0, 1 , 2 ,...,n. In other words, we have aninterval-coefficient
polynomial
Xn
k=0[a−k,a+k]xkThis is a realistic situation where one needs to design controllers to handle
stability under this type of ìuncertaintyî ñ that is, regardless of how the coef-
ficientsakarechosenineach[a−k,a+k]. The controller needs to be robust in the
sense that it will keep the plant stable when the plant parameters vary within
some bounds. For that, we need to be able to check thenroots of members
of an infinite family of polynomials. It seems like an impossible task. But if
that were so, there would be no way to construct controllers to obtain robust
stability.
Mathematically, it looks as though we are facing an ìinfinite problem.î How-
ever, like some other problems that appear to be infinite problems, this is afinite
problem. This discovery, due to Kharitonov in 1978, makes robust control pos-
sible for engineers to design. Here is an outline of his result.
Theorem 2.4 (Kharitonov)Suppose[a−k,a+k],k=0, 1 ,...,nis a family of
intervals. All polynomials of the formPn(x)=
Pn
k=0akxk,whereak∈[a−
k,a- k],
are stable if and only if the following four Kharitonov canonical polynomials are
stable:
K 1 (x)=a− 0 +a− 1 x+a+ 2 x^2 +a+ 3 x^3 +a− 4 x^4 +a− 5 x^5 +a+ 6 x^6 +∑∑∑
K 2 (x)=a+ 0 +a+ 1 x+a− 2 x^2 +a− 3 x^3 +a+ 4 x^4 +a+ 5 x^5 +a− 6 x^6 +∑∑∑
K 3 (x)=a+ 0 +a− 1 x+a− 2 x^2 +a+ 3 x^3 +a+ 4 x^4 +a− 5 x^5 +a− 6 x^6 +∑∑∑
K 4 (x)=a− 0 +a+ 1 x+a+ 2 x^2 +a− 3 x^3 +a− 4 x^4 +a+ 5 x^5 +a+ 6 x^6 +∑∑∑
Note that the pattern for producing these four polynomials is obtained by
repetitions of the symbol pattern