Robust Stability

Sometimes we don’t know the exact parameters of a system but only that they are in a certain range. It is often required to determine whether a system is stable for all possible values of its parameters.

Consider a set of polynomials:

$$\mathcal{P}=\{a(s)=a_0{s}^n+a_1{s}^{n-1}+\cdots +a_{n-1}s+a_n:a_i\in [{\underline{a}}_i,{\bar{a}}_i]\}$$

This set of polynomials is often called an interval polynomial.

Kharitonov Theoreom

All members of $\mathcal{P}$ are stable if and only if the following four polynomials are stable:

$$\begin{array}{rl}{a}_1(s)& ={\underline{a}}_0{s}^n+{\underline{a}}_1{s}^{n-1}+{\bar{a}}_2{s}^{n-3}+{\bar{a}}_3{s}^{n-3}+\underline{a_4}{s}^{n-4}\\ a_2(s)& ={\underline{a}}_0{s}^n+{\bar{a}}_1{s}^{n-1}+{\bar{a}}_2{s}^{n-3}+{\underline{a}}_3{s}^{n-3}+{\underline{a}}_4{s}^{n-4}\\ a_3(s)& ={\bar{a}}_0{s}^n+{\underline{a}}_1{s}^{n-1}+{\underline{a}}_2{s}^{n-3}+{\bar{a}}_3{s}^{n-3}+{\bar{a}}_4{s}^{n-4}\\ a_4(s)& ={\bar{a}}_0{s}^n+{\bar{a}}_1{s}^{n-1}+{\underline{a}}_2{s}^{n-3}+{\underline{a}}_3{s}^{n-3}+{\bar{a}}_4{s}^{n-4}\end{array}$$

Example