Next 3 lectures will be on stability. One HW on stability due next Tuesday As well the special problem.
When dealing with linear systems, we form the characteristic polynomial \(P(s)\). If we know the location of the roots, we can also find other properties.
reader: if \(P(S)\) is stable and we reverse the order of the coefficients to obtain \(\hat{P}(s)\), is \(\hat{P}(s)\) stable?
Next we look at interval polynomial. \(P(s)= s^n + \sum _{i=0}^{n-1} a_i s^i\). We are interested in robust stability where each coefficients \(a_i\) has some range of values it can take \[ a_i^- \leq a_i \leq a_i^+ \] Robust stability: This polynomial is stable no matter what values \(a_i\) takes in this range. There are also robust tracking, robust damping, robust dynamic systems and other area where robustness is applied.
We can ask: How sensitive is system due to change in parameters? For \(n, a_i\) coefficients there is \(2^n\) vertices If the extreme points define stable polynomial (i.e. max and min values of each interval), we would expect it to be stable for values in between.
Kharitonov theorem gave 4 fixed polynomials to check for stability of robust polynomial. We define the four polynomials as
\begin{align*} K_1(s) &= a_0^+ + a_1^+ s + a_2^- s^2 + a_3^- s^3 + \dots \\ K_2(s) &= a_0^- + a_1^- s + a_2^+ s^2 + a_3^+ s^3 + \dots \\ K_3(s) &= a_0^+ + a_1^- s + a_2^- s^2 + a_3^+ s^3 + a_4^+ s^4 \\ K_4(s) &= a_0^- + a_1^+ s + a_2^+ s^2 + a_3^- s^3 +a_4^- s^4 \end{align*}
Kharitonov theorem The uncertain polynomial \(P(s)\) is robustly stable iff the above four polynomials are stable.
The proof of necessity is easy. The sufficiency proof is hard.
Examples: Let \(P(s)=s^3+4s^2+5s+2\). This is stable by construction \((s+1)^2(s+2)\). Roots are negative. Suppose we have interval polynomial \(P(s)=s^3+[3.5,4.5]s^2+[4.5,5.5]s+[1.5,2.5]\) then the four polynomials are
\begin{align*} K_1(s) &= s^3+4.5s^2+5.5s+1.5\\ K_2(s) &= s^3+3.5s^2+4.5s+2.5\\ K_3(s) &= s^3+4.5s^2+4.5s+1.5\\ K_4(s) &= s^3+3.5s^2+5.5s+2.5 \end{align*}
For low order polynomials, sometimes have to check less than 4 polynomials Sometimes only need to check 3 or just 2 as some will come up duplicate.
reader: check stability of the above.
If we start with stable polynomial \(P(s)\), we can ask, by what percentage can we perturb the coefficients while preserving stability? Similar to asking for radius of stability.
How to set it up? How to use percentage?
\begin{align*} P(s) &= s^3 + 4(1+\epsilon )s^2 + 5(1+\epsilon )s+2(1+\epsilon ) \\ &= s^3 + 4[1-\epsilon ,1+\epsilon ]s^2 + 5(1-\epsilon ,1+\epsilon )s+2[1-\epsilon ,1+\epsilon ] \\ \end{align*}
For \(\epsilon \ll 1\) we know it is stable. If \(\epsilon \) denotes the percentage perturbation, we want \[ \epsilon _{max} = \sup{\epsilon : \text{s.t. robust stability is guaranteed}} \]
HW7 assigned.