Finding roots of a polynomial of 5th order in Matlab.

**paulbk108** · February 3rd 2011, 02:00 AM

I have an equation

q(s) = s^5 + s^4 + 4s^3 + 4Ks^2 + 2Ks + K

The answer I have indicated is 0.5361 < K < 0.9326.

Can this be done in Matlab, and if so how?

My apologies I am still feeling my way through Matlab.

Regards.

**pickslides** · February 3rd 2011, 12:23 PM

Try this

Code:

syms s K;
solve('s^5 + s^4 + 4*s^3 + 4*K*s^2 + 2*K*s + K','K')

**Ackbeet** · February 3rd 2011, 12:38 PM

I'm not an expert in MATLAB, but are you sure you want to use the symbolic toolbox? You've got a quintic polynomial there, which likely doesn't have an analytical solution. In fact, the OP specified a root in a particular interval.

**paulbk108** · February 3rd 2011, 03:09 PM

Using your code pickslides I get

ans =

-(s^5 + s^4 + 4*s^3)/(4*s^2 + 2*s + 1)

To me that has not solved for K. The other way to my thinking is to go through the Routh array but for such a complex function this seems daunting and time consuming...and that's assuming it is even possible that way.

Regards.

**CaptainBlack** · February 3rd 2011, 07:30 PM

Originally Posted by paulbk108

I have an equation

q(s) = s^5 + s^4 + 4s^3 + 4Ks^2 + 2Ks + K

The answer I have indicated is 0.5361 < K < 0.9326.

Can this be done in Matlab, and if so how?

My apologies I am still feeling my way through Matlab.

Regards.

Solve what, this is not an equation for solving? State clearly exactly what you want to find and what you have to start with. In other words tell us what the real question is.

For example do you mean find $$$K$ such that:

$s^5 + s^4 + 4s^3 + 4Ks^2 + 2Ks + K=0$ ?

(which is verging on the trivial, so not likely to be what you mean)

CB

**paulbk108** · February 4th 2011, 12:03 AM

s^5 + s^4 + 4s^3 + 2Ks + K = 0 is the denominator of a given transfer function.

The question asks to find out whether the system is stable using the Routh Hurwitz criterion, then determine its range of stability for K > 0. I thought it may be possible to be done on Matlab...

**CaptainBlack** · February 4th 2011, 02:17 AM

Originally Posted by paulbk108

s^5 + s^4 + 4s^3 + 2Ks + K = 0 is the denominator of a given transfer function.

The question asks to find out whether the system is stable using the Routh Hurwitz criterion, then determine its range of stability for K > 0. I thought it may be possible to be done on Matlab...

There is a link to a Matlab script to do this on the Wikipedia page for the Routh Hurwitz criterion.

CB

**chisigma** · February 4th 2011, 04:44 AM

Originally Posted by paulbk108

I have an equation

q(s) = s^5 + s^4 + 4s^3 + 4Ks^2 + 2Ks + K

The answer I have indicated is 0.5361 < K < 0.9326.

Can this be done in Matlab, and if so how?

My apologies I am still feeling my way through Matlab.

Regards.

People without Mathlab may try to solve the problem searching the value of k for which the roots of...

$\displaystyle q(s)=s^{5} + s^{4} + 4\ s^{3} + 4\ k\ s^{2} + 2\ k\ s + k$ (1)

... have all negative real part use the following approach. Let's suppose to search the values of k for which $q(s)$ has a factor $s^{2}+ \alpha$ with $\alpha>0$ , such that is...

$q(s)= (s^{2}+ \alpha)\ (s^{3} + \beta_{2} s^{2} + \beta_{1}\ s + \beta_{0})$ (2)

Comparing (1) and (2) we arrive to the following system of equation...

$\beta_{2}=1$

$\alpha + \beta_{1}=4$

$\beta_{0} + \alpha\ \beta_{2}=4\ k$

$\alpha\ \beta_{1}= 2\ k$

$\alpha\ \beta_{0}=k$

... which leads to the pair of equations...

$\alpha^{2} - 4\ \alpha + 2\ k=0$

$\alpha^{2} - 4\ k\ \alpha + k=0$ (3)

... the solutions of which are...

$\alpha=0\ ,\ k=0$

$\alpha=3.461072192556...\ ,\ k=.932634024069...$

$\alpha=.2889278074438...\ ,\ k=.53611597593...$

Kind regards

$\chi$ $\sigma$

**paulbk108** · February 7th 2011, 12:45 AM

Thanks so much for this solution. Could you please explain it with some greater detail?

By the way - the Matlab script I was referred to is a great piece of script - but does not accept terms such as "k"...

Regards.

**CaptainBlack** · February 7th 2011, 12:54 AM

Originally Posted by paulbk108

By the way - the Matlab script I was referred to is a great piece of script - but does not accept terms such as "k"...

Regards.

Use it inside a bisection search to locate the end point of the interval you are looking for. Alternativly eveluate the thing on k=[0:0.001:10] and plot the result.

CB

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