The Routh-Hurwitz Stability Criterion

In this video we explore the Routh Hurwitz Stability Criterion and investigate how it can be applied to control systems engineering. The Routh Hurwitz Stability Criterion can be used to determine how many roots of a polynomial are in the right half plane.

Topics and time stamps:
0:00 – Introduction and applications
1:23 – Hilarious math joke
3:00 – Procedure/algorithm to build Routh array
14:46 – Example with a 6th order polynomial
33:09 – Example of a special edge case
43:58 – Application of Routh-Hurwitz to root locus
53:09 – Generalization to any polynomial
1:15:54 – Summary of conditions for 2nd, 3rd, and 4th order polynomials

Additional References

Additional videos in this series:
-Introduction to Root Locus (TBD)

#Controls
#ControlTheory

[AimeePak]

AE511: Great summary at the end! I do appreciate the use of the Routh array and the "figure-8" for higher order polynomials, good way to remember how to solve for the stability criterion without having to remember each polynomial's conditions :)

[wohlgemuth_official]

Finally the joke made the year 2019 good. Thank you sir.

[krystoflis]

Thanks Chris! Good overview of Routh Horwitz

[pietjepuk]

Dear Christopher,

As an electrical enigineer I am used to parameters "ai" being built up by formulas containing component values L, C & R. So checking each constraint is quite nasty work in itself.
I am glad to see you talking about redundancy regarding the constraints for a 3rd order equation at time coordinate 1:04:00.
You prove that When a3>0,  “a0>0 & a2>0” both requires  and implies automatically  "b1= a2.a1 - a3.a0 > 0"
I believe that likewise you could prove: When a3>0,  “a0 > 0 and b1= a2.a1 - a3.a0 > 0” both requires  and implies automatically "a2 > 0"
Can you confirm that ?
If that is true, isn't it important to spell that out in a table like the one that you present at time 1:17:05 ?

Best regards Paul

[dunzhichen]

And by the way, nice joke about Polish people on the right side of the plane : )

[ianbernhardsgruetter]

Im prettty sure there is a small mistake at the end when you compute K. The Answer you get is correct in the end, but it should be: 66 > 6 +K —> 60 > K. Not 60 > 6K

[zaneyosif]

AE511: Great video! I'm curious the potential of dealing with higher than 4th order polynomials in the real world and the complexity of solving for the routh hurwitz criterion. Lets say that we wanted to integrate this within software to check for stability, as the system gets a higher order does the throughput/amount of computing power rise exponentially? Is this something that we would ever come across (higher than 4th order)?

[alanlee]

AE511-nice sound effects
Note: (1:19:05) 4th order summary is missing the a0 > 0 check.

[yanghao]

The link of RouthArray.m function is not found. Could you please upload the link again? Thank you

[zafaryabkhan]

In 2nd order-
What will happen if we have :
a2>0 (from assumption),
And from a1 & a0, one of them is equal to zero & another one is greater than zero.

[mohamedalbira]

thank you sir for you effort and please what is the reference that is used in this tutorial

[sethwhittington]

AE511: This seems very useful for quick instability checks once you have your transfer function in the Laplace domain. However, I think that due to my greater familiarity with state space models and my natural understanding of time domain, checking for stability with a state space model feels more comfortable. i.e., wouldn't the method described in this video be equivalent to finding the eigenvalues of the A matrix and ensuring they have negative real parts? (if I'm remembering that correctly)

[gesucristo]

Could you post the link of the routh function again please?

[farhadzaidi]

Sir, what to do if denominator is 5 degree polynomial while the numerator is also a 3 degree polynomial instead of 1

[dvaVivy]

I tried doing this on my homework and i cant seem to get all of them...

[andrewchoi]

lol @ 1:23 hilarious math joke timestamp

[rahulbhavsar]

Which software we should learn for get control system engineering related job in India

[lolerie]

1:05:07 That is horrible idea. All coeff. should be all positive or all negative for any degree to be stable! So, in fact it is more simple to check all coeff. being positive first.