IntermediateTopic 51 of 101Signals, Control & Comms
Stability and Routh-Hurwitz
A criterion to check root locations without solving polynomials.
In depth
Stability needs all closed-loop poles in the left half-plane. Routh-Hurwitz builds an array from characteristic polynomial coefficients; sign changes in the first column equal right-half-plane poles. It finds the range of gain K for stability without solving roots.
Key points
- Necessary: all coefficients same sign
- Row of zeros special case
- Marginal gain
ExampleFor s³+3s²+2s+K, stability requires 0<K<6.
What is Stability and Routh-Hurwitz?
A criterion to check root locations without solving polynomials.
Why is it important to understand stability and Routh-Hurwitz?
The Routh-Hurwitz criterion checks stability from polynomial coefficients without finding the roots, which saves time in early design.
What does the Routh array tell you?
The number of roots in the right half of the s-plane, which indicates the number of unstable poles.
What is a special case in the Routh array?
A zero in the first column or a full row of zeros, which needs special handling such as an auxiliary polynomial.
How is Stability and Routh-Hurwitz used in practice?
Typical applications include checking whether a controller gain keeps a loop stable before simulation, finding the maximum gain that keeps a system stable and testing a high-order characteristic polynomial for stability.
What should I learn before Stability and Routh-Hurwitz?
Study "Steady-state error" first (topic 50 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
What are the key points of Stability and Routh-Hurwitz?
Necessary: all coefficients same sign. Row of zeros special case. Marginal gain.
Can you give an example of Stability and Routh-Hurwitz?
For s³+3s²+2s+K, stability requires 0<K<6.
What comes after Stability and Routh-Hurwitz?
The next topic is "Root locus", which builds directly on this one.
How difficult is Stability and Routh-Hurwitz?
It is classed as Intermediate in Signals, Control & Comms (topic 51 of 101). It is an intermediate topic. Work through the beginner topics in Signals, Control & Comms first.