In control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) control system. The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in 1876 to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts. German mathematician Adolf Hurwitz independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all positive. The two procedures are equivalent, with the Routh test providing a more efficient way to compute the Hurwitz determinants than computing them directly. A polynomial satisfying the Routh–Hurwitz criterion is called a Hurwitz polynomial.

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# Routh-Hurwitz Criterion

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## Routh-Hurwitz Criterion, Special Cases

This video presents two special cases that you can encounter when filling out the Routh Array. The first case is when there is a zero in a row with at least one non-zero element following...

See More## Routh-Hurwitz Criterion, Beyond Stability

This video explains of few uses of the Routh-Hurwitz Criterion that go beyond simply determining how many poles exist in the right half plane. I cover how to determine gain margin and how...

See More## Routh-Hurwitz Criterion, An Introduction

This video gives an introduction into the Routh-Hurwitz Criterion and the Routh Array. I also present a little background information in order to emphasize why the method was developed and...

See More## 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 Stabi...

See More## Routh Array and Stability

I show how we can find the range of allowed controller gains for a system that will allow us to maintain stability using a Routh Array.

See More## Routh Stability Criterion Intro and Example

I introduce and walk through an example problem of how we can use the Routh Stability Criterion to rigorously determine the necessary and sufficient conditio...

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