Routh-Hurwitz Criterion
Suppose we are given an
where
A nonlinear autonomous system can be reduced to the linear system by performing a linearization around an equilibrium point. Then without loss of generality, we may assume that the equilibrium point is at the origin. It is always possible to reach by choosing a suitable coordinate system.
The stability or instability of the equilibrium state is determined by the signs of the real parts of the eigenvalues of
which is reduced to an algebraic equation of the
The roots of this equation can be easily calculated in the case
In such a situation, methods allowing to determine whether all roots have negative real parts and establish the stability of the system without solving the auxiliary equation itself, are of great importance. One of these methods is the Routh-Hurwitz criterion, which contains the necessary and sufficient conditions for the stability of the system.
Consider again the auxiliary equation
describing the dynamic system. Note that the necessary condition for the stability is satisfied if all the coefficients
The main diagonal of the matrix contains elements
The principal diagonal minors
We now formulate the Routh-Hurwitz stability criterion: The roots of the auxiliary equation have negative real parts if and only if all the principal diagonal minors of the Hurwitz matrix are positive provided that
For the most common systems of the
For a second order system, the condition of the stability is given by
or
that is, all coefficients of the quadratic characteristic equation must be positive. In other words, for a system of
For a
or
Similarly, for a
or
If all the
- The coefficient
This corresponds to the case when one of the roots of the auxiliary equation is zero. The system is on the boundary of the aperiodic stability. - The determinant
In this case, there are two complex conjugate imaginary roots. The system is on the boundary of the oscillatory stability.
The Routh-Hurwitz stability criterion belongs to the family of algebraic criteria. It can be conveniently used to analyze the stability of low order systems. The computational complexity grows significantly with the increase of the order. In such cases, it may be preferable to use other criteria such as the Lienard-Shipart theorem or the Nyquist frequency criterion.