routh stability criterion and lyapunov-routh method in control theory

One of the most important issues in the linear control subject is to obtain eigenvalues of the system to study the stability of a system. it is needed to identify the sign of eigenvalues but not the value of it. to fulfil this, there are different methods such as routh stability criterion, lyapunov’s method and nyquist stability criterion. in this research, we will present the most simple one to determine the sign of eigenvalues and we shall discuss and explain different types of stability. also,  we will discuss some special cases that cross a controversial mission, and a new method is proposed to calculate the stability of systems, which we call the lyapunov-routh method (composition of lyapunov and routh method).