routh stability criterion and special cases in routh stability criteria in control theory

When using any system, specifically control systems, it is necessary to make the system stable, and we may wonder what is meant by the stability of the system and when the system is stable and when it is unstable, is the system semi-stable, and what are the ways by which we determine the stability of the system to answer all these questions, we will explainin this article the definition of stability and types of stability. also, we will explain one of the important methods for knowing the stability of the system, which is called the routh method, relative to the mathematician e.j. routh, we will develop a simplified table for calculating a routh table, and we will discuss some special cases (special cases in routh stability criteria) that cross a controversial mission, and give many examples to explain this method more broadly.