The generalized Routh array and a further simplification of the extended Routh array

Abstract

The well-known Routh Array settles the problem of stability of systems of differential equations with rral coefficients. The Extended Routh Array (ERA) is the complex counterpart of the Routh Array and it settles the stability of these systems when the coefficients are complex. Since its construction, the ERA remained more of a theoretical achievement, than a practical tool to test the stability of linear systems. Some attempts were made to overcome the complexity of the ERA. The Modified Extended Routh Array (MERA) was then introduced and it reduced the burden of computations, but still it involved lots of divisions and many operations with complex numbers. In the present work, we use the interlacing property to propose an equivalent criterion to both ERA and MERA, we call the Generalized Routh Array (GRA). The new array has advantages over both ERA and MERA in the sense that neither division algorithm, nor operations of complex numbers are involved. An example is given to illustrate the feasibility of the new test.