superseding of Routh-Hurwitz using stability criterion using MATLAB software

Abstract

The stability of a process control system is extremely important to the overall control process. System stability serves as a key safety issue in most engineering processes. If a control system becomes unstable, it can lead to unsafe conditions. Instability in reaction processes or reactors can lead to runaway reactions, resulting in negative economic and environmental consequences. The absolute stability of a control process can be defined by its response to an external consistent state or set point and returns to steady immediately after a system disturbance. In order to determine the stability of a system, one often must determine the Eigen values of the matrix representing the system’s governing set of differential equations. Unfortunately, Routh Hurwitz method determines whether the system is stable or unstable or critical stable. If there is no change of the sign the system is stable, and all roots lie on the lift half plane, if there is change of sign the system is unstable, and all roots or some of them lie on the right half plane. The number of sign change gives the number of roots of the characteristic equation that lie on the right half plane on the Argand diagrams. If the coefficient of the last row is zero the characteristics equation has one root on the origin. If the coefficient of the last two rows are zeros the characteristics equation has two roots at the origin. If some of the roots lie on the left half plane and other in imaginary axis the system is critically stable. Routh array can be used to determine the ultimate gain only and it is superseded by root solving method as the relative stability determine the region in which the roots are located. I took several different equations and solved them by means of Routh Hortz to determine the system stable or unstable or critical and to solve the same equations used the solution by MATLAB and, in comparison, found that the method of Routh Hurtz only determines that the system is stable or unstable or critical and this is done in several steps and by MATLAB solved in one step and clarified the type of system as well as found the butcher and determined its location on the chart of Argand and thus saved time and effort and reduced costs and also used cases of the theory of relative stability and compared to the method of solution by MATLAB also found that the solution by MATLAB saves time and effort and gave real results.