A Novel Two-Polynomial Criteria for Higher-Order Systems Stability Boundaries Detection and Control

N. Mir-Nasiri, Md. Hazrat Ali


There are many methods for identifying the stability of complex dynamic systems. Routh and Hurwitz’s criterion is one of the earliest and commonly used analytical tools analysing the stability of dynamic systems. However, it requires tedious and lengthy derivations of all components of the Routh array to solve the stability problem. Therefore, it is not a simple method to define analytically, stability boundaries for the coefficients of the system characteristic equation. The proposed brand-new criterion is an effective alternative technique in identifying stability higher-order linear time-invariant dynamic system that binds the coefficients of the system characteristic polynomial at the stability boundaries by means of an additional single constant k. It defines the necessary and sufficient conditions for the absolute stability of higher-order dynamic systems. It also allows the analysing of the system’s precise marginal stability or marginal instability condition when the roots are relocated on imaginary -axis of s-plane. The criterion proposed by the authors, in contrast to Routh criteria, simplifies the identification of maximum and minimum stability limits for any coefficient of the higher-order characteristic equation significantly. The derived in the paper stability boundary formulas for the polynomial coefficients are successfully used for the PID controller with single or multiple gains selections in closed-loop control systems.


Higher-order dynamics; Characteristic polynomial; Stability boundaries; Absolute; Marginal stability;

DOI: http://dx.doi.org/10.12928/telkomnika.v18i6.16256

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