A NOVEL TWO-POLYNOMIALS CRITERIA FOR HIGHER-ORDER SYSTEMS STABILITY BOUNDARIES DETECTION AND CONTROL

There are many methods of identifying general stability of complex dynamic systems. Routh and Hurwitz’s criterion is one of the earliest and commonly used analytical tools analysing stability of a dynamic system. However, it requires redundant calculation of all the elements of the Routh array to identify stability, even the low-order system. Therefore, it is not a simple method to identify, especially analytically, the stability boundaries for the coefficients of the characteristic equation due to tedious and lengthy derivation of all the Routh array elements. The proposed brand-new criterion or algorithm is an effective alternative and a universal technique to identify analytically the stability of up to sixth-order linear time-invariant dynamic system based on the set of unique for all possible system equations (2) and (3) that relate coefficients of the system characteristic polynomial at the stability boundaries by means of a single additional constant k. The expressions derived on this basis for a higher-order dynamic system can be used effectively to identify the boundaries of its stable behaviour spans. It defines the necessary and sufficient conditions for absolute stability of higher-order dynamic systems. It also allows the analysing of the system’s precise marginal stability condition (whether stable or not) and the nature of the system roots at the stability boundaries, i.e. when they are relocated on imaginary jω-axis of s-plane. The criterion proposed by the authors, in contrast to Routh criteria, simplifies significantly the identification of maximum and minimum stability limits for any coefficient of the higher-order characteristic equation. The paper also presents the numerical analysis of stability boundaries for systems with order higher than six based on criteria (2) or (3). The derived stability boundary formulas (2) and (3) for the polynomial coefficients are successfully used for PID controller gains selection in close-loop control systems and this achievement does not have analogy in control theory.