Formulation Method of Gain Calculation at Marginal Stability of a Linear Invariant Control Systems

Hassan Shibly, Orwah H. Shibly


The stability analysis of a linear invariant control system is based mainly on its characteristic equation. There are various methods to examine, to do analysis, and to design a control system. Those methods become less effective and more complicated for use when those methods are used for a high order of an open loop transfer function that has several poles and several zeros.  In this research work a new method was developed to find the gains at marginal stability and the intersection points with the imaginary axis of a single-input and single-output of a linear invariant control system by using two new formulas. First formula is used to construct a new polynomial where its roots are the intersection points with the imaginary axis of the s-plane, and a second formula is used to calculate the gains at the marginal stability of the system. The coefficients of the characteristic equation’s polynomial of the control system are substituted in the first formula to obtain a new polynomial. The roots of the obtained polynomial are substituted in the second formula to obtain the gains at marginal stability. In this research work the derivation of the polynomial’s construction formula, its mathematical proof, and the derivation of the gains formula at marginal stability are presented. The proposed Formulization method is compared with another three common methods in the solution of three examples. The used methods are the proposed method, Routh-Hurwitz criterion, Root Locus technique, and the complex variable s on the imaginary axis. The chosen examples are three control systems where their transfer functions are different in order and in complexity, going from low to high. The comparison shows that the Formulization method is accurate and needs less mathematical operations by the user. It is applicable for any order of a single-input and single-output of invariant control systems. It is an effective method especially for a higher order and for more complicated transfer functions of the control systems.


Gains at marginal stability; Gains calculation at the intersection points; Intersection points with the imaginary axis; Marginal stability; Polynomial of intersection points.

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