Template Method for Exact Value Calculation of Root Locus Break Points

Hassan Shibly, Orwah H. Shibly


One of the methods used to examine stability, to analyze, and to design a linear time invariant control system for a single input and a single output is the Root Locus technique. It is known that Root Locus technique is a powerful and efficient mean used in control of the systems. In addition, it determines the range of gain for a specific type of the time response of a control system. There are several important points on the root locus graph along the real axis of the s-plane. These points are known as the Breakaway, and the Break-in points. In this article those points are called Break points, and the polynomial that some of its roots Break points is called Break polynomial. In this article another method is presented for obtaining the Break polynomial. This method is called the Template method. The mathematical proof of the correctness of the basis of the method is presented. In this method the Break polynomial is obtained by programing or by hand calculation as with other common methods. The hand calculation's part is presented only. The merit of this method is that it is applicable for any system's order, and it can handle complex poles and complex zeros. It is shown that the technique of the template filling is a systematic procedure. So that if a template is used for a specific order of control system it can be expanded to handle higher order system just by adding another row on the top of the existing template. The Template method can be computer programmed and if it is incorporated within the MATLAB root locus graphs program it enhances the graph by presenting the exact values of the Break points on it. The method is compared with other common methods in the solution of examples of control systems to show its simplicity for users, and to show its correctness for its capability of handling various orders of a control systems. The results showed that the method gives accurate results and the required number of mathematical operations of the calculations to obtain the result is about 30% of the mathematical operations required by other methods.


Break-in point calculation; Breakaway points’ calculation; Break polynomial's calculation; Gain at breakaway point; Template method for break points' calculation

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