Array Technique to Calculate the Breakpoints on Root Locus Graph and Related Gains

Year 2021, Volume: 5 Issue: 2, 66 - 63, 30.06.2020

Hassan Shibly , Aalaa Shibly

Abstract

The root locus technique is a powerful and efficient mean to examine stability, and to analyze single input single output linear time invariant system. In addition, the gain range for any response type of the control system be determined. Some of the important points on a root locus graph of control system are the Breakaway, and Break-in points. In this article those points are called Break points, and the polynomial that some of its roots are Break points, is called Break polynomial. After leaving a Break point on root locus graph, the type of some roots of the system characteristic equation changes. The change is from real to a complex at Breakaway point, and from complex to real at break-in point. The type change of roots causes a type change of the system response. The response type of a system is a crucial matter for industrial machines applications. The development of a new method called the Array method is presented. The Array method is a technique to obtain the Break polynomial where several of its roots are the Break points. This technique is based on constructing an array. Then the array is filled by the polynomials' coefficients of the open loop transfer function's denominator and numerator. The mathematical proof of the method bases, and correctness is presented. It shows that the obtained Break polynomial by the proposed method is the same derived polynomial by the most used methods. The proposed method is compared with other methods in solution of examples of control systems to demonstrate its simplicity for the user and its correctness for any order of a single input single output linear invariant control system.

Keywords

Array method for Break points calculation, Breakaway polynomial, Breakaway points' calculation, Breakin points'calculation

Thanks

No supporting

References

• [1] Evans, W.R. “Graphical Analysis of Control System”, AIEE Transactions, vol. 67, pp. 547-551. 1948.
• [2] Evans, W.R., “Control System Synthesis and Root Locus Method”, AIEE Transactions, vol. 69, pp. 66-69, 1950.
• [3] Benjamin C. Kuo, “Automatic Control Systems”, Prentice Hall, 3rd Edition, 1975.
• [4] Richard C. Dorf, Robert H. Bishop, “Modern Control Systems”, Pearson, 13th Edition, 2016.
• [5] Katsuhiko Ogata, “Modern Control Engineering”, Pearson, 5th Edition, 2010.
• [6] Hassan Shibly, Orwah H. Shibly, “Formulation Method for Root Locus Calculation of Breakaway and Break-in Points and the Corresponding Gain”, Advances in Systems Science and Applications, ASSA journal, Vol 21 No 1, 2021-03-31.
• [7] Norman Nise, “Control Systems Engineering”, 7th Edition, 2015.
• [8] Remec M. J., “Saddle-Points of a Complete Root Locus and an Algorithm for Their Easy Location in the Complex Frequency Plane”, Proc. Natl. Electronics Conference, V. 21, pp 605-608, 1965.
• [9] Franklin, G. F.; Powell, J.D.; Emami-Naeini, A., “Feedback Control of Dynamic Systems, 2nd Edition, Addison-Wesley, 1991.
• [10] Krishnan V. “Semi-Analytic Approach to Root Locus”, IEEE Transaction, Automatic Control, V. AC-11, pp102-108, 1966.
• [11] Fitzgerald R.J., “Finding Root Locus Breakaway Points with the Spirule”, IEEE Transaction, Automatic Control, V. AC-11, pp 628-629, 1966.