Enfoque UTE, V.9-N.4, Dec.2018, pp. 69 - 76 http://ingenieria.ute.edu.ec/enfoqueute/ e-ISSN: 1390‐6542 / p-ISSN: 1390-9363 Recibido (Received): 2018/05/15 Aceptado (Accepted): 2018/10/31 CC BY 4.0 Root-Locus Analysis of Delayed First and Second Order Systems (Análisis del Lugar Geométrico de las Raíces de Sistemas de Primer y Segundo Orden Retardados) M. Ríos-Flores 1 , J. F. Marquez-Rubio 1 , B. del Muro-Cuellar 1 , E. Aranda-Bricaire 2 . Abstract: For finite dimensional linear system the root-locus method is well established however for the case of delayed systems the method has some problems due to the transcendental term involved. This work intends to illustrate the problems that arises when a root-locus diagram is performed as well as to develop a Matlab function that provides the root-locus diagram for delayed low order systems. In this way, some comments about the problems that should be tackled to obtain a generalization of the computational method for delayed systems with real m poles and n zeros Keywords: time-delay; root locus diagram; feedback control; poles; zeros. Resumen: Para los sistemas lineales de dimensión finita el lugar geométrico de las raíces es un método bien establecido, de cualquier manera para el caso de los sistemas retardados el método tiene algunos problemas debido al término de retardo involucrado. Este trabajo pretende ilustrar los problemas que surgen cuando se realiza el lugar geométrico, así como al desarrollar una función de Matlab que proporcione el lugar geométrico de las raíces para sistemas de bajo orden con retardo. De igual manera, se realizan comentarios acerca de los problemas que deberán ser considerados para obtener una generalización de un método computacional para sistemas con retardo con m polos y n ceros. Palabras clave: Término de retardo; diagrama del lugar geométrico de las raíces; retroalimentación de control; polos; ceros. 1. Introduction Time delays appearing in the modeling of different classes of systems (chemical processes, manufacturing chains, economy, etc.), become a challenging situation from a control viewpoint that should be affronted to yield acceptable closed-loop stability and performance. Several control strategies as well as stability analysis have been developed to deal with time delays. When the continuous case is considered, the delay operator can be approximated by means of a Taylor or Padé series expansions which could leads to a non-minimum-phase process with rational transfer function representation (Gouaisbaut, 2006). With the same stability purpose analysis, some works have applied the Rekasius substitution; see for instance (Munz, 2009). It should be noticed that using the mentioned strategies and if a closed-loop stability analysis is desirable, the stability results obtained are limited due to the corresponding approximations. On the other hand, (Silva & Bhattacharyya, 2005) provided a complete parametrization of the stabilizing P, PI and PID controllers in the case of first order system plus time-delay (FOPTD) by using a frequency approach analysis. It is important to mention that the used stability analysis is not easy to extend to high order systems due to the fact that many stability conditions should be satisfied and in some cases the problem 1 Instituto Politécnico Nacional, CDMX – México (moy30spl@gmail.com; jfcomr23@yahoo.com.mx; bdelmuro@yahoo.com). 2 CINVESTAV IPN, CDMX – México (earanda@cinvestav.mx).