Analysis of Effects due to Right Half Plane Zeros in PI Controller based Hydro Turbine ⋆ Soumyadeep Bose * Yogesh V. Hote * Sandeep D. Hanwate * * Department of Electrical Engineering, Indian Institute of Technology Roorkee, Roorkee-247667, India(e-mail: soumyadeepb9@gmail.com, yhotefee@iitr.ac.in). Abstract: In this paper, some detrimental time-domain characteristics (zero-crossings, over- shoot(due to zeros) and initial undershoot) are analyzed using the system transfer function model, for identifying the presence of these effects in step response of non-minimum phase linear systems. Moreover, regarding initial undershoot, a theorem for detection using the state- space model, without the need to obtain transfer function, is proposed. In addition to detection, a theorem for estimating the percentage of initial undershoots is also proposed. Applications of these theorems are carried out on load frequency control of hydro-electric power plant which employs hydro-turbine, a non-minimum phase system. Moreover, the effect of a PI controller on initial undershoot is discussed for a general non-minimum phase linear system and illustrated for load frequency control of hydro-electric power plant. Keywords: Initial undershoot, load frequency control, Markov parameters, overshoot, step response, zero-crossings 1. INTRODUCTION Control system engineering has undergone extensive re- search since 1950s, and can be classified in numerous ways. In terms of system properties, a major class of sys- tems are known as Non-Minimum Phase (NMP)systems. Such systems, when formulated in linear terms, contain at least one pole or zero on the Right-Half-Plane (RHP) of the s-plane, as defined in Dorf and Bishop (2014). From asymptotic stability point of view, asymptotically stable systems whose inverse gives unstable impulse response are the stable NMP systems. NMP systems have some drawbacks in contrast to Minimum Phase (MP) systems. For instance, because of an unstable zero, additional phase lead or lag is introduced in frequency domain, causing the closed-loop system become prone to instability. From the controller perspective, presence of a negative term in the denominator of the closed-loop system reduces the stability boundary for tuning the control parameters, thus posing limitations on system performance. Moreover, Qiu and Davison (1993) mentioned that MP systems are ad- vantageous over NMP ones, for instance, regarding the ac- curacy in regulation, tracking and robustness is assured by MP systems possessing the property of right invertability. Authors of this paper also mentioned that some NMP sys- tems perform almost as good as an MP system while some others are impossible to control to give desired response. Although not so prevalent, many practical systems such as drum boiler performance reported by ˚ Astr¨om and Bell (2000), work of Su and Khorasani (2001) on single link flexible manipulator, small-signal performance of a SEPIC inverter in Hegde and Izadian (2013), vertical take-off and ⋆ This work was not supported by any organization. landing system of an aircraft in Boekfah (2017) etc. exist in daily life. Constricting the discussion to the main context of this paper, RHP zeros, besides the effects discussed so far, also impose some unique drawbacks when it comes to the step response of these systems. These effects are zero-crossings, overshoot (due to zeros) and initial undershoot, which are formally defined and described in Hoagg and Bernstein (2007). While zero-crossings occur only for NMP systems’ responses, presence of overshoot (due to zeros) and initial undershoot are indirectly linked to NMP systems. Mul- tiple number of zero-crossings with respect to the initial and steady-state values introduce undesirable ringing and sluggishness in the system’s response, which are often very difficult to nullify. Hence, analysis on the occurrence and tackling these effects is important to improve NMP system’s performance. In this paper, theorems to identify presence of the dis- cussed effects in step response of Continuous-Time, Linear Time-Invariant (CTLTI) NMP systems (using respective transfer functions) are analyzed. The main contribution of this paper is the proposal of theorems to identify and estimate initial undershoot in step response of NMP sys- tems. The major advantage of the proposed theorems is this, that the analysis can be done using both the state- space and transfer function form of the system’s model. This is in contrast to the reported literature on identi- fication of this effect, where the theorems are limited to the transfer function model only. The theorems are used to analyze a practical problem of Load Frequency Control (LFC) of a hydro-electric plant, which involves an NMP Preprints of the 3rd IFAC Conference on Advances in Proportional- Integral-Derivative Control, Ghent, Belgium, May 9-11, 2018 FrAT4.6 © 2018 International Federation of Automatic Control 633