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Licensed Unlicensed Requires Authentication Published by De Gruyter March 20, 2019

Enhanced PID Controller for Non-Minimum Phase Second Order Plus Time Delay System

Purushottam Patil and C. Sankar Rao ORCID logo


A tuning method is developed for the stabilization of the non-minimum phase second order plus time delay systems. It is well known that the presence of positive zeros pose fundamental limitations on the achievable control performance. In the present method, the coefficients of corresponding powers of s, s2 and s3 in the numerator are equated to α, β and γ times those of the denominator of the closed-loop system. The method gives three simple linear equations to get the PID parameter. The optimal tuning parameters α, β and γ are estimated by minimizing the Integral Time weighted Absolute Error (ITAE) for servo problem using fminsearch MATLAB solver aimed at providing lower maximum sensitivity function and keeping in check with the stability. The performance under model uncertainty is also analysed considering perturbation in one model parameter at a time using Kharitonov’s theorem. The closed loop performance of the proposed method is compared with the methods reported in the literature. It is observed that the proposed method successfully stabilizes and improves the performance of the uncertain system under consideration. The simulation results of three case studies show that the proposed method provides enhanced performance for the set-point tracking and disturbance rejection with improved time domain specifications.


A Kharitonov’s Theorem [28]

This theorem gives interval for the coefficient of the characteristic equation for which the system is stable by determining the stability of vortex polynomial obtained for the boundary range of the coefficients.

The closed loop characteristic equation is given by


The characteristic equation obtained by applying Taylor’s series expansion for time delay approximation for the interval system is stated as follows


Where, ai ϵ [ail, aiu],

For i = 1,2, … n., ail is the lower limit and aiu is the upper limit,

The characteristic polynomial is said to be stable only if all four khairtonov polynomial are stable. Their stability is found by applying Routh hurwitz criterion to each equation. The khairtonov’s polynomials are gives as


The coefficient polynomial (36) is stable if and only if all the four vertex polynomials (37–38) are stable. Initial values of kp, a1 and a2 are fixed and perturbation in time delay L is substituted with limits (LΔL)<L<(L+ΔL) in coefficients and Kharitonov’s polynomials are obtained. These polynomial’s stability is checked with Routh–Hurwitz method. In similar procedure, stability regions for kp, a1 and a2 are obtained by varying objective parameter and keeping other parameters constant.


Greek alphabets:

Coefficient for s3


Coefficient for s2


Coefficient for s


Process time constant


Integral time constant


Derivative time constant


Growth rate for Monod kinetics


Proportional, Integral, Derivative controller


Non-Minimum Phase


First Order plus Time Delay


Second Order plus Time Delay


Unstable second order plus time delay


Internal Model Control


Stability Analysis


Integral of Absolute Error


Integral of Time weighted Absolute Error


Integral of Squared Error


Total Variation in manipulated variable


Process transfer function


Controller transfer function


Maximum Sensitivity function


Second Order Plus Time Delay with Zeros


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Received: 2018-10-26
Revised: 2019-02-12
Accepted: 2019-03-04
Published Online: 2019-03-20

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