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

Purushottam Patil and C. Sankar Rao

# Abstract

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.

# Appendix

## 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

(35)G(s)=1+GGc=0

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

(36)G(s)=a0+a1s+a2s2+a3s3+......+ansn

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

(37)G1(s)=a0l+a1ls+a2us2+a3us3+......
(38)G2(s)=a0l+a1us+a2us2+a3ls3+......
(39)G3(s)=a0u+a1ls+a2ls2+a3us3+......
(40)G4(s)=a0u+a1us+a2ls2+a3ls3+......

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.

# Nomenclature

Greek alphabets:
α

Coefficient for s3

β

Coefficient for s2

γ

Coefficient for s

τ

Process time constant

τI

Integral time constant

τD

Derivative time constant

µ

Growth rate for Monod kinetics

Abbreviations:
PID

Proportional, Integral, Derivative controller

NMP

Non-Minimum Phase

FOPTD

First Order plus Time Delay

SOPTD

Second Order plus Time Delay

USOPTD

Unstable second order plus time delay

IMC

Internal Model Control

SA

Stability Analysis

IAE

Integral of Absolute Error

ITAE

Integral of Time weighted Absolute Error

ISE

Integral of Squared Error

TV

Total Variation in manipulated variable

G

Process transfer function

GC

Controller transfer function

MS

Maximum Sensitivity function

SOPTDZ

Second Order Plus Time Delay with Zeros

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