A Novel PID Controller with Second Order Lead/Lag Filter for Stable and Unstable First Order Process with Time Delay

Praveen Kumar Medarametla; Manimozhi Muthukumarasamy

doi:10.1515/cppm-2017-0010

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Article

A Novel PID Controller with Second Order Lead/Lag Filter for Stable and Unstable First Order Process with Time Delay

Praveen Kumar Medarametla and Manimozhi Muthukumarasamy

Published/Copyright: February 24, 2018

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From the journal Chemical Product and Process Modeling Volume 13 Issue 1

Abstract

A novel Proportional-Integral-Derivative (PID) controller is proposed for stable and unstable first order processes with time delay. The controller is cascaded in series with a second order filter. Polynomial approach is employed to derive the controller and filter parameters. Simple tuning rules are derived by analysing the maximum sensitivity of the control loop. Formulae are provided for initial guess of tuning parameter. The range of tuning parameter around the initial guess and the corresponding range of maximum sensitivity is specified based on time delay to time constant ratio. Promising results are obtained with the proposed method is compared against recently proposed methods in the literature. The comparison is made in terms of various performance indices for servo and regulatory responses separately. The proposed method is implemented for an isothermal chemical reactor at an unstable equilibrium point.

Keywords: unstable process; time delay; PID controller; maximum sensitivity; isothermal chemical reactor

Appendix A

Consider an isothermal chemical reactor as shown in Figure 15.

Figure 15:

Block diagram of an isothermal chemical reactor.

Mass balance equation can be written as

{Accumulation of = {Component A in}−{Component A out}+{Generation of A}Component A}

(31)

$(MWAVCA)t+Δt−(MWAVCA)t=(MWAFCAo−MWAFCA)Δt+MWAVrAΔt$

Where

$MWA$ = molecular weight of component A

$V$ =Volume of chemical reactor $(l)$

$CAo$ =Inlet concentration of Component A $(mol/l)$

$CA$ = Out let concentration of component A $(mol/l)$

F=Inflow rate (l/s)

$rA$ = Reaction rate

Dividing the equation with $MWAVΔt$ and considering $Δt→0$

(32)

$dCAdt=FV(CAo−CA)+rA$

Under non ideal mixing conditions [21], $rA$ can be modelled as

(33)

$rA=−k1CA(k2CA+1)2$

Where $k1$ and $k2$ are constants. Substituting eqs (32, 33),

(34)

$dCAdt=FV(CAo−CA)−k1CA(k2CA+1)2$

At steady state

(35)

$dCAdt=0$

Using eq. (34) and (35),

(36)

$FV(CAo−CA)=k1CA(k2CA+1)2$

This isothermal chemical reactor is studied by various researchers [8, 18, 19] for values $F=0.0333l/s$ , $V=1l$ , $CAo=3.288mol/l$ , $k1=10l/s$ , $k2=10l/mol$ . Substituting the above values and solving eq. (36) results three steady states at

$CA=1.7673mol/l,1.3065mol/l,0.0142mol/l$

Linearizing the eq. (34) at steady state $CA=1.3065mol/l$ using Taylor’s series expansion (Neglecting higher order terms) gives following transfer function

(37)

$CA(s)CAo(s)=3.433103.1s−1$

Considering a measurement lag of 20s due to concentration transducer,

(38)

$CA(s)CAo(s)=3.433e−20s103.1s−1$

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Received: 2017-3-31

Revised: 2017-5-27

Accepted: 2017-5-28

Published Online: 2018-2-24

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Articles in the same Issue

https://doi.org/10.1515/cppm-2017-0010

Keywords for this article

unstable process; time delay; PID controller; maximum sensitivity; isothermal chemical reactor