New Simple Controller Tuning Rules for Integrating and Stable or Unstable First Order plus Dead-Time Processes

Transcription

1 Proceeings of the 3th WSEAS nternational Conference on SYSTEMS New Simple Controller Tuning Rules for ntegrating an Stable or Unstable First Orer plus Dea-Time Processes.G.ARVANTS Department of Natural Resources Management an Agricultural Engineering Agricultural University of Athens 75 era Oos Str., Botanikos 855, Athens GREECE A.G.SOLDATOS Department of Electrical an Computer Engineering National Technical University of Athens 9 roon Politechniou Str., Zographou 5773, Athens GREECE asolat@cc.ece.ntua.gr A..BOGLOU Department of nformatics Technology Eucation nstitute of avala Agios Loukas, 6544, avala GREECE akbogl@teikav.eu.gr N..BEARS-LBERS Department of Mechanical an Aerospace Engineering University of California, San Diego EBU Room, La Jolla, CA U.S.A. nbekiari@ucs.eu Abstract: - Three new irect synthesis methos proviing new simple rules of tuning the Pseuo-Derivative Feeback controller for integrating as well as stable or unstable first orer processes with time elay are presente. The propose control structure an tuning methos ensure smooth close-loop response to set-point changes, fast regulatory control an sufficient robustness against parametric uncertainty. Simulation results show that the propose methos is as efficient as the best of the most recent PD controller tuning methos that are known in the extant literature, while their simplicity in eriving the controller settings is a plus point over existing PD controller tuning formulae. ey-wors: - Dea-Time, Process Control, Direct Synthesis Metho, Stable Systems, Unstable Systems, ntegrator Processes, Controller Tuning ntrouction Time elays are common in all process inustries ue to transportation lags, recycle loops, composition analysis loops, etc. For the purpose of esigning controllers, the ynamics of many ea-time processes can be escribe aequately by self-regulating first-orer plus time elay (FOPDT) moels. Furthermore, the ynamic response of many processes is very slow with large ominant time constants. This type of processes can be approximate as integrating plus time elay for the purpose of esigning controllers instea of controlling in the original large time constant form. Designing controllers for integrating plus ea-time (PDT) moels provies superior close-loop performance than performing the esign base on FOPDT moels for nominal an moel uncertainty conitions. On the other han, several important processes an systems SSN: SBN:

2 Proceeings of the 3th WSEAS nternational Conference on SYSTEMS encountere in various fiels of engineering exhibit multiple steay states ue to system nonlinearities, some of them being inherently unstable by esign. For several reasons, like safety, maximization of prouctivity an reuction of economic costs, it is often esirable to operate such processes aroun their unstable steay states. To approximate the open loop ynamics of such systems an for the purpose of esigning controllers, many of these processes can satisfactorily be escribe by unstable first orer plus ea-time (UFOPDT) transfer function moels. Design P an PD controllers for FOPDT, PDT an UFOPDT processes has long been the focus of interest by many process control esigners, an numerous tuning techniques have been reporte in the extant literature. The interesting reaer may refer to [], for an extensive overview of such techniques. The most recent methos of tuning P/PD like controllers for the types of processes investigate in the present paper can be foun in []-[4]. These methos are avantageous over other existing tuning methos, an they will be, in the sequel, our basis for comparisons. The present paper investigates some aspects of the controller configuration propose in [5], an calle the pseuo-erivative feeback controller (PDF), which is put forwar here as an alternative means of tuning three-term controllers for FOPDT, PDT an UFOPDT processes. The aim of the paper is to propose new formulae for tuning the settings of the PDF controller when applie to such processes. The propose methos are alternatives of the irect synthesis tuning metho an they are base on the manipulation of the close-loop transfer function through appropriate approximations of the eatime term in the enominator of the close-loop transfer function as well as appropriate selection of the erivative gain, in orer to obtain a secon orer ea-time close-loop system. On the basis of these methos the settings of the PDF controller are obtaine in terms of two ajustable parameters, one of which can further be appropriately selecte in orer to achieve a esire amping ratio for the closeloop system, while the other is free to esigner an can be selecte in orer to enhance the obtaine regulatory control performance. For assessment of the effectiveness of the propose tuning methos an in orer to provie a comparison with existing tuning methos, a series of simulation examples are presente. Simulation results verify that the PDF control structure an the propose irect synthesis tuning methos ensure smooth close-loop response to set-point changes, fast regulatory control an sufficient robustness in case of moel mismatch. Low Orer Dea-Time Processes & the PDF Controller ntegrating an stable or unstable first orer plus ea-time processes can be escribe by the following general transfer function moel GP () s exp( s)/( q s+ q ) () (, ) for PDT processes ( q, q) ( T, ) for FOPDT processes ( T, ) for UFOPDT processes an,, an T, are the process gain, eatime an time constant respectively. The PDF controller has been propose in [4], an its general feeback configuration is shown in Fig.. The transfer function G () s of the close-loop system is given by G ( s) G () s n (,,, ) s + s s + s+ G ( s) P Dn D D P () The PDF controller is essentially a variation of the conventional PD controller. n contrast to the PD controller, the PDF controller oes not contribute to close-loop zeros, an hence it is expecte that it will not worsen the overshoot of the closeloop response. The two configurations iffer in the way they react to set-point changes (as it can be easily checke, they are equivalent for loa or isturbance changes). The PD controller often has an abrupt response to a step change because the step is amplifie an transmitte irectly to the feeback control element an ownstream blocks. This can inuce a significant overshoot in the response that is unrelate to the close loop system amping. For this reason, it is a common practice to ramp or filter the set-point. The PDF structure avois this because naturally ramps the controller effort, since it internalizes the pre-filter that one woul apply to cancel any zeros introuce in the P/PD control configuration. n this paper, our attention is focuse on the specific PDF configuration for which D, P, R(s) + _ E(s) s + + L(s) _ U(s) s s + PDF control structure G P (s) n D,n D, D, Fig.. The general PDF control structure. Y(s) SSN: SBN:

3 Proceeings of the 3th WSEAS nternational Conference on SYSTEMS D, an D,i, for i,,n-. This feeback scheme is esignate as the PD-F control structure, an its application to the control of ea-time processes of the form () yiels G ( s) exp( s) (3) q q exp( s) P s s s s 3 A Direct Synthesis Tuning Metho for the PD-F Controller A first metho of tuning the PD-F controller (Metho ) is as follows: Suppose that it is esire for the close-loop transfer function to have the form G, es () s exp( s)/ ( λ s + ζλs+ ) (4) Upon equating relations (3) an (4) we obtain ( s + P s+ exp( ) s) (5) ( λ q ) s + ( ζλ q ) s+ Relation (5), further yiels exp( s) ( λ ) ( q s + ζλ q s+ s + P s+ ) (6) Using the first-orer Pae approximation exp( s) (.5 s) / ( +.5s) (7) of the exponential elay term in (6), we obtain ( s + P s + )(.5s + ) (.5s + ) (8) ( λ q ) s + ( ζλ q ) s+ Upon equating like powers of s in both sies of (8), we take the following set of equations λ ( λ ).5.5 q (.5 ) P q +.5( ζλ q ) P ζλ q Then, simple algebra yiels 4q + q q λ, 4λ + 4ζλ + ( ) ( ) ( 4 4λ ζλ) (9) () () P q + q + () Substituting the secon of () in the first of () an in (), we finally obtain 4q + q (3) 4λ + 4ζλ + ( ) P ( ζλ ) q 4 + qλ ( 4λ + 4ζλ+ ) ( ζλ + ) ( λ ) ( 4λ + 4ζλ + ) 4q q 4 (4) (5) Relations (3)-(5) provie the esire settings of the PD-F controller. 4 An Alternative Tuning Metho An alternative metho of tuning the PD-F controller (Metho ) can be obtaine as follows: Observe first that relation (3) may further be written as G ( s) exp( s) (6) q exp ( ) q s P s s s s α exp( αs) α [,) is an ajustable parameter. Using the approximations exp(αs) +αs an exp[-(- α)s]-(-α)s, we obtain G ( s) exp( s) (7) q q [ ( α) s] αs Relation (7) may further be written as exp( s) G () s (8) s q s+ q + P() s P s s s s ( ) p [ ( α) s ] Ps () s+ s+ p ( / ) p s+ + α s Observe now that by selecting αp (9) we obtain Ps () p s+ ( αs+ ) [ ( α) s ] Therefore, relation (8) yiels G ( s) exp( s) q q p s s+ + α s+ ( ) which can further be written as G ( s) [ α s] () exp( s) () q p p q α ( α) s + + s+ Relation (), can be written in the form exp( s) G () s (3) ρ s + ξρs + SSN: SBN:

4 Proceeings of the 3th WSEAS nternational Conference on SYSTEMS ρ q α ( α) (4) ( p ) ( ) + q ( p α ) p ξ (5) q ( α) Observe now that the Routh stability criterion about () yiels p > q, ( ) p < α + q α (6) Therefore, as for P, one can choose the mile value of the range given by inequalities (8). That is ( α) q q( α + + ) p (7) Then, from (7), we obtain P β (8) ( + α) + [ q q( α) ] ( α) which yiels, [ q q( α) ] ( α) (9) [ β ( + α) ] Therefore, P β[ q q( α) ] ( α) (3) [ β ( + α) ] βα [ qq( α) ] ( α) (3) β ( + α) [ ] Clearly, relations (9)-(3) provie the settings of the esire PD-F controller as functions of two ajustable parameters α an β. With regar to the ajustable parameter α, it is pointe out here that in orer to guarantee positive controller settings, the following inequalities must hol q ( ) > q α >, α< an β>(+α)/ (3) For PDT an UFOPDT processes, for which q, q an q T, q -, respectively, the first of (3) always hols true. However for FOPDT processes, for which q T, q, the first of (3) yiels α>(-t)/. For FOPDT processes with /T, the above inequality hols true for any α in the interval [,). n the case /T>, parameter α must be selecte in the interval ((-T)/, ). Note also that, when α is allowe to take the value α, then, an hence the controller oes not inclue any erivative action. Moreover, the ajustable parameter β can, in general, be selecte arbitrarily in the range [(+α)/, ). However, it woul be useful for the esigner to follow some more explicit rule, base on certain criterion relative to the close-loop system performance, in orer to choose the parameter β. Such a criterion is relate to the responsiveness of the close-loop system. n particular, parameter β can be selecte in such a way that a esire amping ratio ξ es is obtaine for the secon orer approximation (3), of the close loop transfer function. n this case, using relations (5), (8), (9), after some trivial algebraic manipulations, one can conclue that parameter β must be selecte as the maximum real root of the quaratic equation β σβ + σ (33) q ( α ) q ( α ) σ + + qq( α) q q( α) q ( α) + ξes ( α) q q( α) (34) q ( α ) + q q( α) an ( α ) ( α ) q q σ + + q q( α) q q( α) q ( α ) 4 ξes α( α) q q( α) ( α ) q + q q( α) Therefore, parameter β is obtaine as (35) β σ + σ σ (36) Once β is obtaine through (36), it remains only one free tuning parameter, i.e. parameter α. However, as it can be easily checke, parameter α can also be constraine, in a certain way. One such way is to select α in orer to obtain a pre-specifie value of ρ, through relation (4). The resulting equation with respect to α is too complicate an of high orer, an it is not presente here. n the present paper, it is preferre to leave the choice of parameter α free to the esigner, in orer to enhance the obtaine regulatory control performance. Note also that, parameters α an β can alternatively be selecte in orer to assign some esire phase an gain margins to the close-loop system. nee, as it can be easily checke, the PD-F controller is equivalent to a series PD controller with a secon orer set-point filter, of the respective forms GC() s C + ( τ Ds+ ) τ Ι s SSN: SBN:

5 Proceeings of the 3th WSEAS nternational Conference on SYSTEMS G SPF () s ( τ s+ )( s+ ) were P C( τ D + τ)/ τ P /( τ D + τ) C / τ Pτ Dτ /( τd + τ) Cτ D Simple examination yiels τd.5 β β 4βα C D τ Ι τι β + β βα.5 4 ( β + β 4 βα)[ q ( q a) ] ( α) [ β ( + a) ] (37a) (37b) (38) provie that β > 4α. The argument an the magnitue of the loop transfer function are then given by qπ φl ( ω) π + ωatan( qqω) (39) + atan τω + atan τ ω ( ) ( D ) + ( τ Ιω) + ( τdω) ( ω) ( ω) τω q + ( qω) A G j L L C Ι (4) Κ C, τ Ι an τ D are given by (37), (38). Then, the phase margin is efine as PM φl( ωg) + π, ω G is the frequency, at which AL( ω G), while the gain margin (or, in the case of UFOPDT systems, the increasing an the ecreasing gain margins) is efine as GM / A L( ω C ), ω C is the frequency (or, in the case of UFOPDT systems, the frequencies) at which φl( ωc) π. t is now obvious, that parameters α an β can be selecte to obtain pre-specifie gain an phase margins. However, the exact analysis is quite complicate an the resulting equations are nonlinear an of high orer. Work on this subject is currently uner progress. 5 A Thir Set of Tuning Rules An alternative set of tuning rules for the PD-F controller can be obtaine as follows: Let us first expan the term ( P ) s + s+ exp( s) in the enominator of (3) in Taylor series. This, yiels ( s + P s+ exp( ) s) +.5 s + s+ (4) ( P ) ( P ) Therefore, (3) takes the form G ( s) exp( s) (4) P q P q + + s + + s+ Now, by selecting γ (43) γ [,) is an ajustable parameter, relation (4) takes the form G ( s) exp( s) (44) P q P q ( γ ) + + s s Relation (44) may further be written as exp( s) G () s (45) μ s + ψμs + μ ( γ) P q (46) P + q ψ (47) ( γ ) P q The Routh stability criterion about (47) yiels q q p >, P < ( ).5 γ + (48) Therefore, as for P, one can choose the mile value of the range given by inequalities (48). That is q +.5 ( γ) + q( γ) P (49) Relation (49) further yiels P θ (5) +.5 ( γ) + q( γ) q Solving (5) with regar to, we obtain q q ( γ) ( γ) [ ] ( γ 3) (5) θ + ( γ ) Using relations (53) an (46), we further conclue that P θ[ qq( γ) ] ( γ) ( γ 3) (5) θ + ( γ ) γθ [ q q( γ ) ] ( γ ) ( γ 3) (53) θ + ( γ ) P SSN: SBN:

6 Proceeings of the 3th WSEAS nternational Conference on SYSTEMS The propose metho (Metho ) provies the settings of the esire PD-F controller through relations (5)-(53), as functions of two ajustable parameters γ an θ that must fulfill the following inequalities q ( ) > q γ >, γ< an θ>(3-γ)/[(-γ)](54) t is obvious that the ajustable parameter γ has the same properties as the ajustable parameter α involve in the tuning rules presente in the previous section. The ajustable parameter θ can, in general, be selecte arbitrarily in the range ((3- γ)/[(-γ)], + ). When a certain amping ratio ψ es is pre-specifie for the secon orer approximation (45), of the close-loop transfer function, then using (47), (5) an (5), an after some trivial algebra, one can conclue that the parameter θ must be selecte as the maximum real root of the quaratic equation ηθ ηθ + η (55) η ( q q( γ ) ) ψ + q (γ 3) es q (γ 3) ( q q ( γ ) ) q ( γ ) η ψ es (γ) q q( γ ) q + q( γ) q(γ 3) q q( γ) q q( ) [ ] [ ( γ ) η q + q ( γ) q q ( γ) ] (56) (57) (58) Therefore, parameter θ is obtaine as θ η η+ η ηη (59) Once θ is obtaine through (59), it remains only γ as a free tuning parameter, which may further constraine, in orer to obtain a pre-specifie value of μ, through relation (46). However, here, it is preferre to leave the choice of the parameter γ free to he esigner, in orer to enhance the obtaine regulatory control performance. Parameters γ an θ can alternatively be selecte to assign esire phase an gain margins to the close-loop system. n the present case, parameters τ D, τ Ι an C, involve in relations, escribing the equivalence between the PD-F an the series form PD controller, are obtaine as τ D.5 θ θ 4θγ (6a) τ Ι.5 θ + θ 4θγ (6b) C ( θ + θ 4 θγ)[ q ( q γ) ] ( γ 3) (6) ( γ) θ + ( γ ) provie that θ > 4γ. Then, following the analysis presente in the previous section, in orer to assign the esire gain an phase margins on has to solve the nonlinear problem consisting of the equations proviing the critical frequencies ω C an ω G, as well as of the couple equations GM / AL( ωc) PM φ ( ω ) + π. L G, 6 Simulation Results For assessment of the effectiveness of the propose tuning methos an in orer to provie a comparison with existing tuning methos, a series of simulation examples are carrie out for ifferent eatime processes. 6. Simulations for PDT processes Consier the PDT process with, 5. Applying Metho with λ es 7.5, ζ es.85, we obtain the PD-F controller settings.6, P.88 an.44. Metho, with α.365 an ζ es, gives β7.75. The controller settings are.9, P.94 an.358. For γ.3 an ψ es, Metho yiels θ8.6. The controller settings are.3, P.98,.369. The metho in [] gives the conventional PD controller settings C.38, τ Ι 7.5 an τ D.97, when the tuning parameter is.. The set point weighte PD controller settings obtaine by using the metho in [4] are C., τ Ι 7.5 an τ D.43. The set point weight is ε.4, an the time constant of the low pass filter of the controller output is T f.74. Figs. an 3 show the comparisons of the servoresponses an the regulatory control responses, respectively, obtaine by Metho an by the methos in [] an [4], in the case of the unperturbe system. A step loa isturbance L. is assume. Figs. 4 an 5 show the respective comparisons, in the case a simultaneous +% uncertainty in both an is consiere. Obviously, the performance of the propose controller is very satisfactory, particularly in the case of regulatory control. The performance is similar to that obtaine by the controller tune accoring to the metho in [4]. The robustness of the PD controller that is tune accoring to [] is critical. Note that, if the tuning parameter use in [] is chosen as.5, the obtaine controller cannot tolerate the consiere simultaneous parametric uncertainty. Lower values of this SSN: SBN:

7 Proceeings of the 3th WSEAS nternational Conference on SYSTEMS.8 Close-Loop Response Fig.. Comparison of servo-responses for the system G P (s)exp(-5s)/s. Black line: Propose metho ; Blue line: Metho in []; Orange line: Metho in [4]. Close-Loop Response Fig. 5. Regulatory control responses for the system G P (s)exp(-5s)/s uner +% simultaneous parametric uncertainty. Other legen as in Fig...4 Close-Loop Response Fig. 3. Comparison of regulatory control responses for the system G P (s)exp(-5s)/s. Other legen as in Fig.. Close-Loop Response Fig. 6. Comparison of servo-responses for the system G P (s)exp(-5s)/s, uner +% simultaneous parametric uncertainty. Dash line metho in [4]. Black soli line: Metho ; Orange line: Metho ; Blue line: Metho. Close-Loop Response Fig. 4. Servo-responses for the system G P (s)exp(- 5s)/s uner +% simultaneous parametric uncertainty. Other legen as in Fig.. Close-Loop Response Fig. 7. Regulatory control responses for the system G P (s)exp(-5s)/s uner +% simultaneous parametric uncertainty. Other legen as in Fig. 6. SSN: SBN:

8 Proceeings of the 3th WSEAS nternational Conference on SYSTEMS tuning parameter, can lea to enhance performance in the case of moel mismatch. However, for these values, the settling time of the nominal close-loop system is very large. So, the metho in [] reveals several rawbacks. Since the metho in [4] outperforms the metho in [], we next perform a comparison of this metho with all the propose methos. The intereste reaer can easily check that in the case of nominal system, all methos show a similar regulatory control performance, while the propose methos show a rather sluggish response to set-point tracking. n the case of moel mismatch, our methos provie more robust controllers, as it can be verifie by Figs. 6 an 7. nee the phase an gain margins of the close-loop system with the controller obtaine by Metho are PM4.998 o an GM-5.35 B, respectively, while Methos an give (PM, GM)( o, B) an (PM, GM) ( o, B), respectively. n comparison, the controller tune accoring to [4] gives (PM, GM)( o, B). Note that the first orer filter of the form /(T f s+), use in [4], in orer to filter the controller output signal, oes not contribute in the stability of the close-loop system. nee, in the absence of the filter, the PD controller esigne as suggeste in [4], gives the closeloop system stability margins (PM, GM)( o, B). For a larger value of T f, say T f., the stability margins are obtaine as (PM, GM) ( o, B). Therefore, the larger the value of T f is, the lesser are the stability margins obtaine. Note also that, in the regulatory control case, for larger values of T f, a worse performance is obtaine, in terms of the obtaine maximum error. Therefore, the filter reners the esign proceure rather complicate without proviing any clear avantage. On the other han, the propose controller is simpler, since it avois the nee for set-point weighting as well the introuction of any filter of the controller output signal. 6. Simulations for FOPDT processes Consier the FOPDT process, with, T,.5. Applying Metho with λ es.5, ζ es, we obtain., P.3333 an Metho for α.5 an β yiels., P.3333 an Finally, for γ.3 an θ.4, Metho yiels.75, P.3947,.359. The PD controller settings obtaine by applying the metho reporte in [3] are C.5; τ Ι.5 an τ D.67. Fig. 8 shows the comparison of the servo-responses an of the regulatory control responses of the propose methos in comparison to those obtaine by the metho in [3], in the case of Close-Loop Response Time (secon) Fig. 8. Servo-responses an regulatory control responses for G P (s)exp(-.5s)/(s+) in case of nominal system parameters. Dot line: Metho in [3]; Soli black line: Metho ; Soli blue line: Metho ; Soli yellow line: Metho. Close-Loop Response Time (secon) Fig. 9. Servo-responses an regulatory control responses for G P (s)exp(-.5s)/(s+) uner +% simultaneous parametric uncertainty. Other legen as in Fig. 8. nominal system parameters. A unit step loa change is introuce at time t sec. Clearly, Methos an provie consierably better performance as compare to the metho in [3], as Metho gives a rather sluggish response. Fig. 9 illustrates the same responses in the case of a +% simultaneous parameter mismatch. n this case the propose methos provie a more robust performance as compare to the metho in [3]. The propose methos provie a very satisfactory performance while its simplicity is another avantage. 6.3 Simulations for UFOPDT processes Consier the typical UFOPDT process moel with, T,.5. Metho for λ es.85, ζ es.85, SSN: SBN:

9 Proceeings of the 3th WSEAS nternational Conference on SYSTEMS yiels.6543, P.76 an.573. Metho for α.38, an β3.84 yiels.646, P.35 an.44. Metho for γ.35 an θ3.8 yiels.67, P.37 an.437. The PD controller settings obtaine by the metho in [3], are C.6, τ Ι 8.375, τ D.45, when one tuning parameter is consiere, an C.9345, τ Ι.779 an τ D.56, when two tuning parameters are use. Figs. an show the comparisons of the servo-responses an the regulatory control responses, respectively, in the case of the unperturbe system. The PD-F controller gives a better performance in case of set-point tracking, while its performance in the case of regulatory control is similar to that of the PD controller tune accoring to [3], in the case two tuning parameters are consiere. Figs. an 3 show the comparisons of the servo-responses an of the responses in case of regulatory control, when a +% mismatch in the process gain is assume. n the case only one tuning parameter is consiere, the metho in [3], cannot tolerate the assume aitive gain uncertainty. The propose methos provie as goo performance as the metho in [3] with two tuning parameters. t is not ifficult to check that, when a -% mismatch in the process gain is assume, the propose methos provie more robust controllers as compare to the metho in [3]. Similar conclusions can be obtaine for mismatch in process time elay or in process time constant. Close-Loop response Time (secon) Fig.. Comparison of servo-responses for system G P (s)exp(-.5s)/(s-). Soli black line: Propose metho ; Soli blue line: Propose Metho ; Soli yellow line: Propose Metho ; Dash line: Metho in [3] with one tuning parameter; Dot line: Metho in [3] with two tuning parameters. Close-Loop response Time (secon) Fig.. Comparison of regulatory control responses for system G P (s)exp(-.5s)/(s-). Other legen as in Fig.. Close-Loop response 3 Fig.. Servo-responses for system G P (s)exp(-.5s)/(s-) uner +% uncertainty in the process gain. Other legen as in Fig.. Metho in [3] with one tuning parameter gives an unstable response that is not illustrate. Close-Loop response Time (secon) Time (secon) Fig. 3. Regulatory control responses for system G P (s)exp(-.5s)/(s-) uner +% uncertainty in the process gain. Other legen as in Fig.. SSN: SBN:

10 Proceeings of the 3th WSEAS nternational Conference on SYSTEMS 7 Conclusions New irect synthesis methos of tuning the PDF controller for integrating as well as for stable or unstable processes with ea time have been presente in this paper. The propose tuning methos ensure smooth close-loop response to set-point changes, fast regulatory control an sufficient robustness against large moel mismatch. The reporte numerical simulation examples verify the effectiveness of the propose methos of tuning three-term controllers for the classes of process moels consiere in the paper. Research on the possible extension of the propose metho, in orer to cover higher orer ea-time processes, as well as eatime processes with stable or unstable zeros, is currently uner progress. References: [] A.O Dwyer, Hanbook of P an PD Control ler Tuning Rules, n Eition, mperial College Press, Lonon, 6. [] M.Chiambaram an R.Pama Sree, A simple metho of tuning PD controllers for integrator/ ea time processes, Computers an Chemical Engineering,, vol. 7, pp. -5, 3. [3] R.Pama Sree, M.N.Srinivas an M.Chiambaram, A simple metho of tuning PD controllers for stable an unstable FOPDT processes, Computers an Chemical Engineering, vol. 8, pp. -8, 4. [4] A.Seshangiri Rao, V.S.R.Rao an M. Chiambaram, Direct synthesis-base controller esign for integrating processes with time elay, Journal of the Franklin nstitute, vol. 346, pp , 9. [5] R.M.Phelan, Automatic Control Systems, New York, Cornell University Press, 978. SSN: SBN: