Analytical and Graphical Design of PID Compensators on the Nyquist plane

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1 Analytical and Graphical Design of PID Compensators on the Nyquist plane R.Zanasi, S.Cuoghi DII-InformationEngineeringDepartment,UniversityofModenaand ggio Emilia, Via Vignolese 95, 4 Modena, Italy, s: roberto.zanasi@unimore.it and stefania.cuoghi@unimore.it. Abstract: This paper focuses on the design of PID compensator to exactly satisfy the gain margin, the phase margin and the gain or phase crossover frequency specifications. The design problem is numerically solved using the so called PID inversion formulae method. A graphical interpretation of the solution on the Nyquist plane is presented. This could be suitable on education environment to deeply understand the design of PID compensators. Simulations results show the effectiveness of the presented method. Keywords: PID Compensators, Linear Control, Compensator Design.. INTRODUCTION C C The proportional-integral-derivative (PID) compensators are widely used in the industrial processes to meet most of the control objectives. The gain and phase margin(gpm) specifications are important measure of the robustness of the controlled systems, and many methods are known in the literature to meet these design requirements. Some of them are described in Lee (4), Kim et al.(5) and Aström and Hagglund(6). These methods are normally based on numerical or graphical trial-and-error solutions which use Bode plots or fuzzy neural network(fnn).inthispaperanewandexactgraphicalprocedure on Nyquist plane to meet the gain margin, the phase margin and the phase or gain crossover frequency specifications is shown. This procedure is similar to ones described in Wang etal.(999)andisbasedonsocalledpidinversionformulae, see Zanasi et al.(), Ntogramatzidis and Ferrante (), Zanasi and Cuoghi (). The paper is organized as follows. In section II, the graphical properties of PID compensator on Nyquist plane are described. In section III a new design method based on the use of the PID inversion formulas is presented. In section IV numerical and graphical solutions of design problems based on GMP specifications are given. Numerical examples and conclusions end the paper.. PID COMPENSATORS: THE GENERAL STRUCTURE Let us consider the classical form of the PID compensator C(s)= +sk D + K I s, () wheretheproportional,derivativeandintegrativeterms,k D and K I are supposed to be real and positive. The frequency response of C(s) ( C(jω)= +j ωk D K ) I, () ω canbewrittenas C(jω)=X+jY(ω), O C(jω) r K D K I a) Y O M ϕ M C(jω) C ϕ X C(jω) R Fig..NyquistplotoffunctionsC(jω)andC - (jω). where X = and Y(ω)= ωk D K I ω. The parameter X is alwayspositive,whilethefunctiony(ω)ispositivefor ω > KI K D andnegativefor<ω < KI K D. Let C( )and C ( )denote,respectively,thesetofallthe PIDcompensatorsC(s)havingthesameparameter { C( )= C(s)asin () K I >,K D >, (3) andthesetofalltheinversefunctionsc(s) - { C ( )= C(s) C( ). (4) C(s) It can be easily shown that the graphical representation of each elementof C( )onthenyquistplaneisaverticalstraightline rwhichpassesthroughpoint(,),seefig.. Property. Theshapeofthefrequencyresponseofeachelementofset C ( )isacirclewithcenterc = K P andradius R = K P whichintersectstherealaxisinpointandpoint KP. This graphical property hinges on the fact that the Nyquist diagramofc - (jω)isacircle X b)

2 D B q C() ω A A B ω A.4.6 C(jω,K I ) K I C ( ) r E = B A C B Fig.. The Nyquist diagrams of frequency responses of set C( )and C ( )for =.4:.:.6. r e m y C(s) G(s) Fig. 3. Unity feedback control structure. C - (jω)= C(jω) =C +R e jθ(ω), (5) where C =R = X, θ(ω) = arctany(ω) X X =k p. [,π] and Proof: the frequency response() can be written in polar form as follows C(jω)=M(ω)e jϕ(ω), wherem(ω)= X cosϕ(ω) and ϕ(ω)=arctany(ω) X.Itfollowsthat C - (jω)canbeexpressedintheform C - (jω)= C(jω) =cosϕ(ω) e jϕ(ω) X = X [+cos(ϕ(ω))] j X sin(ϕ(ω)) = X + X e jϕ(ω), fory(ω) [,+ ].Thelastrelationclearlyshowsthatthe shapeofc - (jω)inthecomplexplaneisacirclewithcenter C = X andradiusr = X. TheNyquistdiagramsoffrequencyresponsesofsets C( ) and C ( )fordifferentvaluesofparameter areshownin Fig.. 3. PIDCOMPENSATORSC(s,K D )ANDC(s,K I )MOVING APOINTATOAPOINTB Consider the block-diagram shown in Fig. 3, where G(s) denotesthetransferfunctionoftheltiplanttobecontrolled.let C(jω ) =M e jϕ denotethevalueofthefrequencyresponse C(jω)=M(ω)e jϕ(ω) atfrequency ω,wherem =M(ω )and ϕ = ϕ(ω ).TostudyhowC(jω )affectsatfrequency ω,letusconsidertwogenericpointsa =M A e j ϕ A andb = M B e j ϕ Bofthecomplexplane.ferringtoFig.4,wesaythat pointaiscontrollabletopointb(orequivalentlythatpointa Fig.4.Controllabledomain D B andgraphicaldesignofcompensatorsc(jω,k I )movingpointatob. can bemoved topointb)ifavaluec(jω )existssuchthat B =C(jω ) A,thatisifandonlyifthefollowingconditions hold: M B =M A M, ϕ B = ϕ A + ϕ. (6) Definition. GivenapointB C,letusdefine controllable domainofthepidcompensatorc(s)topointb theset DB defined as follows: DB {A C =,K I,K D >, ω :C(jω) A=B. Itcanbeeasilyshownthatthedomain D B onnyquistplane is the half-plane which includes pointband is delimited by the straight line q passing through point O and perpendicular to segmentb,seethegrayregioninfig.4. Definition. Given a point B C, let C B ( ) and CB () denote the sets of PID compensators defined as follows { C B ( )= B C(s) C(s) C( ), (7) { B CB (K P)= C(s) C( ), (8) C(s) with C( ) defined in (3). Moreover, let C BKP (s) C B ( ) and CBK P (s) CB ()denoteparticularelementsofthetwo sets C B ( )and CB ()chosenarbitrarily. Definition 3. (PID Inversion Formulae) Given two points A = M A e jϕ A and B =M B e jϕ B of the complex plane C, the PID inversion formulae are defined as follows: X(A,B) = M B cos(ϕ B ϕ A ), M A Y(A,B) = M B M A sin(ϕ B ϕ A ). These formulae are similar to the ones used in Phillips(985) and to the Inversion Formulae introduced and used in Marro andzanasi(998)andzanasiandcuoghi ()forleadand lag compensators design. Property. (FromAtoB)GivenapointB Candchosen apointa =G(jω A ) D B,thesetsC(s,K D)andC(s,K I )of (9)

3 allthepidcompensatorsc(s)thatmovepointatopointbis obtained from() using, respectively, the parameters D =X, K I =K D ω A Y ω A, () forallk D > Y ω A,ortheparameters =X, K D = Y + K I ω A ωa, () forallk I >,wherecoefficientsx =X(A,B)andY =Y(A,B) are obtained using(9). ω g A g ω g A g H C KP H A p Proof: For ω = ω A,relation()canberewrittenas ( C(jω A )= +j ω A K D K ) I. () ω A FromrelationB =C(jω A ) Aand(6)itisevidentthatpoint A =G(jω A ) =M A e j ϕ A canbemovedtopointb =M B e j ϕ B if andonlyif C(jω A )= M B M A e j(ϕ B ϕ A ) =X+jY. (3) Solvingequations(3)withrespecttoXandY,oneobtainsthe PID Inversion Formulae (9). Equations () and () follow directly from() and(3). Property3. The parameter can be determined on the NyquistplaneasshowninFig.4: () drawtheuniquecircle A C assingthroughpointsaand Ohavingitsdiameteronthestraightlinerwhichpasses throughpointsandb; () thecircle A C B intersectsthestraightlinerinpointso ande =B/ ; (3) theparameter isequaltothemodulusofpointbover themodulusofpointe: = B / E. Proof: It follows directly from Property because the circle A CB on the Nyquist plane is the inverse of the frequency response C BKP (jω)offunction C BKP (s) C B ( )andbecause theintersectionsofcircle A CB withthestraightlineroccurin pointsoande =B/. 4. SYNTHESIS OF PID COMPENSATORS Let us consider the case of given steady-state specifications thatimposethevalueoftheintegrativetermk I >.Thiscase occurs for example for type- systems and design specification on velocity error, or for type- systems and design specification onaccelerationerror.thefactthattheintegrativetermk I has beenfixeddonotreducetheadmissibledomain D B topoint B,thatisstillthegrayregionshowninFig.4.Theothertwo degreesoffreedom andk D oftheregulatorcanbeimposed tomeetthephasemargin φ m andthegaincrossoverfrequency. DesignProblemA:(K I,φ m, ). Giventhecontrolschemeof Fig. 3, the transfer function G(s), the steady-state specifications thatimposethevalueoftheintegrativetermk I >anddesign specifications on the phase margin φ m, and gain crossover frequency, design a PID compensatorc(s) such that the loop gain transfer function C( jω)g( jω) passes through point =e j(π+φ m) for ω =. C KP Fig. 5. Graphical solution of Design Problem B on the Nyquist plane. SolutionA: IfpointA =G(jω A )belongstotheadmissible domain D B showninfig.4,thesolutionfollowsdirectlyfrom ()ofpropertywithparameterk I imposedbythesteadystate specifications. Let us now consider the case of steady-state specifications thatdonotconstrainthevalueofk I.Thedegreeoffreedom in the Property can be utilized in order to satisfy another specification. In literature different methods can be found to imposethedegreeoffreedom,suchastoobtainrealzerosof the compensator, Aström and Hagglund(6). Let us consider the following design problems. DesignProblemB:(φ m,g m, ). Giventhecontrolscheme of Fig. 3, the transfer function G(s) and design specifications onthephasemargin φ m,gainmarging m andgaincrossoverfrequency,designapidcompensatorc(s)suchthattheloop gaintransferfunctionc(jω)passesthroughpoint = e j(π+φ m) for ω = andpassesthroughpoint = /G m. SolutionB: LetA p =G(j )denotethevalueofatthe desiredgaincrossoverfrequency ω =,andlet =e j(π+φ m) and = /G m =M Bg e jϕ Bg denotethepointscorresponding tothedesiredphasemargin φ m andgainmarging m,respectively. The set C p (s,ω g ) of all the PID compensators C(s) whichsolvedesignproblembisobtainedfrom()usingthe parameters =X p >, K D = Y p Y g ω g ω p ω g >, (4) K I = Y p ωg Y g ω g ωp ωp ωg >, (5) wherethecoefficientsx p =X(A p, ),Y p =Y(A p, ),X g = X(A g, )andy g =Y(A g, )areobtainedusingtheinversion formulas(9)witha g =G(jω g )=M Ag (ω g )e jϕ Ag (ωg),forallthe frequencies ω g satisfyingtherelation =X g (ω g )= M cos(ϕ Bg ϕ Ag (ω g )). (6) M Ag (ω g ) AsolutionC p (s,ω g )ofdesignproblembexistsonlyif: )thesets ωg ofallthe ω g > satisfying(6)isnotempty;) A p DB p anda g DB g ;3)theparametersK I andk D in(4) and(5) are real and positive.

4 X p,x g D A g ωg A p X p (ω) A g A p A g ω g ω g X g (ω) ω ω g A g C KP Fig.6.FunctionsX g (ω)(blue)andx p (ω)(black). Proof: The design specifications define the position of points =e j(π+φ m), A p =G(j ) and = /G m. According toproperty,thecompensatorsc p (s,k D )whichmovepoint A p D to point are obtained using the parameters and K I in (). The free parameter K D can now be used to force the loop gain frequency responsec p (jω,k D ) to passthroughpoint.thisconditioncanbesatisfiedonlyif afrequency ω g existssuchthatcompensatorc p (s,k D )moves pointa g =G(jω g ) D topoint,thatisonlyif(6)holds. Thefrequencies ω g S ωg satisfying(6)areacceptableonly if the compensatorc g (s,k D ) which moves pointa g topoint, obtained using Property, is equal to the compensator C p (s,k D ).ThisconditionissatisfiedonlyifthetwocompensatorssharethesameK I andk D,thatisonlyif K I = ω pk D Y p = ω gk D Y g ω g. (7) Solving(7)withrespecttoK D oneobtainstheexpressionof K D givenin(4),thatcanbesubstitutedin(7)obtaining(5). Thesolutionsareacceptableonlyif,K D,K I >. The solution of equation(6) can be obtained graphically, by plottingx g (ω)andbyfindingallthefrequencies ω g S ωg for whichx g (ω g )intersectsthehorizontallinex p =,seefig.6. IntheexampleofFig.6itisS ωg = {ω g, ω g andtherefore there are two solutions:c p (s,ω g ) andc p (s,ω g ). The loop gainfrequencyresponsesh (jω) =C p (jω,ω g )(red line)andh (jω)=c p (jω,ω g )(magentaline)onthe NyquistplaneareshowninFig.5.Thetwosolutionssatisfy thedesignspecificationsandareacceptableonlyifk D >and K I >. Property4. Thefrequencies ω g S ωg satisfying(6)canbe graphically determined on the Nyquist plane as shown in Fig. 5: () draw the circle C (jω) on the Nyquist plane and determinetheparameter ofcompensatorc p (s,k D )as describedinproperty3whena=a p andb= ; () draw the circle C (jω) having its diameter on the segmentdefinedbypointsoand ; (3) theintersectionsa g,a g ofcircle C (jω)with correspondtothefrequencies ω g, ω g belongingtoset S ωg. Proof: The circles C (jω) (black line) and C (jω) (blue line) shown in Fig. 6 represent, respectively, the frequency responses of functions C (s) C ( ) and C (s) C ( )with =X g givenin(6).thesetwocirclescanbe easilydeterminedonthenyquistplanebecausethepointsa p, and areknownfromthedesignspecificationsand is given by the graphical construction described in Property 3. A frequency ω g satisfying(6)existsonlyif H (jω) A p C KP Fig. 7. Graphical solution of Design Problem C. G(jω g )C KP (jω g )=, (8) wherec KP (s) is the PID compensator () with the value of parameter determinedasdescribedabove.lation(8)can also be rewritten as follows = C KP (jω) = C KP (jω), (9) with ω = ω g,andthereforeitcanbesolvedgraphicallyonthe Nyquistplanebyfindingtheintersections ω g S ωg of with CB g (jω). DesignProblemC:(φ m,g m, ω g ). Giventhecontrolscheme of Fig. 3, the transfer function G(s) and the design specifications on the phase margin φ m, gain margin G m and phase crossoverfrequency ω g,designapidcompensatorc(s)such that the loop gain transfer function C( jω)g( jω) passes through point = /G m for ω = ω g andpassesthroughpoint = e j(π+φ m). SolutionC: LetA g =G(jω g )denotethevalueofatthe desiredphasecrossoverfrequency ω g,andlet =e j(π+φ m) and = /G m =M Bg e jϕ Bg denotethepointscorresponding tothedesiredphaseandgainmargins.thesetc g (s, )ofall the PID compensators C(s) which solve Design Problem C is obtained from() using the parameters =X g >, K D = Y p Y g ω g ω p ω g >, () K I = Y p ωg Y g ω g ωp ωp ωg >, () where X p =X(A p, ),Y p =Y(A p, ), X g =X(A g, ) and Y g =Y(A g, )areobtainedusingtheinversionformulas(9) witha p =G(j )=M Ap ( )e jϕ Ap (ωp),forallthefrequencies satisfyingtherelation =X p ( )= M cos(ϕ Bp ϕ Ap ( )). () M Ap ( ) AsolutionC g (s, )existsonlyif: )thesets ωp ofallthe > ω g satisfying()isnotempty;) A p D anda g D ;3)theparametersK I andk D in() and() are real and positive. TheproofisquitesimilartotheonegivenforSolutionB.

5 DesignProblemD:(φ m,g m ). Giventhecontrolschemeof Fig. 3, the transfer function G(s) and the design specifications on the phase margin φ m and gain margin G m, design a PID compensator C(s) such that the loop gain transfer function C(jω) passes through points =e j(π+φ m) and = /G m. D A g A p H A g C KP H H SolutionD: Let =e j(π+φ m) and = /G m =M Bg e jϕ Bg denotethepointscorrespondingtothedesiredphasemargin φ m andgainmarging m.thesetc KP (s,,ω g )ofallthecompensatorsc(s)whichsolvethedesignproblemdisobtainedas follows: a)findallthepairs (, ω g ) S γω offrequencieswhichsolve the equation =X p ( )=X g (ω g ), (3) wheretheparameter >ischosenarbitrarily,s KP ω isthe set of all the pairs (, ω g ) satisfying (3) with ω g >, andfunctionsx p ( )andx g (ω g )aredefinedin()and(6) witha p =G(j ) =M Ap ( )e jϕ Ap () anda g =G(jω g ) = M Ag (ω g )e jϕ Ag (ω g). b)foreachpair(, ω g ) S KP ωcompute K D = Y p Y g ω g ωp ωg >, (4) K I = Y p ωg Y g ω g ωp ωp ωg >. A solution exists only if: ) satisfies < <min(max(x p ( )),max(x g (ω g ))) (5) )S KP ωisnotempty;3)a p ( ) DB p anda g (ω g ) DB g ;4) K D andk I in(4)arerealandpositive. Proof: The proof hinges on the fact that C(s) has to be design to move point A p =G(j ) to point and point A g =G(jω g )topoint.asolutionc KP (s,,ω g )existsonly ifthefrequencies and ω g satisfy()and(6),thatisonlyif theysatisfy(3).foreachvalueof satisfying(5),onecan findthesets KP ωofallthesolutions(,ω g )of(3). The solutions of (3) can also be obtained graphically by plotting X p (ω) and X g (ω) and by finding, for each admissible value of, all the pairs (,ω g ) S ωp where X p ( ) andx g (ω g )intersectthehorizontalline,seefig.6.inthe example of Fig. 6 there are three different solutions: S KP ω = {(,ω g ),(,ω g ),(,ω g ). The solution (,ω g ) is not admissible because > ω g. The loop gain frequency responsesh (s),h (s)andh (s)ofthesethreesolutionson thenyquistplaneareshowninfig.8.thesesolutionsareacceptableonlyifparametersk D andk I givenin(4)arepositive. Thesolutionof(3)canalsobeobtainedontheNyquistplane. Givenpoints and andadesiredvaluefor >,the circles CB p (jω)and CB g (jω)canbedrawnonthenyquist plane as the circles having their diameters on the segments defined by points {O, and {O,, respectively, as describedinproperty(3).eachpair(ω p,ω g )correspondingtothe intersectionsofwithcircles CB pkp (jω)and CB gkp (jω) is a possible solution for Design Problem D. If does H C KP A p Fig. 8. Graphical solution of Design Problem D. D B A C B KP B B H(jω) Fig.9.GraphicalsolutionofDesignproblemA. notintersectbothcircles C KP (jω)and C KP (jω),thechosen valueof isnotacceptable. 5. NUMERICAL EXAMPLES Design Problem A: Given the following type- plant 8(s+) G(s)= s(s+.5) (s+3), design a PID compensator C(s) in order to achieve the accelerationconstantk a =,thephasemargin φ m =5 o andthegain crossoverfrequency =.5. Solution: The integral constant K I is determined by steadystate requirement as K a =lims C(s)G(s)= 8K I =, s.5 3 thatleadstok I =.48.ThepointA =G(j ) =.84e j78o belongstotheadmissibledomain DB definedbyb=e j3o, see Fig. 9. From inversion formulae (9) it follows that X =.734andY =.939.Finallyrelations()leadto =.734 andk D =.433.Thedesignedcompensator()is C(s)= s+.48. s The corresponding loop gain transfer function H( jω) = C(jω)isplottedinredinFig.9.Theimprovementof the closed-loop step response is shown in Fig. q

6 .5.5 y (t) y (t) time Fig.. Design problem A: closed-loop step responses without thecontrollery (t)andwiththecontrollery (t). DesignProblemB:GiventhesystemproposedinWangetal. (999) G(s)=.s +.33s+.4 e s, (6) design a PID compensator to meet the following design specifications:phasemargin φ m =6,gainmarginG m =3andgain crossoverfrequency =.335. Solution.Themodulusandthephaseofpoint arem Bp =, ϕ Bp =.Thevalueof obtainedby(4)is =X p =.67,thefrequenciesobtainedsolving(6)are ω g =.5 and ω g =.57, see Fig. 6. The corresponding points on area g =.546e j8 anda g =.56e j58..from (4) and(5), the regulators satisfying the Design Problems B are C (s)=.3449s +.67s+.4, (7) s C (s)=.476s +.67s (8) s ThecorrespondingloopgaintransferfunctionsH (jω)(red line)andh (jω)(magentaline)areplottedinfig.5. DesignProblemC:Giventheplant(6),designaPIDcompensator to meet design specifications on the phase margin φ m =6, the gain margin G m =3 and the phase crossover frequency ω g =.5. Solution.Themodulusandthephaseofpoint arem Bg =.333and ϕ Bg =8.Thevalueof obtainedby()is = X g =.67.Thefrequenciesobtainedsolving()are =.33and =.8,seeFig.6.Thecorrespondingpointson area p =.767e j57.9 anda p =.55e j68.7.since > ω g,thecorrespondingpidisnotacceptable.theunique admissiblesolutionisthepid(7)obtainedsubstituting in () and(). The corresponding loop gain transfer function H (jω)isplottedinredinfig.7. DesignProblemD:Giventheplant(6),designaPIDcompensator to meet design specifications on the phase margin φ m =6 andthegainmarging m =3. Solution. The design specifications define the points = e j, =.333e j8 and =X p =X g =.67. The four solutions (i,ω gj ) of equation (3) can be graphically determined as shown in Fig. 6. The two acceptable regulators(7),(8)areobtainedfor ( =.33,ω g =.5) and ( =.33,ω g =.57).Thesolutiondeterminedby ( =.8,ω g =.5)isnotacceptablebecause > ω g. The solution determined by ( =.8,ω g =.57) isnotacceptablebecausethepidparametersk D andk I are negative and the controlled system is unstable. The obtained loop gain transfer functions are plotted in Fig CONCLUSION The presented graphical and analytical methods for the design ofpidcontrollersarebasedontheuseofpidinversionformulae, which allow to achieve given design specifications on the gain margin, the phase margin and the gain or phase crossover frequency. One of the main advantages of the graphical solution over other graphical approaches, such as Yeung(), is that it can be directly determined in the complex plane by finding the intersections of the frequency response of the plant with circles, that can be easily determined from the design specifications. The simplicity of the method and its graphical interpretation onthenyquistplaneseemtobeusefulbothforeducational and industrial purposes. It is well known that in the process of learning the graphical representation of a numerical solution makes it easier to understand and easier to recall. The drawing and the comparison of the same function plotted in different diagrams, i.e. the loop gain frequency response on the Bode, the Nyquist and the Nichols diagrams, have a great educational value. They emphasize some properties hidden in a single representation and lead to a deeper knowledge of the concepts. In this paper some properties of PID regulators on the Nyquist diagrams are pointed out. These have been used to exactly solve the considered design problem to obtain the system robustness. REFERENCES Aström, K.J., and Hagglund, T. Advanced PID control. Instrument Society of America, search Triangle Park, NC, 6. Kim,K.,Kim,Y.C. ThecompletesetofPIDcontrollerswith guaranteed gain and phase margins. Proc. 44th IEEE Conf. on Decision and Control, and the European Control Conf. 5, Seville, Spain, 5 December 5, pp Lee, C.H. A survey of PID controller design based on gain and phase margins. International Journal of Computational Cognition: (3):63-, 4. Marro, G. and Zanasi, R., New Formulae and Graphics for Compensator Design. IEEE International Conference On Control Applications, Trieste, Italy, September -4, 998. Ntogramatzidis L., and Ferrante A. Exact Tuning of PID Controllers in Control Feedback Design, IET Control Theory & Applications, 5(4): ,. Phillips, C.L. Analytical Bode Design of Controllers. IEEE Transactions on Education, E-8, no., pp , 985. Wang, Q.G., Fung, H.W., Zhang, Y., PID tuning with exact gain and phase margins. ISA Transactions, 38,43-49, 999. Yeung,K.S. and Lee, K.H. A universal design chart for linear time-invariant continuous-time and discrete-time compensators. IEEE Transactions on Education, vol.43, no.3, pp.39-35, Aug. Zanasi R., Cuoghi S., and Ntogramatzidis L. Analytical and Graphical Design of Lead-Lag Compensators, International Journal of Control, 84(): ,. Zanasi R. and Cuoghi S. Design of Lead-Lag compensators for robust control, IEEE ICCA, International Conference on Control and Automation, Santiago, Chile 9-//. ZanasiR.andCuoghiS. Directmethodsforthesynthesisof PID compensators: analytical and graphical design, IEEE IECON, Industrial Electronics Conference, Melbourne, Australia, 7-//.