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热流道温度控制器研究与设计（桂电子）,热流,温度,控制器,研究,设计,电子
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ORG 0000HJMP STARTORG 0070HSTART: MOV SP,#60HCLR TF0CLR AMOV 58H , AMOV 59H,AMOV 5AH ,AMOV 5BH ,AMOV 5CH ,AMOV 5DH ,AMOV 5EH ,AMOV 5FH ,AMOV 58H, A MOV 59H, A MOV 5AH, A MOV 5BH, A MOV 5CH, A MOV 5DH , A MOV 5EH , A MOV 5FH , A MOV 12H, #7 ；设定初始温度为77度 12H为十位，13H为个位,43H为百位MOV 13H, #7MOV 43H, #0 SETB P3.0 SETB P3.1 SETB P3.2 SETB P2.0 SETB P2.1 SETB P2.2 SETB P2.3LCALL DELAY1 ；延时程序全部参考书本MOV TCON ,#00HMOV TMOD ,#01HMOV TH0,#0D8HMOV TL0,#0F0HSETB EASETB ET0SETB TR0LCALL T0ZHONGDUAN ;暂时还不知道要不要这个中断程序这样调用LLL： JNB TF0,$ LCALL GETTEM1MOV 31H, ALCALL DELAY4LCALL GETTEM1MOV 32H, ALCALL DELAY4LCALL GETTEM1MOV 33H,ALCALL DELAY4LCALL GETTEM1MOV 34H ,ALCALL DELAY4LCALL GETTEMMOV 35H,ALCALL DELAY4LCALL GETTEM1MOV 36H, ALCALL DELAY4LCALL GETTEM1MOV 37H , ALCALL DELAY4LCALL FIGUREWAVELCALL PIDPROGRAMLCALL TEMCHANGELCALL SHOWTEM1 ;调温度设定显示子程序，此程序中包含了键盘扫描子程序LCALL SHOWTEM2 ；调实际测量温度显示子程序CLR TF0SETB TR0SJMP LLLT0ZHONGDUAN:PUSH ACCMOV A,5FHJZ LOOPB1DEC AMOV 5FH,ACLR P3.4SJMP LOOPB2LOOPB1: SETB P3.4LOOPB2: INC 58HMOV A,58HCJNE A,#0FAH,LOOPB3MOV 58H,#00HCLR TR0SETB TF0LOOPB3: MOV TH0,#0D8HMOV TL0,#0F0HPOP ACCRETIGETTEM1:SETB P3.3 NOP NOP CLR P3.3 ; 将地址传送到ADC NOP NOP SETB P2.0 NOP NOP CLR P2.0 ; 启动转换 NOP NOPWAIT: JB P3.7, MOVD ; 结束转换 AJMP WAIT ; 等待转换结束MOVD: CLR P3.6 NOP NOP SETB P3.6 ; 在ADC端口输出转换数据 NOP NOP MOV A,P0 ; 将ADC中的数据保存到AccMOV R0,A ; 将Acc暂存到R0CLR P3.6 ; DAC输出浮点数 NOP NOP RETFIGUREWAVE:MOV R2,#06HMOV R1,#31HMOV A, R1LCALL SMALLLCALL BIGLCALL PINGJINZHISMALL:LOOPSMALL: INC R1MOV B,R1CJNE A, B, NEXSMALLNEXSMALL: JC NETSMALLMOV A ,BNETSMALL: DJNZ R1 ,LOOPSMALLMOV 38H, ARETBIG:LOOPBIG: INC R1MOV B, R1CJNE A ,B , NEXBIGNEXBIG : JNC NETBIGMOV A, BNETBIG: DJNZ R2, LOOPBIGMOV 39H, ARETPINGJINZHI:MOV R2 ,#06HMOV R1 ,#31HMOV R0 ,#30HCLR CMOV A, R1LOOPPING: INC R1MOV B,R1ADDC A,BDJNZ R2 ,LOOPPINGMOV R0 ,AMOV A, R0SUBB A,38HSUBB A,39HMOV R0 ,AMOV A,R0MOV B ,5DIV ABMOV 0FH , AMOV A , BCJNE A , #03H , ENDGOMOV A , #01HADD A ,0FHSJMP ENDPINGENDGO:CJNE A , #04H , ENDPINGMOV A , #01HADD A , 0FHENDPING:MOV 30H ,ARETNOPRETTEMCHANGE:MOV 20H ,#02HMOV DPTR ,#0000HMOV 21H, DPLMOV 22H ,DPHMOV A , 30HMOV B ,20HMUL ABADD A,21HMOV 3CH ,AMOV A ,BADDC A ,22HMOV 3DH, ARETSHOWTEM1: JNB P2.4 ,KEYGETTEMFUZHI: MOV A , 13H MOV 41H , A MOV A ,12H MOV 42H ,A MOV 43H ,#0MOV R0, #41HMOV A ,R0INC R0MOV DPTR ,#TAB1MOVC A ,A + DPTRMOV P1 ,A SETB P2.3LCALL DELAY3CLR P2.3LCALL DELAY3MOV A , R0INC R0MOV DPTR ,#TAB1MOVC A ,A + DPTR MOV P1 , A SETB P2.2LCALL DELAY3CLR P2.2LCALL DELAY3MOV A , R0MOV DPTR ,#TAB1MOVC A ,A + DPTR MOV P1 , A SETB P2.1LCALL DELAY3CLR P2.1LCALL DELAY3 CLR P2.4 CLR P2.5CLR P2.6CLR P2.7 SETB P2.4 SETB P2.5SETB P2.6SETB P2.7 LCALL DELAY3TAB1: DB 3FH,06H,5BH,4FH,66H,6DH,7DH,07H,7FH,6FHKEYGET: KEY0: LCALL DELAY4JB P2.5 , KEYK3KEYK2: LCALL DELAY4MOV A , 13HMOV B , #1SUB A ,BLCALL DELAY4CJNE A , #0 , KEYK21MOV 13H ,#9KEY21: JNB P2.5 , KEYK2JB P2.6 , KEYK4KEYK3: LCALL DELAY4MOV A , 12HMOV B , #1SUB A ,BLCALL DELAY4CJNE A , #0 , KEYK31MOV 12H ,#9KEY31: JNB P2.6 , KEYK3JB P2.7 , KEYK5KEYK4 : LCALL DELAY4MOV A , 43HMOV B , #1SUB A ,BLCALL DELAY4CJNE A , #0 , KEYK41MOV 43H ,#9KEY41: JNB P2.7 , KEYK4JB P3.5 , KEY0LOOP TEMFUZHIRETSHOWTEM2:LCALL SHIFT ;二进制转化为十进制 MOV A,R1 INC R1 MOV DPTR,#TAB2 MOVC A,A+DPTR MOV P1,A SETB P3.0 LCALL DELAY3 CLR P3.0 LCALL DELAY3 MOV A,R1 SWAP A ANL A,#0FH MOVC A,A+DPTR MOV P1,A SETB P3.1 LCALL DELAY3 CLR P3.1 LCALL DELAY3 MOV A,R1 INC R1 ANL A,#0FH MOVC A,A+DPTR MOV P1,A SETB P3.2 LCALL DELAY3 CLR P3.2 LCALL DELAY3TAB2: DB 3FH,06H,5BH,4FH,66H,6DH,7DH,07H,7FH,6FH ; 数字代码表 RETSHIFT2:MOV R1,#2BH MOV A,3DH MOV B,#100 DIV AB MOV R1,A INC R1 MOV A,#10 XCH A,B DIV AB SWAP A ADD A,B MOV R1,A RETPIDPROGRAM:MOV R5 ,12HMOV R4, 13HMOV R3 ,30HMOV R2 ,#00HLCALL CPL1LCALL DSUMMOV 1AH ,R7MOV 1BH ,R6MOV R5 ,16HMOV R4 ,17HMOV R0, #26HLCALL MULT1MOV R5 ,1AHMOV R4 ,1BHMOV R3, 1CHMOV R4 ,1DHLCALL DSUMMOV R5,14HMOV R4 ,15HMOV R0 , #22HLCALL MULT1MOV R5 ,25HMOV R4 , 24HMOV R3 ,2EHMOV R2, 2DHLCALL DSUMMOV 26H, R7MOV 27H ,R6MOV R5 , 1AHMOV R4 ,1BHMOV R3, 1EHMOV 1FH, 1DHLCALL DSUMMOV A, R7MOV R5 ,AMOV A,R6MOV R4 , AMOV R3 , 1CHMOV R2 , 1DHLCALL DSUMMOV A, R7MOV R5 , AMOV A ,R6MOV R4 ,AMOV R3 ,1CHMOV R2 ,1DHLCALL DSUMMOV R5 ,18HMOV R4 ,19HMOV R0 ,#22HLCALL MULT1MOV R5 , 25HMOV R4 , 24HMOV R3, 26HMOV R2 , 27HLCALL DSUMMOV A ,R7MOV R3 , AMOV A, R6MOV R2 , AMOV R5 , 10HMOV R4 , 11HLCALL DSUMMOV 10H , R7MOV 11H , R6MOV 1EH,1CHMOV 1FH , 1DHMOV 1CH, 1AHMOV 1DH , 1BHMOV A ,10HJNB ACC.7 , CONT1MOV 2AH ,#00HRETCONT1:MOV A , 11HRLC AMOV A ,10HRLC AMOV R2 ,ASUBB A, #0FAHJNC CONT2MOV 2AH , R2RETCONT2:MOV 2AH ,#0FAHRETCPL1:MOV A ,R2CPL AADD A , #01HMOV R2 ,AMOV A ,R3CPL AADDC A , #00HMOV R3 , ARETDSUM:MOV A ,R4ADD A ,R2MOV R6, AMOV A ,R5ADDC A ,R3MOV R7 ,ARETMULT:MOV A ,R6MOV B ,R4MUL ABMOV R0 ,AMOV R3 , BMOV A ,R4MOV B ,R7MUL ABADD A ,R3MOV R3 , AMOV A ,BADDC A ,#00HMOV R2 ,AMOV A ,R6MOV B ,R5MUL ABADD A ,R3INC R0MOV R0 ,ACLR F0MOV A ,R2ADDC A ,BMOV R2 , AJNC LASTSETB F0LAST:MOV A ,R7MOV B ,R5MUL ABADD A ,R2INC R0MOV R0 ,AMOV A ,BADDC A ,#00HMOV C , F0ADDC A ,#00HINC R0MOV R0 , ARETMULT1:MOV A ,R7RLC AMOV 5CH, CJNC POS1MOV A ,R6CPL AADD A ,#01HMOV R6 , AMOV A ,R7CPL AMOV R2,#00HADDC A , R2MOV R7 ,APOS1:MOV A ,R5RLC AMOV 5DH, CJNC POS2MOV A ,R4CPL AADD A , #01HMOV R4 ,AMOV A ,R5CPL AADDC A , #00HMOV R5 , APOS2:MOV 09H, R0LCALL MULTMOV C ,5CHANL C ,5DHJC TPLMOV C , 5CHORL C ,5DHJNC TPLMOV R0 , 09HMOV A , R0CPL AADD A ,#01HMOV R0 ,AINC R0MOV A ,R0CPL AADDC A ,#00HMOV R0 ,AINC R0MOV A ,R0CPL AADDC A ,#00HMOV R0 , AINC R0MOV A ,R0CPL AADDC A , #00HMOV R0 , ATPL :RETDELAY1: MOV R6,#10DELAY20MS2: MOV R7, #80LOOP20MS: DJNZ R7, LOOP20MSDJNZ R6 , DELAY20MS2RETDELAY3: MOV R5,#30 DE31: MOV R4 , #30 DE32: DJNZ R4 ,DE32 DJNZ R5 , DE31 RETDELAY4: MOV R5 , #100DE41: MOV R4 , #60DE42: DJNZ R4 ,DE42 DJNZ R5 ,DE41 RETEND IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART B: CYBERNETICS, VOL. 35, NO. 1, FEBRUARY 200523Relay Feedback Tuning of Robust PID ControllersWith Iso-Damping PropertyYangQuan Chen, Senior Member, IEEE, and Kevin L. Moore, Senior Member, IEEEAbstractAnewtuningmethodforproportional-inte-gral-derivative (PID) controller design is proposed for a classof unknown, stable, and minimum phase plants. We are able todesign a PID controller to ensure that the phase Bode plot is flat,i.e., the phase derivative w.r.t. the frequency is zero, at a givenfrequency called the “tangent frequency” so that the closed-loopsystem is robust to gain variations and the step responses exhibitan iso-damping property. At the “tangent frequency,” the Nyquistcurve tangentially touches the sensitivity circle. Several relayfeedback tests are used to identify the plant gain and phase at thetangent frequency in an iterative way. The identified plant gainand phase at the desired tangent frequency are used to estimatethe derivatives of amplitude and phase of the plant with respect tofrequency at the same frequency point by Bodes integral relation-ship. Then, these derivatives are used to design a PID controllerfor slope adjustment of the Nyquist plot to achieve the robustnessof the system to gain variations. No plant model is assumed duringthe PID controller design. Only several relay tests are needed.Simulation examples illustrate the effectiveness and the simplicityof the proposed method for robust PID controller design with aniso-damping property.Index TermsBodes integral, flat phase condition, iso-dampingproperty, proportional-integral-derivative (PID) controller, PIDtuning, relay feedback test.I. INTRODUCTIONACCORDING to a survey 1 of the state of process con-trolsystemsin1989conductedbytheJapanElectricMea-suring Instrument Manufacturers Association, more than 90of the control loops were of the proportional-integral-deriva-tive (PID) type. It was also indicated 2 that a typical papermill in Canada has more than 2,000 control loops and that 97%use PI control. Therefore, the industrialist had concentrated onPI/PID controllers and had already developed one-button typerelay auto-tuning techniques for fast, reliable PI/PID controlyet with satisfactory performance 37. Although many dif-ferent methods have been proposed for tuning PID controllers,the ZieglerNichols method 8 is still extensively used for de-terminingtheparametersofPIDcontrollers.ThedesignisbasedManuscript receivedFebruary 27, 2004; revisedon May30, 2004. This paperwas recommended by Associate Editor S. Phoha.Y. Chen is with the Center for Self-Organizing and Intelligent Systems(CSOIS), Department of Electrical and Computer Engineering, College ofEngineering, Utah State University, Logan, UT 84322-4160 USA (e-mail:yqchen).K. L. Moore was with the Center for Self-Organizing and Intelligent Sys-tems (CSOIS), Department of Electrical and Computer Engineering, College ofEngineering, Utah State University, Logan, UT 84322-4160 USA. He is nowwith the Research and Technology Development Center, The Johns HopkinsUniversity Applied Physics Laboratory, Laurel, MD 20723-6099 USA (e-mail:kevin.moore).Digital Object Identifier 10.1109/TSMCB.2004.837950on the measurement of the critical gain and critical frequency ofthe plant and using simple formulae to compute the controllerparameters. In 1984, strm and Hgglund 9 proposed an au-tomatic tuning method based on a simple relay feedback testwhich uses the describing function analysis to give the criticalgain and the critical frequency of the system. This informationcan be used to compute a PID controller with desired gain andphase margins. In relay feedback tests, it is a common practiceto use a relay with hysteresis 9 for noise immunity. Anothercommonlyusedtechniqueis tointroduceanartificialtimedelaywithin the relay closed-loop system, e.g., 10, to change the os-cillation frequency in relay feedback tests.After identifying a point on the Nyquist curve of the plant,the so-called modified ZieglerNichols method 4, 11 canbe used to move this point to another position in the complexplane. Two equations for phase and amplitude assignment canbe obtained to retrieve the parameters of a PI controller. For aPID controller, however, an additional equation should be intro-duced. In the modified ZieglerNichols method, the ratio be-tween the integral timeand the derivative time, is chosento be constant, i.e., in order to obtain a unique solu-tion.The control performance is heavily influenced by the choiceofasobservedin10.Recently,theroleofhasdrawnmuchattention, e.g., 1214. For the ZieglerNichols PID tuningmethod,is generally assigned as a magic number four 4.Walln, strm, and Hgglund proposed that the tradeoff be-tweenthepractical implementationandthesystemperformanceis the major reason for choosing the ratio betweenandasfour 12.The main contribution of this paper is the use of a newtuning rule which gives a new relationship betweenandinstead of the equationproposed in the modifiedZieglerNichols method 4, 11. We propose to add an extracondition that the phase Bode plot at a specified frequencyat the point where sensitivity circle touches Nyquist curve islocally flat which implies that the system will be more robust togain variations. This additional condition can be expressed as, which can be equivalently expressedas(1)whereis the frequency at the point of tangency andis the transfer function of the open loopsystem including the controllerand the plant. Theabove equivalence in (1) is mathematically explained in detail1083-4419/$20.00 2005 IEEE24IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART B: CYBERNETICS, VOL. 35, NO. 1, FEBRUARY 2005Fig. 1.Illustration of the basic idea for isodamping robust PID tuning.in the Appendix. In this paper, we consider the PID controllerof the following form:(2)This “flat phase” idea proposed earlier is illustrated inFig. 1(a) where the Bode diagram of the open loop system isshown with its phase being tuned locally flat around. Wecan expect that, if the gain increases or decreases a certain per-centage, the phase margin will remain unchanged. Therefore,in this case, the step responses under various gains changingaround the nominal gain will exhibit an iso-damping property,i.e., the overshoots of step responses will be almost the same.This can also be explained by Fig. 1(b) where the sensitivitycircle touches the Nyquist curve of the open loop system at theflat phase point. Clearly, since gain variations are unavoidablein the real world due to possible sensor distortion, environmentchange and etc., the iso-damping is a desirable property whichensures that no harmful excessive overshoot is resulted due togain variations.Assume that the phase of the open loop system atis(3)So,thedefinitionofisthephaseangleofat the frequency. Then, the corresponding gain can be ex-pressed by(4)With these two conditions, (3) and (4), and the new condition(1), all the three parameters of PID controller can be calculated.As in the ZieglerNichols method,andare used to tunethe phase condition (3) andis determined by the gain condi-tion(4).However,thecondition(1)givesarelationshipbetweenandinstead of.Note that in this new tuning method,is not necessarilythegain crossoverfrequencyalthoughclose. Precisely,is thefrequency at which the Nyquist curve tangentially touches thesensitivity circle. Similarly, the tangent phase, is not nec-essarily the phase margin usually used in previous PID tuningmethods. According to 4, the phase margin is always selectedfrom 30 to 60 . Due to the flat phase condition (1), the deriva-tive of the phase nearwill be relatively small. Therefore, ifis selected to be around 30 , such as 35 , the phase marginwill be generally within the desired interval.II. SLOPEADJUSTMENT OF THEPHASEBODEPLOTIn this section, we will show howandare related underthe new condition (1).Substitutebyso that the closed loop system can bewritten as, where(5)is the PID controller obtained from (2). The phase of the closedloop system is given by(6)The derivative of the closed loop systemwith respect tocan be written as follows:(7)From (1), the phase of the derivative of the open loop systemcan not obviously be obtained directly from (7). So, we need tosimplify (7).The derivative of the controller with respect tois(8)To calculate, since we have(9)CHEN AND MOORE: RELAY FEEDBACK TUNING OF ROBUST PID CONTROLLERS WITH ISO-DAMPING PROPERTY25differentiating (9) with respect togives(10)Straightforwardly, we arrive at(11)Substituting (5), (8), and (11) into (7) gives(12)Hence, the slope of the Nyquist curve at any specific frequencyis given by(13)where, following the notations introduced in 15, 16,andare used throughout this paper defined as follows:(14)(15)Here,ourtaskistoadjusttheslopeoftheNyquistcurvetomatchthe condition shown in (1). By combining (1), (6), and (13), oneobtains(16)After a straightforward calculation, one obtains the relationshipbetweenandas follows:(17)where.Notethatduetothenatureofthequadraticequation,analternativerelationship,isthat.Weshould discard one to ensure that thegain is a real positivenumber to avoid the right half plane zeros in. In whatfollows, (17) is used. Additional,could be negative ifisnot specified properly.Fig. 2.Relay plus artificial time delay (?) feedback system.The approximation offor stable and minimum phase plantcan be given as follows 17:(18)whereis the static gain of the plant,isthe phase andis the gain of the plant at the specificfrequency.It is obvious thatandare related byalone. For thisnewtuning method,includes alltheinformationthatwe needof the unknown plant. In what follows, we show that theesti-mate formula can be extended to plants with integrators and/ortime delay.Consider the plant withintegrators(19)Clearly, one can not get the static gain of such systems to com-putedirectly. But from (15)(20)for the systems with integrators,should be estimated ac-cording to the systems without any integrator.For the plant with a time delay(21)in the same way(22)Consequently, substituting (18) we obtain(23)Obviously, the time delay will not contribute to the estimationof.So, in general, for the plant with both integrators and a timedelay(24)26IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART B: CYBERNETICS, VOL. 35, NO. 1, FEBRUARY 2005Fig. 3.Frequency responses of? ?and? ? ?(dashed line: the modified ZieglerNichols, solid line: the proposed). (a) Comparison of Bodeplots. (b) Comparison of Nyquist plots.Fig. 4.Comparisons of frequency responses and step responses of? ?and? ? ?(dashed line: the modified ZieglerNichols, solid line: theproposed. For both schemes, gain variations 1, 1.1, 1.3 are considered in step responses). (a) Comparison of Nyquist plots. (b) Comparison of step responses.according to (20) and (23)(25)III. NEWPID CONTROLLERDESIGNFORMULAESupposethatwehaveknownat.Howtoexperimentallymeasurewill be discussed in the next section based onthe measurement ofand.To write down explicitly the formulae for, and, letus summarize what are known at this point. We are given1), the desired tangent frequency;2), the desired tangent phase;3) measurement ofand4) the estimation of.Furthermore, using (3) and (4), the PID controller parameterscan be set as follows:(26)(27)where. Finally,can be computed from(17).Remark III.1: The selection ofheavily depends on thesystem dynamics. For most of plants, there exists an interval forthe selection ofto achieve flat phase condition. If no betteridea about, the desired cutoff frequency can be used as theinitial value. For, a good choice is within 30 to 35 .CHEN AND MOORE: RELAY FEEDBACK TUNING OF ROBUST PID CONTROLLERS WITH ISO-DAMPING PROPERTY27Fig. 5.Comparisons of frequency responses of? ?and? ? ?(dashed line: the modified ZieglerNichols, solid line: the proposed).(a) Comparison of Bode plots. (b) Comparison of Nyquist plots.Fig.6.Comparisonofstepresponsesof? ?and? ? ?(solidline: the proposed modified controller with gain variations 1, 0.9, 0.8; dottedline: the modified ZieglerNichols controller with gain variations 1, 0.9, 0.8).IV. MEASURINGarg,ANDVIARELAYFEEDBACKTESTSFollowing the discussion in the previous section, the param-eters of a PID controller can be calculated straightforwardly ifwe know, and.As indicated in (18),can be obtained from the knowl-edge of the static gain, and. Thestatic gainoris very easy to measure and it is as-sumed to be known. The relay feedback test, shown in Fig. 2,can be used to “measure”and. In the relayfeedback experiments, a relay is connected in closed-loop withthe unknown plant as shown in Fig. 2 which is usually usedto identify one point on the Nyquist diagram of the plant. Tochangetheoscillationfrequencyduetorelayfeedback,anartifi-cialtimedelayisintroducedintheloop.Theartificialtimedelayis the tuning knob here to change the oscillation frequency.Our problem here is how to get the right value ofwhich cor-responds to the tangent frequency. To solve this problem, aniterative method can be used as summarized in the following:1. Start with the desired tangent fre-quency.2. Select two different values (and) for the time delay parameter prop-erly and do the relay feedback test twice.Then, two points on the Nyquist curve ofthe plant can be obtained. The frequen-cies of these points can be represented asandwhich correspond toand,respectively. The iteration begins withthese initial valuesand.3. With the values obtained in the pre-vious iterations, the artificial timedelay parametercan be updated using asimple interpolation/extrapolation schemeas follows:whererepresents the current iterationnumber. With the new, after the relaytest, the corresponding frequencycanbe recorded.4. Comparewith. If, quititeration. Otherwise, go to Step 3. Here,is a small positive number.The iterative method proposed above is feasible because ingeneral the relationship between the delay timeand the oscil-lation frequencyis one-to-one.After the iteration, the final oscillation frequency is quiteclose to the desired oneso that the oscillation frequency isconsidered as. Hence, the amplitude and the phase of the28IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART B: CYBERNETICS, VOL. 35, NO. 1, FEBRUARY 2005Fig. 7.Comparisons of frequency responses of? ?and? ? ?(dashed line: the modified ZieglerNichols, solid line: the proposed).(a) Comparison of Bode plots. (b) Comparison of Nyquist plots.Fig.8.Comparisonofstepresponsesof? ?and? ? ?(solidline: the proposed modified controller with gain variations 1, 1.5, 1.7; dottedline: the modified ZieglerNichols controller with gain variations 1, 1.5, 1.7).plant at the specified frequency can be obtained. Using (18),one can calculate the approximation of.V. ILLUSTRATIVEEXAMPLESThe new PID design method presented above will be illus-trated via some simulation examples. In the simulation, the fol-lowing classes of plants, studied in 12, will be used.(28)(29)(30)(31)A. High-Order PlantConsiderplantin(28).Thisplantwasalsousedin15.The specifications are set asand.The PID controller designed by using the proposed tuning for-mulae is(32)The PID controller designed by the modified ZieglerNicholsmethod is(33)The Bode and the Nyquist plots are compared in Fig. 3. Fromthe Bode plots, it is seen that the phase curve near the fre-quencyis flat. The phase margin roughlyequals 45 . That means the controller moves the pointof the Nyquist curve toon the unit circle witha phase of 135 and at the same time makes the Nyquist curvesatisfy (1).However, in Fig. 3(b), the Nyquist plot of the open loopsystem is not tangential to the sensitivity circle at the flat phasebut to another point on the Nyquist curve. Definethe frequency interval corresponding to the flat phase. So, thegain crossover frequencycan be moved withinby adjustingbywhere.For this example, ifis changed to,the flat phase segment will tangentially touch the sensitivitycircle. The Nyquist plot of the open loop system with themodified proposed PID controller, i.e., is shown inFig. 4(a) and the step responses of the closed loop system arecompared in Fig. 4(b). Comparing the closed-loop system withthe modified proposed PID controller to that with the modifiedZieglerNichols controller, the overshoots of the step responsesfrom the proposed scheme remain almost invariant undergain variations. However, the overshoots using the modifiedZieglerNichols controller change remarkably.CHEN AND MOORE: RELAY FEEDBACK TUNING OF ROBUST PID CONTROLLERS WITH ISO-DAMPING PROPERTY29Fig. 9.Comparisons of frequency responses of? ?and? ? ?(Dashed line: The modified ZieglerNichols, Solid line: The proposed). (a)Comparison of Bode plots. (b) Comparison of Nyquist plots.B. Plant With an IntegratorFor the plant, the proposed controller iswith respect to,and. Thecontroller designed by the modified ZieglerNichols method isTheBodeplotofthissituation,showninFig.5(a),isquitedif-ferent with that of plant. The flat phase occurs at the peakof the phase Bode plot. The Nyquist diagrams are compared inFig. 5(b). The step responses are compared in Fig. 6 where theproposed controller does not exhibit an obviously better per-formance than the modified ZieglerNichols controller for theiso-damping property because of the effect of the integrator.C. Plant With a Time DelayFor the plantthe proposed controller iswith respect to, and. ThecontrollerdesignedbythemodifiedZieglerNicholsmethodisThe Bode plotsand Nyquistplotsare compared inFig. 7.Thestep responses are compared in Fig. 8 where the iso-dampingproperty can be clearly observed.Fig. 10.Comparison of step responses of? ?and? ? ?(Solid line: The proposed modified controller with gain variations 1, 1.5, 1.7;Dotted line: The modified ZieglerNichols controller with gain variations 1,1.5, 1.7).D. Plant With an Integrator and a Time DelayFor the plant, the proposed controller iswith respect to, and. Thecontroller designed by the modified ZieglerNichols method isTheBode plotsand Nyquistplots are compared inFig.9. Thestep responses are compared in Fig. 10 where the iso-dampingproperty can be clearly observed.30IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART B: CYBERNETICS, VOL. 35, NO. 1, FEBRUARY 2005VI. CONCLUSIONAnewPIDtuningmethodisproposedforaclassofunknown,stable and minimum phase plants. Given the tangent frequency, the tangent phaseand with an additional condition thatthe phase Bode plot atis locally flat, we can design thePID controller to ensure that the closed loop system is robustto gain variations and to ensure that the step responses exhibitan iso-damping property. No plant model is assumed during thePID controller design. Only several relay tests are needed. Sim-ulation examples illustrate the effectiveness and the simplicityof theproposed methodfor robustPID controller designwith aniso-damping property for different types of plants.Our further research efforts include1) determining the width and the position of the flat phase soas to achieve the performance of the proposed controllerand simplify the design procedure;2) testing on more types of plants;3) exploring nonminimum phase, open loop unstable sys-tems.APPENDIXDERIVATION OF(1)Assume that. Then,The derivative ofwith respect tois thatFurthermore,onehas,whichmeansthat. Sinceand, therefore,.ACKNOWLEDGMENTThe first author, Y. Chen, wishes to thank L.-C. Fu, Ed-itor-in-Chief of Asian Journal of Control for providing acomplimentary copy of the “Special Issue on Advances in PIDControl,”Asian J. Control (vol. 4, no. 4, 2002). The authors alsowish to thank C. Hu for performing the simulation and B. M.Vinagre for his comments on an earlier version of the paper.Finally, useful comments from the three anonymous reviewersare acknowledged.REFERENCES1 S. Yamamoto and I. Hashimoto, “Recent status and future needs: Theview from Japanese industry,” in Proc. 4th Int. Conf. Chemical ProcessControl, Chemical Process Control CPCIV, 1991.2 W.L. Bialkowski,“Dreams versus reality: A view from both sidesof thegap,” Pulp Paper Canada, vol. 11, pp. 1927, 1994.3 A. Leva, “PID autotuning algorithm based on relay feedback,” Inst.Elect. Eng.Proc. Part-D, vol. 140, no. 5, pp. 328338, 1993.4 T. Hagglund and K. J. Astrom, PID Controllers: Theory, Design, andTuning, 2nd ed: ISA The Instrumentation, Systems, and AutomationSociety, 1995.5 C.-C. Yu, “Autotuning of PID controllers: Relay feedback approach,” inAdvancesinIndustrialControl.London,U.K.:Springer-Verlag,1999.6 K. K. Tan, W. Qing-Guo, H. C. Chieh, and T. Hagglund, Advances inPID controllers.London, U.K.: Springer-Verlag, 2000, Advances inIndustrial Control.7 S. P. Bhattacharyya, A. Datta, and M. T. Ho, Structure and Synthesis ofPID Controllers.London, U.K.: Springer-Verlag, 2000.8 J. G. Ziegler and N. B. Nichols, “Optimum settings for automatic con-trollers,” Trans. ASME, vol. 64, pp. 759768, 1942.9 K. J. strm and T. Hgglund, “Automatic tuning of simple regulatorswith specifications on phase and amplitude margins,” Automatica, vol.20, no. 5, pp. 645651, 1984.10 K. K. Tan, T. H. Lee, and Q. G. Wang, “Enhanced automatic tuningprocedure for process control of PI/PID controllers,” AlChE J., vol. 42,no. 9, pp. 25552562, 1996.11 C. C. Hang, K. J. strm, and W. K. Ho, “Refinements of the Ziegler-Nichols tuning formula,” Proc. Inst. Elect. Eng.Pt. D, vol. 138, no. 2,pp. 111118, 1991.12 A.Walln,K.J.strm,andT. Hgglund,“Loop-shapingdesignofPIDcontrollers with constant? ?ratio,” Asian J. Contr., vol. 4, no. 4, pp.403409, 2002.13 H. Panagopoulos, K. J. strm, and T. Hgglund, “Design of PID con-trollers based on constrained optimization,” in Proc. American ControlConf., San Diego, CA, 1999.14 B. Kristiansson and B. Lennartsson, “Optimal PID controllers includingroll off and Schmidt predictor structure,” in Proc. IFAC 14th WorldCongr., vol. F, Beijing, China, 1999, pp. 297302.15 A. Karimi, D. Garcia, and R. Longchamp, “PID controller design usingBodes integrals,” in Proc. American Control Conf., Anchorage, AK,2002, pp. 50075012.16, “Iterativ
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