Human-Simulating Intelligent PID Control.

Transcription

1 Internatonal Journal of Moern Nonlnear Theory an Applcaton, 017, 6, ISSN Onlne: ISSN Prnt: Human-Smulatng Intellgent PID Control Zhunng Lu Qngao No. Mle School of Shanong Provnce, Qngao, Chna How to cte ths paper: Lu, Z.N. (017) Human-Smulatng Intellgent PID Control. Internatonal Journal of Moern Nonlnear Theory an Applcaton, 6, Receve: Aprl 9, 017 Accepte: June 18, 017 Publshe: June 1, 017 Copyrght 017 by author an Scentfc Research Publshng Inc. Ths work s lcense uner the Creatve Commons Attrbuton Internatonal Lcense (CC BY 4.0). Open Access Abstract Motvate by PID control smplcty, robustness an valty to eal wth the nonlnearty an uncertantes of ynamcs, through smulatng the ntellgent behavor of human manual control, an only usng the elementary nformaton on han, ths paper ntrouces a smple formulaton to represent pror knowlege an experences of human manual control, an proposes a smple an practcable control law, name Human-Smulatng Intellgent PID control (HSI-PID), an the smple tunng rules wth the explct physcal meanng. HSI-PID control can not only easly ncorporate pror knowlege an experences of experts control nto the controller but also automatcally acqure knowlege of control experences from the past control behavor to correct the control acton onlne. The theoretcal analyss an smulaton results show that: HSI-PID control has the better flexblty, stronger robustness, an especally the faster self-learnng ablty, an t can make the moton of system entcally track the esre response, whether the controlle system has the strong nonlnearty an uncertantes of ynamcs or not, even uner the actons of uncertan, varyng-tme an strong sturbances. Keywors Nonlnear Control, PID Control, Manual Control, Knowlege-Base Control, Intellgent Control 1. Introucton It s well known that the PID controllers [1] [] [3] are stll use extensvely n nustral control an stue ntensvely n current control area because the PID control not only has exceeng smplcty an strong robustness but also can effectvely eal wth nonlnearty an uncertantes of ynamcs an asymptotc stablty can be acheve accorngly. A major rawback of PID control s that t often suffers a serous loss of performance, that s, causes large overshoot an long settlng tme, even may lea to nstablty ue to unlmte ntegral acton. To sguse ths rawback, varous DOI: /jmnta June 1, 017

2 PID-lke control laws have been propose to mprove the transent performance. For example, a saturate-p, an fferental feeback plus a PI controller rven by a boune nonlnear functon of poston errors [4], a lnear PD plus an ntegral acton of a nonlnear functon of poston errors [5], an a lnear PD plus a ouble ntegral acton rven by the postons error an the fltere poston [6] are presente recently. An unconquerable rawback of PID-lke controller above s that ts tunng proceure s very complex, teous an ffcult to obtan the satsfe transent control performance because these PID-lke controllers often prouce surge an bg overshoot, even may lea to nstablty. It s obvous that all the PID-lke control laws above can only mprove the control performance n some extent but the ntrnsc shortcomngs of PID control are not absolutely elmnate from ts root. In 009, general ntegral control [7] appeare. After that varous general ntegral control laws were presente. For example, general concave ntegral control [8], general convex ntegral control [9], general boune ntegral control [10] an the generalzaton of the ntegrator an ntegral control acton [11] were all evelope by resortng to an ornary control along wth a known Lyapunov functon. Although these general ntegral control laws above can effectvely eal wth the ntrnsc shortcomngs of PID control an have the better control performance; these PID-lke an general ntegral controllers above are all unntellgent. Hence, t s very nterestng an challengng to seek for PID control laws wth ntellgence, especally self-learnng ablty onlne. Base on the statement above, t s obvous that the error, e, an ts ervatve, e are the funamental elements of PID control, whch are the essental nformaton use n manual control, an then the human manual control law can be escrbe, as follows: where f ( ) an ( ) (,, σ ) ( ee, ) u = f ee σ = γ ( ) γ 0, 0 = 0 γ can be arbtrary lnear or nonlnear functon. Base on the cognton above, through smulatng the ntellgent behavor of human manual control, ths paper proposes a smple an practcable control law, name Human-Smulatng Intellgent PID control (HSI-PID). The man contrbutons are as follows: 1) a smple an practcal formulaton to represent pror knowlege an experences of manual control s evelope, an then t s easy to be use to esgn a controller; ) the smple an practcal tunng rules wth the explct physcal meanng are ntrouce, an then t s easy to tune a hgh performance controller; 3) two kns of fre new an smple ntegrator an ntegral control acton s propose, an then the ntrnsc shortcomngs of PID control s remove n prncple; 4) HSI-PID control not only can easly ncorporate pror knowlege an experences of experts control nto the controller but also can automatcally acqure knowlege of control experences from the past control behavor to correct the control acton onlne, an then can make the moton of system entcally track the esre response. Moreover, smulaton results verfy (1) 75

3 our conclusons agan. The remaner of the paper s organze as follows: Secton escrbes Human-Smulatng PID control law an efntons. Secton 3 presents the tunng rules of human-smulatng ntellgent PID control. Example an smulaton are prove n Secton 4. Conclusons are presente n Secton 5.. Human-Smulatng PID Control Law An experence operator wll antcpate all types of sturbances to the system. It woul be very ffcult to reprouce n an automatc controller the many jugments that an average person makes aly an unconscously. So t s more realstc for us to smulate the control proceure of the human manual control n orer to obtan as nearly perfect operaton as possble. A general human manual control proceure can be escrbe as follows: 1) when the error s large an the error tens to ncrease, the human controller prouces a largest control acton, an attempt to restran the error to ncrease contnuously; ) when the error ecreases, the human controller wll graually reuce the control acton to lmt the egressve rate of error for avong the system out of control; 3) when the error s small an ten to zero, the human controller prouces a small or even nverte control acton, an attempt to prevent the overshoot; 4) when the error s large, the human controller prouces an naccuracy control acton, f the error s small, the human controller prouces a precse control acton; 5) the ampltue of control acton s etermne by the sze of error an tenency that the error s changng. Base on the statements above, t s easy to see that the human manual control law s n a goo agreement wth the sgmo-type functon. So, the sgmo-type functon can be use to smulatng the proceure of human manual control, that s, approxmatng Equaton (1). For the purpose of ths paper, t s convenent to ntrouce the followng efnton. Defnton 1. Tow Error [1], e In human manual control proceure: f e e or e e, an then e = e or e = e ; If e e, an then e = e. For smulatng the behavor that human eals wth the nformaton, Tow Error, e s efne by, e e e e = e e < e< e () e e e where e s a postve constant, mum of Tow Error. The schematc graph of Tow Error s shown n Fgure 1. Defnton. Desre Acceleraton [1], e In human manual control proceure, one usually has an explct expectaton for the spee of response an overshoot. Namely, uner the control acton, the error, e wll change n conformty to the esre response. It cannot too fast, also cannot too slow, eventually the error, e an ts ervatve, e all ten to 76

4 Fgure 1. Tow error. zero n the meantme. In another wors, there s some kns of the cooperatve relatonshp between the error an ts rate n human manual control proceure. For smulatng the behavor of human manual control, t s nspensable to ntrouce the concept of Desre Acceleraton, e, an whch can be taken as the sgmo-type functon, ( ) kp e+ ke e = e sgmo e where e s a postve constant, mum of Desre Acceleraton; k p an k are all postve constants. Dscusson 1. When the ntal conton an parameter of Equaton (3) s gven, the esre velocty, e, an esre error, e can be obtan. Ths shows that Equaton (3) not only escrbes the esre response spee an esre response error of human manual control proceure but also constructs a kn of cooperatve relatonshp between the error an ts rate. So, Equaton (3) can be vewe as a representaton of pror knowlege an experences of human manual control. The schematc graph of Desre Acceleraton s shown n Fgure..1. PD-Type Human-Smulatng Intellgent Control Law From Equaton (3), f the controller output s assmlate to the force actng on the moton object, an usng Newton s laws of moton, the controller output, u can be gven by the followng equaton: u = k e (4) m where k m s a postve constant, f we carry on the moton control, an then t s the mass or nerta, f we carry on the process control, an then t s a constant lke the mass. Combnng Equatons () an (3) nto Equaton (4) results n PD-type Human-smulatng ntellgent control law, as follows: ( ) kp e+ ke u = kme sgmo e (3) (5) 77

5 Fgure. Desre acceleraton. Dscusson. From Fgure, Equaton (), an control law (5), t s easy to see that: 1) when e > e an the recton of e an e s entcal, the control law (5) prouce a largest an naccuracy control acton, u to restran the error to ncrease contnuously; ) when e > e, an the recton of e an e s reverse, u wll graually ecrease to lmt the egressve rate of error for avong the system out of control; 3) when e < e, an the recton of e an e s reverse, u wll become a small or even nverte control acton to prevent the overshoot; 4) when e an e all ten to zero, u ten to zero, too. In another wors, the control acton, u can make the error reuce along the esre law, eventually the error, ts rate an control acton all ten to zero n the meantme. These show that the PD-type Human-smulatng ntellgent control law; (5) s n a goo agreement wth the optmal proceure of manual operaton... PID-Type Human-Smulatng Intellgent Control Law It s obvous that PD-type Human-smulatng ntellgent control law escrbe by Equaton (5) s not capable of reucng the error to zero an makng the error strctly change along the esre law when a constant sturbances act on the controlle system. However, an experence operator can automatcally ajust the control output an prouce an accurate control acton to elmnate or reject all kns of sturbances. Ths proceure can be escrbe as follows: accorng to the current state of the controlle system an the former control experences, the human controller prouces a control acton, f the control acton just counteracts the sturbance acton, an then he mantans the control acton; f the control acton s large, an then he ecreases the control acton, vce versa. In other wors, through correctng the control acton agan an agan, eventually the operator can make the error change along the esre law an stablze the error at zero or acceptable lmt, an then the control acton tens to a constant or changes along wth the change of sturbances. Base on the statements above, t s easy to know that: 1) smulatng the control proceure that human manual control elmnates or rejects all kns of the 78

6 sturbances s very challengng because the sturbances are often uncertan an varyng-tme; ) human manual control s a proceure of automatcally accumulatng the control experences, an self-aaptng the acton of all kns of sturbances an uncertantes of ynamcs by exploraton. However, ntegral control just s of ths kn of ablty. So, for smulatng the manual control behavor above, the ntegral acton must be ntrouce nto the control law, as follows: ( ) k e+ ke u = k e sgmo + k σ p m e kp( e + ke ) σ = kme sgmo e ( ) k e+ ke kσ p u = kmesgmo e σ = e + ke where k s a postve constant. Dscusson 3. Compare to PID-lke control an general ntegral control laws reporte by [1]-[11], the strkng features of the control laws (6) an (7) are: 1) The pror knowlege an experences of manual control can be ncorporate nto the controller easly; ) When e + ke = 0, the ntegrator output remans constant; f the ntegral acton s larger than the one neee, e consequentally ncreases, an then the ntegrator output wll nstantly ecrease, vce versa. However, the ntegrator output of PID-lke control [1]-[6] contnues to ncrease unless the error passes through zero, an then for makng the ntegral control acton tens to a constant, the error s usually neee to pass through zero repeately. Therefore, the ntrnsc shortcomng of PID control can be remove n prncple; 3) Just as the statement above, the basc prncple of the ntegrator n (6) an (7) s smlar wth the general ntegrator n [7] [8] [9] [10] [11], but ther man fference s that Tow error s ntrouce nto the ntegrator here. Therefore, the ntegral control acton propose here can constrant the response rate. However, the ntegral control acton n [7] [8] [9] [10] [11] o not place restrctons on the response rate, an then t s easy to lea to nstablty snce the response rate coul be too rap. Therefore, two kns of the ntegrator an ntegral acton n (6) an (7) are fre new; 4) The ntegral control acton s a compensaton for the shortcomng of PD control acton, or an accumulaton of the past PD control acton. Ths metho to ajust ntegral acton can be vewe as the accumulaton of the past control experences or learnng from past control experences; 5) Combnng the emonstraton above an Dscusson, t s easy to see that the control laws (6) an (7) can strctly constrant the moton of system along the trajectores of esre response. All these show that the control laws propose here not only can easly ncor- (6) (7) 79

7 porate pror knowlege an experences of experts control nto the controller but also can automatcally acqure knowlege of control experences from the past control behavor to correct the control acton onlne, an then can make the moton of system entcally track the esre response. Therefore, the control laws (6) an (7) shoul have more rap aaptve or self-learnng ablty, better flexblty an stronger robustness, an can easly yel hgher control performance. Ths s why t s calle human-smulatng ntellgent PID control. 3. Tunng Controller For practcal applcatons n nustral control, t s very necessary an nterestng to evelop a smple an effcent metho to tune the controller propose here. So, the purpose of ths secton s to aress the tunng rules of human-smulatng ntellgent PID control. In the control laws (6) an (7), there are sx parameters. The parameters, k m an e have the explct physcal meanng, an often are all fxe constant, so easy to be etermne. In practce, f k m or e s uncertan an the bggest control acton, u s known, the prouct, kme as a whole can rectly be taken as u or an approprate value n contrast to u. e s a characterstc varable, whch ncates the controller shoul carry on a brakng acton when e = e an e s slghtly less than e n orer to avo bg overshoot, oscllaton an nstablty, so e can be etermne by the experence of human manual control or be measure rectly. Hence, only three parameters, k p, k an k nee be etermne. From the control law (5), t s easy to know that: 1) when e = 0 an e e, the output of sgmo-type functon shoul amount to the mum, an then k p can be obtane; ) when e = 0 an e = e, the controller output shoul amount to the mum, too, an then k can be etermne by the followng formula: k = e e (8) where e s the mum of the acceptable spee of response, t can be measure rectly when the bggest control acton acts on the controlle system. The purpose that we ntrouce the ntegral acton s to complete rejecton of uncertan sturbances an ynamcs of controlle system, an yel zero steaystate error. So, the ntegral acton n the control laws s nspensable for most nustral control applcatons. Ths leas to that k become a very key parameter, whch tunng rules are very ffcult, even mpossble to be escrbe by an accurate formulaton. In general, f k s too small, the ntegral acton ncreases the correcton slowly, ths easly results n a slower spee of response an a longer settlng tme; f k s too large, the ntegral acton ncreases the correcton more raply, ths coul cause large overshoot, even may lea to nstablty. The optmum response s always acheve through some sort of compromse. In general, the settngs above are usually close to the fnal values. However, a realstc expectaton s that some tweakng of the parameters wll be requre to obtan a transent performance of hgh control. In practce, f the overshoot s 80

8 large, whch means that spee of response s too rap or mum of Tow Error s too small, an then we can fx e an ncrease k, or smultanety ncrease e an k, vce versa. 4. Example an Smulaton To llustrate the effect of the control laws, a two lnk manpulators shown n Fgure 3 s consere. Its ynamcs are of the followng form: M11q 1 + M1q + F11q + F1qq 1 + G1 = u1 M1q 1 + Mq + F11q 1 + F1qq 1 + G = u M = m + m l + ml + mll cos q, where: 11 ( 1 ) 1 1 ( ) M = ml, M1 = ml + mll 1 cos ( q ), F11 = F1 = mll 1 sn ( q ), G = mglsn ( q1 + q) G = ( m + m ) gl sn ( q ) + m gl sn ( q + q ) The normal parameter values of system are selecte as: m 1 = 3 kg, m = kg, l1 = l = 1m, g = 10 m s. The esre (set pont) postons are chosen as: when t < 10 s, q 1 = 1, q = 1; when 10 s t < 0 s, q 1 = 3, q = ; when t 0 s, q1 = q = 0. The tangent hyperbolc functon, tanh ( ) s use as the sgmo functon. The parameters of control law (6) are gven as: km 1e 1 = 300, kme = 00, k 1, = 5, k p1, = 3, e 1, = k 1, = 1. The smulatons wth samplng pero of 1ms are mplemente. Uner the normal an perturbe parameters, the moton trajectores of Lnk1 an Lnk are shown n Fgure 4 an Fgure 5, respectvely. The otte lnes are the smulaton results uner normal parameters. The real lnes are the smulaton results uner perturbe parameters, that s, m = kg s substtute for m = 7 kg when 10 s t < 0 s, corresponng to movng payloa of 5 kg. From the smulaton results, t s easy to see that: 1) the optmum response can be acheve by a set of control parameters n the whole oman of nterest, even uner the case that the payloa s change abruptly; ) the moton trajectores shown n Fgure 4 an Fgure 5 are almost completely entcal, ths llustrates Fgure 3. The two lnk robot manpulators. 81

9 Fgure 4. Smulaton results of Lnk 1. Fgure 5. Smulaton results of Lnk. that the moton of the system can entcally track the esre response, whether the controlle system has the strong nonlnearty an uncertantes of ynamcs or not, even uner the actons of uncertan, varyng-tme an strong sturbances. All these emonstrates that: 1) HIS-PID controller not only can remove the ntrnsc shortcomngs of PID control but also has the faster self-learnng ablty, better flexblty an stronger robustness wth respect to uncertan nonlnear system; ) HIS-PID control not only can effectvely eal wth uncertan nonlnear system but also t s a powerful an practcal tool to solve the control esgn problem of ynamcs wth the nonlnear an uncertan actons. 5. Conclusons In ths paper, a novel human-smulatng ntellgent PID control law s foune by smulatng the ntellgent behavor of human manual control. The man contrbutons are as follows: 1) the formulaton to represent pror knowlege an experences of manual control are very smple an practcal, an then t s easy to be use to esgn a controller; ) the smple an practcal tunng rules wth the explct physcal meanng are ntrouce, an then t s easy to tune a hgh performance controller; 3) two kns of fre new an smple ntegrator an ntegral acton are propose, an then the ntrnsc shortcomngs of PID control are re- 8

10 move n prncple; 4) the control laws propose here not only can easly ncorporate pror knowlege an experences of experts control nto the controller but also can automatcally acqure knowlege of control experences from the past control behavor to correct the control acton onlne, an then can make the moton of system entcally track the esre response. Moreover, smulaton results verfy our conclusons agan. For HSI-PID control, ths paper s only a startng pont an further cooperatve research s neee to make progress n ths area. For example, the esgn theory to ensure system stablty s not evelope. References [1] Lu, J.K. (004) Avance PID Control MATLAB Smulaton. Publshng House of Electroncs Inustry, Bejng. [] Knospe, C.R. (006) PID Control. IEEE Control Systems Magazne, 6, [3] Hu, B.G. (006) A Stuy on Nnonlnear PID Controllers Proportonal Component Approach. Acta Automatca Snca, 3, [4] Armoto, S. (1994) A Class of Quas-Natural Potentals an Hyper-Stable PID Servo-Loops for Nonlnear Robotc System, Transactons of the Socety of Instrument an Control Engneers, 30, [5] Kelly, R. (1998) Global Postonng of Robotc Manpulators va PD Control plus a Class of Nonlnear Integral Actons. IEEE Transactons on Automatc Control, 43, [6] Ortega, R., Lora, A. an Kelly, R. (1995) A Semglobally Stable Output Feeback PID Regulator for Robot Manpulators. IEEE Transactons on Automatc Control, 40, [7] Lu, B.S. an Tan, B.L. (009) General Integral Control. Proceengs of the Internatonal Conference on Avance Computer Control, Sngapore, -4 January 009, [8] Lu, B.S., Luo, X.Q. an L, J.H. (013) General Concave Integral Control. Intellgent Control an Automaton, 4, [9] Lu, B.S., Luo, X.Q. an L, J.H. (014) General Convex Integral Control. Internatonal Journal of Automaton an Computng, 11, [10] Lu, B.S. (014) Constructve General Boune Integral Control. Intellgent Control an Automaton, 5, [11] Lu, B.S. (014) On the Generalzaton of Integrator an Integral Control Acton. Internatonal Journal of Moern Nonlnear Theory an Applcaton, 3, [1] Peng, L. (1995) Neural Network Control of the Intellgent Unerwater Vehcle, Ocean Engneerng, 13,

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