Intelligently Tuned Weights Based Robust H 4 Controller Design For Pneumatic Servo Actuator System With Parametric Uncertainty

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1 Austrlin Journl of Bsic nd Applied Sciences, 5(4): , 2011 ISSN Intelligently Tuned Weights Bsed Robust H 4 Controller Design For Pneumtic Servo Actutor System With Prmetric Uncertinty Hzem I. Ali, Smsul Bhri Mohd Noor, S. M. Bshi, Mohmmd Hmiruce Mrhbn Deprtment of Electricl nd Electronic Engineering, Fculty of Engineering, University Putr Mlysi, Mlysi Abstrct: This pper presents new method for tuning the weighing functions to design n H 4 controller. Bsed on prticle swrm optimiztion (PSO) lgorithm the, weighting functions re tuned. The PSO lgorithm is used to minimize the infinity norm of the trnsfer functions mtrix of the nominl closed loop system to obtin the optiml prmeters of the weighting function. This method is pplied to typicl industril pneumtic servo ctutor controlled by jet pipe vlve. The pneumtic system nonlinerity nd system prmeters uncertinty re the min problems in the design of desired controller for this plnt. A liner model of the plnt t certin operting point is derived nd the structured (prmetric) perturbtions in the plnt coeficients re tken into ccount. This method ensures n optiml robust stbility nd robust performnce for the pneumtic servo ctutor system. Simultion results re presented to verify the objectives of this method. Key words: Robust control, H 4 control, Pneumtic ctutor, Nonliner system, Uncertin system, PSO. INTRODUCTION A control system is robust if it remins stble nd chieves certin performnce criteri in the presence of possible uncertinties. The robust design is to find controller, for given system, such tht the closed loop system is robust. The H 4 optimiztion pproch nd its relted pproches, being developed in the lst two decdes nd still n ctive reserch re, hve been shown to be effective nd efficient robust design methods for liner, time invrint control systems (D-Wei et l, 2005). The gol of robust systems design is to retin ssurnce of system performnce in spite of model inccurcies nd chnges. A system is robust when it hs cceptble chnges in performnce due to model chnges or inccurcies. A control system is robust when (i) it hs low sensitivity, (ii) it is stble over the rnge of prmeter vritions nd (iii) the performnce continues to meet the specifiction in the presence of set of chnges in the system prmeters (Dorf, 2001). H 4 is one of the most known techniques vilble nowdys for robust control. It is method in control theory for the design of optiml controllers. Bsiclly, it is n optimiztion method tht tkes into considertion strong definition of the mthemticl wy to express the bility to include both clssicl nd robust control concepts within single design frmework. It is known tht H 4 control is n effective method for ttenuting disturbnces nd noise tht pper in the system. It hs been proven to be one of the best techniques in liner control system (Alok, 2007). Robust control techniques such s modern H 4 nd clssicl quntittive feedbck theory (QFT) hve received comprtively little ttention in the fluid power literture, especilly with regrd to pneumtic systems (Mrk nd Nrimn, 2004). High performnce position control of pneumtic ctutors remins difficult tsk. In most industril pplictions, sfety requires tht the pressure of the ir supply be kept low, which mkes it difficult to design high bndwidth systems. Moreover, low supply pressure tends to limit the chievble ctutor stiffness, which ffects the bility of the servomechnism to reject disturbing lods. Nonliner control vlve flows nd uncertinties in the plnt prmeters lso complicte the design of high performnce pneumtic servos. On the other hnd, the pneumtic ctutors re widely employed in position nd speed control pplictions when chep, clen, simple, nd sfe operting conditions re required. In recent yers, low cost microprocessors nd pneumtic components becme vilble in the mrket, which mde it possible to dopt more sophisticted control strtegies in pneumtic system control (Jihong et l, 2007). The pneumtic cylinders cn offer better lterntive to electricl or hydrulic ctutors for certin types of pplictions nd the pneumtic ctutors Corresponding Author: Hzem I. Ali, Deprtment of Electricl nd Electronic Engineering, Fculty of Engineering, University Putr Mlysi, Mlysi E-mil: hzemcontrol2001@yhoo.com 207

2 provide the previously enumerted qulities t low cost. They re lso suitble for clen environments nd sfer nd esier to work with. However, position nd force control of these ctutors in pplictions tht require high bndwidth is some how difficult, becuse of compressibility of ir nd highly nonliner flow through pneumtic system components. A typicl pneumtic system includes force element (pneumtic cylinder), commnd device (vlve), connecting tubes, nd piston, pressure nd force sensors. The externl lod consists of the mss of externl mechnicl elements connected to the piston nd perhps force produced by environmentl interction (Edmond nd Yildirim, 2001). A schemtic digrm of the pneumtic ctutor system is shown in Figure 1. Fig. 1: Schemtic digrm of pneumtic servo ctutor system. In this pper method for tuning n H 4 controller weights is presented. An optiml H 4 controller will be obtined ccording to minimiztion of the required cost functions using prticle swrm optimiztion (PSO), which is used to minimize the cost function s powerful optimiztion method with high efficiency in comprison to other methods. This method tries to find the optiml vlues in wide rnge of serching spce through the interction of prticles in the popultion. The PSO hs become one of the most powerful methods for solving optimiztion problems. The method is proved to be robust in solving problems feturing nonlinerity nd nondifferentibility, multiple optim, nd high dimensionlity. The dvntges of the PSO re its reltive simplicity nd stble convergence chrcteristic with good computtionl efficiency (Mjid et l, 2009). 2. Pneumtic Servo Actutor:. System Description: The purpose of the servo ctutor unit is to move the lod by displcement (y P ) in complince with commnd signls from the control section. Figure 1 shows the bsic construction nd opertion of the electropneumtic servo ctutor. The source of power used in this type of ctutor is compressed ir supplied to the jet pipe. An electromgnetic force generted by the flowing electric current rottes the jet pipe. Recting to the pressure differentil in the cylinder cvities, the piston together with the rod moves with speed dependent on the irflow, ir pressure nd lod. The dimeter nd stroke of the piston re 80mm nd 50mm respectively. Output piston feedbck is provided by liner potentiometer, the slider of which is driven by the piston. The servo unit consists of control surfce ctutor, feedbck trnsmitter, polrized jet rely, nd power mplifier (Hzem et l, 2008). The minor loop of the servo shown in Figure 2 is used to ensure proportionl movement with respect to input commnds. b. System Mthemticl Model nd Dynmics: The nlysis of pneumtic ctutors requires combintion of thermodynmics, fluid dynmics nd the dynmics of motion. For constructing mthemticl model, three mjor considertions must be, involved (Hzem et l, 2009). 1. The mss flow rtes through the vlve. 2. The pressure, volume nd temperture of the ir, in cylinder. 3. The dynmics of the lod. 208

3 Fig. 2: Block digrm of servo ctutor system. The vlve is four port pneumtic jet pipe vlve. This vlve is treted s equivlent to two three-port vlves, one of ech side of the cylinder. Considering the left hnd side of the cylinder, Figure 1, nd the thermodynmic system is enclosed in the box, or control volume. Mny studies hve shown tht for dequte models for controller design n isotherml behvior of the ir my be ssumed. Strting with the definition of the density, using the idel gs eqution nd ssuming n isotherml process the mss flow rtes equtions cn be written s (Peter, 2007): M V V (1) M V V b b b b b (2) M M Where nd b re the mss flow rtes in (chmbers () nd ((b)) respectively, ρ is the density of the ir, V nd V b re the volumes in chmbers nd b. For symmetric cylinder the volumes in chmbers nd b re given by: V V A o PYP V V A b o PYP (3) (4) where y p is the position displcement, A p is the piston re, V o is the ir volume in cylinder when the piston in mid point, V nd V b re the volumes in chmbers nd b, respectively. Differentiting equtions (3) nd (4) nd substitute them in (1) nd (2) the following equtions will be obtined: 1 1 M P ( V A ) P Ay RT RT o yp P 1 1 M P ( V A ) P Ay RT RT b b o pyp b P b b (5) (6) where R is the gs constnt, D is the specific het rtio, T nd T b re the tempertures in chmbers nd b respectively, P nd P b re the pressures in chmbers nd b respectively. Rerrnging the equtions (5) nd (6) nd dding the lod dynmics eqution tht influences the overll performnce of the piston motion, the system equtions will be: 209

4 RT PA p P M y ( V A y ) ( V A y ) P o p p o p p (7) RT PA b p P M y ( V A ) ( V A ) b b P o pyp o pyp (8) Ap Ap 1 1 y P P F F M M M M p b L f (9) where M, F L nd F f re the lod mss, disturbing force nd friction force respectively. The eqution tht governs the mss flow rte of ir through ech control vlve orifice is nonliner eqution nd not suited for controller design. If fst servo vlve is used, the dynmics of the vlve cn be neglected. Assuming for the moment positive input signl to the vlve, short line between vlve nd cylinder nd chmber pressures of bout hlf the supply pressure, the control vlve equtions will be simplified to be (Peter, 2007): PS M KV u 2 PS M b KV u 2 (10) (11) where K V is the vlve coefficient, P S is the supply pressure nd u is the electric vlve input signl respectively. Substituting equtions (10) nd (11) in equtions (7) nd (8) nd combining the Lplce trnsformtions of equtions (7), (8) nd (9) llows the operting point dependnt trnsfer function model of the open loop system to be written s: yp(s)= G(s)U(s) - G d (s)f d (s) (12) where Gs () RTA 2K MV o 2 2 ( ) 2 f Ap Pi ( s ) ss M p MV o (13) nd G () s d 1 M 2 ( A ) 2 2 f p i ( s ) ss M MV o P (14) 210

5 where K K V PS 2 This system considered, four-port vlve is used to control double cting through rod cylinder. There re three nonlinerities in the pneumtic servo system. The first one is the nonliner chrcteristic of the vlve, nd the other two nonlinerities re the volume nd bulk modulus when they re used s coefficients in the equtions. The nominl vlues of system prmeters re given in Tble 1 (Hzem et l, 2008). Tble 1: The nominl system model prmeters. 3. Uncertin Pneumtic Actutor System Model: The pneumtic servo ctutor system cn be represented s shown in Figure 3. The prmeters, b nd c cn be defined s: MVi fvi Api, b nd c 2K KART 2K KART K KRT p p p (15) The three physicl prmeters, b nd c re not known exctly therefore, they cn be ssumed tht their vlues re within certin rnge. Tht is, (1 p ), bb(1 p ), cc(1 p ) b b c c (16) 211

6 Fig. 3: Block digrm of pneumtic servo ctutor system with its min prmeters. where = , b = , c = re the nominl vlues of, b nd c, p, p b nd p c nd δ, δ b nd δ c represent the possible perturbtions on these prmeters. In this pper we let p = , p b =0.64, p c =0.174 nd -1# δ, δ b, δ c #1. Not tht this represent up to % uncertinty in the prmeter, 64% uncertinty in the prmeter b nd 17.4% uncertinty in the prmeter c. The three constnts blocks in Figure 3 cn be replced by block digrms in terms of, p, δ, etc., in unified pproch. The quntity 1/ cn be represented s n upper liner trnsformtion (LFT) in δ s: P p F M (1 p ) M (1 ) u(, ) with (17) p p bb(1 p ), cc(1 p ) Similrly, the prmeters nd cn be represented s n upper LFT in δ b, nd δ c where b=f u (M b,δ b ) nd c= F u (M c, δ c ) b b c c M 0 b 0 c, nd M c pb b pc c with (18) b Figure 4 shows the block digrm of the pneumtic system with uncertin prmeters. The inputs nd outputs of δ, δ b nd δ c re denoted s y, y b, y c nd u,u b, u c respectively. Let G p denotes the input/output dynmics of the pneumtic system, which tkes into ccount the uncertinty of prmeters. G p hs four inputs (u,u b, u c, u), four outputs (y, y b, y c, y) nd three sttes (x 1,x 2,x 3 ). 212

7 Fig. 4: Block digrm of the pneumtic system with uncertin prmeters. The stte spce representtion of G p is (Alok, 2007): A B1 B2 Gp C1 D11 D12 C2 D21 D22 (19) where c b c b p pc 1 0 c 0 0 p A 0 0 1, B , B2 0, C1 0 0 b 1 0 D11 0, D12 0, C2 [1 0 0], D21 [0 0 0], D22 0 (20) 1 0 It is cler tht the system mtrix G p hs no ny uncertin prmeter nd depends only on bc, p, p b, p c nd on the originl system prmeters. The uncertin behvior of the originl system cn be described by n upper LFT representtion y = F u (G p,δ)u with digonl uncertinty mtrix Δ s shown in Figure 5. The open loop frequency response of the perturbed system is shown in Figure 6. These chrcteristics show tht the bndwidth of the system decreses 213

8 Fig. 5: Upper LFT representtion of the system with prmeters vritions until the system becomes slower. On the other hnd, the phse mrgin lso decreses tht mke the system tends to oscillte nd be unstble in some cses. 4. Controller Design: 4.1 Biliner trnsform nd Weighting functions Selection: Since the proposed system hs jw-xis pole, the H 4 controller, if it is relibly computed, would hve mrginlly stble closed loop pole t the corresponding jw-xis loction. This problem led to singulrities in the equtions tht determine the stte spce reliztion of H 4 control lw. So simple biliner trnsform hs been found to be extremely useful when used with robust control synthesis. This trnsformtion cn be formulted s jw-xis pole shifting trnsformtion (Richrd nd Michel, 1997): s s p1 s 1 p 2 where p 1 < 0 nd selected to be 0.1, p 2 is selected to be infinity. This is equivlent to simply shifting the jw-xis by p 1 units to the left. The H 4 controller is obtined for the shifted system then it is shifted bck to the right with the sme units. The design requirements nd objectives for pneumtic servo ctutor system in this pper is to find liner, output feedbck control u(s) K 4 (s)y(s) which ensures tht the closed loop system will be internlly stble. Also, the required closed loop system performnce should be chieved for the nominl plnt G p. The selection of weighting functions nd weighting gins for specific design problem is not n esy procedure nd often needs mny itertions nd fine-tuning nd it is hrd to find generl formul for the weighting functions tht will work in every cse (Bittr nd Sles, 1998). So to obtin good control design, it is necessry to select suitble weighting functions. The performnce nd control weighting functions formuls used in this pper re (Richrd nd Michel, 1997; Zhou nd Doyle, 1998): (21) ( s 2 w sw ) Wp () s ( s 2 w s w ) c c c c (22) wbc s k M Wu ( S) k s w u bc k (23) where β is the d.c. gin of the function which controls the disturbnce rejection, α is the high frequency gin which controls the response pek overshoot, w c is the function crossover frequency, ζ 1 nd ζ 2 re the dmping rtios of crossover frequency, w bc is the controller bndwidth, Mu is the mgnitude of K 4 S, nd g is 214

9 smll vlue. For the control weighting function in eqution (23), the vlue of k ws selected to be 2 for getting second order function. 4.2 H 4 Controller Design: The H 4 controller ws designed so tht H 4 norm from input W =F d = dist to output e Z e minimized. Where dist is the input disturbnce signl, e p, e u re the weighted error nd control signls. Figure 7 shows the stndrd feedbck digrm of the system with weights. The generlized plnt P is expressed by: et WG p d WG p d dist e u 0 Wu u e Gd G p where u is the control signl. p u is (24) Fig. 6: Open loop frequency response chrcteristics with prmeters uncertinty. Fig. 7: The stndrd feedbck digrm of the system with weights. 215

10 The lower liner frctionl trnsformtion of the generlized plnt P nd controller K 4 cn be described by: (, ) WGS p d F1 P K WG u dks (25) where S= (1+G p K 4 ) -1 is the sensitivity function of the nominl plnt. The objective of H 4 control is to find the controller K 4 tht internlly stbilizes the system such tht Tzw is minimized (Zhou nd Doyle, 1998). where Tzw is the trnsfer function of the system from input W to output Z nd cn be expressed s: T zw WGS p d WG u dks The H 4 control minimizes the cost function in (26) using γ -itertion (Skogestd nd Postlethwite, 2005) to find the stbilizing controller such tht T zw. So in this pper we hve tried to find the optiml vlue of γ by using PSO lgorithm for tuning the weighting functions tht hve gret effect on the overll design of H 4 control technique. The optiml vlue of γ is the infimum overll γ such tht the H 4 control conditions stted in (Zhou nd Doyle, 1998) re stisfied. A suboptiml H 4 controller ws obtined using the following Mtlb commnd: [ K, T, ] hinf syn( P, n,, tol) zw suboptiml y where n y nd n u re the dimensions of y nd u, γ min nd γ mx re the lower nd upper bound for γ optiml, nd tol is the tolernce to the optiml vlue. 4.3 Tuning of Weights Using PSO: The PSO lgorithm is used to tune the selected weighting functions to obtin the optiml vlues of their prmeters tht ensure controlled system with good disturbnce rejection, good trnsient response nd low control signl. The fitness function used in PSO lgorithm is the performnce criteri stted in eqution (26). The lgorithm obtins the minimum vlue of the infinity norm of the performnce criteri. The velocity nd position equtions of PSO lgorithm re (Sheng-Fu et l, 2007): mx (26) (27) v wv c rnd ( x x ) c rnd ( x x ) k1 k b k g k i i 1 i i 2 i i x x v k1 k k1 i i i (28) (29) k k where v i is the prticle velocity, x i is the current prticle position, w is the inerti weight nd it is b g selected to be 1.5, x i nd x i re the best vlue nd the globl best vlue, rnd is rndom function, c 1 nd c 2 re lerning fctors nd re selected to be c 1 = c 2 = 2. The swrm size is (100) with (7-dimensions) (vribles to be optimized) nd the number of genertions is (100). The proposed PSO lgorithm for finding the optiml vlues of the weighting functions prmeters is described by the flowchrt shown in Figure 8. The overll block digrm of the system with PSO tuning lgorithm is shown in Figure

11 Fig. 8: Flowchrt for tuning the weighting functions using PSO. Fig. 9: Block digrm of the overll controlled system. The intervl for γ itertion is chosen between 0.1 nd 10. The obtined controller is: - 217

12 K s 68.7s5840 s 2373s s (30) The optiml weighting functions prmeters obtined using PSO lgorithm re s follows: - β = , α = 0.01, w c = 4.93, ζ 1 = 0.38, ζ 2 = , M u = , g = nd w bc = 1.7 with γ = Simultion Results: Figure 10 shows the singulr vlues of closed loop system with the controller K 4. It is seen tht the mximum vlue of the closed loop system is less thn one, tht s men tht the condition W G K 1 p(1 p ) 1 is stisfied. Figure 11 shows the frequency chrcteristics of the sensitivity function compred with the inverse of the performnce weighting function. It is cler tht the sensitivity function lies below the inverse of W p tht is mens tht the performnce criterion ws stisfied. Figure 12 shows the frequency response of the open loop perturbed system with the controller. It is cler tht the system is lwys stble with ll prmeters uncertinty tht is robust stbility ws stisfied. Figure 13 nd b show the time response chrcteristics of the closed loop nominl system nd perturbed system. The time response chrcteristic of the system subjected to disturbnce is shown in Figure 14. For prcticl requirements, it is required the control signl to be smll to void the problem of the sturtion. Figure 15 shows the frequency chrcteristics of the control signl where smll mximum vlue ws obtined. Fig. 10: The lrgest singulr Vlue of the closed loop controlled system T ZW. Conclusion: A robust H 4 controller hs been designed to ssure robust stbility nd robust performnce of pneumtic servo ctutor system with prmetric uncertinty. The prticle swrm optimiztion method (PSO) ws used for tuning the weighting functions. The optiml prmeters of the weighting functions tht re obtined using PSO lgorithm led to obtin robust controller tht chieves the position control of pneumtic ctutor. It ws lso shown tht the proposed method for tuning the weighting functions is esy to implement nd better results could be obtined using this method. Finlly simple nd low order robust controller ws obtined. 218

13 Fig. 11: Frequency chrcteristics of sensitivity function S (solid line) nd the inverse of the weighting function Wp (dshed line). Fig. 12: Frequency response chrcteristics of perturbed controlled system. 219

14 (A) (B) Fig. 13: Closed loop time response chrcteristics of the closed loop controlled system ) nominl system b) perturbed system. 220

15 Fig. 14: Time response chrcteristics of the closed Loop controlled system subjected to disturbnce. Fig. 15: Frequency response chrcteristics of the control signl. REFERENCES Alok, S., liner systems optiml nd robust control, Tylor nd Frncis Group, USA. Bittr, A. nd R.M. Sles, H 2 nd H 4 control for Mglev vehicles, IEEE Control Systems, 18: D-Wei, G., H.P. Petkov nd M.M. Konstntinov, Robust control design with Mtlb, Springer. Dorf, R.C., Modern control system, Prentice Hll, Inc. Upper Sddle River, NJ. Edmond, R. nd H. Yildirim, A high performnce pneumtic force ctutor system, ASME, Journl of Dynmic Systems, Mesurement nd Control, 122(3): Hzem, I.A., B.M. Smsul, S.M. Bshi nd M.H. Mrhbn, Robust QFT controller design for positioning pneumtic servo ctutor, Proceedings of Student Conference on Reserch nd Development, Johor, Mlysi, 128-1, Hzem, I.A., B.M. Smsul, S.M. Bshi nd M.H. Mrhbn, A review of pneumtic ctutors (modeling nd control), Austrlin Journl of Bsic nd Applied Sciences, 3(2): Jihong, W., U. Kott nd K. Ji, Trcking control of nonliner pneumtic ctutor systems using sttic stte feedbck lineriztion of the input-output mp, Proceedings Estonin Acdemic Science of Physics nd Mthemtics, Mjid, Z., S. Nsser nd K.G. Msoud, Design of n H 4 PID controller using prticle swrm optimiztion, Interntionl Journl of Control, Automtion, nd Systems, 7(2):

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