Non Linear Modelling and Control of Hydraulic Actuators B. Šulc, J. A. Jan

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1 Acta Polytechnica Vol. 4 No. 3/ Non Linear Moelling an Control of Hyraulic Actuators B. Šulc, J. A. Jan This aer eals with non-linear moelling an control of a ifferential hyraulic actuator. The nonlinear state sace equations are erive from basic hysical laws. They are more owerful than the transfer function in the case of linear moels, an they allow the alication of an object oriente aroach in simulation rograms. The effects of all friction forces (static, Coulomb an viscous) have been moelle, an many henomena that are usually neglecte are taken into account, e.g., the static term of friction, the leakage between the two chambers an external sace. Proortional Differential (PD) an Fuzzy Logic Controllers (FLC) have been alie in orer to make a comarison by means of simulation. Simulation is erforme using Matlab/Simulink, an some of the results are comare grahically. FLC is tune in a such way that it rouces a constant control signal close to its maximum (or minimum), where ossible. In the case of PD control the occurrence of eaks cannot be avoie. These eaks rouce a very high velocity that overstes the allowe values. Keywors: moelling, simulation, flui ynamics, robotics, fuzzy control, roortional-erivative control. Introuction Hyraulic actuators are use for elivering high actuation forces an high ower ensity. Due to their simle construction an low cost, hyraulic actuators are wiely use. These actuators have highly nonlinear moel characteristics. Two tyes of actuators are use, ifferential (single-ro) an synchronizing (ouble-ro). From the control engineering oint of view, synchronizing or symmetric actuators are referre because there is no iston area ifference an this fact reuces non-linearities, but on the other han, the construction of these tyes of actuators is ifficult an exensive. Also in some situations, e.g., robots, cranes, etc., ue to limite sace symmetric actuators cannot be use. In the literature [3, 6,, etc.] various moels of hyraulicactuatorsareresente.somearelinearmoels,mostly alie to synchronizing cyliners. Many authors have neglecte the leakage; also the friction forces are not comletely u A V F F Q A s x s Q int Q ext, F ist Fig. : Schematic iagram of a ifferential hyraulic actuator x c V Q moele. These factors are all inclue in the erivation of our moel that rovies the basis for creating object oriente library blocks. The schematic iagram of a ifferential actuator, as shown in Fig., consists of a constant ressure suly um, a magnetically controlle sool valve an a ifferential hyraulic cyliner. All the variables are exlaine in the list of symbols (Table ). The moel is erive in the next section. The motivation is to reare the relations an arameter efinitions for object-oriente blocks that are suitable for rogramming in secial Matlab S-functions. Mathematical Moelling Let the flow areas to the suly an return ort of the sool valve be roortional to the sool islacement x s,(i.e., A Kxs).Thentheflowrateofoilacrossthesoolvalve can be given as: Q C x () where C C K A A At the nominal ressure ro ( N ) an nominal flow (Q N ), this constant can be estimate as QN C () xs,max 5. N Alying the continuity equation to both chambers an oilv, Axc oilq Qint (3) oilv, Axc oil Qint Q ext Q an consiering the comressibility of oil E oil as oil Eoil oil Czech Technical University Publishing House htt://ctn.cvut.cz/a/ 4

2 Acta Polytechnica Vol. 4 No. 3/ after utting this formula into Eq. (3) we obtain the equations c oil oila x V, Axc oilq Qint (4) c oil oil A x V, Axc oil Qint Qext Q oil V, Axc oil V, A xc where ( c Q A x Q int sgn xc Qint Qext A Q Q C sg xs s s sgxssgn (6) Q C( sgxssgn sgxssgn s s ) Qint kint Qext, kext, where coefficients k int an k ext, can be evaluate by means of the general formula by [] i r crc 3 kleak 6 L QN C 5. N E oil [] can be given as: Eoil Eoil,max log C C (7) max an the following notification is use x sgnx x x x x sgx x The forces on the cyliner can be exresse as (using Newton s Secon Law of Motion) m x c A A FFvc F ist (8) xc where vc. The friction forces FFvc are viscous friction, Coulomb friction an static friction, i.e., vc FFvc kvvc sgnvcfc Fsex C s The lot of friction versus velocity is shown in Fig.. The factors of friction forces F v (viscous), F c (Coulomb) an F s ) (5) (static) can be ientifie from the total friction (F F )anvelocity (v c ). These (F F an v c ) can be estimate from the values of the osition, an ressures in both chambers of the cyliners exerimentally with zero loa, an using then numerical integration of the osition for eterming velocity. Sool valve Sool valve ynamics can be erive in a similar way as for a cyliner, but the following linear secon orer ifferential equation is a wiely use an sufficient aroximation: Friction Force [N] Velocity [m/s] Fig. : Friction verses velocity rofile xs t s n B x t n n ut (9) Tyical values of sool valve arameters are: natural frequency n 3 5 s an aming factor B.7.. Correct moelling of a sool valve requires resetting of the integrators by bringing their inuts to zero as soon as the en ositions of the sool valve are overstee. This recaution is shown in Fig. 4. State sace form of the hyraulic actuator xc xs Let x xc, x, x3, x4, x5 xs, x6 then from Eq. (5 9) we get: x x x m x 3 A x 4 A F F x F Loa E oilx3 x3 V, Ax Qx3, x5 Ax Qintx3, x4 () Eoilx x 4 4 V, A x Q intx4, x5 Qextx4 Ax Qx4, x5 x 5 x6 x 6 n u Bnx6 nx 5 4 Czech Technical University Publishing House htt://ctn.cvut.cz/a/

3 Acta Polytechnica Vol. 4 No. 3/ Fig. 3: Simulink block iagram of the hyraulic actuator moel 3 Moelling in MATLAB/Simulink The moel has been imlemente in versatile software Matlab/Simulink, which is wiely use in control engineering communities aroun the worl. Cyliner, sool valve an controller moels are shown in Fig. 3. These blocks were create using the above erive equations, an a GUI (grahical user interface) will aear by clicking the block. Parameters can be entere through these menus. Fig. 4 shows the sool valve moel, where anti-winu has been inclue. Fig. 5 shows the cyliner moel, which is moele using four state variables, an the cyliner stos at the ens have been introuce. In /w/w Constant Sign Prouct Fig. 4: Simulink sool valve moel with en osition stos Fig. 5: Simulink block iagram of the hyraulic cyliner moel Czech Technical University Publishing House htt://ctn.cvut.cz/a/ 43

4 Acta Polytechnica Vol. 4 No. 3/ 4 Fuzzy Logic Controller (FLC) Looking for an algorithm that woul be closer to the time otimal controller (bang-bang control), we trie using non- -linear Fuzzy Logic Control esign techniques. The basic structure of a Fuzzy Logic Controller is shown in Fig. 6. From the many tyes of fuzzy controllers, e.g., PID-like, sliing moe, Takagi-Sugeno etc., we selecte Mamani tye Fuzzy Logic Controller. This controller uses two inuts (control error an its erivate) an one outut (control signal). For both inuts, three fuzzy sets (two Sigmoi an one Gaussian membershi functions), an for outut five fuzzy sets (two Sigmoi an three Gaussian membershi functions) have been taken, as shown in Fig. 6. Rules are efine such that fortheregionwherewehavelargeerrors,aconstantan maximum allowable control signal will be elivere, whereas in the small error region it will behaves as a non-linear PD-like controller, see Table. Cris rocess-state values Normalization Fig. 7: Inserting FLC into the Simulink Moel constants. The controller arameters can be otimise heuristically, but goo suort for otimising these arameters is rovie by the Matlab toolbox calle the NCD Block set. Even in the resente case of a fuzzy logic controller when a minimum number of fuzzy variables an rules have been use, better results than with a PD controller have been achieve. As a fuzzifier, three fuzzy sets (two sigmoi an one Gaussian membershi functions) have been use to convert the cris values of the inuts into fuzzy linguistic variables, see Fig. 9. Similarly five fuzzy sets (two sigmoi an three Gaussian membershi functions) have been use to convert fuzzy linguistic variables into cris values of the oututs as a control signal. Fig. 6: The structure of a Fuzzy Knowlege Base Controller Table : Looku Table (exlaining rules) Change of control error Control error N Z P N.... P Z.. Z.. P N NL.. PL Two tyes of Inference Engines are use; one is comosition-base inference an the other is iniviual-rule-base inference. In the efuzzification, the set of moifie control outut values in the form of fuzzy sets is converte into single oint-wise values or cris values. Various efuzzification methos can be use; we use the centre of gravity or centre of area metho. Fuzzy controllers have the isavantage that it is ifficult to tune the large number of arameters, in contrast to the roortional erivative (PD) controller, which can be exresse by the equation: upd KP e KD e () where u PD is the control signal, e the control error, an the constants K P, K D reresent roortional an erivate Fig. 8: GUI for Outut in Matlab for FLC Fig. 9: Schematic iagram of fuzzy inference system 44 Czech Technical University Publishing House htt://ctn.cvut.cz/a/

5 Acta Polytechnica Vol. 4 No. 3/ Fig. : GUI for efining Rules in Matlab for FLC The whole rocess of fuzzy control can be exresse in the following stes, consiering x as inut an y as outut for each ste. Also two tyes of membershi functions are use as given below: Sigmoi membershi function: fx; a, c e ax c Gaussian membershi function: xc fx; a, c e a Ste : Fuzzifier: Conversion of cris values to fuzzy variables: For Inut, i.e., error, conversion has been one using three fuzzy variables (Negative (N) using the sigmoi membershi function of arameters [5.], Zero (Z) using the Gaussian membershi function of arameters [.5 ], an Positive (P) using the sigmoi membershi function of arameters [5.]). ThesameroceurewasfolloweforInut, i.e., change of error with arameters ([9.5], [.4 ], [9.5]). Ste : Logical Oerators The logical oerator AND use in the rules is escribe as: y min x, x, Ste 3: Imlication The imlication use in rules is as follows: if error is P AND change of error is N then control signal is P. Ste 4: Aggregation: In Aggregation, the rules irecte towars one outut are combine together. We use the logical oerator OR with max function as given below: y max x, x, Inourcasewehaoneoututanfiverules. Ste 5: Defuzzification: Defuzzification is the rocess of converting fuzzy variables to cris values. The center of gravity metho has been use to obtain cris values of the outut from the reference outut fuzzy sets. The reference fuzzy sets consist of five fuzzy variables (negative large (NL) using the sigmoi membershi function [75.], negative (N) using the Gaussian membershi function [.8.5], zero (Z) using the Gaussian membershi function [.8 ], ositive (P) using the Gaussian membershi function [.8.5] an ositive large (PL) using the sigmoi membershi function [75.]). The following formula has been use to convert the fuzzy variables to cris values. y cijaij xi i aij x i i As shown in Fig. 8, GUIs has been use in Matlab, where all the inut an outut fuzzy variables, inference system, fuzzifier an efuzzifier can be efine very easily by setting the corresoning arameters. Then the controller is connecte to the moel in Simulink, as shown in Fig Results an conclusions This work has two aims: the first is to elaborate a way of non-linear moelling of a ifferential hyraulic actuator suit- Position [mm] with Small Gain Position [mm] with Large Gain X - Desire X-PD X-FLC X - Desire X-PD X-FLC Fig. : Position trajectory of hyraulic actuator for FLC an PD control

6 Acta Polytechnica Vol. 4 No. 3/ 45 4 Velocity [mm/s] with Large Gain V-PD V-FLC 45 4 Velocity [mm/s] with Small Gain V-PD V-FLC Fig. : Velocity trajectory of hyraulic actuator for FLC an PD control 3 5 Control Signal [mm] with Small Gain U-PD U-FLC 3 5 Control Signal [mm] with Large Gain U-PD U-FLC 5 Limit (6 mm) 5 Limit (6 mm) Fig. 3: Control signal for a hyraulic actuator alying FLC an PD control able for rogramming in Simulink. The secon goal is to use this moel for comaring a classical an a non-linear fuzzy logic control loo esign. The obtaine results o not show any iscreancy with our exerience or with the available exerimental ata. The resonses that we obtaine confirm the general avantage of FLC in quicker resonses, ue to the fact that only FLC is able to rouce control signals that are not only roortionally eenent on control error asinthecaseofapdcontroller.thefunamentalimortance of a roer choice of the gain in the PD controller is emonstrate in by the columns in Fig. 3, where the left column shows resonses with half of the gain use in the exerimentseicteintherightcolumn.pdforsmallste changes surrisingly achieve worse control resonses than for larger ste changes. The anti-winu roblem has been solve, as shown in Fig. 5, which exlains the imortance of Velocity [mm/s] with an without anti-winu With Anti-winu without Anti-winu with no Matlab limits Position [mm] with an without Anti-winu With Anti-winu without Anti-winu with no Matlab limits Fig. 4: Velocity rofile with winu roblems Fig. 5: Position curves with winu roblems 46 Czech Technical University Publishing House htt://ctn.cvut.cz/a/

7 Acta Polytechnica Vol. 4 No. 3/ resecting the limits in the moel. However in this case less ifference is shown between with an without antiwinu, ue to the unsuitability of the arameters. Table : Lists of Symbols x, x c x, v c x 3, x 4, x 5, x s x 6, v s Q / Acknowlegement This work has been suorte by GA ČR, Research Project No: //347; Numerical methos for otimal control system esign. References osition of cyliner iston [m] velocity of cyliner iston [m/s] ressure of oil in chamber [Pa] ressure of oil in chamber [Pa] osition of valve iston [m] velocity of valve iston [m/s] flow in ie an [m 3 /s] V,/ volume of Oil in chamber at x c [m 3 ] F F E oil friction force [N] bulk moulus of oil [Pa] A iston area in chamber [m ] A iston area in chamber [m ] r c L iameter of iston of cyliner [m] raial clearance between iston an cyliner [m] length of iston hea [m] B sool valve aming [] n sool valve natural frequency [s ] ynamic viscosity [Pas] [] Berzen, W.: Nonlinear Control of Hyraulic Cyliners. Proceeings of the 6 th IEEE Meiterranean Conference on Control an Systems (MCCS 98), Italy, 998. [] Blackburn, J. F., Reethof, G., Shearer J. L.: Flui Power Control. Massachusetts, (U.S.A.): The MIT Press, 96. [3] Costa Branco, P. J., Dente, J. A.: The Alication of Fuzzy Logic in Automatic Moelling Electromechanical Systems. Fuzzy Sets an Systems, 998, Vol. 95, No. 3, [4] Daugherty,R.L.,Franzini,J.B.,Finnemore,E.J.:Flui Mechanics with Engineering Alications. Singaore:Mc Graw-Hill Book Comany, 989. [5] Dimiter, D., Hans, H., Michael, R.: An Introuction to Fuzzy Control. Berlin: Sringer-Verlag, 996, ISBN [6] Horacek, P., Biner, Z.: An Aroach for Design an Imlementation of Fuzzy Controller. In: Proceeings of Euroean Congress on Fuzzy an Intelligent Technologies EUFIT 93, Germany, Setember 993, Vol., , ISBN [7] Jan,J.A.,Šulc,B.:Conventional an Fuzzy Logic Control of Non Linear Hyraulic Actuators For Flexible Robots. IASTED International Conference Alie Simulation an Moelling (ASM ), Marbella, Sain,,. 3 37, ISBN [8] Jan,J.A.,Šulc,B.:Fuzzy Logic Control in Hyraulic Actuator Moelle from Own-built Simulink Block Library. Proceeings of the Conference on Information Engineering an Process Control, CTU, Prague, Czech Reublic,, , ISBN [9] Jan,J.A.,Šulc,B.:Moelling an Simulation of Nonlinear Hyraulic Actuator Using Conventional an Fuzzy Logic Control. 6 th International Conference on Prouction Research (ICPR-6), Prague, Czech Reublic,, ISBN [] Jan, J. A., Šulc, B.: Fuzzy Logic Control in Differential Hyraulic Cyliners riven by Servo-valves: Simulation an Visualization. 5 th International Stuent Conference, CTU in Prague, Prague (Czech Reublic): Poster. [] Kugi, A., Schlacher, K., Keintzel, G.: Position Control an Active Eccentricity Comensation in Rolling Mills. Automatisierungstechnik Germany, 999, at 8/99, ISBN [] Merrit, H. E.: Hyraulic Control Systems. NewYork:John Wiley & Sons, ] Noskievič, P.: Moelling an System Ientification. (In Czech), Ostrava (Czech Reublic): Montanex a.s., 999, ISBN [4] Staler, W.: Analytical Robotics an Mechatronics. New York: McGraw-Hill Inc., 995, ISBN [5]Sugeno,M.,Yasukawa,T.:A Fuzzy-Logic-Base Aroach to Qualitative Moeling. IEEE Trans. on Fuzzy Systems, February 993, Vol., No., [6] Šulc, B.: Integral Win-U in Control an System Simulation. In: Control Engineering Solutions. A Practical Aroach (P. Albertos, R. Strietzel an N. Mort (Eitors)), IEE, Lonon 997,. 6 76, ISBN Bohumil Šulc hone: fax: sulc@fsi.cvut.cz Ing. Jave Alam Jan, Ph.D. jave@stuent.fsi.cvut.cz Deartment of Instrumentation an Control Engineering Czech Technical University in Prague Faculty of Mechanical Engineering Technická Prague 6, Czech Reublic Czech Technical University Publishing House htt://ctn.cvut.cz/a/ 47