CACSD for hydraulic cylinders.pdf

1、Proceedings of the 2000 IEEE International Symposium on Computer-Aided Control System Design Anchorage, Alaska, USA September 25-27, 2000 CACSD for hydraulic cylinders MP1-2 340 Markus Lemmen, Markus Brocker, Department of Measurement and Control, FB7/FG8, University of Duisburg, Germany. lemmenkni-

2、duisburg.de, broeckeruni-duisburg.de Bram de Jager, Harm van Essen Systems & Control Group, Faculty of Mechanical Engineering, Eindhoven University of Technology, The Netherlands. A.G.de.Jagerwfw.wtb.tue.nl, H.A.v.Essenwfw.wtb.tue.nl Abstract Hydraulic drives such as synchronizing cylinders (also kn

3、own as double-rod cylinders) or differential cylin- ders (also known as single-rod cylinders) have nonlinear plant dynamics. Consequently, one can obtain higher control performance for tracking control with nonlin- ear controllers. The required calculations for the con- troller design may be carried

4、 out by hand but are te- dious and error-prone. Hence, this paper demonstrates the advantage of using the computer algebra / sym- bolic computation system Maple to compute nonlinear controllers for the two different kind of plants treated here. Thus, we compute an exact linearizing controller for sy

5、nchronizing cylinders with means of (nonlin- ear) static state feedback with Maple. The controller shows good performance in experiments. Hydraulically driven differential cylinders, however, are not exact lin- earizable by static feedback with respect to the piston rod position as output. Thus, we

6、design an input- output-linearizing controller for this kind of plant with the same software package. Again, this controller is tested in experiments and shows good performance. 1 Introduction In practice, hydraulic cylinders are often used as drives in industry due to their good power-to-weight-rat

7、io and low life-cycle costs. Within cylinder drives synchroniz- ing (also known as double-rod cylinders represented by dashed and solid lines in fig. 1) and differential cylin- ders (also known as single-rod cylinders, solid lines only in fig. 1) are distinguished. These drives, however, in- herit n

8、onlinear dynamics which a controller has to cope with. Consequently, a nonlinear controller can signif- icantly improve the tracking performance for position control with respect to standard linear control tech- niques. The required calculations for designing such a nonlinear controller may be done

9、by hand but are te- 0-7803-6566-6/00$10.0002000 IEEE dious and error-prone. Hence, the goal of this paper is to demonstrate the advantages of using Maple to com- pute nonlinear controllers (e.g. l, 2, 31) for the two different kind of plants treated here. Xcyl QA,PA QB, PB - I! Figure 1: Layout of h

10、ydraulic driven cylinders: - differ- ential cylinder, - - and - synchronizing cylin- der. The paper is structured as follows: First, we derive a state-space model for the two different cylinder drives in section 2. Then, the controller is computed in sec- tion 3. Therefore, we make use of convienien

11、t CACSD toolboxes like the NonLinCon-package l for Maple 4. We show, that with respect to the piston rod position as system output, hydraulically driven synchronizing cylinders are exact linearizable by static state feed- back: The relative degree T equals the order of the system model n and the exa

12、ct linearizing controller according to 5, 6, 7 1 is computed symbolically. The controller shows good performance in experiments as is shown in section 4. Hydraulically driven differential cylinders, however, are not exact linearizable by static 101 Authorized licensed use limited to: GUILIN UNIVERSI

13、TY OF ELECTRONIC TECHNOLOGY. Downloaded on March 13,2010 at 07:27:54 EST from IEEE Xplore. Restrictions apply. feedback 8 with respect to the piston rod position as output. Nevertheless, an input-output-linearizing static state feedback may be computed as is shown in section 3. This input-output-lin

14、earizing controller is embedded within a standard linear outer control loop to place eigenvalues for the i/o-linearized system. This controller set-up is tested in experiments and shows good performance, also. Finally, some conclusions are drawn and perspectives are given. 5001 8 8 7 7 I I I 0 -500

15、-1000 -1500 -2000 -2500 -3000 -3500 -4M)o -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x, WSl Figure 2: Derivative of the friction force w.r.t. 5cyl 2 Modelling Hydraulically driven actuators mainly consist of three parts: a power supply, a power regulation and a power transformation unit (c.f. fig. 1

16、). Hydraulic power is supplied by means of an hydraulic pump with a con- stant supply pressure PO. The tank pressure pt in most cases is equal to the environmental pressure. A servo valve or proportional valve serves as a power regulator. Here, the amount of hydraulic flow QA into chamber A and QB f

17、or chamber B, respectively, is regulated by means of the valve position, so it is finally controlled by its input voltage U,. For modelling, we have to dis- tinct between the two cases U, 3 0. This hydraulic flow affects the pressures PA and p and hence, an ac- celeration of the piston rod Zcyl can

18、be observed for the total accelerated mass mtl, taking the friction forces Ff and disturbances Fd into account. According to fig. 1, one important property of synchro- nizing cylinders is that they posess the same pressure area in chamber A and B. Hence, let us introduce the pressure area ratio A A

19、AB cp:= -. Thus, we have cp = 1 for synchronizing cylinders and cp # 1 for differential cylinders. For hydraulic cylin- ders we obtain a four-dimensional state space model by defining XI := xCyl =: 9, 2 2 := Xcyl, 23 := PA and 2 4 := pg in input affine form (c.f. 5, 6, 9, 71) j . = f(x) +&)U Y = h(x

20、), where f ( . ) = and 0 0 1 for U 0 and 0 0 for U 0 (see lo, 8, 111 for further details). Within (2) we have mti = mp + mi + pfl (V(x,i) + V(x,l) for the total load, vA(Xcy1) = KA + VB(xcy1) = KB + ( : - - zcy) AB for the oil vol- 1 2 umes, E*) = -Efl,max log10 heuristically obtained equation-for t

21、he bulk modu- 1 , where lus of oil elasticity, B, = Qn is some supplier dependent nominal value and The friction force Ff is sgn(x) := modelled as a combination of viscous friction F, static friction F , and coulomb friction Fc: Ff(kcyi) = Fvkcyl+sgn(kCyl) (Fc+ Fsexp(-y)+FoEset. (3) Vn - Jm %,ax x -

22、1 ,x 0 and kexcpVBEfl(x3)Jm sgn(X3 - Pt) a second step eigenvalues for the i/o-linearized trans- U , = kexv,Efl(X4$?% sgnbo - 24) + . (8) mtlVAVB 1 G(s) = sr + pr-lsr-l + . . . + PlS + Po which is obtained by the i/c-linearizing controller for U h(X(t) - 5 Pk-lL?-lh(z(t) k=l L,LTf-1 h( x( t) W t ) )

23、 = 7 (9) where n, are assigned by pole-placement. F our system model (5) indeed we obtain r = 3 . i . i i t - 0 0 . 5 1 1 . 5 2 2.5 3 3 . 5 4 - t - S . . . . . . . . . . . . The experimental results shown in fig. 6(a) prove our exact linearization approach (7) with a linear output feedback o = K(y,t

24、-y) (c.f. section 3) to be suitable for solving the tracking control problem. The behaviour is satisfactory even if the initial position has been esti- mated incorrectly (c.f. fig. 6(b) and the tracking per- formance is much better than could be obtained by a simple linear control technique applied

25、directly to hte plant. The controlled synchronizing cylinder reaches the desired trajectory within t E = 0.181 s and reason- 105 0 0,s 1 1,5 2 2,s 3 33 4 - t S b) Figure 6: Desired and measured cylinder position of the exact linearizing control: (a) correct (b) incor- rect initial position (0.4 m) A

26、uthorized licensed use limited to: GUILIN UNIVERSITY OF ELECTRONIC TECHNOLOGY. Downloaded on March 13,2010 at 07:27:54 EST from IEEE Xplore. Restrictions apply. 0,36 1 034 y 032 0,30 0,28 0,26 0,24 m a) 0,36 034 y 0,32 0,30 0,28 0,26 0,24 m 0 0,5 1 1,5 2 2,5 3 3 3 4 self may be done with the help of

27、 symbolic or compu- tational algebra packages like Maple 4. Hence, we demonstrate how helpful computer algebra packages like the NonLinCon l or NSAS 2 package become in this context: the calculations are carried out error-free almost automatically and the controller source code may be exported in C.

28、 Hence, the controller design and implementation issue becomes fast and cheap with ex- cellent quality and performance. 0 0,5 1 1,5 2 2,5 3 3,5 4 J- S Figure 7: Desired and measured cylinder position of the i/o-linearizing control: (a) correct and (b) in- correct initial position (0.25 m) able overs

29、hoot. The results from the experiments performed with the differential cylinder can be seen in fig. 7(a). Here, the input-output-linearization (9), (10) is shown to be a suitable approach for tracking. Again, the simplest lin- ear controller for the a additional linear output feed- back U = K(yrt -

30、y) (c.f. section 3) has been used. The behaviour is satisfactory also, even if the initial posi- tion has been estimated incorrectly (c.f. fig. 7(b) and is much better than a purely linear controller designed for the original plant. The controlled synchronizing cylin- der reaches the desired traject

31、ory within t E = 0.215 s and reasonable overshoot. 5 Conclusions In this paper we show that nonlinear controller design may be used for position tracking purposes within hy- draulic driven cylinders to improve the control per- formance compared to standard linear control tech- niques. The calculatio

32、ns which have to be carried out become cumbersome and error-prone, since we are fac- ing nonlinear plant dynamics. The computations of Lie derivates, the verification of exact or i/o-linearizability as well as the design of the linearizing controller it- References l H. van Essen and B. de Jager, “A

33、nalysis and design of nonlinear control systems with the symbolic computation system maple,” in European Control Con- ference ECC93, (Groningen/Netherlands) , pp. 2081- 2085,1993. 2 M. Lemmen, T. Wey, and M. Jelali, “NSAS - ein Computer-Algebra-Paket zur Analyse und Synthese nichtlinearer Systeme,”

34、Forschungsbericht (Technical Report) 20/95, MSRT, University of Duisburg, 1995. 3 B. de Jager, “Symbolic aids for modelling, anal- ysis and synthesis of nonlinear control systems,” in SYMBOLIC METHODS in control system analysis and design (N. Munro, ed.), IEE Control Engineering Series 56, pp. 297-320, London: IEE, 1999. 4 B. W. Char, K. 0. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan, and S. Watt, Maple V - Lan- gua