Nonlinear pressure control of self-supplied variable displacement axial piston pumps

Transcription

1 This ocument contains a post-print version of the paper Nonlinear pressure control of self-supplie variable isplacement axial piston pumps authore by W. Kemmetmüller, F. Fuchshumer, an A. Kugi an publishe in Control Engineering Practice. The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting. Please, scroll own for the article. Cite this article as: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac BibTex entry: % This file was create with JabRef.9.. % Encoing: author = {Kemmetmüller, W. an Fuchshumer, F. an Kugi, A.}, title = {Nonlinear pressure control of self-supplie variable isplacement axial piston pumps}, journal = {Control Engineering Practice}, year = {}, volume = {8}, pages = {84-93}, oi = {.6/j.conengprac.9.9.6}, url = { } Link to original paper: Rea more ACIN papers or get this ocument: Contact: Automation an Control Institute (ACIN Internet: Vienna University of Technology office@acin.tuwien.ac.at Gusshausstrasse 7-9/E376 Phone: Vienna, Austria Fax:

2 Copyright notice: This is the authors version of a work that was accepte for publication in Control Engineering Practice. Changes resulting from the publishing process, such as peer review, eiting, corrections, structural formatting, an other quality control mechanisms may not be reflecte in this ocument. Changes may have been mae to this work since it was submitte for publication. A efinitive version was subsequently publishe in W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac.9.9.6

3 Nonlinear pressure control of self-supplie variable isplacement axial piston pumps W. Kemmetmüller,a, F. Fuchshumer b, A. Kugi a a Automation an Control Institute, Vienna University of Technology, Gusshausstr. 7 9, 4 Vienna, Austria b Hyac Electronic GmbH, Hauptstr. 7, 668 Saarbrücken, Germany Abstract The present paper eals with the pressure control of self-supplie variable isplacement axial piston pumps subject to fast changing, unknown loas. First, the setup of the system an the mathematical moel are escribe. As the pump is self-supplie, the mathematical moel exhibits a switching right-han sie which makes the control esign a challenging task. A nonlinear two egrees-of-freeom control strategy, comprising a feeforwar an a feeback control, in combination with a loa estimator is propose for the pressure control. The proof of the stability of the overall close-loop system is base on Lyapunov s theory. The performance of the control concept is verifie by means of experiments. The results show that the propose control concept has an excellent an robust behavior. Key wors: axial piston pumps, nonlinear control, switche systems, loa estimation, loa-sensing, pressure control. Introuction Electrohyraulic systems are wiely sprea in many inustrial plants an mobile machines like excavators, cranes, etc., which is mainly ue to the very high power ensity of hyraulic systems compare to electrical or mechanical rives. The generally poor energetic efficiency constitutes one of the major rawbacks of electrohyraulic systems. Conventional hyraulic supply systems typically provie a constant supply pressure or a constant supply volume flow, inepenent of the actual emans of the loa. Thus, the worst energetic efficiency occurs in the case when no energy is neee by the loa. The increasing emans on the energetic efficiency requires the implementation of hyraulic supply systems which can be ajuste to the actual requirements of the loa (loa-sensing, see, e.g., Wu, Burton, Schoenau & Bitner (; Fineisen (6. Basically, two approaches o exist to control the supply volume flow. If a fixe isplacement pump is use, then the input spee of the pump can be utilize to change the output volume flow. In many applications, fixe isplacement pumps are riven by electric motors which allow an easy control of the spee. The ynamics, however, are very limite such that in general the emans on the ynamical performance cannot be met with this concept. This is even more obvious if the pump is riven by a combustion engine. The secon possibility to control the volume flow of a pump is to change the isplacement of the pump. In this context, variable-isplacement axial piston pumps are often use, whereby the isplacement of the pump (i.e. the volume flow Corresponing Author: Tel./Fax: +43( / aresses: kemmetmueller@acin.tuwien.ac.at (W. Kemmetmüller, franz.fuchshumer@hyac.com (F. Fuchshumer, kugi@acin.tuwien.ac.at (A. Kugi can be change by tilting a swash plate. This can be one fast enough to meet the ynamical emans of many loas. The present paper eals with the supply pressure control of electrohyraulic systems comprising a variable isplacement axial piston pump an a variable loa. Typically, linear control strategies are use in such applications, see, e.g., Grabbel & Ivantysynova (5; Wu, Burton, Schoenau & Bitner (. Since electrohyraulic systems exhibit a significant nonlinear behavior the performance of the close-loop system is normally rather limite. Furthermore, a rigorous stability proof is lacking in most cases an the tuning of the controller parameters turns out to be very time-consuming. In this work, a new moelbase nonlinear control strategy is erive, which, on the one han, takes into account the essential nonlinearities of the system an, on the other han, can be easily ajuste to pumps of ifferent installation sizes in the same moel range. A general problem in esigning a loa-sensing system is to fin out the actual emans of the loa, since in most cases the loa is neither known nor can it be measure. This problem also occurs in the application consiere in this paper where the loa not only is unknown but may also change in a very fast manner. In orer to eal with this fact, the nonlinear control strategy has to be augmente by a loa estimator. This is a challenging task since it is well known that the separation principle of the controller an the estimator esign oes not hol for nonlinear systems. In this contribution, the stability of the close-loop system consisting of the nonlinear controller, the nonlinear loa estimator an the plant moel is proven by means of Lyapunov s theory. In orer to meet the high emans both on the tracking behavior an the robustness of the close-loop system a two egrees-of-freeom control structure, comprising a feeforwar an a feeback part, is propose in the controller esign. Thereby, the esign of the control strategy becomes very chal- Preprint submitte to Control Engineering Practice September, 9 Post-print version of the article: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

4 lenging in the consiere application ue to the fact that the axial piston pump is self-supplie, i.e. the volume flow which is necessary to control the pump is taken from the output volume flow of the pump. This, as will be outline in etail in the next section, yiels a switching mathematical moel of the system. It is well known from literature that the controller esign an the proof of the stability is much more emaning for switching systems (Branicky, 995, 998; DeCarlo, Branicky, Pettersson & Lennartson, ; Liberzon, 3. The paper is organize as follows: In the next section, the electrohyraulic system uner consieration is escribe in etail an a short overview of the mathematical moeling as well as a precise efinition of the control task is given. Section 3 is concerne with the control esign, whereby a feeforwar controller is esigne in the first part an is augmente by a feeback controller in the secon part. The esign of a loa estimator is the topic of Section 4, where a simple an an extene estimator for the loa is presente. Finally, the feasibility of the overall control concept is shown by means of measurement results in Section 5.. Mathematical Moeling an Control Task The variable isplacement axial piston pump uner consieration is to be use in injection moling machines, where liqui plastics is injecte into a mol by a screw conveyor. The isplacement of the screw conveyor is controlle by a hyraulic piston actuator, which, in turn, is controlle by the axial piston pump. Basically, the injection process can be ivie into two phases: i. In the first phase, the mol is fille with liqui plastics. To accomplish this task, the screw conveyor has to be move with a constant velocity. This means that the axial piston pump has to provie a constant volume flow. ii. In the secon phase, the mol is completely fille. In orer to compensate for the shrinking of the cooling plastics, liqui plastics has to be supplie to the mol with constant pressure. Thus, the axial piston pump has to control the pressure in the piston actuator. The control task for the first phase is rather simple since the volume flow q p of the pump is basically proportional to the swash plate angle ϕ p. Thus, a simple (linear controller for the swash plate angle turns out to be sufficient in terms of accuracy an ynamic performance, see, e.g., Fuchshumer (9. On the other han, the control of the pressure in the piston actuator is a challenging task since (i there are very high emans on the ynamics an the accuracy of the pressure an (ii the characteristics of the loa can change ramatically with very high ynamics. The control task is further complicate by the fact that the pump is self-supplie, which means that the volume flow necessary to control the swash plate angle of the pump is taken from the pump volume flow q p, cf. Fig.. This in turn entails a switching character of the mathematical moel. In Fig. the schematic iagram of a variable isplacement axial piston pump is shown. The pump uner consieration consists of 9 pistons which are place in the barrel. The barrel, riven by an inuction machine, rotates with the (almost constant angular velocity ω p an is force against the valve plate, which alternately connects the pistons to the tank an to the loa pressure. The pistons themselves are born against the swash plate by means of slippers. A tilt (angle ϕ p of the swash plate results in an axial isplacement of the pistons. Thereby, oil is taken from the tank via the intake port an elivere to the loa via the ischarge port. The volume flow of the pump q p can be change continuously by changing ϕ p. swash plate ϕ p p t q a actuator p a intake port ω p slipper ischarge port piston barrel valve plate Figure : Schematic iagram of the variable isplacement axial piston pump. Fig. epicts the schematic iagram of the overall electrohyraulic system uner consieration. It comprises the variable isplacement pump which elivers the volume flow q p to the loa volume. The loa volume is connecte to the tank with tank pressure p t via a variable loa orifice. In orer to change the volume flow q p of the pump, the angle ϕ p of the swash plate has to be ajuste. For this task a single acting hyraulic actuator is use, whereby the restoring force is generate by a spring. The actuator pressure p a is controlle by a 3 ways / 3 lans (3/3 proportional irectional valve generating the actuator volume flow q a. As shown in Fig., the control valve is supplie by the loa pressure p l, i.e., the pump is self-supplie. Thereby, only positive actuator volume flows q a are taken from the loa, while negative volume flows are ischarge to the tank. The avantage of the chosen experimental setup is that all relevant situations occurring in injection mouling machines can be emulate uner well-efine conitions. The mathematical moeling of electrohyraulic systems was the topic of numerous works, see, e.g., Blackburn, Reethof & Shearer (96, Merritt (967 an McCloy & Martin (98 for a general overview. Especially, etaile works are available for the mathematical moeling of variable isplacement pumps (Manring & Johnson, 996; Manring, 5; Ivantysyn J & Ivantysynova M, 993; Fineisen, 6. Since the etaile mathematical moels capturing all the ynamical effects are in general rather complicate they are not suitable for a moel Post-print version of the article: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

5 control valve q a loa volume p a η(q a actuator p t q p p l p t ϕ p loa orifice q l pump Figure : Schematic iagram of the electrohyraulic system. base controller esign. Therefore, an analysis of the ynamics of the system base on the singular perturbation theory was performe in Fuchshumer (9 to systematically reuce the overall complexity of the mathematical moel. The resulting moel, which will be use in the subsequent control esign, is given by t ϕ p = q a t p l = β ( kp ϕ p k l pl η(q a, p t (a (b where ϕ p is the swash plate angle, p l is the loa pressure an q a is the volume flow into the actuator. The effective area of the actuator is enote by A a an the effective raius is given by r a. Further, β is the bulk moulus of the oil, is the volume of the loa an k l enotes the unknown coefficient of the loa orifice (loa coefficient. The function η(q a escribes the volume flow taken from the loa in orer to tilt the swash plate { qa for q η(q a = a > ( else. Finally, the volume flow q p of the pump is given by q p = k p ϕ p with the pump coefficient k p = n pa p r p ω p, (3 π with the number of pistons n p, the cross-sectional area A p of a piston, the effective raius of rotation r p of the pistons an the constant angular velocity ω p of the barrel. As was shown in Fuchshumer (9, this consierably simplifie mathematical moel of the electrohyraulic system in Fig. covers the essential (nonlinear behavior of the real system an thus serves as a goo basis for the controller esign. Remark. Henceforth it is assume that the volume flow q a into the actuator is the control input of the system. In reality, of course, only the position of the spool of the 3/3 proportional irectional valve can be irectly controlle. However, a servocompensation is implemente in the system which calculates the spool position s v necessary to achieve a esire actuator volume flow q a. More etails on this topic will be given in Section 5. 3 The control esign task can now be summarize as follows: Given the (nonlinear mathematical moel of the system (, (, esign a (nonlinear controller (with q a as the control input for the loa pressure p l which is capable of following high ynamic trajectories p l, (t without exact knowlege of the loa. The control task is complicate by the following facts: From (, ( it can be seen that the mathematical moel of the electrohyraulic system constitutes a switching system, since the right-han sie is changing epenent on the sign of q a. This means, of course, that many classical stability results an control esign methos for nonlinear systems cannot be irectly applie. In aition to the fact that the loa coefficient k l is unknown in the real application, it can even change very rapily. The controller has to be robust with respect to moel uncertainties an measurement noise. 3. Control Design This section is concerne with the evelopment of a nonlinear moel base control strategy for the electrohyraulic system (, (. In this work, a two egrees-of-freeom control structure comprising a feeforwar an a feeback part is use to solve the aforementione control task. In orer to take into account the unknown loa coefficient k l, the controller is augmente by an estimator for k l. For the subsequent consierations, system (, ( is formally split into two systems: system Σ I which is vali for q a, Σ I : t ϕ p = q a (4a t p l = β ( kp ϕ p k l pl, (4b an system Σ II which hols for q a > Σ II : t ϕ p = q a (5a t p l = β ( kp ϕ p k l pl q a. (5b 3.. Feeforwar Control System Σ I A simple investigation of (4 shows that the system Σ I is ifferentially flat with the loa pressure p l as a possible flat output, see, e.g., Fliess, Lévine, Martin & Rouchon (995 for an introuction to the concept of flatness for nonlinear systems. Defining a sufficiently smooth (at least twice continuously ifferentiable esire trajectory p l, of the loa pressure in (4b yiels ṗ l, = β ( kp ϕ p, k l pl,. (6 Post-print version of the article: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

6 In orer to calculate the esire swash plate angle ϕ p, from (6 the knowlege of the loa coefficient k l woul be necessary. To circumvent this problem at this point, an estimation ˆk l is use in the feeforwar control instea of k l. The esign of the estimator an the proof of the overall close-loop stability will be treate later on in this paper. For the time being only the existence of an appropriate estimator for k l is presume. This irectly gives the esire swash plate angle ϕ p, ϕ p, = k p ( Vl β ṗl, + ˆk l pl,. (7 The secon erivative of the esire loa pressure p l, = β ( k p q a, ˆk l ṗ l, ˆk l pl, pl, inclues the esire actuator volume flow q a,. Thus, the feeforwar control for Σ I reas as FF I : q a, = A ( ar a Vl k p β p l, + ˆk l ṗ l, + ˆk l pl,. (9 pl, System Σ II In contrast to Σ I, the loa pressure p l is no longer a flat output for Σ II. Nevertheless, an inversion-base feeforwar control strategy can be applie to this system. Thus, the esire loa pressure p l, is use in (5b (8 ṗ l, = β ( kp ϕ p, k l pl, q a,, ( yieling the esire volume flow q a, for Σ II with k l replace by ˆk l in the form FF II : q a, = β ṗl, + k p ϕ p, ˆk l pl,. ( The corresponing esire swash plate angle ϕ p, in ( is etermine by the ifferential equation for the internal ynamics t ϕ p, = ( k p ϕ p, + ˆk l pl, + V l β ṗl,. ( Obviously, the internal ynamics is exponentially stable. 3.. Feeback Control The feeforwar control esigne in the last subsection has to be augmente by a feeback control in orer to stabilize the tracking error in case of parameter variations or moel uncertainties. Therefore, the pressure error e p = p l p l, an the error in the swash plate angle e ϕ = ϕ p ϕ p, are introuce. Applying FF I to the system Σ I yiels the error system for q a t e ϕ = q a,c t e p = β ( kp e ϕ k l ep + p l, + ˆk l pl,, (3a (3b where q a,c = q a q a, enotes the feeback part of the control input. Similarly for q a >, using FF II in Σ II results in t e ϕ = q a,c t e p = β ( kp e ϕ k l ep + p l, + ˆk l pl, q a,c. (4a (4b Before a controller can be esigne for the switche system (3, (4, some important facts on the stability of switche systems have to be iscusse. First of all, it is well known from literature that a switche system may be unstable even if the iniviual systems are all stable for themselves (Branicky, 995, 998; DeCarlo, Branicky, Pettersson & Lennartson, ; Liberzon, 3. Therefore, it is not sufficient to esign a stabilizing controller for the error systems (3 an (4 separately. One possibility to achieve a systematic proof of the stability of switche systems is given by the metho of multiple Lyapunov functions as propose by Branicky (998. Thereby, the stability of each system has to be proven with a Lyapunov function an in aition it has to be shown that each Lyapunov function is strictly non-increasing uring switching. While the proof of the first conition is rather straightforwar for many systems, the proof of the secon conition is in general ifficult. One way to avoi the proof of the secon conition is to use a common Lyapunov function for all systems. Although the esign of a common Lyapunov function turns out to be a rather elicate issue for general nonlinear switching systems, this approach will be pursue in the following. For the time being, it is assume that a common Lyapunov function an feeback controllers FB I an FB II for the error systems I an II, respectively, have alreay been foun such that the stability of each close-loop system is guarantee. At this point the question arises when an how the control law consisting of the feeforwar an the feeback control is switche. The intuitive approach woul be that FF I + FB I are active for q a an FF II + FB II for q a >. Then, however, two problems occur: First, switching the feeforwar control FF I an FF II base on q a = yiels to iscontinuities in the esire value of the swash plate angle ϕ p, an thus in e ϕ. In orer to make this more obvious, consier at the beginning that q a = q a, + q a,c < an therefore FF I an FB I are active. If at time t s the volume flow q a equals zero, switching to FF II an FB II woul occur. In this case, the initial value ϕ p, (t s of the ifferential equation ( woul be set to the actual value of ϕ p, at t = t s in FF I ue to (7 an thus the trajectory is continuous. However, switching from q a > (i.e. FF II an FB II are active to FF I an FB I at time t = t s when q a =, the esire swash plate angle ϕ p, has to satisfy the following relations at t = t s ϕ p, = ( Vl k p β ṗl, + ˆk l pl, + q a, (5 Subsequently, the feeforwar control for Σ I an Σ II will be referre to by FF I an FF II, respectively. 4 Note that the actual esign of the feeback controllers will be performe in the next subsection. Post-print version of the article: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

7 accoring to ( for FF II an ϕ p, = ( Vl k p β ṗl, + ˆk l pl, (6 accoring to (7 for FF I. Of course there is no reason for q a, to be zero at time t = t s, since the switching conition q a (t s = only leas to q a, (t s = q a,c (t s. Consequently, switching at q a = in general provokes a iscontinuous time evolution of ϕ p, an thus of e ϕ. Note that this behavior originates from the fact that the relative egree of the output to be controlle p l is changing when switching between the systems Σ I an Σ II ue to (4 an (5, respectively. See, e.g., Isiori ( for more etails on the notion of relative egree of a nonlinear system. There is, however, a secon problem which occurs in connection with a switching base on the conition q a =. If FF I + FB I yiels q a = this oes not necessarily imply that FF II + FB II also yiels q a =. In orer to clarify this, let us consier the situation where q a < with FF I an FB I active an switching takes place if q a from FF I + FB I crosses zero, i.e. q a =. In this case, q a calculate from FF II + FB II may also be negative which woul cause immeiate switching back to FF I an FB I. Thus, there is a set where neither FF I an FB I nor FF II an FB II are vali. As a result (an for perfect switching, a sliing motion along the (sliing submanifol q a = of FF I + FB I woul take place. The first problem, i.e. the iscontinuity of the esire trajectories, can be solve by switching the feeforwar control FF I an FF II an the feeback control FB I an FB II inepenently. Therefore, the zero crossing of the esire volume flow q a, is use as a switching criterion for the feeforwar control instea of the actuator volume flow q a. Since q a, = yiels the same ϕ p, for FF I an FF II, cf. (5 an (6, this switching criterion avois the aforementione problems when switching from FF II to FF I. The sliing motion of the controller can also be circumvente by switching the feeback control FB I an FB II inepenently of the system. In contrast to the feeforwar control the choice of a suitable switching criterion is much more complicate in this case since the feeback control may consist of arbitrary nonlinear functions of the states e p an e ϕ. Furthermore, the inepenent switching of the feeforwar an the feeback control requires the proof of the stability of the close-loop system of all eight possible combinations of feeforwar control (FF I, FF II, feeback control (FB I, FB II an systems (Σ I, Σ II with one common Lyapunov function. In orer to simplify matters, in this work a common feeback law FB I = FB II will be use. The general proceure of the feeback controller esign is as follows: First, a feeback controller an a control Lyapunov function are esigne for the error system (3, resulting from the application of FF I to Σ I. Afterwars, the stability of the close-loop system for the other three combinations of feeforwar control an system (FF II, Σ II, (FF I, Σ II an (FF II, Σ I, respectively, is proven using a common Lyapunov function an feeback law. This, of course, implies the stability of the overall switche close-loop control system. 5 Feeforwar FF I with System Σ I Applying FF I to Σ I results in the error system (3. For the esign of the feeback controller it is assume that the estimate value ˆk l is exactly equal to the real value k l (certainty equivalence conition, see, e.g., Krstić, Kanellakopoulos & Kokotović (995. The esign of an estimator for k l an the proof of the stability of the overall close-loop system comprising the feeforwar control, the feeback control an the estimator will be given in the next section. As a starting point the positive efinite function W c W c = δ e p + δ e ϕ, (7 with positive constants δ, δ >, is chosen as a possible caniate for a control Lyapunov function (CLF. The change of W c along a solution of the error system (3 reas as t W c = δ βk l ( ep + p l, p l, ep + δ βk p e p e ϕ δ e ϕ q a,c. (8 For the consiere application a simple feeback control law of the form q a,c = λ p e p + λ ϕ e ϕ (9 with constant controller parameters λ p, λ ϕ >, is chosen. At this point one may woner why a linear feeback controller suffices in terms of the emans on the close-loop ynamics. Note that the excellent performance of the overall close-loop system (cf. Section 5 is mainly ue to (i the feeforwar controller, which systematically accounts for the nonlinearities in the tracking case an (ii the nonlinear loa estimator for k l, to be esigne in the next section, in the isturbance case in combination with (iii the propose switching strategy. Substituting the feeback control law (9 into (8 an setting δ to δ = results in δ k p β λ p ( t W c = δ λ p k l k p ( ep + p l, p l, ep δ λ ϕ e ϕ. ( Clearly, since k p, k l > the right-han sie of ( is negative efinite an this proves the asymptotic stability of the closeloop system (3, i.e. FF I with Σ I, an (9. Similar results can be obtaine for the three other combinations of feeforwar control an system (FF II, Σ II, (FF I, Σ II an (FF II, Σ I, see the Appenix A. Thus, the stability of the close-loop system consisting of the switche feeforwar control, the common feeback control an the switche system is proven if the certainty equivalence conition k l = ˆk l hols. In the next section an estimation of ˆk l will be erive an the stability of the overall close-loop system will be proven. Post-print version of the article: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

8 4. Estimation of the loa coefficient k l The esign of the estimator for the loa coefficient k l is base on the assumption that k l is unknown but constant. In this contribution, two ifferent estimators will be erive. The first rather simple approach is straightforwar an well known from literature but has the rawback that it can harly be tune to meet the emans on the ynamic performance an on the robustness. In particular these eficiencies become apparent when applying this simple estimator to the experimental setup. For this reason, an extene estimator also will be presente where the whole measurement information is exploite within the esign process. 4.. Simple estimator The simple estimator is suppose to take the form t ˆk l = χ k (e p, e ϕ, t, ( where the right-han sie χ k of ( has to be etermine. In orer to o so, the CLF (7 is extene by a quaratic term in the estimation error ê k = k l ˆk l W tot = W c + W e = δ e p + δ e ϕ + ˆλ k ê k, (3 with the tuning parameter ˆλ k > of the estimator. Before calculating the change of W tot along a solution of the error system (3, (4, (4 or (47, respectively, it is useful to rewrite the right-han sies in such a way that only expressions with k l an k l ˆk l = ê k o appear but no ones which are explicitly weighte with ˆk l. Note that this can always be achieve since the righthan sies of the error systems (3, (4, (4 an (47 are all affine in the loa coefficient k l. For the error system (3 this rearrangement of the right-han sie exemplarily yiels t e ϕ = q a,c t e p = β ( kp e ϕ k l ep + p l, + k l pl, β ( kl ˆk l pl,. } {{ } ê k (4a (4b Analogously, all other error systems can be rewritten to exhibit a similar structure. The first term of (4b equals the error system (3 if the certainty equivalence conition hols an the secon term of (4b accounts for the estimation error. Now, the change of the overall Lyapunov function W tot along a trajectory of the close-loop system (4 with (9 an ( can be calculate as t W c = δ k l λ p ( ep + p l, p l, ep δ λ ϕ e ϕ k p δ λ p pl, e p ê k + ˆλ ê k χ k. k p k (5 This result correspons to ( except for the last two terms. In orer to rener Ẇ tot negative semi-efinite, the thir term in 6 (5 is cancelle out by the last term. Thus, the estimator ue to ( reas as t ˆk δ λ p l = ˆλ k pl, e p (6 k p Obviously, using the same approach for the other three error systems yiels the same result. Since the calculations are straightforwar they are omitte here. With this, the stability of the close-loop system comprising the switche feeforwar control, the common feeback control, the estimator an the switche system has been proven. Simulation stuies an experimental results with the simple estimator, however, show that (i a suitable choice of ˆλ k is very ifficult to fin an that (ii the emans on the ynamic performance an accuracy cannot be achieve. Furthermore, the estimator shows a weak robustness to moel uncertainties. Therefore, the simple estimator is not feasible for practical implementation. 4.. Extene estimator The basic iea in the evelopment of the extene estimator for the loa coefficient k l is to aitionally estimate the loa pressure p l, although this quantity is available by measurement. The main reason for this is to provie aitional egrees-offreeom for the esign an parametrization of the estimator. The estimator for the loa pressure p l is compose of a preiction an a correction part, where the preictor is basically a copy of the mathematical moel (4b, (5b an the corrector term χ p (e p, e ϕ, t is use to stabilize an ajust the estimator ynamics, namely t ˆp l = β ( kp ϕ p ˆk l pl χp, for q a (7a t ˆp l = β ( kp ϕ p ˆk l pl q a χp, for q a >. (7b As it can be seen the switching between (7a an (7b relies on the zero-crossing of the actuator volume flow q a. Thus, the estimator (7 is switche synchronously to the system (4, (5. For the estimation of the loa coefficient k l the same approach as in ( is use t ˆk l = χ k (e p, e ϕ, t. (8 Introucing the estimation errors ê p = p l ˆp l an ê k = k l ˆk l it can be easily seen that the error system for both systems Σ I an Σ II has the ientical form t êp = β pl ê k + χ p (e p, e ϕ, t t êk = χ k (e p, e ϕ, t. (9a (9b The corrector terms χ p (e p, e ϕ, t an χ k (e p, e ϕ, t in (9 have to be esigne in orer to allow for a proof of the stability of the overall close-loop system comprising the feeforwar an the feeback controller, the system an the extene estimator. Post-print version of the article: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

9 However, before proving the stability of the overall close-loop system, the stability of the extene estimator itself will be analyze. For this purpose the Lyapunov function caniate W e = ê p + ˆλ k ê k (3 with the estimator parameter ˆλ k > is chosen. The change of W e along a solution of (9 is then given by t W e = β pl ê p ê k + ê p χ p + ˆλ k ê k χ k. (3 In orer to compensate for the first inefinite term, χ k is fixe as χ k (e p, e ϕ, t = ˆλ k β pl ê p. (3 Then, the choice of the corrector term χ p in the form 5. Measurement Results In this section, the properties of the propose control strategy comprising the feeforwar controller FF I (7, (9 an FF II ( an (, the common feeback controller (9 an the extene estimator (37 an (38, are analyze by means of measurement results of a test stan. The test stan was esigne an built by the company HYDAC Electronic GmbH, see Fig. 3. The main components of this test stan are the variable isplacement axial piston pump riven by an inuction machine an controlle by the control valve, the loa volume an the loa orifice. The schematic iagram of the hyraulic circuit of the test stan is given in Fig. an the parameters of the system are summarize in Table. control valve axial piston pump χ p (e p, e ϕ, t = ˆλ p ê p (33 with the estimation parameter ˆλ p > reners Ẇ e in (3 negative semi-efinite. This implies stability in the sense of Lyapunov of the extene estimator. Up to now, the estimator has been analyze separately from the rest of the system. In orer to stuy the stability of the overall close-loop system with the extene estimator the overall Lyapunov function, cf. (7 an (3 W tot = W c + W e = δ e p + δ e ϕ + ê p + ˆλ k ê k (34 is use. Keeping the analysis of the simple estimator, especially (4, (5 an (6, in min, it can be seen that the time erivative Ẇ tot along a solution of the overall close-loop system is negative except for the term δ λ p k p pl, e p ê k. (35 This term can be cancelle out by augmenting χ k (e p, e ϕ, t from (3 in the form ( β χ k = ˆλ k pl ê p + δ λ p pl, e p. (36 k p Summarizing, the extene estimator reas as t ˆp l = β ( kp ϕ p ˆk l pl + ˆλ p ê p ( t ˆk β l = ˆλ k pl ê p + δ λ p pl, e p k p for q a an t ˆp l = β ( kp ϕ p ˆk l pl q a + ˆλ p ê p ( t ˆk β l = ˆλ k pl ê p + δ λ p pl, e p k p for q a >. (37a (37b (38a (38b 7 loa orifice inuction machine Figure 3: Experimental setup of the test stan for the axial piston pump. parameter symbol value unit bulk moulus β.6 9 Pa eff. area of actuator A a 3 mm eff. raius of actuator r a 5 mm number of pistons n p 9 area of piston A p 65 mm raius of rot. of pistons r p 3 mm angular vel. of pump ω p 5π s 3 m3 s pump coefficient k p.3 min. swash plate angle ϕ p,min.5 max. swash plate angle ϕ p,max 8 loa volume.5 l min. loa coeff. k l,min nom. loa coeff. k l,nom 9 max. loa coeff. k l,max 4 Table : Parameters of the pump an the loa. 9 m3 s Pa 9 m3 s Pa 9 m3 s Pa The actuator for tilting the swash plate is controlle by a (3/3 proportional irectional valve, cf. Fig.. In contrast to the previous assumption, the volume flow q a into the actuator cannot be irectly assigne by means of this valve. In fact, only the position s v of the spool of the valve can be controlle irectly. Post-print version of the article: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

10 The volume flow q a is given in the form q a = α ( Ala (s v p l p a A at (s v p a p t, (39 ρ where α enotes the constant ischarge coefficient, ρ is the mass ensity of the oil an A la (s v an A at (s v are the opening areas of the valve from the loa to the actuator an from the actuator to the tank, respectively. Furthermore, p l is the loa pressure, p a enotes the actuator pressure an p t = is the tank pressure. In the system uner consieration a valve with a small negative overlap is use, whose opening characteristics are epicte in Fig. 4. area in mm A at A la s v in mm Figure 4: Opening characteristics A la (s v an A at (s v of the proportional valve. In orer to calculate the real control input, i.e. the valve spool position s v, from the virtual control input q a, (39 is solve for s v. A unique solution of this equation always exists, although it can only be evaluate numerically. With this it is possible to compensate for the nonlinearities of the valve (servocompensation such that the spool position s v can be calculate from the volume flow q a. Thereby, as alreay mentione before, the ynamics of the valve is neglecte since it is consierably faster (rise time of approximately 5 ms than the ynamics of the system. The control strategy, extene by the servo-compensation, was realize in form of a Simulink C-coe s-function, compile using Matlab Real-Time Workshop an implemente on a SPACE realtime harware DS3. Thereby, a sampling time of T s = ms is use. Furthermore, the parameters of the controller an the estimator are chosen accoring to Table. In the first measurement result the tracking behavior of the loa pressure p l is analyze. Therefore, two measurements are performe, one with a small loa coefficient k l = k l,min (see the left-han sie of Fig. 5 an one with a larger, nominal loa coefficient k l = k l,nom (see the right-han sie of Fig. 5. As can be seen from the time-evolution of the loa pressure p l an excellent tracking performance is achieve inepenent of the actual value of the loa coefficient. On the other han, the ifferent 8 parameter value λ p 8 λ ϕ 5 3 ˆλ p 6 ˆλ k δ Table : Parameters of the controller an extene estimator. loa coefficients have a large influence on the trajectories of the swash plate angle ϕ p. Obviously, this is ue to the fact that only a small volume flow q p of the pump is necessary to provie the (small loa volume flow q l = k l,min pl for the small loa coefficient while a much higher volume flow q p is necessary for the larger loa coefficient k l,nom. The influence of the ifferent loa coefficients can also be seen in the plots of the actuator volume flow q a an the real control input s v, which are given at the bottom of Fig. 5. The secon measurement result, given in Fig. 6, shows the behavior of the system for rapi changes of the loa coefficient. Here, the loa orifice is close an opene as fast as possible while a esire trajectory p l, (t in the loa pressure is tracke. Of course, the fast change of k l yiels to significant errors in the loa pressure but these errors are compensate in a very fast way. At this point it is worth mentioning that the stability proof of the overall close-loop system, cf. Section 4., relies on the assumption that the loa coefficient is unknown but constant. Clearly, the case of rapily changing loas is not covere by the stability proof but the measurment results show that the control strategy is reliable also in this situation. The ynamical behavior of the estimation of the loa coefficient, as given on the right-han sie of Fig. 6, shows that the estimation ˆk l tracks the rapily changing loa coefficient in an excellent manner. Thereby, it has to mentione that the rather large overshoot in the estimation of the loa coefficient can be reuce by ajusting the parameters of the estimator. However, the main focus of the control strategy is goo tracking of the loa pressure p l an not the exact estimation of the loa coefficient. For this task, the chosen parameters of the controller an estimator have proven to be feasible in practical application an turne out to be a goo compromise between tracking performance of the loa pressure an a goo estimation of the loa coefficient. In the final measurement result, the tracking behavior of the loa pressure is analyze for slowly varying loa coefficients. Here, the loa coefficient k l is slowly increase while the loa pressure shoul track a rectangular like reference trajectory, cf. Fig. 7. For this case again an excellent tracking performance can be achieve, while at the same time a goo estimation of the loa coefficient is obtaine. To sum it up, the measurement results show a very goo performance of the overall control strategy an thus prove the practical feasibility of the propose control strategy comprising the feeforwar control, the feeback control an the extene estimator. Post-print version of the article: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

11 pressure in bar 8 6 p l, p l pressure in bar 8 6 p l, p l ϕp in.5.5 ϕp in qa in l/min qa in l/min sv in mm.. sv in mm Figure 5: Measurement results for the tracking behavior of the loa pressure p l for a small loa coefficient k l = k l,min on the left-han sie an a larger, nominal loa coefficient k l = k l,nom on the right-han sie. 9 Post-print version of the article: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

12 pressure in bar p l, p l ˆkl in m3 s Pa Figure 6: Measurement results for a rapi change of the loa coefficient k l while tracking a trajectory in the loa pressure p l pressure in bar ˆkl in m3 s Pa 5 4 p l, p l 3 l Figure 7: Measurement results for the slow change of the loa coefficient k l while tracking a trajectory in the loa pressure p l. 6. Conclusion the existing inustrial solution. In this work a new (nonlinear control concept for the pressure control of self-supplie variable isplacement axial piston pumps with variable loa was esigne. First, the basic setup of the electrohyraulic system an its mathematical moel was escribe. Therein it was pointe out that the switching character of the mathematical moel, which is ue to the self-supply mechanism of the pump, an the fast changing unknown loas constitute the main challenges for the controller esign. In orer to solve this control task, a two egrees-of-freeom control structure comprising a feeforwar an a feeback controller in combination with a loa estimator was propose. The avantages of this approach are (i the systematic proof of the close-loop stability for unknown but constant loa coefficients base on Lyapunov s stability theory, (ii the moel-base esign, which allows an easy implementation of the control concept to other installation sizes in the same moel range, an (iii the simple parameterization by means of a few controller parameters. The feasibility of the control strategy was shown by measurement results, whereby an excellent robustness behavior an a superior tracking performance coul be achieve. Furthermore, the practical use of the propose control concept is affirme by the inustrial partner who also stresses the significant improvement of the propose control concept compare to A. Proof of stability A.. Feeforwar FF II with System Σ II The error system for the feeforwar controller FF II applie to the system Σ II is given in (4. By using the certainty equivalence conition, the CLF from (7 an the feeback control (9 with ( yiels the change of the CLF W c along the solution of (4 t W c = δ k l λ p ( ep + p l, p l, ep k p δ λ p k p e p + λ pλ ϕ k p (4 e p e ϕ + λ ϕ e ϕ. The right-han sie of (4 is negative efinite if the conition 4k p > λ ϕ > (4 is fulfille, which also implies the asymptotic stability of the close-loop system (4, (9. Post-print version of the article: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.

13 A.. Feeforwar FF I with System Σ II Due to the inepenent switching of the feeforwar part an the system it may happen that the feeforwar control FF I is applie to the system Σ II an vice versa FF II is applie to Σ I. Using FF I in combination with Σ II results in the error system t e ϕ = q a,c t e p = β ( kp e ϕ k l ep + p l, + ˆk l pl, q a. (4a (4b Inserting the control law (9 into (4 an presuming the certainty equivalence conition, the change of the CLF W c along a solution of the error system (4 reas as t W c = δ k l λ p k p ( ep + p l, p l, ep δ λ ϕ e ϕ δ λ p e p q a. k p This result equals ( except for the last inefinite term (43 δ λ p k p e p q a. (44 Since the actuator volume flow q a is positive for the system Σ II, the inefinite term (44 is negative for e p >. Thus, Ẇ c is negative efinite for e p >. In orer to show a similar relation for e p <, the quantity q a is replace by q a = q a, +q a,c in (43, with q a,c accoring to (9. Doing so, (43 can be rewritten in the form t W c = δ k l λ p ( ep + p l, p l, ep δ k p λ p e p + λ pλ ϕ e p e ϕ + λ ϕ e ϕ k p k δ λ p e p q a,, p k p which correspons to (4 except for the last inefinite term (45 δ λ p k p e p q a,. (46 Similar arguments as in the previous subsection show that also in this case the change of the common Lyapunov function (7 along a solution of (47 is negative efinite for λ p > an λ ϕ satisfying the inequality conition (4. References Blackburn J.F., Reethof G. & Shearer J.L. (96. Flui Power Control. New York: John Wiley & Sons. Branicky M.S. (995 Stuies in hybri systems: Moeling, analysis an control. PhD Thesis Massachusetts Institute of Technology. Branicky M.S. (998 Multiple Lyapunov functions an other analysis tools for switche an hybri systems. IEEE Trans. on Automatic Control, 43(4, DeCarlo R.A., Branicky M.S., Pettersson S. & Lennartson B. (. Perspectives an results on stability an stabilizability of hybri systems. Proc. of the IEEE, 88(7, Fineisen F. (6 Oil-Hyraulics (in German. Berlin, Germany: Springer. Fliess M., Lévine J., Martin P. & Rouchon P. (995 Flatness an Defect of Non-linear Systems: Introuctory Theory an Examples. International J. of Control, 6(6, Fuchshumer F. (9 Moeling, analysis an nonlinear moel-base control of variable isplacement axial piston pumps (in German. PhD Thesis Vienna University of Technology. Grabbel J. & Ivantysynova M. (5 An investigation of swash plate control concepts for isplacement controlle actuators. Int. Journal of Flui Power, 6(, Ivantysyn J. & Ivantysynova M. (993 Hyrostatic pumps an rives (in German. Würzburg, Germany: Vogel. Isiori A. ( Nonlinear Control Systems. Lonon, UK: Springer. Khalil H.K. ( Nonlinear Systems, 3r E. Upper Sael River: Prentice Hall. Krstić M., Kanellakopoulos I. & Kokotović P. (995 Nonlinear an Aaptive Control Design. New York: John Wiley & Sons. Liberzon D. (3 Switching in Systems an Control. Boston, USA: Birkhäuser. Manring N.D. & Johnson R.E. (996 Moeling an esigning a variableisplacement open-loop pump. ASME J. of Dynamic Systems, Measurement an Control, 8(, Manring N.D. (5 Hyraulic Control Systems. New Jersey, USA: John Wiley & Sons. McCloy D. & Martin H.R. (98 Control of Flui Power: Analysis an Design. New York: John Wiley & Sons. Merritt H.E. (967 Hyraulic Control Systems. New York: John Wiley & Sons. Wu D., Burton R., Schoenau G. & Bitner D. ( Establishing operating points for a linearize moel of a loa sensing system. Int. Journal of Flui Power, 3(, Now, since FF I is only active if q a, is negative, the expression (46 is negative for e p < which also proves the negative efiniteness of Ẇ c for e p <, provie that (4 hols. Summarizing, it has been shown that Ẇ c is negative efinite an thus the stability of the close-loop system (4, where FF I is applie to Σ II, with the control law (9 subject to the inequality (4 is proven. A.3. Feeforwar FF II with System Σ I The last case to be consiere is the feeforwar controller FF II applie to the system Σ I. In this case the error system reas as t e ϕ = q a,c t e p = β ( kp e ϕ k l ep + p l, + ˆk l pl, + q a,. (47a (47b Post-print version of the article: W. Kemmetmüller, F. Fuchshumer, an A. Kugi, Nonlinear pressure control of self-supplie variable isplacement axial piston pumps, Control Engineering Practice, vol. 8, pp ,. oi:.6/j.conengprac The content of this post-print version is ientical to the publishe paper but without the publisher s final layout or copy eiting.