On Using Unstable Electrohydraulic Valves for Control

Transcription

1 Kailash Krishnaswamy Perry Y. Li Department of Mechanical Engineering, University of Minnesota, 111 Church St. SE, Minneapolis, MN On Using Unstable Electrohyraulic Valves for Control High banwith, high flow rate electrohyraulic valves typically have two or more stages. Most multi-stage valves are expensive, require meticulously clean flui, an introuce higher orer ynamics. On the other han, single-stage spool valves are cheaper an more reliable. However, a majority of them are not suitable for high banwith, high flow rate applications ue to limitations of the electromechanical/solenoi spool-stroking actuators. In this paper, we investigate the feasibility of reucing this limitation by exploiting the transient flow forces in the valve so as to achieve spool ynamics that are intrinsically open-loop unstable. While conventional valves are esigne to be open-loop stable, the unstable valve esign has to be stabilize via close-loop feeback. Simulation case stuies are conucte to examine the potential ynamic an energetic avantages that an unstable valve may offer. These stuies inicate that unstable valves provie faster response than the stable counterparts when stroking forces are limite. Moreover, unstable valves ten to require less positive power an energy to operate. DOI: / Keywors: Flow Instability, Solenoi Actuator, Electrohyraulic Valves, Transient Forces, Unstable Flow Forces, Unstable Valves I Introuction High banwith, high flow rate electrohyraulic valves typically have two or more stages, one of which is usually a nozzle flapper pilot valve. Although highly popular, multi-stage electrohyraulic valves are more expensive than single-stage valves an a majority of them have the rawbacks that 1 they require meticulously clean flui, as irt eposition will cause the pilot valve to malfunction; 2 they increase the orer of the ynamics of the system, thus potentially introucing unesirable time lags, an making control esign more challenging. Single-stage, irect-acting control valves are valves in which the spools are irectly stroke by an electromechanical or solenoi actuator. They are less expensive an less sensitive to irt. They are also easier to manufacture an have lower orer ynamics than multistage valves. Proportional control valves are examples of this type of valve. Unfortunately, most of the commercially available single-stage irect-acting valves are not suitable for high performance, high flow rate applications. It is because at high banwith an large flow rate, the force an power require of the electromechanical actuator to stroke the spool become very significant, thus limiting the performance of the single-stage valve. The iscussion above inicates that single-stage irect-acting control valves may become more practical for high performance, high flow rate applications if it is possible to reuce the force an/or power eman on the electromechanical actuator that strokes the spool. With avances in control theory an technologies, one iea is to esign the valve spool so that they are openloop unstable an to utilize the flow forces associate with the instability avantageously. Spool valves can be mae unstable by appropriately manipulating the transient flow forces. The unstable valve will be stabilize subsequently via close loop feeback. The iea is similar to the esign of high performance fly-bywire fighter aircrafts in which the aeroynamics are sometimes Contribute by the Dynamic Systems an Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receive by the Dynamic Systems an Control Division February 9, Associate Eitor: N. Manring. eliberately esigne to be open-loop unstable so as to enhance their agility. Past stuies on valve instability have been restricte to ensuring that instabilities o not occur 1 3. In this paper, we investigate whether a single-stage valve which is esigne to be open-loop unstable offers any avantages in terms of performance improvements or requirements on the electromechanical/solenoi actuator. The rest of the paper is organize as follows. In Section II, we iscuss how flow forces etermine the stability of a four way irectional spool valve. In section III we present two numerical simulation experiments to quantify the potential benefits of unstable valves. Performance is quantifie in terms of step responses, an in terms of power an effort require to track sinusoial signals. Section IV contains iscussion an some concluing remarks. II Flow Forces an Spool Stability A four way irectional flow control valve is shown in Fig. 1. We assume that it is matche an critically centere. The isplacement x v of the spool controls the flow into a hyraulic evice connecte to the two ports on the right. In aition to the stroking force u provie by the electromechanical/solenoi actuator, the spool of the valve experiences both pressure forces an flow inuce forces or Bernoulli forces 1. Since the same pressure acts on the opposing surfaces of the lans of equal areas, they have no net effect on the ynamics of the spool. Flow inuce forces are of two types, 1 steay-state flow forces an 2 transient flow forces. Invisci, incompressible flow is assume in the following erivations. Steay-state flow forces are the reaction forces on the spool ue to the changes in the momenta of the flui entering an leaving the valve chamber. Referring to Figs. 2 a an 2 b, as the spool meters flow into out of the valve chamber, the vena contracta in which the flui enters leaves the chamber is at an angle to the spool axis. The flui however leaves enters perpenicular to the spool axis. Thus, flui entering an leaving the valve chamber can have ifferent lateral an axial momenta. This necessitates reaction forces on the spool in both lateral an axial irections. By locating the ports symmetrically on the circumference of the valve Journal of Dynamic Systems, Measurement, an Control MARCH 2002, Vol. 124 Õ 183 Copyright 2002 by ASME

2 sleeve, net lateral forces can generally be eliminate. The axial force is however, significant an contributes to net steay state flow forces on the spool. If the flow rate through an orifice is Q, then the steay-state flow force for one metering orifice is given by Q2 cos (1) A c where is the flui ensity, A c is the area of the vena contracta an, which ranges between 21 eg to 69 eg 1, is the angle of the vena contracta. Notice that the irection of this reaction force acts to close the orifice, regarless of whether the spool is metering into or metering out of the valve chamber. Let the area graient of the valve be w in 2 /in. Then the orifice size is given by A o w x v. Assuming Q 1 Q 2 Q in Fig. 1, it can be shown that for a matche, critically centere, symmetric valve, Fig. 1 A single-stage critical-centere spool valve connecte to a ouble-ene actuator Q Q 1 Q 2 C wx v 1 P s x v x v L P (2) where C is the ischarge coefficient, P s is the hyraulic supply pressure, an P L P 1 P 2 is the loa pressure, i.e., the ifferential pressure between the two valve chambers or the work ports. Since the area of the vena contracta is proportional to the orifice area, then in 1, A c C c A 0 C c w x v for some C c 0. Hence, the total steay state flow force for both the meter-in an meterout orifices in the irection of positive x v is given by 1 : F st 2C C v w P s P L cos x v (3) where C v C /C c. Notice from 3 that the steay-state flow force acts like a spool-centering spring an is proportional to x v. Hence the magnitue of the steay-state flow force will be large when the valve operates in applications that require high flow rate. The transient flow forces are reaction forces on the spool as the flui in the annular valve chamber accelerates in response to variation in flow rate. To erive the transient flow forces, we utilize the principle of conservation of momentum an a quasisteay analysis, i.e., we assume that the expression for flow rate in 2 hols for time varying flow. Consier the control volumes CV 1 an CV 2 enclose in the valve chambers in Figs. 2 a an 2 b, respectively. The longituinal momenta in the irection of positive x v of the incompressible flui in chambers i 1, 2 are given by: M i CV i V x V where V x is the longituinal flui velocity of the flui element. Notice that in Fig. 2 a, V x is negative for positive Q(t) as shown. Let the flow rate be given by Q(t) Q in (t) Q out (t). Then, by the continuity of incompressible flow, for any cross sectional area normal to the spool axis, we have Q t A 1 V x A 1 A 2 V x A 2 (4) where A 1 is any normal cross-sectional area in chamber 1 an A 2 is any normal cross-sectional area in chamber 2. Therefore, the longituinal momentum of the flui in the control volumes CV 1 an CV 2 are given by: Fig. 2 Valve configuration an transient flow forces when QÕtÌ0 an QÌ0. a Unstable transient flow force occurs when flow is metere into the valve chamber; b stable transient flow force occurs when flow is metere out of the valve chamber. Figures a an b correspon to the upper an lower chambers in the spool configuration in Fig. 1 L 1 M 1 x 0 M 2 x 0 V x A 1 A1 x x 0 L 2 V x A 2 A2 x x 0 L 1 Qx L1 Q L 2 Qx L2 Q 184 Õ Vol. 124, MARCH 2002 Transactions of the ASME

3 where L 1 an L 2 are the lengths of the flui columns for chambers 1 an 2 measure from the port centers. Conservation of momentum ictates that: M 1 M 2 net longituinal momentum flux t longituinal force applie by the valve. (5) Thus, we see that the force acting on the flui consists of a steay component, ue to the momentum flux, which is exactly the steay state flow forces alreay consiere, an the transient component ue to (/t)(m 1 M 2 ). Following 1,4,2,5,3, an applying a quasi-steay assumption, i.e. 4 is vali even for transient flow, the transient flow force acting on the flui is t M 1 M 2 L 2 L 1 Q t. (6) By Newton s thir law, a longituinal force of magnitue given by 6 must act on the valve in the irection opposite to that of 6. For invisci flow, all of this force acts on the spool. For the valve chamber which is metering flow out of the valve Fig. 2 b, the transient flow force acts to close (Q/t 0) or open (Q/t 0) the orifice so as to resist Q/t. On the other han, for a valve chamber that is metering flow into the valve Fig. 2 a, the transient flow force acts to open (Q/t 0) or close (Q/t 0) the orifice so as to encourage Q/t. Hence, the transient flow forces are stabilizing when flow is being metere out, an it is estabilizing when flow is being metere in. If Q is given by the orifice equation 2, we see that in general Q/t an hence the transient flow force, epen on P L /t an x v /t. However, as emonstrate experimentally in 6, the effects of pressure variation are small, an are normally neglecte 1. The four way irectional valve in Fig. 1 has both a meter-in an a meter-out chamber, therefore, combining 2 an 6, the net transient flow forces acting on the spool in the irection of the positive x v is: F tr M 1 M 2 L 1 L 2 Q t t L 1 L 2 C w P s x v x v L P x v t, (7) where L 1 is the istance between the center of the supply pressure port an the center of the outler loa port; an L 2 is the istance between the center of the return port an the center of the input loa see Fig. 1. Let us efine LªL 2 L 1 to be the amping length of the valve. L etermines whether the transient flow forces are stabilizing (L 0) or estabilizing (L 0). Note that the magnitue of the transient flow force increases as the spool moves quickly, such as in high banwith applications. In the absence of viscous amping effects an external centering springs, the spool ynamics are given by: M s ẍ v F st F tr u (8) where M s is the mass of the spool, u is the stroking force prouce by the electromechanical/solenoi actuator, F st is the steay state flow force in 3, an F tr is the unsteay flow force in 7. If we efine the effective spring rate an amping coefficient to be: B f ªLC w P s x v x v L P (9) then, Eq. 8 becomes: K f ª2C C v w cos P s x v x v P L, (10) M s ẍ v B f ẋ v K f x v u. (11) Because of the stabilizing effect of the steay-state flow force, the spring rate K f is always positive. However, the sign of the amping coefficient B f epens on the amping length L. In a conventional single-stage flow-control valve, L 0 by esign so that B f 0. The electromechanical/solenoi actuator must be powerful enough to overcome both the steay state an the transient flow forces to move the spool. As remarke earlier, these forces become more significant as both flow rate an banwith increase. Because the force an power capabilities of the electromechanical/solenoi spool actuator are limite, the performance in terms of flow rating an banwith of single stage valves may be limite. The basic assumptions use in the erivation of the transient flow force, i.e., incompressible flow, quasi-static analysis are also use by past authors 1,4,2,5,3. The moel that uses these assumptions an the orifice equation 2 has been use to preict spool instability in 3. In 5, quasi-static analysis is also use to stuy the relative contribution of transient an steay-state flow forces, except that CFD techniques are use to moel, more accurately, the steay-state flow/spool isplacement relationship for the pilot stage of a relief valve. It shoul be note that given the quasi-steay assumption, the information provie by a etaile flow pattern ue to complex valve geometry plays the role of moifying 1 the steay flow force in 1 an 3, an hence the equivalent spring constant K f in 10 ; an 2 the flow/spool isplacement relationship 2. As long as the flow/spool isplacement relationship is monotonic, the effect of transient flow force will still take the form of a amping term. An orifice equation that takes into account transient flow is propose in 7. This moel is compare to the usual orifice equation 2 for an electrohyraulic valve in 8. However, it is foun in 8 that the transient moification has negligible effect on the overall performance of that particular valve. In the erivation above, the flow is assume to be non-viscous. Flui viscosity plays two roles in the above analysis. First, in 5, the spool sleeve in aition to the lans contributes to the force on the RHS. If the viscous force on the spool sleeve is moele to be proportional to the flow rate which via 2 is in turn proportional to the spool isplacement, then flui viscosity will increase or ecrease the effective spring constant K f in 10 epening on the amping length. Secon, viscosity contributes to the rag on the lans as the spool moves in the sleeve. This effect contributes to an increase in the amping coefficient B f in 9. Intereste reaers may therefore interpret the simulation results below by mentally offsetting the values for K f an B f accoringly in orer to unerstan the effects of complex flow pattern, an of flui viscosity. In the next section, we present simulation case stuies to investigate the potential performance improvement if the single-stage valve is mae unstable by choosing L 0, B f 0 in 9. The hypothesis is that the transient flow forces associate with the instability can be utilize to overcome the steay state flow forces, thus alleviating the limitation of the electromechanical/solenoi actuator. This will help expan the use of the inherently cheaper, an more reliable single-stage valves into higher performance an higher flow rate applications. III Simulation Case Stuies To investigate the hypothesis above, simulation case stuies are conucte in which the force exerte by a ouble-ene cyliner actuator is to be controlle. We consier the situation in which the Journal of Dynamic Systems, Measurement, an Control MARCH 2002, Vol. 124 Õ 185

4 cyliner is constraine e.g., pushing against a rigi wall. The setup is illustrate in Fig. 3. A single-stage irect-acting control valve, configure either in the stable configuration (L 0,B f 0) or in the unstable moe (L 0,B f 0) is use to control the actuator. The ynamics of the system are given by Ṗ L Q t C w V cyl P s sign x v P L x v (12) V cyl M s ẍ v B f ẋ v K f x v u (13) where is the compressibility, V cyl is volume of the cyliner, M s is the mass of the spool, P L is the loa pressure, B f an K f are the flow force inuce effective amping an spring rate given in 9 an 10. In these stuies P s 20.6 MPa 3000 psi, 1.03 GPa 150,000 psi an 872 kg m lbm in. 3. Coefficients in 9 10 are w m 1.35 in., C 0.6, C c 1. These are typical values in applications. The amping lengths of L m 1 in. were teste. We assume that the input to the system is the stroking force u(t) for the valve provie by an electromechanical actuator. Since the cyliner is constraine, the piston force is F p A p P L where A p is the cyliner area so that the control problem is equivalent to controlling the ifferential pressure P L across the two valve chambers. The stable an unstable valve configurations are compare in two settings: What is the optimal step response that the system is capable of achieving when the electromechanical actuator u(t) is limite? What are the force an power requirements of the electromechanical actuator in orer for the piston force to track the esire sinusoial profiles? The first situation aims to illustrate the ifferences of the capabilities of the stable an unstable valves when the electromechanical/solenoi actuators have limite capabilities. The secon situation aims to illuminate the ifferences in the requirements for the electromechanical/solenoi actuators for the two valves to achieve the same level of banwith performance. It is emphasize that the simulation stuies are intene to illustrate only the potential performances of the two valves an their requirements for electromechanical actuators. The actual closeloop performances, which woul epen on the controller esign, is not consiere here. A Step Responses Uner Input Constraints. Suppose that the stroking force u(t) in 13 is constraine. We wish to compare the optimal feasible responses for the system given by 12, 13 for various stable an unstable configurations specifie by amping Fig. 3 Actuator-valve setup consiere in the simulation stuies Fig. 4 Optimal step responses for the stable LÄ m 1 in an unstable configuration LÄÀ m À1 in with upper actuator constraint ū of N 50 lb top an N 200 lb bottom lengths L to achieve a esire pressure of P es 6.89 MPa 1000 psi quickly, from an initial pressure of 0 Pa. The optimal control problem is to fin u(t), t 0,T f so that min J u ª 0T f Pes P L t 2 t subject to u u t ū (14) where u an ū are, respectively, the impose lower an upper saturation limits on the stroking force. All initial conitions are assume to be 0. Minimizing J minimizes inirectly the response time. Other cost functions that reflect the general iea of a goo step response can also be use. To compute the optimal solutions, the optimal control problem was iscretize using a sampling time of 2.5 ms an converte to a parameter optimization problem. The final time T f 1.5 s in 14 was juge aequate for this problem. Matlab s Ver R11 Mathworks Inc., MA constraine optimization function, fmincon was then use to compute the 1.5/ optimal control values. Note that the optimal responses are open-loop solutions which are intene only to illustrate the capabilities of the valves. To achieve this actual optimal response, close-loop control is neee. The optimal controls were compute for both the stable (L m 1 in an unstable (L m 1 in configurations uner two ifferent actuator saturation limits (u,ū) ( 1.3 kn ( 300 lbf), N (50 lbf)) an (u,ū) ( 1.3 kn ( 300 lbf), N (200 lbf)). A less severe negative constraint on the electromechanical actuator is impose to emphasize the cost of positive energy generating actuation. For this particular task, negative actuation correspons to braking, which may have potentially less limiting ways of implementation. The respective optimal step responses are shown in Fig. 4. With the unstable valve, P L rises to the esire level P es faster than with the stable valve. The 100 percent rise times are 17 percent an 24 percent shorter for the unstable valve than for the stable valve for the two sets of saturation limits Table 1. The optimal stroking forces u opt (t) are shown in Fig. 5. For both the stable an unstable configurations, the optimal control is initially saturate at the upper limits (ū). This is followe by a perio of eceleration. In the case of the unstable configuration, the maximum allowable braking force occurs at some times for both sets of upper actuator limits ū N 50 lbf an N 186 Õ Vol. 124, MARCH 2002 Transactions of the ASME

5 Table 1 100% use time for the stable an unstable valve for two sets of ū Fig. 7 Power Consume by the valves Fig. 5 Optimal control effort for stable an unstable valve configurations 200 lbf. The spool s phase portraits of (x,ẋ ) Fig. 6 reveal that the spool inee attains higher velocities in the unstable case than in the stable case. This is especially the case when the actuator limitation is more severe (ū N (50 lbf)). In this situation, the spool spee is not high enough to require full braking. The higher achievable spool spees contribute to the faster responses. This confirms the expectation that the spool is more agile in the unstable configuration than in the stable configuration. The electromechanical actuator power inputs (u ẋ ) an the work one by the stroking actuator are shown in Fig. 7 an Table 2 respectively. Notice that using the unstable valve, the positive work performe by the stroking actuator is reuce to less than 1/3 require for the stable valve. In terms of negative braking power an work, for both saturation limits, these are larger in magnitue for the unstable valve than for the stable valve. The peak positive power for the unstable valve is larger than for the stable valve when ū N 50 lbf. This is attribute to the fact that the peak power for the unstable valve occurs when the actuator is in saturation an when the spool spee is very high. It is interesting to note that the peak power for the stable configuration when ū N 50 lbf is use for closing the valve. Note also that both the rise time improvement Table 1 an the positive work reuction Table 2 are more pronounce when the saturation limit is more severe. The effect of amping length L on the rise times of both the unstable an the stable valve are investigate next. The optimal stroking force, u(t) uner saturation limits of (u,ū) ( 1.3 KN ( 300 lbf), N(50 lbf)) for various amping lengths were compute Fig. 8. We observe from Fig. 8 that as the amping length increases, the valve takes longer to reach the esire pressure, P es. In the stable configuration (L 0), the 100 percent rise times increase rapily with larger positive amping lengths. In the unstable configuration (L 0), the rise times ecrease as the amping length becomes more negative. However, a lower limit is achieve for very large negative amping lengths e.g., compare L m 7 in an L m 4 in. The limite return in benefit is ue to the fact that for large negative amping lengths, the optimal control is limite by the lower braking capability of the electromechanical actuator. B Tracking Sinusoial Force Profiles. We now consier the task of controlling the system in Eqs to track various sinusoial ifferential pressure or equivalently actuator force profiles. The goal here is to compare the force an power requirements for the electromechanical stroking actuators to achieve a certain level of performance in terms of banwith. To o this, we Table 2 Work one by the stroking actuator Fig. 6 Phase Plots, x võs ẋ Journal of Dynamic Systems, Measurement, an Control MARCH 2002, Vol. 124 Õ 187

6 Fig. 8 Damping length versus rise times 19 is true, then we achieve our esire pressure tracking as S 1 (P L ) woul converge to 0. To achieve this, we introuce a secon sliing surface, S 2 (x ) as follows: S 2 x ẋ ẋ es x x es. (20) As the ynamics of the spool 13 are secon orer, it is necessary to ifferentiate 20 once to etermine the control, u. However, this is not possible as 19 is iscontinuous in x ue to the sign function. Therefore, we propose a uniformly boune C smooth approximation to the sign function, ŝ(x ), which is given below: ŝ x ª 2 arctan c x (21) where c 0. It can be verifie that the approximation error, ŝ(x ) sign(x ) is uniform in x an can be mae arbitrarily small by choosing c to be sufficiently large. Now, the ynamics of the esire spool position, x es in Eq. 19 is moifie to be: x es V cyl C w 1 S 1 Ṗ es L. (22) 1 P s ŝ x v P L o not assume that the electromechanical stroking actuator is limite, an etermine an compare the control u(t) for both the stable an unstable valves so that P L t P es L t P sin 2 f t (15) for various frequencies f Hz. To etermine the require control effort, a multiple sliing surface control law 9 was esigne to achieve near perfect tracking of the esire force ifferential pressure trajectories. This is just a convenient way to obtain the control effort or the so calle inverse ynamics require to achieve tracking of 15. In principle, any controller that achieves asymptotic tracking shoul utilize the same control effort after convergence. The multiple sliing surface controller is esigne as follows. Define the loa pressure sliing surface as follows: S 1 P L P L P L es. (16) We etermine the esire spool position trajectory, x es, so that the surface efine in 16 is exponentially convergent. Assume for the time being that we can irectly control the spool isplacement x. This assumption will be remove later by introucing a secon sliing surface. Hence, we nee to choose x so that S 1 (P L ) converges to zero, exponentially. Utilizing the pressure ynamics, 12, we obtain: t S 1 P L C w V cyl If we choose x to be: x x es V cyl t S 1 P L Ṗ L Ṗ L es, (17) C w P s sign x P L x Ṗ es L. (18) 1 S 1 Ṗ es L, (19) 1 P s sign x P L where 1 R is the gain to the sliing surface, it is easy to see that the surface S 1 (P L (t)) an hence P L (t) P L es (t) converge exponentially to 0. Unfortunately, the actual control is u(t), the spool ynamics are given by 13, an so x cannot be controlle irectly. Notice from 13 that if x is controlle in such a way, using u, so that Differentiating 20, we have, t S 2 x ẍ ẍ es ẋ ẋ es. (23) Using the spool ynamics 13, we have: t S 2 x 1 B M f ẋ K f x u ẍ es ẋ ẋ es. s (24) Therefore, the control u, u B f ẋ K f x M s ẍ es ẋ ẋ es 2 S 2 x, (25) where 2 0, guarantees that the secon sliing surface, S 2 (x )is exponentially convergent. In Eq. 25, the erivatives ẋ es es an ẍ were analytically etermine from Eq. 22. Because the ynamics of the first sliing surface 16 is exponentially stable, by choosing c in 21 appropriately, it can be shown that the P L (t) P es L (t) can be arbitrarily close to 0. Having esigne the controller, simulations were performe to compare the power consumption of the valve in the stable configuration amping length L m 1 in an in the unstable configuration (L m ( 1 in)) at ifferent frequencies. We emphasize that as long as the esire force pressure profile is tracke, the control effort require after convergence shoul not epen on the particular control law. The multiple sliing surface controller is use simply because it is easy to esign an implement. Notice from Figs. 9 an 10 that the multiple sliing surface controller is able to achieve loa pressure tracking at both low an high frequencies for both the stable an the unstable valves. The trajectories for control effort an the power input to track a 10 Hz an 140 Hz sinusoial pressure profile are shown in Figs. 11 an 12. To track the sinusoial signals, both the stable an the unstable valve require near sinusoial control inputs. It is interesting to note that the magnitues of the control effort are the same for both the stable an the unstable configurations. The magnitues of the control effort in both cases increase with frequency. There is however a phase ifference between the stable an unstable configurations. It can be shown that compare to the stable configuration, the unstable configuration emans a smaller control effort than the stable configuration when x an ẋ are of the same sign, an it emans more force when x an ẋ are of ifferent signs. This is consistent with the power input profiles in Figs. 11 an 12, which show that in the unstable configuration, the power input 188 Õ Vol. 124, MARCH 2002 Transactions of the ASME

7 Fig. 9 Sinusoial tracking at 10 Hz Fig. 12 Effort an power neee to track a 140 Hz sinusoial trajectory Fig. 10 Sinusoial tracking at 140 Hz is mainly negative electromechanical actuator absorbing energy ; whereas for the stable configuration, the power input is mainly positive electromechanical actuator generating energy. This observation persists for all frequencies Figs. 13 an 14. At high frequencies, the positive power requirement for the unstable valve is only about 20 percent of that of the stable valve. At low frequency e.g., 10 Hz, see Fig. 11, negligible positive power is require for the unstable valve. On the other han, the magnitue of the negative power an work by the electromechanical actuator are larger for the unstable valve than for the stable valve. Thus the electromechanical actuator acts like a brake Figs. 13 an 14. As the frequency increases, such as in high banwith applications, the ifferences in the positive an negative power requirements between the stable an unstable configurations become very prominent. IV Discussion an Conclusions The simulation case stuies presente in Section III inicate that unstable valves are more agile an can potentially achieve a significantly faster response than the stable valve uner the same Fig. 11 Effort an power neee to track a 10 Hz sinusoial trajectory Fig. 13 Maximum positive power consume by the stable an unstable valves at various frequencies Journal of Dynamic Systems, Measurement, an Control MARCH 2002, Vol. 124 Õ 189

8 be use, close-loop feeback control is a necessity. With avances in control theory an technologies, an the fact that embee sensing an feeback control are alreay prevalent in electrohyraulics, this might not be a significant impeiment. Although, as emonstrate in the sinusoial responses, the force magnitue require of the stroking actuator may not necessarily be small, unstable valves ten to require smaller positive power. When larger stroking forces are require, they are power absorbing an are associate with braking. These results suggest that if valve instability is to be exploite, the stroking actuators for the unstable valves shoul be esigne to act preominantly as controllable brakes. Fig. 14 Maximum negative power consume by the stable an unstable valves at various frequencies stroke actuator force limitation. This offers the opportunities for single-stage valves to improve on their performances towar those offere by multistage servo valves. The unstable valve also requires significantly less positive power input than what a stable valve oes. Moreover, these benefits become more significant at high frequencies or at low actuator capabilities. These are exactly the situations where performance improvements are especially neee of single-stage valves. Of course, if unstable valves are to References 1 Hebert E. Merritt, 1967, Hyraulic Control Systems, Wiley, NY. 2 D. McCloy an H. R. Martin, 1973, The Control of Flui Power, Wiley, NY. 3 J. F. Blackburn, G. Reethof, an L. L. Shearer 1960, Flui Power Control, MIT Press, Cambrige, MA. 4 Lee, S-Y., an Blackburn, J. F., 1952, Contributions to hyraulic control 2-transient flow forces an valve instability, ASME Machine Design Division, Vol. 74, pp M. Borghi, M. Milani, an R. Paoluzzi, 1998, Transient flow force estimation on the pilot stage of a hyraulic valve, Proceeings of the ASME-IMECE FPST-Flui Power Systems & Tech., Vol.5, pp D. Wang, R. Doli, M. Donath, an J. Albright, 1995, Development an verification of a two-stage flow control servovalve moel, Proceeings of the ASME-IMECE FPST-Flui Power Systems & Tech., Vol.2, pp Funk, J. E., Woo, D. J., an Chao, S. P., 1972, The transient response of orifices an very short lines, ASME J. Basic Eng., 94, No. 2, pp Martin, D. J., an Burrows, C. R., 1976, The ynamic characteristics of an electrohyraulic servovalve, ASME J. Dyn. Syst., Meas., Control, 98, No. 4, pp Won, M., an Herick, J. K., 1996, Multiple-surface sliing control of a class of uncertain nonlinear systems, Int. J. Control, 64, pp Õ Vol. 124, MARCH 2002 Transactions of the ASME