Dynamic Redesign of a Flow Control Servovalve Using a Pressure Control Pilot 1

Transcription

1 Perry Y. Li Department of Mechanical Engineering, University of Minnesota, 111 Church St. SE, Minneapolis, MN Dynamic Redesign of a Flow Control Servovalve Using a Pressure Control Pilot 1 In this paper, the dynamic performance of an unconventional two-spool flow control servovalve using a pressure control pilot is analyzed. Such valves are less expensive than typical servovalves but also tend to be limited in their dynamic performance. Based on a previously developed eight state nonlinear model, we develop a simplified linear model which is able to capture the essential dynamics of the valve. Using root locus analysis method, the limitation in dynamic performance is shown to be due to a zero introduced by the structure of the interconnection of the subsystems. Design parameters that move the zero further to the left half plane, and do not adversely affect other steady-state criteria are identified. The effectiveness of these parameters to improve the dynamic performance is demonstrated. This analysis demonstrates how the structure of the interactions between subsystems in a dynamic component, such as a hydraulic valve, can critically limit the dynamic performance of the component. DOI: / I Introduction Most designs of servo-flow control valves 1 consist of a single spool boost stage, a nozzle flapper pilot, and a feedback wire. These valves have very high performance but tend to be expensive because of the stringent manufacturing tolerances and the complicated assembly process. A less common, commercially available alternate design Fig. 1 consists of a pressure control pilot stage and a boost stage that uses two separate spools to independently meter flow into and out of the valve. Since the critical dimensions are easier to adjust and a feedback wire is not used, such valves are easier to manufacture and assemble. Consequently, they tend to be cheaper. Readers are also referred to 2, in which an experimentally validated complete physical model is presented, for a more detailed discussion of the advantages of the unconventional two-spool servovalve. Despite these advantages, the unconventional two-spool servovalve design tends to have lower dynamic performance in terms of bandwidth compared to the conventional servovalve design utilizing a single spool and a feedback wire. For example, the valves studied in 2,3 have bandwidths between Hz whereas conventional servovalves of similar flow ratings can have bandwidths of over 100 Hz. It would therefore be advantageous if dynamic response of the two-spool design can be improved. In this paper, we study the unconventional two-spool design so as to understand the nature of the performance limitation, and to suggest design modifications for potential performance improvements. The experimentally validated eight state nonlinear physical model derived in 2 consists of the interconnection of three subsystems. This model is similar to the one constructed by Lin and Akers previously 3. Using this model, we develop a simplified five state linear model that retains the interconnection structure as well as the dominant dynamics. The reduced model reveals a puzzling aspect of the valve dynamics in that each of the three subsystems has bandwidth at least an order of magnitude higher than the bandwidth of the complete model. Using simple root locus arguments, it is found that the way in which the subsystems are interconnected creates a zero which causes the bandwidth of the 1 Most of this work was presented at the ASME IMECE-2001 Symposium on Control and Modeling of Fluid Power Systems DSC Division 2B-1 in New York City, November Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the Dynamic Systems and Control Division February 1, Associate Editor: N. Manring. interconnected system to be significantly lower than the individual subsystems. Based on this insight, several system parameters that can potentially improve the dynamic performance without adversely affecting the steady-state performance such as flow gain are identified, and their effects demonstrated. The rest of the paper is organized as follows. In Section II, we formulate a simplified model of the two-spool flow control servovalve. The interconnections of the three linearized subsystems are studied using root locus techniques in Section III. Section IV presents the effort to optimize the performance by applying the insights gained in Section III. Sections V and VI contain discussion and concluding remarks, respectively. II Simplified Model of the Two-Spool Flow Control Servovalve The unconventional flow control servovalve shown in Fig. 1 uses a two-spool boost stage and a pressure control flapper-nozzle pilot stage. The two stages are separated by a simple transition plate and connected via two pressure chambers. The design philosophy of the valve is as follows. The pressure control pilot stage generates a differential pressure between the two fluid chambers adjacent to the flapper, determined by the current input to and the torque generated by the electromagnetic torque motor. For example, if the torque motor applies a counter clockwise torque to the flapper, the flapper displaces to the right. This tends to increase the pressure P 2 and to decrease the pressure P 1. The differential pressure acts on the two ends of the two spools in the boost stage. Since the spools are spring centered, their equilibrium displacements will be roughly proportional to the differential pilot pressure and inversely proportional to the combined mechanical and flow force induced spring stiffness. Flows into and out of the valve are separately metered according to the displacements of the two spools. A Review of Full State Model. The servovalve can be considered an interconnection of three subsystems: 1 the pilot subsystem whose states are the flapper displacement x f left to right positive and velocity ẋ f ; 2 the pressure chambers whose states are the chamber pressures P 1 and P 2 ; and 3 the boost stage spool dynamics whose states are the displacements and velocities of the two spools x a, ẋ a and x b, ẋ b. Therefore, the total number of states is eight. Following 2, the dynamics of the pilot subsystem can be represented by: 428 Õ Vol. 124, SEPTEMBER 2002 Copyright 2002 by ASME Transactions of the ASME

2 well defined when x 0. Therefore, depending on the sign of the spool displacement, the transient flow force may introduce negative damping effects. The third subsystem is associated with the dynamics of the pressures P 1, P 2 in, respectively, the upper and the lower fluid chambers connecting the pilot stage and the boost stage spools. Its dynamics are given by: Ṗ 1 Q 1 P 1,x f V 1 t V 1 t ; Ṗ 2 Q 2 P 2,x f V 2 t. (4) V 2 t Here, Q 1 (P 1,x f ) and Q 2 (P 2,x f ) are the total flows into the upper and lower chambers, V 1 (t) and V 2 (t) are the volumes in the chambers, and is the compressibility of the fluid. Q 1 (P 1,x f ) and Q 2 (P 2,x f ) are comprised of the flows from the pilot supply orifice, leakage past the nozzle, and to a small extent, leakage past the spools: for i 1 and 2, Fig. 1 A two-spool flow control servovalve using a pressure control pilot state. M p ẍ f B p ẋ f K px f A n P 1 P 2 4 C 2 df x fo x f 2 P 1 x fo x f 2 P 2 g x f,i (1) where M p, B p, and K p are, respectively, the combined inertia, damping, and mechanical stiffness of the flapper, x fo is the null nozzle-flapper gap when x f 0, and C df is the discharge coefficient of the flapper-nozzle, A n is the nozzle area. The first and second terms on the right-hand side correspond, respectively, to the pressure and the flow induced forces at the nozzle, and g(x f,i), which is a highly nonlinear function see 2 for details, represents the force on the flapper generated by the electromagnetic torque motor with input current i. The dynamics of the two boost stage spools are given by: where x a and x b represent the upward displacements of spools A and B, respectively, M s is the spool inertia, B s is the viscous damping coefficient, 2K s is the total stiffness of the two springs above and below the spools Fig. 1, A s is the spool area. The steady-state and transient flow forces manifest themselves as spring forces and positive/negative damping forces with K f(, ) 0, and B f(x,p) 0 when x 0, B f(x,p) 0 when x 0, and not (2) (3) Q i C d0 A 0 2 P sp P i C df D n x fo x f 2 P i leakage i where sign is used for i 1 and sign is used for i 2, P sp is the pilot supply pressure which is usually lower than the supply pressure for the boost stage, A o is the area of the orifice to the supply pressure, C d0 and C df are the discharge coefficients of the orifice to the supply and the gap between the flapper and nozzle. The first two terms in 5 are monotonically decreasing functions of P i. Thus, they provide at least local exponential stability for the pressure dynamics 4. Notice also the pilot stage communicates with the chamber pressures via Q 1 and Q 2 since they depend on the flapper displacement x f. On the other hand, the pressure chambers are affected by the boost stage spool dynamics via V 1 (t), V 2 (t), V 1(t), and V 2(t) in 4 since (5) V 1 V 1o A s x a A s x b, (6) V 2 V 2o A s x a A s x b (7) V 1 V 2 A s ẋ a A s ẋ b (8) where V 1o and V 2o are the chamber volumes when the spools are centered (x a x b 0). For details of the model, readers are referred to 2. B Reduced Order Linear Model. In order to obtain meaningful design information, we consider a reduced fifth-order linear model. This is achieved as follows: 1 In the spool system, we consider only the total spool displacement not the displacements of the individual spools t x a t x b t. 2 In the pressure chamber system, we consider only the differential pressure P t P 1 t P s t. 3 The transient component of the flow forces and leakage flows past the spools are ignored. 4 The pilot stage dynamics in 1, the spool dynamics in 2 3 and the differential pressure dynamics obtained from 4 are linearized at the equilibrium condition given by flapper displacement x f 0, chamber pressures P 1 P 2.P, spool displacements x a x b 0, spool velocities ẋ a ẋ b 0, chamber volumes V 1 V 2 (V 10 V 20 )/2.V, and work pressures P a P b P s /2 where P s is the boost stage supply pressure. The rationale for ignoring the mean pressure (P 1 P 2 )/2 is that near the equilibrium, the chamber pressures P 1 and P 2 affect the pilot and spool dynamics 1, 2, 3, predominantly through (P 2 P 1 ), and individually only through second order effects. Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 2002, Vol. 124 Õ 429

3 Because the linearized dynamics of the spools differential displacement ( xªx a x b ) are given by an exponentially stable system driven by the difference in the flow forces, M s x B s x 2K s K f x difference in flow forces (9) x would be small since the difference in flow forces are small. The dynamics of x are therefore also neglected. The resulting reduced linear models for the pilot, chamber pressure and spool subsystems are, respectively: Pilot subsystem M p ẍ f B p ẋ f K p x f A n B P G i (10) Pressure chamber subsystem Spool subsystem where P V P 2 V x f 2 A s V t (11) M s B s 2K s K fs 2A s P t (12) Gª g i x f 0,i 0, K fs ªK f 0,P s /2, K p K p 16 C 2 df x fo P g, Bª C x f x 2 df x 2 fo, f 0,i 0 Q 1 x f Q 2 x P,x f 0 f, P,x f 0 Q 1 P 1 Q 2 P P,x f 0 2 P,x f 0 (13) B,, and are all positive quantities. The expression for K p shows that the nozzle flow forces and the magnetics tend to offset the mechanical stiffness K p of the flapper. The term A n B in 10 is the apparent nozzle area of the flapper-nozzle upon which the pressure and the nozzle flow forces act. From 11, is the convergence rate of the pressure chamber normalized by the inverse of the chamber capacitance, /V. To verify that the linearized model in indeed captures the dominant dynamics of the valve, the responses to the step current input are simulated for an input step size of i 20 ma 50% full range using the complete nonlinear model in 2 and the linearized model in The step responses of the differential pressure P(t) are very close Fig. 2. The 64% risetime for the linearized and nonlinear models are 8.1 ms and 8.2 ms, respectively. The similarity among the responses of the model in and of the full nonlinear model in 2 suggests that the dynamics of the system represented by the interconnection between the pilot, pressure, and spool dynamics are well captured by the reduced order linear dynamics. Each of the pilot 10, differential pressure 11, and the boost spool 12 subsystems are stable. Using physical parameters of the valve that are verified in 2, it can be shown that the pilot subsystem has a natural frequency of n,p 3037 rad/s, and a damping ratio of p 0.91; the differential chamber pressure subsystem has an eigenvalue of /V 1599 rad/s; the boost spool subsystem has a natural frequency of n,s 1861 rad/s with a negligible damping ratio of s The eigenvalues of the combined system are at rad/s, jrad/s, jrad/s. We point out that the dominant pole is at rad/s which is consistent with the fact that the 64% rise-time of the linearized model is 8.2 ms. In order to improve the dynamic performance of the valve, the dominant pole must be moved further into the left half plane. Fig. 2 Differential pressure response to a 20 ma step input current: full nonlinear model and simplified linearized model. Also shown is a first order simplified model to be discussed in Section III-B. III Analysis for Performance Limitation We now proceed to analyze the fifth-order linearized model to understand why the bandwidth of the valve is relatively low, whereas the natural frequency of each individual subsystem is at least an order of magnitude higher. Is the limited performance due to the fact that the spools are too lightly damped ( s 0.025)? Or, is the fluid capacitance in the pressure chamber the reason? As we shall see, neither the spool damping nor the chamber capacitance is important. The key turns out to be the structure of the interconnection between the pressure chamber, the pilot, and the spool subsystems. A Fifth-Order Root Locus. The pilot, chamber pressure, and the boost spool subsystems are connected in a closed-loop manner as shown in Fig. 3 with K 1 K 2 1. To understand the effect of the interconnection, we apply Evan s root locus technique 4 to investigate how the closed-loop eigenvalues migrate as the parameters in the system are varied. Consider first the inner loop in Fig. 3 which is the interconnection between the pilot and the differential pressure subsystems. Figure 4 shows the loci of the closed-loop poles of the inner loop as the fictitious gain K 1 is varied from 0. K 1 1 corresponds to the gain in the actual loop in the present valve design. Figure 5 shows the locus of the closed-loop poles of the outer loop system in Fig. 3 as the fictitious gain K 2 is increased. The set of poles at K 2 1 are the actual poles in the valve. As expected, when K 2 1, the dominant pole is Fig. 3 Block diagram for root locus analysis with the upper feedback loop as the inner loop. The actual valve dynamics are obtained when K 1 ÄK 2 Ä Õ Vol. 124, SEPTEMBER 2002 Transactions of the ASME

4 Fig. 4 Root locus diagram of the pilotõchamber differential pressure subsystem as K 1 increases from 0\ in Fig. 3 Fig. 6 Partial root locus diagram of outer loop with inner loop being the spoolõchamber system with K 2 Ä1 as K 1 increases from 0\. Two other poles on the far left are not included. at p rad/s which is the reason why the dynamic performance is limited. Notice that the real parts of all the other eigenvalue locations are significantly more negative. Figure 5 shows that for the present valve design i.e., K 2 1, the pole locations are well approximated by the asymptotic behaviors of the root locus. These are governed by the open-loop pole and zero configurations. In particular, the dominant pole at p is being attracted to the zero at 0. Two other poles are close to the zeros at the pilot s open-loop pole locations. The remaining two poles are also close to the two 90 deg asymptotic branches. Since the damping in the spools can only affect the asymptotes slightly, contrary to our initial speculation, the negligible damping of the spool does not contribute significantly to the relatively poor dynamic performance of the valve. Rather, the reason is due to the zero at the origin. This zero is present because the spool subsystem interacts with the chamber pressure dynamics via. As far as the loop gains K 1 and K 2 are concerned, from Fig. 5, decreasing K 2 delays the migration of the dominant pole to the zero at the origin. Similarly, if we had interconnected the differential pressure and the spool lower loop first before connecting the pilot system in Fig. 3 and derived the corresponding root loci, then we would have noticed that by increasing K 1 in Fig. 3, the partial root locus is shown in Fig. 6. Notice that the dominant pole migrates from an open loop pole near the origin toward the left half plane as K 1 increases. B Reduced Order Root Locus. The root locus analysis above indicates that the four complex poles of the valve design can be approximated by their asymptotic behaviors. In addition, the complex poles originate from the poles associated with the pilot and the spool subsystems. This suggests that we may approximate the behavior of the dominant eigenvalue of the valve by considering the pilot and the spool systems as quasi-static systems, i.e., by assuming that at each instant, the spools and the flapper are in static equilibria with the instantaneous differential pressure. The resulting configuration is given in Fig. 7. Figure 2 shows that the step response for the differential pressure dynamics matches well with the eight-state nonlinear model and the fivestate linear model. Indeed, the closed-loop pole of the reduced order quasi-static system in Fig. 7 is p rad/s which is very close to the actual dominant pole rad/s. The characteristic equation for the system in Fig. 7 is given by: 1 V 2 A n B K p S 4A 2 s S 2K s K fs 0. (14) The root locus for positive /V inverse chamber capacitance, i.e., the ratio between the fluid compressibility and the chamber volume is given in Fig. 8 which shows that the performance of the system would be limited by a zero at Fig. 5 Root locus diagram of the spool subsystem and the pilotõchamber differential pressure subsystem as K 2 increases from 0\ and K 1 Ä1inFig.3 Fig. 7 Block diagram with the pilot and the spool subsystems approximated by their static systems Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 2002, Vol. 124 Õ 431

5 2 A n B K p 2 4A / s 2K s K fs. (15) In the current valve design, the zero of the quasi-static model is at rad/s. Since the pole of the quasi-static model is already at rad/s, the performance cannot be significantly improved by increasing /V. Although the presence of finite pressure chamber dynamics is essential for the existence of the slow valve dynamics, the values of the chamber volume, the compressibility, or capacitance i.e., V / do not matter significantly. Rather, the performance limitation is determined by the feedback structure itself. The performance limitation can be alleviated if the zero in 15 can be moved further to the left. Consistent with the previous analysis, this can be achieved by increasing the pilot loop gain and by decreasing the spool loop gain i.e., K 1 and K 2, respectively, in Fig. 3. IV Fig. 8 Root locus of the reduced order model in Fig. 7 Dynamic Redesign A Dynamic Performance. Since the dominant pole of the system limits the performance of the valve, it must be moved further to the left half plane to improve dynamic performance. According to the root locus analysis in Section III-A, this can be achieved if the loop gain in the upper loop is increased, and the loop gain in the lower loop is decreased. From the reduced order root locus analysis in Section III-B, performance is limited by the open-loop zero of the reduced order system. Since the dominant pole location of the valve will be close to the zero location, the zero location must be moved to the left if the dynamic performance is to improve. The expression for this blocking zero is: 2 G p, (16) 2A s G s where and are defined in 13 and are associated with the convergence rate of and the flapper input gain to the pressure chamber dynamics 11 ; G p ªA n B/K p and G s ª2A s /2K s K fs are, respectively, the steady-state ratio of the flapper displacement x f to the differential pressure P, and steady-state ratio between the total spool displacement and P. Therefore, from 16, the key design parameters in the various subsystems are: Nozzle-flapper: The apparent nozzle area A n B, which can be modified by changing the physical size of the nozzle and the gap between nozzle-flapper. is the sensitivity of the nozzle flow to flapper displacement which can also be modified by changing the nozzle diameter. The apparent flapper stiffness K p is affected by the mechanical stiffness, the negative magnetic stiffness, and the negative nozzle flow induced stiffness. Boost stage spools: The spool area A s, and the centering spring stiffness K s. Pressure Chambers:, which is the convergence rate of the differential pressure normalized by the inverse chamber capacitance, /V. Shifting the zero in 16 to the left can be achieved by 1 modifying the flapper nozzle design so as to increase G p increase A n B, decrease K p, increase ; 2 modifying the boost stage Fig. 9 Consequences on the static criteria when various design parameters are used to improve the dynamic performance. Direction of the arrow indicates the direction of proposed change. X represents significant degradation, represents some improvement. For each column, the variables in all other columns are assumed to be constant. spool design so as to decrease A s G s decrease A s, increase K s ; or 3 by modifying the open-loop convergence rate of the pressure chamber dynamics so as to increase. B Steady-State Criteria. In addition to their effects on the dynamic response of the valve, it is important also to evaluate the effects of these design parameters on the operating pressure and motion ranges, and ultimately the flow gain of the valve. We determine these from the D.C. components of the transfer functions assuming 2 / is large. Differential pressure gain: P s 2 / G p G I s 1 2 / G s 0 p A n B G A n B Flapper displacement gain: x f s 2 P s G I s 2 A I s s 0 s 0 n B Spool displacement gain: Flow gain: Q L s I s s 0 s I s s 0 P s G s G sg I s A s 0 n B C dw P s s 2 I s s 0 C dw P s GG s 2 A n B where G g/ i for the torque motor, P s is the supply pressure for the boost stage. For a given input current, it is generally preferable that P and x f be small, and the flow Q L be large. Because the pilot supply pressure P sp is limited, large P excursion reduces the operating range. On the other hand, large x f generally requires large magnetic air gap, which can complicate the design of the torque motor. A large flow gain is desirable so that a small input current can be used to control large flows. The consequences on these criteria when the various parameters are used to improve the dynamic performance are summarized in Fig. 9. If the apparent nozzle area A n B is increased, for the same input current, the differential pressure and the steady-state flapper displacement will be decreased. Unfortunately, since the flow gain has also been proportionately reduced, more force is required from the torque motor to achieve the same flow. Similarly, decreasing G s of the spool system will decrease the flow gain significantly. Increasing will have the adverse effect of increasing flapper motion. The design parameters that do not adversely affect the steady state criteria significantly are the apparent flapper stiffness K p, the spool area A s while maintaining G s constant and of the nozzle-flapper. Of these three parameters, in fact only K p and A s can be used independently to improve dynamic performance without affecting the steady state performance., which is the sensitivity of the chamber flow to flapper displacement, cannot be varied to any significant degree without affecting the apparent nozzle area A n B. Indeed, even a 5% increase in necessitates an increase in 432 Õ Vol. 124, SEPTEMBER 2002 Transactions of the ASME

6 Fig. 11 Step response of the valve modified by K p]0.75k p K p ]0.54K p. Note that 64% rise time has been reduced from 8.2msto4.9ms. Fig. 10 Step response top and current-flow relationship bottom of the valve modified by A s ]A s Õ2, K s ]K s Õ2 the actual nozzle area A n by over 140%! This can potentially increase the apparent nozzle area A n B which in turn decreases the steady-state flow gain. C Reducing the Spool Area A s. According to the expression of the zero in 16, if spool area A s is reduced by 50% and G s is maintained constant, the bandwidth of the system should double and the flow gain should remain the same. Ignoring K fs spring constant due to steady-state flow force in G s, K s is halved. Figure 10 a shows that the rise time of the modified 20 ma step response has been reduced from 8.2 ms to 4.2 ms. The quasi-static flow-current relationship in Fig. 10 b, generated using a5hz sinusoidal current input, shows that the flow-gain of the modified valve is only slightly smaller than in the original design. The small decrease is due to the fact that G s is not exactly kept constant by ignoring K fs. The modified valve also shows a decreased hysteresis, which is consistent with improved dynamic response. Interestingly, the step response for the reduced A s case also shows some slight ripples. This is due to the spools differential displacement xªx a x b which is ignored in the linear analysis. The dynamics of x are given by 9 which are stable but underdampled. As K s is reduced to compensate for reduction in A s, however, its effect on the flow become slightly more prominent with a larger amplitude and a lower ripple frequency. To reduce this effect, damping of the main spool is required. Fig A o Step response of the valve modified by changing A o to D Reducing the Flapper Stiffness K p. K p is the apparent flapper stiffness given by: K p K p 16 C 2 df x f 0 P g. x f x f 0,i 0 It can be reduced by reducing the mechanical stiffness K p or by increasing the nozzle flapper gap x fo, or by increasing the magnitude of the negative stiffness due to the permanent magnet in the torque motor. We consider reducing the mechanical stiffness K p 0.75K p so that K p is reduced to 54% of the original value. The location of the zero is expected to migrate from rad/s to rad/s. Figure 11 shows that the 64% rise time has indeed been reduced from 8.2 ms to 4.9 ms. V Discussion The original design philosophy of the two spool flow control servo-valve using a pressure control pilot is that the pilot stage first establishes a differential pressure, which in turn positions the two boost stage spools according to the stiffness of the centering Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 2002, Vol. 124 Õ 433

7 spring. This assumes that the pilot stage and the boost spool stage are in a cascade configuration. In the actual configuration Fig. 3, the pilot stage and the boost stage are in fact in a closed-loop configuration. As the differential pressure moves the spools, the differential pressure itself is changed. This upsets the effectiveness of the pilot stage to establish the intended differential pressure. As this feedback effect from the boost stage to the pilot stage is reduced, we recover the original intention for a cascade configuration. This is exactly what reducing G s e.g., by decreasing the spool area A s, or increasing K s does. Similarly, the relative importance of the feedback effect of the spool motion is ameliorated if the effectiveness of the pilot stage is improved. This is achieved by increasing or G p. The zero in Fig. 7 which limits the dynamic performance exists because the open-loop spool subsystem itself has a zero at the origin see Fig. 3. An interesting avenue of future investigation is to remove this zero in the spool system. Since the spools affect the pressure dynamics via the spool velocities, removing the zero implies that the spools should be damped, not spring loaded as is presently the situation. In this case, the current input would control the spool speed, and hence the time derivative of flow rate (dq L /dt) rather than the flow rate itself. Such a device would be analogous to electromechanical motors under current control which are typically considered torque or acceleration devices Fig. 12. VI Conclusions The dynamic response of a unconventional two-spool flow control servo-valve using a pressure control pilot stage has been analyzed. Using a reduced order linear model and simple root locus analysis, the limited dynamic performance of the valve has been shown to be related to the existence of a zero in the interconnection of the subsystems. Based on the analysis, reducing spool area or the apparent flapper stiffness have been identified and shown to be effective in improving dynamic performance, while maintaining steady state performance such as flow gain. In this paper, simple techniques taught at the undergraduate level, such as local linearization and root locus are used. Although the valve model is nonlinear, these simple techniques are still useful in providing fundamental insights into the design of the valves, especially how the structure of subsystem interconnection can affect the system performance. These techniques should be amenable to the analysis and optimization of other hydraulic components as well. References 1 Merritt, Herbert E., 1967, Hydraulic Control Systems, J Wiley, NY. 2 Anderson, Randall, T., and Li., Perry, Y., 2002, Mathematical Modeling of a Two Spool Flow Control Servovalve, published in this issue, ASME J Dyn. Syst., Meas., Control, 124 3, pp Also in Proceedings of the ASME Dynamic Systems and Control Division, IMECE Orlando, FL., DSC-Vol. 69-1, pp Lin, S-C. J., and Akers, A., 1990, Modeling and Analysis of the Dynamics of a Flow Control Servovalve That Uses a Two-Spool Configuration, Proceedings of the ASME Winter Annual Meeting, Vol. WA90/FPST-3. 4 Franklin, Gene F., Powell, J., David, and Emami-Naeini, Abbas, 1995, Feedback Control of Dynamic Systems, Addison Wesley, third edition, NY. 434 Õ Vol. 124, SEPTEMBER 2002 Transactions of the ASME