DYNAMIC REDESIGN OF A FLOW CONTROL SERVO-VALVE USING A PRESSURE CONTROL PILOT

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1 Proceeding of IMECE ASME International Mechanical Engineering Congre & Exhibition November -6,, New York, New York, USA IMECE/DSC-B- DYNAMIC REDESIGN OF A FLOW CONTROL SERVO-VALVE USING A PRESSURE CONTROL PILOT Perry Y. Li Department of Mechanical Engineering Univerity of Minneota Church St. SE, Minneapoli, Minneota pli@me.umn.edu ABSTRACT In thi paper, the dynamic performance of an unconventional two-pool flow control ervo valve uing a preure control pilot i analyzed. Such valve are le expenive than typical ervovalve but alo tend to be limited in their dynamic performance. Baed on a previouly developed eight tate nonlinear model, we develop a implied linear model which i able to capture the eential dynamic of the valve. Uing root locu analyi method, the limitation in dynamic performance i hown to be due to a zero introduced by the tructure of the interconnection of the ubytem. Deign parameter that move the zero further to the left half plane, and do not adverely affect other teady tate criteria are identied. The effectivene of thee parameter to improve the dynamic performance i demontrated. Introduction Mot deign of ervo flow control valve [4] conit of a ingle pool boot tage, a nozzle flapper pilot, and a feedback wire. Thee valve have very high performance but tend to be expenive becaue of the tringent manufacturing tolerance and the complicated aembly proce. A le common, commercially available alternate deign (Fig. ) conit of a preure control pilot tage and a boot tage that ue two eparate pool to independently meter flow into and out of the valve. Since the critical dimenion are eaier to adjut, pot aembly and feedback wire i not ued, uch valve are eaier to manufacture and to aemble. Conequently they tend to be cheaper. Reader are alo referred to [], in which an experimentally validated complete phyical model i preented, for a more detailed dicuion of the advantage of the unconventional two-pool ervo valve. Depite thee advantage, the unconventional two-pool ervo valve deign tend to have lower dynamic performance in term of bandwidth compared to the conventional ervo valve deign utilizing a ingle pool and a feedback wire. For example, the valve tudied in [], [3] have bandwidth between 5-4Hz wherea conventional ervo-valve of imilar rating can have bandwidth of over Hz. It would therefore be advantageou if dynamic repone of the two-pool deign can be improved. In thi paper, we tudy the unconventional two-pool deign o a to undertand the nature of the performance limitation, and to ugget deign modication for potential performance improvement. The experimentally validated eight tate nonlinear phyical model derived in [] conit of the interconnection of three ubytem. Thi model i imilar to the one contructed by Lin and Aker previouly [3]. Uing thi model, we develop a implied ve tate linear model that retain the interconnection tructure a well a the predominant dynamic. The reduced model reveal a puzzling apect of the valve dynamic in that each of the three ubytem ha bandwidth at leat an order of magnitude higher than the bandwidth of the complete model. Uing imple root locu argument, it i found that the way in which the ubytem are interconnected create a zero which caue the bandwidth of the interconnected ytem to be ignicantly lower than the individual ubytem. Baed on thi inight, everal ytem parameter that can potentially improve the dynamic performance without adverely affecting the teady tate perfor- Copyright by ASME

2 Thi tend to increae the preure P and to decreae the preure P. The differential preure act on the two end of the two pool in the boot tage. Since the pool are pring centered, their equilibrium diplacement will be roughly proportional to the differential pilot preure and inverely proportional to the pring tiffne. Flow into and out of the valve are eparately metered in and out according to the diplacement of the two pool.. Review of full tate model The ervo valve can be conidered an interconnection of three ubytem, ) the pilot ubytem whoe tate are the flapper diplacement x f (left to right poitive) and velocity ẋ f ; ) the preure chamber whoe tate are the chamber preure P and P ; and the boot tage pool dynamic whoe tate are the diplacement and velocitie of the two pool x a,ẋ a and x b, ẋ b. Therefore, the total number of tate i eight. Following [], the dynamic of the pilot ubytem can be repreented by: M p ẍ f + B p ẋ f + K p x f = A n (P P )+ (x fo + x f ) P (x fo x f ) Λ P + g(x f ;i) () 4πC df Figure. A two-pool flow control ervo-valve uing a preure control pilot tate. mance (uch a flow gain) are identied, and their effect demontrated. The ret of the paper i organized a follow. In ection, we formulate a implied model of the two-pool flow control ervo valve. The interconnection of the three linearized ubytem are tudied uing root locu technique in ection 3. Section 4 preent the effort to optimize the performance by applying the inight gained in ection 3. Section 5 and 6 contain dicuion and concluding remark repectively. Simplied model of the two-pool flow control ervo valve The unconventional flow control ervo valve hown in Fig. ue a two-pool boot tage and a preure control flappernozzle pilot tage. The two tage are eparated by a imple tranition plate and connected via two preure chamber. The deign philoophy of the valve i a follow. The preure control pilot tage generate a differential preure between the two fluid chamber 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 applie a counter clockwie torque to the flapper, the flapper diplace to the right. where M p, B p and K p are repectively the combined inertia, damping and mechanical tiffne of the flapper, x fo i the null nozzle-flapper gap when x f =, and C df i the dicharge coefcient of the flapper-nozzle, A n i the nozzle area. The rt and econd term on the right hand ide correpond repectively to the preure and the flow induced force at the nozzle, and g(x f ;i), which i a highly nonlinear function (ee [] for detail), repreent the force on the flapper generated by the electromagnetic torque motor with input current i. The dynamic of the two boot tage pool are given by: M ẍ a + B ẋ a + K x a =(P P )A M ẍ b + B ẋ b + K x b =(P P )A + B f (x a ;P a )ẋ a K f (x a ;P a )x a z } z } tranient flow force teady tate flow force B f (x b ;P b )ẋ b K f (x b ;P b )x b z } z } tranient flow force teady tate flow force where x a and x b repreent the upward diplacement of pool A and B repectively, M i the pool inertia, B i the vicou damping coefcient, K i the total tiffne of the two pring above and below the pool (Fig. () (3) ), A i the pool area. The teady tate and tranient flow force manifet themelve a pring force and poitive/negative damping force with K f ( ; ) >, and B f (x;p) > when x >, B f (x;p) < when Copyright by ASME

3 x <, and not well dened when x =. Therefore, depending on the ign of the pool diplacement, the tranient flow force may introduce negative damping effect. The third ubytem i aociated with the dynamic of the preure P, P in repectively the upper and the lower fluid chamber connecting the pilot tage and the boot tage pool. Ṗ = β Q (P ;x f ) V (t) V (t) ; Ṗ = β Q (P ;x f ) V (t) : (4) V (t) Here, Q (P ;x f ) and Q (P ;x f ) are the total flow into the upper and lower chamber, V (t) and V (t) are the volume in the chamber, and β i the compreibility of the fluid. Q (P ;x f ) and Q (P ;x f ) are compried of the flow from the pilot upply orice, leakage pat the nozzle, and to a mall extent, leakage pat the pool: for i = and, Q i = C d A ρ (P p P i ) C df πd n (x f ± x f ) ρ P i leakage i where + ign i ued for i = and ign i ued for i =, P p i the pilot upply preure (which i uually lower than the upply preure for the boot tage), A o i the area of the orice to the upply preure, C d and C df are the dicharge coefcient of the orice to the upply and the gap between the flapper and nozzle. The rt two term in (5) are monotonically decreaing function of P i. Thu, they provide at leat local exponential tability for the preure dynamic (4). Notice alo the pilot tage communicate with the chamber preure via Q and Q ince they depend on the flapper diplacement x f. On the other hand, the preure chamber are affected by the boot tage pool dynamic via V (t), V (t), V (t) and V (t) in (4) ince (5) V = V o A x a A x b ; (6) V = V o + A x a + A x b (7) V = V = A ẋ a A ẋ b (8) where V o and V o are the chamber volume when the pool are centered (x a = x b = ). For detail of the model, reader are referred to [].. Reduced order linear model In order to obtain meaningful deign information, we conider a reduced 5th order linear model. Thi i achieved a follow:. In the pool ytem, we conider only the total pool diplacement (not the diplacement of the individual pool) Σ(t)= (x a (t)+x b (t)):. In the preure chamber ytem, we conider only the differential preure P(t)=P (t) P (t): 3. The tranient component of the flow force and leakage flow pat the pool are ignored. 4. The pilot tage dynamic in (), the pool dynamic in ()- (3) and the differential preure dynamic obtained from (4) are linearized at the equilibrium condition given by flapper diplacement x f =, chamber preure P = P =: P, pool diplacement x a = x b =, pool velocitie ẋ a = ẋ b =, chamber volume V = V =(V +V ) = =: V, and work preure P a = P b = P = where P i the boot tage upply preure. The reulting reduced linear model for the pilot, chamber preure and pool ubytem are repectively: Pilot ubytem M p ẍ f + B p ẋ f + K p x f =(A n + B) P+ G i (9) Preure chamber ubytem Spool ubytem where P = α β V P γ β V x f + βa V Σ(t) () M Σ+B Σ+(K + K f )Σ = A P(t) () G := g i x f =;i= K p = K p 6πCdf x fo P g x f x f = Q P;x f = x f α = Q = Q P P B := 4πC df x fo ; γ = Q P;x f = ; K f := K f (;P =) x f =;i= P;x f = ; P;x f = B, γ and α are all poitive quantitie. The expreion for K p how that that nozzle flow force and the magnetic tend to offet the mechanical tiffne K p of the flapper. The term A n +B in 3 Copyright by ASME

4 Step repone of nonlinear and linear model Chamber/Pilot ytem Step occur here Nonlinear Linear K γ x f A n+ B + Mp +Bp + Kp + A n+ B Pilot G i Differential preure pi % rie time Time ec Figure. Differential preure repone to a ma tep input current: full nonlinear model and implied linearized model. K β A V + α β / V Preure chamber Σ A M +B + Spool K + K f Figure 3. Block diagram for root locu analyi with the upper feedback loop a the inner loop. The actual valve dynamic are obtained when K = K =. 5 4 P Root locu of the differential preure / pilot interconnection 3 K= (9) i the apparent nozzle area of the flapper-nozzle upon which the preure and the nozzle flow force act. From (), α i the convergence rate of the preure chamber normalized by the invere of the chamber capacitance, β= V. To verify that the linearized model in (9)-() indeed capture the dominant dynamic of the valve, the repone to the tep current input are imulated for an input tep ize of i = ma (5% full range) uing the complete nonlinear model in [] and the linearized model in (9)-(). The tep repone of the differential preure P(t) are very cloe (Fig. ). The 64% rie-time for the linearized and nonlinear model are 8: m and 8:m repectively. The imilarity among the repone of the model in (9)-() and of the full nonlinear model in [] ugget that the dynamic of the ytem repreented by the interconnection between the pilot, preure and pool dynamic are well captured by the reduced order linear dynamic. Each of the pilot (9), differential preure (), and the boot pool (9) ubytem are table. Uing phyical parameter of the valve that are veried in [], it can be hown that the pilot ubytem ha a natural frequency of ω n;p = 337 rad/, and a damping ratio of ζ p = :9; the differential chamber preure ubytem ha an eigenvalue of α = 599rad/; the boot β V pool ubytem ha a natural frequency of ω n; = 86rad/ with a negligible damping ratio of ζ = :5. The eigenvalue of the combined ytem are at 37:rad/, 7 ± 6475 jrad/, 837 ± 7 jrad/. We point out that the dominant pole i at 37:rad/ which i conitent with the fact that the 64% rietime of the fully linearized model i 8: m. In order to improve the dynamic performance of the valve, the dominant pole mut be moved further into the left half plane. Imag Axi 3 4 K= Real Axi Figure 4. Root locu diagram of the pilot / chamber differential preure ubytem a K increae from! in Fig Analyi for performance limitation We now proceed to analyze the linearized model (9)-() to undertand why the bandwidth of the valve i relatively low, wherea the natural frequency of each individual ubytem i at leat an order of magnitude higher. I the limited performance due to the fact that the pool are too lightly damped (ζ = :5)? Or, i the fluid capacitance in the preure chamber the reaon? A we hall ee, neither the pool damping nor the chamber capacitance i important. The key turn out to be the tructure of the interconnection between the preure chamber, the pilot and the pool ubytem. 3. Full order root locu The pilot, chamber preure and the boot pool ubytem are connected in a cloed loop manner a hown in Fig. 3 with 4 Copyright by ASME

5 x Root locu of the interconnection between pool and differential preure / pilot ubytem 4 x Root locu of the interconnection between pilot and differential preure / pool ubytem 4.5 K =.5 K = Imag Axi.5 K = K = Imag Axi.5.5 K = Open loop pole.5.5 K = Real Axi Figure 5. Root locu diagram of the pool ubytem and the pilot / chamber differential preure ubytem a K increae from! and K = in Fig. 3. K = Real Axi Figure 6. Partial root locu diagram of outer loop with inner loop being the pool / chamber ytem with K = a K increae from!. Two other pole on the far left are not included. K = K =. To undertand the effect of the interconnection, we apply Evan root locu technique [] to invetigate how the cloed loop eigenvalue migrate a the parameter in the ytem are varied. Conider rt the inner loop in Fig. 3 which i the interconnection between the pilot and the differential preure ubytem. Fig. 4 how the loci of the cloed loop pole of the inner loop a the ctitiou gain K i varied from!. K = correpond to the gain in the actual loop in the preent valve deign. Figure 5 how the locu of the cloed loop pole of the outer loop ytem in Figure 3 a the ctitiou gain K i increaed. The et of pole at K = are the actual pole in the valve. A expected, when K =, the dominant pole i at p = 37:rad/ (which i the reaon why the dynamic performance i limited). Notice that the real part of all the other eigenvalue location are ignicantly more negative. Figure 5 how that for the preent valve deign (i.e. K ß ), the pole location are well approximated by the aymptotic behavior of the root locu. Thee are governed by the open loop pole and zero conguration. In particular, the dominant pole at p = 37: i being attracted to the zero at. Two other pole are cloe to the zero at the pilot open loop pole location. The remaining two pole are alo cloe to the aymptote. Since the damping in the pool can only affect the aymptote lightly, contrary to our initial peculation, the negligible damping of the pool doe not contribute ignicantly to the relatively poor dynamic performance of the valve. Rather, the reaon i due to the zero at the origin. Thi zero i preent becaue the pool ubytem interact with the chamber preure dynamic via Σ. A far a the loop gain K and K are concerned, from Fig. 5, decreaing K delay the migration of the dominant pole to the zero at the origin. Similarly, if we had interconnected the differential preure and the pool (lower loop) rt before connecting the pilot ytem in Fig. 3 and derived the correponding root loci, then we would have noticed that by increaing K in Fig. 3, the partial root locu i hown in Fig. 6. Notice that the dominant pole migrate from an open loop pole near the origin toward the left half plane a K increae. 3. Reduced order root locu The root locu analyi above indicate that the four complex pole of the valve deign can be approximated by their aymptotic behavior. In addition, the complex pole originate from the pole aociated with the pilot and the pool ubytem. Thi ugget that we may approximate the behavior of the dominant eigenvalue of the valve by conidering the pilot and the pool ytem a quai-tatic ytem, i.e. by auming that at each intant, the pool and the flapper are in tatic equilibria with the intantaneou differential preure. The reulting conguration i given in Fig. 7. Indeed, the cloed loop pole of the reduced order quai-tatic ytem in Fig. 7 i p = 7:7rad/ which i very cloe to the actual dominant pole ( 37:rad/). The characteritic equation for the ytem in Fig. 7 i given by: + β V α+γ A n+b K p + 4A K +K f = : () 5 Copyright by ASME

6 i γg K p β V + α β V Preure chamber γ(a n +B) K p + 4A K +K f Simplied pool/pilot ytem Figure 7. Block diagram with the pilot and the pool ubytem approximated by their tatic ytem. pole location Zero at α+γg p A G Figure 8. Root locu of the reduced order model in Fig.7. The root locu for poitive i given in Fig. 8 which how that β V the performance of the ytem would be limited by a zero at α+γ A n + B 4A = : (3) K p K + K f In the current valve deign, the zero of the quai-tatic model i at 9:3 rad/. Since the pole (of the quai-tatic model) i already at 7:7 rad/, the performance cannot be ignicantly improved by increaing β= V which i the ratio between the fluid compreibility and the chamber volume. Although the preence of nite preure chamber dynamic i eential for the exitence of the low valve dynamic, the value of the chamber volume, the compreibility, or capacitance (i.e. V =β) do not matter ignicantly. Rather, the performance limitation i determined by the feedback tructure itelf. The performance limitation can be alleviated if the zero in (3) can be moved further to the left. Conitent with the previou analyi, thi can be achieved by increaing the pilot loop gain and by decreaing the pool loop gain (i.e. K and K repectively in Fig. 3). 4 Dynamic Redeign 4. Dynamic performance Since the dominant pole of the ytem limit the performance of the valve, it mut be moved further to the left half plane to improve dynamic performance. According to the root locu analyi in Section 3., thi can be achieved if the loop gain in the upper loop i increaed, and the loop gain in the lower loop i decreaed. From the reduced order root locu analyi X P in Section 3., performance i limited by the open loop zero of the reduced order ytem. Since the dominant pole location of the valve will be cloe to the zero location, the zero location mut be moved to the left if the dynamic performance i to improve. The expreion for thi blocking zero i: α+γg p A G ; (4) where G p := A n+b K p and G := A K +K f are repectively the teady tate ratio of the flapper diplacement x f to the differential preure P, and teady tate ratio between the total pool diplacement Σ and P. Therefore, from (4), the key deign parameter in the variou ubytem are: Nozzle-flapper: The apparent nozzle area A n + B, which can be modied by changing the phyical ize of the nozzle and the gap between nozzle-flapper. γ i the enitivity of the nozzle flow to flapper diplacement which can alo be modied by changing the nozzle diameter. The apparent flapper tiffne K p i affected by the mechanical tiffne, the negative magnetic tiffne and the negative nozzle flow induced tiffne. Boot tage pool: The pool area A, and the centering pring tiffne K. Preure Chamber: α, which i the convergence rate of the differential preure normalized by the invere chamber capacitance, β= V. Shifting the zero in (4) to the left can be achieved by ) modifying the flapper nozzle deign o a to increae γg p (increae A n + B, decreae K p, increae γ); ) modifying the boot tage pool deign o a to decreae A G (decreae A, increae K ); or 3) by modifying the open loop convergence rate of the preure chamber dynamic o a to increae α. 4. Steady State Criteria In addition to their effect on the dynamic repone of the valve, it i important alo to evaluate the effect of thee deign parameter on the operating preure and motion range, and ultimately the flow gain of the valve. We determine thee from the D.C. component of the tranfer function auming γ=α i large. Differential preure gain: P() I() = (γ=α)g p G = +(γ=α)g p A n + B ß G A n + B 6 Copyright by ASME

7 Flapper diplacement gain: x f () I() = = α γ Spool diplacement gain: Σ() I() Flow gain: Q L () I() P() I() P() = G ß = = = C dw = I() P ρ Σ() I() ß α G = γ A n + B G G A n + B ß C dw = P ρ GG A n + B where G = g i for the torque motor, P i the upply preure for the boot tage. For a given input current, it i generally preferable that P and x f be mall, and the flow Q L be large. Becaue the pilot upply preure P p i limited, large P excurion reduce the operating range. On the other hand, large x f generally require large magnetic airgap, which can complicate the deign of the torque motor. A large flow gain i deirable o that a mall input current can be ued to control large flow. The conequence on thee criteria when the variou parameter are ued to improve the dynamic performance are ummarized in Figure 9. If the apparent nozzle area A n + B i increaed, for the ame input current, the differential preure and the teady tate flapper diplacement will be decreaed. Unfortunately, ince the flow gain ha alo been proportionately reduced, more force i required from the torque motor to achieve the ame flow. Similarly, decreaing G of the pool ytem will decreae the flow gain ignicantly. Increaing α will have the advere effect of increaing flapper motion. The deign parameter that do not adverely affect the teady tate criteria ignicantly are the apparent flapper tiffne K p, the pool area A, (while maintaining G contant) and γ of the nozzle-flapper. Of thee three parameter, in fact only K p and A can be ued independently to improve dynamic performance without affecting the teady tate performance. γ, which i the enitivity of the chamber flow to flapper diplacement, cannot be varied to any ignicant degree without affecting the apparent nozzle area A n + B. Indeed, even a 5% increae in γ neceitate an increae in the actual nozzle area A n by over 4%! Thi can potentially increae the apparent nozzle area A n + B which in turn decreae the teady tate flow gain. 4.3 Reducing the Spool Area A According to the expreion of the zero in (4), if pool area A i reduced by 5% and G i maintained contant, the K p # (A n + B) " γ " G # A # α " p P p p x f X Q L - X - X - - Figure 9. Conequence on the tatic criteria when variou deign parameter are ued to improve the dynamic performance. Direction of the arrow indicate the p direction of propoed change. X repreent ignicant degradation, repreent ome improvement. For each column, the variable in all other column are aumed to be contant. bandwidth of the ytem hould double and the flow gain hould remain the ame. Ignoring K f (pring contant due to teady tate flow force) in G, K i halved. Figure a) how that the rie time of the modied ma tep repone ha been reduced from 8:m to 4:m. The quai-tatic flow-current relationhip in Fig. b), generated uing a 5Hz inuoidal current input, how that the flow-gain of the modied valve i only lightly maller than in the original deign. The mall decreae i due to the fact that G i not exactly kept contant by ignoring K f. The modied valve alo how a decreaed hyterei, which i conitent with improved dynamic repone. 4.4 Reducing the flapper tiffne K p K p i the apparent flapper tiffne given by: K p = K p 6πC df x fo P g x f : x f =;i= It can be reduced by reducing the mechanical tiffne K p or by increaing the nozzle flapper gap x fo, or by increaing the magnitude of the negative tiffne due to the permanent magnet in the torque motor. We conider reducing the mechanical tiffne K p! :75 K p o that K p i reduced to 54% of the original value. The location of the zero i expected to migrate from 9:3 rad/ to :3 rad/. Figure how that the 64% rie time ha indeed been reduced from 8:m to 4:9m. 5 Dicuion The original deign philoophy of the two pool flow control ervo-valve uing a preure control pilot i that the pilot tage rt etablihe a differential preure, which in turn poition the two boot tage pool according to the tiffne of the centering pring. Thi aume that the pilot tage and the boot pool tage are in a cacade conguration. In the actual conguration (Fig. 3), the pilot tage and the boot tage are in fact in a cloed loop conguration. A the differential preure move the pool, the differential preure itelf i changed. Thi upet 7 Copyright by ASME

8 Flow gpm % rie time Reduced pool area Original deign Reduced pool area Original deign Time 3 Current flow relationhip when A i halved Reduced pool area Original deign the effectivene of the pilot tage to etablih the intended differential preure. A thi feedback effect from the boot tage to the pilot tage i reduced, we recover the original intention for a cacade conguration. Thi i exactly what reducing G (e.g. by decreaing the pool area A, or increaing K ) doe. Similarly, the relative importance of the feedback effect of the pool motion i ameliorated if the effectivene of the pilot tage i improved. Thi i achieved by increaing γ or G p. The zero in Fig. 7 (which limit the dynamic performance) exit becaue the open-loop pool ubytem itelf ha a zero at the origin (ee Fig. 3). An intereting avenue of future invetigation i to remove thi zero in the pool ytem. Since the pool affect the preure dynamic via the pool velocitie, removing the zero implie that the pool hould be damped, not pring loaded a i preently the ituation. In thi cae, the current input would control the pool peed, and hence the flow acceleration ( dq L dt ) rather than the flow rate itelf. Such a device would be conitent with current controlled electromechanical motor which are typically conidered torque or acceleration device. Flow gpm Current ma Figure. Step repone (left) and current-flow relationhip of the valve modied by A ψ A =, K ψ K =. Flow gpm Original deign Reduced K p deign Time ec Figure. Step repone of the valve modied by K p ψ :75 K p (K p ψ :54K p ). Note that 64% rie time ha been reduced from 8.m to 4.9m. 6 Concluion The dynamic repone of a unconventional two-pool flow control ervo-valve uing a preure control pilot tage ha been analyzed. Uing a reduced order linear model and imple root locu analyi, the limited dynamic performance of the valve ha been hown to be related to the exitence of a zero in the interconnection of the ubytem. Baed on the analyi, reducing pool area or the apparent flapper tiffne have been identied and hown to be effective in improving dynamic performance, while maintaining teady tate performance uch a flow gain. In thi paper, imple technique taught at the undergraduate level, uch a local linearization and root locu are ued. Although the valve model i highly nonlinear, thee imple technique are till ueful in providing fundamental inight into the deign of the valve, epecially how the tructure of ubytem interconnection can affect the ytem performance. Thee technique hould be amenable to the analyi and optimization of other hydraulic component a well. Reference [] Randall T. Anderon and Perry Y. Li. Mathematical modeling of a two pool flow control ervovalve. In Proceeding of the ASME Dynamic Sytem and Control Diviion, IMECE Orlando, FL., volume DSC-Vol. 69-, page 3 38,. Alo ubmitted to the ASME Journal of Dynamic Sytem, Meaurement and Control. [] Gene F. Franklin, J. David Powell, and Abba Emami-Naeini. Feedback control of dynamic ytem. Addion Weley, third edition, 995. [3] S-C. J. Lin and A. Aker. Modeling and analyi of the dynamic of a flow control ervovalve that ue a two-pool conguration. In Proceeding of the ASME Winter Annual Meeting, volume WA9/FPST-3, 99. [4] Hebert E. Merritt. Hydraulic Control Sytem. John Wiley and Son, Copyright by ASME