Robust Optimal Design of Unstable Valves - CiteSeerX

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 6, NOVEMBER 2007

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Robust Optimal Design of Unstable Valves QingHui Yuan, Member, IEEE, and Perry Y. Li, Member, IEEE

Abstract—This paper is concerned with the application of robust control design concepts for the physical geometric design of electrohydraulic valves. Currently, limitations of solenoid actuators have prevented single stage electrohydraulic valves which are simpler and more cost effective from being utilized in high flow rate and high bandwidth applications. Fluid flow force induced instability has been proposed as a means to alleviate the demand on the solenoid actuators. Previous research has demonstrated that simple changes in the valve geometry can be used to manipulate both the transient flow force as well as the steady flow force for this purpose. This paper considers the dimensional design of such “unstable” valves to minimize the net steady flow force. The robust optimal design method, in which the design must be robust to uncertainties such as variations in operating pressure ranges and dynamic viscosity, etc., is proposed. By representing the original problem as a linear fractional transformation interconnection, the robust design problem is formulated into one of synthesizing an optimal controller for an appropriate static plant with a structured uncertainty. An algorithm for solving this design synthesis problem is proposed. A case study is conducted to compare the nominal optimal (without considering uncertainty) and the robust optimal designs. It is shown that viscosity effect is exclusively utilized in the nominal optimal design, whereas both the viscosity effect and the nonorifice flux effect are needed in the robust optimal design. The robust optimal design imposes smaller steady flow force on the spool than the nominal optimal design under perturbed situations. Based on the robust design method, an actual prototype design of the unstable valve has been developed. Index Terms—Electrohydraulics, flow force, linear fractional transformation (LFT), proportional valves, robust design, structured uncertainty, unstable valve.

I. INTRODUCTION

N SINGLE stage direct acting electrohydraulic valves, the main spools are stroked directly by solenoid actuators. They have advantages over multistage electrohydraulic valves in being low cost, easy to maintain, and insensitive to contamination. However, in high flow rate and high frequency applications, the force and power requirement for the solenoid actuators become prohibitive due to the significant flow induced forces. The approach we adopt is to propose new valve geometries that utilize the fluid flow force induced instability to enhance the spool agility without increasing the requirements on the solenoid actuators. By subsequently stabilizing the spool

I

Manuscript received March 21, 2005; revised May 8, 2006. Manuscript received in final form January 3, 2007. Recommended by Associate Editor A. Isaksson. This work was supported by the National Science Foundation under Grant ENG/CMS-0088964. The authors are with the Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minneapolis, MN 55455 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCST.2007.908080

via feedback control, both the dynamic response and flow rating of single stage valves can be practically improved. Previous works have demonstrated that simple changes in valve geometry can be used to manipulate both the transient flow force as well as the steady flow force to induce spool instability [1]–[5]. The unstable phenomena are largely controlled by the following design parameters: 1) the damping length ; 2) a geometry constant determined by the inner and outer radii of the valve chamber; and 3) the nonmetering orifice momentum flux coefficient which is determined by the angles of the nonorifice ports to the spool axis. Both the sign and the magnitude of the transient flow force induced damping coefficient are directly related to those of [1], [2], [6], the viscosity induced stable/unstable steady flow force is affected by [3], and the nonorifice momentum flux component of the stable/unstable steady flow force is affected by [4], [5]. This paper addresses how to choose these design parameters , and such that the steady flow force (that the solenoid actuator must act against) is minimized while keeping the valve size reasonable. In addition, the design must be robust to model uncertainties, flow ranges, and variation in operating conditions such as operating pressures, temperature, and fluid viscosity. The robust optimal design problem is posed as a min–max optimization similar to that in robust control theory [7]. By showing that the relationship between design parameters, perturbations, and the objective function can roughly be represented as a linear fractional transformation (LFT) interconnection, the valve design problem of finding the optimal set of geometric parameters becomes equivalent to one of designing an optimal robust performance controller for an appropriate static plant with a structured uncertainty. This reformulation then enables us to apply to our problem, an extension of the result in [8] for synthesizing robust optimal controllers for static plants. The optimal solution based on this approach is then compared to the nominal optimal solution in a case study. The study shows that in the presence of perturbations, the unstable valve should utilize both the viscosity effect and the nonorifice momentum effect to minimize the steady flow force. This results in a different answer from the perturbation free case, in which the viscosity effect is exclusively utilized to zero out the steady flow forces over the full range of the orifice opening. The rest of this paper is organized as follows. Modeling and the design problem formulation are presented in Section II. In Section III, the nominal optimal design over the space of the design parameters is addressed, without considering perturbations. Section IV presents the robust optimal design methodology. In Section V, the comparison case study of nominal and robust optimal solutions is discussed. The physical implementation based on the robust design scheme is presented in Section VI. Section VII contains some concluding remarks.

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bility of both the transient [1], [2] and the steady [4], [5] flow forces. is a geometric constant, which together with determine the viscosity effect of the steady flow forces. For the valve in Fig. 1 [4], [5] (2)

Fig. 1. Typical configuration of a four-way direction flow control valve.

II. MODELING AND PROBLEM FORMULATION Fig. 1 shows a typical critically centered, matched, and symmetric four-way flow control valve [1]. The A and B ports are connected to the load (hydraulic actuator), the is connected to the supply pressure, and ghe ports are connected to the reservoir. In a single stage valve, the spool is stroked directly in turn conby solenoid actuators. The spool displacement of the meter-in and meter-out orifices (the trols the area assumption that the valve is matched ensures that both orifices is defined to take on the sign of . have the same area). . As the For instance, Fig. 1 shows the case where fluid flows through the valve, the fluid flow imparts flow forces on the spool, which must be overcome by the solenoid actuators. The flow force is the reaction force from the spool on the fluid volume. The latter can be computed, for a given flow and valve opening, by considering the axial momentum balance of the fluid volume in each valve chamber [4], [5]. The flow force consists of the following: 1) fluid momentum flux at the two metering orifices; 2) fluid momentum flux at the two nonmetering ports; 3) viscous force between the fluid and the valve sleeve; 4) D’Alembert force due to the acceleration of the fluid volume. The flow force component due to the last item is the transient flow force since it arises when the flow rate is varying. The first three items constitute the so-called steady flow force. This paper is concerned with the steady flow force only. The resultant steady flow force acting on the valve spool is given by [4], [5] (1) where the three terms correspond to the momenta at the metering orifices and at the nonmetering ports and to the viscous effect, respectively. In (1), is the flow rate, is the fluid den69 and are the effective vena contracta angle, and sity, the contraction coefficient of the metering-orifice, is the fluid is the combined coefficient for the viscosity. and nonmetering port momentum flux, where are the momentum fluxes at the nonmetering inlet and outlet ports. In [4] and [5], it is shown that can be manipulated to some extent by varying the angles of the nonmetering ports to the spool axis. is the damping length which is defined to be (see Fig. 1). It plays important roles in the sta-

where and are the spool stem and the sleeve radii. Thus, , i.e., a narrow flow channel. small requires In conventional valves, the metering orifice momentum [first term in (1)] is the most significant component of the steady flow force that the solenoid actuator must overcome. This term is especially significant when flow is large. The nonorifice port momentum and the viscous components are generally ignored. The damping length is typically designed to be positive in consideration of the stability of the transient flow forces [1], [2]. Our approach is to so design the valve geometry so as to utilize the nonorifice momentum and viscous components to cancel out the metering-orifice momentum component. Since the metering orifice momentum component is stable (in the sense that it tends to restore the orifice to the closed position), the nonorifice port momentum flux and the viscous components should be unstable. In particular, negative damping length will be needed. Incidentally, would also make the transient flow force to behave like negative damping, which is consistent with the strategy in [2] to use the unstable transient flow force to improve the spool’s agility. , and can be thought of as design parameters determined by the valve geometry [3]–[5]. In addition, should not be too large, nor too small, which would otherwise require that the valve be very long or the fluid channel to be very narrow. The coefficient for the nonmetering momentum flux must also lie . within a feasible range. Physical constraints restrict that Substituting the orifice equation

where is the pressure drop across each orifice into (1), and defining the normalized orifice area to be , where is the maximum orifice area, we have (3) where . Notice that are constants for fixed pressure drop and viscosity . Ideally, the steady flow force should be identically zero over the full range of the orifice opening (to minimize the load on the solenoid actuator). Therefore, the magnitude of the steady flow force in (3) should be minimized by the design. Because (i.e., it is an odd function), . Therefore, the valve design problem is to (4) with given by (3). itive because of physical constraint.

is restricted to be posis constrained to be

YUAN AND LI: ROBUST OPTIMAL DESIGN OF UNSTABLE VALVES

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negative to ensure that negative damping lengths are used in order that the transient flow force is also unstable [2]. Since and are always multiplied together in (3), optimization can be done with respect to . Once is and (hence obtained, the values of the valve geometry, ), and [hence in (2)] can be assigned to keep the overall valve dimension reasonable. One can also imand so as to observe the restricpose feasible ranges for tion on the valve dimensions and on our ability to manipulate the nonmetering momentum flux coefficient .

, and

where are constants. Let , and

. , by the mean value theorem

Then, for each

(8) for some

and, where

III. NOMINAL OPTIMAL DESIGN

for some

When the operating pressure and the fluid viscosity are constants and well known, the min–max problem in (4) is solved by

(5) resulting in

for any

(9)

is introduced as an additional perturbation because is typically unknown when applying the mean value theorem. Howfrom is expected to be small, ever, if deviation of can be neglected. then Using (8), (7) can be expanded so that it is clearly affine in the products of the perturbations

.

IV. ROBUST OPTIMAL DESIGN In practice, operating conditions such as the pressure across the orifices and the dynamic viscosity in drop (3) vary within certain ranges. varies with varying load is constant. Viscosity even if the supply pressure varies significantly with temperature of the hydraulic system. The valve design must cope with these variations. Since , i.e., we need only consider . Thus, the robust optimal design objective becomes (10) (6) where , operating pressure over the ranges of orifice opening and viscosity . In (6), is given by (3) but now vary with and . A. Perturbation Formulation The orifice opening and the actual operating conditions can be written in terms of the nominal condition and weighted perturbations as

, with . For example,

for and

,

.

B. LFT Representation Equation (10) can be represented as a linear fractional transformation (LFT) as shown in Fig. 2. • The perturbations are modeled in a structured uncertainty matrix

(11)

where and are the nominal pressure and the nominal dynamic viscosity, are the corresponding are the normalized pressure and weighting functions, is the variation in orifice area. viscosity perturbations, Substituting these into (3), we have

where and . Notice for , but that the th diagonal element yet, ’s are not all independent. • The “controller” is defined to contain the design parameters and (12) • The “plant”

is a real matrix of the form (13)

(7)

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should not depend on . Unfortunately, for Ideally, cannot be completely pulled out. This the problem at hand, issue can be dealt with in the synthesis step of determining the optimal . ), its worst case Given a “controller” (or a design performance is given by

(15) is taken over the set of uncertainties . For the where the where perturbations considered in Section IV-A, in Eq. (11) s.t. (16) is a conservative Because of the introduction of estimate of the worst case for a given design . The robust optimal design problem (6) can thus be approximated by one of finding the with the smallest worst case performance for the uncertainty set under consideration

Fig. 2. LFT interconnection of the perturbed closed-loop system.

where given by

can be arbitrarily chosen, and

is (17) The robust optimal geometry design problem has thus been converted into a robust optimal control synthesis problem for which synthesis are availsynthesis tools such as synthesis or able for certain uncertainty structures. The optimal design pacan be retrieved back from the oprameters . timal “controller” Unfortunately, synthesis tools are not available for the nonin (16). To proceed, linearly structured uncertainty set can be embedded in the larger uncertainty sets of diagonally structured uncertainties, and full block unstructured uncertainties

where Diagonal: (18) Full block: s.t.

Then (10) will be exactly represented by the output of the closed-loop perturbed system in Fig. 2 given by (14) and are, respectively, the lower and where the is a function of due upper LFT connections [7]. Note that and . to the elements

(19)

While the over-parameterization of the perturbations will result in conservative designs, synthesis tools are nevertheless more amenable. . In this paper, we focus on minimizing (15) with This is a convex optimization problem so that a tight optimal solution can be obtained by utilizing and extending the result in [8] for static systems with full block uncertainty. On the other hand, the multiple scalar diagonally structured uncertainty problem is significantly more complicated. Since most synthesis tools rely on minimizing an upper bound of , and is known not to be tight (see the upper bound for [8, Lemma 1] and [7, Th. 10.4] for more details), the synthesis


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problem can only be solved approximately. In addition, minimizing the upper bound itself is also known to be nonconvex. Interested readers may refer to [9, pp.53–54] for more information on the nonconvex property of multiple scalar structured uncertainty. in either or Embedding the uncertainty set involves significant over-parameterization. Thus, it is advantageous to reduce the dimension of . In the case study [introduced in Section V, we neglect all terms that include , since variations in in (9)], i.e., being related to the second derivative of are not significant. This reduces the dimensions of from 13 to 9 and that of the “plant” from to . Further reduction can be achieved by considering the probability distribution of ’s. If the ’s are normally distributed with zero means, then high order product terms such as or are likely to be small and can, therefore, be neglected as well. C. Synthesizing the Optimal “Controller” In this section, we present a synthesis algorithm for the robust optimal design problem (17) for the full-block uncertainty . In [8], the authors present a method to synthesize set a robust controller for the similar static system as in Fig. 2. This algorithm was also utilized in [10] for the robust control of xerographic printing process. Nevertheless, the algorithm proposed in [8] cannot be used directly to solve robust design problem due to the following. 1) The robust performance index in [8] is of the form

where

denotes the maximum eigenvalue of its argument

are chosen so that

are both invertible and . exists such Theorem 1 implies that given , to check if a , we need only check if exists satisthat fying (22) and (23). The latter is a convex optimization problem. To see this, notice that (22) and (23) can be written as

where

(20) which is different from (15) in that the size of the uncertainty set is not fixed. 2) The assumption in [8] that the static matrix must be independent of is not valid in our problem. A new algorithm for synthesizing the controller for the present situation is proposed as follows. Theorem 1: Consider the robust optimal design problem (17) for the full block uncertainty set ( originally, and if terms are neglected). Then, given , there exists satisfying

The maximum eigenvalue is a convex function of its argument that is affine in or . It has been proven in [9] that the sets satisfying (22) and (23) are open intervals on the real exists so that line. Therefore, to check if a for a given , we need only check if the intervals intersect. The minimum is obtained when the two intervals intersect at exactly one point. This minimization can be found via a bisection search. Once a corresponding to a feasible is obtained, the controller can be synthesized by the following algorithm from [8, Lemma 2]:

(21) is given by (15) with where exists so that a positive scalar

if and only if

(22) and (23)

(24) Proof (Theorem 1): First, we transform the performance criteria in (15) into a form similar to the one proposed in [8].

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Given a , we choose accordingly as


and modify

in (13) to obtain

for

Pre- and postmultiplying (a) and (b) gives (32)

(25)

Then, it is easy to show that . The robust performance criteria becomes of the form in [8] given by (20)

(26) , (26) is For uncertainty matrices being full blocks, problem [8, Lemma equal to its upper bound given by the 1] (27) where is of the form . To deal with the fact that being a function of , we and gather terms with together expand

(33) The proof is complete. Remark 1: The previous algorithm can be modified to find a suboptimal solution to the optimal robust design problem with . In (27), the diagonally structured uncertainties should be replaced by , and an alternate structure involving is needed. As mentioned earlier, multiple scalars the RHS of (27) is not a tight bound of . Nevertheless, an approximate solution can be obtained by minimizing the upper bound. Although the upper bound minimization problem with involving multiple scalars is not convex, an algorithm can be derived by successively keeping of the fixed, and by minimizing the remaining . Since this subcan be applied. problem is convex, the procedure for The quality of this solution can be evaluated by comparing it to a lower bound estimate of the upper bound in (27). For details of this approach, please see [11] and [12].

V. CASE STUDY

(28) Combining (28) and the expressions for in the form of yields

and

in (13)

(29) Hence, have

is affine in

. Then for a

, we

(30)

(a) (b) (31)

A case study is conducted to compare the designs from the nominal optimal and the robust optimal design methods. Typical valve parameters are assumed. The maximum orifice area 10 m , discharge coefficient is and is 69 . The valve is assumed to the vena contracta angle is be connected to a double ended cylinder so that the flows into and out of the actuator are identical (see Fig. 1). The nominal 10.3 MPa (1500 psi). We supply pressure is assumed to be to be in the range, consider the load pressure 0–7.13 MPa (0–1000 Psi). Therefore, the pressure difference 3.42 MPa across each metering orifice is 1.71 MPa (500 Psi 250 Psi) [1]. Hence, the nominal pres3.42 MPa (500 psi), sure and the uncertainty weight are . and A typical hydraulic fluid (Mobil DTE Oil 970391) has a nom0.0375 kg/m/s at 40 C and inal dynamic viscosity 871 kg/m . The approximate relationship between temperature and viscosity is [1] (34) 40 C is the reference temperature, is the dynamic where 0.0311 C for the Mobil DTE Oil. viscosity at , and Assuming a temperature variation of 27 C–60 C, there is a . 50% variation in , thus, 0 Ns m The nominal optimal design is found to be and 1.476 10 m . The maximum steady flow force under the nominal operating condition is 5.16 10 Pa. and for the various The robust optimal designs uncertainty weights, for both the full block uncertainty computed using the algorithm in Section IV-B, and for the diagonally structured uncertainty using the


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TABLE I NOMINAL AND ROBUST OPTIMAL DESIGNS FOR VARIOUS LEVELS OF PERTURBATION AND THE CORRESPONDING PREDICTED WORST CASE STEADY FLOW FORCE REPRESENTS THE FULL-BLOCK UNCERTAINTY AND (K ) IS THE OPTIMAL WORST CASE PERFORMANCE. S CORRESPONDS TO MAGNITUDES. S DIAGONALLY STRUCTURED UNCERTAINTIES AND (K ) IS UPPER BOUND ESTIMATE OF THE OPTIMAL WORST CASE PERFORMANCE

approximate algorithm in Remark 1, are shown in Table I together with the predicted worst case steady flow force magnifor and the upper bound tude (as computed from for ). of The following observations can be made. 1) The nominal optimal design uses the viscosity effect exclusively to zero out the steady flow force component due to the metering momentum flux. The nonmetering port momentum flux is not used. The resulting maximum steady flow force is nearly 0. and ), 2) In the absence of uncertainty ( the robust optimal design is the same as the nominal optimal design and the predicted worst case steady flow force . The discrepancy is due to the numerical accuracy of the iterative nature of robust design algorithm. 3) When uncertainties are accounted for, the robust optimal designs (computed for either the full block uncertainties or diagonally structured uncertainties) diverge from the nominal optimal design. The extent the designs diverge from the nominal design increases with the level of uncertainty. As the uncertainties in the orifice pressure drop and the viscosity increases from 0% to 50% of the nominal values, the robust optimal design gradually relies less on the viscosity and more on the nonmetering port momentum effect flux effect to reduce the overall steady flow force. 4) For zero uncertainty, the steady flow forces are nearly zero for the full range of orifice opening. As the level of uncertainty increases, it is more difficult to reduce the steady flow force under all circumstances. This is demonstrated and as the level of by the increasing uncertainty increases. 5) The various robust designs for both the full block uncertainty and diagonally structured uncertainty lie roughly on the same straight line (see Fig. 3). Thus, the optimal design for a certain level of uncertainty that assumes a full block uncertainty is similar to the optimal design that assumes a diagonally structured uncertainty but for a different level of uncertainty. To directly evaluate the “unstable” nominal optimal design and the two “unstable” robust optimal designs that assume either

Fig. 3. Optimal robust design parameters for various uncertainties and using ) and diagonally structured uncertainties full block uncertainties (squares, S ). The design parameters of the prototype valve is also shown. (triangle S

full block uncertainty or diagonally structured uncertainty, exhaustive simulations are conducted. The orifice pressure drops are varied by as much as 37.5%. The and the viscosity and uncertainty weights of are used for the robust optimal designs. The contour plots of as a function of for the three designs are plotted in Figs. 4–6. With the nominal optimal deare exactly the nominal values, sign (see Fig. 4), when dethe steady flow force is exactly zero. However, when viates from the nominal values by 37.5%, the steady flow force magnitude can be as large as 173 N. For the robust optimal design with full block uncertainty (see Fig. 5), and the same amount of perturbations, the worst case steady flow force magnitude is less than 94.9 N. This is only 40% of that predicted by 238.2 N in Table I (for the 37.5% uncertainty case). The discrepancy is due to the fact that the uncertainty is considered instead of . This structure

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j

j

Fig. 4. Contour of max F with 37.5% uncertainty in (P; ) for the nominal optimal design (C; L) = (0; 1:467 10 ). The worst case steady flow force magnitude is 173 N.

j

j

0

2

Fig. 5. Contour of max F with 37.5% uncertainty in (P; ) for the robust optimal design (C; L) = (7:03 10 ; 5:66 10 ) computed assuming 37.5% full block uncertainty (S = S ). The worst case steady flow force magnitude is 94.9 N.

2

0

2

is conservative because correlations between some uncertainty variables are ignored. Thus, some perturbation considered by the robust optimal method are in fact infeasible. For the robust optimal design with diagonally structured uncertainty (see Fig. 6), for the same amount of perturbations, the worst case steady flow force magnitude is less than 92.4 N. The upper bound for the predicted worst case is 187.8 N in Table I. Thus, the robust design that assumes a diagonally structured uncertainty, although still very conservative, is more realistic than the estimate by assuming a full block uncertainty. The optimal design for diagonally structured uncertainty has similar actual performance as the optimal design for the full block uncertainty. The difference between actual performance and estimated performance suggests that a more careful weeding out of the over-parameterization would be worthwhile. to the To compare the “unstable” valve designs where more traditional design where , the contour plot of the

j

j

Fig. 6. Contour of max F with 37.5% uncertainty in (P; ) for the robust optimal design (C; L) = (10:9 10 ; 0:58 10 ) computed as). The worst case suming 37.5% diagonally structured uncertainty (S = S := sup ( F ) 92:4. performance is sup

jj

j

2

0 2 jj

j

Fig. 7. Contour of max F with 37.5% uncertainty in (P; ) for the neutral valve design that corresponds to 90 deg port angle (C = 4:0 10 ) and L = 0. The worst case steady flow force magnitude is 239.8 N.

2

maximum steady flow force for a stable valve with and port angle is 0 deg is shown in Fig. 7. The worst case steady flow force magnitude is as large as 239.8 N which is more than double that for the robust “unstable” valve designs. This validates the usefulness of using the “unstable” steady flow force to reduce the overall steady flow force. VI. PHYSICAL IMPLEMENTATION Based on the robust optimal designs in Fig. 3, we design the prototype of an unstable valve. Fig. 8 shows the prototype valve assembly with a standard CETOP5 port interface for a flow rating of 24 L/min (rated at 10 MPa ( 1500 psi) across the valve, i.e., 5 MPa drop across each metering orifice). The nonmetering ports consist of eight circular 6-mm diameter holes distributed over the circumference of the valve sleeve. The valve differs from a conventional design in that it is configured to


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TABLE II INLET, OUTLET, AND TOTAL MOMENTUM FLUX COEFFICIENTS c ; c AND C (:= c +c ) FOR VARIOUS NONMETERING PORT ANGLES AS COMPUTED FROM CFD COMPUTATIONS

of the prototype valve, a size three solenoid is used to drive the valve. The valve is compared to a conventional commercial valve with a neutral/stable damping length which has a similar flow rating but is driven by a larger size five solenoid. The detailed description of the design, modeling, and testing of the prototype valve can be found in [11]. It was experimentally demonstrated that despite the smaller solenoid, the prototype valve has a dc gain which is higher and a step response which is faster than the commercial valve.

Fig. 8. Pro/engineering model of the unstable valve prototype.

have: 1) a negative damping length , with and determined by the robust optimal design; 2) the nonmetering port angle (set to be 0 in conventional designs) is set so that the desired nonmetering port momentum flux coefficient is which deachieved; and 3) the diameter of the spool stem termines will also be obtained from the robust optimal design. Notice that the robust designs for the various uncertainties lie roughly on a straight line (see Fig. 3). The design parameters cannot be too large to be physically implementable. We choose 5.6 10 Ns m 8 10 m. This corresponds roughly to the robust design for the 31.5% diagonally structured uncertainty case (or 27% full block uncertainty case). We need to determine the geometric parameters , and (see Figs. 1 and 8). that The length of the valve is roughly equal to 70 mm and should not be too large. By choosing 10 mm, we achieve a damping length of 60 mm and a total valve length of 0.16 m. is then computed from m

(35)

18 mm to conform to The valve sleeve diameter is set to and (2), the spool stem a typical conventional valve. From diameter is 16.6 mm. Since the relationship between the nonmetering port angle and the inlet/outlet/total nonmetering port momentum flux , and cannot be determined analytically, coefficients a series of computational fluid dynamic (CFD) experiments are conducted. The relationship between and the momentum flux coefficients is shown in Table II. From Table II, the desired is achieved with a port angle of 30 . For optimal nonextreme values of , one can linearly interpolating the values in Table II. In summary, the robust design method gives the geometric pa18 mm 16.6 mm 70 mm rameters: 10 mm and 30 . A prototype valve that incorporates the unstable flow force effects was manufactured. Due to other manufacturing constraints, the exact geometry has to be modified somewhat from the optimal design. To demonstrate the improved performance

VII. CONCLUSION In this paper, the robust control design methodologies are applied to the robust physical design of systems. Specifically, the dimensions of an “unstable” valve where induced unstable steady flow forces are utilized to reduce the overall steady flow force that limit performance of single stage electrohydraulic valves. After an appropriate over-parameterization of the uncertainties, the robust optimal design problem is solved by designing an optimal controller for some static plant with structured uncertainty. The algorithm extends previous result in that the static plant to be controlled and the controller parameters do not need to be completely separated in the LFT representation. Comparison of the nominal optimal and robust optimal designs shows that under the nominal conditions, the steady forces can be totally neutralized by using the viscosity effect. In the presence of uncertainties, the robust optimal design utilizes both the viscosity effect and the nonorifice flux effect to optimize the steady flow force. The robust optimal design is then used for specifying the parameters of a valve prototype. The design is shown to outperform both the nominal optimal unstable valve design and a convention stable valve design. Many hardware design problems need to address robustness issues that control systems design routinely deal with. This paper is a successful demonstration of how robust control design methodology can be used to solve robust physical design problems. REFERENCES [1] H. E. Merritt, Hydraulic Control System. New York: Wiley, 1967. [2] K. Krishnaswamy and P. Y. Li, “On using unstable electrolydraulic valves for control,” ASME J. Dyn. Syst., Meas., Control, vol. 124, no. 1, pp. 182–190, Mar. 2002. [3] Q. Yuan and P. Y. Li, “Using viscosity to improve spool agility,” in Proc. Power Transmission Motion Control (PTMC), 2002, pp. 263–286. [4] Q. Yuan and P. Y. Li, “Using steady flow forces for unstable valve design: Modeling and experiments,” ASME Fluid System Technol. Div. Publication—2003 Int. Mech. Eng. Congr., vol. 10, no. 42924, pp. 27–37, 2003. [5] Q. Yuan and P. Y. Li, “Using steady flow forces for unstable valve design: Modeling and experiments,” ASME J. Dyn. Syst., Meas., Control, vol. 127, no. 3, pp. 451–462, 2005.

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[6] Q. Yuan and P. Y. Li, “An experimental study on the use of unstable electrolydraulic valves for control,” in Proc. Amer. Control Conf., 2002, pp. 4843–4848. [7] K. Zhou, Essentials of Robust Control. Englewood Cliffs, NJ: Prentice-Hall, 1998. [8] R. Smith and A. Packard, “Optimal control of perturbed linear static systems,” IEEE Trans. Autom. Control, vol. 41, no. 4, pp. 579–584, Apr. 1996. [9] A. Packard and J. C. Doyle, “The complex structured singular value,” Automatica, vol. 29, no. 1, pp. 71–109, 1993. [10] P. Y. Li and S. A. Dianat, “Robust stabilization of tone reproduction curves for the xerographic printing process,” IEEE Trans. Control Syst. Technol., vol. 9, no. 2, pp. 407–415, Mar. 2001. [11] Q. Yuan, “Design of high performance electrohydraulic valves that utilize unstable flow forces,” Ph.D. dissertation, Dept. Mech. Eng., Univ. Minnesota, , 2006. [12] Q. Yuan and P. Y. Li, “Extension of robust design of unstable valve: Diagonally structured uncertainty,” in Proc. Amer. Control Conf., 2006, pp. 4617–4618. QingHui Yuan (M’07) received the M.S. degree in electrical engineering and the Ph.D. degree in mechanical engineering from the University of Minnesota, Minneapolis, MN, in 2005 and 2006, respectively. Since 2005, He has been a Research Scientist with the Decision and Control Division, Eaton Corporation Innovation Center, Eden Praire, MN. His current research interests include modeling and control of electrohydraulic system and vehicle stability control.

Perry Y. Li (S’87–M’95) received the B.A. degree (hons.) and the M.A. degree in electrical and information sciences from Cambridge University, Cambridge, U.K., in 1987, the M.S. degree in biomedical engineering from Boston University, Boston, MA, and the Ph.D. degree in mechanical engineering from the University of California, Berkeley, in 1995. After graduation, he was part of the Research Staff with the Xerox Corporation, Webster, NY. In 1997, he joined the University of Minnesota, Minneapolis, where he is currently an Associate Professor and Deputy Director of the ERC for Compact and Efficient Fluid Power. His research interests include control systems, mechatronics, and fluid power systems. Dr. Li was a recipient of the 2000 Japan-USA Symposium for Flexible Automation Young Investigator Award.