MODELING AND CONTROL OF A DUAL-SOLENOID ACTUATOR FOR UNSTABLE VALVE

Transcription

1 MODELING AND CONTROL OF A DUAL-SOLENOID ACTUATOR FOR UNSTABLE VALVE Qinghui Yuan and Perry Y. Li Dept. of Mechanical Engineering University of Minnesota 111 Church ST. SE Minneapolis, MN 55455, USA. {qhyuan,pli}@me.umn.edu ABSTRACT Unstable valve is a new concept that is concerned with implementing fast responsive and cost effective single stage direct acting electrohydraulic valves. The main issue is that in high flow rate and high bandwidth operations, the force and power required of the solenoid actuator for stroking the spool become prohibitive. Our approach is to utilize flow force induced instability to enhance the spool agility, so that the demand on the solenoids can be alleviated. However, the open loop instability and the nonlinear nature of the valve make the controller design more challenging. In this paper, a sliding mode controller is designed to stabilize the system and to achieve the displacement trajectory tracking. A case study validates the controller. The simulation also shows that this control scheme provides unstable valves with a benefit of saving the electrical energy. NOMENCLATURE c d Discharge coefficient d Parameter of solenoid i Current through the coil of solenoid m Mass of spool and armatures of solenoids u Voltage across the coil of solenoid v Spool velocity w Orifice area gradient x Spool displacement B f Damping coefficient of spool dynamics C Geometry constant of valve F Solenoid force F steady Steady flow forces F transient Transient flow forces K f Spring coefficient of spool dynamics L Damping length L Inductance of solenoid R Resistance P l Load pressure P s Supply pressure α Geometry constant of valve β Parameter of solenoid λ Flux linkage µ Dynamic viscosity of fluid θ Vena contracta angle ρ Density of fluid THIS RESEARCH IS SUPPORTED BY THE NATIONAL SCIENCE FOUNDATION ENG/CMS

2 1 Introduction Electrohydraulics has widespread applications due to its unique advantages: high power density, high efficiency, ease of control, high stiffness, and flexibility. Electrohydaulic valves can be classified into multi-stage servo valves and single-stage proportional valves. Multi-stage servo valves include the pilot stage and the second stage. They use the electromagnetic actuator, like solenoid and torque motor, to directly drive the first stage. The pilot stage is then to drive the second spool stage with differential pressures. In general, multi-stage electrohydraulic valves are designed for the high performance applications. However, they are expensive both to acquire and to maintain, and not reliable due to the susceptibility to contamination. Single-stage proportional valves usually consist of a sliding spool, which is directly actuated by a pair of solenoids. Compared with multi-stage servo valves, single-stage valves have simpler configuration, thus making the design and manufacturing inexpensive. In addition, they have larger orifice area, and more loose clearance between spool and sleeve. Therefore, single-stage valves are less susceptible to the contamination. However, single-stage proportional valves are not suitable in higher performance applications, where the flow induced forces are significant and the electromagnetic actuators are not sufficient to provide large enough powers and forces to stroke the spool. Therefore, even though single-stage valves have the advantages of low cost and reliability, they are not currently in the forefront of the electrohydraulic valve field. The objective of the research aims at designing a single-stage valve to have the low or medium performance of multi-stage valves on the market. Actually, this idea gains increasing acceptance in industry [1]. The approach we adopt is to design the valve spools so that they are open loop unstable. The open loop unstable systems will be stabilized subsequently using feedback control. The same research method has been successfully applied on unstable aircraft design to improve the maneuverability [2]. Naturally, however, we still have two relevant concerns: (1) possibility of distabilizing the valve. Addressing to this question, our previous researches confirm that the open loop instability can be obtained by utilizing the flow induced forces [7] [8]; (2) the challenges for the controller design. We know that the intrinsic nonlinearity in the valve, as well as the open loop instability, would make it harder to find an appropriate controller. In this paper, we will focus on designing a nonlinear controller to stabilize the system. The rest of the paper are organized as follows. In section 2, we present the flow force models and their relationship with the open loop stability of the spool dynamics. Based on a popular solenoid model and flow force models, a sliding mode controller is presented to decompose the actions of two solenoids in section 3. In section 4, a case study is conducted to validate the controller. Section 5 contains some concluding remarks. 2 Flow induced forces and spool stability In this section, we briefly review the flow force models that incorporate some new components. Then we discuss how to affect the spool stability utilizing the flow induce forces. 2.1 Flow induced force models The origin of the flow induced forces has been investigated in the 195 s and 196 s, both theoretically and experimentally [5]. The flow induced forces can be classified into steady flow forces and transient flow forces. Steady flow forces are those that persist as the spool is at standstill and the flow rate across the valve is constant. The conventional analysis assume that the fluid is incompressible and inviscid, and that the flow directions in non-orifice ports are perpendicular to the spool axis. Then steady flow forces are mainly associated with the flow directions at the orifices. On the assumption of viscous incompressible fluid, however, the control volume analysis, 2

3 Actuator Q Q Sleeve Ps Pt Land Xv Rod L1 L2 Figure 1. Typical configuration of a four way direction flow control valve. The two Q ports are connected to the load (hydraulic actuator), and P s is connected to the supply pressure, and P t is connected to the return. In a single stage valve the spool is stroked directly by solenoid actuators. Damping length L is defined to be L := L 2 L 1. in which we choose the control volumes to encompass the fluid inside the chambers in Fig. 1, shows that the steady flow force models should also incorporate two new components, viscosity effect and non-orifice momentum flux (see [8] for more details). Then we have F steady (x) = 2c d cos(θ)(p s P l )wx+sign(x)cc 2 P s P l }{{} d w 2 x 2 P s P l αµlc d wx ρ ρ orifice flux }{{}}{{} non-orifice flux viscosity effect (1) where x is the spool displacement, L is the damping length that is defined by L := L 2 L 1 in Fig. 1, w is the orifice area gradient, α,c are geometry constants, µ is the fluid dynamic viscosity, ρ is the fluid density, θ is the Vena Contracta angle, c d is the discharge coefficient, P s is the supply pressure, and P l is the pressure across the load. These effects are also verified both by computational fluid dynamics (CFD) analysis and by the experimental study [8]. In particular, the viscosity effect can be used to successfully explain the experimental results in [6], in which the variation of the steady flow forces according to the damping length has been observed, but not well explained. On the other hand, transient flow forces are those responsible for the bulk acceleration of the flow rate when the spool is in motion and when the flow rate across the valve varies. Therefore, transient flow forces are proportional to the volume in the chamber and the rate of change of flow rate. We have F transient (ẋ) = ρlc d P s P l wẋ (2) ρ 2.2 Flow induced forces and spool stability The dynamics of the spool of a proportional valve is given by mẍ + B f ẋ + K f x = F (3) 3

4 k1 k2 k3 Bf m F Figure 2. The diagram of the spool dynamics where m the mass of spool and the armatures of the solenoid actuators in the valve, F is the electromagnetic forces of the solenoids in single-stage valve. B f and K f are the equivalent damping and spring coefficients contributed by transient and steady flow forces. According to Eqs. (1) (2), they are given by B f = Lc d w ρ(p s P l ) K f = 2c d cos(θ)(p s P l )w Cc 2 P s P l }{{} d w 2 P s P l x +αµlc d w ρ ρ K 1 } {{ } K 2 } {{ } K 3 (4) where K 1,K 2,K 3 represent the spring effect contributed by the orifice flux, non-orifice flux, and the viscosity effect respectively. Obviously, the stability of the system can be determined by the sign of B f and K f. Straightforwardly, the sign of B f is actually determined by the damping length L. Nevertheless, it is more complex when it comes to the sign of K f, which depends on the sum of K 1,K 2,K 3. We know that K 1 > since orifice momentum flux always tends to close the orifice regardless of whether the flow meters into the chamber or the flow meters out of the chamber [5]. K 2 < illustrates that non-orifice flux always tends to reduce the steady flow forces. K 2 < K 1 is implied in [8]. Similar to B f, the sign of K 3 has to be determined by the damping length L. In the current commercial valves, the geometry is designed to be L 2 > L 1, namely L >, then we know K 3 >, i.e. K f >, and B f >. The spool dynamics is stable. The solenoid actuators have to overcome the stable damping and spring effects by flow induced forces, which would increase significantly in the high flow rate and high frequency situations. In contrast, our research aims at designing the negative damping length L < so that the spool is open loop unstable. The open loop instability is due to B f <. In addition, the open loop gain x(s) F(s) is enhanced by the reduced K f (K 3 < ). The benefits of configuring the spool open loop unstable are to improve the spool agility and to alleviate the significant power/force requirement of solenoid actuators. In [4], the simulation study verified that (1) the spool in an unstable configuration result in faster under the limitation of actuation. (2) The unstable valves require less positive powers than the stable ones at the higher frequency. The preliminary experimental result validate the proposal as well [7]. 3 Modeling and control of a dual-solenoid actuator For an open loop unstable proportional valve, a challenging task is to design a controller to stabilize the system via feedback because of the nonlinear nature of the solenoid actuator. In this section, we first model a dual-solenoid actuator, and then propose a nonlinear controller to decompose the action of two solenoids. 4

5 i1 d+x Spool d-x +u1- +u2- i2 F1 F2 x Solenoid 1 Solenoid 2 Figure 3. Configuration of a dual-solenoid actuator in a directional valve. The spool also functions as a rigid connection of the armatures of two solenoids. 3.1 Modeling of a dual-solenoid actuator The nonlinear model is developed based on the differential configuration of a dual-solenoid actuator, as depicted in Fig. 3. Two solenoids produce contractile forces F 1 and F 2 which are acting on the spool. Under the operating condition, the spool also experiences the flow induced forces that are well modeled in [8]. The voltages over the coils are u 1 and u 2, and the current through the coils are i 1 and i 2. For simplicity, assume that two solenoids are identical, so two coils have the same resistance R. Then the electromagnetic forces are F 1 = β 2 F 2 = β 2 i 2 1 (d + x) 2 = λ2 1 2β i 2 2 (d x) 2 = λ2 2 2β (5) (6) where β and d are two parameters determined by the geometries and material properties of the solenoids, and x denote the displacement of the spool, in which x = as the spool is at the central position. The system dynamics of the dual-solenoid actuator is modeled as λ 1 = R d + x β λ 1 + u 1 λ 2 = R d x β λ 2 + u 2 ẋ = v v = 1 m (F 2 F 1 + F steady + F transient ) (7) where λ 1 and λ 2 are flux linkages, v is the velocity of the spool, m is the lumped mass of the spool and the armatures, F steady,f transient are given by Eqs (1) (2). The friction is neglected in the models. The currents through two coils are i 1 = λ 1 / L 1 i 2 = λ 2 / L 2 (8) where the inductances of coils are given by L 1 = β/(d + x) and L 2 = β/(d x). 5

6 3.2 Sliding mode controller design for a dual-solenoid actuator in an unstable valve The diffemorphism of Eq. (7) can be written in the form z 1 = y = x z 2 = ż 1 = v z 3 = ż 2 = 1 m [ ] 1 2β (λ2 2 λ 2 1) + F steady (x) + F transient (v) (9) And ż 3 = 1 [ 1 m β (λ 2 λ 2 λ 1 λ1 ) + df steady ẋ + df ] transient v dx dv = a(ξ) + b(ξ)[ λ 1 λ 2 ]u }{{} ũ where ξ = [ λ 1 λ 2 x v ] T is the state of Eq. (7), the control signal u = coils of the solenoids, a(ξ) = 1 df steady dx mβ ( R d x β λ2 2 + R d+x β λ2 1 )+ 1 m and df transient dv can be easily obtained from Eqs. (1) (2). [ u1 u 2 df steady dx v + m 1 (1) ] is the voltages across the df transient dv z 3, and b(ξ) = 1 mβ. Note that the coefficient of u, b(ξ)[ λ 1 λ 2 ], is not a square matrix, hence non-invertible. Fortunately, b(ξ) is invertible. Therefore, we want to manipulate the formula by defining ũ = [ λ 1 λ 2 ]u in Eq. (1) as an ad hoc variable, so that the problem is feedback linearizable. For instance, if ũ = b(ξ) 1 ( a(ξ) 3z 3 3z 2 z 1 ), we will obtain the closed-loop system... x = 3ẍ 3ẋ x (11) whose origin is globally exponentially stable since its poles are 1. Since state feedback linearization is sensitive to the process noise, we would like to design a controller that has better robustness performance. We will consider sliding mode control. In the sliding model control, trajectories are forced to reach a sliding manifold in finite time and to stay on the manifold for all future time. Motion on the manifold is independent of uncertainties. First of all, we consider stabilization of the system. We choose the sliding manifold s = a 1 z 1 + a 2 z 2 + z 3 = (12) On the manifold, the motion is governed by a 1 x+a 2 ẋ+ẍ =. Carefully choosing a 1,a 2 guarantees that x(t) reaches zero as t tends to infinity. We want to bring the trajectory to s = and maintain it there. The variable s satisfies ṡ = a 1 z 2 + a 2 z 3 + a(ξ) + b(ξ)ũ (13) Suppose the uncertainty of a 1 z 2 + a 2 z 3 + a(ξ,z 2,z 3 ) is bonded by ε, then we take ũ = b(ξ) 1 [ a 1 z 2 a 2 z 3 a(ξ,z 2,z 3 ) 2εsgn(s)] (14) 6

7 With V = (1/2)s 2 as a Lyapunov function candidate, we have V = sṡ ε s (15) It is clear that the trajectory reaches the manifold s = in finite time and, once on the manifold, it cannot leave it [3]. Once obtaining ũ, u can be determined by solving the following optimal problem min u u 2 2 st [ λ 1 λ 2 ]u = ũ (16) It is easily shown that the optimal solution yields the control signal u = λ 1 λ 2 1 +λ2 2 λ 2 λ 2 1 +λ2 2 ũ (17) The function of trajectory tracking can be easily implemented by modifying the definition of the sliding manifold s. Suppose x des (t) is the desired trajectory that is sufficiently smooth. Let z 1,des = x des,z 2,des = ẋ des,z 3,des = ẍ des. Then, chose the sliding manifold s = a 1 e 1 + a 2 e 2 + e 3 = (18) where e 1 = z 1 z des,1,e 2 = z 2 z des,2,e 3 = z 3 z des,3. Similar to the foregoing stabilization controller design, the control signal is given by Eq. (17), in which ũ = b(ξ) 1 [ a 1 (z 2 z 2,des ) a 2 (z 3 z 3,des ) + ż 3,des a(ξ) 2εsgn(s)]. 4 Case study A case study is conducted to verify the controller. Suppose the pressure supply is 2 psi and no load is connected with the valve. α,c are assigned the values according to [8], i.e., α = m 2, C = Ns 2 m 6. The diameter of the chamber D =.127 m(.5 in). Let the orifices be axisymmetric, then the gradient w = πd. Suppose the desired trajectory is a sinusoidal waveform with the magnitude m and the frequency 3 Hz. In addition, the parameters of the solenoid are identified experimentally to be β = NA 2 m 2,d = m. We use the controller proposed in Section 3.2 to implement the trajectory tracking for both a stable valve in which L =.2m, and a unstable valve in which L =.2m. In this simulation study, we assume P l =, so there is no hydraulic actuator dynamics. Therefore, we focus on investigating the trajectory tracking of the spool displacement. Without loss of generality, we choose the the desired displacement trajectory to be sinusoid. It is worth mentioning that the systems can be easily augmented to include the hydraulic actuator for the sake of pressure tracking and motion tracking though. For the unstable valve with L =.2 m, Figs. 4(a) 5(a) show the actual trajectory of the displacement, and various electrical and mechanical signals. It can be seen regardless of the open loop instability and nonlinearity of the system, the sliding mode controller successfully converges 7

8 Voltage(V) Position (m) 5 x 1 3 Desired Actual Current (A) Power (W) time (s) P 1 P 2 u 1 u 2 i 1 i 2 (a) L =.2 m Voltage(V) Position (m) 5 x 1 3 Desired Actual u 1 u Current (A) Power (W) i 1 i P 1 P time (s) (b) L =.2 m Figure 4. Sinusoidal tracking implemented by sliding mode controller for L =.2 m and L =.2 m. From top to bottom: the desired and actual displacements of spool; the voltage waveforms of two solenoids (control input); the currents through two solenoids; the electrical powers provided by amplifiers. F steady F transient F solenoid P solenoid (W) time (s) (a) L =.2 m F steady F transient F solenoid P solenoid (W) time (s) (b) L =.2 m Figure 5. Various mechanical signals corresponding to the displacement trajectory in Fig. 4 for L =.2m and L =.2m. From top to bottom: the steady flow forces; the transient flow forces; the solenoid forces; the mechanical power provided by the dual-solenoid actuator. 8

9 the actual displacement trajectory to the sinusoid with the desired amplitude and phase. In addition, even though the controller design is based on voltage control scheme, the optimal criterion in Eq. (16) ensures the complementary current waveforms, as shown in Fig. 4(a). Undoubtedly, this is an optimal decomposition of two solenoid actions from the angle of energy saving, because the electromagnetic forces are directly related to the currents as shown in Eqs. (5) (6), hence the current complementarity guarantees that no force counteraction occurs between two solenoids. The same controller scheme also works for the stable valve with L =.2 m, as can shown in Figs. 4(b) 5(b). Now we want to compare the power consumptions for the stable and unstable valves. First we consider the mechanical powers provided by the solenoids. In Fig. 5, the stable valve with L =.2m consumes the positive mechanical power, which is in the normal operation mode. However, the similar behavior is not observed for the unstable valve with L =.2m. On the contrary, the negative mechanical power for the unstable valve implies that the actuator works mostly in braking mode. It is also clear that the mechanical power fed into the solenoids is mainly provided by the transient flow forces in this case. In Fig. 5, the waveforms of F transient have the reserve phases. The mechanical power analysis coincides with the result in [4]. Next we are interested in the electrical power exchanges between the actuator and the power amplifier. The electrical powers for L =.2 m and L =.2 m are plotted in Fig. 4. it is not surprising to see that compared with the case where L =.2m, the electrical power for L =.2m tends to be more negative, i.e., the amplifier acts more like a power absorber than like a power supplier. However, it is worth noting that the electric power profiles are different from the mechanical power profiles, as can be seen in the figures. The disagreement can be explained by the powerdissipative and energy-storing components inside the solenoids. For each solenoid, we know u = Ri + λ where u is the voltage, i is the current, R is the resistance, and λ is the flux linkage. Since λ = Li where L is the inductance, we have u = Ri + L i + i d L dx v (19) The rate of flow of energy can be obtained by multiplying Eq. (19) with i and can be written as ui = Ri 2 + d dt ( ) 1 2 Li d L i2 2 dx v (2) where ui is the power provided by electric amplifier, 2 1 d L i2 ( dx v is the mechanical output, Ri 2 represents the power dissipated on the resistor, and dt d 12 Li 2) represents the rate of increase in the stored magnetic field energy. Obviously Ri 2 + d dt ( 12 Li 2) results in the instantaneous difference between the electric and mechanical power. In addition, note that the power is dissipated on the resistors. For stable valves, the dissipated energy, as well as the mechanical output, is offered by the electric amplifier. Nevertheless, for unstable valves, the energy dissipated by the resistor can be provided by the mechanical input. Therefore, unstable valves have the advantage of saving the electrical power. 9

10 5 Conclusion Unstable valve is a new concept of improving the single-stage proportional valve performance, by taking advantage of the open loop instability contributed by the flow induced forces. As a result, the open loop instability and the nonlinear nature of the valve result in more challenges in controller design. In this paper, a nonlinear sliding mode controller is presented, for the sake of stabilizing the system and implementing the trajectory tracking. The controller is then validated by the simulation study, in which the sliding mode controllers are designed to implement the sinusoidal displacement trajectory tracking. The controller can decompose two solenoid actions with the benefit of energy saving in that there is no counteraction between two solenoids. In addition, compared with the stable valve, the amplifier of the unstable valve behaves more like a energy absorber than a energy supplier. It implies that unstable valves have benefits of electrical energy effectiveness. These features will make unstable valves very competitive in the high performance applications. However, note that the mechanical powers are dominantly negative in Fig. 5(a), while some of the negative power is observably converted to the positive electrical power in Fig. 4(a). It implies that the braking operation provided by the solenoids does not effectively transform the mechanical energy into electrical energy due to the dissipation on the resistors. In order to enhance the energy saving property of unstable valve, we will investigate a new actuator with a fast responsive magneto-rheological (MR) brake in parallel with dual solenoids. The controller that can decompose the actions of two solenoids and the MR brake will also be studied in future. REFERENCES [1] J Cullman. Servo-performance in proportional valves. Machine Design, 65(12):4 42, June [2] D. Enns, H. Ozbay, and A. Tannenbaum. Abstract model and controller design for an unstable aircraft. Journal of Guidance, Control, and Dynamics, 15(2), March-April [3] Hassan K. Khalil. Nonlinear System, Third Edition. Prentice Hall, Upper Saddle River, New Jersey 7458, 22. [4] Kailash Krishnaswamy and Perry Y. Li. On using unstable electrolydraulic valves for control. Transactions of ASME. Journal of Dynamic Systems, Measurement and Control, 124(1):182 19, March 22. [5] Hebert E Merritt. Hydraulic Control System. John Wiley and Sons, [6] T. Nakada and Y. Ikebe. Measurement of the unsteady axial flow force on a spool valve. In Proceedings of the IFAC Symposium, Warsaw,Poland, May 198. Editors: IFAC Syposium staff with H.J. Leskiewcz, pp , Pneumatic and Hydraulic Components and Instruments in Automatic Control: Proceedings of the IFAC symposium, Warsaw, Pland, May, 198. [7] Qinghui Yuan and Perry Y. Li. An experimental study on the use of unstable electrolydraulic valves for control. In Proceedings of the 22 American Control Conference, Anchorage, Alaska;, pages , May 22. [8] Qinghui Yuan and Perry Y. Li. Modeling and experimental study of flow forces for unstable valve design. In Proceedings of the 23 ASME international mechanical engineering congress, Washington DC., USA, number IMECE , Nov