DSCC2012-MOVIC

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1 ASME 5th Annual Dynamic Systems and Control Conference joint with the JSME th Motion and Vibration Conference DSCC-MOVIC October 7-9,, Fort Lauderdale, Florida, USA DSCC-MOVIC-8784 DISPLACEMENT CONTROL OF HYDRAULIC ACTUATORS USING A PASSIVITY BASED NONLINEAR CONTROLLER Meng Wang ERC for Compact and Efficient Fluid Power Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota wangx833@umn.edu Perry Y. Li ERC for Compact and Efficient Fluid Power Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota pli@me.umn.edu ABSTRACT To increase the efficiency of hydraulic systems by eliminating valve throttling losses, a direct displacement open circuit is proposed to control a single rod hydraulic actuator. The circuit provides three control inputs, including the displacement of a variable displacement pump, the opening area of a proportional valve, and the position of a directional valve. Pump control has a low bandwidth, but the efficiency is high due to the lack of throttling losses. Valve control has a high bandwidth, but the throttling loss is high. A novel approach has been proposed to distribute the control efforts between the pump and the proportional valve considering both control bandwidth balancing and throttling loss reduction. The proportional valve will follow a high frequency opening profile, while the nominal valve opening is large, and the pump output flow will follow a low frequency demand. Experimental results validate the effectiveness of the proposed approach. INTRODUCTION Methods to improving the efficiency of hydraulic systems have received a lot of attention in recent years. Hydraulic actuators are typically controlled using two approaches, throttling valve control and displacement control. Valve control has the advantage of high precision and high bandwidth due to the small inertial being moved. One power source can drive multiple actuators at different pressure levels by using throttling valves to regulate the pressure of each branch to the desired level. The disadvantage of this approach lies in the fact that a significant amount of energy is wasted through the throttling valves as heat, Address all correspondence to this author. which drastically degrades the system efficiency. To partially solve the problem, a load sensing LS circuit is created. In this circuit, the flow source is controlled to adapt to the pressure of the highest load branch, so that only a small pressure difference is present between the pressure source and the required pressure level, which reduces the throttling loss. However, the throttling loss is still significant unless the required pressure levels of the different load branches are similar to one another; otherwise only the load branch that requests the highest pressure can be controlled efficiently. Another drawback is the challenge to maintaining the stability of a LS system []. In displacement control, the amount of flow is controlled to feed the actuator at the required pressure directly, no excess flow or extra pressure drop are provided. In this way, no throttling valve is required theoretically, which reduces the system loss considerably. However, the control bandwidth of a variable displacement pump/motor is lower than the control bandwidth of a valve. In addition, one variable flow source can drive only one circuit, and multiple variable displacement pumps VDPs are required for a multiactuator system. There are two principally different hydraulic circuit configurations to implement direct displacement control. One is an open circuit, in which the pump inlet and the actuator return line are connected to the hydraulic tank separately. A typical configuration is shown in Fig., which includes a variable displacement pump/motor and four -way valves. The circuit can realize fourquadrant operation. Since the pump can be driven as a motor, energy regeneration can be achieved. The other configuration is a closed circuit, in which the actuator return line is connected to the inlet of the pump inlet. Typical hydraulic actuators are asymmetric e.g. cylinder, when us- Copyright by ASME

2 Figure. Simplified circuit diagram of the open circuit solution implemented on two drives [] ing close circuits, the unequal fluid volume needs to be compensated [] [3]. Compared with open circuits, close circuits require extra components besides the variable flow source to compensate the unequal flow volume, which makes the hydraulic circuits complex. Notice that not all the hydraulic pumps can be driven as motors, we propose an open circuit to accomplish direct displacement control of a single rod hydraulic actuator, as shown in Fig.. The circuit consists of a variable displacement pump, a 4-way directional valve, and a proportional valve. The 4-way directional valve is operated in an open-loop manner, so no valve position feedback is required. reducing throttling loss point of view, we would like to reduce the usage of the throttling valve, and to maintain it as fully open as possible. From the control bandwidth point of view, we would like the throttling valve to provide a control effort of high frequency, and the variable flow pump to provide a control effort of low frequency. In this paper, we proposed an approach to distributing the total control efforts between the pump and the valve, so that the valve has a large nominal opening with a high frequency low amplitude varying opening on top of the nominal opening and the pump provides a flow with low frequency. In sec, the displacement control open circuit will be described. In sec 3, a passivity based nonlinear controller will define the total control efforts, which can drive the hydraulic actuator to track a pre-defined trajectory. The novel approach to distributing the control efforts between the pump and the throttling valve will be discussed in sec 4. Experimental validation of the proposed hydraulic circuit and the control approach will be presented in sec 5. Finally, some concluding remarks and the future work will be covered in sec 6. DIRECT DISPLACEMENT CONTROL IN AN OPEN CIRCUIT. System Dynamics In our modeling, the dynamics of the actuator, and the pressure dynamics inside the actuator chambers are considered. The dynamics of the directional valve and the proportional throttling valve are ignored. Figure. Direct displacement control open circuit Although the circuit includes simple and limited amount of hydraulic components, it provides the control authority to achieve hydraulic actuator manipulation and chamber pressure regulation. The directional valve is utilized to change the flow direction, and it remains fully open to reduce throttling loss. In this circuit, no charge pump or accumulator is available on the return line to maintain the return chamber pressure; instead, a one-way proportional valve is introduced on the return line. The variable flow source, the throttling valve and the directional valve are utilized together to achieve the actuator trajectory tracking. A novel approach to distributing the control efforts between the variable flow pump and the proportional valve is proposed. From Figure 3. Hydraulic configurations for different directional valve operations The hydraulic cylinder is modeled as a mass acted upon by the pressure forces from the two actuator chambers, the linear viscous friction force, and the load force. mẍ = P A P A bẋ + F L + d where m is the mass of the cylinder rod, and x is the position Copyright by ASME

3 of the cylinder rod. A and A are the areas of the cylinder cap end and the rod end. b is the viscous friction coefficient,f L is the carrying load force, and d represents the unknown disturbance force. Depending on the position of the directional valve, the circuit has two configurations, as shown in Fig. 3. Q is the flow rate entering the supply chamber. V and V are the volumes in the respective chambers and hoses when x =. The throttling valve on the return line is modeled as an orifice. The orifice coefficient is defined as K v = C d A max ρ, with A max being the maximum valve opening area, and u [, ] being the normalized throttling valve command. ρ is the fluid density, and C d =.6 is the coefficient of discharge. The two positions of the directional valve are denoted by u d = {, }, and the respective circuits are shown in Fig. 3a and Fig. 3b. Let the tank pressure be P t, the chamber pressure dynamics are modeled as: when u d = Fig.3a: = V + A x Q A ẋ a = V A x A ẋ uk v P P t signp P t b when u d = Fig.3b: = V + A x A ẋ uk v P P t signp P t = V A x A ẋ + Q c d with U total = Hx, u d Q + Gx, u d ΨP, P, u d u Hx, u d = Gx, u d = { A ΨP, P, u d = V +A x u d = A u d = V { A x A V A x u d = A V +A x u d = { K v P P t signp P t u d = K v P P t signp P t u d = 4 This section will focus on designing a U total to achieve the cylinder trajectory tracking. Firstly, we will develop a desired compensation control law DCCL [4] for cylinder trajectory tracking from the mechanical side in sec 3.. Secondly, an h function will be proposed to support the design of a pressure hydraulic force tracking error based compressible energy storage function in sec 3. to account for the pressure tracking error. Finally, both effects are considered together to develop U total in sec Trajectory tracking using a DCCL controller Assume x d t, ẋ d t, ẍ d t, and... x d t are available and smooth. Following the typical mechanical robot motion controller design procedure, a DCCL controller was developed to accomplish cylinder position tracking. Let e = x x d, and define the following reference velocity r, and the reference velocity error e v : r = ẋ d λe; e v = ẋ r = ė + λe 5 with λ >, the dynamic of e v becomes: mė v = mẍ d λė bė + ẋ d + F H + F L + d 6 3 Passivity based controller The control objective is for the actuator position xt to track a reference trajectory x d t. Introducing one variable as F H = P A P A to model the hydraulic force acting on the actuator, the cylinder dynamics become: The Lyapunov function considering the position and velocity tracking errors is defined as: W mech = me v + K pe 7 An ideal hydraulic force F v is proposed as: F v = mẍ d λė + bė + ẋ d F L K v e v K p e 8 mẍ =F H bẋ + F L + d F A H =U total V + A x + A ẋ 3 V A x }{{} Lx With this hydraulic force, the time derivative of W mech becomes: Ẇ mech = K v e v λk p e e v d + e v F 9 where F = F H F v. 3 Copyright by ASME

4 3. Pressure Error Storage Function The Lyapunov function defined in Eqn 7 needs to be augmented with the pressure error or hydraulic force error. For a two chamber single rod hydraulic actuator, we proposed a storage function in terms of pressure error or hydraulic force error. The storage function is formulated via a proposed monotonic function h : R R [5]. hσ := A lnv + A σ + A lnv A σ Let F H = A P A P, x be the current actuator force and position, and F v be the desired actuator force. Define x fd and x from: hx = h x F H, hx fd = h x F v Consider the pressure error storage function: x W F = [hσ hx fd ] dσ x fd which is derived considering the fluid compressible energy in both chambers. Recall the definitions in Eqn. 6 and Eqn. 8, Ẇ F becomes x d dt W F = [hx hx fd ] ẋ ḣx fd dσ x fd = F ẋ U total F v x x fd 3 Notice that h is continuous and monotonic, from mean value theorem: hx hx fd = Lαx x fd, for α x fd, x 4 where L is defined in Eqn. 3. Eqn. 3, Ẇ F is manipulated into: Ẇ F = F ẋ + Plug this relationship into F U total F Lα v Passive Control Law Now augment W mech defined in Eq.7 with W F defined in Eq. to define the total energy storage function: W total = me v + x K pe + [hσ hx fd ] dσ 6 x fd The logic of defining h is based on the course note from ME887. Topics in Control: Passivity and Control of Interactive Mechanical and Fluid Powered Systems and a paper in progress The time derivative Ẇtotal becomes: Ẇ total = K v e v λk p e e v d r F + F Lα Propose a control law U total as: Ẇ total becomes: U total F v 7 U total = Lxr + F v λ 3 F + u rob 8 Ẇ total = K v e v λk p e e v d λ 3 Lα F + F Lr + u rob Lα 9 where L = Lx Lα. Analysis in Ref [6] has shown that L is bounded by F as: L γ F F, for some finite positive γ F. Therefore, by selecting u rob = K rob F, with a large feedback gain K rob > γ F, Ẇ total becomes: Ẇ total K v e v λk p e e v d λ n F with λ n = λ 3 Lα + K rob γ F > The Lyapunov function W total is positive definite in e, e v, and P. When d =, the time derivative of the Lyapunov function Ẇtotal is negative definite in e, e v, and P. Therefore, the tracking errors e, e v, and P will converge to zero exponentially. When d is bounded, by selecting K v large enough, all the tracking errors can converge to an arbitrarily small value exponentially, as far as the control efforts do not saturate [6]. 4 Control Effort Distribution The previous section has defined the desired total control effort Utotal in Eq. 8. This section will discuss how to distribute the three control inputs Q, u, and u d, so that:. The trajectory tracking performance is guaranteed: U totalt :=Hx, u d Q + Gx, u d ΨP, P, u d u. The pressure in both actuator chambers stay bounded, with the upper bounded denoted by P i, the lower bounded denoted by P i, and the tank pressure denoted by P t : P i P i t P i > P t, i, 3. The usage of the the proportional valve u is minimized to reduce the throttling loss. 4 Copyright by ASME

5 4. The control effort Utotal is achieved in the following manner: Gx, u d ΨP, P, u d u is used to track the high frequency component of Utotal t, and Hx, u d Q is used to track the low frequency component of Utotal from Eqn.. 4. Directional Valve As defined in Eqn.4, the sign of Hx, u d and Gx, u d are consistent with the sign of u d. If the pressures in both chambers stay above the atmosphere pressure, ΨP, P, u d is positive. Therefore, u d is determined by the sign of U total : u d t = { when U total when U total < 3 4. Flow Q and Throttling Valve u To distribute the total control efforts between Q and u after determining the directional valve, two factors are considered. On one hand, to minimize the throttling loss, the usage of the proportional valve u should be minimized. In other words, we would like u to be as close to as possible. On the other hand, Q and u have different control bandwidths, and we would like to track the high frequency component of Utotal t using u, and to track the low frequency component of Utotal using Q. To achieve this control efforts distribution, we proposed the following method. We assume the dynamics of the desired pump flow Q des follows: Q des = λ Q Q des + λ Q w + Q des U total = H Q des + GΨ ū des + GΨ ũ des 4 where Q des and ū are some pre-defined nominal values of Q des and u des. Here, we would like ū to be as close to as possible, but ū is constrained by the variation of ũ des, because u des = û des + ũ des. A similar argument can be applied to Q des. Since the flow from the variable displacement pump is always greater than, a positive nominal flow Q des is introduced to account for this effect so that the process noise w of the desired pump flow dynamics can be modeled as a zero-mean white noise. λ Q is the time constant of the desired pump flow dynamics, which quantifies the control bandwidth of the desired pump flow. The second equation in Eqn.4 is manipulated into: Utotal GΨū des = H } GΨ {{} GΨ Q des + ũ des 5 ν where ν plays the role of a measurement, and ũ des plays the role of a measurement noise. Here we model ũ des as a zero-mean white noise. Next, we would like to estimate Q est by considering the pump flow dynamics in Eqn.4 and the measurement ν, while minimizing the variation of w and ũ des. The variance of the process noise w is denoted by Q J, and the variance of the measurement noise ũ des is denoted by R J. we propose an estimator as: ˆQ des = λ Q ˆQdes + L est ν H GΨ ˆQ est and the following objective function will be minimized: J = tf t wq J 6 w + vr v dτ 7 With this objective function, the design of the estimator feedback gain L est follows the design of a typical deterministic Kalman filter [7]: e = λ Q P e + λ Qq J H Pe R J GΨ L est = P eth 8 GΨR J ˆQ est t is the desired flow we would like the variable displacement pump output flow Q to track. To re-produce U total, the desired proportional valve opening fraction is: u des = ū des + ũ des J = U total H ˆQ est GΨ 9 Note that, the pump flow ˆQ e st in Eqn. 6 and the throttling valve opening u des in Eqn. 9 are feasible only when the pressure in both chambers stay bounded. If the pressure is higher or lower than the pressure bound defined in Eqn., ˆQ est and u des needs to be adjusted to account for the pressure compensation. A framework of manipulating the chamber pressures while maintaining the actuator trajectory tracking was presented in [8]. 5 EXPERIMENTAL RESULTS The proposed direct displacement control open circuit, and the control strategy was implemented experimentally. In the experimental set up, instead of implementing a variable displacement pump, a virtually variable displacement pump VVDP was used as the variable flow source [9]. Figure 4 shows the hydraulic configuration of the VVDP. A constant flow source a fixed displacement pump is pulse width modulated PWMed by a high speed on/off valve, and the average flow going to load 5 Copyright by ASME

6 is smoothed out by an accumulatoror the hoses and the connectors. By controlling the PWM duty ratio, we can vary the mean flow going to the load branch. This is equivalent to varying the displacement of a variable displacement pump. Since the on/off valve has low loss in either on or off states, this approach potentially can produce a variable flow with minimum throttling losses. Actuator Champer Pressure [psi] Pcap Prod Figure 6. Chamber Pressures Figure 4. Software enabled VVDP circuit low frequency, and the high frequency component of Utotal is realized via the proportional valve. The proportional valve is controlled to be as open as possible, limited by Q and u being saturated. The distribution of Utotal between Q and u is shown in Fig. 8. Since the VVDP cannot function as a hydraulic motor, no negative flow can be provided, and therefore the lowest bound of Q is lpm. When Q = lpm, all the control action is accomplished via the throttling valve. The hydraulic actuator reference trajectory is a triangle wave, which corresponds to a full stroke traveling within 4sec. As shown in Fig. 5 and Fig. 6, the actuator can achieve a good trajectory tracking, and both chamber pressures stay bounded. actuator position [cm] position tracking error [cm] 3 Figure 5. reference position measurement Actuator position reference and the tracking error Direction Valve Ud Proportional Valve u [%] Pump Flow Q [lpm] The directional valve command, the proportional valve command, and the supply flow command are shown in Fig. 7. In this experiment, ū des is set to be.8, which means the nominal opening of the proportional valve is 8%. The Kalman filter presented in sec 4. produces a smooth flow command with Figure 7. and the VVDP Control efforts from the directional valve, the throttling valve, 6 Copyright by ASME

7 U * total x 5 Figure 8. U * total H Qd GΨ u U total distribution between Q and u [6] Wang, M., and Li, P. Y.,. Passivity based adaptive control of a two chamber single rod hydraulic actuator. In Proc. of The ACC, Montreal, Canada. [7] Goodwin, G. C., Graebe, S. F., and Salqado, M. E.,. Control System Design. Prentice-Hall, Inc., Upper Saddle River, NJ. [8] Wang, M., Li, P. Y., Tu, H., Rannow, M., and Chase, T. R.. [9] Tu, H., Rannow, M. B., Wang, M., Li, P. Y., Chase, T. R., and de Ven, J. V. Design, modeling, and validation of a highspeed rotary pwm on/off hydraulic valve. In submitted to ASME JDSMC. 6 CONCLUSION AND FUTURE WORK In this paper, an open circuit is proposed to achieve direct displacement control of a single rod hydraulic actuator. The circuit provides three control inputs, including the output flow from a virtually variable displacement pump, the opening area of a proportional valve, and the position of a directional valve. A passivity based nonlinear controller was developed to define the total control efforts, so that the actuator can achieve trajectory tracking. A novel control effort distribution method was proposed to distribute the total control efforts between the variable displacement pump and the proportional valve, so that the pump flow follows a low frequency demand, and the valve opening area follows a high frequency but nominally large open profile. The approach can coordinate the difference on control bandwidth from the pump and the valve, while minimizing the throttling loss. Experimental tests prove the feasibility and the effectiveness of the approach. ACKNOWLEDGMENT This material is based upon work performed within CCEFP, supported by the National Science Foundation under Grant No. EEC REFERENCES [] Hewett, A., 994. Hydraulic circuit flow control. US Patent No [] Heybroek, K., 8. Saving energy in construction machinery using displacement control hydraulics-concept realization and validation. PhD Thesis, Linkoping University. [3] Rahmfeld, R., and Ivantysynova, M.,. Development and control of energy saving hydraulic servo drives. In Proc. of the st FPNI-PhD Symposium, Hamburg Germany, pp [4] Horowitz, R., and Sadegh, N., 99. Stability and robustness analysis for a class of adaptive controllers for robotic manipulators. Int. J. of Robotics Research, 9, pp [5] Li, P. Y., and Wang, M.,. Passivity based nonlinear control of hydraulic actuators based on an euler-lagrange formulation. In Proc. of the ASME Dynamic Systems and Control Conference, Arlington, VA. 7 Copyright by ASME