HYDRAULIC LINEAR ACTUATOR VELOCITY CONTROL USING A FEEDFORWARD-PLUS-PID CONTROL

Transcription

1 HYDRAULIC LINEAR ACTUATOR VELOCITY CONTROL UING A FEEDFORWARD-PLU-PID CONTROL Qin Zhang Department of Agricultural Engineering University of Illinois at Urbana-Champaign, Urbana, IL 68 ABTRACT: A practical approach to design a feedforward-plus-proportional-integral-derivative (FPID) controller for accurate and smooth velocity control on a hydraulic linear actuator is presented. The integrated controller consists of a feedforward loop and a PID loop. The feedforward loop is designed to compensate for the nonlinearity of the electrohydraulic system, including the deadband of the system and the nonlinear flow gain of the control valve. A feedforward gain schedule determines the basic control input based on the command velocity. The PID loop complements the feedforward control via velocity tracking error compensation. Also presented are comparisons of the experimental results from an open-loop feedforward controller, a conventional PID controller, and an integrated feedforward-pid (FPID) controller. In each experiment, the controllers were tuned to provide optimal responses. Results demonstrate that the FPID controller is capable of improving the accuracy and dynamic performance of velocity control of a hydraulic linear actuator. INTRODUCTION Hydraulic actuators are widely used on mobile equipment and robots, due to their high power density, environment tolerance, and compact size (Backe, 993). One of the fundamental tasks in designing hydraulic actuating systems is the development of effective velocity control of the actuator using a control valve (Burrows, 994). The adoption of electrohydraulic (E/H) proportional valves, which are usually 4-way infinite position directional control valves, increased the efficiency and performance of hydraulic actuating systems (Caputo, 994). Most E/H proportional valves have undesirable inherent characteristics including high nonlinearity, asymmetric flow gain, and hysteresis. The nonlinear behavior may vary with changes in system load and operating conditions. Nonlinear effects can cause performance deterioration in terms of response speed, control accuracy, and stability of the system (Anderson, 988). Nonlinear characteristics make it difficult to adequately control an E/H actuating system using a classical proportional-integral-derivative (PID) controller (Zhang et al. 996). Many researchers attempted to develop adequate controls for accurate and efficient control of nonlinear systems. Edge and Figueredo (987) developed a model reference adaptive control (MRAC) for a hydraulic servosystem utilizing a linear model. The MRAC algorithm modified the controller parameters to make the plant track the system model. Researchers have successfully developed various neural networkbased controllers for E/H systems (Newton, 994; Watton and Kwon, 996; Qian et al. 998). Neural network-based control is capable of offering good nonlinear control and an ability to learn. Fuzzy logic control is another capable control technology for handling the high nonlinearity inherent in hydraulic actuating systems (Tessier and Kinsner, 995; Wang and Chang, 998; Zhang et al.

2 999). Zheng et al. (998) applied an adaptive learning approach to control the position of a hydraulic linear actuator using an E/H proportional valve. The system is capable of compensating for major nonlinearities, including deadband, saturation, and friction, to achieve satisfactory performance. This paper presents a practical approach to design a feedforward-plus-proportional-integralderivative (FPID) controller for accurate and smooth velocity control on a hydraulic linear actuator. The integrated FPID controller consists of a feedforward loop and a PID loop. The feedforward loop is designed to compensate for the nonlinearity of the electrohydraulic system, including the deadband of the system and the nonlinear flow gain of the control valve. A feedforward gain schedule determines the basic control input based on the command velocity. The PID loop complements the feedforward control via the velocity tracking error compensation. DECRIPTION OF THE HYDRAULIC LINEAR ACTUATOR YTEM Hydraulic cylinder actuators are widely used in many fluid power systems, such as robots, aircraft, construction machinery, and agricultural machinery. To develop an adequate velocity controller for hydraulic cylinder actuators without loss of generality, a hardware-in-the-loop E/H linear actuating system simulator was developed. This interactive simulator consists of a computer-based controller, a pulse width modulation (PWM) valve control driver, an electrohydraulic proportional directional control valve, a hydraulic linear actuator, and a load adjustable cylinder (Fig.). The load adjustable cylinder is linked to the actuator and allowed a positive load to resist the motion of the actuator. The linear actuator is a single-rod, double-acting hydraulic cylinder; its velocity is determined by the flow rate supplied to the cylinder. A Packer-Hannifin (Elyria, OH) four-way proportional E/H directional control valve is used to regulate the flow rate (Fig. ). In this system, the orifice area of the cylinder-to-tank (C-T) port is always larger than that of the pump-to-cylinder (P-C) port, and a positive load is always applied to the actuator. It is reasonable to assume that the actuator velocity is controlled solely by the P-C orifice. Based on flow continuity theory, the actuator extending motion can be described using a flow continuity theory when the friction and leakage are neglected. The system momentum can be determined by actuating force, system load, and system mass. The actuator retracting motion can be modeled in a similar manner to the extending motion. dy V dp = A + () dt β dt Q d y P A P A = m + F () dt Linear ECU M

3 where, Q is the supplying flow rate to the actuator, A and A are the cylinder areas of the headend and the rod-end, P and are cylinder pressures in the head-end and the rod-end, V is the cylinder chamber volume at the head-end, β is the fluid bulk modulus, F is the external force acting on the cylinder, m is the mass of the actuator, y is the displacement of the actuator, and t is the time. A directional control valve regulates the flow rate by adding flow restriction to the pathways through an adjustable orifice. Therefore, the supplying flow rate to the head-end chamber of the cylinder can be determined by orifice equations. Q = C d A ori ρ ( P ) P where, A ori is the valve orifice area, C d is the orifice coefficient, ρ is the fluid mass density, and P is the pump supply pressure. For a solenoid driven spool valve, the orifice areas are determined by the displacement of the spool, which is controlled by the solenoid actuating force against the centering springs if the friction and flow forces are neglected. The solenoid actuating force is regulated by the control input voltage signal, u, to the PWM driver of the solenoid valve. Aori = g du (4) where, g is the gain coefficient of the solenoid drive, and u is the control input voltage signal. d Furthermore, the flow discharge coefficient C d is also a function of spool displacement x, which is a function of control signal u. Introducing a flow gain coefficient g F to the orifice equation, the valve orifice equation can be represented as follows. Q = g u P (5) F P g F = Cd g d (6) ρ where, g F is the flow gain coefficient. Modifying the orifice equation using Q, modifying the momentum equation using actuator motion mass, and neglecting cylinder friction and back pressure, eqn. (7) and (8) give the nonlinear state (3) P Q P-C Q C-H P P V F A V A, V, Q P-C Q C-T P T Figure. Asymmetric hydraulic linear actuator system with a directional control valve.

4 model of extending motion. imilarly, the nonlinear state model of retracting motion can be developed. These models provide the fundamentals for accurate velocity control on the hydraulic linear actuator. V dp g Fu P P = Av + β dt (7) dv P A = m + F dt (8) where, v is the velocity of the actuator. Equations (7) and (8) were linearized about an operating point to create a set of differential equations with constant coefficients. The solutions of these equations can be used to design the control system for the E/H linear actuator. and, ( δp ) d kuδ u k pδp = Aδ v + C (9) dt d( δv) δ P A = m () dt Q ku = () u k Q u= u P = () P P = P V C = (3) β where, k P and k u are, respectively, the gain factors of cylinder chamber pressure and input control signal, C is the hydraulic damping factor, δ P, δ Q, and δ v are, respectively, the increment of cylinder pressure, pump flow supply, and actuating velocity, and δ u is the increment of control input signal. The velocity of the actuator is governed by the flow rate of the supplied fluid, which is determined by the pressure drop across the valve orifice. The dynamic model of the system studied is given as eqn. (4). C m A & = & (4) δp kuδu Aδ x k pδp where, δ P & is the increment of pressure changing rate in the cylinder, x& is the velocity of the linear actuator, and δ x& is the increment of cylinder velocity. CONTROLLER DEIGN The form of eqn. (4) falls under the category of systems that is feedback linearizable (Isidori, 989) as follows. ( x) g( x)u ( x) x & = f + (5) y = h (6)

5 G FF (s) u ff v d + e - G PID (s) u pid u G HYD (s) v a v fa F(s) v s d s Disp. ensor Figure 3. chematic block diagram of the feedforward-plus-proportional-integralderivative (FPID) controller. G FF (s) is the feedforward gain, G PID (s) is the overall gain of the feedback PID controller, G HYD (s) is hydraulic system gain, and F(s) is the filter. n m p where x R is the state of the system, u R is the control variable, and y R is the system output. The resulting equation of the linear plant of the nonlinear system is: v ( s) = ku ωn ( ) u( s) A s s + ξω s + ω. (7) n n The model indicates that the velocity control system for an E/H linear actuator is a third-order system. The dynamic behavior of such a system is affected by the spool valve dynamics, the system pressure, and the size of the hydraulic cylinder. A model-based controller was developed based on the transfer function of this linearized system for the feedforward-plus-feedback control system. This controller uses a simple proportional-integral-derivative (PID) algorithm to stabilize the system. G where, K D, K I, and base controller. K P D (8) s K are, respectively, the derivative, integral, and proportional gains of the I ( s) = K + + K s P The feedforward loop in this FPID controller (Fig. 3) consists of an inverse valve transform, which provides the steady-state control characteristics of the E/H valve in terms of actuator velocity and control voltage to the PWM driver. The valve transform was developed by applying a series of excitation signals to the system and measuring the response. The valve transform was then converted into an inverse valve transform using the following equation. u = f ( v) (9) The inverse valve transform was used as the gain scheduled in the feedforward control, which was integrated into the feedforward loop of the FPID controller. The resulting controller is capable of compensating for valve deadband and asymmetric flow gain via the feedforward gain schedule, and compensating for velocity tracking error via the feedback PID gains. The FPID controller was implemented on the hardware-in-the-loop E/H linear actuator system simulator. The robustness of the FPID control was evaluated based on its performance and stability. Performance robustness deals with unexpected external disturbances and stability robustness deals with internal structural or parametric changes in the system. The design of this FPID controller was

6 based on the worst case scenario of the system operating conditions in tuning both the PID gains and the feedforward gain. REULT AND DICUION In implementing the developed controller on the hardware-in-the-loop E/H actuating system simulator, various tests have been conducted for evaluating the performance of the FPID controller versus a feedforward controller and a PID controller. The tests were performed by disconnecting either the PID loop or the feedforward loop of the FPID controller, and carefully tuning the parameters in the remaining loop to optimal levels. Experimental results indicate a superior performance of the FPID controller over both the feedforward controller and the PID controller. Results also indicate that the controller is capable of compensating for external disturbances and internal gain changes, which verified the robustness of the FPID controller. Figure 4 shows the velocity tracking accuracy of a feedforward controller with rough-tuned (Fig. 4A) and well-tuned (Fig. 4B.) inverse valve transforms. The rough-tuned inverse valve transform consists of five characteristic points, including the neutral point, the deadband points and the Time (sec) (A) Roughly tuned inverse valve transform (consists of 5 points) Time (sec) (B) Well tuned inverse valve transform (consists of 35 points) Figure 4. Velocity control performance on the hydraulic actuator from feedforward controls with inverse valve transforms. The dashed line is the desired velocity and the solid line is the actual obtained velocity.

7 saturation points for both positive and negative motion of the cylinder actuator. Test results indicate that a roughly tuned feedforward controller can effectively compensate for the system deadband. However, it resulted in noticeable errors in velocity control, especially when the external load changed, because it is not capable of compensating for the nonlinear flow gain, and its characteristic points do not reflect load variations. The mean square root (RM) error in the velocity tracking from a ramp test was. mm/s (Fig. 4A). The velocity tracking accuracy can be greatly improved by adding more points to the inverse valve transform, which provides the capability of compensating for the asymmetric nonlinear flow gain of the control valve. Figure 4B presents an almost perfect velocity tracking control by using a 35 points inverse valve transform. The RM error in velocity tracking was 6.7 mm/s, which is a 5% reduction in velocity tracking error compared to the five-point valve transform. Notice that the valve transform is external load and flow rate sensitive, which results in variations in the optimal valve transform. One well-tuned inverse valve transform would not be sufficient to ensure accurate velocity tracking under all operation conditions in an open-loop feedforward controller. A PID controller is capable of compensating for the tracking error to achieve more accurate velocity control. The evaluation tests investigated the velocity tracking performance using a well-tuned PID controller. Figure 5A shows the velocity tracking accuracy using a controller with a proportional Time (s) (A) P control with K P = Time (s) (B) PI control with K P =. and K I =.5 Figure 5. Velocity control performance on the hydraulic actuator from different PID controllers. The dashed line is the desired velocity and the solid line is the actual obtained velocity.

8 gain factor of.. The controller resulted in a noticeable error in velocity control (RM error of.mm/s) without loss its stability. Adding an integral term to the controller improved the tracking accuracy in the low velocity region. However, higher gains would result in larger errors in velocity tracking (RM error of.7 mm/s) and cause a visible jerky motion at higher velocities (Fig. 5B). uch problems resulted from using high feedback gains to compensate for the large deadband with the PID controller. By integrating both the feedforward and the PID loops, an integrated FPID controller was created. The FPID controller compensated for the large deadband and nonlinear flow gain (Fig. 6) using the feedforward loop and compensated for the disturbance generated tracking errors using the PID loop Time (s) 6 (A) Ramp input Time (s) (B) ine wave input Time (s) (C) tep input Figure 6. Velocity control performance on the hydraulic actuator from the FPID control with a roughly tuned inverse valve transforms. The dashed line is the desired velocity and the solid line is the actual obtained velocity.

9 Figure shows the velocity tracking accuracy of the cylinder actuator with either ramp, sinusoidal, or step control inputs from the FPID controller. The integrated controller consists of a five-point inverse valve transform in the feedforward loop and reduced proportional and integral gains of.5 and. in the PID loop. The reduction in the PID feedback gains improved the stability of the system and assured the robustness of the controller. The RM errors were 5.6 mm/s, 5.7 mm/s, and 7.8 mm/s from a ramp, a sinusoidal, and a step-input test, respectively. The RM error from the FPID controller was about 6.4% and 45.% lower than that from the well-tuned feedforward controller and PID controller, respectively, under ramp input conditions. It is concluded that the FPID controller is capable of compensating for system deadband and nonlinear gain via an inverse valve transform, and compensating for velocity tracking error via reduced PID gains. CONCLUION Electrohydraulic linear actuators are highly nonlinear systems, with load- and flow-sensitive system deadband, asymmetric flow gain hysteresis, and saturation. The dynamic behavior of the nonlinear system could be described using a fluid continuity equation, a valve orifice equation, and an actuator momentum equation with assumptions of no flow leakage and no friction. It is reasonable to assume that actuator velocity is controlled solely by the valve orifice of the pump-to-cylinder (P-C) port, and if the load on the actuator is positive. A feedforward-plus-proportional-integral-derivative (FPID) control was developed to achieve highperformance velocity control on a hydraulic linear actuator. The FPID controller consists of a feedforward loop for nonlinearity compensation and PID loop for tracking error compensation. The controller is capable of compensating for system deadband and nonlinear gain via an inverse valve transform, and compensating for the velocity tracking error via reduced PID gains. Experimental results indicate that the RM error from the FPID controller is about 5% and 45% lower than that from a well-tuned feedforward controller and a well-tuned PID controller, respectively, under ramp input conditions. Test results demonstrate the effectiveness of using a FPID controller to achieve accurate velocity control on linear actuators, and verify that FPID control is capable of improving the dynamic performance and the stability of hydraulic linear actuator systems. REFERENCE. Anderson, W.R Controlling Electrohydraulic ystems. Marcel Dekker, New York.. Backe, W The present and future of fluid power. Proc Instn Mech Engrs, Part I: Journal of ystems and Control Engineering, 7: Burrows, C.R Fluid power progress in a key technology. JME International Journal, eries B, 37: Caputo, D Electrohydraulic proportional valves increase system efficiency. Hydraulics & Pneumatics, November 994, pp Edge, K.A. and K.R.A. Figueredo 987. An adaptively controlled electro-hydraulic servo system. Proc Instn Mech Engrs, Part B: Journal of Management & Engineering Manufacture, : Isidori, A Nonlinear Control ystems. pringer Verlag, New York, NY.

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