Water Hydraulic Servo Motor Velocity Control Using PID Funnel Control with Future Distance Estimation *

Transcription

1 JFPS International Journal of Fluid Power System -, /8, 7 Water Hydraulic Servo Motor Velocity Control Using PID Funnel Control with Future Distance Estimation * Chanyut KHAJORNTRAIDET **, Kazuhisa ITO *** Because of the environmental problems, water hydraulic system is introduced an alternative driving source for many applications. However, typical properties of water make the water hydraulic system difficult to be controlled; for example, the system contains parameters uncertainty and strong nonlinearity. This paper proposed the proportionalintegral-derivative funnel control () with future distance estimation for the water hydraulic servo motor velocity control. The controller utilizes the evident merit of future distance estimation for control signal generation. This control strategy requires no system identification, and can ensure not only the transient response behavior but also the steadystate error of the system using predefined arbitrary funnel boundaries. The simulation results show that the proposed controller has high efficiency to control the velocity of the water hydraulic servo motor system. The proposed controller shows better control performance especially during the transient operation compared with the conventional. Additionally, the controller is stable and robust for the case that dead-zone effects of servo valve are included. Keywords: Water Hydraulic System, Funnel Controller, Internal Model, Future Distance Estimation. Introduction A water hydraulic servo motor system is known as an environmental friendly system. The system uses water as working fluid because water has outstanding properties such as clean, nonflammable, inexpensive, readily available, and easily disposable. More importantly, if compared with oil hydraulics, water hydraulics has a quicker response ). These properties show high potential to application for food processing, chemical industry, medical industry, forestry, rehabilitation, and so on. Hence, water hydraulic system has the original market which is different from oil hydraulics. However, the applications of water hydraulic system are still quite restricted because of some reasons, for instance, expensive devices, nonlinearities for low viscosity of working fluid which leads to considerable leakage and larger friction torque for strong seal. To improve the control performance, many effective controllers are developed so far. In practical applications, the efficiency of conventional intuitive PID controller is quite limited because this is the fixed-gain controller. To improve the performance of this controller, disturbance observer and/or time-consuming system identification strategies are necessary ). The effective controller, funnel controller, is developed based on idea of high-gain feedback control 3). The controller is based on high-gain (time-varying) control concept. This * Manuscript received September 6, 6 ** King Mongkut's University of Technology North Bangkok (9 M, Nong lalok, Bankhai, Rayong Thailand) *** Shibaura Institute of Technology (37 Fukasaku, Minuma-ku Saitama, Japan) nb354@shibaura-it.ac.jp controller is originally restricted to the systems of relative degree one with the positive high frequency gain. For the system having the relative degree two, a funnel controller with derivative feedback to achieve output tracking of the systems is proposed 4). The controller adjusts its time-varying proportional gain by the measured control error and its timederivative. Applying funnel controller, we were able not only to ensure stability and asymptotic tracking of the system, but also to pre-specify the transient behavior of the plant by funnel boundaries. The funnel controller employs an adjustable proportional gain to control system with unknown parameters. The reciprocal vertical distance or future distance between the value of control error and funnel boundary function is used to calculate the time varying gain 5)-7). The prescribed transient accuracy of the closed-loop system can be achieved by appropriate boundary design. The control performance was robust to parameter uncertainty or variation not affected by the system structure 8). Additionally, the funnel controller neither identifies nor estimates the system under control and was applicable both for linear and nonlinear systems with satisfied structural system properties ). In application, because of power limit and system protection, a saturation limit should be considered. This research presents the improvement of performance using future distance (minimal distance) estimation. The modified is applied for the velocity control of a water hydraulic servo motor system. The saturation of control signal is also considered and an antiwindup using back calculation method 9) is designed for the water hydraulic control system. The performance of the - -

2 JFPS International Journal of Fluid Power System -, /8, 7 conventional and the modified FID-FC are compared by simulation. Moreover, the effects of a servo valve dead-zone for the controller performance are investigated. Obviously, the future distance estimation can improve the efficiency of the system to be controlled especially during transient operation.. Nomenclature left dead-zone of servo valve right dead-zone of servo valve discharge coefficient leakage coefficient parameter for static friction : motor displacement volume servo valve piston diameter : bulk modulus moment of inertia of flywheel servo gain supply pressure atmospheric pressure inlet pressure outlet pressure load torque motor torque friction torque viscous friction coefficient static friction torque Coulomb friction torque valve displacement valve displacement with dead-zone : volume of pipe Λ initial value of funnel boundary asymptotic accuracy for error asymptotic accuracy for error derivative water density funnel prescribed time constant valve time constant rotational velocity of motor shaft 3. Water Hydraulic Motor System Modeling In this section, the mathematical modeling of the water hydraulic servo motor and the servo valve for the velocity control system are introduced. The relationship of the spool displacement and the control input voltage can be written as The simplified sketch of the water hydraulic motor and the servo valve are shown in Fig.. Because of low viscosity of water, an overlapped valve is necessary for preventing some leakage flows in the valve actuator system. Actually, the servo valve is complicated equipment which contains the effects of many dynamics and nonlinearities. However, in this research, we consider mainly on the servo valve overlapped then the effects of valve dead-zone should be considered. In this system, the effects of dead-zone in both sides of servo valve are modeled as Eq. (). xv br, xv br xve =, bl < xv < br xv bl, xv bl This equation represents the effects of dead-zone affected on the valve displacement. Fig. Water hydraulic motor with servo valve The dead-zone of the servo valve applied in the control system is presented in Fig.. bl x ve Fig. Dead-zone of servo valve The water hydraulic motor can turn in two directions (CW and CCW), and the flow equations of water hydraulic motor are presented as kqxve Ps P,forxve Q = kq xve P Pr,forxve < br () (3) τ x () t = x () t + k u() t () v v v v - -

3 JFPS International Journal of Fluid Power System -, /8, 7 kqxve P Pr,for xve Q = kqxve Ps P,forxve < where = =. A torque balance equation and a friction torque equation of the water hydraulic motor can be represented in Eq. (5) and (6), respectively. (4) Iω + Tf + TL = T (5) M The friction torque is calculated by the relation as follows: Tf = Tvω+ sign( ω) Tc + Tse ω cs The torque generated by the water hydraulic motor is computed from the load pressure ( - ) and the relation as follows: DM TM = ( P P) (7) π Additionally, each pressure dynamics in motor chamber can be written as E DM P = ( Q ω Cli ( P P)) (8) V π E D P = ( Q + ω + C ( P P)) (9) M li V π The efficiency of the hydraulic motor depends on many factors such as manufacturing precision, maintenance of close tolerances, internal leakage, friction between sliding parts and internal fluid turbulence. The internal leakage leads to worse volumetric efficiency, friction and also fluid turbulence deteriorating mechanical efficiency. Based on the mathematical modeling of the water hydraulic motor system with servo valve, we will consider the velocity control of the system. The linearized model of the considered system is minimum phase, relative degree two, and positive high frequency gain. Hence, the funnel control can be applied for the water hydraulic motor velocity control. 4. Using Future Distance Estimation We consider the funnel controller for general relativedegree-two systems that uses two pre-specified funnel boundaries. This controller adjusts its time-varying proportional gain by the measured error between desired reference and the plant output. Additionally, the error timederivative is considered for calculation of another control gain in this case. The funnel boundaries for the error and its error time-derivative are predefined to ensure the transient (6) behavior. However, the drawback of the introduced controller for the relative degree two systems is on the convergence of tracking error because the controller does not has an integrating component. To solve this problem, the controller is combined with the internal model ). This internal model, which uses the conventional PI controller structure, does not deteriorate this class of high-gain stabilizable systems. The prescribed transient accuracy of the closed-loop system is achieved by appropriate boundaries design. The important structural knowledge of the plant is required for the funnel controller implementation. Moreover, to guarantee that the error evolves within the funnel region, the control input must be sufficiently dimensioned. In application, the control performance both transient and steady-state accuracy might not hold true if the plant only roughly known in structure; there just exists vague criteria to pre-specified funnel boundary. Therefore, the control performance investigation by simulation is important. In this research, the effective exponential funnel boundaries for the error and the error time-derivative are implemented. Two funnel boundaries are shown in Fig. 3 and the equations of these exponential funnel boundaries are presented in Eqs. () and (), respectively. ( ) ( ) F () t = F () t = Λ λ exp t/ τ + λ, λ > () exp E (( ) E) ( E) F() t = F () t + λ = λ Λ / τ exp t/ τ + λ, λ > exp + (), + () () (t), t t t () (t) () () () t Fig. 3 Exponential funnels for error and error time-derivative There are two more conditions for the application of the funnel controller for the relative-degree-two plant. First, the reference input signal must be bounded and continuously differentiable. Then the function of the funnel boundary for the error time-derivative must be satisfied the condition as follows: () /( ())+ for a positive constant δ. Therefore, the error time-derivative funnel boundary introduced in Eq. () is permitted. d F() t = F() t + λ = ( Λ λ) / τeexp ( t/ τe) + λ () dt - 3 -

4 JFPS International Journal of Fluid Power System -, /8, 7 The funnel controller for relative degree two cannot ensure asymptotic tracking of the system under controlled ). Hence, the combination of an internal model (PI controller) is required. The funnel controller for relative-degree-two systems with the internal model, combined in series, is called PID Funnel Controller (). In general, this controller usually applies the vertical distances between error and its boundary (also between norm of the error time-derivative and its boundary) for generating the control signal. This research introduces the future distance and exploits the ability of looking ahead by predefined funnel boundary applying with the that mentioned above. The future distance can be called as the minimal distance because it represents the minimal predicted distance in the future based on the predefined funnel boundaries. The application of future distance leads to higher control signal compared with the conventional. As indicated by Hackl et al. 4), the extended FC can be consider as a combination of proportional and derivative controller (PD), and the structure of this controller can be presented as follows: u() t = k () t e() t + k () t k ()() t e t (3) () (t) () M, M, M, where, = (), () = (),,, ( =, )., () ( =,) are the time varying gains calculated from estimated distance between norm of the error () and its corresponding funnel as well norm of the error timederivative () and its funnel boundary. The structure of the PDFC shown in Eq. (3) is one of the proposed structures and this structure can be adjusted. For the calculation of, (), the detail is illustrated in Fig (),, (,), () Fig. 4 Idea of future distance estimation The function () is scaling functions (for =,) which are chosen as ( λ ) s () t = cf () t + c ( i=,) (4) i i i, t i i + (), (,), (t) () Actually, the appropriated scaling function also affects the control signal generation. Many researches on funnel control, the structure of the scaling function as presented in the, previous equation is always applied. Moreover, the value of the scaling function will convert to a certain value of the time go to infinity. For the time-varying gain,, (), the calculation process will consider the error time-derivative and its funnel boundary. The proposed controller has higher control gains especially during the transient response because the minimal distance is smaller than the vertical distance. The smaller distance leads to the higher controller gains. The approximated future distance is calculated by derivative method by assuming that during the small time interval, the funnel boundary can be estimated as a linear function. This method is suitable for the real time application because it requires less calculation time compared with the numerical method. The approximated future distance can be calculated as follows: ( ˆ () i ( ) ()) ( ) ˆ ˆ ˆ Fi, Fi, i Fi, Fi, dˆ (, t t ) = F t e t + t t ( i=,) (5) The derivative of Eq. (5) with respect to the approximated future time is considered while is constant (at the current state). For prediction of the future distance, the linear approximated funnel boundary is expressed in Eq. (6); ( ) ( ) ( ) ( ) Fˆ T = F t T t + F t ( i =,) forallt t (6) i i i Based on the above equation, when is equal,, the function (,) and its derivative can be computed. Substituting these results to the derivative of Eq. (5) and consider when the derivative of the obtained equation is equal zero, the approximated future time is obtained as follows: ( ) (), () () + F () t t F t d t F t t tˆ = ( i=,) i V i i Fi, i (7) Additionally, the structure of the controller expressed in Eq. (3) does not contain the integral term. It is possible that the steady-state error may not be vanishing. Hence, this drawback can be able to overcome by introducing a proportional plus integral (PI) like extension given by I ( τ) um() t = u() t + k u dτ t (8) then connecting it in series to the original FC. This structure can ensure asymptotic accuracy and disturbance rejection. Based on this presented idea, the with minimal distance estimation is expressed as - 4 -

5 JFPS International Journal of Fluid Power System -, /8, 7 k M,() t um() t = km,() t et () + et () km,() t t km,( τ ) + ki km,( τ) e( τ) + e( τ) dτ k M,( τ ) (9) where a constant is an integral gain. The mathematical model of water the hydraulic servo motor system is identified and presented in the work of Pham et al. ). The system with the velocity controller is shown in Fig. 5. The second-order under damped system, as the reference model, is used to generate reference speed and its derivative. There are two considered s, the conventional and the proposed with minimal distance estimation. Reference Reference model Fig. 5 Block diagram of water hydraulic servo motor with Moreover, because of the power limitation and system protection, a saturation block should be implemented to the control system application. The effect of saturation causes the controller performance issue. Hence, the controller antiwindup is required to deal with this issue while the control signal reaches the saturation limit. 5. Simulation Results The water hydraulic servo motor system is satisfied following properties: (a) relative degree two, (b) known sign of high frequency gain (positive), and (c) minimum phase, therefore, the funnel control can be applied. The simulation results of water hydraulic motor system with the introduced with minimal distance estimation and the conventional are investigated. All simulations are performed with MATLAB/Simulink with a sampling time h =.5 ms. Additionally, the requires a continuous reference velocity and velocity derivative, therefore, we apply the standard second-order system as shown in Eq. () for generating the reference velocity and velocity derivative for s; Cs () K ω = Rs () s s n + ζωn + ωn - Water with antiwindup hydraulic motor - () where =, = 57.3 min -, and Ϛ =.86. The controller parameters of the conventional and with minimal distance estimation are shown in Table. Table : Controller parameters Proposed Conventional = = = 4 = 4 = 4. = 4. =.5 =.5 =. =. The parameters for reference model are chosen to generate the smooth velocity output with low oscillation. The controller parameters and funnel boundary parameters of these two controllers are designed to be same for the performance comparison in the next step. The scaling functions () and s () are funnel boundaries of the error and error time-derivative, respectively. The specifications of the water hydraulic motor system for this simulated system are obtained from the work of Pham et al. ). The values of system parameters are listed in Table. This set of parameter are chosen to simulate the water hydraulic motor with. Subsequently, the results of the system with two controller are compared. Table : System parameters Parameter Unit Value valve time constant ( ) s. servo gain ( ) m/v.53x -4 right dead-zone ( ) m.3x -4 left dead-zone ( ) m.4x -4 discharge coefficient ( ) -.6 valve piston diameter () m 8x -3 density of water () kg/m supply pressure ( ) Pa 4x 6 atmospheric pressure ( ) Pa.abs.x 6 flywheel moment of inertia () kg m 4.5 viscous friction coefficient ( ) N s/m.3 static friction torque ( ) N m. Coulomb friction torque ( ) N m.5 static friction coefficient ( ) -. displacement volume ( ) m 3 5x -6 bulk modulus () Pa.x 9 volume of pipe ( ) m 3 5x - leakage coefficient ( ) m 3 /s Pa x - The simulation results of the water hydraulic velocity control using the with minimal distance estimation and the conventional are shown in Fig

6 JFPS International Journal of Fluid Power System -, /8, 7 Angular velocity (min - ) Error (min - ) 8 6 Reference Modified Upper boundary Lower boundary e e Fig. 6 Velocity response and error in the case of reference tracking Based on the velocity control investigation in the case of reference tracking, the velocity responses for both controllers show effective performance and each velocity error is bounded within the specified funnel boundary. Obviously, the proposed has faster tracking performance compared with the conventional, especially during transient response. Additionally, the proposed controller shows smaller velocity error magnitude during the transient period. However, the steady-state response of these controllers is very closely because the effects of the approximated future distance are relatively small. During the steady-state operation, it means that the approximated future distance value is close to the vertical distance because of constant funnel boundary. Subsequently, the control signal and control gains for this case are indicated in Fig. 7. Other important factors that should be considered to improve the controller performance are the funnel boundary design and the selection of the scaling function. 5-5 Modified k k k MD, k MD, During the transient period, the control gains of the proposed are higher than the conventional. Therefore, the control signal of the proposed controller is higher than the conventional and this effect leads to the faster response of the system. On the other hand, when the time goes by, the time-varying control gains of both controllers converge to the same value. Moreover, the results show that the control signal fluctuation is inevitable in the case of the fast control response requirement for the high gain feedback control system. The enlarged results of the control signal and generated control gains of both controllers during the transient operation are shown in Fig Modified k k 8 k MD, 6 k MD, Fig. 8 Comparison of control signals and control gains of The enlarge results of control signals and control gains at.-.5 seconds and at.-. seconds are shown in Fig. 9 and, respectively. Additionally, the enlarged results from 33. to 35. seconds are exhibited in Fig.. - Modified k 7 k 6 k MD, 5 k MD, Fig. 9 Enlarged control signals and control gains at.-.5 seconds Fig. 7 Control signal and control gain in the case of reference tracking - 6 -

7 JFPS International Journal of Fluid Power System -, /8, k k.8 k MD,.6 k MD,.4. Fig. Enlarged control signal and control gains at.-. seconds Fig. Enlarged control signal and control gains at seconds Modified Modified k k.8 k MD,.6 k MD, Based on the results indicated in Fig. 8, 9,, and, we can observe that the proposed has high control signal because of the high gain effects. While the controller has high control gain, it will lead to large oscillation during the transient response. The fluctuation is mainly from two reasons. First, the proposed presented in Eq. (9) utilizes the calculated gain power for computation of the control signal. While the calculated gains has high value because of the effect of the minimal distance estimation, the high gains affect directly to the variation of control signal. Second, as presented by the proposed structure, the high values of control gain are multiply by the error derivative which also contains some oscillation. In addition, the specified boundaries of the funnel influence the control signal and the response of the system to be controlled. Moreover, while the reference velocity closes to zero, the velocity response of the system with both controllers suffers from the effects of dead-zone of servo valve. Because of this reason, the control signal has some oscillations around seconds. As time goes by, the prescribed funnel boundary size become narrow. This effect leads to the rapid change of the control signal and the control signal because the control signal is proportional to the distance between the error and the funnel boundary. The designed final value of the funnel boundary also impacts the system response. In addition, the control gains of the conventional may be negative while the value of absolute error bigger than the funnel boundary. In contrast, the control gains of the proposed cannot be negative because of the estimated calculation process of the future distance which is presented in Eq. (5). The enlarge velocity response, error, and control gain calculation results of both controllers during the desired velocity closed to zero are shown in Fig. and 3, respectively. Angular velocity (min - ) Error (min - ) - Reference Modified Upper boundary 5 Lower boundary 4 e 3 e Fig. Enlarged velocity responses and error at seconds Modified k. k k MD,.8 k MD, Fig. 3 Dead-zone effects on control signal generation The results obtained from both and with minimal distance estimation show small different, this is - 7 -

8 JFPS International Journal of Fluid Power System -, /8, 7 because the funnel boundary of both controllers are much closed during this period. In this simulation, the final value of velocity error are specified at 4. rpm for both controllers, therefore, the controllers will not respond quickly while the error is far from the boundary. Additionally, the dead-zone effects allow the control signal variation with no action when the desired velocity is closed to zero. For the control strategy improvement, we can focus on the design of the additional funnel conditions during steadystate operation. Because the funnel boundary during the steady-state operation is very narrow, hence, the system with funnel controller will not be robust for the suddenly external disturbances and it may lead to the serious stability problems. Based on the flexible funnel boundary, the response of the system in the case of disturbances can be improved. Additionally, the design of controller structure and the selection of scaling function affect the funnel controller performance. Therefore, the appropriate structure and the effective scaling function should be also considered. 6. Conclusion Based on the simulation results, the idea of future distance estimation can improve the efficiency of the. Additionally, the calculation process of the future distance estimation can ensure the positive control gains. However, the efficiency of both controllers is still affected and limited by saturation. This problem can be solved by implementation of anti-windup controller. The modified controller using the future distance shows the better response especially during transient period compared with the conventional. References Yang Y. S., Semini C., Tsagarakis N. G., Caldwell D. G., and Yuquan Z.: A novel design of spool-type valves for enhanced dynamic performance," Proceedings of IEEE International Conference on Advanced Intelligent Mechatronics (AIM), pp , Jul. -5 (8) Hackl C. M., Hofmann A. G., and Kennel R. M.: Funnel control in mechatronics: An overview, Proceeding of the 5 th IEEE Conference on Decision and Control and European Control Conference, pp () 3 Ilchmann A., Ryan E.P. and Sangwin C.J.: Tracking with prescribed transient behaviour, Journal of ESIAM Control, Optimization & Calculus of Variations, vol.7, pp () 4 Hackl C. M., Hopfe N., Ilchmann A., Mueller M., and Trenn S.: Funnel control for systems with relative degree two, Journal of SIAM on Control and Optimization, vol. 5, no., pp () 5 Hackl C. M., Ji Y., and Schrder D.: Enhanced funnelcontrol with improved performance, Proceeding of the 5 th Mediterranean Conference on Control and Automation, pp. -6, Jun. 7-9 (7) 6 Schuster H., Hackl C. M., Westermaier C., and Schröder D.: Funnel-control for electrical drives with uncertain parameters, Proceeding of the 7th International Power Engineering Conference pp. 5-, Nov. 9-Dec. (5) 7 Ilchmann A. and Ryan E. P.: High-gain control without identification: A survey: vol. 3, issue, pp. 5-5, (8) 8 Hackl C. M., Hofmann A. G., De Doncker R. W., and Kennel R. M.: Funnel control for speed & position control of electrical drives: A survey, Proceeding of the 9 th Mediterranean Conference on Control and Automation, Corfu, Greece, pp () 9 A. Visioli, Practical PID Control. Springer-Verlag, London (6) ) Hackl C. M., Endisch C., and Schroder D.: Funnelcontrol in robotics: An introduction, Proceeding of 6 th Mediterranean Conference on Control and Automation, pp (8) ) Pham N. P., Ito K., and Ikeo S.: The application of simple adaptive control for simulated water hydraulic servo motor system, Proceeding of IEEE Conference on Industrial Technology (ICIT), pp. 4-9 (3) - 8 -