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Control 2004, University of Bath, UK, September 2004 ID-210 A FIXED CASCADE CONTROLLER WITH AN ADAPTIVE DEAD-ZONE COMPENSATION SCHEME APPLIED TO A HYDRAULIC ACTUATOR M.A.B. Cunha*, R. Guenther# and E.R. De Pieri# * Pelotas Federal Centre for Technological Education Praça Vinte de Setembro 455, 96015-360, Pelotas – Brazil – Fax (55) (53) 284-5006 # Federal University of Santa Catarina – Robotics Laboratory - Campus Universitário – 88040-900 - Florianópolis - Brazil mauro@cefetrs.tche.br, guenther@emc.ufsc.br, edson@das.ufsc.br Keywords: adaptive control, cascade control, hydraulic such a way that it can work with low price proportional actuator, dead-zone compensation. valves. Dead-zone is a static input-output relationship which for a Abstract range of input values gives no output [8]. The range of a This paper addresses the position trajectory control of a dead-zone is stated by its left and right breakpoints. If the hydraulic actuator by combining a fixed cascade controller breakpoints are known, one can cancel the valve dead-zone including the valve dynamic with an adaptive dead-zone by using a dead-zone inverse [8]. Most of valve electronic compensation scheme. Amongst the hydraulic actuators cards have a dead-zone compensation scheme manually nonlinearities, the valve dead-zone is one of the main position adjusted that is based on the dead-zone inverse. In [3], a fixed error sources in the hydraulic actuator closed loop control. scheme similar to the used in electronic cards is described. Bu The dead-zone compensation scheme used here is based on and Yao [1] propose a dead-zone compensation scheme by the dead-zone inverse. As the dead-zone breakpoints depend measuring the relationship between the valve spool position on the system operation point and they vary for each valve, and the flowrate, approaching this flowrate curve by straight one uses an adaptation law to estimate those points, in such a lines and using the inverse of that curve to compensate the way that the proposed algorithm does not need manual valve dead-zone. These fixed schemes need experimental procedures to identify the dead-zone breakpoints. In addition, tests and are effective in a certain region around the point the adaptation scheme allows the system to work in different where the dead-zone compensation was adjusted. In order to points of operation. These estimated breakpoints are also used overcome this problem, algorithms that on-line estimate the in the dead-zone which has to be intentionally added to the breakpoint values must be used [7,8]. valve spool position measured in the electronic card. In this work, one combines an adaptive scheme based on the Simulation results illustrate the main characteristics of the adaptive dead-zone compensation algorithms proposed in [8] proposed scheme. combined with NFCC [3]. By using this algorithm, manual adjusts are not needed. 1 Introduction In this work, section 2 presents the hydraulic actuator Hydraulic actuators are widely used in industrial applications, mathematical model; section 3 describes the fixed cascade mainly due to their ability to generate high forces (or torques) controller (NFCC); section 4 addresses the adaptive dead- with small dimension actuators. Unfortunately, these zone compensation scheme; section 5 discusses the actuators present some undesirable characteristics: lightly simulation results and, in section 6, one presents the damped dynamics, highly nonlinear behaviour, difficulties in conclusions. obtaining the parameter values, amongst others. Such undesirable characteristics limit the closed loop performance 2 Hydraulic actuator mathematical model that can be obtained with a classical controller. In order to improve the hydraulic actuators closed loop performance, Consider the hydraulic actuator shown in Fig. 1, where M many different control techniques have been proposed in the represents the system total mass, B is the viscous friction literature. These work’s authors have been developing coefficient, ps is the supply pressure, p0 is the return pressure, cascade controllers to overcome such limitations [2]. p1 and p2 are the pressure in lines 1 and 2, v1 and v2 are the volume in lines 1 and 2, A is the cylinder piston cross Amongst the hydraulic actuators nonlinearities, the valve sectional area, Q1 is the flowrate from the valve to chamber 1, dead-zone and the dry friction force are the main sources of Q2 is the flowrate from chamber 2 to the valve and u is the position error in the hydraulic actuator closed loop control. electrical voltage applied to the electronic card. The amount of dead-zone and friction force are inversely related to the components price. This work deals with the use In this work, the valve under consideration is an overlapped of the fixed cascade controller (NFCC) proposed in [3] with four-way valve. In these kinds of valves, the lands of the an adaptive algorithm that compensates the dead-zone [8]; in spool are greater than the annular parts of the valve body, in Control 2004, University of Bath, UK, September 2004 ID-210 such a way that when the spool is displaced from the central The cascade strategy consists in interpreting the hydraulic position, there will be a region where there is not flowrate actuator mathematical model as two subsystems: a (dead-zone). mechanical subsystem driven by a hydraulic one. From this interpretation, the control law is calculated in two steps [6]: p1 v1 p2 v2 y (i) Compute a control law p ∆d (desired pressure difference) M for the mechanical subsystem such that the output “y” tracks B A the desired trajectory y d as close as possible; Q1 Q2 (ii) Compute a control law “u” for the hydraulic subsystem such that p ∆ tracks p ∆d as close as possible. u The NFCC mechanical subsystem control law is given by p ∆d = (M&& r + By r − K D z ) 1 (6) y & ps p0 A where ~ = y − y d is the position trajectory tracking error, y H y d ra u lic P o w e r U n it Figure 1 – Hydraulic Actuator y = y − λ~ , z = y − y = ~ + λ~ and K & r & yd & & & y y r and λ are D positive constants. Assuming the valve dynamic as a first order system, the hydraulic actuator mathematical model is given by The control law u = uNFCC for the hydraulic subsystem is M&& + By = Ap ∆ y & (1) given by & ∆ = −fAy + fK h gx v (2) 1 (p ∆d − K P ~ ∆ ) & p (7) p & x vd = + Ay & x v = DZ 1 (x vb ) (3) Khg f 1 φ (8) x vb = −ω v x vb + K em ω v u & (4) u NFCC = ~ x vd + ωv x v − 1 fK h g p ∆ − K V ~ v & x βv ωv K em φ2 where f = f ( y) = , x v is the valve spool (0.5v ) − (Ay) 2 2 where ~ v = x v − x vd is the valve spool position trajectory x position, p∆ = p1 - p2 is the cylinder chambers pressure tracking error, x vd is the desired spool position trajectory, ~ = p − p is the pressure difference trajectory tracking difference, β is the bulk modulus, v = v1 + v2, Kh is the p∆ ∆ ∆d hydraulic constant, g = g( x v , p ∆ ) = p S − sgn( x v )p ∆ , K em error and KP, KV, φ1 and φ2 are positive constants. is the valve constant, ω v is the valve bandwidth, x vb is the Remark 1 – In [3], without considering the valve dead-zone, valve spool position before the dead-zone. The signal x vb is i.e. x v = x vb , one demonstrates that the closed loop system the signal that is measured by an internal transducer in the {(1)(2)(4)(6)(7)(8)} is exponentially stable when the system valve and is available in the electronic card. The relationship parameters are known. A methodology to tune the controller’s between x v and x vb is given by gains was presented in [5]. Remark 2 – If the valve has a dead-zone, ~ does not tend to p ∆ x vb − b r , x vb > b r zero, i.e. there is a trajectory tracking error in the hydraulic x v = DZ1 (x vb ) = 0, (5) b l ≤ x vb ≤ b r subsystem and, consequently, the position trajectory tracking x −b , x < b error ~ will tend to a bounded set. vb l vb l y where b r is the right breakpoint and b l is the left breakpoint. 4 Adaptive dead-zone compensation Figure 2 shows a block diagram of the Equations (3), (4) and (5). In order to propose the schemes to compensate the dead-zone and to read the valve spool position, one assumes a static xv relation between xv and xvb, such that x vb = K em u . Therefore, u ωv xvb xv Equations (3), (4) and (5) can be written as x v = DZ 1 (K em u ) = DZ 2 (u ) K em (9) s + ωv b l b r x vb br K em u − b r , u > Figure 2 – Valve with dead-zone block diagram K em x v = DZ 2 (u ) = 0, l ≤ u ≤ r b b (10) 3 New fixed cascade control (NFCC) K em K em bl The NFCC was proposed in [3]. Here, it is briefly presented K em u − b l , u < K for completeness. em A dead-zone inverse (DZI)[8] that can be used to compensate the valve dead-zone is given by Control 2004, University of Bath, UK, September 2004 ID-210 ~ , p p r + u NFCC , u NFCC > 0 & η u ≤ u NFCC < 0 p (15) p l = 2 NFCC ∆ l min ˆ u = DZI(u NFCC ) = 0, u NFCC = 0 (11) 0, otherwise p + u , u NFCC < 0 l NFCC where η1, η2 and prmax are positive constants and plmin is a br bl negative constant. where p r = and p l = . K em K em The adaptation laws, Equations (14) and (15), depend on the Substituting Equation (11) into Equation (10), one obtains signals ~∆ and uNFCC. From remarks 1 and 2, one concludes p x v = K em u NFCC (12) that when the hydraulic actuator parameters are known, the Figure 3 illustrates the proposed scheme compensation in hydraulic subsystem trajectory error ~∆ tends to a residual set p Equation (11). As it was mentioned before, the valve spool due to the valve dead-zone. In addition, if the direct inverse position that is read in the electronic card is the signal xvb. dead-zone is implemented with the true values pr and pl, the Thus, in order to obtain the signal xv that is used in the control dead-zone is completely compensated [8]. Therefore, while law uNFCC, a dead-zone equal to the valve dead-zone must be ~ is different from zero, the estimated parameters p and p∆ ˆr included as shown in figure 4 [3]. ˆ ~ in the p l are different from pr and pl. It justifies the use of p∆ xv xv proposed adaptation laws. The use of uNFCC is to consider the uNFCC pr xv sign of uNFCC and to decrease the adaptation rate when the u xvb Kem valve is closing. pl xvb bl br xvb 4.2 Smoothing the dead-zone inverse Figure 3 – Dead-zone fixed compensation When the estimated breakpoints approach to the true values, the control signal tends to present high-frequency components. Although the valve works like a filter, this high- valve xv xv frequency signal can excite unmodelled dynamics and can also cause actuator vibration. uNFCC pr u xvb xv xvb Kem bl br xvb To overcome this problem, one can smooth the dead-zone pl inverse by using trigonometric functions. Another way is to change Equations (13), (14) and (15) by including a small measured signal dead-zone to avoid excessive switching: xv in the valve p r + u NFCC , u NFCC > l rs ˆ electronic card signal xv u = DZI(u NFCC ) = 0, ˆ l ls ≤ u NFCC ≤ l rs (16) used to calculate bl br xvb p + u ˆl , u NFCC < l ls the control law NFCC − η u ~ , l <u NFCC ≤ p r max uNFCC p p r = 1 NFCC ∆ rs & ˆ (17) Figure 4 – Inclusion of a dead-zone to obtain the valve spool 0, otherwise position ~ , p & η u ˆ p p l = 2 NFCC ∆ l min ≤ u NFCC < l ls (18) 4.1 Adaptation law for the dead-zone breakpoints 0, otherwise where lls is a negative constant and lrs is a positive constant. In The dead-zone breakpoints can be approximated by using both alternatives, smoothing the control signal yields to experimental tests. However, besides the necessity to adjust position errors greater than the errors when a dead-zone direct the compensation scheme for each valve, the values of br and inverse is used. bl vary according to the operation point (temperature, pressure, amongst others). To overcome this problem, one 4.3 Obtaining the valve spool position proposes an adaptive algorithm based on an adaptation law proposed in [8]. To obtain the valve spool position signal xV that is used in the In the proposed scheme, the adaptation and control laws are control law, one uses the same scheme proposed in figure 4, respectively given by with br and bl substituted by p r + u NFCC , u NFCC > 0 ˆ (13) b r = K em .p r ˆ ˆ (19) u = DZI(u NFCC ) = 0, ˆ u NFCC = 0 (20) b = K .p ˆ ˆ p + u ˆl , u NFCC < 0 l em l NFCC − η u ~ , 0<u NFCC ≤ p r max p (14) 5 Simulation results p r = 1 NFCC ∆ & ˆ 0, otherwise In this section, the simulation results of the closed loop system with the NFCC with the dead-zone adaptive Control 2004, University of Bath, UK, September 2004 ID-210 compensation are presented. Initially, the dead-zone is not 4 x 10 -3 Trajectory tracking error compensated. Then, after 12 seconds the adaptive 3 compensation scheme is turned on. 2 dead-zone adaptive compensation on The system parameters [4] are assumed to be known: y - y (m ) 1 M = 20.66 Kg, B = 316.2 Nsm-1, β = 109 Pa, ps = 100x105 Pa, 0 d A = 7.6576x10-4 m2, |u|max = ± 10 V, v = 9.5583x10-4 m3, -1 K h = 6.55x10 −8 m 4 V −1s −1 N −0.5 , ωv = 147 rads-1 and -2 Kem = 0.76. The controller gains are KP = 500, KV = 90, -3 KD = 11000 and λ = 30 [4]. The valve dead-zone was set -4 0 5 10 15 20 25 30 with br = 0.5 V (5%) and bl = -0.8 V (8%). The adaptation time(s) gains η1 = η2 = 5x10-6 were adjusted in simulations and Estimated right breakpoint prmax = 2 V and plmin = -2V. 1 The desired trajectory yd is based on a 7th order polynomial, br_ est i m at ed 0.8 Equation (21) , and on straight lines and is given by Equation 0.6 (22) [2,4]. Figure 5 illustrates the desired trajectory. dead-zone adaptive compensation on 0.4 _ y d1 ( t ) = −6t + 21t − 25.2t + 10.5t 7 6 5 4 (21) 0.2 y d1 (t) if t < 1 0 0.3 if 1 ≤ t ≤ 2 -0.2 0 5 10 15 time(s) 20 25 30 − y ( t − 2) + .3 if 2 < t < 3 (22) y d ( t ) = d1 Estimated left breakpoint m 0.2 − y d1 ( t − 3) if 3 ≤ t ≤ 4 0 dead-zone adaptive compensation on − .3 if 4 < t < 5 bl_ es t i m at ed -0.2 y d1 ( t − 5) − .3 if 5 ≤ t ≤ 6 -0.4 Figure 6 shows the system response when the dead-zone -0.6 _ adaptive inverse is used. Note that when the compensation -0.8 adaptive scheme is turned on, the trajectory tracking error ~ y -1 tends to zero. However, the control signal presents high- frequency components that can cause vibrations in the 0 5 10 15 time(s) 20 25 30 actuator. u N F C C 5 Figure 7 shows the system response when the inverse dead- 4 dead-zone adaptive compensation on zone is smoothed by Equations (16), (17) and (18) with 3 lrs = 0.05 and lls = -0.05. One observes that in this case the 2 (V ) trajectory tracking error decreases when the compensation 1 starts to work, but it does tend to zero. In practice, the values NFCC 0 -1 of lrs and lls must be adjusted so that there are not vibrations in u -2 the actuator. -3 Simulation results (not showed here) considering parametric -4 uncertainties showed an increase in the trajectory tracking -5 0 5 10 15 20 25 30 time(s) error. At the moment, a cascade controller taking into account Control signal after compensation those uncertainties is underdevelopment. 5 dead-zone adaptive compensation on 4 Desired Trajectory 3 0.4 2 0.3 1 u(V ) 0.2 0 0.1 -1 -2 d 0 y -3 -0.1 -4 -0.2 -5 0 5 10 15 20 25 30 time(s) -0.3 -0.4 0 5 10 15 20 25 30 Figure 6 – Response for NFCC with an adaptive dead- time(s) zone compensation Figure 5 – Desired Trajectory Control 2004, University of Bath, UK, September 2004 ID-210 4 x 10 -3 Trajectory tracking error 6 Conclusions 3 dead-zone adaptive compensation on In this work, a fixed cascade controller (NFCC) was 2 combined with an adaptive dead-zone compensation scheme. Simulation results showed that when the hydraulic actuator y - y (m ) 1 0 parameters are known, the proposed scheme yields to null d -1 trajectory tracking error and yields to small errors when the -2 dead-zone inverse is smoothed. With this desired trajectory, the estimated dead-zone breakpoints tended to the true values, -3 showing that the employed scheme does not need previous or -4 0 5 10 15 time(s) 20 25 30 on-line manual adjustments. Estimated right breakpoint Future works include the theoretical proof and experimental 1 implementation of the proposed combination and the development of an algorithm that takes into account the br_ est i m at ed 0.8 parametric uncertainties in the hydraulic and mechanical 0.6 dead-zone adaptive compensation on subsystems. 0.4 _ 0.2 References 0 [1] F. Bu, B. Yao. “Nonlinear adaptive robust control of -0.2 hydraulic actuators regulated by proportional directional 0 5 10 15 20 25 30 time(s) control valves with deadband and nonlinear flow gains”, 0.2 Estimated left breakpoint American Control Conference, 4129-4133, (2000). 0 dead-zone adaptive compensation on [2] M.A.B. Cunha, R. Guenther, E.R. De Pieri,, V.J. De Negri. “A cascade strategy using nonlinear control techniques bl_ es t i m at ed -0.2 applied to a hydraulic actuator”, Fluid Power Net -0.4 International – PhD Symposium, 1, 57-70, (2000). -0.6 [3] M.A.B. Cunha, R. Guenther, E.R. De Pieri,, V.J. De _ -0.8 Negri. “A fixed cascade controller applied to a hydraulic -1 actuator including the servovalve dynamic”, Power Transmission and Motion Control, Sulfok: Professional 0 5 10 15 20 25 30 time(s) Engineering Publishing, 59-72, (2000). u N F C C 5 [4] M.A.B. Cunha. “Cascade control of a hydraulic actuator: 4 dead-zone adaptive compensation on theoretical and experimental contributions”, In Portuguese, 3 Ph.D. Thesis, Federal University of Santa Catarina, (2001). 2 [5] M.A.B. Cunha, R. Guenther, E.R. De Pieri,, V.J. De (V ) 1 Negri. “Design of cascade controllers for a hydraulic NFCC 0 -1 actuator”, International Journal of Fluid Power, 3, 2, 35-46, u -2 (2002). -3 -4 [6] R. Guenther, E.R. De Pieri. “Cascade Control of -5 0 5 10 15 20 25 30 Hydraulic Actuators”, Journal of the Brazilian Society of time(s) Mechanical Sciences, 19, 2, 108-120, (1997). Control signal after compensation 5 4 dead-zone adaptive compensation on [7] R.R. Selemic, F.L. Lewis. “Deadzone Compensation in 3 Motion Control Systems Using Neural Networks”, IEEE 2 Transactions on Automatic Control, 45, 4, 602-613, (2000). 1 [8] G. Tao, P.V. Kokotovic. “Adaptive Control of Systems u( V ) 0 with Actuator and Sensor Nonlinearities”, New York: John -1 -2 Wiley & Sons, Inc., (1996). -3 -4 -5 0 5 10 15 20 25 30 time(s) Figure 7 – Response for NFCC with a smoothed adaptive dead-zone compensation

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