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A FIXED CASCADE CONTROLLER WITH AN ADAPTIVE DEAD-ZONE COMPENSATION

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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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