Provident control of an electro-hydraulic servo with experimental results

Mechatronics Vol. 6, No. 3, pp. 249-260, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0957-4158/96 $15.00 + 0.00

Pergamon

PROVIDENT CONTROL OF AN ELECTROHYDRAULIC SERVO WITH EXPERIMENTAL RESULTS

iLKER TUNAY" and OKYAY KAYNAK* *Departmentof Systems Scienceand Mathematics,WashingtonUniversity,One BrookingsDrive, St. Louis,Missouri, 63130, USA and 1"UNESCOChair on Mechatronics,BogaziqiUniversity, Bebek, Istanbul, 80815, Turkey (Received25 October1995;accepted21 December1995)

Abstract - In this study, we propose a novel variable structure control method for output tracking of uncertain nonlinear systems, with experimental application to an electro-hydraulic servo system. First we perform a state transformation to convert the system to the normal form. Then we apply a method that we have termed provident control, so called because of its frugal but judicious use of the input when compared to other variable structure methods which use large control authority to drive the state to the sliding surface. This method combines variable structure control with variable structure adaptation and performs switching with hysteresis between the structures so as to avoid a sliding mode. By completely separating the reaching and tracking phases, a two-degree-of-freedom controller is obtained. Experimental results indicate significant performance improvement over the well-known boundary layer sliding control, together with robustness against parametric uncertainty and external disturbances. Copyright © 1996 Elsevier Science Ltd

1. INTRODUCTION The second half of twentieth century has been an age of advances in precision motion control. Developments in the areas of efficient electrical power amplification, microprocessors, control theory and high quality actuators, such as direct drive motors, have made electrical control systems the primary choice in most sectors of the industry. However, hydraulic control systems have several unique features which make them indispensable in many fields, especially mobile, airborne and high power applications. To name a few: actuators with high torque to inertia ratios, readily available power supply in mobile applications, ability to be operated under continuous, intermittent, reversing and stalled conditions without damage. Furthermore, the torque developed by an electrical actuator is limited by magnetic saturation and high power requirements can only be met with bulky machines, whereas the torque developed by a hydraulic actuator is only limited by mechanical safe stress levels. 249

250

1. TUNAY and O. KAYNAK

On the other hand, the nonlinear nature of the components and intriguing aspects of fluid dynamics which are difficult to model from a control point of view, makes tracking controls difficult. Linear control theory can be utilized if the control objective is set-point control, but for tracking tasks, performance with a linear controller will degrade significantly over a large region of operation. Besides, in such demanding tasks as robotic manipulators or machine tools control, the system may experience sudden and large changes in the load, so that robust and/or adaptive methods may be necessary for both stability and tracking performance. Nonlinear state transformation theory combined with a variable structure controller can be a viable alternative for high performance tusks [ 1,2]. In this study, we propose a novel variable structure control method for output tracking of uncertain nonlinear systems, with experimental application to an electro-hydraulic servo system. First we perform a state transformation to convert the system to the normal form [3]. Then we apply a method that we have termed provident control, so called because of its frugal but judicious use of the input when compared to other variable structure methods which use large control authority to drive the state to the sliding surface. This method combines variable structure control with variable structure adaptation and performs switching with hysteresis between the structures so as to avoid a sliding mode. By completely separating the reaching and tracking phases, a two-degree-of-freedom controller is obtained. In the following section, we develop the model for the servo system. In section three the model is transformed to the normal form and internal dynamics are investigated. Section four describes the proposed control algorithm, while the next one presents the experimental results and possible improvements. The appendix contains the units and descriptions of various symbols used in the model, together with their estimated values.

2. THE MODEL FOR THE ELECTRO-HYDRAULIC SERVO

The electro-hydraulic servo system that we consider uses an axial multi-piston type, symmetrical action, rotary hydraulic motor with a fixed swashplate, which can deliver up to 4.5 Nm of torque to an inertial disk attached to the motor shaft [4]. Oil flow to the motor is controlled by a four port, two stage, proportional flow type valve. The spool type second stage is driven by the flapper-nozzle type first stage, which is in turn controlled by an electrical torque motor, with armature attached to the flapper. The hydraulic power supply, filter and accumulator complete the hydraulic setup. Valve port pressures can be measured by strain gauge type transducers and the shaft position is measured by an incremental encoder with 635 pulses/rev. However, there is no provision for the measurement of the port flows, nor the valve spool position. Let us take the flow out of the valve to be positive for both output ports. Using the square root law for turbulent flow across an orifice [5], the valve port flows can be written

QI = +s'gn(Ps- Plv)Koxs 1 - ~

xs _

(1) Ql=+Koxs

Q2 = - s i g n ( P s - P i v ) K o x s J ] l - ~ e~

vl

" Xs <

s r

The motor flows consist of active flow, compressibility of oil (together with the expansion of hoses), internal leakage from the high pressure port to the low pressure one and external leakage

Control of an electro-hydraulicservo system

251

from each motor chamber past the pistons to the drain, the latter being the prominent leakage factor in rotary motors [6]. Assuming both leakage flows to be laminar, the motor port flows are Q1 : +Vrn(~+ Vc dPl ÷ KIi(PI - P2) + KIePI 2K b dt Q2:_Vm0_ ~ Vc dP2 2K b clt

(2)

Kli(Pl_P2)+KleP 2

The disparity between the valve and motor pressures is due to the transport lag of the hoses. The most significant dynamics associated with the valve arise from the inertia of the spool in response to an input signal, which can be described by a first order lag. We can safely ignore the dynamics of the valve electrical motor and the kinetic losses of the oil as it flows through constrictions, compared to the above terms. Hence, the valve dynamics are Ksf x s + Kvf v i Xs = - As

(3)

As

To simplify the analysis, let us ignore the hose lag (which is about 1 ms in the experimental setup), hence Ply ~ P1, P2v ~ P2 • Since the valve is much faster compared to the other modes (with a pole at approximately 70 Hz in the setup), we can replace xs in (1) by its steady-state _ Kvf v i . The torque generated by the motor is 10Vm(P1 -P2), where the factor 10 value: x s _Ksf serves to balance the units. Besides viscous friction, the main source of friction losses is the force opposing motion of the piston in its bore that is proportional to the pressure acting on the piston area. Other motor elements, like bearings, are also loaded proportional to the motor pressures and cause friction torques [6].These are represented by Tf. Coulomb friction caused by the seals is small compared to these effects and hence is ignored. Then the state equations for the system become I~I=2v~-[QI-Vm0-(KIi+KIe)PI+KIiP2] 1 P2--2v~[Q2+Vm0-(KIi+KIe)P2+KIiPI] t

(4)

= }I'OVm(P, - P2)-"O- s,g.(O)Tf(h + P2)]J 3. T R A N S F O R M A T I O N TO T H E N O R M A L F O R M

Defining the shaft position as the output, y~O, and differentiating the output until the input, u--vi, appears, we arrive at

252

I. TUNAY and O. KAYNAK r ~2

=

=

I[10Vm(PI

0 6 P2)-B0 slgn(0)Tf(Pl+P2)]

(5)

where Z(-) is an arbitrary differentiable function. The normal form state equations become (~2 =

i

~3 Ab(x)]uI a(x) Aa(x) +[b(x) i

w(x)

-

m0-(Kli +KIe)P1 +KIiP2 + Vm0+KliPl-(Kli +KIe)P2

a(x)- (2JKbKlc + BVc)sign(0)Tf (PI ~ P~ ) j 2 Vc [20JKbVm(Kle+2Kli)j2Vc 10BVcVm](pl- - P2) ~ ( B2J~ 40KbVm2]0jV c )

b(x)

KbKoKvfJvcKsf2(10V m sign(0)Tf)~'l

~P1 + 2(10Vm + sign(0,)TfF~p s]/P2 J (6)

Here [~, q] = ~(x) has not been substituted for brevity. The term Aa represents both parametric uncertainty and possible external disturbances, whereas Ab comes only from parametric uncertainty. We can parameterize the input gain b(x)-Ab(x) as b(x) = p. [3(x)

Ab(x) = p. p(x) - p*J3(x) ~ p. [3(x)

P~ KbKoKvfJvcKsf" 13(x)=(10Vm slgn(0)Tf)21'-PIps +(10Vm+sign(0)Tfl2' ! UPs/P2/i (7) where p'denotes the true value of the parameter p, p is the estimation error and 13(x) is a known signal. The significant part of the uncertainty in p comes from the unknown load inertia J, which is subject to frequent sudden changes in the course of operation with varying payloads. The other parameters in the model (4) may also be uncertain and external disturbances (which may be timevarying but do not depend on the states) may be acting on the system. In that case the transformation (5) can still be used, however to arrive at a slightly different form than (6), see [7]. Before designing the tracking controls, we have to investigate the stability of the internal dynamics. This can be accomplished by analyzing the zero-dynamics, which can be described as the dynamics of the system when the output is kept at identically zero by the control signal [3]. A Sl(ffic~ent condition for the stability of the internal dynamics in output tracking is the exponential stability of the zero-dynamics [7].

Control of an electro-hydraulic servo system

253

In order to avoid the complications which arise from the defmition of rl = dpa(X) in (5), we can choose a simpler function for the internal state to complete the transformation (5). For instance, let rl = ~b4(x) = P2 .It can be checked that this choice yields a state transformation with a globally nonsingular Jacobian. In this case we will lose the special structure of the normal form (6), the input will enter the differential equation for the internal state, but stability properties will remain the same. If the output is kept at zero, the states evolve on a one dimensional "manifold" of the state space given by M*-- {[~ rl]•9t4: ~ = 0 } - - { x • 9 1 4 : 0 = 6 = 0 , PI--P2}. Physically, this corresponds to a stationary motor and the only event going on is the external leakage from motor chambers to the drain. The unique input that keeps the state on M*, assuming that the state is initially on M*, is given by

x•M"

=

b(;,11)

; =0

--

0Vt

since b(x)> 0 Vx • M*. With this input and states constrained to M*, the zero-dynamics are simply il = t'2 -

2KbKle "O , which is exponentially stable around the origin, hence the desired Vc

stability result.

4. THE PROVIDENT CONTROL APPROACH

The controller for the model (6) should better have the following merits: Robustness to modeling errors and external disturbances, a fast transient to return to tracking in the case of a load change or a large external disturbance, good tracking performance and overall stability. Moreover, the a priori information required for control design should be minimal, i.e. one should be able to design a feasible controller in the lack of precise information on the bounds of the model parameters and/or disturbances. These goals can be attained fairly closely by a variable structure controller in which there are two separate controllers for the reaching phase and tracking phase; the first one designed for stability and the second for good tracking performance. Let us define vectors consisting of the derivatives of the desired and actual outputs respectively

Yr -= Yd--'Y

Ya -= y...y(r-1) r . Then the tracking error vector is e=ya-yr.

Let us define a scalar measure of tracking error, o, following the philosophy of sliding control as treated in [3] O~- @lciei' i

(r-1)! ~r-i ' i = l ..... r ci=l)l,rr-i'!li-l'!)

(8)

where ~ is a positive constant. In this way, the r'th order output tracking problem reduces to a first order stabilization problem in ¢~. The dynamics of the scalar ¢~are given by • r 6= Eciei i=l

r-1 r-1 = y(r) -y(dr) + E c i e i + l = a - A a + ( b - A b ) u - y (r) + E c i e i + l i=l i=l

(9)

254

i. TUNAY and O. KAYNAK We select the control input as: r-I

)

I -a(x)+y(dr)-'~ciei+l u-- b(x) i=l

uc

_

+b(x)

I

I

b(x) Un+b(x~Uc

(10)

The first term above is the common "nonlinearity cancellation action" or nominal control, while u¢ is a corrective term put into service to compensate for the uncertain part of the system dynamics. With this control the ~ dynamics become 6 - - A a ( x ) - ~. 13(x). u + u c • The main problem is how to design u c and to combine it with parameter adaptation. When 6 is large, u c must drive the state trajectory towards 6=0 as fast as possible (the reaching phase); when 6 is small, u c must act to keep it as small as possible (the tracking phase). Along the same line of thought; in the reaching phase, the adaptation law may be driven by the combined tracking error 6. However, when 6 is small the adaptation must be based on prediction errors. Prediction error based adaptation requires a reasonably accurate regressor structure, which makes it unsuitable for use in the reaching phase in which large initial errors dominate. This suggests a variable structure controller combined with variable structure adaptation. Let us denote the controller structures in the reaching and tracking phases respectively as Zr and Zr and the time spent in the structures as tr and tt. Note that tr and tt need not be contiguous intervals. They are unions of left closed right open intervals such that t r ~ J t I : ~ , t r c ~ t t ~ . In order to avoid a sliding mode at the instant of structure switching, we use a relay with hysteresis with input 6. The switching points are ql (from Z~ to E~ ) and q2 (from Z~ to Zt ), with q~>q2. The control structures are given by : Uc(t Er

-'

-Um.sat(~(t)/ql

~(t) = G o . or(t). 13(x)- u(t)

(I la)

G(t) : 0

-R(s)
(1 lb)

Here G(t) is the adaptation gain (with initial value Go ) and R(s) stands for a linear control law (s:Laplace variable). Um is a constant chosen such that it will always be possible to drive the state towards 6=0. Let us assume that a roughly estimated bound on the modeling errors exist, i.e.]Aa(t)l < b a Vt. Then, selecting U m as U m ~ b a +6 u, where 8 n is a small positive constant, will ensure this. Since U m has no effect on the steady-state performance, a large value (if allowed by the hardware), can be chosen for faster convergence to the tracking phase. The constant q l should be selected sufficiently large so that unless the system experiences a large unexpected disturbance (such as a transient power supply failure), the structure remains at E t. The value of q2 should be selected such that 6 is small enough at the instant of structure switching, but it should not be so small that the 6=0 crossing will not be detected by the computer. The adaptation law in the reaching phase is the common gradient rule which is derived to satisfy negative definiteness of the derivative of a Lyapunov function, thus it ensures stability.

Control of an electro-hydraulic servo system

255

Note that we do not need a priori information on the set of allowable parameters, although a good initial estimate would provide faster reaching. On the other hand, adaptation in the tracking phase may be based on the least squares law, which is well-known enough that we omit the details. Since the two structures are independent and the adaptation is indirect, desired modifications to the estimation law in ~ (such as exponential forgetting or covariance resetting) are possible. For the linear controller R(s), many design methods are available and the designer can choose whatever method he favors as long as the close loop ~ dynamics are stable. If this is the case, then it can be shown that the overall control system is stable in the sense that all the signals involved remain bounded, since the zero dynamics are exponentially stable [7]. For smooth transition to the tracking phase, the initial conditions in the realization of R(s) should be set appropriately.

5. E X P E R I M E N T A L R E S U L T S AND D I S C U S S I O N

5.1 Experimental Verification The estimates of the model parameters are seen in the appendix. While some of these are actual constants (e.g. Vm, As), the others are functions of temperature and operating conditions (e.g. Kb, Ps). However, the largest variation during operation is expected in the inertia J. To simulate an actual payload change, initially J was taken as 0.43, therefore the initial value of p was 18, while the actual value should be around 23. For the linear controller we carried out the design in discrete-time. We chose a second order controller with integral action and low pass characteristics to stabilize the cy dynamics, and determined the gain using the root-locus technique. Since the procedure is well-known, we omit the details here. The resulting discrete-time controller was z 2 1.4z + 0.49 uc(z) - -R(z) = -120 O(z) z 2 - 0.3z - 0.7 -

(12)

The other controller parameters are as follows: X=60, Urn=7000, G0=4xl0 "6 , q1=800, q2=40. The design was implemented on a 80486 based PC with a sampling period of 2.5 ms. For comparison purposes we also designed a fixed boundary layer sliding controller [3], for the same model. The boundary layer thickness was 2000, the minimum value determined experimentally to yield a non-oscillatory control signal. This time we employed precise timevarying bounds on the uncertainties and assumed that J could vary between 0.3 and 0.5. The experimental results are seen in Figs 1 to 8. The reference velocity trajectory is a sequence of ramp, constant and sinusoidals (Fig. 1). Comparing the ~ trajectories (Figs. 2,3) and position errors (Figs. 4,5), we see that the provident controller yields a significant performance improvement over the sliding controller. The first reason is the addition of parameter estimation. The second one is that the sliding controller is designed to ensure stability in the reaching phase, therefore uses conservative upper bounds for the uncertainties, a fact which degrades steady-state tracking performance, although it yields a fast transient. On the other hand, the provident controller, being a two-degree-of-freedom design, does not suffer from this shortcoming. After an initial period in which parameter estimation prevails, good tracking is achieved as seen in Figs.5 and 6. In the tracking phase the position error was less than 0.007 rad (-0.4 degrees) and the velocity error was less than 0.2 rad/s. The parameter estimation, seen in Fig. 7, is satisfactory and illustrates the idea behind variable structure adaptation: a fast transient in the reaching phase to obtain quick correction, followed by fine tuning and smooth convergence in the tracking phase.

256

[. TUNAY and O. KAYNAK

The control signal is shown in Fig. 8 and is somewhat noisy because of the differentiation of the velocity signal to obtain the acceleration.

5.2 Discussion and Possible Improvements

Several contributors in the tracking error are as follows: The pulsating motion of the pump creates ripples in the supply pressure which cannot be filtered adequately by the accumulator. The torque generated by the motor has ripples due to the action of the five axial pistons (fundamental component at about 80 Hz at 100 rad/s). The motor shaft is slightly bent and hence causes torque disturbance. All these disatrbances are at frequencies higher than the bandwidth of the valve, therefore they cannot be effectively canceled by the action of control. However it is possible to improve the signal quality using Kalman filtering. Another possible way of improving the performance in the tracking phase may be to spend more time on the linear controller R(s). Instead of the simple root-locus based design above, one can employ the wide variety of linear control design methods to reduce the effect of disturbances mentioned. If it were possible to measure the valve spool position, one could include the valve model in (4) and obtain a fifth order model. At the expense of a more complex controller, it could be possible to overcome the aforementioned limitations of the valve and to gain additional insight as to the nonlinearities (e.g. hysteresis and flow gain characteristics) exhibited by the valve. Also it would be beneficial to use an accurate flowmeter to determine the leakage flow characteristics. Finally, utilizing a higher precision incremental encoder will always improve signal quality.

1 O0

i

i

i

f

9O 8o 7O

6O 2 5O

413

I

2

4 time (s)

6

Fig. 1. Reference velocit~ trajectory.

10

Control of an electro-hydraulic servo system

257

2000 1500 1000

L$ 500

~

o

~ I S T 3 L F I V p T T W w ' T T I ' ~ I T I I ' ' I I I " | I W F I l I T I r ~ I y l H m I I I W I I ~ F V I T , ~ --- I~ 1 ]

i-,,. I ~l.ii~5-W"Twrrl

I~T¢%

-500 -100(

0

t

i

e

e

2

4

6

8

time (s)

10

Fig. 2. ~ trajectory with boundary layer sliding controller. 600 5OO

~" 400

~

3oo

~

200

,j

1 O0

0

I

0

2

4

time (s)

I

I

6

8

10

Fig. 3. cr trajectory with provident controller. 0.4

0.3

~0.2

:=_

o.1

0

-0.1 0

e 2

i 4

time (s)

e 6

i 8

Fig. 4. Position error with boundary layer sliding controUer,

10

258

[. TUNAY and O. KAYNAK

_2 ~0 ~• -2 -4 -6

0

I

I

I

I

2

4

6

8

10

limc (s) Fig. 5. Position error with provident controller. 24

,

1

w

i

23 22 "~ 21

2(1 •~ 19 18 I

I

i

I

2

4

6

8

lime

Fig. 6 1500

10

(~,)

Velocity error ~ l t h provident controller,

i

i

' 2

' 4

i

i

' 6

' 8

1000 50O

0 .~ -500 -1000 -1506

0

lime (~)

Fig. 7. Parameter estimation.

10

Control of an electro-hydraulicservo system 0.4

i

i

i 2

, 4

i

i

i 6

i 8

259

0.3

0.2 0.1 :=

o I -o.1 -0.2 0

time(s)

10

Fig. 8. Controlvoltage. 6. CONCLUSION In this paper we have proposed a new variable structure control method combined with variable structure adaptation. Although the application at hand was the output tracking problem of an electro-hydraulic servo system, the method is general enough to be applicable to single input single output affine nonlinear systems. The experimental results indicate fairly good performance and stable behavior in the face of sudden disturbances and/or load changes. However, the effect of unmodelled dynamics (such as the valve torque motor, valve spool, supply pump, hose delay etc.) on stability and performance is a question that remains to be answered. It is possible to incorporate these dynamics into the above analysis at the cost of increased complexity, but as long as the state variables corresponding to these dynamics are not being measured, a significant performance improvement would not be expected. Finally, we believe that it is possible to extend the presented method to MIMO nonlinear systems. REFERENCES

1. Tunay, | . and Kaynak, O., Provident Control of an Electro-Hydraulic Servo System, Proc. of IEEE Industrial Electronics Conference; 1ECON '93, Hawaii, Nov. 15-19, 1993, vol. 1, 91-96, (1993) 2. Hwang, C.L. and Lan, C.H., The Position Control of Electrohydranlic Servomechanism via a Novel Variable Structure Control, Mechatronics, Vol 4, No 4, 369-391, (1994) 3. Slotine, J.J. and Li, W., Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ (1991) 4. Electro-Hydraulic Servomechanism Type EHS 160 Manual, Feedback Instrmnents Ltd., Sussex, UK 5. McCloy, D. and Martin, H.R., The Control of FuidPower, John Wiley & Sons, NY, (1973) 6. Merritt, H.E., Hydraulic Control Systems, John Wiley & Sons, NY, (1967) 7. Tunay, i. and Kaynak, O., A New Variable Smacture Controller for Affme Nonlinear Systems with Non-matching Uncertainties, Int. Journal of Control, v.62, 917-939, (1995)

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i, TUNAY and O. KAYNAK

APPENDIX

Quantity Value As

B J K0 Kb Kle

K~, Kst

2 0.35 0.34 97.2 3500 0.24 0.03 870 1.7

PI,P2 Plv, P2~

P~

66

Q1,Q2

Tf V¢ Vm Xs Vi

0

0.26 85 0.72

Unit cm 2

Explanation

Valve spool orifice area Ncm~a&s Lumped viscous fric. coef. of motor and load Ncms2/rad Rotational inertia of motor and load Valve flow gain ml/cm The bulk modulus of oil bar ml/bar External leakage coefficient ml/bar Internal leakage coefficient ml/s/cm Spool flapper feedback constant Valve voltage gain ml/s/mV bat Motor input port pressures Valve output port pressures bar bar Nominal supply pressure ml/s Valve output port flows Ncm/bar Motor piston friction coefficient ml Effective volume of motor and hoses Motor displacement per radian ml/rad cm Valve spool displacement Valve input voltage mV Angular displacement of motor shaft rad Positive square root