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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 5, NO. 1, MARCH 2000 79

Adaptive Robust Motion Control of Single-RodHydraulic Actuators: Theory and Experiments

Bin Yao, Member, IEEE, Fanping Bu, John Reedy, and George T.-C. Chiu

Abstract—High-performance robust motion control ofsingle-rod hydraulic actuators with constant unknown in-ertia load is considered. In contrast to the double-rod hydraulicactuators studied previously, the two chambers of a single-rodhydraulic actuator have different areas. As a result, the dynamicequations describing the pressure changes in the two chamberscannot be combined into a single load pressure equation. Thiscomplicates the controller design since it not only increases thedimension of the system to be dealt with but also brings in thestability issue of the added internal dynamics. A discontinuousprojection-based adaptive robust controller (ARC) is constructed.The controller is able to take into account not only the effect ofparameter variations coming from the inertia load and varioushydraulic parameters but also the effect of hard-to-model non-linearities such as uncompensated friction forces and externaldisturbances. The controller guarantees a prescribed outputtracking transient performance and final tracking accuracy ingeneral while achieving asymptotic output tracking in the presenceof parametric uncertainties. In addition, the zero error dynamicsfor tracking any nonzero constant velocity trajectory is shown tobe globally uniformly stable. Extensive experimental results areobtained for the swing motion control of a hydraulic arm andverify the high-performance nature of the proposed ARC controlstrategy. In comparison to a state-of-the-art industrial motioncontroller, the proposed ARC algorithm achieves more than amagnitude reduction of tracking errors. Furthermore, duringthe constant velocity portion of the motion, the ARC controllerreduces the tracking errors almost down to the measurementresolution level.

Index Terms—Adaptive control, electrohydraulic system,motion control, robust control, servo control.

I. INTRODUCTION

H YDRAULIC systems have been used in industry in a widenumber of applications by virtue of their small size-to-

power ratios and the ability to apply very large force and torque;examples like electrohydraulic positioning systems [1], [2], ac-tive suspension control [3], and industrial hydraulic machines[4]. However, hydraulic systems also have a number of char-acteristics which complicate the development of high-perfor-

Manuscript received April 15, 1999; revised January 4, 2000. Recommendedby Technical Editor K. Ohnishi. This work was supported in part by the NationalScience Foundation under CAREER Grant CMS-9734345 and in part by a grantfrom the Purdue Research Foundation. This paper was presented in part at the1999 American Control Conference, San Diego, CA, June 2–4.

B. Yao, F. Bu, and G. T.-C. Chiu are with the Ray W. Herrick Laboratory,School of Mechanical Engineering, Purdue University, West Lafayette,IN 47907 USA (e-mail: [email protected]; [email protected];[email protected]).

J. Reedy is with the Ray W. Herrick Laboratory, School of MechanicalEngineering, Purdue University, West Lafayette, IN 47907 USA (e-mail:[email protected]) and is also with Caterpillar Inc., Peoria, IL 61656USA.

Publisher Item Identifier S 1083-4435(00)02468-6.

mance closed-loop controllers. The dynamics of hydraulic sys-tems are highly nonlinear [5]. Furthermore, the system may besubjected to nonsmooth and discontinuous nonlinearities due tocontrol input saturation, directional change of valve opening,friction, and valve overlap. Aside from the nonlinear nature ofhydraulic dynamics, hydraulic systems also have a large extentof model uncertainties. The uncertainties can be classified intotwo categories:parametric uncertaintiesanduncertain nonlin-earities. Examples of parametric uncertainties include the largechanges in load seen by the system in industrial use and thelarge variations in the hydraulic parameters (e.g., bulk mod-ulus) due to the change of temperature and component wear [6].Other general uncertainties, such as the external disturbances,leakage, and friction, cannot be modeled exactly and the non-linear functions that describe them are not known. These kindsof uncertainties are called uncertain nonlinearities. These modeluncertainties may cause the controlled system, designed on thenominal model, to be unstable or have a much degraded perfor-mance. Nonlinear robust control techniques, which can deliverhigh performance in spite of both parametric uncertainties anduncertain nonlinearities, are essential for successful operationsof high-performance hydraulic systems.

In the past, much of the work in the control of hydraulic sys-tems has used linear control theory [1], [2], [7]–[9] and feedbacklinearization techniques [10], [11]. In [3], Alleyne and Hedrickapplied the nonlinear adaptive control to the force control ofan active suspension driven by a double-rod cylinder, in whichonly parametric uncertainties of the cylinder are considered.They demonstrated that nonlinear control schemes can achievea better performance than conventional linear controllers.

In [12], the adaptive robust control (ARC) approach proposedby Yao and Tomizuka in [13]–[15] was generalized to providea rigorous theoretic framework for the high performance robustcontrol of a double-rod electrohydraulic servo system by takinginto account the particular nonlinearities and model uncertain-ties of the electrohydraulic servo systems. The presented ARCscheme uses smooth projections [13], [16] to solve the designconflicts between adaptive control technique and robust con-trol technique, which is technical and may not be convenientfor practical implementation. In [17], the recently proposed dis-continuous projection-method-based ARC design [15] is gen-eralized to solve the practical problems associated with smoothprojections [12].

This paper continues the work done in [17] and will gen-eralize the results to the electrohydraulic systems driven bysingle-rod actuators. In contrast to the double-rod hydraulicactuators studied in [17], the areas of the two chambers of asingle-rod hydraulic actuator are different. As a result, the two

1083-4435/00$10.00 © 2000 IEEE

80 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 5, NO. 1, MARCH 2000

dynamic equations relating the pressure changes in the twochambers to the servovalve opening cannot be combined intoa single equation that relates the load pressure to the valveopening as in [3], [5], [12], and [17]. This complicates thecontroller design since it not only increases the dimension ofthe system to be dealt with but also brings in the stability issueof the resulting internal dynamics. A discontinuous projectionbased ARC controller will be constructed to handle variousparametric uncertainties and uncertain nonlinearities. Stabilityof the resulting internal dynamics will be discussed and it willbe shown that the zero error dynamics under nonzero constantvelocity tracking is globally uniformly stable, which is thefirsttheoretical result available in the literature on the stability ofzero error dynamics for single-rod hydraulic actuators.

To test the proposed advanced nonlinear ARC strategy, athree-link robot arm (a scaled-down version of the industrialhydraulic machine arm) driven by three single-rod hydrauliccylinders has been set up. Extensive comparative experimentalresults have been obtained for the swing motion control ofthe hydraulic arm. Experimental results verify the high-per-formance nature of the proposed nonlinear ARC approach; incomparison to a state-of-the-art industrial motion controller,the proposed ARC algorithm achieves more than a magnitudereduction of tracking errors. Furthermore, during the constantvelocity portion of the motion, the ARC controller reduces thetracking errors almost down to the measurement resolutionlevel.

This paper is organized as follows. Problem formulation anddynamic models are presented in Section II. The proposed non-linear ARC strategy is given in Section III. Experimental setupand results are presented in Section IV, and conclusions aredrawn in Section V.

II. PROBLEM FORMULATION AND DYNAMIC MODELS

The system under consideration is the same as that in [12] and[17], but with a single-rod hydraulic cylinder. The schematic ofthe system is depicted in Fig. 1. The goal is to have the inertiaload to track any specified motion trajectory as closely as pos-sible; an examples would be a machine tool axis [18].

The dynamics of the inertia load can be described by

(1)

wheredisplacement of the load;mass of the load;

and pressures inside the two chambers of thecylinder;

and ram areas of the two chambers;combined coefficient of the modeleddamping and viscous friction forces onthe load and the cylinder rod;modeled Coulomb friction force;lumped uncertain nonlinearities due to ex-ternal disturbances, the unmodeled frictionforces, and other hard-to-model terms.

Fig. 1. Single-rod electrohydraulic servo systems.

The actuator (or the cylinder) dynamics can be written as [5]

(2)

wheretotal control volume of the firstchamber;total control volume of the secondchamber;

and two chamber volumes when ;effective bulk modulus;coefficient of the internal leakage of thecylinder;coefficient of the external leakage ofthe chamber;coefficient of the external leakage ofthe chamber;supplied flow rate to the forward (orcylinder-end) chamber;return flow rate of the return (orrod-end) chamber.

and are related to the spool valve displacement of theservovalve by [5]

for

for(3)

where, flow gain coefficients of the servovalve;

supply pressure of the fluid;tank or reference pressure.

In the experiments, the valve dynamics are neglected and theservovalve opening is directly related to the control inputby a known static mapping. However, for the completeness ofthe theory development, the valve dynamics used by other re-searchers [19] is included in the following to illustrate how totake into account the effect of valve dynamics if necessary. Asin [19], the spool valve displacement is related to the controlinput by a first-order system given by

(4)

YAO et al.: ADAPTIVE ROBUST MOTION CONTROL OF SINGLE-ROD HYDRAULIC ACTUATORS: THEORY AND EXPERIMENTS 81

where and are the time constant and gain of the servovalvedynamics, respectively.

As in [17], to minimize the numerical error and facilitate thegain-tuning process, constant scaling factors and areintroduced to the pressures and the valve opening, respectively;the scaled pressures are , ,

, , and the scaled valveopening is . Define the state variables

, theentire system, (1)–(4), can be expressed in a state-space form as

(5)

where , ,, ,

, , ,, ,

, , ,and the nonlinear functions and are defined by

for

for (6)

in which and is the sign function.Given the desired motion trajectory , the objective is

to synthesize a bounded control inputsuch that the outputtracks as closely as possible in spite of various

model uncertainties.

III. ARC OF SINGLE-ROD HYDRAULIC ACUTATORS

A. Design Model and Issues to be Addressed

In general, the system is subjected to parametric uncertain-ties due to the variations of , , , , , , ,

, and . For simplicity, in this paper, we only consider theparametric uncertainties due to, , the leakage coefficients

, , and , and (the nominal value of the distur-bance ). Other parametric uncertainties can be dealt with in thesame way if necessary. In order to use parameter adaptation toreduce parametric uncertainties for an improved performance,

it is necessary to linearly parametrize the state-space equation(5) in terms of a set of unknown parameters. To achieve this, de-fine the unknown parameter set as

, , ,, , . The

state-space equation (5) can, thus, be linearly parametrized interms of as

(7)

where .For simplicity, following notations are used throughout the

paper: is used for theth component of the vectorand theoperation for two vectors is performed in terms of the cor-responding elements of the vectors. The following practical as-sumption is made.

Assumption 1:The extent of parametric uncertainties anduncertain nonlinearities are known, i.e.,

(8)

whereand are known.

Physically, and . So it is also assumed thatand .

At this stage, it is readily seen that the main difficulties in con-trolling (7) are: 1) the system has unmatched model uncertain-ties since parametric uncertainties and uncertain nonlinearitiesappear in equations that do not contain control input; this diffi-culty can be overcome by employing backstepping ARC designas done in the following; 2) the nonlinear static flow mappings

and are functions of also and arenonsmoothsince theydepend on the sign of ; this prohibits the direct application ofthe general results in [15] to obtain an ARC controller, whichneed the differentiability of all terms; and 3) as will becomeclear later, the “relative degree” [14] of the system is four. Sincethe system state has a dimension of five, there exists a one-di-mensional internal dynamics after an ARC controller is synthe-sized via backstepping [20]. It is, thus, necessary to check thestability of the resulting internal dynamics, which is a uniquefeature associated with the control of single-rod hydraulic actu-ators.

In the following, the discontinuous projection-based ARC de-sign for double-rod hydraulic cylinders [17] will be generalizedto overcome the first two difficulties to obtain an ARC controllerfor (7). To this end, the following notations are introduced.


B. Notations and Discontinuous Projection Mapping

Let denote the estimate ofand the estimation error (i.e.,). Viewing (8), a simple discontinuous projection can

be defined [21], [22] as

if andif andotherwise

(9)

where . By using an adaptation law given by

(10)

where , is adiagonal adaptation rate matrix, andis an adaptation functionto be synthesized later. It can be shown [13] that for any adapta-tion function , the projection mapping used in (10) guarantees

(11)

For simplicity, let , , and represent the calculable partof the , , and , respectively, which are given by

(12)

C. ARC Controller Design

The design parallels the recursive backstepping ARC designin [15] and [17] with an added difficulty as follows.

Step 1: Noting that the first equation of (7) does not have anyuncertainties, an ARC Lyapunov function can be constructedfor the first two equations of (7) directly. Define a switching-function-like quantity as

(13)

where is the output tracking error, isthe desired trajectory to be tracked by, and is any positivefeedback gain. Since is astable transfer function, making small or converging to zero isequivalent to making small or converging to zero. Therefore,the remainder of the design is to makeas small as possiblewith a guaranteed transient performance. Differentiating (13)and noting (7)

(14)

Define the load pressure as . It is, thus, clearfrom (14) that can be thought as the virtual control input to(14), and the resulting equation has almost the same form as thedesign equation in Step 1 of the ARC algorithm for double-rod

actuators in [17]. Thus, the same ARC design technique can beused to construct an ARC control functionfor the virtual control input such that the output trackingerror converges to zero or a small value witha guaranteed transient performance. The resulting control func-tion and adaptation function are givenby

(15)

where a positive-definite (p.d.) constant diagonal matrixto be specified later, is any positive scalar, is a positiveweighting factor, and

(16)

In (15), functions as an adaptive control law used to achievean improved model compensation through online parameteradaptation given by (10), and as a robust control law, inwhich is any function satisfying the following conditions:

condition 1

condition 2(17)

where is a positive design parameter which can be arbitrarilysmall. Essentially, condition 1 of (17) represents the fact that

is synthesized to dominate the model uncertainties comingfrom both parametric uncertaintiesand uncertain nonlineari-ties with a control accuracy measured by the design parameter

, and condition 2 is to make sure that is dissipating innature so that it does not interfere with the functionality of theadaptive control part . How to choose to satisfy con-straints like (17) can be found in [14] and [15].

Let denote the input discrepancy. Forthe positive-semidefinite (p.s.d.) function defined by

, using the same technique as in [17], it canbe shown that

(18)

Step 2: In Step 1, as seen from (18), if , outputtracking would be achieved by noting (17) and using the stan-dard ARC arguments [15]. Therefore, Step 2 is to synthesize avirtual control function such that converges to zero or a smallvalue with a guaranteed transient performance as follows. From(7) and (15)

(19)


where

(20)

In (20), represents the calculable part ofand can be usedin the control function design while is the incalculable partdue to various uncertainties and has to be dealt with by certainrobust feedback as follows.

In (19), the valve opening cannot be treated as the virtualcontrol input for dynamics since both and

contain ; this important fact has beenneglected in most of the previous research due to certain tech-nical requirements. Here, we use a similar strategy as in [12] and[17] to overcome this technical difficulty. Namely, a new virtualcontrol input is introduced as

(21)

By doing so, it is readily seen that (19) has essentially the samestructure as the design equation in Step 2 of the ARC design [17]for double-rod actuators (although (19) is much more compli-cated in form). Thus, the same design strategy can be appliedto construct a virtual ARC control lawfor . The resulting ARC control function and adaptationfunction are given by

(22)

where is a positive weighting factor, is a constant,and are p.d. constant diagonal matrices, and

(23)

Similar to (17), is a robust control function satisfying thefollowing two conditions:

condition 1

condition 2

(24)

in which is a positive design parameter.Let be the input discrepancy. Consider the

augmented p.s.d. function given by .Noting (18), (19), and (22), by straightforward substitutions asin [17], it can be shown that

(25)

where denotes under the condition that (or).

Step 3: Noting the last equation of (7), Step 3 is to synthesizean actual control law for such that tracks the virtual con-trol function with a guaranteed transient performance. Theproblem here is that is not differentiable at since itcontains . Fortunately, noting that is differentiableanywhere except at the singular point of and iscontin-uous everywhere, its left and right derivatives at existand are finite. Thus, an actual control inputcan still be syn-thesized to accomplish the job as follows.1 By the definitions of

, , and , it can be checked out that the derivative ofis given by

(26)

where ,, and

. Noting (7) and (12), can begrouped into three terms as

(27)

where

1Such an input may experience a finite magnitude jump atx = 0, but isallowed in implementation.


(28)

In (27), and represent the calculable and incalculableparts of , respectively, except the terms involving the controlinput . Similarly, can be grouped into two terms as

(29)

where

(30)

Thus, similar to (19), the dynamics can be obtained as

(31)

To design an ARC law for the control inputsuch that, converge to zero or small values, let us consider the

augmented p.s.d. function given by

(32)

where . Noting (25) and (31) with (28) and (30), it isstraightforward to show that

(33)

where denotes under the condition that (or), and

.In viewing (33), the following ARC control law

and the associated adaptation function are proposed:

(34)

where , and are p.d. constant diagonal matrices,and is a robust control function satisfying

1)

2)

(35)

in which is a design parameter.

D. Main Results

Theorem 1: If controller parameters, and

in (15), (22), and (34) are chosen such that, then, the control

law (34) with the adaptation law (10) guarantees the following.

1) In general, the output tracking error and the trans-formed states are bounded. Fur-thermore, given by (32) is bounded above by

(36)

where and.

2) If after a finite time , , i.e., in the presence ofparametric uncertainties only, then, in addition to resultsin 1), asymptotic output tracking (or zero final trackingerror) is achieved, i.e., as .

Remark 1: Results in 1) of Theorem 1 indicate that the pro-posed controller has an exponentially converging transient per-formance with the exponentially converging rate and thefinal tracking error being able to be adjusted via certain con-troller parameters freely in aknownform; it is seen from (36)that can be made arbitrarily large, and , the boundof (an index for the final tracking errors), can be made ar-bitrarily small by increasing feedback gainsand/or decreasing controller parameters . Sucha guaranteed transient performance is especially important forpractical applications since execute time of a run is short. The-oretically, this result is what a well-designed robust controllercan achieve. In fact, when the parameter adaptation law (10) isswitched off, the proposed ARC law becomes a deterministicrobust control law and results 1) of the Theorem remain valid[13], [14].

Result 2) of Theorem 1 implies that the effect of parametricuncertainties can be eliminated through parameter adaptationand an improved performance is obtained. Theoretically, result2) is what a well-designed adaptive controller can achieve.

Remark 2: It is seen from (36) that the transient outputtracking error may be affected by the initial value , whichmay depend on the controller parameters, also. To further


reduce transient tracking error, the reference trajectory initial-ization can be used as in [14], [15], and [20]. Namely, insteadof simply letting the reference trajectory for the controller bethe actual desired trajectory or position [i.e., ],the reference trajectory can be generated using a stablefourth-order filter with four initials chosenas

(37)

Such a trajectory initialization guarantees thatand . Thus, the transient output tracking

error is reduced. It is shown in [14] and [20] that such a trajec-tory initialization is independent from the choice of controllerparameters and and can be performed offline once the initialstate of the system is determined.

Proof of Theorem 1:Substituting the control law (34) into(33), it is straightforward to show that

(38)

Thus, following the standard discontinuous projection-basedARC arguments as in [15] and [17], the theorem can beproved.2

E. Internal Dynamics and Zero Dynamics

Theorem 1 shows that, under the proposed ARC con-trol law, the four transformed system state variables,

, are bounded, from whichone can easily prove that the position, the velocity , andthe load pressure are bounded. However,since the original system (7) has a dimension offive, thereexists a one-dimensional internal dynamics, which reflects thephysical phenomenon that there exist infinite number sets ofpressures in the two chambers of the single-rod actuator toproduce the same required load pressure . Inother words, the fact that the load pressureand the valve opening are bounded does not necessarilyimply that the pressures and in the two chambers of thesingle-rod cylinder are bounded. It is, thus, necessary to checkthe stability of the resulting internal dynamics to make surethat the two pressures and are bounded for a bounded

2The details are quite tedious and can be obtained from the authors.

control input in implementation. Although simulation andexperimental results seem to verify that the two pressuresare indeed bounded when tracking a nonzero speed motiontrajectory, it is of both practical and theoretical interest to see ifwe can prove this fact, which is the focus in this section.

Rigorous theoretical proof of the stability of the internal dy-namics of a nonlinear system tracking an arbitrary time-varyingtrajectory is normally very hard and tedious, if not impossible[23]. Bearing this fact in mind, in the following, we take thefollowing pragmatic approach, which is a standard practice inthe nonlinear control literature [23]. Namely, instead of lookingat the stability of the general internal dynamics directly, we willcheck the stability of the zero error dynamics for tracking certaintypical motion trajectories. For the particular problem studiedin this paper, it is easy to verify that the zero error dynamics fortracking a desired trajectory is the same as the internaldynamics when . Thus, if the zero error dynamicsis proven to be globally uniformly asymptotically stable, then,it is reasonable to expect that all internal variables are boundedwhen the proposed ARC law is applied, sinceis guaranteed toconverge to a small value very quickly by Theorem 1.

Since most industrial operations of hydraulic cylinders in-volve the tracking of a constant velocity trajectory (e.g., thelarge portion of the typical point-to-point movement is underconstant velocity tracking as shown in the experiments), we willfocus on the zero error dynamics associated with the trackingof a nonzero constant velocity trajectory. For these operations,compared with the large control flows needed to maintain therequired speed, the leakage flows are typically very small andcan be neglected in the analysis for simplicity. The results aresummarized in the following theorem.

Theorem 2: Assume that the leakage flows can be neglectedand the system is absent of uncertain nonlinearities, i.e.,

and in (7). When tracking any nonzeroconstant velocity trajectory (i.e., is a nonzeroconstant), the following results hold.

1) The corresponding pressures of the two chambers and thevalve opening have a unique equilibrium; when ,the unique equilibrium point is given by

(39)

and when ,


(40)

where , , and are the equilibrium values ofthe two pressures and and the valve opening ,respectively.

2) The resulting zero error dynamics is globally3 uniformlyasymptotically stable with respect to the equilibriumvalues given by (39) or (40).

Proof of Theorem 2:When tracking a trajectory with aconstant velocity of perfectly, the actual position and ve-locity would be and , respectively.From (7), under the assumption made in Theorem 2, the equi-librium values of the two pressures and the valve opening shouldsatisfy the following equations:

(41)

When , , which is physically intuitive(for positive constant velocity, the valve opening has tobe positive). Thus, and

. It is, thus, straightfor-ward to verify that (39) is the unique solution to (41). Similarly,when , and it can be verified that (40) is theunique solution to (41). This completes the proof of 1) of thetheorem.

To prove 2) of the theorem, in the following, we will considerthe case when only since the case for canbe worked out in the same way. The zero error dynamics—theinternal dynamics for tracking such a trajectory perfectly (i.e.,

and )—is described by

(42)

From the first equation of (42),

(43)

and

(44)

Substituting the last two equations of (42) into (44), the valveopening of the zero error dynamics is obtained as

(45)

Noting that , , andall have to be positive to be physically

meaningful,4 from (45), for positive . Thus,

and (46)

3For all pressures within physical limits.

Let and be the pressure pertur-bations from their equilibrium values. Substituting (45) into thelast equation of (42), we have

(47)

Noting (46), (43), and (39) for ,

(48)

Thus, (47) becomes

(49)

where

(50)

Noting that , , , and are all positive func-tions, defined by (50) is a positive function and is uniformlybounded below by a positive number, i.e., there exists asuch that . Thus, from (49), the derivative of the Lya-punov function is

(51)

which is negative definite with respect to . Thus, the originof the dynamics is globally uniformly asymptotically stable,and is bounded. Similarly, it can be proved that the originof the dynamics is also globally uniformly stable. This com-pletes the proof of 2) of the theorem.

Remark 3: For the regulation problem (i.e., the desired tra-jectory is a constant), , and it is seen from (41)that the equilibrium pressures are not unique; in fact, any pres-sures and which satisfy

are a set of equilibrium values. Thus, the zero dynamicsfor the regulation problem will be at most marginally stable butnot asymptotically stable. Further study is needed for the sta-bility of the internal dynamics for the regulation problem, whichis one of the focuses of our future research.

IV. COMPARATIVE EXPERIMENTS

A. Experiment Setup

To test the proposed nonlinear ARC strategy and study funda-mental problems associated with the control of electrohydraulicsystems, a three-link robot arm (a scaled-down version of anindustrial hydraulic machine arm) driven by three single-rodhydraulic cylinders has been set up at the Ray W. HerrickLaboratory of the School of Mechanical Engineering, PurdueUniversity, West Lafayette, IN. The three hydraulic cylinders

4h (x ) andh (x ) represent the effect of the control volumes of the twochambers, which are positive for any operations.g andg represents the flowgains due to the square roots of pressure drops, which are positive.


Fig. 2. Experimental setup.

(Parker D2HXTS23A, DB2HXTS23A, and DB2HXT23A)are controlled by two proportional directional control valves(Parker D3FXE01HCNBJ0011) and one servovalve (ParkerBD760AAAN10) manufactured by Parker Hannifan Company.Currently, experiments are performed on the swing motion con-trol of the arm (or the first joint) with the other two joints fixed.The schematic of the system is shown in Fig. 2. The swingcircuit is driven by a single-rod cylinder (Parker D2HXTS23Awith a stroke of 11 in) and controlled by the servovalve.The cylinder has a built-in LVDT sensor, which provides theposition and velocity information of the cylinder movement.Pressure sensors (Entran’s EPXH-X01-2.5KP) are installed oneach chamber of the cylinder. Due to the contaminative noisesin the analog signals, the effective measurement resolution forthe cylinder position is around 1 mm and the resolution for thepressure is around 15 lbf/in. Since the range of the velocityprovided by the LVDT sensor is too small (less than 0.09 m/s),backward difference plus filter is used to obtain the neededvelocity information at high-speed movement. All analogmeasurement signals (the cylinder position, velocity, forwardand return chamber pressures, and the supplied pressure) arefed back to a Pentium II PC through a plugged-in 16-bit A/Dand D/A board. The supplied pressure is 1000 lbf/in.

B. System Identification

For the swing motion shown in Fig. 2, due to the nonlineartransformation between the joint swing angleand the cylinderposition , the equivalent mass of a constant swing in-ertia seeing at the swing cylinder coordinate depends on theswing angle or the cylinder position as well. However, byrestricting the movement of the swing cylinder in its middlerange, this equivalent mass will not change much and, thus,can be treated as constant as assumed in the paper; the equiva-lent mass is around 2200 kg for the no-load situation. Althoughtests have been done to obtain the actual friction curve shownin Fig. 3, for simplicity, friction compensation is not used inthe following experiments [i.e., let in (1)] to testthe robustness of the proposed algorithm to thesehard-to-model

Fig. 3. Friction force.

terms. The cylinder physical parameters are in ,in , in , and in .

By assuming no leakage flows and moving the cylinder at dif-ferent constant velocities, we can back out the static flow map-ping of the servovalve (3). The estimated flow gains are

in psi s and in psi s ,where the unit for the servovalve opening is normalized in termsof the control voltage supplied to the servovalve at the steadystate. Dynamic tests are also performed and reveal that the ser-vovalve dynamics is of second order with a bandwidth around 10Hz. The effective bulk modulus is estimated aroundPa.

C. Controller Simplifications

Some simplifications suitable for our experiments have beenmade in implementing the proposed ARC control strategy.First, unlike other experiments [3], [19] where high bandwidthservovalves with spool position feedback is used, our systemis designed to mimic typical industrial use of electrohydraulicsystems. As such, our system is not equipped with costlysensors to measure the spool displacement of the servovalve forfeedback. Furthermore, since a standard industrial servovalveis used, the valve dynamics is actually of a second order witha not-so-high bandwidth around 10 Hz (as opposed to thefirst-order dynamics (4) used by other researchers [19]). Thisgreatly increases the difficulties in implementing advancedcontrol algorithms if valve dynamics were to be consideredin the controller design. To deal with this practical issue, thefollowing pragmatic approach is taken: the valve dynamics areneglected in the controller design stage (i.e., lettingin the previous ARC controller design) and the closed-loopstability will be achieved by placing the closed-loop bandwidthof the ARC controller below the valve bandwidth with a certainamount of margin as done in the following experiments. Withthis simplification strategy, the valve opening is related tothe control input by a static gain given by . Thus,the ARC control design in Step 3 in Section III-C is not neededand the actual control inputis directly calculated based on thecontrol function given by (22) and the static flow mapping(21) with replaced by . The corresponding adaptationfunction is given by in (22). By doing so, the resultingARC controller is quite simple and yet sufficient, as shown inthe comparative experimental results.


The second simplification is made to the selection of the spe-cific robust control term in (15) and in (22). Let and

be any nonlinear feedback gains satisfying

(52)

in which and is defined in (8). Then,using similar arguments as in [15] and [17], one can show thatthe robust control function of satisfies (15) and(17), and satisfies (22) and (24).

Knowing the above theoretical results, we may imple-ment the needed robust control terms in the following twoways. The first method is to pick up a set of values for

and to calculate the right-handside of (52). and can then be determined so that (52)is satisfied for a guaranteed global stability and a guaranteedcontrol accuracy. This approach is rigorous and should bethe formal approach to choose. However, it increases thecomplexity of the resulting control law considerably since itmay need significant amount of computation time to calculatethe lower bound. As an alternative, a pragmatic approach isto simply choose and large enough without worryingabout the specific values of , and .By doing so, (52) will be satisfied for certain sets of values of

, and , at least locally around thedesired trajectory to be tracked. In this paper, the second ap-proach is used since it not only reduces the online computationtime significantly, but also facilitates the gain tuning processin implementation.

D. Comparative Experimental Results

Three controllers are tested for comparison:1) ARC: This is the controller proposed in this paper and de-

scribed in previous sections with the simplification outlined inSection IV-C. For simplicity, in the experiments, only two pa-rameters, , and , are adapted; the first parameter representsthe effect of the equivalent mass and the second parameter repre-sents the effect of the nominal value of the lumped disturbance.The effect of leakage flows is neglected to test the performancerobustness of the proposed algorithm to these terms. Since thevalve dynamics is neglected and their bandwidth is not so high(around 10 Hz), not so large feedback gains are used to avoidinstability; the control gains used are .The scaling factors are andrespectively. Weighting factors are . Adaptationrates are set at .

2) Deterministic Robust Control (DRC):This is the samecontrol law as ARC but without using parameter adaptation, i.e.,letting . In such a case, the proposed ARC con-trol law becomes a deterministic robust control law [15].

3) Motion Controller: This is the state-of-the-art industrialmotion controller (Parker’s PMC6270ANI two-axis motioncontroller) that was bought from Parker Hannifin Companyalong with the Parker’s cylinder and valves used for theexperiments. The controller is essentially a proportional plusintegral plus derivative (PID) controller with velocity andacceleration feedforward compensation. Controller gains areobtained by strictly following the gain tuning process stated intheServo Tuner User Guidecoming with the motion controller[24]. The tuned gains are

, which represent theP-gain, I-gain, D-gain, the gain for velocity feedforwardcompensation, and the gain for acceleration feedforwardcompensation, respectively.

The three controllers are first tested for a slow point-to-pointmotion trajectory shown in Fig. 4, which has a maximumvelocity of m/s and a maximum acceleration of

m/s . The tracking errors are shown in Fig. 5. Asseen, the proposed DRC and ARC have a better performancethan the motion controller in terms of both transient and finaltracking errors. Due to the use of parameter adaptation as shownin Fig. 8, the final tracking error of ARC is reduced almostdown to the position measurement noise level of 1 mm whileDRC still has a slight offset. This illustrates the effectivenessof using parameter adaptation, although the estimates do notconverge to their true values due to other modeling errors, suchas the neglected friction force. The pressures of ARC are shownin Fig. 6, which are regular and, as predicted, are bounded. Thecontrol input of ARC is shown in Fig. 7, which is regular.

To test the performance robustness of the proposed algo-rithms to parameter variations, a 45-kg load is added at theend of the robot arm, which increases the equivalent massof the cylinder to 4000 kg. The tracking errors are shown inFig. 9. As seen, even for such a short one-run experiment, theadaptation algorithm of the ARC controller is able to pick upthe change of the inertial load (the parameter estimateshownin Fig. 10 drops quicker than the one in Fig. 8) and an improvedperformance is achieved in comparison to the nonadaptiveDRC. Again, both ARC and DRC exhibit better performancethan the motion controller.

The three controllers are then run for a fast point–point mo-tion trajectory shown in Fig. 4, which has a maximum velocityof m/s and an acceleration of m/s ; bothare near their physical limits. The tracking errors are shown inFig. 11. As seen, the motion controller cannot handle such an ag-gressive movement well and a large tracking error around 15–20mm is exhibited during the constant high-speed movement. Incontrast, the tracking error of the proposed ARC during the en-tire run is kept within 5 mm. Furthermore, the tracking errorgoes back to the measurement noise level of 1 mm very quicklyafter the short large acceleration and deceleration periods. Asseen from the transient pressures of ARC shown in Fig. 12 andthe control input shown in Fig. 13, the system is actually at itsfull capacity during the short large acceleration and decelera-tion periods. During the acceleration period, the control inputexceeds the maximal output voltage of 10 V while the pressure

reaches close to the supplied pressure of 6.897 MPa, andduring the deceleration period, is down to reference pressure


Fig. 4. Point-to-point motion trajectories.

Fig. 5. Tracking errors for slow point-to-point motion without load.

Fig. 6. Pressures for slow point-to-point motion without load.

Fig. 7. Control input for slow point-to-point motion without load.

Fig. 8. Parameter estimation for slow point-to-point motion without load.

Fig. 9. Tracking errors for slow point-to-point motion with load.

Fig. 10. Parameter estimation for slow point-to-point motion with load.

Fig. 11. Tracking errors for fast point-to-point motion without load.


Fig. 12. Pressures for fast point-to-point motion without load.

Fig. 13. Control input for fast point-to-point motion without load.

Fig. 14. Parameter estimation for fast point-to-point motion without load.

Fig. 15. Tracking errors for fast sine-wave motion with load.

Fig. 16. Pressures for fast sine-wave motion without load.

while reaches the supplied pressure of 6.897 MPa. Despitethese difficulties, ARC performs very well as seen in Fig. 11.The parameter estimates are shown in Fig. 14.

Comparative experiments between ARC and DRC are alsorun for tracking sinusoidal motion trajectories with different fre-quencies. For example, the tracking errors for tracking a 1/4-Hzsinusoidal trajectory of with load are shownin Fig. 15. As seen, ARC performs better than DRC, which il-lustrates the effectiveness of using parameter adaptation. Thetracking error of ARC is very small, mostly around the measure-ment resolution of 1 mm, which verifies the high-performancenature of the proposed ARC control strategy. The pressures ofARC are shown in Fig. 16, which are bounded but experiencean abrupt change around when the de-sired velocity changes the direction at those instances. This re-sult qualitatively agrees with the result predicted in 1) of The-orem 2, where the equilibrium values for pressures for positivedesired velocity tracking are much different from the equilib-rium values for negative desired velocity tracking.

V. CONCLUSIONS

In this paper, a discontinuous projection-based ARC con-troller has been developed for high-performance robust controlof electrohydraulic systems driven by single-rod actuators.The proposed ARC controller takes into account the particularnonlinearities associated with the dynamics of single-rod hy-draulic actuators and uses parameter adaptation to eliminate theeffect of unavoidable parametric uncertainties due to variationsof inertia load and various hydraulic parameters. Uncertainnonlinearities such as external disturbances and uncompen-sated friction forces are effectively handled via certain robustfeedback for a guaranteed robust performance. The controllerachieves a guaranteed transient performance and final trackingaccuracy for the output tracking while achieving asymptoticoutput tracking in the presence of parametric uncertaintiesonly. Furthermore, it is shown that the zero error dynamics fortracking any nonzero constant velocity trajectory is globallyuniformly stable. Extensive comparative experimental resultsare obtained for the swing motion of a hydraulic arm. Ex-perimental results verify the high-performance nature of theproposed ARC strategy.


REFERENCES

[1] P. M. FitzSimons and J. J. Palazzolo, “Part I: Modeling of a one-de-gree-of-freedom active hydraulic mount,”ASME J. Dynam. Syst., Meas.,Contr., vol. 118, no. 3, pp. 439–442, 1996.

[2] P. M. FitzSimons and J. J. Palazzolo, “Part II: Control,”ASME J. Dynam.Syst., Meas., Contr., vol. 118, no. 3, pp. 443–448, 1996.

[3] A. Alleyne and J. K. Hedrick, “Nonlinear adaptive control of active sus-pension,”IEEE Trans. Contr. Syst. Technol., vol. 3, pp. 94–101, Jan.1995.

[4] B. Yao, J. Zhang, D. Koehler, and J. Litherland, “High performanceswing velocity tracking control of hydraulic excavators,” inProc. Amer-ican Control Conf., 1998, pp. 818–822.

[5] H. E. Merritt, Hydraulic Control Systems. New York: Wiley, 1967.[6] J. Whatton, Fluid Power Systems. Englewood Cliffs, NJ: Pren-

tice-Hall, 1989.[7] T. C. Tsao and M. Tomizuka, “Robust adaptive and repetitive digital

control and application to hydraulic servo for noncircular machining,”ASME J. Dynam. Syst., Meas., Contr., vol. 116, no. 1, pp. 24–32, 1994.

[8] A. R. Plummer and N. D. Vaughan, “Robust adaptive control for hy-draulic servosystems,”ASME J. Dynam. Syst., Meas., Contr., vol. 118,no. 2, pp. 237–244, 1996.

[9] J. E. Bobrow and K. Lum, “Adaptive, high bandwidth control of a hy-draulic actuator,”ASME J. Dynam. Syst., Meas., Contr., vol. 118, no. 4,pp. 714–720, 1996.

[10] R. Vossoughi and M. Donath, “Dynamic feedback linearization forelectro-hydraulically actuated control systems,”ASME J. Dynam. Syst.,Meas., Contr., vol. 117, no. 4, pp. 468–477, 1995.

[11] L. D. Re and A. Isidori, “Performance enhancement of nonlinear drivesby feedback linearization of linear-bilinear cascade models,”IEEETrans. Contr. Syst. Technol., vol. 3, pp. 299–308, May 1995.

[12] B. Yao, G. T. C. Chiu, and J. T. Reedy, “Nonlinear adaptive robust controlof one-dof electro-hydraulic servo systems,” inProc. ASME IMECE’97,vol. 4, Dallas, TX, 1997, pp. 191–197.

[13] B. Yao and M. Tomizuka, “Smooth robust adaptive sliding mode controlof robot manipulators with guaranteed transient performance,” inProc.American Control Conf., 1994, pp. 1176–1180.

[14] B. Yao and M. Tomizuka, “Adaptive robust control of SISO nonlinearsystems in a semi-strict feedback form,”Automatica, vol. 33, no. 5, pp.893–900, 1997.

[15] B. Yao, “High performance adaptive robust control of nonlinear systems:A general framework and new schemes,” inProc. IEEE Conf. Decisionand Control, 1997, pp. 2489–2494.

[16] B. Yao and M. Tomizuka, “Adaptive robust control of MIMO nonlinearsystems,” in Proc. IEEE Conf. Decision and Control, 1995, pp.2346–2351.

[17] B. Yao, F. Bu, and G. T. C. Chiu, “Nonlinear adaptive robust controlof electro-hydraulic servo systems with discontinuous projections,” inProc. IEEE Conf. Decision and Control, 1998, pp. 2265–2270.

[18] B. Yao, M. Al-Majed, and M. Tomizuka, “High performance robust mo-tion control of machine tools: An adaptive robust control approach andcomparative experiments,”IEEE/ASME Trans. Mechatron., vol. 2, pp.63–76, June 1997.

[19] A. Alleyne, “Nonlinear force control of an electro-hydraulic actuator,”in Proc. Japan/USA Symp. Flexible Automation, Boston, MA, 1996, pp.193–200.

[20] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic,Nonlinear and Adap-tive Control Design. New York: Wiley, 1995.

[21] S. Sastry and M. Bodson,Adaptive Control: Stability, Convergence andRobustness. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1989.

[22] G. C. Goodwin and D. Q. Mayne, “A parameter estimation perspectiveof continuous time model reference adaptive control,”Automatica, vol.23, no. 1, pp. 57–70, 1989.

[23] H. K. Khalil, Nonlinear Systems, 2nd ed. Englewood Cliffs, NJ: Pren-tice-Hall, Inc., 1996.

[24] Servo Tuner User Guide, Parker Hannifin Company, Cleveland, OH,Nov. 1994.

[25] B. Yao and M. Tomizuka, “Smooth robust adaptive sliding mode controlof robot manipulators with guaranteed transient performance,”ASME J.Dynam. Syst., Meas., Contr., vol. 118, no. 4, pp. 764–775, 1996.

Bin Yao (S’92–M’96) received the B.Eng. degree inapplied mechanics from Beijing University of Aero-nautics and Astronautics, Beijing, China, the M.Eng.degree in electrical engineering from Nanyang Tech-nological University, Singapore, and the Ph.D. de-gree in mechanical engineering from the Universityof California, Berkeley, in 1987, 1992, and 1996, re-spectively.

Since 1996, he has been an Assistant Professor inthe School of Mechanical Engineering, Purdue Uni-versity, West Lafayette, IN. His research interests in-

clude design and control of intelligent high-performance coordinated control ofelectromechanical/electrohydraulic systems, optimal adaptive and robust con-trol, nonlinear observer design and neural networks for virtual sensing, mod-eling, fault detection, diagnostics, and adaptive fault-tolerant control, and datafusion.

Dr. Yao was awarded a Faculty Early Career Development (CAREER) Awardfrom the National Science Foundation (NSF) in 1998 for his work on the engi-neering synthesis of high-performance adaptive robust controllers for mechan-ical systems and manufacturing processes. He received the Caterpillar Engi-neering Young Faculty Development Award in 1997.

Fanping Bu received the B. Eng. and M.Eng. degreesin mechanical engineering from Tsinghua University,Beijing, China, in 1994 and 1997, respectively. He iscurrently working toward the Ph.D. degree at PurdueUniversity, West Lafayette, IN.

His current research interests include adaptive con-trol, robust control, robotics, and control of electro-hydraulic systems.

John Reedy was born in Los Alamos, NM, in1973. He received the B.S.M.E. degree from theUniversity of California, San Diego, in 1996. Heis currently working toward the Master’s degree atPurdue University, West Lafayette, IN, specializingin the design and control of high-performanceelectrohydraulic systems.

He is also currently with Caterpillar Inc., Peoria,IL, analyzing and designing electrohydraulic controlsystems for mobile equipment.

George T.-C. Chiu received the B.S.M.E. degreefrom National Taiwan University, Taipei, Taiwan,R.O.C., and the M.S. and Ph.D. degrees from theUniversity of California, Berkeley, in 1985, 1990,and 1994, respectively.

From 1994 to 1996, he was an R&D Engineer withHewlett-Packard Company, developing high-per-formance color inkjet printers and multifunctionmachines. Since 1996, he has been an AssistantProfessor and a Feddersen Fellow in the Schoolof Mechanical Engineering, Purdue University,

West Lafayette, IN. His current research interests are design and control ofimaging and printing systems, signal and image processing, adaptive andoptimal control, integrated design and control of electromechanical systems,and mechatronics.

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