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A simplified approach to force control for electro-hydraulic systems
Control Engineering Practice 8 (2000) 1347}1356
A simpli"ed approach to force control for electro-hydraulic systems夽 Andrew Alleyne*, Rui Liu Department of Mechanical and Industrial Engineering, University of Illinois, Urbana Champaign, 140 Mechanical Engineering Building, MC-244, 1206 West Green Street, Urbana, IL 61801, USA Received 30 August 1999; accepted 4 May 2000
Abstract In this paper, a Lyapunov-based control algorithm is developed for force tracking control of an electro-hydraulic actuator. The developed controller relies on an accurate model of the system. To compensate for the parametric uncertainties, a Lyapunov-based parameter adaptation is applied. The adaptation uses a variable structure approach to account for asymmetries present in the system. The coupled control law and the adaptation scheme are applied to an experimental valve-controlled cylinder. Friction modeling and compensation are also discussed. The experimental results show that the nonlinear control algorithm, together with the adaptation scheme, gives a good performance for the speci"ed tracking task. The original adaptive control law is then simpli"ed in several stages with an examination of the output tracking at each stage of simpli"cation. It is shown that the original algorithm can be signi"cantly simpli"ed without too signi"cant a loss of performance. The simplest algorithm corresponds to an adaptive velocity feedback term coupled with a simple force error feedback. 2000 Elsevier Science ¸td. All rights reserved. Keywords: Nonlinear control; Electro-hydraulic systems; Force control; Pressure control; Lyapunov methods; Adaptive algorithms; Friction; Hydraulic actuator
1. Introduction Hydraulic systems are important actuators in modern industry, principally because they have a high power/ mass ratio, fast response, and high sti!ness: a combination unmatched by any other commercial technology. Therefore, investigating the control of position or force outputs of hydraulic actuators should be of great interest to both the academic and industrial "elds. In particular, force and pressure tracking are important for some applications, such as vibration isolation and automotive active suspension, where an (almost) ideal force actuator is assumed in current research and application. In Alleyne and Liu (1999a), it is shown that fundamental limits exist on simple controllers for force or pressure tracking with hydraulic systems. Therefore, advanced controllers are a necessity. The cylinder velocity acts as a feedback
夽
An early and shorter version of this paper by the authors, entitled &&Nonlinear force/pressure tracking of an electro-hydraulic actuator'' was presented at the 1999 IFAC World Congress in Beijing, P.R. China, July 1999. * Corresponding author. Tel.: #1-217-244-9993; fax: #1-217-2446534. E-mail address: [email protected] (A. Alleyne).
term from the position output of the cylinder to the pressure di!erential across the piston. This is illustrated in Fig. 1 which shows a block diagram of such a system's linearized dynamics. Therefore, for a hydraulic servosystem, the poles of the load will manifest themselves as the zeros of the open loop force transfer function. If the system poles are lightly damped, this implies that the linearized dynamics of the hydraulic force/pressure loop will have zeros near the imaginary axis. The present work utilizes a particular controller structure to address this challenging problem. The reader is referred to McCloy and Martin (1973) for a more detailed explanation of the linearized system dynamics in Fig. 1. In Alleyne (1996) a Lyapunov-based force control approach was developed for a model of a hydraulic servo system and a gradient parameter estimation scheme was introduced to account for modeling uncertainty. The present paper uses a similar nonlinear control approach; however, it extends the results to include an important friction compensation scheme and presents experimental results verifying the approach. The experimental results are key to eliciting important issues in force control that were not evident in previous simulation studies. In addition, it will be shown that a few elements of the overall nonlinear adaptive controller make the majority of the
0967-0661/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 6 1 ( 0 0 ) 0 0 0 8 1 - 2
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Fig. 1. Linearized dynamics.
contributions to the "nal overall control signal. Therefore, in an attempt at algorithmic e$ciency and simplicity the control law can be greatly simpli"ed with little reduction in performance. The rest of the paper is organized as follows. Section 2 describes the detailed nonlinear model equations and also provides experimental veri"cation of the model. In Section 3, the control law and adaptive algorithm are derived and system stability proved. Section 4 gives the experimental results for force tracking on the experimental system. After the original algorithm has been demonstrated, Section 5 then introduces a series of simpli"cations with corresponding results. A conclusion then summarizes the main points of the paper.
2. System model The problem to be studied is depicted in Fig. 2. The goal is to have the actuator track a speci"ed force or pressure trajectory. The force-producing element is a double-ended hydraulic cylinder driven by a singlestage, four-way spool valve. The valve is operated via a direct drive linear motor and has its own closed loop analog controller. Due to this inner control loop the dynamics of the valve approximate a "rst-order model with a rate limit. The rate limit is fast enough to be ignored for our purposes here. This model matches well with actual time responses given by manufacturer's data shown in Fig. 3.The di!erential equations governing the dynamics of the actuator are given in Merritt (1967) for a symmetric actuator as follows: < R PQ "!Ax !C P #Q , RK * * 4b * C P !sgn(x )P Q T *, Q "C wx * B T o
(1) (2)
where < is the total actuator volume, b the e!ective R C bulk modulus, P the load pressure, A the actuator ram *
Fig. 2. A schematic of the experimental system.
Fig. 3. Valve step responses (Moog Product Literature).
area, x the actuator piston position, C coe$cient of RK leakage, Q the load #ow, C the discharge coe$cient, * B w the spool valve area gradient, x the spool valve posiT tion, P the supply pressure, and o the #uid density. Q Combining (1) and (2) with other system parameters results in the system state equations given below (Alleyne, 1996): x "x , 1 x " (!kx #Ax !Friction), m (3) x "!ax !bx #(c(P !sgn(x )x )x , Q 1 K x "! x # u, q q where x is the actuator piston position (x), x the actuator piston velocity (x ), x the load pressure (P ), * x the valve position (x ), u the input current to servo T valve, a"4Ab /< , b"4C b /< , and c"4C b w/ C R RK C R B C (< (o). R Eq. (3) is a greatly simpli"ed representation of the actual system dynamics. In order to check if this state space model captures the key component of the system dynamics, a detailed Matlab/SIMULINK model including friction was also derived. The friction model is a static, memoryless friction model incorporating static friction, Coulomb friction and Stribeck e!ects. A typical velocity-friction plot of such friction model is shown in Fig. 4. The friction model used in the modeling of the system is a novel one: it includes Karnopp's stick}slip model (Karnopp, 1985) and the Stribeck e!ect (Armstrong-Helouvry, Dupont & Canudas de Wit, 1994). In Karnopp's friction model, there are two key points: (1) a &stick' phase occurs when velocity is within a small critical velocity range, instead of only when velocity is exactly zero and (2) there is a maximum value that friction can have when the mass under consideration sticks. Let this maximum value be denoted as F . This DQR?RGA is the same as common Coulomb friction. However, within this stick region, the amplitude of friction is not
A. Alleyne, R. Liu / Control Engineering Practice 8 (2000) 1347}1356
Fig. 4. Friction model used in the system model, including Karnopp and Stribeck models.
just a constant value multiplied by the sign of the velocity which is the case for a common Coulomb friction model. Instead, it is such that the sum of all forces including friction is zero, i.e., the friction balances the other forces. Once the amplitude of the sum of other forces exceeds that of F , the &stick' phase cannot be maintained and DQR?RGA the mass under consideration will move (slip). The Stribeck e!ect is also observed in the experiment and included in the friction model. If the range of the Stribeck e!ect is small, the velocity}friction relation could be approximated as linear, with a negative slope. Experiments were conducted to test the model. Fig. 5 shows the estimated friction}velocity relation, with Coulomb and Stribeck friction, during tracking of a 0.5 Hz square wave. The value of friction was obtained as F "P A!mxK . (4) D * Since the friction in Fig. 5 was not directly measured but rather obtained from the measured system variables, the obtained relation between velocity and friction is called the estimated friction. The modeled data matched well with the experimental data. The data in Fig. 5 compare favorably with established data in the literature. For example, the friction measured by Amin, Friedland and Harnoy (1997) for a DC motor application is shown in Fig. 6. Although the magnitudes on the axes are quite di!erent, the friction characteristics are very similar to those of Fig. 5 at lower velocities with a clear Stribeck e!ect plus a hysteretic e!ect. Fig. 7 demonstrates the simulation model "delity by comparing the measured and modeled pressure (x ) state when the system was subjected to a 15 Hz sinusoidal open-loop input to the valve. The pressure data shown here has been post processed to "lter noise on the signal using a zero-phase error, non-causal "lter. As can be seen in the "gure, the modeled system matches well with the experimental data.
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Fig. 5. Estimated friction}velocity relation for 0.5 Hz square wave.
Fig. 6. Measured friction characteristics (Amin et al., 1997).
Fig. 7. Measured and modeled load pressure (- - -"model).
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3. Controller design Temporarily ignoring the e!ects of friction, for reasons to be given later, the system output of interest in Eq. (3) is the force (or pressure) from the actuator, thus y"Ax (or x ). Therefore, the relative degree of the nonlinear system is 2 and the zero dynamics of the system are the "rst two state equations with x "0. It can be easily veri"ed that the system is nonlinear minimum phase (Khalil, 1996), and thus the zero dynamics are stable. For ease of following analysis, the third and fourth state equations can be represented as x "f #g x , x "f #g u, where
(5)
f "!ax !bx , g "c(P !sgn(x )x , Q f "!x /q and g "K/q. An examination of Eq. (3) will indicate that this system falls into the class of systems known as Strict Feedback Form systems de"ned in Krstic, Kannellakopoulos and Kokotovic (1995). Therefore, a recursive Lyapunov design such as Integrator Backstepping would be appropriate. The following is a derivation of such a recursive law for the hydraulic force control problem. The desired force trajectory F can be divided by the actuator ram area BCQGPCB A to determine the desired load pressure pro"le x . BCQGPCB De"ne the following pressure and valve position errors: e "x !x , e "x !x , (6) BCQGPCB BCQGPCB where the desired valve position will be determined shortly. De"ne the positive-de"nite Lyapunov function < "o e #o e , (7) where o '0, o '0. Di!erentiating (7) along the systems trajectories (5) yields
clearly indicates convergence of the output pressure to its desired value. The desired value of the servovalve is dependent on the desired pressure (x ) as well as the system states. It will be up to the actual control signal u, to be de"ned shortly, to ensure the valve tracks its desired value. Substituting (9) into (8) results in
1 o u" !f #x !k e ! g e , (11) BCQGPCB o g where k '0, then
(12)
In practice, (P !sgn(x )x is seldom zero when the Q system is operating smoothly, since "x " is seldom close to P . In the rare case (P !sgn(x )x equals zero (e.g., Q Q due to the noise in x ), it is set to a small positive number to avoid the problem of dividing by zero. The cascaded nature of the control given in Eq. (12) is quite common to Strict Feedback Form systems. A series of states, in this case only x , are used as a &synthetic' controllable vari ables to form the "nal control, u. The interested reader is referred to Krstic et al. (1995) for further details of the Backstepping types of controller designs. The equations of motion in (3) describe a very sti! dynamic system due to low oil compressibility. Therefore, the performance of the controller relies fairly heavily on high model accuracy. In an automotive Active Suspension application, Alleyne and Hedrick (1995) demonstrate that poor tracking performance will occur when parameters are not known su$ciently accurately. Therefore, to account for parametric uncertainties, an estimation scheme is essential. In particular, the hydraulic parameters in the third state equation are very di$cult to measure o!-line, and their values may be slowly varying
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Fig. 8. 10 Hz sine wave force tracking result. Actual force: light line, desired: dark line.
during operation. Previous experience has shown that the performance is most sensitive to the ratio of hydraulic parameters a/c. In Alleyne and Liu (1999b), the second hydraulic parameter, c, was adjusted on-line using a gradient-based adaptation algorithm to determine the appropriate controller parameters. The results were quite convincing as shown in the sinusoidal and step tracking responses of Figs. 8 and 9. The results in the "gures show signi"cant improvement over other attempts at the same problem shown, for example, in Heinrichs, Sepheri and Thornton-Trump (1997). However, the Lyapunov-based gradient adaptation used to obtain these results did not lend itself to the ready simpli"cation that will be demonstrated in Section 5. Therefore, here we choose to adapt on the parameter a using the following scheme that is similar to Alleyne and Liu (1999b). Denote the estimated value of a as a( . De"ne a new Lyapunov function as < "< #o e, (13) ? ? where e "a!a( , o '0. Di!erentiating (13) yields ? ?
(15)
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Fig. 9. 0.5 Hz square wave force tracking result. Actual force: light line, desired: dark line.
where fK "!a( x !bx . Substituting (15) into (14) gives,
1 o u" !f #x !k e ! g e . BCQGPCB g o Substituting (18) into (17) gives
(17)
(18)
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By LaSalle's theorem (Khalil, 1996), the tracking errors e and e converge to zero globally. In addition, if the condition of persistent excitation is satis"ed, the parameter estimate will converge to its true value. The original control algorithm of Eq. (12) can now be augmented as follows: x BCQGPCB 1 " (a( x #bx #x !k e ), BCQGPCB c(P !sgn(x )x Q q x o #x u" !k e ! ce (P !sgn(x )x , BCQGPCB o Q K q o a( "! x e . o ? (22)
To compensate for the e!ect of friction, which had not been included at the beginning of this section, the preceding analysis is coupled to a feedforward friction cancellation scheme. Therefore, the desired value of the pressure state (x ) would be given as (F #Friction) . (23) x " BCQGPCB BCQGPCB A The friction estimate in (23) would be based on the identi"ed friction model. The friction cancellation is not included in the previous feedback controller synthesis since it would involve terms that are not di!erentiable, which is a requirement of the previous Lyapunov analysis. The Coulomb plus Stribeck model of friction was developed, veri"ed, and implemented successfully as will be demonstrated in Section 4. Previous experience with hydraulic force control (Alleyne & Hedrick, 1995) has shown that hydraulic force control systems can be quite asymmetric with respect to spool valve opening. Some of the hydraulic parameters for Eq. (3) can di!er depending on whether the valve is open in the positive or negative direction. The reasons for these di!erences are most likely a variation in #ow characteristics through the valve depending on which valve port is open to the pump. To compensate for a possible asymmetry, the adaptive update law of Eq. (20) can be modi"ed as follows.
a( " >
o ! x e , o ? 0,
∀x *0, ∀x (0, ∀x *0,
(24) 0, o ! x e , ∀x (0. o ? The adaptive update law for the &positive' a( set is activated whenever the spool position is positive. At the same time, the &negative' a( set is frozen at some value. a( " \
When the valve transitions from one direction to the other, the roles of the respective parameter sets are reversed. The interested reader can "nd a stability proof of this variable structure adaptation algorithm in Alleyne (1998). For the experimental results given in later sections, this switching adaptation was found to be quite important to match the output force trajectory precisely in this system. This would also hold true for systems with an intentional asymmetry such as valve-controlled single-ended cylinders.
4. Experimental results The following is an application of the control law and adaptation scheme given in Eq. (22) to a typical electrohydraulic system. The system consists of a double-ended hydraulic actuator and a DDV-D633 valve, both made by Moog, Inc. A dSpace DS1102 DSP controller handles all of the control functions with a TI TMS320 digital signal processor running at 1 kHz. The derivatives in the control law formulation are obtained by numerical di!erentiation. With a low-pass "lter, in this case a 70 Hz cuto!, second-order Butterworth "lter, the numerical di!erentiation is accomplished with satisfactory noise reduction. In Eq. (22), to attenuate the noise inherent in the pressure measurement used to determine x , the BCQGPCB analog pressure sensor signals are also processed through the same low-pass "lter. The numerical integration of a( is done by a standard fourth-order Runge Kutta numerical integration method starting with nonzero initial conditions based on an estimate of the hydraulic parameter. For this system, only the P ("x ) measurement is * available and there is no direct load cell measurement. Since the force cannot be directly measured with the system, the formulated data (kx#bx #mxK ) was used to estimate the force signal, as, by Newton's second law mxK #kx#bx "P A!Friction * and the output force is de"ned as P A!Friction. The * values of the physical and controller parameters of the system used in the experiments are shown in Table 1. The control gains k and k were obtained via online tuning of the controller. Fig. 10 shows the result of force tracking for a 1 Hz sine wave. The desired force trajectory is in the form F "A #B sin(2p f t)#F (NewtoBCQGPCB D ns) where A "1000, B "131, f"1. The controller is activated at approximately 2.0 s. Fig. 11 shows the adapted parameters that were used to obtain the force tracking of Fig. 10. Shown in the "gure are two separate traces representing independently estimated parameters that depend on the system states. As discussed in Section 3, the convergence of the estimated parameter to two separate values justi"es the variable structure adaptation approach implemented in Eq. (24).
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Table 1 Values of system and controller parameters I
II
m k b
24 (kg) 16010 (N/m) 310 (N/m/s)
P Q A a b c
III 1.0344;10(Pa) 3.2673;10\ (m) 1.513;10(N/m) 1.0 (1/s) 8.0;10
IV
F DQR?RGA F DIGLCRGA Stribeck slope Stribeck velocity range
k k o o o ?
200 (N) 131 (N) 1144 (N s/m) $0.1 (m/s)
2000 500 1;10\ 0.8;10 5;10\
I: mechanical constants. II: hydraulic constants. III: friction parameters. IV: controller parameters. 1.0344;10 Pascal is equivalent to 1500 psi.
be achieved. However, a close examination of the x portion of Eq. (22) indicates a wide variation in BCQGPCB the order of magnitude of the individual terms. The term a( x is signi"cantly larger in magnitude than the two terms bx #x , therefore those two terms could be BCQGPCB omitted in a simpli"ed version of the control algorithm. Additionally, the last term ((o /o )ce (P !sgn(x )x ) Q in the input signal u is relatively small compared to the other signals. If we approximate (P !gn(x )x as Q roughly a constant value, or even a slowly varying parameter, then the overall control law of (22) can be simpli"ed to yield
Fig. 10. 1 Hz force tracking output. Actual force: dark line, desired: light line.
x "KK x !K e , BCQGPCB q x #x u" !k e , BCQGPCB K q
KKQ " >
!o x e , ) 0,
0, KKQ " \ !o x e , )
Fig. 11. a( parameter adaptation.
5. Simpli5ed force tracking control The control algorithm of Eq. (22), along with the switching modi"cation of Eq. (24), does a very good job of tracking a force signal. Figs. 8 and 9 give a further indication of the level of tracking performance that can
∀x *0, ∀x (0, ∀x *0, ∀x (0,
(25)
where two new control parameters (KK , K ) are intro duced. The new parameters ensure that the order of magnitude of the system feedback gains remains the same. For example, K is approximately the same order of magnitude as k divided by c(P /2. Fig. 12 demon Q strates the force tracking capability of this modi"ed algorithm. The force tracking is quite favorable in comparison with the full algorithm's response in Fig. 10. Some care must be taken when implementing this "rst simpli"ed algorithm. The gains are now modi"ed to be quite di!erent from the original algorithm. The adapted value of KK is shown in Fig. 13 to give an idea of the > new estimated parameter magnitude. K "1.5;10\, o "5e!7, k "600. ) The algorithm in Eq. (25) can be even further simpli"ed if the valve dynamics are fast enough relative to the required tracking frequency. Given a fast, stable valve response such as with the Moog 633 DDV used in this
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Fig. 12. 1 Hz force tracking (simpli"cation C1). Actual force: dark line, desired: light line.
Fig. 13. KK
>
parameter adaptation (simpli"cation C1).
work, the valve dynamics can actually be ignored and the controller can be written directly in terms of the valve input. Eq. (26) shows this second simpli"cation where the gain from input voltage to valve position is included. 1 (KK x !K e ), K GHI !o x e , ∀x *0, ) KQK " > 0, ∀x (0, 0, ∀x *0, KKQ " \ !o x e , ∀x (0. ) u"
(26)
Fig. 14. 1 Hz force tracking (simpli"cation C2). Actual force: dark line, desired: light line.
Fig. 14 demonstrates the tracking performance of this second simpli"cation which uses the same adaptation and control gains as Simpli"cation C1. As can be seen in the "gure, the second simpli"cation now endures a fairly large phase lag in comparison to the original algorithm or the "rst simpli"cation. This may be important for applications that need high tracking bandwidth. However, if the performance requirements are not as strict, this extremely simpli"ed algorithm may prove su$cient. For example, Fig. 15 demonstrates the algorithm may be suitable for step responses that do not require a very short rise time. An examination of the simpli"ed algorithm in (26) reveals an interesting insight. The control signal, u, consists of essentially a velocity feedback term ((KK /K)x ) and a pressure error feedback term (!(K /K)e ). In essence, the velocity feedback term of the control law acts to cancel the natural feedback that occurs between cylinder displacement and net #ow into the cylinder. This is depicted in Fig. 16 on a linearized system representation which shows a modi"cation of the original linearized open-loop system given in Fig. 1. The adaptive portion of Eq. (23) is essential for accurately determining the appropriate parameter value that will give the most exact cancellation. If the cancellation is exact then the overall force control system, which was previously quite di$cult, becomes that of controlling a simple "rst-order system if the valve dynamics are neglected. If the valve dynamics are included then the force feedback controller design must take them into account. The concept of a velocity cancellation feedback coupled with a pressure error feedback was previously proposed by Heintze (1997) for a robotic application. Heintze (1997) describes a successful `Cascade *P inner-loopa which has a structure that is almost identical to the input signal (u) in Eq. (26). The
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and velocity of the cylinder can also be calculated a priori. If this is the case, the feedback velocity cancellation can actually be replaced with a feedforward velocity cancellation scheme which would have a reduced noise component in it.
1 KK u" K GHI
KQK " >
x !K e , BCQGPCB GFFHFFI
!o x e , ) 0, 0,
KKQ " \ !o x e , ) Fig. 15. Step force tracking (simpli"cation C2). Actual force: dark line, desired: light line.
major di!erence is that the control law in Heintze (1997), including control gains, was obtained primarily via intuition and tuning. Additionally, there was no use of a parameter estimation scheme as given in (26). The simpli"ed algorithms given in this work were determined from reduction of a controller that was systematically designed via Lyapunov-based analysis. As mentioned before, the parameter estimation, particularly the variable structure adaptation, was essential for determining the correct cancellation parameter KK . Although not explicitly shown here, the parameter would actually vary during experimental testing as the hydraulic oil in the tank heated up from room temperature to a steady-state operating value. If one wished to have proper force tracking over a wide range of environmental conditions, the adaptive loop would be very important. If the force pro"le is known a priori and the dynamics of the environment are well known, the desired position
∀x *0, ∀x (0, ∀x *0, ∀x (0.
(27)
Additionally, using the desired value of x would give a certain `anticipatorya nature, or phase lead in a linear sense, to the controller which could help to o!set the phase lag shown in Fig. 14.
6. Conclusion This paper developed and implemented a Lyapunovbased nonlinear controller for the task of force tracking of an electro-hydraulic actuator with a single-stage servo valve. The controller relies on an accurate model of the system. Parameter uncertainty in the system model was then compensated with an adaptation scheme based on Lyapunov analysis. A variable structure nature was introduced into the adaptation scheme to compensate for parametric asymmetry. The coupled control law and adaptation schemes were implemented on an experimental system. The experimental results showed that the proposed control law and adaptation schemes are e!ective for force tracking of signal trajectories. Previous results (Alleyne & Liu, 1999b) showed performance with
Fig. 16. Linearized schematic of simpli"cation C2.
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signi"cantly high frequencies. The control algorithm was then simpli"ed by retaining only those components which contributed the majority of e!ort to the "nal input signal. Two separate levels of simpli"cation were given. The second simpli"cation results in a controller with a velocity cancellation feedback along with a simple error feedback. While not having as high a performance as the original controller in terms of bandwidth, the simpli"cation of the algorithm may be attractive in applications where higher bandwidth is not as crucial. Acknowledgements This work was supported by the National Science Foundation under Grant DMI 96-24837CAREER and the O$ce of Naval Research under Contract N00014-961-0754. We would like to thank Moog Inc. for their support. References Alleyne, A., & Hedrick, J. K. (1995). Nonlinear adaptive control of active suspensions. IEEE Transactions on Control Systems Technology, 3(1), 94}101. Alleyne, A. (1996). Nonlinear force control of an electro-hydraulic actuator. Proceedings of the Japan/USA symposium on yexible automation, vol. 1 (pp. 193}200). New York: ASME.
Alleyne, A. (1998). A variable structure gradient adaptive algorithm for a class of dynamical systems. Systems and Control Letters, 33, 171}186. Alleyne, A., & Liu, R. (1999a). On the limitations of force tracking control for hydraulic active suspensions. ASME Journal of Dynamic Systems Measurement and Control, 121(2), 184}190. Alleyne, A., & Liu, R. (1999b). Nonlinear force/pressure tracking of an electrohydraulic actuator, Proceedings of 1999 IFAC world congress, Beijing, China, vol. B (pp. 469}474). Amin, J., Friedland, B., & Harnoy, A. (1997). Implementation of a friction estimation and compensation technique. IEEE Control Systems Magazine, 17(4), 71}76. Armstrong-Helouvry, B., Dupont, P., & Canudas de Wit, C. (1994). A survey of analysis tools and compensation methods for the control of machines with friction. Automatica, 30, 1083}1138. Heintze, H. (1997). Design and control of a hydraulically actuated industrial brick laying robot. Ph.D. dissertation, Mechanical Engineering, TU Delft, The Netherlands. Heinrichs, B., Sepheri, N., & Thornton-Trump, A. B. (1997). Positionbased impedance control of an industrial hydraulic manipulator. IEEE Control Systems Magazine, 17(1), 46}52. Karnopp, D. (1985). Computer simulation of stick-slip friction in mechanical dynamic systems. ASME Journal of Dynamic Systems, Measurement, and Control, 107(1), 100}103. Khalil, H. K. (1996). Nonlinear systems (2nd ed). Upper Saddle River, NJ: Prentice-Hall. Krstic, M., Kannellakopoulos, I., & Kokotovic, P. V. (1995). Nonlinear and adaptive control design. New York: Wiley. McCloy, D., & Martin, H. R. (1973). The control of yuid power. New York: Wiley. Merritt, H. E. (1967). Hydraulic control systems. New York: Wiley. Moog Product Literature. DDV: Proportional control valves with integrated electronics.
A simpli"ed approach to force control for electro-hydraulic systems夽 Andrew Alleyne*, Rui Liu Department of Mechanical and Industrial Engineering, University of Illinois, Urbana Champaign, 140 Mechanical Engineering Building, MC-244, 1206 West Green Street, Urbana, IL 61801, USA Received 30 August 1999; accepted 4 May 2000
Abstract In this paper, a Lyapunov-based control algorithm is developed for force tracking control of an electro-hydraulic actuator. The developed controller relies on an accurate model of the system. To compensate for the parametric uncertainties, a Lyapunov-based parameter adaptation is applied. The adaptation uses a variable structure approach to account for asymmetries present in the system. The coupled control law and the adaptation scheme are applied to an experimental valve-controlled cylinder. Friction modeling and compensation are also discussed. The experimental results show that the nonlinear control algorithm, together with the adaptation scheme, gives a good performance for the speci"ed tracking task. The original adaptive control law is then simpli"ed in several stages with an examination of the output tracking at each stage of simpli"cation. It is shown that the original algorithm can be signi"cantly simpli"ed without too signi"cant a loss of performance. The simplest algorithm corresponds to an adaptive velocity feedback term coupled with a simple force error feedback. 2000 Elsevier Science ¸td. All rights reserved. Keywords: Nonlinear control; Electro-hydraulic systems; Force control; Pressure control; Lyapunov methods; Adaptive algorithms; Friction; Hydraulic actuator
1. Introduction Hydraulic systems are important actuators in modern industry, principally because they have a high power/ mass ratio, fast response, and high sti!ness: a combination unmatched by any other commercial technology. Therefore, investigating the control of position or force outputs of hydraulic actuators should be of great interest to both the academic and industrial "elds. In particular, force and pressure tracking are important for some applications, such as vibration isolation and automotive active suspension, where an (almost) ideal force actuator is assumed in current research and application. In Alleyne and Liu (1999a), it is shown that fundamental limits exist on simple controllers for force or pressure tracking with hydraulic systems. Therefore, advanced controllers are a necessity. The cylinder velocity acts as a feedback
夽
An early and shorter version of this paper by the authors, entitled &&Nonlinear force/pressure tracking of an electro-hydraulic actuator'' was presented at the 1999 IFAC World Congress in Beijing, P.R. China, July 1999. * Corresponding author. Tel.: #1-217-244-9993; fax: #1-217-2446534. E-mail address: [email protected] (A. Alleyne).
term from the position output of the cylinder to the pressure di!erential across the piston. This is illustrated in Fig. 1 which shows a block diagram of such a system's linearized dynamics. Therefore, for a hydraulic servosystem, the poles of the load will manifest themselves as the zeros of the open loop force transfer function. If the system poles are lightly damped, this implies that the linearized dynamics of the hydraulic force/pressure loop will have zeros near the imaginary axis. The present work utilizes a particular controller structure to address this challenging problem. The reader is referred to McCloy and Martin (1973) for a more detailed explanation of the linearized system dynamics in Fig. 1. In Alleyne (1996) a Lyapunov-based force control approach was developed for a model of a hydraulic servo system and a gradient parameter estimation scheme was introduced to account for modeling uncertainty. The present paper uses a similar nonlinear control approach; however, it extends the results to include an important friction compensation scheme and presents experimental results verifying the approach. The experimental results are key to eliciting important issues in force control that were not evident in previous simulation studies. In addition, it will be shown that a few elements of the overall nonlinear adaptive controller make the majority of the
0967-0661/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 6 1 ( 0 0 ) 0 0 0 8 1 - 2
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Fig. 1. Linearized dynamics.
contributions to the "nal overall control signal. Therefore, in an attempt at algorithmic e$ciency and simplicity the control law can be greatly simpli"ed with little reduction in performance. The rest of the paper is organized as follows. Section 2 describes the detailed nonlinear model equations and also provides experimental veri"cation of the model. In Section 3, the control law and adaptive algorithm are derived and system stability proved. Section 4 gives the experimental results for force tracking on the experimental system. After the original algorithm has been demonstrated, Section 5 then introduces a series of simpli"cations with corresponding results. A conclusion then summarizes the main points of the paper.
2. System model The problem to be studied is depicted in Fig. 2. The goal is to have the actuator track a speci"ed force or pressure trajectory. The force-producing element is a double-ended hydraulic cylinder driven by a singlestage, four-way spool valve. The valve is operated via a direct drive linear motor and has its own closed loop analog controller. Due to this inner control loop the dynamics of the valve approximate a "rst-order model with a rate limit. The rate limit is fast enough to be ignored for our purposes here. This model matches well with actual time responses given by manufacturer's data shown in Fig. 3.The di!erential equations governing the dynamics of the actuator are given in Merritt (1967) for a symmetric actuator as follows: < R PQ "!Ax !C P #Q , RK * * 4b * C P !sgn(x )P Q T *, Q "C wx * B T o
(1) (2)
where < is the total actuator volume, b the e!ective R C bulk modulus, P the load pressure, A the actuator ram *
Fig. 2. A schematic of the experimental system.
Fig. 3. Valve step responses (Moog Product Literature).
area, x the actuator piston position, C coe$cient of RK leakage, Q the load #ow, C the discharge coe$cient, * B w the spool valve area gradient, x the spool valve posiT tion, P the supply pressure, and o the #uid density. Q Combining (1) and (2) with other system parameters results in the system state equations given below (Alleyne, 1996): x "x , 1 x " (!kx #Ax !Friction), m (3) x "!ax !bx #(c(P !sgn(x )x )x , Q 1 K x "! x # u, q q where x is the actuator piston position (x), x the actuator piston velocity (x ), x the load pressure (P ), * x the valve position (x ), u the input current to servo T valve, a"4Ab /< , b"4C b /< , and c"4C b w/ C R RK C R B C (< (o). R Eq. (3) is a greatly simpli"ed representation of the actual system dynamics. In order to check if this state space model captures the key component of the system dynamics, a detailed Matlab/SIMULINK model including friction was also derived. The friction model is a static, memoryless friction model incorporating static friction, Coulomb friction and Stribeck e!ects. A typical velocity-friction plot of such friction model is shown in Fig. 4. The friction model used in the modeling of the system is a novel one: it includes Karnopp's stick}slip model (Karnopp, 1985) and the Stribeck e!ect (Armstrong-Helouvry, Dupont & Canudas de Wit, 1994). In Karnopp's friction model, there are two key points: (1) a &stick' phase occurs when velocity is within a small critical velocity range, instead of only when velocity is exactly zero and (2) there is a maximum value that friction can have when the mass under consideration sticks. Let this maximum value be denoted as F . This DQR?RGA is the same as common Coulomb friction. However, within this stick region, the amplitude of friction is not
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Fig. 4. Friction model used in the system model, including Karnopp and Stribeck models.
just a constant value multiplied by the sign of the velocity which is the case for a common Coulomb friction model. Instead, it is such that the sum of all forces including friction is zero, i.e., the friction balances the other forces. Once the amplitude of the sum of other forces exceeds that of F , the &stick' phase cannot be maintained and DQR?RGA the mass under consideration will move (slip). The Stribeck e!ect is also observed in the experiment and included in the friction model. If the range of the Stribeck e!ect is small, the velocity}friction relation could be approximated as linear, with a negative slope. Experiments were conducted to test the model. Fig. 5 shows the estimated friction}velocity relation, with Coulomb and Stribeck friction, during tracking of a 0.5 Hz square wave. The value of friction was obtained as F "P A!mxK . (4) D * Since the friction in Fig. 5 was not directly measured but rather obtained from the measured system variables, the obtained relation between velocity and friction is called the estimated friction. The modeled data matched well with the experimental data. The data in Fig. 5 compare favorably with established data in the literature. For example, the friction measured by Amin, Friedland and Harnoy (1997) for a DC motor application is shown in Fig. 6. Although the magnitudes on the axes are quite di!erent, the friction characteristics are very similar to those of Fig. 5 at lower velocities with a clear Stribeck e!ect plus a hysteretic e!ect. Fig. 7 demonstrates the simulation model "delity by comparing the measured and modeled pressure (x ) state when the system was subjected to a 15 Hz sinusoidal open-loop input to the valve. The pressure data shown here has been post processed to "lter noise on the signal using a zero-phase error, non-causal "lter. As can be seen in the "gure, the modeled system matches well with the experimental data.
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Fig. 5. Estimated friction}velocity relation for 0.5 Hz square wave.
Fig. 6. Measured friction characteristics (Amin et al., 1997).
Fig. 7. Measured and modeled load pressure (- - -"model).
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3. Controller design Temporarily ignoring the e!ects of friction, for reasons to be given later, the system output of interest in Eq. (3) is the force (or pressure) from the actuator, thus y"Ax (or x ). Therefore, the relative degree of the nonlinear system is 2 and the zero dynamics of the system are the "rst two state equations with x "0. It can be easily veri"ed that the system is nonlinear minimum phase (Khalil, 1996), and thus the zero dynamics are stable. For ease of following analysis, the third and fourth state equations can be represented as x "f #g x , x "f #g u, where
(5)
f "!ax !bx , g "c(P !sgn(x )x , Q f "!x /q and g "K/q. An examination of Eq. (3) will indicate that this system falls into the class of systems known as Strict Feedback Form systems de"ned in Krstic, Kannellakopoulos and Kokotovic (1995). Therefore, a recursive Lyapunov design such as Integrator Backstepping would be appropriate. The following is a derivation of such a recursive law for the hydraulic force control problem. The desired force trajectory F can be divided by the actuator ram area BCQGPCB A to determine the desired load pressure pro"le x . BCQGPCB De"ne the following pressure and valve position errors: e "x !x , e "x !x , (6) BCQGPCB BCQGPCB where the desired valve position will be determined shortly. De"ne the positive-de"nite Lyapunov function < "o e #o e , (7) where o '0, o '0. Di!erentiating (7) along the systems trajectories (5) yields
clearly indicates convergence of the output pressure to its desired value. The desired value of the servovalve is dependent on the desired pressure (x ) as well as the system states. It will be up to the actual control signal u, to be de"ned shortly, to ensure the valve tracks its desired value. Substituting (9) into (8) results in
1 o u" !f #x !k e ! g e , (11) BCQGPCB o g where k '0, then
(12)
In practice, (P !sgn(x )x is seldom zero when the Q system is operating smoothly, since "x " is seldom close to P . In the rare case (P !sgn(x )x equals zero (e.g., Q Q due to the noise in x ), it is set to a small positive number to avoid the problem of dividing by zero. The cascaded nature of the control given in Eq. (12) is quite common to Strict Feedback Form systems. A series of states, in this case only x , are used as a &synthetic' controllable vari ables to form the "nal control, u. The interested reader is referred to Krstic et al. (1995) for further details of the Backstepping types of controller designs. The equations of motion in (3) describe a very sti! dynamic system due to low oil compressibility. Therefore, the performance of the controller relies fairly heavily on high model accuracy. In an automotive Active Suspension application, Alleyne and Hedrick (1995) demonstrate that poor tracking performance will occur when parameters are not known su$ciently accurately. Therefore, to account for parametric uncertainties, an estimation scheme is essential. In particular, the hydraulic parameters in the third state equation are very di$cult to measure o!-line, and their values may be slowly varying
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Fig. 8. 10 Hz sine wave force tracking result. Actual force: light line, desired: dark line.
during operation. Previous experience has shown that the performance is most sensitive to the ratio of hydraulic parameters a/c. In Alleyne and Liu (1999b), the second hydraulic parameter, c, was adjusted on-line using a gradient-based adaptation algorithm to determine the appropriate controller parameters. The results were quite convincing as shown in the sinusoidal and step tracking responses of Figs. 8 and 9. The results in the "gures show signi"cant improvement over other attempts at the same problem shown, for example, in Heinrichs, Sepheri and Thornton-Trump (1997). However, the Lyapunov-based gradient adaptation used to obtain these results did not lend itself to the ready simpli"cation that will be demonstrated in Section 5. Therefore, here we choose to adapt on the parameter a using the following scheme that is similar to Alleyne and Liu (1999b). Denote the estimated value of a as a( . De"ne a new Lyapunov function as < "< #o e, (13) ? ? where e "a!a( , o '0. Di!erentiating (13) yields ? ?
(15)
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Fig. 9. 0.5 Hz square wave force tracking result. Actual force: light line, desired: dark line.
where fK "!a( x !bx . Substituting (15) into (14) gives,
1 o u" !f #x !k e ! g e . BCQGPCB g o Substituting (18) into (17) gives
(17)
(18)
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By LaSalle's theorem (Khalil, 1996), the tracking errors e and e converge to zero globally. In addition, if the condition of persistent excitation is satis"ed, the parameter estimate will converge to its true value. The original control algorithm of Eq. (12) can now be augmented as follows: x BCQGPCB 1 " (a( x #bx #x !k e ), BCQGPCB c(P !sgn(x )x Q q x o #x u" !k e ! ce (P !sgn(x )x , BCQGPCB o Q K q o a( "! x e . o ? (22)
To compensate for the e!ect of friction, which had not been included at the beginning of this section, the preceding analysis is coupled to a feedforward friction cancellation scheme. Therefore, the desired value of the pressure state (x ) would be given as (F #Friction) . (23) x " BCQGPCB BCQGPCB A The friction estimate in (23) would be based on the identi"ed friction model. The friction cancellation is not included in the previous feedback controller synthesis since it would involve terms that are not di!erentiable, which is a requirement of the previous Lyapunov analysis. The Coulomb plus Stribeck model of friction was developed, veri"ed, and implemented successfully as will be demonstrated in Section 4. Previous experience with hydraulic force control (Alleyne & Hedrick, 1995) has shown that hydraulic force control systems can be quite asymmetric with respect to spool valve opening. Some of the hydraulic parameters for Eq. (3) can di!er depending on whether the valve is open in the positive or negative direction. The reasons for these di!erences are most likely a variation in #ow characteristics through the valve depending on which valve port is open to the pump. To compensate for a possible asymmetry, the adaptive update law of Eq. (20) can be modi"ed as follows.
a( " >
o ! x e , o ? 0,
∀x *0, ∀x (0, ∀x *0,
(24) 0, o ! x e , ∀x (0. o ? The adaptive update law for the &positive' a( set is activated whenever the spool position is positive. At the same time, the &negative' a( set is frozen at some value. a( " \
When the valve transitions from one direction to the other, the roles of the respective parameter sets are reversed. The interested reader can "nd a stability proof of this variable structure adaptation algorithm in Alleyne (1998). For the experimental results given in later sections, this switching adaptation was found to be quite important to match the output force trajectory precisely in this system. This would also hold true for systems with an intentional asymmetry such as valve-controlled single-ended cylinders.
4. Experimental results The following is an application of the control law and adaptation scheme given in Eq. (22) to a typical electrohydraulic system. The system consists of a double-ended hydraulic actuator and a DDV-D633 valve, both made by Moog, Inc. A dSpace DS1102 DSP controller handles all of the control functions with a TI TMS320 digital signal processor running at 1 kHz. The derivatives in the control law formulation are obtained by numerical di!erentiation. With a low-pass "lter, in this case a 70 Hz cuto!, second-order Butterworth "lter, the numerical di!erentiation is accomplished with satisfactory noise reduction. In Eq. (22), to attenuate the noise inherent in the pressure measurement used to determine x , the BCQGPCB analog pressure sensor signals are also processed through the same low-pass "lter. The numerical integration of a( is done by a standard fourth-order Runge Kutta numerical integration method starting with nonzero initial conditions based on an estimate of the hydraulic parameter. For this system, only the P ("x ) measurement is * available and there is no direct load cell measurement. Since the force cannot be directly measured with the system, the formulated data (kx#bx #mxK ) was used to estimate the force signal, as, by Newton's second law mxK #kx#bx "P A!Friction * and the output force is de"ned as P A!Friction. The * values of the physical and controller parameters of the system used in the experiments are shown in Table 1. The control gains k and k were obtained via online tuning of the controller. Fig. 10 shows the result of force tracking for a 1 Hz sine wave. The desired force trajectory is in the form F "A #B sin(2p f t)#F (NewtoBCQGPCB D ns) where A "1000, B "131, f"1. The controller is activated at approximately 2.0 s. Fig. 11 shows the adapted parameters that were used to obtain the force tracking of Fig. 10. Shown in the "gure are two separate traces representing independently estimated parameters that depend on the system states. As discussed in Section 3, the convergence of the estimated parameter to two separate values justi"es the variable structure adaptation approach implemented in Eq. (24).
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Table 1 Values of system and controller parameters I
II
m k b
24 (kg) 16010 (N/m) 310 (N/m/s)
P Q A a b c
III 1.0344;10(Pa) 3.2673;10\ (m) 1.513;10(N/m) 1.0 (1/s) 8.0;10
IV
F DQR?RGA F DIGLCRGA Stribeck slope Stribeck velocity range
k k o o o ?
200 (N) 131 (N) 1144 (N s/m) $0.1 (m/s)
2000 500 1;10\ 0.8;10 5;10\
I: mechanical constants. II: hydraulic constants. III: friction parameters. IV: controller parameters. 1.0344;10 Pascal is equivalent to 1500 psi.
be achieved. However, a close examination of the x portion of Eq. (22) indicates a wide variation in BCQGPCB the order of magnitude of the individual terms. The term a( x is signi"cantly larger in magnitude than the two terms bx #x , therefore those two terms could be BCQGPCB omitted in a simpli"ed version of the control algorithm. Additionally, the last term ((o /o )ce (P !sgn(x )x ) Q in the input signal u is relatively small compared to the other signals. If we approximate (P !gn(x )x as Q roughly a constant value, or even a slowly varying parameter, then the overall control law of (22) can be simpli"ed to yield
Fig. 10. 1 Hz force tracking output. Actual force: dark line, desired: light line.
x "KK x !K e , BCQGPCB q x #x u" !k e , BCQGPCB K q
KKQ " >
!o x e , ) 0,
0, KKQ " \ !o x e , )
Fig. 11. a( parameter adaptation.
5. Simpli5ed force tracking control The control algorithm of Eq. (22), along with the switching modi"cation of Eq. (24), does a very good job of tracking a force signal. Figs. 8 and 9 give a further indication of the level of tracking performance that can
∀x *0, ∀x (0, ∀x *0, ∀x (0,
(25)
where two new control parameters (KK , K ) are intro duced. The new parameters ensure that the order of magnitude of the system feedback gains remains the same. For example, K is approximately the same order of magnitude as k divided by c(P /2. Fig. 12 demon Q strates the force tracking capability of this modi"ed algorithm. The force tracking is quite favorable in comparison with the full algorithm's response in Fig. 10. Some care must be taken when implementing this "rst simpli"ed algorithm. The gains are now modi"ed to be quite di!erent from the original algorithm. The adapted value of KK is shown in Fig. 13 to give an idea of the > new estimated parameter magnitude. K "1.5;10\, o "5e!7, k "600. ) The algorithm in Eq. (25) can be even further simpli"ed if the valve dynamics are fast enough relative to the required tracking frequency. Given a fast, stable valve response such as with the Moog 633 DDV used in this
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Fig. 12. 1 Hz force tracking (simpli"cation C1). Actual force: dark line, desired: light line.
Fig. 13. KK
>
parameter adaptation (simpli"cation C1).
work, the valve dynamics can actually be ignored and the controller can be written directly in terms of the valve input. Eq. (26) shows this second simpli"cation where the gain from input voltage to valve position is included. 1 (KK x !K e ), K GHI !o x e , ∀x *0, ) KQK " > 0, ∀x (0, 0, ∀x *0, KKQ " \ !o x e , ∀x (0. ) u"
(26)
Fig. 14. 1 Hz force tracking (simpli"cation C2). Actual force: dark line, desired: light line.
Fig. 14 demonstrates the tracking performance of this second simpli"cation which uses the same adaptation and control gains as Simpli"cation C1. As can be seen in the "gure, the second simpli"cation now endures a fairly large phase lag in comparison to the original algorithm or the "rst simpli"cation. This may be important for applications that need high tracking bandwidth. However, if the performance requirements are not as strict, this extremely simpli"ed algorithm may prove su$cient. For example, Fig. 15 demonstrates the algorithm may be suitable for step responses that do not require a very short rise time. An examination of the simpli"ed algorithm in (26) reveals an interesting insight. The control signal, u, consists of essentially a velocity feedback term ((KK /K)x ) and a pressure error feedback term (!(K /K)e ). In essence, the velocity feedback term of the control law acts to cancel the natural feedback that occurs between cylinder displacement and net #ow into the cylinder. This is depicted in Fig. 16 on a linearized system representation which shows a modi"cation of the original linearized open-loop system given in Fig. 1. The adaptive portion of Eq. (23) is essential for accurately determining the appropriate parameter value that will give the most exact cancellation. If the cancellation is exact then the overall force control system, which was previously quite di$cult, becomes that of controlling a simple "rst-order system if the valve dynamics are neglected. If the valve dynamics are included then the force feedback controller design must take them into account. The concept of a velocity cancellation feedback coupled with a pressure error feedback was previously proposed by Heintze (1997) for a robotic application. Heintze (1997) describes a successful `Cascade *P inner-loopa which has a structure that is almost identical to the input signal (u) in Eq. (26). The
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and velocity of the cylinder can also be calculated a priori. If this is the case, the feedback velocity cancellation can actually be replaced with a feedforward velocity cancellation scheme which would have a reduced noise component in it.
1 KK u" K GHI
KQK " >
x !K e , BCQGPCB GFFHFFI
!o x e , ) 0, 0,
KKQ " \ !o x e , ) Fig. 15. Step force tracking (simpli"cation C2). Actual force: dark line, desired: light line.
major di!erence is that the control law in Heintze (1997), including control gains, was obtained primarily via intuition and tuning. Additionally, there was no use of a parameter estimation scheme as given in (26). The simpli"ed algorithms given in this work were determined from reduction of a controller that was systematically designed via Lyapunov-based analysis. As mentioned before, the parameter estimation, particularly the variable structure adaptation, was essential for determining the correct cancellation parameter KK . Although not explicitly shown here, the parameter would actually vary during experimental testing as the hydraulic oil in the tank heated up from room temperature to a steady-state operating value. If one wished to have proper force tracking over a wide range of environmental conditions, the adaptive loop would be very important. If the force pro"le is known a priori and the dynamics of the environment are well known, the desired position
∀x *0, ∀x (0, ∀x *0, ∀x (0.
(27)
Additionally, using the desired value of x would give a certain `anticipatorya nature, or phase lead in a linear sense, to the controller which could help to o!set the phase lag shown in Fig. 14.
6. Conclusion This paper developed and implemented a Lyapunovbased nonlinear controller for the task of force tracking of an electro-hydraulic actuator with a single-stage servo valve. The controller relies on an accurate model of the system. Parameter uncertainty in the system model was then compensated with an adaptation scheme based on Lyapunov analysis. A variable structure nature was introduced into the adaptation scheme to compensate for parametric asymmetry. The coupled control law and adaptation schemes were implemented on an experimental system. The experimental results showed that the proposed control law and adaptation schemes are e!ective for force tracking of signal trajectories. Previous results (Alleyne & Liu, 1999b) showed performance with
Fig. 16. Linearized schematic of simpli"cation C2.
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signi"cantly high frequencies. The control algorithm was then simpli"ed by retaining only those components which contributed the majority of e!ort to the "nal input signal. Two separate levels of simpli"cation were given. The second simpli"cation results in a controller with a velocity cancellation feedback along with a simple error feedback. While not having as high a performance as the original controller in terms of bandwidth, the simpli"cation of the algorithm may be attractive in applications where higher bandwidth is not as crucial. Acknowledgements This work was supported by the National Science Foundation under Grant DMI 96-24837CAREER and the O$ce of Naval Research under Contract N00014-961-0754. We would like to thank Moog Inc. for their support. References Alleyne, A., & Hedrick, J. K. (1995). Nonlinear adaptive control of active suspensions. IEEE Transactions on Control Systems Technology, 3(1), 94}101. Alleyne, A. (1996). Nonlinear force control of an electro-hydraulic actuator. Proceedings of the Japan/USA symposium on yexible automation, vol. 1 (pp. 193}200). New York: ASME.
Alleyne, A. (1998). A variable structure gradient adaptive algorithm for a class of dynamical systems. Systems and Control Letters, 33, 171}186. Alleyne, A., & Liu, R. (1999a). On the limitations of force tracking control for hydraulic active suspensions. ASME Journal of Dynamic Systems Measurement and Control, 121(2), 184}190. Alleyne, A., & Liu, R. (1999b). Nonlinear force/pressure tracking of an electrohydraulic actuator, Proceedings of 1999 IFAC world congress, Beijing, China, vol. B (pp. 469}474). Amin, J., Friedland, B., & Harnoy, A. (1997). Implementation of a friction estimation and compensation technique. IEEE Control Systems Magazine, 17(4), 71}76. Armstrong-Helouvry, B., Dupont, P., & Canudas de Wit, C. (1994). A survey of analysis tools and compensation methods for the control of machines with friction. Automatica, 30, 1083}1138. Heintze, H. (1997). Design and control of a hydraulically actuated industrial brick laying robot. Ph.D. dissertation, Mechanical Engineering, TU Delft, The Netherlands. Heinrichs, B., Sepheri, N., & Thornton-Trump, A. B. (1997). Positionbased impedance control of an industrial hydraulic manipulator. IEEE Control Systems Magazine, 17(1), 46}52. Karnopp, D. (1985). Computer simulation of stick-slip friction in mechanical dynamic systems. ASME Journal of Dynamic Systems, Measurement, and Control, 107(1), 100}103. Khalil, H. K. (1996). Nonlinear systems (2nd ed). Upper Saddle River, NJ: Prentice-Hall. Krstic, M., Kannellakopoulos, I., & Kokotovic, P. V. (1995). Nonlinear and adaptive control design. New York: Wiley. McCloy, D., & Martin, H. R. (1973). The control of yuid power. New York: Wiley. Merritt, H. E. (1967). Hydraulic control systems. New York: Wiley. Moog Product Literature. DDV: Proportional control valves with integrated electronics.