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Adaptive robust control of linear motors for precision manufacturing
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECT...
14th World Congress ofTFAC
A-la-0l-5
Copyright <© 1999 IFAC 14th Triennial World Congress, Beijing, P,R. China
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECISION MANUFACTURING Bin Vao
and
Li XU
School of Mechanical Engineering Purdue University. West Lafayette. IN 47906. USA [email protected]
Abstract: Linear motors are sensitive to disturbances and parameter variations. Certain types of linear motors such as the iron core are also subject to nonlinear effects due to periodic cogging force and force ripple. An adaptive robust controller based on the discontinuous projection method is constructed to address these issues. In particular. based on the particular structures of various periodic non linear forces, design models consisting of known basis functions with unknown weights are used to approximate those unknown nonlinear forces. On-line parameter adaptarion is then utilized to reduce the effect of various parametric uncertainties such as unknown weights, inertia, and motor parameters while certain robust control laws are used to handle the uncompensated uncertain nonlinearities effectively for high performance. Simulation results are shown to illustrate the effectiveness of the proposed algorithm. Copyright ©19991FAC Keywords: Linear Motors. Motion Control, Adaptive Control. Robust Control, Servo Control 1. INTRODUCTION Linear motor gains high-speedjhigh-accuracy potential by eliminating mechanical transmission (Alter and Tsao. 1996; Braembussche et al.. 1996). However, it also loses the advantage of using mechanical transmissions - gear reduction reduces the effect of model uncertainties such as parameter variations(e.g., uncertain payloads) and external disturbance (e.g., cutting forces in machining). Furthermore, certain types of linear motors such as the iron core have significant uncertain nonlinearitiesdue to force ripple and cogging force (Braembussche et al.. 1996). These uncertain nonlinearitis are directly transmitted to the load and thus have significant effect on the motion of the load. In the past, a great deal of effort has been devoted to solving these difficulties (Alter and Tsao, 1996; Braembussche et al.. 1996; Otten et al.. 1997). In (Alter and Tsao, 1996), Hoo optimal feedback control is used to provide high dynamic stiffness to external disturbances (e.g .. cutting forces in machining). To reduce the nonlinear effect of force ripple and cogging force, in (Braembussche et al.. 1996), feedforward compensation terms were added to the position controller. In (Otten et al., 1997), a neural-networkbased learning feedforward controller was proposed to reduce positional inaccuracy due to reproducible ripple forces or any other reproducible and slowly varying disturbances.
Recently, Vao and Tomizuka proposed an adaptive robust control (ARC) approach (Vao and Tomizuka.
1994b; Vao and Tomizuka, 1997b), for the robust control of uncertain nonlinear systems in the presence of both parametric uncertainties and uncertain nonlinearities. The approach effectively combines the design techniques of adaptive control (AC) and those of deterministic robust control (DRC) and improves performance by preserving the advantages of both AC and DRe. A general theoretical framework is recently formalized by Yao in (Yao, 1997). Applications include the high-speedjhigh-accuracy motion control of machine tools (Vao et al., 1997), the control of electro-hydraulic servo systems (Yao et al., 1998), and the control of robot manipulators (Yao and Tomizuka, 1994a).
This paper generalizes the ARC approach to the high-speed/high-accuracy motion control of iron core linear motors. In addition to the usual design difficulties due to large variation of inertia load, unknown friction, and disturbance forces, there are several particular problems associated with the control of iron core linear motors. First, the iron core linear motors have a large magnitude of ripple forces. Second, the electrical current feedback loop cannot be neglected. This added electrical dynamics increases the controller design difficulties since the resulting system is of higher" relative degree" (Yao and Tomizuka, 1997b) and backstepping design via ARC Lyapunov functions (Yao, 1997) has to be employed. Third, due to variations of the inertia load and the motor force constant, parametric uncertainties will appear in the input channels of the
25
Copyright 1999 IF AC
ISBN: 008 0432484
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECT...
intermediate design steps. All these issues will be addressed in the paper. The paper thus advances the control of linear motors considerably since none of the published papers deal with all these aspects all together at once. 2. PROBLEM FORMULATION AND DYNAMIC MODELS
The dynamic equation of the inertia load driven by an iron core brush less DC linear motor (LCK series in (Anorad Linear Motors, 1997») can be described by
14th World Congress ofTFAC
identical and are equally spaced at a pitch of P, then, is a periodic function with a period of P, i.e., fCQgging(XL +P} = !cogging(XL). (2)Force ripple is developed due to the periodic variation of the force constant. When current flows through the coils, a position dependent periodic force is resulted. To take this effect into account, a position dependent term can be added to modify the force constant in (3) to Kp(xd = KFO + KFx(XL), in which Kp.,(xL) is a periodic function with a period of P, i.e., KF.,(XL +
!coggina(XL)
P) = KF",(XL).
By taking into account the effect of cogging force and force ripple, the motor force is given by where XL represents the displacement of the inertia load, M is the mass of the inertia load plus the iron core translator, Fm is the total electra-magnetic force generated by the linear motor, B is the combined coefficient of the damping and viscous friction on the load, J friction represents the combination of stiction and Coulomb friction, and f,.lis(t, XL, XL) represents the external disturbances such as cutting force in machining. A simple mathematical model for friction ft .. iction that takes into account of stiction, Coulomb friction, and Stribeck effect can be written as (de Wit et al., 1995) f!richon(:h) = -[le
+ Us -
Fm = KF(xL)i + /c()gging(xL) KF(xL} = KFO + KF.,(XL)
where i is related to the control input u by (3). Defining the position, velocity, and motor current as the state variables, Le., x = [xl,x2,x31 T ~ [XL,XL,i]T, from Eqs. (1), (3), and (4), the entire system can be expressed in state space form as: :h =X2 X2
Je)e-I';LI;,.I~lsgn(xL) (2)
where 13 is the level of static friction, le is the minimum level of Coulomb friction, and Xs and are empirical parameters used to describe the Stribeck effect.
e
Without going into the details about the working mechanisms of the amplifier and the motor, a popular simplified model that relates the generated Fm to the control input voltage u of the amplifier is given by (Krause, 1986)
(4)
.
"'3
= ~[(KFO + KF",(Xl))X3 -
BX2
+ /iriction(X2)
(5)
+ fdi8(t,XI, X 2)] 1 ""L"'2 + "L'"
+!cogging(xd R KE
= -];"'3 -
Given the desired motion trajectory XLd(t), the objective is to synthesize a control input u such that the output y = Xl tracks XLd(t) as closely as possible in spite of various model uncertainties. 3. ADAPTIVE ROBUST CONTROL (ARC) OF LINEAR MOTOR SYSTEMS
3.1 Design Models and Assumptions (3)
where K FO is the average force constant, i is the motor current amplitude, Rand L are the armature resistance and inductance respectively, KE is the electromotive force coefficient. Such a simplified model has been successfully used by many researchers in the control of rotary motors and certain types of linear motors (Le-Huy et al., 1986). For iron-core linear motors, the simplified linear motor model (3) may not be enough since these types of motors have relatively larger magnitude of nonlinear cogging force and force ripple, which are neglected in the simplified model (3). These nonlinear forces are generated due to the particular structure of the ironcore linear motors (Braembussche et al., 1996; CUen et al., 1997): (l)Cogging or dentent force is a magnetic force developed due to the attraction between the permanent magnets and the iron cores of the translator. It depends only on the relative position of the motor coils with respect to the magnets. Thus, this force can be described as !cogging(XL). If the permanent magnets of the same linear motor are all
To begin the controller design, practical and reasonable assumptions on the system have to be made. For the reproducible cogging, force ripple, and friction force, although feedforward compensation techniques based on off-line identified models can be used to cancel their effects (e.g., force ripple compensation in (Braembussche et al., 1996) and friction compensation in (Lee and Tomizuka, 1996)), such a procedure may be time consuming and sometimes may not be so practical. In the following, we will use some simple models with unknown weights to approximate them; the unknown weights will be adjusted on-line via certain parameter adaptation law to achieve an improved model compensation and the model approximation error will be handled via certain robust control law to achieve a guaranteed robust performance. Specifically, since fcogging(XI) and KF.,(ZI) are periodic functions with a known period P, they can be approximated quite accurately by their first several harmonics. which are denoted as !collging(Xt) and KFx(Zt) respectively and represented by !cogging(Xt}
= AJ'Sc(xt)
(6)
Kp,,(xt} = AkSK(Xt)
where Ac = [Acls,Acl c ,.'. ,AC918,AcQ1C)T AK
= [AKIs,AKIC, .. ' ,AK'I28,AKQ2clT E
E R2q, and R2q2 are the
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Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECl...
unknown weights, 8 c (xd = [sin{~xI),COS(~XI)' ... ' sinepz'xl),COs(2PZ'xl)V and 8K(X!) = [Bin(~xl)' C08(~Xt), ... , sin(l~2xl),COs(~xd]T are the known basis shape functions, and ql and q2 are the numbers of harmonics that we would like to compensate for each term. The friction (2) is discontinuous at XI = o. Thus one cannot use the model (2) for friction compensation since, as evident from (5), there is no way that one can generate a discontinuous motor force to accomplish it. To by-pass this technical difficulty, we will use a simple continuous friction model to approximate the actual discontinuous friction model (2) for model compensation; the model used in the paper is given by hdction = Aj8j(X2), where the amplitude AI may be known or unknown. and 8/(X2) may be chosen as certain differentiable function. The second equation of (5) can thus be written as: :i:z = -.!..[(Kpo M +AJ8/(xz)
+ AkSK(Xl»X3 - BX2 + AcT Sc(X1) + f(t , xl,X2,X3)]
positive. i.e., there exists
+ 0~8K(xd
KPmin
is
> 0 such that
3.2 Discontinuous Projection Let {j denot~ the~ estimate of 9 and jj the estimation (i.e., 9 = 9 - 9). By using an adaptation law given by
e~ror
(11)
where r > 0 is a diagonal matrix, T is an adaptation function to be synthesized later, and Projjj(e) is the widely used discontinuous projection mapping (Sastry and Bodson, 1989; Goodwin and Mayne, 1989). It can be shown (Yao and Tomizuka, 1994b) that for any adaptation function T. the projection mapping used in (11) guarantees (Pl) (P2)
(7)
3.3 Controller Design
i
where represents the lumped external disturbances and model approximation errors. which is given by j
A2 . V6 E [Omi", Om...",] , KF = 61
=:. (KF .. (XI) - !friction)
KF.,(Xl»X3
+ (leogging -
+
(8)
(f/riction lcog9ing) Id •• (t, Xl ,X2)
+
The design parallels the recursive backstepping design procedure via ARC Lyapunov functions in (Vao and Tomizuka, 1997a; Vao. 1997) as follows.
Step 1 In general, the system is subjected to parametric uncertainties due to the variations of M. B, Ac. AK, AI, KE, L. R, and the nominal value of the lumped disturbance In order to use parameter adaptation to reduce parametric uncertainties to improve performance, it is necessary to linearly parameterize the state space equations in terms of a set of unknown parameters. For this purpose, define the unknown parameter set 8 = [(h, 02b, 03, 64, 0Sb, 86, (h , Og, egJT E RP=1+2q, +2q2 as 01 =
i. in.
=
= --5-. = t.
41-. -to and
04 = Kt?,6 z b kAK E R 2n, 63 85b = i"Ac E R 2q2. 06 ~, 07 9a = 99 The state space equations (5) and
=
¥.
=(7) can thus be linearly parameterized in terms of 9 as :h
Noting that the first equation of (9) does not have any uncertainties, an ARC Lyapunov function can thus be constructed for the first two equations of (9) directly. Define a switching-function-like quantity as Zll
= et
X2 -
%2
= kFX~ + 93X2 +66 +d-
+ 63X2 + 848/(X2)
X2eq.
X2cq
~
:CId -
k"el
(13)
where el = Xl - Xld(t). in which the desired trajectory Xld(t) will be specified later to achieve a guaranteed transient performance, kp is any positive feedback gain. If Z2 is small or converges to zero exponentially. then the output tracking error Cl will be small or converge to zero. Differentiating (13) and noting (9)
=X2
X2 = [01 + 9~SK(Xt)JX3 +6r~Sc(xI) + 96 + d
+ kpet =
+ 948/(X2) + e~Sc(xtl
X2cq
(14)
(9)
xs = 88X3 + 8gx2 + 87'1.t
where d(t,xl,X2,X3) = i,,[! - 1nl represents the uncompensated disturbances and modeling errors. For simplicity. in the following. the following notations are used: ei for the i-th component of the vector e, em.in for the minimum value of e, and e maz for the maximum value of e. The operation < for two vectors is performed in terms of the corresponding elements of the vectors. The following practical assumptions are made
where X2eq ~ Xld - kpei. In (14), if we treat X3 as the input. we can synthesize a virtual control law 0:2 for Xs such that Z2 is as small as possible. Since (14) has both parametric uncertainties 9 and uncertain non linearity d, the ARC approach proposed in (Yao, 1997) will be generalized to accomplish the objective. The control function by
0:2
consists of two parts given
Al . The extents of the parametric uncertainties
and uncertain nonlinearities are known, i.e.. (10)
= (lhmin, ... , 90711;,,]', 6ma ~ = [61711"",
where Omi" ... , O-.,..,jT,
and dd(t,Xl,X2) are known.
or
where Kp = 81 + SK(Xl) is the estimated motor force constant, 0::2 .. tunctions as an adaptive control law used to achieve an improved model compensation through on-line parameter adaptation given by (11). and 0:28 is a robust control law to be synthesized later. Noting Pi of (12) and Assumption 27
Copyright 1999 IF AC
ISBN : 0 08 043248 4
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECT...
A2, KF > O. Thus the control function (15) is well defined no matter what type of adaptation function to be used. Let Z3 = X3 - 0<2 denote the input discrepancy. Substituting (15) into (14) leads to Z2
=:
+ 02.) -
KF(Z3
eT 4>2
+ ii
14th World Congress ofTFAC
In (22), 02c is calculable and can b'7 used in the design of control functions but a2u cannot due to various uncertainties. Therefore, a2u has to be dealt with in this step design. From (9), is
(16)
where tP2=[a2a, S'k(Xl)02a, X2. S/(X2), S;(xI). 1, 0, 0, O)T The robust control function 02s consists of two terms given by
= 98X3 + 99X2 + 97U -
0:2c -
0:21>
(24)
Consider the augmented p.s.d. function V given by W3
>
(25)
0
Noting (20) and (24),
17=172 1"'2 where 0282 is a robust control function designed in the following and k281 is any non linear feedback gain satisfying
+Oru -
Xs
KF(Xl)
---k2s1Z2 KFmin
+ KF(X1)0282
112
= -~ [W2 KFZ2 + 08X3 +Ogx2 87
(19)
u.
. V2
= KF(X.}W2 _ Z 2Z 3 -
K F (Xl)
W 2 - - - k 2,l Z 2
+W2Z2{KF(x.}02.2 -
!f/:Tnin _
(J
4>2
2
(20)
+ d}
W3
= 1£81 + 1£.2.
kSB1
Define a positive semi-definite (p.s.d.) function V2 as V2 = !w2zi where W2 > 0 is a weighting factor. From(19), its time derivative is
.=.V2 1"'2
condition ii
Z20!2.2:5
OT
~
0
1':2
+W3%3
{-T (J7U. -
where C2 is a design parameter which can be arbitrarily small. How to choose 0282 to satisfy constraints like {21} can be found in (Yao, 1997). One example is given by 0:2.2 = - 2K F "2. 22, where h2 is any wun e'2 function satisfying h2 2: 118Mll z llrP211 2 + o~ , in which (JM
= (Jmaz -
9m i'"
Noting the last equation of (9), Step 2 is to synthesize an actual control law for u such that i tracks the desired control function 02 synthesized in Step 1 with a guaranteed transient performance, which can be done by using the same ARC design techniques as in Step 1. From (I5) and {9}, 002
8U2
8U2 .
80:2(J~
8U2
= -8X1 X 2 + - X 2 + -- + 8X2 8(J 8t = O,2c +62..
(22)
-
8002 80<2 ~} -8 d - - _ (J (28)
[!!!4Z2 WS
~X3, U~2
80
T (X.)("!:4Z2 SK W3
~s ( ) - ~ST( 8x2" f:1:2, 8X:l c Xl ) , - ~ 8:1:'2 ' U a ,
~X3), ux~ ::E'3t 3:'2 ]T
Similar to (21), the robust control function chosen to satisfy
U s2
is
condition i
condition ii
where
is a design parameter. For example, let in which h3 is any function satisfying ha 2: 118M 1J211<1>3; IJ2 + 6~
U.2
Step 2
•
-
~ 8Z2 Z'2,
(J
X2
where <1>3 = (21)
(27)
= - -(J7t11in -k38123 2 2 2: ka + 11 80 Cesll + II c 4ts r r/lsll 1£.1
where ks > 0 is a constant, C e3 , and Cq,3 are positive definite constant diagonal matrixes , and U;t2 is a robust control function to be chosen later. Substituting (27) and (23) into (26) and using similar techniques as in (16) V
-
0:2<0]
1
8 U2
The robust control function 0282 is now chosen to satisfy the following conditions condition i ZZ{KF(Xl )0:2.2
(26)
= a2
1£"
-
""2 = KF('''1)Z3 -r -8 4>2 + d
+ (JgX3 + 09X2
1<>2 denotes V2 under the condition that (or Z3 = 0). Similar to (15), the actual control input u to be synthesized consists of two parts given by 1£ = u,,(x,e,t) + u.(x,O,t)
where
in which Crp2 is a positive definite constant diagonal matrix to be specified later and k2 is a positive design parameter. Substituting (17) into (16),
+W3ZS{ W2 KFZ2 W3 62c - 0:2u}
C3
= -297':;n~3h3Z3'
Theorem 1. Let the parameter estimates be updated by the adaptation law (11) in which T is chosen as (30)
If the control parameters C.p2 = diag{C.p21 •... ' C,p2p}, = diag{Cq,31,'" ,Cq'>3p}, and C83 = diag{C831," ., C83p} are chosen such that. VI. Cq,21C/131 2: and Cq,31C631 2: ..jfW3. where Cq,21. Cq,31 and C931 are the l-th diagonal elements of c 4t2 , c 4t3 and C(J3 respectively. Then, the control law (27) guarantees that Cq,3
A.
v'tw2wa
In general, the control input is bounded and the closed-loop system is globa lIy stable with V bounded above by Vet) :$ IlXp(-Avt)V(O)
cV + AV [1 -
exp( -AV t )]
(31)
28
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECl...
where Av = 2min{k2 , k3} and tV =
W2 C 2
+
W3 C 3·
B If after a finite time to, d = 0, i.e., in the presence of parametric uncertainties only, then, in addition to results in A, asymptotic output tracking (or 6 zero final tracking error) is also achieved. Proof: The Theorem can be proved in the same way 3S in (Vao, 1997).
Desired Trajectory Initialization and Generation It is seen from (31) that transient tracking error is affected by the initial value V(O), which may depend on the controller parameters also. To further reduce transient tracking error, the desired trajectory initialization can be used as in (Yao, 1997JJ. Namely, instead of simply letting the desired trajectory for the controller be the actual desired trajectory or position (i.e., Xld(t) = XLd(t», we can generate Xld(t) using any stable 3rd order filter with initial conditions $ld(O), . .. , x~~)(O) chosen as XlcJ(O) = Xl(O), XId(O) = X2(0) 21ld(O) = {KFX3 + OaX2 + 04S, + (j'f"sc + 06}lt=o
(32)
Then. el(O) = o,z;(O) = O,i = 2,3 and V(O) = o. Thus transient tracking error is minimized. It is seen from (32) that the trajectory initialization is independent from the choice of controller parameters such as k = [k 2 , k3jT and c = (c2, e3]T. Thus. once the initial position of the linear motor is determined, the above trajectory initialization can be performed offline. 4. SIMULATION RESULTS To illustrate the above designs, simulation results are obtained for a linear motor system with motor physical parameters obtained from the specifications of the iron core LCK-S-l linear motor by Anaroad Company. Nominal values of the system are: an inertia of M = 10kg, a damping coefficient of B = O.SN/m/s, a motor back EMF of KE = 18.5V/m/s, a force constant of KFO = 5S.5N/A. a coil resistance of R = 3.911, an inductance of L = 30mH, and a magnet pitch of p = 30mm. The actual friction force used for simulation is given by (2), where /. = 10, fe = 6, X. = 0.001 and ~ = 1. The following situation is considered: The shape function for friction compensation is chosen as S,(X2) = tanh(lOOOx2) and the friction amplitude At is allowed to vary from 5N to ION. Cogging amplitude is allowed to be any value less than SON, and the amplitude of KFx(:rt) is less than 1. 11 N/A. For simplicity, in the simulation, only the first harmonics are used in the controller to approximate the periodic cogging force and KF",(Xl), i.e., ql = q2 := 1 in (6). A sampling rate of O.2ms is used in the simulation. The following two controllers are compared: (i) ARC, the ARC law presented in section Ill. The controller parameters are: kp = 200,k2ol = 200,W2 = 1, .02 = S x
104 ; ks.1
rate is
= 300,ws = 0.1, .os = 1 x 107 . The adaptation
I' = dia.g[ 342,0.39,0.39, 3.S x 10- 3 , 0.61, 288, 288,
X 10 4 ]; and (ii) DRC, Deterministic Robust Control - the same control law as the above ARC but without parameter adaptation.
S1.2. 125. 8 x 10 3 , 1.8
To test the tracking capability of the proposed algorithm, a sinusoidal desired trajectory given by XLd = O.01sin{21rt) and the trajectory initialization (32) are used . Simulations are first run for the case that model uncertainties are dominated by parametric uncertainties as follows. Actual cogging force /cogging(Xt} and the position dependent force constant KF::: used for plant simulation are assumed to be of first harmonics only, Le., Fcagging = 25sin(21rXI/P + 1r/4) and KFx = 1.1lsin(21rXl/P+7r/4), and there is no external disturbance (fai. = 0). Tracking errors are shown in Fig. 1. As seen, both controllers have very small tracking errors but the final errors of ARC are much smaller than that of DRC due to parameter adaptation. The simulation is then run for the case that actual models of icc99 in 9(Xl) and KF.,(Xl) are much different from their design models (6). Specifically, both /coggin9(xt} and KF .. (Xl) are assumed consisting of the first and the third harmonics (Braembussche et al., 1996) as Fcog9 ing = 15sin(27rXI/ P+n /4)+20sin(67rxl/ P+O.09"1r) and KF% = KFO(0.016sin(21rxl / P
+ 11'/4) +0.02sin(6l\"Xl / P +O.091r}).
Tracking errors are shown in Fig.2. As seen, the mismatch between the design models and the actual models for ripple forces does not degrade the performance of the proposed ARC much. This verifies the performance robustness of the proposed ARC to uncompensated uncertain nonlinearities. To test performance robustness ofthe proposed ARC to external disturbances, an external disturbance force idi. = 30+5rand (N) is then added to the system during the first second . As shown in Fig.3. the added disturbance force has a large effect on DRC but not much on ARC. This is because that the proposed ARC can adapt to the nominal value of the lumped disturbance (116) very quickly. 5. CONCLUSIONS An ARC controller has been constructed for the adaptive robust motion control of an iron core linear motor. The controller takes into account the efrect of parametric uncertainties and uncertain nonlinearities due to external disturbances, uncompensated friction and force ripple. In particular, design models consisting of known basis functions with unknown weights are used to approximate the unknown nonlinear ripple forces. On-line parameter adaptation is then utilized to reduce the effect of various parametric uncertainties while the uncompensated uncertain nonlinearities are handled via certain robust control laws. As a result, time-consuming and costly rigorous off-line identification of friction and ripple forces 29
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECT...
is avoided while without sacrificing tracking performance. Simulation results are provided to illustrate the proposed scheme. ACKNOWLEDGMENT
14th World Congress ofTFAC
VilO, Bin, M. AI-Majed and M. Tomizuka (1997). High performance robust motion control of machine tools: An adaptive robust control approach ilnd comparative experiments. IEEEjASME Trans. on Mechatronics 2(2), 6376.
This work is supported by the National Science Foundation under the CAREER grant CMS-9734345. The authors would like to thank Dr. Y. Shin for helpful discussions on the subject. 6. REFERENCES
Alter, D. M. and T. C. Tsao (1996). Control of linear motors for machine tool feed drives: design and implementation of hoo optimal feedback control. ASME J. of Dynamic
systems, Measurement, and Contro/IIS, 649-656. Anorad Linear Motors (1997). Anorad Corporation. 110 Oser Avenue, Hauppauge, NY11788. Braembussche, P. V., J. Swevers, H. Van Brussel and P. Vanherck (1996). Accurate Tracking control of Linear Synchronous Motor Machine tool Axes. Mechatronics 6(5), 507-52l. de Wit, C. Canudas, H. 01s5On, K. J. Astrom and P. Lischinsky (1995). A new model for control of systems with friction. IEEE Trans. on Automatic Control 40(3), 419-425. Goodwin, G, C. and D. Q. Mayne (1989). A parameter estimation perspective of continuous time model reference adaptive control. Automatica 23(1), 57-70. Krause, P. C. (1986). Analysis of Electric Machinery. McG rilw-HiII , Inc. Le-Huy, H .. R. Perret and R. Feuil1et (1986). Minimiaztion of Torque Ripple in Brushless DC Motor Drives. IEEE Transactions on Industrial Applications 22(4).748-755. Lee, H. S. and M. Tomizuka (1996). Robust motion contaller design for high-accuracy positioning systems. IEEE Trans. on Industrial Electronics 43(1). 48-55. OUen, G., T. Vries, J. Amerongen, A. Rankers and E. Gaal (1997). Linear Motor Motion Control Using a Learning Feedforward Controller. IEEEjASME Transactions on Mechatronics 2(3), 179-187. Sastry, S. and M. Bodson (1989). Adaptive Control: Stability. Convergence and Robustness. Prentice Hall, Inc .. Englewood Cliffs, NJ 07632. USA. Yao. Bin (1997). High performance adaptive robust control of nonlinear systems: a general framework and new schemes. In: Proc. of IEEE Conference on Decision and Control. pp. 2489-2494. Yao, Bin and M. Tomizuka (1994a). Comparative experiments of robust and adaptive control with new robust adaptive controllers for robot manipulators. In: Proc. of IEEE Conf. on Decision and Control. pp. 1290-1295. Yao, Bin and M. Tomizuka (1994b). Smooth robust adaptive sliding mode control of robot manipulators with guaranteed transient performance. In: Proc. of American Control Conference. pp. 1176-1180. The full paper appeared in ASME Journal' of Dynamic Systems, Measurement and Control, Vol. 118, No.4. pp764-775, 1996. Yao. Bin ilnd M. Tomizuka (1997a). Adaptive robust control of MIMO nonlinear systems. Submitted to Automatica (revised in 1998). Parts of the paper were presented in the IEEE Conf. on Decision and Control, pp2346-235 1 , 1995. and the IFAC World Congress, VoL F. pp335-340. 1996. Yao, Bin and M. Tomizuka (1997b). Adaptive robust control of SISO nOn linea. systems in a semi-strict feedback form. Automatica 33(5). 893-900. (Part of the paper appeared in Proc. of 1995 American Control Conference, pp2500-2505). Yao, Bin, Fanping Bu and George T. C. Chiu (1998). Nonlinear adaptive robust control of electrtrhydraulic servo systems with discontinuous projections. In: Proc. of IEEE Conf on Decision and Control. pp. 2265-2270.
··C~~--'C::;.5'---"""',,.----,,c< .•--~ Time (S6C)
Fig. 1. Tracking errors in the presence of parametric uncertainty
.•
".~---.;c':c----,,:----,,:':.,c-------!
Time ,sac)
Fig. 2. Tracking errors with model mismatch
141(10" 12. '.
~~--~.----,,:---~c----~ 11"", (sac)
Fig. 3. Tracking errors in the presence of parametric uncertainty and external disturbances 30
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress ofTFAC
A-la-0l-5
Copyright <© 1999 IFAC 14th Triennial World Congress, Beijing, P,R. China
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECISION MANUFACTURING Bin Vao
and
Li XU
School of Mechanical Engineering Purdue University. West Lafayette. IN 47906. USA [email protected]
Abstract: Linear motors are sensitive to disturbances and parameter variations. Certain types of linear motors such as the iron core are also subject to nonlinear effects due to periodic cogging force and force ripple. An adaptive robust controller based on the discontinuous projection method is constructed to address these issues. In particular. based on the particular structures of various periodic non linear forces, design models consisting of known basis functions with unknown weights are used to approximate those unknown nonlinear forces. On-line parameter adaptarion is then utilized to reduce the effect of various parametric uncertainties such as unknown weights, inertia, and motor parameters while certain robust control laws are used to handle the uncompensated uncertain nonlinearities effectively for high performance. Simulation results are shown to illustrate the effectiveness of the proposed algorithm. Copyright ©19991FAC Keywords: Linear Motors. Motion Control, Adaptive Control. Robust Control, Servo Control 1. INTRODUCTION Linear motor gains high-speedjhigh-accuracy potential by eliminating mechanical transmission (Alter and Tsao. 1996; Braembussche et al.. 1996). However, it also loses the advantage of using mechanical transmissions - gear reduction reduces the effect of model uncertainties such as parameter variations(e.g., uncertain payloads) and external disturbance (e.g., cutting forces in machining). Furthermore, certain types of linear motors such as the iron core have significant uncertain nonlinearitiesdue to force ripple and cogging force (Braembussche et al.. 1996). These uncertain nonlinearitis are directly transmitted to the load and thus have significant effect on the motion of the load. In the past, a great deal of effort has been devoted to solving these difficulties (Alter and Tsao, 1996; Braembussche et al.. 1996; Otten et al.. 1997). In (Alter and Tsao, 1996), Hoo optimal feedback control is used to provide high dynamic stiffness to external disturbances (e.g .. cutting forces in machining). To reduce the nonlinear effect of force ripple and cogging force, in (Braembussche et al.. 1996), feedforward compensation terms were added to the position controller. In (Otten et al., 1997), a neural-networkbased learning feedforward controller was proposed to reduce positional inaccuracy due to reproducible ripple forces or any other reproducible and slowly varying disturbances.
Recently, Vao and Tomizuka proposed an adaptive robust control (ARC) approach (Vao and Tomizuka.
1994b; Vao and Tomizuka, 1997b), for the robust control of uncertain nonlinear systems in the presence of both parametric uncertainties and uncertain nonlinearities. The approach effectively combines the design techniques of adaptive control (AC) and those of deterministic robust control (DRC) and improves performance by preserving the advantages of both AC and DRe. A general theoretical framework is recently formalized by Yao in (Yao, 1997). Applications include the high-speedjhigh-accuracy motion control of machine tools (Vao et al., 1997), the control of electro-hydraulic servo systems (Yao et al., 1998), and the control of robot manipulators (Yao and Tomizuka, 1994a).
This paper generalizes the ARC approach to the high-speed/high-accuracy motion control of iron core linear motors. In addition to the usual design difficulties due to large variation of inertia load, unknown friction, and disturbance forces, there are several particular problems associated with the control of iron core linear motors. First, the iron core linear motors have a large magnitude of ripple forces. Second, the electrical current feedback loop cannot be neglected. This added electrical dynamics increases the controller design difficulties since the resulting system is of higher" relative degree" (Yao and Tomizuka, 1997b) and backstepping design via ARC Lyapunov functions (Yao, 1997) has to be employed. Third, due to variations of the inertia load and the motor force constant, parametric uncertainties will appear in the input channels of the
25
Copyright 1999 IF AC
ISBN: 008 0432484
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECT...
intermediate design steps. All these issues will be addressed in the paper. The paper thus advances the control of linear motors considerably since none of the published papers deal with all these aspects all together at once. 2. PROBLEM FORMULATION AND DYNAMIC MODELS
The dynamic equation of the inertia load driven by an iron core brush less DC linear motor (LCK series in (Anorad Linear Motors, 1997») can be described by
14th World Congress ofTFAC
identical and are equally spaced at a pitch of P, then, is a periodic function with a period of P, i.e., fCQgging(XL +P} = !cogging(XL). (2)Force ripple is developed due to the periodic variation of the force constant. When current flows through the coils, a position dependent periodic force is resulted. To take this effect into account, a position dependent term can be added to modify the force constant in (3) to Kp(xd = KFO + KFx(XL), in which Kp.,(xL) is a periodic function with a period of P, i.e., KF.,(XL +
!coggina(XL)
P) = KF",(XL).
By taking into account the effect of cogging force and force ripple, the motor force is given by where XL represents the displacement of the inertia load, M is the mass of the inertia load plus the iron core translator, Fm is the total electra-magnetic force generated by the linear motor, B is the combined coefficient of the damping and viscous friction on the load, J friction represents the combination of stiction and Coulomb friction, and f,.lis(t, XL, XL) represents the external disturbances such as cutting force in machining. A simple mathematical model for friction ft .. iction that takes into account of stiction, Coulomb friction, and Stribeck effect can be written as (de Wit et al., 1995) f!richon(:h) = -[le
+ Us -
Fm = KF(xL)i + /c()gging(xL) KF(xL} = KFO + KF.,(XL)
where i is related to the control input u by (3). Defining the position, velocity, and motor current as the state variables, Le., x = [xl,x2,x31 T ~ [XL,XL,i]T, from Eqs. (1), (3), and (4), the entire system can be expressed in state space form as: :h =X2 X2
Je)e-I';LI;,.I~lsgn(xL) (2)
where 13 is the level of static friction, le is the minimum level of Coulomb friction, and Xs and are empirical parameters used to describe the Stribeck effect.
e
Without going into the details about the working mechanisms of the amplifier and the motor, a popular simplified model that relates the generated Fm to the control input voltage u of the amplifier is given by (Krause, 1986)
(4)
.
"'3
= ~[(KFO + KF",(Xl))X3 -
BX2
+ /iriction(X2)
(5)
+ fdi8(t,XI, X 2)] 1 ""L"'2 + "L'"
+!cogging(xd R KE
= -];"'3 -
Given the desired motion trajectory XLd(t), the objective is to synthesize a control input u such that the output y = Xl tracks XLd(t) as closely as possible in spite of various model uncertainties. 3. ADAPTIVE ROBUST CONTROL (ARC) OF LINEAR MOTOR SYSTEMS
3.1 Design Models and Assumptions (3)
where K FO is the average force constant, i is the motor current amplitude, Rand L are the armature resistance and inductance respectively, KE is the electromotive force coefficient. Such a simplified model has been successfully used by many researchers in the control of rotary motors and certain types of linear motors (Le-Huy et al., 1986). For iron-core linear motors, the simplified linear motor model (3) may not be enough since these types of motors have relatively larger magnitude of nonlinear cogging force and force ripple, which are neglected in the simplified model (3). These nonlinear forces are generated due to the particular structure of the ironcore linear motors (Braembussche et al., 1996; CUen et al., 1997): (l)Cogging or dentent force is a magnetic force developed due to the attraction between the permanent magnets and the iron cores of the translator. It depends only on the relative position of the motor coils with respect to the magnets. Thus, this force can be described as !cogging(XL). If the permanent magnets of the same linear motor are all
To begin the controller design, practical and reasonable assumptions on the system have to be made. For the reproducible cogging, force ripple, and friction force, although feedforward compensation techniques based on off-line identified models can be used to cancel their effects (e.g., force ripple compensation in (Braembussche et al., 1996) and friction compensation in (Lee and Tomizuka, 1996)), such a procedure may be time consuming and sometimes may not be so practical. In the following, we will use some simple models with unknown weights to approximate them; the unknown weights will be adjusted on-line via certain parameter adaptation law to achieve an improved model compensation and the model approximation error will be handled via certain robust control law to achieve a guaranteed robust performance. Specifically, since fcogging(XI) and KF.,(ZI) are periodic functions with a known period P, they can be approximated quite accurately by their first several harmonics. which are denoted as !collging(Xt) and KFx(Zt) respectively and represented by !cogging(Xt}
= AJ'Sc(xt)
(6)
Kp,,(xt} = AkSK(Xt)
where Ac = [Acls,Acl c ,.'. ,AC918,AcQ1C)T AK
= [AKIs,AKIC, .. ' ,AK'I28,AKQ2clT E
E R2q, and R2q2 are the
26
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECl...
unknown weights, 8 c (xd = [sin{~xI),COS(~XI)' ... ' sinepz'xl),COs(2PZ'xl)V and 8K(X!) = [Bin(~xl)' C08(~Xt), ... , sin(l~2xl),COs(~xd]T are the known basis shape functions, and ql and q2 are the numbers of harmonics that we would like to compensate for each term. The friction (2) is discontinuous at XI = o. Thus one cannot use the model (2) for friction compensation since, as evident from (5), there is no way that one can generate a discontinuous motor force to accomplish it. To by-pass this technical difficulty, we will use a simple continuous friction model to approximate the actual discontinuous friction model (2) for model compensation; the model used in the paper is given by hdction = Aj8j(X2), where the amplitude AI may be known or unknown. and 8/(X2) may be chosen as certain differentiable function. The second equation of (5) can thus be written as: :i:z = -.!..[(Kpo M +AJ8/(xz)
+ AkSK(Xl»X3 - BX2 + AcT Sc(X1) + f(t , xl,X2,X3)]
positive. i.e., there exists
+ 0~8K(xd
KPmin
is
> 0 such that
3.2 Discontinuous Projection Let {j denot~ the~ estimate of 9 and jj the estimation (i.e., 9 = 9 - 9). By using an adaptation law given by
e~ror
(11)
where r > 0 is a diagonal matrix, T is an adaptation function to be synthesized later, and Projjj(e) is the widely used discontinuous projection mapping (Sastry and Bodson, 1989; Goodwin and Mayne, 1989). It can be shown (Yao and Tomizuka, 1994b) that for any adaptation function T. the projection mapping used in (11) guarantees (Pl) (P2)
(7)
3.3 Controller Design
i
where represents the lumped external disturbances and model approximation errors. which is given by j
A2 . V6 E [Omi", Om...",] , KF = 61
=:. (KF .. (XI) - !friction)
KF.,(Xl»X3
+ (leogging -
+
(8)
(f/riction lcog9ing) Id •• (t, Xl ,X2)
+
The design parallels the recursive backstepping design procedure via ARC Lyapunov functions in (Vao and Tomizuka, 1997a; Vao. 1997) as follows.
Step 1 In general, the system is subjected to parametric uncertainties due to the variations of M. B, Ac. AK, AI, KE, L. R, and the nominal value of the lumped disturbance In order to use parameter adaptation to reduce parametric uncertainties to improve performance, it is necessary to linearly parameterize the state space equations in terms of a set of unknown parameters. For this purpose, define the unknown parameter set 8 = [(h, 02b, 03, 64, 0Sb, 86, (h , Og, egJT E RP=1+2q, +2q2 as 01 =
i. in.
=
= --5-. = t.
41-. -to and
04 = Kt?,6 z b kAK E R 2n, 63 85b = i"Ac E R 2q2. 06 ~, 07 9a = 99 The state space equations (5) and
=
¥.
=(7) can thus be linearly parameterized in terms of 9 as :h
Noting that the first equation of (9) does not have any uncertainties, an ARC Lyapunov function can thus be constructed for the first two equations of (9) directly. Define a switching-function-like quantity as Zll
= et
X2 -
%2
= kFX~ + 93X2 +66 +d-
+ 63X2 + 848/(X2)
X2eq.
X2cq
~
:CId -
k"el
(13)
where el = Xl - Xld(t). in which the desired trajectory Xld(t) will be specified later to achieve a guaranteed transient performance, kp is any positive feedback gain. If Z2 is small or converges to zero exponentially. then the output tracking error Cl will be small or converge to zero. Differentiating (13) and noting (9)
=X2
X2 = [01 + 9~SK(Xt)JX3 +6r~Sc(xI) + 96 + d
+ kpet =
+ 948/(X2) + e~Sc(xtl
X2cq
(14)
(9)
xs = 88X3 + 8gx2 + 87'1.t
where d(t,xl,X2,X3) = i,,[! - 1nl represents the uncompensated disturbances and modeling errors. For simplicity. in the following. the following notations are used: ei for the i-th component of the vector e, em.in for the minimum value of e, and e maz for the maximum value of e. The operation < for two vectors is performed in terms of the corresponding elements of the vectors. The following practical assumptions are made
where X2eq ~ Xld - kpei. In (14), if we treat X3 as the input. we can synthesize a virtual control law 0:2 for Xs such that Z2 is as small as possible. Since (14) has both parametric uncertainties 9 and uncertain non linearity d, the ARC approach proposed in (Yao, 1997) will be generalized to accomplish the objective. The control function by
0:2
consists of two parts given
Al . The extents of the parametric uncertainties
and uncertain nonlinearities are known, i.e.. (10)
= (lhmin, ... , 90711;,,]', 6ma ~ = [61711"",
where Omi" ... , O-.,..,jT,
and dd(t,Xl,X2) are known.
or
where Kp = 81 + SK(Xl) is the estimated motor force constant, 0::2 .. tunctions as an adaptive control law used to achieve an improved model compensation through on-line parameter adaptation given by (11). and 0:28 is a robust control law to be synthesized later. Noting Pi of (12) and Assumption 27
Copyright 1999 IF AC
ISBN : 0 08 043248 4
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECT...
A2, KF > O. Thus the control function (15) is well defined no matter what type of adaptation function to be used. Let Z3 = X3 - 0<2 denote the input discrepancy. Substituting (15) into (14) leads to Z2
=:
+ 02.) -
KF(Z3
eT 4>2
+ ii
14th World Congress ofTFAC
In (22), 02c is calculable and can b'7 used in the design of control functions but a2u cannot due to various uncertainties. Therefore, a2u has to be dealt with in this step design. From (9), is
(16)
where tP2=[a2a, S'k(Xl)02a, X2. S/(X2), S;(xI). 1, 0, 0, O)T The robust control function 02s consists of two terms given by
= 98X3 + 99X2 + 97U -
0:2c -
0:21>
(24)
Consider the augmented p.s.d. function V given by W3
>
(25)
0
Noting (20) and (24),
17=172 1"'2 where 0282 is a robust control function designed in the following and k281 is any non linear feedback gain satisfying
+Oru -
Xs
KF(Xl)
---k2s1Z2 KFmin
+ KF(X1)0282
112
= -~ [W2 KFZ2 + 08X3 +Ogx2 87
(19)
u.
. V2
= KF(X.}W2 _ Z 2Z 3 -
K F (Xl)
W 2 - - - k 2,l Z 2
+W2Z2{KF(x.}02.2 -
!f/:Tnin _
(J
4>2
2
(20)
+ d}
W3
= 1£81 + 1£.2.
kSB1
Define a positive semi-definite (p.s.d.) function V2 as V2 = !w2zi where W2 > 0 is a weighting factor. From(19), its time derivative is
.=.V2 1"'2
condition ii
Z20!2.2:5
OT
~
0
1':2
+W3%3
{-T (J7U. -
where C2 is a design parameter which can be arbitrarily small. How to choose 0282 to satisfy constraints like {21} can be found in (Yao, 1997). One example is given by 0:2.2 = - 2K F "2. 22, where h2 is any wun e'2 function satisfying h2 2: 118Mll z llrP211 2 + o~ , in which (JM
= (Jmaz -
9m i'"
Noting the last equation of (9), Step 2 is to synthesize an actual control law for u such that i tracks the desired control function 02 synthesized in Step 1 with a guaranteed transient performance, which can be done by using the same ARC design techniques as in Step 1. From (I5) and {9}, 002
8U2
8U2 .
80:2(J~
8U2
= -8X1 X 2 + - X 2 + -- + 8X2 8(J 8t = O,2c +62..
(22)
-
8002 80<2 ~} -8 d - - _ (J (28)
[!!!4Z2 WS
~X3, U~2
80
T (X.)("!:4Z2 SK W3
~s ( ) - ~ST( 8x2" f:1:2, 8X:l c Xl ) , - ~ 8:1:'2 ' U a ,
~X3), ux~ ::E'3t 3:'2 ]T
Similar to (21), the robust control function chosen to satisfy
U s2
is
condition i
condition ii
where
is a design parameter. For example, let in which h3 is any function satisfying ha 2: 118M 1J211<1>3; IJ2 + 6~
U.2
Step 2
•
-
~ 8Z2 Z'2,
(J
X2
where <1>3 = (21)
(27)
= - -(J7t11in -k38123 2 2 2: ka + 11 80 Cesll + II c 4ts r r/lsll 1£.1
where ks > 0 is a constant, C e3 , and Cq,3 are positive definite constant diagonal matrixes , and U;t2 is a robust control function to be chosen later. Substituting (27) and (23) into (26) and using similar techniques as in (16) V
-
0:2<0]
1
8 U2
The robust control function 0282 is now chosen to satisfy the following conditions condition i ZZ{KF(Xl )0:2.2
(26)
= a2
1£"
-
""2 = KF('''1)Z3 -r -8 4>2 + d
+ (JgX3 + 09X2
1<>2 denotes V2 under the condition that (or Z3 = 0). Similar to (15), the actual control input u to be synthesized consists of two parts given by 1£ = u,,(x,e,t) + u.(x,O,t)
where
in which Crp2 is a positive definite constant diagonal matrix to be specified later and k2 is a positive design parameter. Substituting (17) into (16),
+W3ZS{ W2 KFZ2 W3 62c - 0:2u}
C3
= -297':;n~3h3Z3'
Theorem 1. Let the parameter estimates be updated by the adaptation law (11) in which T is chosen as (30)
If the control parameters C.p2 = diag{C.p21 •... ' C,p2p}, = diag{Cq,31,'" ,Cq'>3p}, and C83 = diag{C831," ., C83p} are chosen such that. VI. Cq,21C/131 2: and Cq,31C631 2: ..jfW3. where Cq,21. Cq,31 and C931 are the l-th diagonal elements of c 4t2 , c 4t3 and C(J3 respectively. Then, the control law (27) guarantees that Cq,3
A.
v'tw2wa
In general, the control input is bounded and the closed-loop system is globa lIy stable with V bounded above by Vet) :$ IlXp(-Avt)V(O)
cV + AV [1 -
exp( -AV t )]
(31)
28
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECl...
where Av = 2min{k2 , k3} and tV =
W2 C 2
+
W3 C 3·
B If after a finite time to, d = 0, i.e., in the presence of parametric uncertainties only, then, in addition to results in A, asymptotic output tracking (or 6 zero final tracking error) is also achieved. Proof: The Theorem can be proved in the same way 3S in (Vao, 1997).
Desired Trajectory Initialization and Generation It is seen from (31) that transient tracking error is affected by the initial value V(O), which may depend on the controller parameters also. To further reduce transient tracking error, the desired trajectory initialization can be used as in (Yao, 1997JJ. Namely, instead of simply letting the desired trajectory for the controller be the actual desired trajectory or position (i.e., Xld(t) = XLd(t», we can generate Xld(t) using any stable 3rd order filter with initial conditions $ld(O), . .. , x~~)(O) chosen as XlcJ(O) = Xl(O), XId(O) = X2(0) 21ld(O) = {KFX3 + OaX2 + 04S, + (j'f"sc + 06}lt=o
(32)
Then. el(O) = o,z;(O) = O,i = 2,3 and V(O) = o. Thus transient tracking error is minimized. It is seen from (32) that the trajectory initialization is independent from the choice of controller parameters such as k = [k 2 , k3jT and c = (c2, e3]T. Thus. once the initial position of the linear motor is determined, the above trajectory initialization can be performed offline. 4. SIMULATION RESULTS To illustrate the above designs, simulation results are obtained for a linear motor system with motor physical parameters obtained from the specifications of the iron core LCK-S-l linear motor by Anaroad Company. Nominal values of the system are: an inertia of M = 10kg, a damping coefficient of B = O.SN/m/s, a motor back EMF of KE = 18.5V/m/s, a force constant of KFO = 5S.5N/A. a coil resistance of R = 3.911, an inductance of L = 30mH, and a magnet pitch of p = 30mm. The actual friction force used for simulation is given by (2), where /. = 10, fe = 6, X. = 0.001 and ~ = 1. The following situation is considered: The shape function for friction compensation is chosen as S,(X2) = tanh(lOOOx2) and the friction amplitude At is allowed to vary from 5N to ION. Cogging amplitude is allowed to be any value less than SON, and the amplitude of KFx(:rt) is less than 1. 11 N/A. For simplicity, in the simulation, only the first harmonics are used in the controller to approximate the periodic cogging force and KF",(Xl), i.e., ql = q2 := 1 in (6). A sampling rate of O.2ms is used in the simulation. The following two controllers are compared: (i) ARC, the ARC law presented in section Ill. The controller parameters are: kp = 200,k2ol = 200,W2 = 1, .02 = S x
104 ; ks.1
rate is
= 300,ws = 0.1, .os = 1 x 107 . The adaptation
I' = dia.g[ 342,0.39,0.39, 3.S x 10- 3 , 0.61, 288, 288,
X 10 4 ]; and (ii) DRC, Deterministic Robust Control - the same control law as the above ARC but without parameter adaptation.
S1.2. 125. 8 x 10 3 , 1.8
To test the tracking capability of the proposed algorithm, a sinusoidal desired trajectory given by XLd = O.01sin{21rt) and the trajectory initialization (32) are used . Simulations are first run for the case that model uncertainties are dominated by parametric uncertainties as follows. Actual cogging force /cogging(Xt} and the position dependent force constant KF::: used for plant simulation are assumed to be of first harmonics only, Le., Fcagging = 25sin(21rXI/P + 1r/4) and KFx = 1.1lsin(21rXl/P+7r/4), and there is no external disturbance (fai. = 0). Tracking errors are shown in Fig. 1. As seen, both controllers have very small tracking errors but the final errors of ARC are much smaller than that of DRC due to parameter adaptation. The simulation is then run for the case that actual models of icc99 in 9(Xl) and KF.,(Xl) are much different from their design models (6). Specifically, both /coggin9(xt} and KF .. (Xl) are assumed consisting of the first and the third harmonics (Braembussche et al., 1996) as Fcog9 ing = 15sin(27rXI/ P+n /4)+20sin(67rxl/ P+O.09"1r) and KF% = KFO(0.016sin(21rxl / P
+ 11'/4) +0.02sin(6l\"Xl / P +O.091r}).
Tracking errors are shown in Fig.2. As seen, the mismatch between the design models and the actual models for ripple forces does not degrade the performance of the proposed ARC much. This verifies the performance robustness of the proposed ARC to uncompensated uncertain nonlinearities. To test performance robustness ofthe proposed ARC to external disturbances, an external disturbance force idi. = 30+5rand (N) is then added to the system during the first second . As shown in Fig.3. the added disturbance force has a large effect on DRC but not much on ARC. This is because that the proposed ARC can adapt to the nominal value of the lumped disturbance (116) very quickly. 5. CONCLUSIONS An ARC controller has been constructed for the adaptive robust motion control of an iron core linear motor. The controller takes into account the efrect of parametric uncertainties and uncertain nonlinearities due to external disturbances, uncompensated friction and force ripple. In particular, design models consisting of known basis functions with unknown weights are used to approximate the unknown nonlinear ripple forces. On-line parameter adaptation is then utilized to reduce the effect of various parametric uncertainties while the uncompensated uncertain nonlinearities are handled via certain robust control laws. As a result, time-consuming and costly rigorous off-line identification of friction and ripple forces 29
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ADAPTIVE ROBUST CONTROL OF LINEAR MOTORS FOR PRECT...
is avoided while without sacrificing tracking performance. Simulation results are provided to illustrate the proposed scheme. ACKNOWLEDGMENT
14th World Congress ofTFAC
VilO, Bin, M. AI-Majed and M. Tomizuka (1997). High performance robust motion control of machine tools: An adaptive robust control approach ilnd comparative experiments. IEEEjASME Trans. on Mechatronics 2(2), 6376.
This work is supported by the National Science Foundation under the CAREER grant CMS-9734345. The authors would like to thank Dr. Y. Shin for helpful discussions on the subject. 6. REFERENCES
Alter, D. M. and T. C. Tsao (1996). Control of linear motors for machine tool feed drives: design and implementation of hoo optimal feedback control. ASME J. of Dynamic
systems, Measurement, and Contro/IIS, 649-656. Anorad Linear Motors (1997). Anorad Corporation. 110 Oser Avenue, Hauppauge, NY11788. Braembussche, P. V., J. Swevers, H. Van Brussel and P. Vanherck (1996). Accurate Tracking control of Linear Synchronous Motor Machine tool Axes. Mechatronics 6(5), 507-52l. de Wit, C. Canudas, H. 01s5On, K. J. Astrom and P. Lischinsky (1995). A new model for control of systems with friction. IEEE Trans. on Automatic Control 40(3), 419-425. Goodwin, G, C. and D. Q. Mayne (1989). A parameter estimation perspective of continuous time model reference adaptive control. Automatica 23(1), 57-70. Krause, P. C. (1986). Analysis of Electric Machinery. McG rilw-HiII , Inc. Le-Huy, H .. R. Perret and R. Feuil1et (1986). Minimiaztion of Torque Ripple in Brushless DC Motor Drives. IEEE Transactions on Industrial Applications 22(4).748-755. Lee, H. S. and M. Tomizuka (1996). Robust motion contaller design for high-accuracy positioning systems. IEEE Trans. on Industrial Electronics 43(1). 48-55. OUen, G., T. Vries, J. Amerongen, A. Rankers and E. Gaal (1997). Linear Motor Motion Control Using a Learning Feedforward Controller. IEEEjASME Transactions on Mechatronics 2(3), 179-187. Sastry, S. and M. Bodson (1989). Adaptive Control: Stability. Convergence and Robustness. Prentice Hall, Inc .. Englewood Cliffs, NJ 07632. USA. Yao. Bin (1997). High performance adaptive robust control of nonlinear systems: a general framework and new schemes. In: Proc. of IEEE Conference on Decision and Control. pp. 2489-2494. Yao, Bin and M. Tomizuka (1994a). Comparative experiments of robust and adaptive control with new robust adaptive controllers for robot manipulators. In: Proc. of IEEE Conf. on Decision and Control. pp. 1290-1295. Yao, Bin and M. Tomizuka (1994b). Smooth robust adaptive sliding mode control of robot manipulators with guaranteed transient performance. In: Proc. of American Control Conference. pp. 1176-1180. The full paper appeared in ASME Journal' of Dynamic Systems, Measurement and Control, Vol. 118, No.4. pp764-775, 1996. Yao. Bin ilnd M. Tomizuka (1997a). Adaptive robust control of MIMO nonlinear systems. Submitted to Automatica (revised in 1998). Parts of the paper were presented in the IEEE Conf. on Decision and Control, pp2346-235 1 , 1995. and the IFAC World Congress, VoL F. pp335-340. 1996. Yao, Bin and M. Tomizuka (1997b). Adaptive robust control of SISO nOn linea. systems in a semi-strict feedback form. Automatica 33(5). 893-900. (Part of the paper appeared in Proc. of 1995 American Control Conference, pp2500-2505). Yao, Bin, Fanping Bu and George T. C. Chiu (1998). Nonlinear adaptive robust control of electrtrhydraulic servo systems with discontinuous projections. In: Proc. of IEEE Conf on Decision and Control. pp. 2265-2270.
··C~~--'C::;.5'---"""',,.----,,c< .•--~ Time (S6C)
Fig. 1. Tracking errors in the presence of parametric uncertainty
.•
".~---.;c':c----,,:----,,:':.,c-------!
Time ,sac)
Fig. 2. Tracking errors with model mismatch
141(10" 12. '.
~~--~.----,,:---~c----~ 11"", (sac)
Fig. 3. Tracking errors in the presence of parametric uncertainty and external disturbances 30
Copyright 1999 IF AC
ISBN: 008 0432484