Adaptive Approach to Motion Controller of Linear
Induction Motor with Friction Compensation
Chin-I Huang, Student Member, IEEE, and Li-Chen Fu, Fellow, IEEE
Abstract—In this paper, we propose a nonlinear adaptive
con-troller and an adaptive backstepping concon-troller for linear induction motors to achieve position tracking. A nonlinear transformation is proposed to facilitate controller design. In addition, the very unique end effect of the linear induction motor is also considered and is well taken care of in our controller design. We also consider the effect of friction dynamics and employ observer-based compensa-tion to cope with the friccompensa-tion force. A stability analysis based on Lyapunov theory is also performed to guarantee that the controller design here can stabilize the system. Also, the computer simulations and experiments are conducted to demonstrate the performance of our various controller design.
Index Terms—Adaptive control, induction motor drives.
NOMENCLATURE
a1 = (RsDLr)+βLLmrRr. a2 = βRLrr.
a3 = LmLrRr. a4 = RLrr
B Viscous friction coefficient.
β Lm D . c Lr D D = LsLr− L2m. Fe Electromagnetic force.
FL All mechanical load force.
¯
FL Mechanical load force without the end effect.
Fr Friction force.
iq(id) q-(d-) axis input stator current.
Kf Force constant (= 3P Lmπ/2τ Lr).
Lm Mutual inductance.
Ls(Lr) Primary (secondary) inductance.
Mm Primary mass.
P Number of pole pairs.
pr Position of the primary.
p = Pπ
τ.
Rs(Rr) Primary (secondary) resistance.
vr presaged espeed of the primary.
Vqs(Vds) q-(d-) axis input stator voltage. λq(λd) q-(d-) axis secondary flux.
τ Pole pitch.
Manuscript received May 12, 2005; revised December 7, 2006. Recom-mended by Technical Editor P. R. Pagilla
C. I. Huang is with the Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: [email protected]). L. C. Fu is with the Department of Electrical Engineering and Computer Sci-ence and Information Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: [email protected]).
Digital Object Identifier 10.1109/TMECH.2007.901945
I. INTRODUCTION
N
OWADAYS, linear induction motors (LIMs) are widely used in many industrial applications, including transporta-tion, conveyor systems, actuators, material handling, pumping of liquid metal, sliding door closers, etc., with satisfactory per-formance. The most obvious advantage of linear motors is that they have no gears, and require no mechanical rotary-to-linear converters when translation of payload is concerned. The LIM has many advantages such as simple structure in place of the gear between the motor and the motion devices, reduction of mechan-ical losses and the size of motion devices, silence, high starting thrust force, and easy maintenance, repairing, and replacement. But for high precision motion performance, the friction problem is one of the significant limitations.In the early works, Yamamura has first discovered a particular phenomenon of the end effect on LIM [1]. A control method, de-coupling the control of thrust and the attractive force of an LIM using a space vector control inverter, was presented in [2], i.e., by selecting voltage vectors of PWM inverters appropriately.
Although the parameters of the simplified equivalent circuit model of an LIM can be measured by conventional methods (no-load and locked secondary tests), due to limited length of the machine the realization of the no-load test is almost impossible. Thus, the applicability of conventional methods for calculating the parameters of the equivalent model is limited. In order to measure the parameters, application of the finite element (FE) method for determining the parameters of a two-axis model of a three-phase linear induction motor has been proposed in [3] and [4]. Another method is proposed by removing the secondary [5]. To resolve the unique end-effect problem, speed-dependent scaling factors are introduced to the magnetizing inductance and series resistance in the d-axis equivalent circuit of the ro-tary induction motor (RIM) [6] to correct the deviation caused by the “end effect”. On the other hand, there is a thrust correc-tion coefficient introduced by [7] and [8] to calculate the actual thrust to compensate for the end effect. A related method to deal with the problem is that an external force corresponding to the end effect is introduced into the RIM model to provide a more accurate modeling of an LIM under consideration of the end effect as shown in [9]. In another work [10], [21], [22], extra compensating-winding was proposed to alleviate such problem.
Although the end effect is an important issue of the LIM control, there are still many works in the literature that fail to consider it such as [11], [12], and [23]–[25]. In this paper, we will take this as an important issue which cannot be ignored. In addition, as the contact area of bearing in LIM is much larger than that of RIM, the friction term cannot be neglected, inspite
of the fact that such “friction problem” may not be that critical in high speed applications.
On the other hand, for high precision motion performance, the friction problem leads to one of the significant limitations. Friction varies with temperature, age, and is known to dominate in the low velocity region and during velocity reversal [20]. Fail-ure to compensate for friction in applications may lead to large tracking errors, undesired stick-slip motion, and limit cycles when velocity reversals in the trajectory are required [15], [19]. In order to predict and compensate for the underlying friction, Canudas de Wit has proposed the LuGre model [15], which includes the Stribeck effect, hysteresis, and spring-like char-acteristics of the stiction, as well as variant break-away force. Furthermore, the adaptive scheme suggested in [16]–[18] has been extended to handle the nonuniform parametric variations of the friction force, and adaptive dynamic friction compensa-tion [20] be applied for trajectory tracking control of robots.
The remainder of this paper is organized as follows. In Sec-tion II, we introduce the system model and state the present control objectives. Section III will develop a nonlinear adaptive control of the LIMs presuming that the acceleration signal is measurable. However, in Section IV, we propose an adaptive backstepping approach without measurement of the accelera-tion signal. To demonstrate the effectiveness of the hereby de-veloped controllers, Section V and VI will, respectively, provide thorough simulations and experiments with satisfactory perfor-mance. Finally, Section VII gives some concluding remarks.
II. PROBLEMFORMULATION
To formulate the dynamic model of a linear induction motor, we consider the following assumptions to make the analysis more tractable.
Assumption 1: three phases are balanced.
Assumption 2: the magnetic circuit is unsaturated.
Assumption 3: it is without end effect (we will relax this assumption later in controller design); then the dynamics of the entire system can be rearranged into the following more compact form: ˙iq =−a1iq+ a2λq− βpvrλd+ cVqs ˙id =−a1id+ βpvrλq+ a2λdcVds ˙ λq = a3iq− a4λq+ pvrλd ˙ λd = a3id− a4λd+ pvrλd Mm˙vr= Kf(λdiq− λqid)− ¯FL− Fr. (1) For the meanings of various symbols, readers should refer to the Nomenclature. In this paper, we try to design the speed and position controller for the LIM whose parameters are assumed to be known except the payload. However, some knowledge about the payload structure is available, which is expressed in terms of a second-order differential equation as shown by
¯ FL= M
L˙vr+ bL0+ bL1vr+ bL2vr2. (2) Furthermore, the friction force Fr in (1) is modeled by the LuGre friction model [20], which uses the following dynamics
to express the variation of the friction force variation: dl dt = vr− |vr| n(vr) l (3) Fr= ζ0l + ζ1 dl dt + ζ2vr (4)
where l is the friction state that physically stands for the aver-age deflection of the bristles between the two contact surfaces. The friction force parameters ζ0, ζ1, and ζ2can be physically
explained as the stiffness of bristles, damping coefficient, and viscous coefficient, respectively. In our design, we assume that these three parameters are unknown positive constants. A pa-rameterization of n(vr) is assumed to be known, and has been proposed to describe the Stribeck effect [20], i.e.,
n(vr) = Fc+ (Fs− Fc)e−(
v r v s)
2
(5) where Fc, Fs, and vsare the Coulomb friction value, stiction force value, and the Stribeck velocity, respectively.
III. OBSERVER ANDNONLINEARADAPTIVE CONTROLLERDESIGN
In this section, the controller which achieves position track-ing of the LIM is proposed. In order to relax the need of flux measurement and to cope with friction, the flux and friction ob-server will be proposed in Section III-B and C, respectively. The controller should overcome the unknown payload of the LIM while the reasonable assumptions are made. In Section III-D, we propose a position controller to achieve the objective of position tracking with knowledge of the mutual inductance. In Section III-E, an adaptive position controller is proposed to deal with uncertain inductance which achieves the same objective. A. Analysis of Mechanical Load and End Effect
The fundamental difference between a rotary induction motor and an LIM is the finite length of the magnetic and electric circuit of the LIM along the direction of the travelling field. The open magnetic circuit causes an initiation of the so-called longitudinal end effects.
In an LIM, as the primary past moves, the secondary past seemingly is continuously shifting along the same moving di-rection. This linear shift will induce a resistance to a sudden increase in flux penetration at the leading front because by na-ture only a gradual build-up of the flux density in the air gap is permitted. More specifically, as the primary coil set of the LIM moves, a new field penetrates into the reaction rail in the entry area, whereas the existing filed disappears at the exit area of the primary coil as shown below in Fig. 1.
We should note that when the speed is higher, the air-gap flux is more unbalanced. Because the mutual flux between the primary and the secondary is decreased by the end effect, we can see that the equivalence of the end effect is a reduction force, which is a function of speed. As we know that most functions can be described in Taylor series reasonably, we assume that the end effect can be regarded as an external force which can be expressed as∞n=0bnvn
r + Me˙vr. In the paper, we will truncate the series into the first three terms.
Fig. 1. Air-gap average flux distribution due to end effect.
As a result, for an LIM, the end effect plus the load force and the friction effect can be represented as a function of the speed vr, which can normally be expressed by the following form:
FL+ Fr= 2 n=0 bnvnr+ Me˙vr + ¯FL + ζ0l + ζ1 vr− |vr| n(vr) l + ζ2vr = [Me˙vr+ b0+ b 1vr+ b 2vr2] + [M L˙vr+ bL0 +bL1vr+bL2vr2]+ ζ0l+(ζ1+ ζ2)vr− ζ1 |vr| n(vr)l = ML˙vr+ bo + b1vr+ b2vr2+ ζ0l− ζ1 |vr| n(vr)l with ML= Me+ M L L b0= b +bL 0 0 b1= b +bL 1 1 + ζ1+ ζ2 b2= b +b L 2 2
where FL is denoted as the mechanical payload account-ing for the end effect and can be expressed in a compact form as FL= ΘT
LVr with the unknown constant parame-ters ΘL= [ML b0 b1 b2]T and a known function vector Vr= [ ˙vr v0r vr1 v2r]T.
To proceed further, we introduce an additional assumptions as shown below:
Assumption 4: x2=λ2q+λ2d< 0
and then further simplify the dynamics shown in (1) by intro-ducing a nonlinear coordinate transformation given as follows:
x1= i2q+ i2d x2=λ2q+λ2d x3= iqλq+ idλd x4= iqλd− idλq x5= vr.
Remark: The transformation tries to redefine the secondary flux norm, the electric force, and the primary speed as individual variables x2, x4, and x5, respectively, and certainly the nonlinear
transformation is nonunique.
Initially, let the stator voltage inputs be chosen as cVds= −λq λ2 d+λ 2 q V and cVqs= λd λ2 d+λ 2 q V [13]
where V is an exogenous function to be defined later. Given such transformation and the chosen input, then the dynamical equations shown in (1) can thus be converted into the following dynamic model: ˙x1=−2a1x1+ 2a2x3+ 2x4V /√x2 ˙x2=−2a4x2+ 2a3x3 ˙x3= a3x1+ a2x2− (a1+ a4)x3+ px5x4 ˙x4=−px5x3− βpx5x2− (a1+ a4)x4+ V M ˙x5= Kfx4− 2 n=0 bnxn5− ζ0l + ζ1 |x5| n(x5) l (6)
where M = Mm+ ML. To control the system (6), we develop the position controller to achieve the goal of position tracking, namely, pr= 0tx5dt→ pdas t→ ∞ will be explained in the following section.
B. Flux Observer Design and Analysis
Under our assumptions, all the states are measurable except the secondary flux. Therefore, we have to build a set of ob-servers to estimate the secondary flux. The obob-servers then can be designed as follows:
˙ˆλq = a3iq− a4ˆλq+ pvrˆλd ˙ˆλd= a3id− a4λˆd− pvrλˆq.
Theroem 1: [13] If the dynamical equations are described as in system (6) with all the states being measurable except the secondary flux. Moreover, all the parameters are assumed known beforehand. Then, the flux observers designed can guarantee that both ˆλd− λd→ 0 and ˆλq− λq → 0 as t → ∞.
C. Two Nonlinear Observer Designs for Friction Estimation In this paper, we consider the frictional effect represented by a LuGre model as described by (3) and (4). But we know that the friction state l is not measurable. In order to handle different nonlinearities due to l which are present in the system dynamics, we employ two nonlinear observers to estimate the unmea-surable state l and replace l with its estimates ˆl0 and ˆl1 [16]
and [17], of which the dynamics are respectively given by dˆl0 dt = x5− |x5| n(x5) ˆ l0+ η0 dˆl1 dt = x5− |x5| n(x5) ˆ l1+ η1 (7)
where η0, η1 are compensation terms yet to be determined
computed as d˜l0 dt =− |x5| n(x5) ˜ l0− η0 d˜l1 dt =− |x5| n(x5) ˜ l1− η1
where ˜l0= l− ˆl0and ˜l1= l− ˆl1are estimation errors. D. Nonlinear Adaptive Position Controller With Friction Compensation Design
Now, we introduce another state
x6= pr (8)
to facilitate investigation of the development of a position con-troller. Then, we define the tracking errors as follows:
ep= pr− pd= e6. (9)
Normally, while the position tracking error is driven to zero, the speed is also regulated to zero. Thus, we naturally define a joint error signal S as follows:
S = ˙ep+ aep = e5+ ae6
where a is a positive scalar gain. The resulting error dynamics are derived as follows:
M ˙S = Kfx4− 2 n=0 bnxn5 − M( ˙vd− ae5)− ζ0l + ζ1 |x5| n(x5) l = Kfx4− 2 n=0 bnxn5 − M( ˙vd− ae5)− ζ0(˜l0+ ˆl0) + ζ1 |x5| n(x5) (˜l1+ ˆl1).
Based on this equation, we will propose a position tracking controller, and the following theorem summarizes the design procedure and the resulting control effect.
Theorem 2: Consider a linear induction motor whose dynam-ics are governed by system (6) under the assumptions (A.1)– (A.4). Given friction observer (7) and a smooth desired position trajectory pd with pd, ˙pd, ¨pd and p¨d being all bounded, then the following control input can achieve the control objective pr→ pdast→ ∞(i.e., x6= prwill followpdasymptotically) with the control input
Vqs= λd λ2 q+λ2d V c Vds= −λq λ2 q+λ2d V c and V = √1 x2 [(a1+ a4)x4+ βpx2x5+ px3x5 + ˙x4d− ρ2e4− KfS] where x4d= 1 Kf 2 n=0 ˆbnxn 5+ ˆM (˙vd− ae5) + ˆζ0ˆl0− ˆζ1 |x5| n(x5) ˆ l1− ρ1S
withρ1, ρ2> 0, and e5= x5− vd, e4= x4− x4d, and the pa-rameter adaptation laws
˙ˆb0=−S,˙ˆb1=−Sx5 ˙ˆb2=−Sx25 ˙ˆ M =−S( ˙vd− ae5) ˙ˆζ0=−Sˆl0 ˙ˆζ1=−S |x5| n(x5) ˆ l1
subject to the design of the compensation terms for the friction observer as follows:
η0=−S η1=− |x5| n(x5)
S while keeping all the internal signals bounded.
Proof: In order to show the boundedness of all the parameter estimates and the tracking errors e4, e5, we choose a
Lyapunov-like function Veas shown by: Ve= 1 2[M S 2+ e2 4+ ˜b20+ ˜b21+ ˜b22+ ˜M2+ ˜ζ02+ ˜ζ12+ ˜l20+ ˜l12]. (10) According to the proposed adaptive laws as
˙ˆb0=−S ˙ˆb1=−Sx5 ˙ˆb2=−Sx25 ˙ˆ M =−S( ˙vd− ae5) ˙ˆζ0=−Sˆl0 ˙ˆζ1=−S |x5| n(x5) ˆ l1
and the suggested compensation terms for the friction observer as:
η0=−S η1=− |x5| n(x5)
S. If one designs the auxiliary signalx4das
x4d= 1 Kf 2 n=0 ˆ bnxn5+ ˆM ( ˙vd−ae5) + ˆζ0ˆl0−ˆζ1 |x5| n(x5) ˆ l1−ρ1S
then the time derivative of the function Vebecomes ˙ Ve= e4[−(a1+ a4)x4− βpx2x5− px3x5+√x2V − ˙x4d+ KfS]− ρ1S2− ζ0 |x5| n(x5) ˜ l20− ζ1 |x5| n(x5) ˜ l21. (11) Now, by hypothesis the actual input is designed as
V =√1 x2
which apparently leads to the result that ˙ Ve=−ρ1S2− ρ2e24− ζ0 |x5| n(x5) ˜ l20− ζ1 |x5| n(x5) ˜ l12≤ 0 where ρ1, ρ2> 0, ζ0, ζ1are positive constants, and the friction
characteristic function n(x5) is chosen to be a positive function,
which readily implies boundedness of all parameter estimates as well as of both signals x4and x5. The reason is as follows:
Since ˙Vein (11) is nonpositive, we conclude that all the error signals in Veand in particular, x5and x4dare bounded, which
in turn implies that x4 and hence, [from system (6)] are both
bounded. This further implies that the estimation errors ˜l0, ˜l1∈ L∞, the parametric errors ˜ζ0, ˜ζ1∈ L∞, and hence, the parameter estimates ˆζ0, ˆζ1∈ L∞. From the friction dynamics in (3) and the boundedness of speed x5, boundedness of the friction state l is
clear, which then implies the observer states ˆl0, ˆl1are bounded.
Now, we are ready to show that all the internal signals are kept bounded. By using a fact that if Isis bounded, then all signals xi, limi=1, ., 5. are then guaranteed to be bounded.
By the power formula, Ps= Kfx4x5= 3VsIscan be shown bounded from the above reasoning. We now want to show that Iswill be bounded via argument of contradiction: say, Is even-tually grows unbounded, then Vs and hence, V will diminish eventually; however, if Isdoes grow unbound, then it implies that V will tend to px5x3/√x2 eventually. But, from the
dy-namics of x2in (6), we have x2and x3grow at the same rate,
which readily says thatV form the hypothesis will also grow un-bounded. This obviously leads to a contradiction and therefore, Isis bounded.
Furthermore, we can show that ˙x4d is bounded, and hence,
˙e4and ˙S are also bounded, which implies the convergence of e4
and S due to Barbalat’s Lemma. Therefore, the control scheme with the properly designed input V from the hypothesis will drive the output prto the desired pdasymptotically. E. Consideration of Uncertain Inductance
From the previous LIM dynamics, the parameters
a1,β,c and Kf depend on the mutual inductance, but as we know the mutual inductance is hard to identify due to its intri-cate structure and undesirable end effect. In particular,
a1= RsLr+ RrL2m/Lr LsLr− L2m = a10+ α, c = Lr LsLr− L2m = c0+ σ
where α and σare uncertainty terms of a1and c, respectively.
We rewrite the dynamic (6) as follows: ˙x1=−2(a10+ α)x1+ 2a2x3+ 2(c0+ σ)x4 √ x2 V ˙x2=−2a4x2+ 2a3x3 ˙x3= a3x1+ a2x2− (a10+ α + a4)x3+ px5x4 ˙x4=−px5x3− βpx5x2− (a10+ α + a4)x4 + (c0+ σ)√x2V M Kf ˙x5= x4− 2 n=0 bn Kf xn5 − ζ0 Kf l + ζ1 Kf |x5| n(x5) l ˙x6= x5 (12)
and design the control input Vqs= λd λ2 q+λ2d V Vds= −λq λ2 q+λ2d V.
To facilitate subsequent investigation, we define several vari-ables as follows: ˜ α = α− ˆα ˜ β = β− ˆβ dn= bn Kf H = M Kf ξ0= ξ0 Kf ξ1= ξ1 Kf
where ˆα is the estimate of α, ˆβ is the estimate of β.
In order to show the boundedness of all the parameter esti-mates and the tracking errors e4, S, we choose a Lyapunov like
function Veas shown by Ve= 1 2[HS 2+ e2 4+ ˜d20+ ˜d21+ ˜d22 + ˜H2+ ˜ξ02+ ˜ξ12+ ˜l02+ ˜l21+ ˜α2+ ˜β2] (13) where ˜ H = ˆH− H ˜ ξ0= ˆξ0− ξ0 ˜ ξ1= ˆξ1− ξ1 ˜ d0= ˆd0− d0 ˜ d1= ˆd1− d1 ˜ d2= ˆd2− d2.
If we employ friction observer (7) and design the parameter adaptive laws as ˙ˆ d0=−S ˙ˆ d1=−Sx5 ˙ˆ d2=−Sx25 ˙ˆ H =−S( ˙vd+ ae5) ˙ˆ α =−e4x4β =˙ˆ −e4px5x2
˙ˆ ξ0=−Sˆl0 ˙ˆ ξ1=−S |x5| n(x5) ˆ l1
with the friction observer compensation terms being defined as η0=−S η1=− |x5|
n(x5) S together with the proper design of
x4d= 2 n=0 ˆ dnxn5+ ˆH( ˙vd− ae5) + ˆξ0ˆl0− ˆξ1 |x5| n(x5) ˆ l1− ρ1S
then the time derivative of the Lyapunov-like function ˙Ve becomes ˙ Ve=−ρ1S2+ e4[g(x) + (c0+ σ) √ x2V ] − ξ0 |x5| n(x5) ˜ l20− ξ1 |x5| n(x5) ˜ l21 where g(x) =−px5x3− βpx5x2− (a10+ α + a4)x4− ˙x4d.
After we substitute the properly designed input V as
V = 1
c0√x2{−g(x) − η sgn(e4
)}
where sgn(·) is the sign function, then the time derivative ˙Ve can be reexpressed as ˙ Ve=−ρ1S2− ξ0 |x5| n(x5) ˜ l20− ξ1 |x5| n(x5) ˜ l12 − c0+ σ c0 e4[η sgn(e4) + c0 c0+ σ g(x, S)} ≤ −ρ1S2− ξ0 |x5| n(x5) ˜ l20− ξ1 |x5| n(x5) ˜ l12 − c0+ σ c0 [η− c0 c0+ σ |g(x, S)|} |e4| .
Now, if η is chosen to satisfy η≥ |g(x)| + k for some k> 0, then we have ˙ Ve≤ −ρ1S2− ξ0 |x5| n(x5) ˜ l02− ξ1 |x5| n(x5) ˜ l21− ρ2|e4|
for some ρ2> 0, which again implies boundedness of all internal
signals and convergence of the position tracking error by the argument similar to that in Section III and IV.
IV. ADAPTIVEBACKSTEPPINGCONTROLLERDESIGN
Remark: In the previous section, we have proposed an adap-tive controller for the LIMs, which will require acceleration sig-nals of the motor. Although this signal can be obtained through numerical differencing and digital filtering, it is more suscep-tible to noise. In order to avoid such problem, we propose the following nonlinear backstepping position controller without need of acceleration signal in this section.
Theorem 3: Consider a linear induction motor whose dynam-ics are governed by system (6) under the assumption (A.1)– (A.4). Given a friction observer (7) a smooth desired position
trajectory pd with pd, ˙pd, ¨pdand ¨pd being all bounded, then the following control input can achieve the control objective pr→ pd as (i.e., x6= pr will follow pd asymptotically) with the control input
Vqs= λd λ2 q+λ2d V c Vds= −λq λ2 q+λ2d V c, and V = √1 x2 [g1(x) + ˆΘT1Vr1− Kfz1− ρ2z2] with adaptation law given as
˙ˆ Θ =−Γ1z1Vr ˙ˆ Θ1=−Γ2z2Vr1 ˙ˆ ξ0=−z1ˆl0 ˙ˆ ξ1=−z1 |x5| n(x5) ˆl 1
and compensation terms of the friction observer specified as η0=−z1 η1=− |x5| n(x5) z1 whereΘ = [M b0b1b2]T, Vr= [ ˙vrv0rvr1vr2]T, Γ1, Γ2> 0, z1= S, z2= x4− α1, α1=−ρ1M z1+ 1 Kf ˆ ΘTVr− a Kf M e5+ ˆξ0ˆl0− ˆξ1 |x5| n(x5) ˆ l1 for someρ1, ρ2> 0, and
g1(x) = px3x5+ βx2x5+ (a1+ a4− a)x4 − ρ1(Kfx4+ κJ e6), ΘT1Vr1= (ρ1+ a Kf )ΘTVr+ 1 Kf ΘTV˙r = (ρ1+ a Kf )ΘTVr+ 1 Kf ΘTVr
with the parameter vectorΘas well as the known function vector Vr satisfying
ΘTV˙r= ΘTVr
keeping all the internal system signals bounded. Proof:
Step 1. Choose a different stabilizing function α1as follows:
α1=−ρ1M z1+ 1 Kf ˆ ΘTVr− a Kf M e5+ ˆξ0ˆl0− ˆξ1 |x5| n(x5) ˆ l1 (14) where ˆΘ denotes the online parameter estimate. Also, redefine the new error variables z1= S, z2= x4− α1.
Evaluate the time derivative of the Lyapunov-like function shown by V1= 1 2M z 2 1+ 1 2Γ1 ˜ ΘTΘ +˜ 1 2ξ˜ 2 0+ 1 2ξ˜ 2 1+ 1 2˜l 2 0+ 1 2˜l 2 1 (15)
along the solution trajectories to obtain ˙ V1=− ρ1KfM z12+ Kfz1z2 + ˜ΘT 1 Γ1 ˙˜ Θ + z1Vr + ˜ξ0(ξ˙˜0− z1ˆl0) + ˜ξ1 ˙˜ ξ1− z1 |x5| n(x5) ˆ l1 − ξ0˜l0(η0+ z1) + ξ1˜l1(η1+ |x5| n(x5) z1) − ξ0 |x5| n(x5) ˜ l20− ξ1 |x5| n(x5) ˜ l12 (16)
which motivates us to choose the adaptation law as ˙˜ Θ =Θ =˙ˆ −Γ1z1Vr ˙ˆ ξ0=−z1ˆl0 ˙ˆ ξ1=−z1 |x5| n(x5) ˆ l1 η0=−z1 η1=− |x5| n(x5) z1 (17)
for some proper positive adaptation gain Γ1. As a result, (16)
can be slightly simplified as: ˙ V1=−ρ1KfM z21+ Kfz1z2− ξ0 |x5| n(x5) ˜ l20− ξ1 |x5| n(x5) ˜ l21. (18)
Step 2. The time derivative of z2is now expressed as
˙z2= ˙x4− ˙α1=−g1(x)− ΘT1Vr1+√x2V (19)
where the functions involved above are as previously defined. Thus, we need to select a Lyapunov-like function and design V to render its time derivative nonpositive. We want to apply the augmented Lyapunov-like function candidate as
V2= V1+
1 2z
2
2 (20)
whose time derivative is found to be ˙ V2=−ρ1KfM z22+ Kfz1z2+ z2[−g1(x)− ΘT1Vr1+ √ x2V ] − ξ0 |x5| n(x5) ˜ l02− ξ1 |x5| n(x5) ˜ l21. (21)
The control law V should be able to cancel the indefinite term in (21). On the other hand, to deal with the unknown parameters Θ1, we will try to employ the current estimates ˆΘ1, i.e.,
V = √1 x2
[g1(x) + ˆΘ
T
1Vr1− Kfz1− ρ2z2]. (22)
From this resulting derivative ˙ V2=−ρ1KfM z12+ z2Θˆ1Vr1− ρ2z22 − ξ0 |x5| n(x5) ˜ l02− ξ1 |x5| n(x5) ˜ l21 (23)
in order to cancel the last term in (19), we modify the Lyapunov-like function as follows
V3= V2+ 1 2z 2 2+ 1 2Θ˜ T 1Θ˜1 (24)
and the time derivative of V3, hence, is
˙ V3=−ρ1KfM z12+ ˆΘT1(z2Vr1+ 1 Γ2 ˙˜ Θ1) − ξ0 |x5| n(x5) ˜ l20− ξ1 |x5| n(x5) ˜ l21. (25) Now, the term with ˜Θ1can be eliminated completely with the
update law
˙˜
Θ1=Θ˙ˆ1=−Γ2z2Vr1 (26)
for some positive adaptation gain Γ2,which thus yields
˙ V3=−ρ1KfM z12− ρ2z22− ξ0 |x5| n(x5) ˜ l20− ξ1 |x5| n(x5) ˜ l21 (27)
which guarantees boundedness of all parameter estimates ˆΘ, ˆ
Θ1and z1, z2, and z1∈ L2∩ L∞. To show boundedness of the
rest of states, we can rearrange the dynamical equations from system (6) as shown by ˙ X = ˙x˙x12 ˙x3 = −2a0 1 −2a0 4 2a2a23 a3 a2 −(a1+ a4) xx12 x3 + 2x4 √x 2V 0 px5x4 = AX + u
where A can be shown to be Hurwitz. After reviewing definitions of x3and V , repectively, we found that the first entry of u will
be bounded because x2 grows no slower than x3 if x3 does
grow unbounded [due to the second of (6)]. As a result, u is apparently bounded, and hence, X will be bounded. This then proves the boundedness of all the states. We note that ˙z1 is
also bounded, and hence, by Barbalat’s lemma we can conclude that
lim
l→∞z1→ 0, i.e., pr→ pd as t→ ∞. V. SIMULATIONRESULTS
In our simulation, the parameters of the linear induction mo-tor we use are according to Table I. All the results will be demonstrated in the following sections.
A. Results of Nonlinear Adaptive Controller
In order to confirm the performance of the controller design in Section III. We proposed several kinds of desired position trajectories, including exponential function, sinusoidal function, etc. All the results are shown in Figs. 2 and 3. It is found that both the observed states and the estimated parameters are
TABLE I
SPECIFICATIONS ANDPARAMETERS OF THEMOTOR
Fig. 2. Desired position trajectories, exponential function, sinusuidal function,
pd= 10(1− e−2t) cm
Fig. 3. pd= 10 sin(2t) cm.
shown bounded, and the position tracking errors do converge. The control gains and initial values are chosen in Table II.
TABLE II
CONTROLLERGAIN ANDINITIALVALUES
Fig. 4. pd= 10(1− e−2t) cm with 10-N disturbance.
B. Results of Adaptive Backstepping Controller Design In order to confirm the performance of the controller design in Section IV. We proposed several kinds of desired position trajectories, including exponential command with disturbance, and sinusoidal function, respectively. All the results are shown in Figs. 4 and 5. All these position tracking errors will approach to zero when time goes to infinity.
Remark: Compare the simulated results of the two kinds of approaches. The adaptive backstepping controller has better per-formance, e.g., lower position error, save power (effectively ex-cited current and flux), etc., because the adaptive backstepping controller does not require acceleration signals.
VI. EXPERIMENTALRESULTS
In this section, extensive experiments are done with a 4-ploe, 3-phase linear induction motor with a Y -connected primary, manufactured by NORMAG Co. Detailed parameters and spec-ification will be found in Table I and Fig. 6 shows a photograph of the constituted experimental equipment. The power stage of the motor driver uses a IGBT module, and the PWM drive sig-nals are generated by a 10 KHz SPWM with a 2.5µs dead-time protection circuit. The software we adopt in Simulink and Mat-lab, an excellent product of The MathWorks Inc., Natick, MA. In addition, we use Simu-Drive to combine the motor control card with the Simulink/Real Time Workshop.
Fig. 5. pd= 10 sin(2t) cm.
Fig. 6. Experimental equipment of LIM.
Fig. 7. pd= 10(1− e−2t)cm/.
Fig. 8. pd= 10 sin(2t)cm/sec.
Fig. 9. pd= 10(1− e−2t) cm.
A. Results of Nonlinear Adaptive Controller Design for Position Tracking
In this section, we make a series of experiments on the adap-tive controller design proposed in Section III. For the exponen-tial type of desired trajectory in Fig. 7, the error less than±1µm. In Fig. 8, we adopt sinusoidal trajectories, and the steady error is within±0.5. mm
Remark: For experiment of the exponential type of desired trajectory in Fig. 7, we found chattering phenomenon in the steady state. In order to eliminate such phenomenon, we propose a PI control scheme, which, like a low-pass filter, can cut off high frequency behavior so that the position tracking error is kept within±100µm. But eventually, such an arrangement will truly render the error less than±1µm.
Fig. 10. pd= 10 sin(2t)cm.
B. Results of Adaptive Backstepping Controller for Position Tracking
For the same position tracking problem, to validate the effec-tiveness of the adaptive backstepping controller, we provide sev-eral kinds of desired position trajectories as before. For the expo-nential trajectories in Fig. 9, we found chattering phenomenon in the steady-state error within±50µm. In Fig. 10, we adopt sinusoidal trajectories, the steady error is within±0.7 mm and the similar kind of error convergence can still be guaranteed.
Remark: From above experimental results, we found they are very consistent with simulated results. The adaptive backstep-ping approach has lower position error and lower power loss than nonlinear adaptive approach.
VII. CONCLUSION
In this paper, we have proposed two kinds of adaptive position tracking controller for the linear induction motor with sixth-order nonlinear dynamic model which is controlled by the primary voltage source. To cope with the uncertainty part of the linear induction motor, i.e., friction, end effect, payload, and inductance, we design our controller based on an appropriate nonlinear transformation. Due to inaccessibility to the flux in general, flux observers have been introduced. A stability analy-sis based on Lyapunov theory is performed to guarantee that the closed-loop system is stable. Finally, both the simulation and experimental results confirm the effectiveness of our control design.
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Chin-I Huang (S’01) was born in Hsinchu, Taiwan, R.O.C., in 1973. He received the M.S. degree in electrical engineering from Chung Yuan Christian University, Chung Li, Taiwan, in 2000. He is cur-rently working toward the Ph.D. degree in electrical engineering at National Taiwan University, Taipei, Taiwan.
His research interests include electrical drives, robotics, virtual reality, and nonlinear control theory and applications.
Li-Chen Fu (S’85–M’88–SM’02–F’04) received the B.S. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 1981, and the M.S. and Ph.D. de-grees from the University of California, Berkeley, in 1985 and 1987, respectively.
Since 1987, he has been a member of the faculty, and is currently a Professor in the Department of Elec-trical Engineering and Department of Computer Sci-ence and Information Engineering, National Taiwan University, where he is also currently the Secretary General of the University. His current research inter-ests include robotics, FMS scheduling, shop floor control, home automation, visual detection and tracking, e-commerce, and control theory and applications. He has been the Editor of the Journal of Control and Systems Technology and the Associate Editor of the prestigious control journal, Automatica. In 1999, he became the Editor-in-Chief of the Asian Journal of Control.
Prof. Fu is a Senior Member of the IEEE Robotics and Automation and IEEE Automatic Control Societies. From 2004 to 2005, he was an AdCom Member of the IEEE Robotics and Automation Society. He was the Program Chair of the IEEE International Conference on Robotics and Automation (ICRA) in 2003 and the Program Chair of the IEEE International Conference on Control Appli-cations (CCA) in 2004.
