Adaptive Control Approach for Speed Motion-Sensorless of Linear Induction
Motor Unknown Resistance and Payload
Chin-I Huang1* and Li-Chen, Fu2
1Department of Automation Engineering, National Formosa University, 2Department of Electrical Engineering, National Taiwan University
(e-mail: [email protected], [email protected])
Abstract: In this paper, we will propose a nonlinear adaptive controller for a linear induction motor to achieve speed tracking. A nonlinear transformation is proposed to facilitate controller design. In this controller, only the primary currents are assumed to be measured. The secondary flux and speed observers are designed to relax the need of flux and speed measurement. Besides, the very unique end effect of the linear induction motor is also considered and is well taken care of in our controller design. Stability analysis based on Lyapunov theory is also performed to guarantee that the controller design here is stable. Also, the computer simulations and experiments are done to demonstrate the performance of our various controller design.
Nomenclature
2 1 , , / , / , ( )/ / s r m m r s r m r r D L L L p Pπ β L D c L D a R L D βL R L τ = − = = = = + 2 r/ ,r 3 m r/ ,r 4 r/ r a =βR L a L R L a R L= = q-(d-) axis input stator voltage( )
qs ds
V V i iq d( ) q-(d-) axis input stator current Primary (secondary) resistance
( )
s r
R R L Ls( )r Primary (secondary) inductance
q-(d-) axis rotor flux ( )
qr dr
λ λ vr Linear speed of the primary
Position of the primary
r
p Mm Primary mass
Viscous friction coefficient
B Electromagnetic force
e F
Mechanical load force
P Lm Mutual inductance
Force constant( 3= PLmπ τ/2 )Lr f
K
1. INTRODUCTION
Sensorless control of rotary induction motor (RIM) or linear induction motor (LIM) drives is now receiving wide attention. The main reason is that the speed sensor spoils the ruggedness and simplicity of induction motors (IM). In a hostile environment, speed sensors cannot even be mounted. However, due to the high order and nonlinearity of the dynamics of an IM, estimate the states of speed and rotor flux without measurement becomes a challenging problem [23]. There are many works concerning the sensorless control problem, in which the vector control technique is utilized, but the research results there on sensorless vector control, e.g. [18-19], base their analysis mainly on the steady-state behavior and only supply approximate proofs. In [20], the speed observer is designed and analyzed based on the Lyapunov stability theory. Both observer and controller apply the direct adaptive control scheme to cope with the unknown rotor resistance. In [21-22], an indirect adaptive scheme instead is proposed.
Nowadays, LIMs are now widely used in many industrial applications including transportation, conveyor systems, actuators, material handling, pumping of liquid metal, and sliding door closers, etc., with satisfactory performance. The most obvious advantage of linear motor is that it has no gears and requires no mechanical rotary-to-linear converters. The linear electric motors can be classified into the following: D.C. motors, induction motors, synchronous motors and
stepping motors, etc. Among these, the LIM has many advantages such as simple structure replacement of the gear between motor and motion devices, reduction of mechanical losses and the size of motion devices, silence, high starting thrust force, and easy maintenance, repairing and replacement. In the early works, Yamamura has first discovered a particular phenomenon of the end effect on LIM [1]. A control method, decoupling the control of thrust and the attractive force of a 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]. Another method is proposed by removing the secondary [4].
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 rotary induction motor [5] to correct the deviation caused by the “end effect”. On the other hand, there is a thrust correction coefficient introduced by [6,7] to calculate an 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 end effect as shown in [8]. In another work [9], extra compensating-winding was proposed to compensate such problem.
Although the end effect is an important issue of the LIM control, but there are still many works in the literature without considering it, such as [10-16]. In this paper, we will take this as an important issue which can not be ignored.
2. PROBLEM FORMULATION
To formulate the dynamic model of a LIM as shown in Fig 2.1, we consider the following assumptions to simplify the analysis:
(A.1) Three phases are balanced;
(A.2) The magnetic circuit is unsaturated;
(A.3) It is without end effect (we will relax this assumption later in controller design);
(A.4) All parameters of the induction motor are known, except the secondary resistanceRr;
(A.5) Among all states, only the stator currents are measurable,
then the dynamics of the entire system can be rearranged into the following more compact form
1 2 1 2 3 4 3 4 ' ( ) q q q r d qs d d r q d ds q q q r d d d d r q m r f d q q d L i a i a pv cV i a i pv a cV a i a pv a i a pv M v K i i F λ β λ β λ λ λ λ λ λ λ λ λ λ = − + − + = − + + + = − + = − − = − − & & & & & (1)
In this paper, we try to design the speed and position controller for the LIM. All the parameters are assumed known except the payload. The only information about the payload is its structure, and we use a second-order equation to represent it, i.e., the payload is expressed in terms of
' 2
0 1 2
L L r L L r L r
F =M v& +b +b v +b v . (2) 3. OBSERVER DESIGN
3.1 Analysis of mechanical load and end effect
The fundamental difference between a rotary induction motor and a 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 a LIM, as the primary moves, the secondary is continuously replaced by a new material. This new material will tend to resist a sudden increase in flux penetration and only allow a gradual build up of the flux density in the air gap. 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 core as shown below:
Fig 3.1:Airgap average flux distribution due to end effect [5] 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 reduced 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 hence can assume that the end effect can be regarded as an external force which may be expressed as
' 0 n n r e r n b v M v ∞ = +
∑
& . In the paper, we will truncate the series into the first three terms.For a LIM, the end effect with the load force can be represented as a function of the speed
v
r, which can benormally simplified into the form
2 ' ' 0 2 ' 2 0 1 2 0 1 2 2 1 2 = n L n n r e r L e r r r L r L L r L r o L r r r F b v M v F M v b b v b v M v b b v b v M v b b v b v = = + + = + + + + + + + + + +
∑
& & & &In this paper, the mechanical load with end effect is assumed
in the aforementioned form as T
L r
F = ΘV with the unknown constant parameters Θ = ⎣⎡ML b0 b1 b2⎤⎦ , and a known
function vector T 0 1 2
r r r r r
V = ⎣⎡v& v v v ⎤⎦ . The joint mass
m L
M =M +M is therefore also unknown, which leads to the total mechanical load with motor itself as T
r F= Θ V , where 0 1 2 [ ] T M b b b Θ = .
To proceed further, we introduce some more assumption as shown below:
(A.6) 2 2
2 q d
x =λ +λ > 0,
(A.7) The desired speed should be a bounded smooth function with known first and second order time derivatives, and then further simplify the dynamics shown in (1) by introducing a nonlinear coordinate transformation given as follows: 2 2 1 2 2 2 3 4 5 q d q d q q d d q d d q r x i i x x i i x i i x v λ λ λ λ λ λ = + = + = + = − =
Initially, we adopt the stator voltage inputs as
2 2 2 2
( / ) , ( / )
ds q d q qs d d q
cV = − λ λ +λ V cV = λ λ +λ V [17],
with such transformation, then the dynamical equations shown in (1) can thus be transformed into the following dynamic model: 1 1 1 2 3 4 2 2 4 2 3 3 3 3 1 2 2 1 4 3 5 4 4 5 3 5 2 1 4 4 2 2 5 4 5 0 2 2 (2 / ) 2 2 ( ) ( ) - n f n n x a x a x x x V x a x a x x a x a x a a x px x x px x px x a a x x V Mx K x b x β = = − + + = − + = + − + + = − − − + + =
∑
& & & & & (3)To control the system (3) we develop the position controller to achieve the goal pr →pdas introduced in the following section.
3.2 Observer Design and Analysis
To facilitate observer design to be easy, the dynamics of a linear induction motor can be expressed as
1 2 3 1 2 3 2 2 o q m r q q r d r q qs o d m r d d r d r q ds r q r q m r q r d r d r d m r d r q L i L R i i v R V L i L R i i R v V L R L R i v L R L R i v α α λ λ α α λ α λ α λ λ α λ λ λ α λ = − − − + + = − − + + + = − + + = − + − & & & & (4)
where the meanings of all the variables are listed in the Nomenclature.
As has been mentioned earlier, the secondary resistanceRr is assumed unknown due to its fluctuation with temperature and the primary speed
v
r is not measurable since no speed sensor is mounted. To cope with this, we first rewrite Rr andr v as
r rn r
R =R + and θ vr =vrd+ θv
where Rrn is the nominal value of secondary resistance and rd
v is the desired primary speed. Apparently, θ and r θ v stand for the discrepancies from what we know about Rr and
r
v . For subsequent simplicity of the problem, we will assume θ is a constant and r θ , although it is not a constant, v will be slowly varying, i.e., θ is small enough. According to v the structure of the dynamics in (4), the observers are proposed as follows: 0 1 2 3 1 5 0 1 2 3 2 6 2 3 2 4 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ o q q m r q q r d r q qs o d d m r d d r d r q ds r q r q m r q r d r d r d m r d r q L i k i L R i i v R V u u L i k i L R i i R v V u u L R L R i v u L R L R i v u α α λ λ α α λ α λ α λ λ α λ λ λ α λ = − − − + + + + = − − + + + + + = − + + + = − + − + & % & % & & (5)
where Rˆr =Rrn+θˆr, vˆr =vrd +θˆv, k0 >0and u1~u6are the
auxiliary control signals to be designed later. Note that the symbol ^ above indicates an observed value, whereas the symbol ~ denotes the associated observation error. The design of observers can be detailed as in Theorem 1 shown below.
Theorem 1: Consider the linear induction motor whose
dynamics are governed by (4) under the assumptions (A1)– (A7). If the stator current observers and the secondary flux observers are designed as in (5), where
11 21 12 22 ˆ (ˆ ˆ ) ˆ (ˆ ˆ ), r R q d v v q d k i i k i i θ ϕ ϕ θ ϕ ϕ = − + = − + & % % & % %
then the observed speed and flux of the secondary will be driven to the actual speed and flux and the estimate of secondary resistance will also converge to the actual one subject to the control signals designed as follows:
2 ˆ ˆ 1 ry( 0 ) rr( 0 ) R v q q d d L L u = L i% −r +α L i% −γ 2 ˆ ˆ 2 ry( 0 ) rr( 0 ) R v d d q q L L u = L i% −r +α L i% −γ
[
]
[
]
0 0 3 0 2 4 0 2 1 5 1 2 2 1 6 3 2 4 ˆ ( ) ˆ ( ) sgn( ) ( ) ( ) sgn( ) ( ) ( ) , r r r r L q L r q r d L d L r d r q q L d L u k i R i v i u k i R i v i u i F t F t u i F t F t α α α α = − + − = − + − = + = + % % %% % % %% % %where the auxiliary signals ( , )γ γ are designed as follows: q d
2 2 ˆ ˆ 1 3 5 2 4 6 ( ) ( ) r r r r r r r r R v q q L q L d R v d d L d L q q o q d o d K i i K i i k i u u u k i u u u α ψ α ψ γ χ γ χ χ χ = + + = + − = − − − − = − − − − % % & % % % & % % % & % % & % %
for some constant k k k kR, , ,v o ψ > . 0
Proof: The proof can be referred to in [22].
Lemma 1: The upper bound on (ψ ψ% %q, d) can be explicitly derived by proper design of ˆ(ψ ψq,ˆd) as some positive constants δ and q δ . d
Proof: The proof can be referred to in [20]. Since the upper bound ( , )δ δ is acquired from Lemma 1, q d we can design the additional functions F t1( )≥δq R%r , F t2( )≥δd v%r , F t3( )≥δd R%r
andF t4( )≥δq v%r and devise the control signals as follows: 1 5 1 2 2 1 6 3 2 4 sgn( )[ ( ) ( )] sgn( )[ ( ) ( )] r r q L d L u i F t F t u i F t F t α α = + = + % % where 1, 0 sgn( ) , . 1, 0 if x x x R if x ≥ ⎧ =⎨− ∀ ∈ < ⎩
Seemingly,
u
5 andu
6 may not be realizable due to lack of knowledge of R%r and v% . However, in reality, one can r always choose higher gains F t ii( ), =1...4 , initially to establish close observations first, i.e., R%r and v% tend to r small residual values and then gradually reduce the gains. As the result of the above designs, V&o becomes2 2.
o o q o d
V& = −k i% −k i% (6)
it immediately follows that
2
( , ,i i% %q d ψ ψq, d, ,v R%r %r) is bounded and ( , )i i% %q d ∈L. (7) In order to confirm the above claim on the proper design of
5
u
andu
6, we soon show that R%r and v% will converge to r zero under appropriate conditions besides the high gain condition in the following. Before that, we first present the following working lemmas.Lemma 2: The states ( ,λ λ are bounded provided ( , )q d) i iq d are bounded in finite time.
Proof: The proof can be referred to in [20]. Now, we need to establish another result, which guarantees boundedness of vr presuming boundedness of ( ,λ λ in q d) Lemma 3.3 as follows.
Lemma 3: The primary speed vr is bounded if ( , )i iq d is bounded.
Proof: The proof can be referred to in [20]. Since R%r is bounded from the previous argument, it follows that ˆRr is also bounded. That will lead to the following Lemma 4 to guarantee the boundedness of the estimated rotor flux.
Lemma 4: The error dynamics of the secondary flux in (5) are rearranged in vector form as
2 3 4 2 ˆ ˆ ˆ 0 ˆ . ˆ ˆ ˆ ˆ 0 q r r q m r q r d m r d r r d L R R v i u L u i L R v R λ α λ λ α λ ⎡ ⎤ ⎡− ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢ ⎥ =⎢ ⎥⎢ ⎥+⎢ ⎥⎢ ⎥ ⎢ ⎥+ ⎢ ⎥ ⎢⎣− − ⎥⎣ ⎦⎦ ⎢⎣ ⎥⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦ &% % % &% (8)
Thus, if the estimate of the secondary resistance, ˆRr, is kept positive and bounded away from the origin, then ˆ ˆ( ,λ λ will q d) be bounded provided ( , )i iq d are bounded.
Proof: The proof can be referred to in [20]. From Lemma 3.4, ( ,λ λ% % are bounded, which together with q d) the boundedness of i%q and i%d from (7) readily implies that ( ,χ χ% % are also bounded. Then, that ( , )q d) γ γ are bounded. q d Now, ( , )i i& &% % are proved bounded as well since signals on the q d RHS are all bounded, which together with the property that is revealed in (7) implies
lim ( ) 0q
t→∞i t% = and lim ( ) 0t→∞i t%d =
via Barbalat’s Lemma. Now, we have proved the convergence of the observation errors of the stator currents under the premises of Lemma 3.4. The next problem is to prove if the estimation errors of the secondary resistance and the primary speed have the same convergence property.
To solve this problem, we first make the following definitions: T q d T r r q d o R o v o o I i i X R v diag L k L k L kψ L kψ ψ ψ ⎡ ⎤ = ⎣ ⎦ ⎡ ⎤ = ⎣ ⎦ ⎡ ⎤ Λ = ⎣ ⎦ % % % % %
and rearrange the error dynamics equations ( ) ( ) ( ) T I AI W t X B t X W t I CI = + + = −Λ + Λ &% % & % % (9) where 11 12 2 21 22 2 5 2 6 2 ˆ ˆ ˆ ˆ 0 , ( ) , 0 0 0 ( ) , . 0 0 o r r o o o o r o r o r r o o o o r o r q d o o r o r q d o o r o r k R v L T L L L L L L k v R L L L L L L L T u L L L L L u L L L L L A W t B t C ϕ ϕ α ϕ ϕ α ψ α ψ α ψ ψ ⎡ ⎤ ⎡− ⎤ − ⎢ ⎥ ⎢ ⎥ = =⎢ ⎥ ⎢ − ⎥ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦
Now, to prove X tends to zero asymptotically, we have to use the conclusion stated in Lemma 3.5 given below.
Lemma 3.5: The system with the form ( ) ( ) T I AI W t X X W t I = + = −Λ &% % & % (10)
is exponentially stable if and only if W t( )is persistently exciting (PE), i.e., there exist two positive constant and such that ( ) ( ) 0 0. t T T t W τ W τ τ εd I t + ≥ > ∀ ≥
∫
Proof: The proof can be referred to in [20]. By Lemma 5, we conclude that the system (10) is exponentially stable provided W t( )is PE. However, from (9) the asymptotic convergence property of X and I% may no
longer hold due to the forcing term added to the homogeneous system (10). To cope with this, we first note that the forcing term in (9) with respect to the homogeneous part in (10) will be ultimately in the order of the magnitude of
( )
B t under the premises of Lemma 3.4, since ( , )i i% %q d will then tend to zero. Thus, in our observer design, we gradually decrease the values of the control signals ( , )u u5 6 to zero as
I% tends to zero. Under such design, the equilibrium point (I%=0,X =0) of (9) remains asymptotically stable so that ( , ,R v% %r r ψ ψq, d) tends to zero asymptotically. In turn, this will confirm the hypothesis where the rotor resistance estimate
ˆ r
R is kept positive and bounded away from the origin in Lemma 3.4. Finally, the fact that R%r and v%r tend to zero asymptotically and that the control signals u3 and u4
converge to zero, λ%q and λ%d will also tend to zero asymptotically. Up to now, we can conclude the satisfactory convergence property provided PE condition holds and
( , )i i are bounded. q d
4. ADAPTIVE CONTROLLER DESIGN
The controller can overcome the unknown payload of the LIM under the reasonable assumptions. In subsection 4.1, we will propose a speed controller to achieve the objective of speed tracking. A nonlinear adaptive speed controller is proposed to deal with parameters understanding of the mutual inductance and in the uncertain inductance is also considered in our controller design subsection 4.2.
4.1 Non-adaptive speed controller design
Theorem 2 Consider a linear induction motor whose dynamics are governed by system (3) under the assumptions (A.1~A.7). Given Flux and Speed Observers (5) and a smooth desired speed trajectory vd with v vd, and&d v&&d being all bounded, then the following controller can achieve the control objective vr →vd (i.e., x5=vr will follow vd asymptotically) with the control input
2 2 d qs q d V V c λ λ λ = + , 2 2 q ds q d V V c λ λ λ − = + , and 1 4 4 3 5 4 2 4 2 1 [( ) d ] V a a x px x x e x ρ = + + +& − , where 2 4 0 1 5 2 5 1 5 1 ( ) d d f x b b x b x Mv e K ρ = + + + & −
with ρ ρ > , and 1, 2 0 e5=x5− , vd e4 =x4−x4d, while all the
internal signals are kept bounded.
Proof:
In order to show the boundedness the tracking errors e e4, 5,
we choose a Lyapunov like function Ve as shown below:
2 2 5 4 1[ ] 2 e V = Me +e , (11)
1 5 2 5 4 5 0 2 5 5 4 4 4 ( ) ( ) e f d d V K e x e b b x b x Me v e x x = − + + − + − & & & . If one design the auxiliary signal x4d as
2 4 0 1 5 2 5 1 5 1 ( ) d d f x b b x b x Mv e K ρ = + + + & − ,
then the time derivative of the function
V
e becomes2 1 5 4[ ( 1 4) 4 5 2 5 3 2 4 ] e d V e e a a x px x px x x V x ρ β = − + − + − − + − & & . Now, design the actual input
1 4 4 5 2 5 3 4 2 4 2 1 [( ) d ] V a a x px x px x x e x β ρ = + + + +& − ,
then it will lead to the result that
2 2
1 5 2 4 0
e
V& = −ρe −ρ e ≤ with ρ ρ1, 2 >0. (12)
Since V&ein (12) is nonpositive, we conclude all the error signals in Ve and, in particular, x5and x4d are bounded, which in turn implies that x4 and hence x&5 (from system (3))
are both bounded. By Lemma 2, we thus conclude that all the internal signals are kept bounded. Now, since Is is bounded, then Lemma 2 guarantees that of all signals ,x ii =1,..,5., are hence bounded.
By the power formula, Ps =a x x5 4 5 =3V Is s, which can be shown bounded from the above. We now show that Is will be bounded via argument of contradiction. Say, Is eventually grows unbounded, then Vs and, hence, V will diminish eventually. However, if Is does grow unbound, then it implies that V will tend topx x5 3/ x2 eventually. However,
from the dynamics of x2 in (3), we have x2 and x3 grow at the same rate, which readily says that V will also grow unbounded. This obviously leads to a contradiction and therefore Is is bounded.
Furthermore, we can show that x4d is bounded, and hence
4
e and e5 are also bounded, which implies the convergence of e4 and e5 due to Barbalat’s Lemma. Therefore, the control scheme with the properly designed input V will drive the output vr to the desired vdasymptotically.
Actually, the parameters M b b, , 0 1 and b2 in the system (3)
are unknown, and therefore the adaptive speed controller design will be proposed in the following subsection.
4.2 Nonlinear adaptive speed controller design
From the previous LIM dynamics, the parameters
1, , and f
a β c K depend on the mutual inductance, but as we know the mutual inductance is hard to identify due to its intricate structure and undesirable end effect. In particular,
2 1 2 10 0 2 / , s r r m r s r m r s r m R L R L L a a L L L L c c L L L α σ + = = + − = = + −
where α and σ are uncertainty terms of a1 and c, respectively. We rewrite the dynamic equations (3) as follows:
0 4 1 1 1 2 3 2 2 4 2 3 3 3 3 1 2 2 10 4 3 5 4 4 5 3 5 2 10 4 4 0 2 2 5 4 5 0 2( ) 2 2 2 2 ( ) ( ) ( ) n n n f f c x x a x a x V x x a x a x x a x a x a a x px x x px x px x a a x c x V b M x x x K K σ α β α σ = + = − + + = − + = + − + + + = − − − + + + + = −
∑
& & & & & (26)and design the control input
2 2 d qs q d V V c λ λ λ = + , 2 2 q ds q d V V c λ λ λ − = +
To facilitate subsequent investigation, we define several variables as follows: ˆ α α α%= − , ˆ, n and n f f b M d H K K β β β%= − = =
where ˆα is the estimate of α , β is the estimate of β . ˆ In order to show the boundedness of all the parameter estimates and the tracking errors e e4, 5 , we choose a Lyapunov like function Ve as shown below:
2 2 2 2 2 2 2 2 5 4 0 1 2 1 [ ] 2 e V = He +e +d% +d% +d% +H% +α% +β% (27) whose time derivative can be evaluated as follows.
If we design the parameter adaptive laws as
0 5 ˆd = −e & , 1 5 5 ˆd = −e x & , 2 2 5 5 ˆd = −e x & , 5 ˆ d H& = −e v& , 4 4 ˆ 4 5 2 ˆ e x , e px x α&= − β&= −
together with the proper design of x4d as
2 4 5 1 5 0 ˆ ˆ [ n ] d n d n x d x Hv ρe = =
∑
+ & − ,then the time derivative of the Lyapunov-like function V e becomes
2
1 5 4[ ( ) ( 0 ) 2 ] e
V& = −ρe +e g x + c +σ x V
where g x( )= −px x5 3−βpx x5 2−(a10+ +α a x4) 4− &x4d . After we substitute the properly designed input
V
as:4 0 2 1 { ( ) sgn( )} V g x e c x η = − −
where sgn( )⋅ is the sign function, then the time derivative V&e can be re-expressed as 0 0 1 4 4 0 0 2 0 0 1 5 4 0 0 ( ) [ sgn( ) ( ) ( )} ( )[ ( ) ( ) } e c c V e e g x c c c c e g x e c c σ ρ η σ σ ρ η σ + = − − + + + ≤ − − − + &
Now, if η is chosen to satisfy η≥ g x( ) +kfor some k>0, then we have
2
1 5 2 4
e
V& ≤ −ρe −ρ e
for some ρ >0, which again implies boundedness of all 2 internal signals and convergence of the speed tracking error by the argument similar to that in subsection 4.2.
5. EXPERIMENTAL RESULTS
Table 5.1: Specification and parameter of the motor
Specification Parameters normal value
3 Phase (Y-connected) Rated Power 1HP Rated Air gap 0.125in Rated Current 5A Rated Voltage 240V Rated Poles 4 Pitch 46.5mm Secondary length 82cm s
R
= 13.2Ω
rR
= 11.78Ω
sL
= 0.42H rL
= 0.42H mL
= 0.4H M = 4.775 Kg B = 53Kg/secThe experiment are done with a 4-ploe, 3-phase LIM with a Y-connected primary, and is manufactured by NORMAG Co.. Detailed parameters and specification will be found in Table 5.1. The power stage of the motor driver uses a IGBT module, and the PWM drive signals are generated by a 10 KHz SPWM with a 2.5
μ
s
dead-time protection circuit. For the exponential desired speed trajectory in Figure 5.1, the speed error is nearly limited within ±10% of the command magnitude. In Figure 5.2, we adopt external disturbance, a 1.6Kg book, at about 4sec and remove it at 8sec, we see that the performance is also good.7. CONCLUSIONS
In this paper, we have proposed an adaptive speed sensorless controller for the LIM. In addition, to cope with situation where some states, such as flux and primary speed, are not available and the uncertainty part of the LIM, i.e., secondary resistance, end effect, payload, and inductance, we first construct the state observers and the secondary resistance estimator to provide the asymptotic accurate value of the states and the parameters. Then, we design our controller based on an appropriate nonlinear transformation. Stability analysis based on Lyapunov theory is performed to guarantee the controller design is stable. Finally, the experimental results confirm the effectiveness of our control design.
REFERENCES
[1] S. Yamamura, Theory of Linear Induction Motors, John Wiley & Sons, 1972.
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