* Correspondence to: Li-Chen Fu, Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan,
R.O.C.
Non-linear sensorless indirect adaptive speed control of
induction motor with unknown rotor resistance and load
Yu-Chao Lin, Li-Chen Fu * and Chin-Yu Tsai Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C.
Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan, R.O.C.
SUMMARY
In this paper, a nonlinear indirect adaptive sensorless speed and #ux tracking controller for induction motors is proposed. In the controller, only the stator currents are assumed to be measurable. Flux observers and rotor speed estimator are designed to relax the need to #ux and speed measurement. Besides, two estimators are also designed to overcome drifting problem of the rotor resistance and unknown load torque. Rigorous stability analysis based on Lyapunov theory is also performed to guarantee that the controller designed here is stable. Computer simulations and experiments are done to demonstrate the performance of our design. Copyright 2000 John Wiley & Sons, Ltd.
KEY WORDS: indirect adaptive control; induction motor; sensorless
1. INTRODUCTION
1.1. Motivation and related researches
The induction motor is a coupled system with highly non-linear dynamics. In its "ve states, only stator currents are measurable in sensorless control problem making it di$cult to be solved by the engineers. Besides, drift of system parameter drifting is also another problem to cope with. Due to the highly non-linear dynamics, the control of induction motors becomes a popular problem for investigating the relevant non-linear theory and adaptive control over the past decade. In the early years, all system states are assumed to be measurable and parameters are assumed to be known. Under these assumptions, techniques such as classical "eld orientation [1] and input}output linearization [2, 3] are utilized to design the controller. Especially, the controller in Reference [3] was adaptive with respect to both the load and the rotor resistances. Since in these schemes the #ux sensors are required, this makes it impractical for implementation. Therefore, #ux observers are then designed to relax the need of #ux measurement [4, 5]. These #ux observers are designed under the assumption that the rotor resistance is known. Generally, the value of the rotor resistance may drift due to the heating of the rotor, and observers proposed
above are sensitive to such resistance value. Therefore, e!orts have been made to design estimators of the rotor resistance and/or other system parameters [6, 7]. Following these researches, further e!orts were then to design controllers and #ux observers which are adaptive with respect to both system parameters and/or the load [8}11].
All the schemes above require speed measurement. However, the speed measuring device is rather costly relative to the price of an induction motor in general. Besides, the measured signals are usually noisy and di$cult to deal with. Therefore, controllers that do not require speed measurement are obviously preferable for practical implementation. The problem which designs a speed controller of induction motors without rotational transducers then becomes widely interesting and is usually referred to as sensorless control problem. The "rst paper regarding sensorless control problem can be traced back to 1981. In that paper, the vector control technique is utilized. After that many research results on sensorless vector control have been proposed [12, 13], in which analyses are mainly based on the steady-state behaviour and only rough proofs are supplied. On the other hand, in References [14, 15], many researches on sensorless control were discussed and compared, including vector control and other modern control theory, such as robust and model reference adaptive system (MRAS). Here, we follow this trend to design a fully non-linear adaptive sensorless speed controller for induction motors based on #ux observer.
1.2. Organization of this paper
In this paper, an introduction of induction motors and related researches are discussed in Section 1. In Section 2, the mathematical model of an induction motor will be presented. The main part of this paper is Section 3 in which observers and controller are designed and proved in detail. The computer simulations and experimental results are presented in Section 4. Finally, we will make some conclusions in Section 5.
2. PRELIMINARIES AND PROBLEM FORMULATION In this chapter, we introduce the induction motor model and our control problem.
2.1. Motor model
If the induction motor never goes into the saturation region, and the air-gap MMF is sinusoidal, then it can be characterized by the following dynamic equations [16]:
¸IQ ?"!MRI?!bI?#Rt?#but@#b<? ¸IQ @"!MRI@!bI@!but?#Rt@#b<@ ¸tQ?"!Rt?#M RI?!but@
¸tQ@"!Rt@#M RI@#but?
¹"k2 (t?I@!t@I?) (1)
where ¸, b, b, b are constants de"ned in the nomenclature. The mathematical model listed above is referred to the well-known stator "xed reference frame. The "ve equations derived above
can fully describe the dynamics of the electrical part of the motor-load system. Meanwhile, we can derive the dynamics of the mechanical part through some analysis based on Newton's law
JuR#Du#¹*"¹ (2)
where J'0 is the rotor inertia, D'0 is the damping coe$cient and ¹* is the load torque. After successfully deriving the dynamics of the whole system, we begin to introduce our control problems and objects.
2.2. Problem formulation
In this paper, we try to design a speed controller for induction motors, which does not rely on any shaft sensor outputs and only measures the stator currents. All the parameters of the motor except the rotor resistance and the load torque are assumed to be known. The only knowledge about the rotor resistance we have is its nominal value. Given this information about the motor, for any speed and #ux commands which are second-order continuously di!erentiable bounded functions, this controller should be able to drive the motor to track the commands.
3. INDUCTION MOTOR CONTROL
Before a thorough investigation on the observers and controllers, several assumptions will be presented below to make the problem more precise.
Assumption
(A1) All parameters of the motor are known, except the rotor resistance R. (A2) The stator currents are measurable.
(A3) The load torque ¹* is an unknown constant.
(A4) The desired rotor speed should be a bounded smooth function with known "rst and second-order time derivatives.
Assumption (A1) comes from the uncertainty of the rotor resistance. Its value may vary up to 100 percent during the operation due to the change in the temperature and/or working condition. Assumption (A2) is a realistic consideration in the actual application of the induction motors. The reason why we only measure the stator currents is stated in the introduction of this thesis. Finally, assumptions (A3) and (A4) are also quite realistic in practical applications.
Control objective: Given the desired rotor speed trajectoryu(t), our control goal is to design
control laws such that the rotor speedu and the rotor #ux ( can track their respective desired trajectories asymptotically in time with all internal signals being bounded, subject to assumptions (A1)}(A4).
3.1. Observer design
In order to measure the rotor #uxes, sensors have to be inserted into the air gap. This requires re-designing or re-assembling of the motor. Obviously, this contradicts to the advantages of the
induction motors, i.e. low cost and easy fabrication. Therefore, here we design observers to observe the rotor #ux and relax the need to #ux sensors. Besides, as mentioned in the introduction of this paper, the rotor speed is not measured, either. This leads us to construct a speed observer as well. In addition to the rotor #ux observers and the rotor speed observer, we build observers to observe the stator currents despite they being are readily measurable signals. These additional current observers provide us extra information useful for designing our non-linear adaptive subsequently. We then have "ve observers in total in our system to reconstruct the "ve system states. Moreover, since variation of the rotor resistance may cause undesirable deterioration of performance, a rotor resistance estimator is built as well. These observers and estimators will be discussed in the following contents.
The system model is shown in Section 2. We set R"R#h, u"u#h where h stands for (unknown) di!erence between the actual resistance value and its nominal value, and h denotes the speed tracking error. For easy reference, we de"ne the notations for the observed values and the observation errors as II ?"I?!IK?, II@"I@!IK@, tI?"t?!tK?, tI@"t@!tK@,
RI "R!RK, uJ"u!uL where the symbol ) denotes that it is an observed value and the
symbol&
denotes an observation error. According to the structure of the dynamics in (1), the observers are proposed as in Reference [8]:
¸IKQ ?"kII?!(MRK#b)I?#RKtK #buLtK@#b<?#v#v ¸IKQ @"kII@!(MRK#b)I@!buLtK?#RKtK@#b<@#v#v ¸tKQ?"!RK tK?#MRKI?!buLtK@#v
¸tKQ@"!RK tK@#MRKI@#buLtK?#v (3) where RK "RL#hK, uL"u#hK and the constant k'0 is a control gain. Finally, the speed observer and the rotor resistance estimator, which are not shown here, will be presented toward the end of this section based on Lyapunov stability analysis.
3.2. Analysis of the observer
With the observers designed above, in this section we will proceed to analyse their stability. First, we derive the observation error dynamics by subtracting (3) from (1) to yield
¸IIQ ?"!kII?#RtI?#RItK?!MRII?#butI@#buJtI@!v!v ¸IIQ @"!kII@#RtI@#RItK@!MRII@!butI?!buJtI?!v!v ¸tIQ?"!RtI?!RItK?!butI@!buJtK@#MRII?!v
¸tIQ@"!RtI@!RItK@#butK?#buJtK?#MRII@!v (4) After a careful review of the above equations, an interesting property of these observation errors can be found, i.e. there are a lot of common terms in the error dynamics of the stator currents and in those of the rotor #uxes, but they are with opposite signs. In fact, this phenomenon also appears in the system model (1). In order to utilize this property to cancel the unmeasurable
terms, we design two auxiliary observation errors [8]:
ZI ?"¸II?#¸tI?
ZI @"¸II@#¸tI@ (5)
Another interesting characteristic is studied here. Although the auxiliary observation errors, ZI ? and ZI @, are not measurable (because they are composed of II and tI, and tI is not measurable), their "rst-order time derivatives are measurable. This special characteristic comes from the cancella-tion of the common terms mencancella-tioned above.
Two additional error signalsg? and g@ are then de"ned as follows: g?"ZI?!f?
g@"ZI@!f@ (6)
wheref? and f@ are auxiliary control signals. Again, g? and g@ are unmeasurable errors but with measurable "rst-order time derivatives. It will be clear later that the use of these two signals will bring help to cope with the coupling terms of the unmeasurable signals, such asut.
Motivated by how the coupling terms can be cancelled, we design the observer inputs v and v to be
v"!R¸K (¸II?!f?)!buL¸ (¸II@!f@)
v"!R¸K (¸II@!f@)#buL¸ (¸II?!f?) (7) by (5) and (6), which can be further rearranged into
v"R¸K (¸II?!f?)#buL¸ (¸II@!f@)!R¸g?!bu¸ g@#RtI?#butI@
v"R¸K (¸II!f)!buJ¸ (¸II?!f?)!R¸g@#bu¸ g?#RtI@!butI? (8) Substituting (8) into the "rst two equations of (4), we get the error dynamics about current observation as ¸IIQ ?"!kII?#R¸K g?#buL¸ g@ #RI
tK?!MI?!¸ ¸II ?# 1 ¸f?# 1 ¸g? #uJbtK@!b¸ ¸ II @#b¸f@#b¸g@ ! v¸IIQ @"!kII@#R¸K g@!buL¸ g? #RI
tK@!MI@!¸ ¸II @# 1 ¸f@# 1 ¸g@ #uJ!btK?#b¸ ¸ II ?!b¸f?!b¸g? ! v (9) and we de"ne )"tK?!MI?!¸¸II ?#1 ¸f?# 1 ¸g? )"tK@!MI@!¸¸II @#1 ¸f@# 1 ¸g@ )"btK@!b¸¸ II @#b ¸ f@#b¸g@ )"!btK?#b¸¸ II ?!b ¸f?!b¸g? (10) After investigation on the error dynamics of the stator current observer, we now turn to study of the error dynamics of the rotor #ux observer. It is noteworthy that the rotor #ux error dynamics will not be analysed directly, but instead the analysis on ZI ? and ZI@ will be made.Let v and v be designed as
v"!kII?#¸¸ (RK II?#buLII@)
v"!kII@#¸¸ (RK II@!buLII?) (11) So far, we have derived the error dynamics of the observed stator current and the auxiliary errors. Now, we are ready to perform the Lyapunov stability analysis to examine the stability condition of these observers. First, we de"ne a Lyapunov function candidate as
<" II?#II@# kLg?# kEg@# kS u# k0RIP where control gains kE, kS and k0 should satisfy
Taking the time derivative of < along the system trajectories, we obtain <Q "!kII?!kII@!II?v!II@v# RI
¸¸(II ?g?#II@gJ@)#buJ¸¸(II @g?#II?gJ@) #g?
RK¸¸II ?!¸¸buLI@#kLgR?
#g@
RK¸¸II @#buL¸¸II ?#kLgR@
#RI¸[))II ?#))IK@#k0¸RIQ]#uJ¸[))II ?#))IK@#kS¸uJQ] (12) via de"nitions of four additional functions))
,)) , )) and)) : )) "tK?!MI?!¸ ¸II ?# 1 ¸f?# 1 ¸g(? )) "tK@!MI@!¸ ¸II @# 1 ¸f@# 1 ¸g(@ )) "btK@!b¸ ¸ II @#b¸ f@#b¸gL@ )K"!btK?#b¸¸ II ?!b ¸f?!b¸gL? (13) Note that)) ,)) ,)) and))
are combination of all measurable errors, observed values and auxiliary control signals, and hence are known functions. Here, to complete the "nal analysis an additional assumption is required.
(A5) The speed tracking error and the change of the rotor resistance from its nominal value, i.e. h and h, varies slowly so that its "rst-order time derivative can be negligible, i.e. hQ+0, hQ+0
Again, ifgR?, gR@, RKQ and u( are designed properly, those terms in the last four brackets in (12) can be cancelled out, which then leads to the design of those signal as
gR?"!kE1
¸¸RK II ?!bu( ¸¸II @ gR@"!kE1 ¸¸RK II @#buL ¸¸II ? RIQ "!1 k0 )) ¸ II ?# )) ¸ II @ uJQ"!kS1 )) ¸ II ?# )) ¸ II @ (14)wheregL? and gL@ are estimated values of g? and g@, respectively, satisfying gLQ?"g?, g(?(0)"0
gLQ@"g@, gL@(0)"0
Evidently, if the estimatorsgL? and gL@ are designed as above, the estimation errors can be easily found to be
gJ?"g?(0) gJ@"g@(0)
By substituting the rotor resistance estimator and the rotor speed observer, we can re-assess <Q as <Q "!kII?!kII@!II?v!II@v# RI
¸¸ (II ?gJ?#II@gJ(@)#buJ¸¸(II @gJ?#II@gJ(@) )!
kII?!kII@!II?v!II@v#¸¸1 " II?""RIgJ?"#¸¸b " II?""uJgJ@"
# 1
¸¸" II@""RIgJ@"#¸¸b " II@""uJgJ?"
Clearly, the upper bound on <Q on the right-hand side (RHS) of Equation (15) still does not have de"nite negative sign. To cope with that, we incorporate the variable structure design (VSD) into our controller. Let us "rst assume
gJ?"g?(0))d?
gJ@"g@(0))d@ (15)
and derive the four functions FG(t), i"1, 2, 3, 4 which satisfy
F(t)*d? " RIP", F(t)*d@ " uJ" F(t)*d@ " RIP", F(t)*d? " uJ"
It is not hard to choose the functions F(t) and F(t), whereas in Appendix A we show that the bounds F(t) and F(t) are calculable. Accordingly, we design the four control signal v and v as
v"¸1 sgn (II ?) (F(t)#bF(t))
where sgn( ) ) is the sign function de"ned as
sgn(x)"
1 if x*00 if x(0 ∀x3R
From (A5) and (14), we can in turn design an observer for the rotor speed tracking error, an estimator for rotor resistance change, and the auxiliary signals as
hKQ"k01
)) ¸ II ?# )) ¸ II @, hK(0)*0 hKQ"kS1 )) ¸ II ?# )) ¸ II @ fQ?"ZIQ?!gR? fQ@"ZIQ@!gR@ (17) As a result, <Q becomes <Q )!k ¸(II ?#II@)It follows that II ?, II@, RI, uJ, g? and g@ are all bounded. We now impose that hK(t)#RL*
RL/ 2'0, ∀t*0, by means of incorporating a projection algorithm into the adaptation law
given by the "rst equation in (17). Denoting by hK(t) the modi"cation of hK(t) given by the projection algorithm,hK(t) is chosen such that
[(RL#h)!(RL#hK(t))]*[(RL#h)!(RL#hK(t))] (18) which guarantees that the value of < does not increase when hK is replaced by hK given by the projection algorithm. Since R"RL#h is positive, a simple projection algorithm which guarantees that (18) is satis"ed is obtained by reinitializing the estimatehK(t)
hK(t)"hK(t)"!2RL#e if hK(t))RL withe'0 a positive constant such that
e)2 (RL#h)"2R
Let us for the time being assume that (I?, I@) are bounded in "nite time, i.e. (I?, I@3¸C), then from Appendix A, (t?, t@, u) are also bounded in "nite time. Since (t?, t@, I?, I, u) are bounded, if
RK "RL#hK*RL/2'0, it follows that tK? and tK@ are also bounded according to (3), and
therefore ZI ? and ZI@ are bounded from (5). This implies that f? and f@ are bounded and that v, v,
IIQ @ are ¸ signals, since
R (II ?(q)#II@(q)) dq"k ¸(<(t)!<(0)) )k ¸ <(0), ∀t*0 By Barbalat's lemma, it follows thatlim
R II ?(t)"0, limR II @(t)"0, (19)
We can rewrite (9) and (14) in matrix from as
IIQ "AII#=2(t)X#B(t)
XQ "!"=(t)II#"CII (20) in which II "[II?, II@]2, X"[RI, uJ, g?, g@], ""diag[1/kE, 1/kE, 1/k0, 1/kS] and
A"
!k ¸ 0 0 !k ¸ =2(t)" ) ¸ )¸ ¸¸R buL¸¸ ) ¸ )¸ !buL ¸¸ RK ¸¸ B(t)"v v C2" gJ? ¸¸ ¸¸gJ@ 0 0 gJ@ ¸¸ ¸¸gJ? 0 0 (21)To analyse the asymptotic condition of system (20), we introduce the following important working lemma.
¸emma 1.
If there exist two positive constants ¹ ande such that
R>2 Rthen the equilibrium state X"0, II "0 of the following system:
IIQ "A(t) II#=2(t)X
XQ "!=(t)II (23)
are uniformly asymptotically stable. The proof is shown in Appendix B.
We can get II asymptotically tending to zero from (19), and the matrix B(t) in (20) is bounded. Motivated by Lemma 1, when II is small, we supply small control signals (v, v) in matrix B(t), but if II tend close to zero, we set v, v zero. As a result, we can make Equations (20) and (23) virtually the same. Then, the equilibrium point (II "0, X"0) remains asymptotically stable for such kind of hybrid set-up according to Lemma 1, namely RI , uJ, g?, g@, II? and II@ tend asymptotically to zero. By now, let us rewrite the last two terms in Equation (4).
tKQ? tKQ@" !R/¸ bu/¸ !bu/¸ !R/¸ t?t@ #RI (MI?!tK?)/¸ (MI@!tK@)/¸#bu !t)@ t)? #vvwhere v, v are the functions of II?, II@ as shown in (11). Since I? and I@ are assumed bounded in "nite time and the fact thattK? and tK@ are bounded (to be clear in the following remark), we can conclude thattI? and tI@ also tend asymptotically to zero. In conclusions, Equation (3) with (v, v,
v, v, v, v) given in (7), (11), and (16) and hK, hK, g? and g@ updated according to (17) and (14)
constitute a convergent adaptive observer provided that (22) is satis"ed.
Remark
According to (3), we can rewrite the dynamics of the observed #ux in the form
tKQ? tKQ@" !RK buL !buL !RK tK?tK@ #MRK 0 0 MRK I?I@#vvBecause (I?, I@, v, v) are bounded, we only need to check the convergence of the form as follows:
tKQ? tKQ@" !RK buL !buL !RK tK?tK@ Consider that <tK" (tK?#tK@) then <QtK"tK?tKQ?#tK@tKQ@ "!RK (tK?#tKQ@))0which implies the exponential stability condition of the homogeneous system and hence the bounded input bounded state (BIBS) property of the forced (non-homogeneous) system.
3.3. Controller design
In the previous section, observers and parameter estimator are designed and their stability properties are analysed based on Lyapunov stability theory. We then use the observed or estimated values in designing a proper controller as if they were true values. In this section, we propose a state feedback control for induction motor system which is adaptive with respect to the unknown constant load torque ¹*, assuming all states (I?, I@, t?, t@) are measurable and the rotor resistance (R) is a known constant. The controller objective is to guarantee asymptotical zero convergence of rotor speed tracking error and rotor #ux tracking error. The dynamics model of the induction motor is expressed as (1). And, we de"ne the speed tacking error, #ux tacking error and load torque estimation error as eS"u!u, e("t?#tB!(B, e2"¹*!¹K* where ("t?#t@, ("t#t, u and ( are the reference signals and ¹K* is a time-varying estimation of ¹*.
With the tracking errors and load torque estimation error de"ned above, their dynamics can be derived from (1) as
JeR S"k2(t?I@!t@I?)!Du!¹*!JuR
¸eR("!2R(#2MR(t?I?#t@I@)!2¸( (24) By designing the input signal I? and I@, we "rst consider a Lyapunov function candidate de"ned as
<" (rJeS#r¸et#
re2)
where gains r, r and r are positive.
Taking the time derivative of <, we then get its time derivative according to (24) as <Q "reS(k2(t?I@!t@I?)!Du!¹*!JuR)
#
re((!2R(#2MR(t?I?#t@I@)!2¸()#re2eR2 (25)
If we let I? and I@ be chosen such that
k2 (t?I@!t@I?)!Du!JuR!JuR"!KSeS#¹K*
!
2R(#2MR(t?I?#t@I@)!2¸("!K(e( (26)
with KS'0 and K('
0. From (26), the choice of the currents I? and I@ are equivalent to the following:
I?"(1 ()t?!)t@)
where
)"2R(#2¸(!K(e(
2MR
)"DuB#JuR!(KS!D) eS#¹K*k2 (28) But, the currents are not directly available as control inputs. They are related to the actual controls (stator voltages) by the "rst two equations in (1). To design the actual control inputs <? and <@, we thus let currents in (27) to be the desired currents.
I?"(1 ()t?!)t@)
I@"(1 ()t?!)t@) (29) De"ne the current errors, namely, the di!erences between the actual currents and their desired values, as follows:
e?"I?!I? e@"I@!I@
Then, the error dynamics involving (eS, e(, e?, e@) can be summerized as follows:
JeR S"k2(t?e@!t@e?)!KSeS!e2
¸eR("2MR(t?e?#t@e@)!K(e(
¸eR ?"!MRI?!bI?#Rt?#but@#b<?!¸IQ?
¸eR @"!MRI@!bI@!but #Rt@#b<@!¸IQ@ (30) The derivatives of the reference currents computed from (29) are found as
IQ ?")!¸t@(KS!D)
(k2J e2
IQ @")#¸t?(KS!D)
where )"¸
dtd (t?)#(t? )!¸ d dt (t@)!(t@ ) )"¸dtd (t?)#(t? )#¸ d dt (t@)#(t@ ) (32) in which d dt(t?"tQ?(!2t?(( d dt(t@"tQ@(!2t@(( )"4R((#2¸((#2¸((G!K(eR( 2MR )"(KS!D)k2J e2!) (33) with )"!DuR!JuK#[(KS!D)/J] (k2(t?e@!t@e?)!KSeS)!¹KQ*k2Only the expression of) depends on the unknown signal e2, but others are all made up of known signal. Finally, we chose the control input terms <? and <@ as
<?"b1 (MRI?#bI?!Rt?!but@#)!K?e?#u?)
<@"b1 (MRI@#bI@!Rt@#but?#)!K@e@#u@) (34) where K?, K@*0, and u? and u@ are to be designed later. Given such design arrangement, the error dynamics of I? and I@ become
¸eR ?"!K?e?#u?#¸t@(k2J(KS!D)e2
3.4. Analysis of the controller
Note that the dynamics of the state tracking error (e?, e@, e(, eS) have been explicitly derived in (30)
and (35), where the desired signals( and u, are a priori known, and the desired currents are speci"ed in (29). Then, under these circumstance, we can proceed to check the performance of the controller.
The analysis is also based on the Lyapunov stability theory. First, we de"ne a Lyapunov function candidate for the controller as
<" (rJeS#r¸e(#
re2#r¸ (e?#e@)#ee2eS) (36) for some positive constants r, r, r and r and for some su$ciently small 3('0) such that the quadratic function
<(eS, e2)" (r JeS#re2#ee2eS)
is positive de"nite. So, the Lyapunov candidate function < is indeed positive de"nite. Using the state tracking error dynamics (30) and (35), we can obtain the time derivative of < as
<Q "!rKSeS!rK(e(! rK?e?!rK@e@ # e2
r!eS#re? (KS!D)(k2J¸t@ ! re@ (KS!D)(k2J¸t? # reR2#eeRS # e? (!reSk2t@#2re(MRt?#ru?) #e@(reS k2t@#2re(MRt@#ru@)#eeR2eS (37)
In order to make <Q )0, what is left to be decided is eR2 and <?, <@. After a careful choice, we follow the payload estimation algorithm for eR 2 as
¹KQ *"!eR2 "1
r
!reS#re?¸t@(k2J(KS!D)!re@¸t?(k2J(KS!D)#e
which updates <Q into a slightly di!erent form as <Q "!rKSeS!rK(e(!rK?e?!rK@e@!e Je2# e r
eKSJ #reS # e?ru?!reSk2t@#2re(MRt?!ereS r ¸t@(KS!D)(k2J #rJeeSek2t@ # e@ru@#reSk2t?#2re(MRt@#ereS r ¸t?(KS!D)(k2J !ereSrJ ek2t? (39) By examining the structure of (39) and completing the stability analysis, we now design the auxiliary control terms u? and u@ inside the control input terms in (34) to cancel the sign-inde"nite terms in (39). If u? and u@ are chosen asu?"r1
reSk2t@!2re(MRt?#reeSr ¸t@(k2J(KS!D)
!eeSrJek2t@ u@"r1 !reSk2t?!2re(MRt@!reeS
r ¸t@(k2J(KS!D)
#eeSrJek2t? (40)Then the control laws given by (34) and (40) together with the payload update law given by (38) will readily imply
<Q "!
rKS!re eKSJ #err
eS!rK(e(!rK?e?!rK@e@!e
J e2 (41)
Lete be su$ciently small such that
rKS!re eKSJ #err'0then there exists KT'0 such that (41) implies the following: <Q )!KT#[eU, e(
, e?, e@, e2]2#
Consequently, we know the signals (eS, e(, e?, e@, e2) in the closed-loop system approach to zero
3.5. Stability of the overall system
According to the above observer and control design, the following theorems will formally state all the detailed design procedure seen so far. Also, the consequences of the developed mechanism are clearly summarized and all the proofs can in fact be referenced from the previous sections.
¹heorem 1
Consider an induction motor whose dynamics is governed by (1) under assumptions (A1)}(A5) If the stator current observers and the rotor #ux observers are designed as
¸IKQ ?"kII?!(MRK#b) I?#RKtK?#buLtK@#b<?#v ¸IKQ @"kII@!(MRK#b) I@!buLtK?#RKtK@#b<@#v ¸tKQ ?"!RK tK?!MRKI?!buLtK@#v ¸tKQ @"!RK tK@#MRKI@#buLtK?#v where RK "RL#hK uL"u#hK
then the observed speed and #uxes of the rotoruL will be driven to approach to actual speed u and #uxes (t?, t@), and the estimated rotor resistance will be also driven to approach to actual rotor resistance by the following primary law:
v"!R¸K (¸II?!f?)!buL¸ (¸II@!f@)
v"!R¸K (¸II@!f@)#buL¸ (¸II?!f?)
v"!kII?#¸¸(RK II?#buLII@)
v"!kII@#¸¸(RK II@!buLII?) And the auxiliary control law is as follows:
gR?"!kE1
¸¸RK II ?!buL¸¸II @
gR@"!kE1 ¸¸RK II @!buLwith kE'0, and the adaptation laws are designed as hKQ"
1 k0 )¸ II ?#)¸ II @ e if hK(t)'!RL if hK(t))!RL hKQ"kS1 )¸ II ?#) ¸ II @ for some constantskS, k0, e'0
If we uset?, t@, u, and R estimated values provided by adaptive observer in Theorem 1, we obtain the adaptive feedback control theorem as follows:
¹heorem 2
Consider an induction motor whose dynamics are governed by (1) under assumptions (A1)}(A5). If the rotor #ux observers and the rotor resistance estimator are designed as in Theorem 1, then the mechanical angular speed of the rotor uLP and the rotor #uxes () will be driven to approach to a bounded smooth speed command u and #uxes command ( with unknown load torque by the following primary control law:
<?"b1 (MRK I?#bI?!RKtK?!buLtK@#)!K?eL?#u?) <@"b1 (MRK I@#bI@!RKtK@#buLtK?#)!K@eL@#u@)
for some positive constants KS, K(, K? and K@, where ) and ) are de"ned as in (32), the
auxiliary control inputs are designed as follows:
u?"r1
reLSk2tK@!2reL(MRK tK?#reeLSr ¸tK@()(KS!D)k2J
!eeLS rJek2tK@ u@"r1 ! reLSk2tK?!2reL(MRK tK@!reeLS r ¸tK?()(KS!D)k2J #eeLS rJek2tK?with some positive constants r, r, r and r, and the payload adaptation law as ¹K *"1 r
!reLS#reL?¸tK@ (() KS!D)k2J ! reL@¸tK? (() k2JKS!D) #e J (k2(t)?eL@!tK@eL?)!KSeLS))Note that eL S"uL!u, eL("() !(, eL?"IK?!I?, eL@"IK@!I@, ()"tK?#tK@, i.e. all the variables which are not measurable are all replaced by their observed values or estimates.
Remark
Referring to the error dynamics (30), (38) with proposed control (34), we can express the closed-loop system dynamics in a very concise manner as
XQ "F(X, y) (42)
where X"[eS, e(, e2, e?, e@]2, and y"[R, t?, t@, u], and X converges to zero exponentially.
However, since y is not known a priori, the realistic controller shown in Theorem 2 adopts the estimate of y, namely, yL so that the closed-loop system (42) becomes
XQ "F(X, yL)
and# F(X, yL)!F(X, y) #)kJ#X##yJ#, where yJ"yL!y is always bounded from analysis of the observer. Therefore, by the following rearrangement
XQ "F(X, y)#(F (X, yL)!F(X, y))
and the fact that X satis"es exponential convergence in (42), one can easily verify that X remains to satisfy exponential stability provided# yJ # is small enough.
Through the observer and controller design procedure, we have the actual currents approach to the desired ones, i.e. I?PI?, I@PI@. Because the desired #ux (() and the desired speed (u) are bounded trajectory, the desired currents are bounded. So, it can be veri"ed through arguments of contradiction that the actual currents (I?, I@) are indeed bounded at any time.
4. SIMULATION AND EXPERIMENTAL RESULTS
4.1. Simulation results
In this section, computer simulations of the controller designed above are done by SIM-NON2+PCW 2.0. The induction motor is set initially at rest with load torque (¹*"6#2 * sin(0.5t)) and is required to follow the various desired speed and #ux trajectories with uncertainty in rotor resistance. The desired #ux trajectory is described by the function (("0.5(1!e!t) Wb), and the nominal value of rotor resistance is 0.53). We assume that the
Figure 1. u"8.8496 sin(0.5t)rad\/s with 6#2sin(t)Nt)m load.
variation of the rotor resistance is up to 50 percent. The simulation results are shown in Figures 1 and 2. In each "gure, part (a) shows the trajectories of the desired and the actual speed, whereas part (b) demonstrates the speed tracking error between the desired and actual speed. In a similar
Figure 2. u"!128.496 sin(0.5t)rad\/s with 6#2sin(t)Nt)m load.
way, parts (c) and (d) show the trajectories of the desired and the actual #ux and their error value, respectively. In addition, parts (e) and (f ) respectively, depict the two phase stator currents and input voltages. Finally, part (g) and part (h) demonstrate the real and the estimated values of load
Figure 3. Benchmark test with 6#2 * sin(t) Nt ) m load.
torque and the rotor resistance, respectively. From the simulation results, we can see that the controller can control both the rotor speed and the rotor #uxes simultaneously without measur-ing the rotor #ux and rotor speed signals.
In Figures 1 and 2, one can "nd that the tracking error and the estimation error do grow a little bit larger when the peak speed is higher. The reason is that the speed tracking error h"(u!u) is actually not changing slowly.
Table I. Speci"cations and parameters of the motor. Speci"cation Parameter Poles 4 R 0.83) Rated current 8.6 A R 0.53) Rated voltage 220 V ¸ 86.01 mH Rated frequency 60 Hz ¸ 86.01 mH Rated Speed 1720 RPM M 82.59 mH Rated power 2.2 kW J 0.033 kg m D 0.00825 N m s
4.2. Simulations with benchmark specixcation
We also perform computer simulations for the benchmark example. The rotor speed is required to change between those values:u , 0.1 u , 0.25 u , 1.5u during t"[0, 10] sec, where u "700 rev min\. The rotor #ux is required to change from #t# to 0.5#t# when the rotor speed is 1.5u , where #t# "0.5 Wb.
The modi"ed benchmark assumption is shown below: (B1) Measurable signals are the stator currents (I?, I@). (B2) Load torque ¹* is constant, though unknown.
(B3) All parameters are known, except the rotor resistance R.
(B4) Stator and rotor currents and stator voltages are constrained by 12 A and 300 V. (B5) The system begins with all zero initial conditions.
The speed tracking of the rotor speed and rotor #ux for the benchmark test are shown in parts (c) and (d) of Figure 3. The parameter adaptation is shown in parts (g) and (h). The adaptation is achieved under the restriction of the stator currents and the stator input voltages from assump-tion (B4) of the benchmark test, which is also shown in parts (e) and (f ) of Figure 3.
4.3. Experimental results
Experiments are done with a three horse power induction motor which is manufactured by TECO Co. Ltd. Taiwan. Parameters of the motor are listed in Table I.
The 3-HP induction motor used here is mounted with a 2000 pulse rev\ encoder. The controller is implemented by C language on a Pentium2+ based PC. And the motor driver is implemented by MOSFET whose gate signals are generated by a 10 kHz SPWM with 2)2ls dead-time. The currents are measured by two NANA2+ hall sensors. In order to check the performance, we have conducted three experiments with various kinds of sinusoidal speed commands as shown in Figure 4}6.
From the three "gures, one can see that the speed tracking objective is achieved with di!erent kinds of precision. In Figure 4, since the peak value is low and hence our previous assumption
Figure 4. u"80 sin(t) RPM with no load.
(A5) is more easily satis"ed, the tracking error is quite moderate. But, in Figures 5 and 6, the tracking error is clearly larger but acceptable because the peak speed is higher. But it is noteworthy that Figure 6 demonstrates that the controller developed here indeed can serve as a speed servo, i.e. it is able to drive the motor to follow any kind of smooth speed command.
5. CONCLUSION
In this paper, we have presented a partial-state feedback adaptive sensorless speed and #ux tracking controller for induction motors with "fth-order non-linear dynamic model which is
Figure 5. u"1200 sin(0.5t) RPM with no load.
actuated by a voltage source. We use an indirect adaptive control algorithm. First, we design an observer to get the actual values of #uxes, speed and rotor resistance. Then we design controller to achieve speed and #ux tracking. The main contribution of the controller is that asymptotic tracking of rotor speed and rotor #uxes are achieved without the measurement of both the rotor #uxes and the rotor speed. Moreover, the variations of the rotor resistance and load torque are also taken into account. That is, only with the measurement of stator currents, this controller drives the induction motor to track given twice di!erentiable bounded speed and #ux commands.
Figure 6. u"Hybrid function with no load.
APPENDIX A
Here, we derive an upper bound for the observation error of the rotor speed. First, from the mechanical part of the motor}load system
uR#DJ u"k2J (t?I@!t@I?)!¹*J
e(D/ J)
u#DJ e
D/J
u"J1[k2 (t?I@!t@I?)!¹*] e
then eD/J u"1J
R [k2(t?I@!t@I?)!¹*] e (D /J)qd q#u(0) "¹* J J D(1!e (D/J)R)#k2 J R (t?I@!t@I?) e(D/ J)qdq or u"¹*D (e !(D/J)t !1)#k2 J R (t?I@!t@I?)e!(D/J)(t!q)dq Since ¹* D (e !(D/J)t !1))0, ∀t*0 and the torque can be expressed in the vector form(t?I@!t@I?)"UJI where the matrices are de"ned as
U"[t?, t@] J"
0 !1 1 0 I"I? I@we can obtain the upper bound of the rotor speed as "u")k2J
R " (t?I@!t@I?)"e !(D/J)(t!q)dq )k2 J R e!(D/J)(t!q)dq' (#U# ' #I#) )k2D (#U# ' #I#) (A1)
where the subscript t denotes a truncated function which is de"ned as
In the above, we see that the upper bound on the rotor speed is determined by the norm of rotor #uxes and the norm of stator currents. However, the rotor #ux is unmeasurable, and therefore we try to "nd its upper bound associated with the currents. From the mathematical model of the motor in (1), we can write the dynamics of the rotor #ux in the form
t? t@" !R/¸ !bu/¸ bu/¸ !R/¸t?t@#MR/¸0 0 M R/¸
I?I@ Consider <t" (t?#t@) then <Q t"t?t?#t@t@ "t?!R ¸t?!bu¸ t@#M R¸ I? #t@!R ¸t@#bu¸ t?#MR¸ I@ "!R ¸ (t?#t@)#MR¸ (I?t?#I@t@) )!R ¸ (t?#t@)#2¸R (t?#t@)# MR 2¸ (I?#I@) "!R 2¸(t?#t@)# MR 2¸ (I?#I@) so that <Qt#R ¸ <t)MR ¸ (I?#I@)
Following the procedure as that in the derivation for the upper bound on the rotor speed, the upper bound on <tis <t)MR 2¸
R e!(R/¸) (t!q) (I?#I@) dq#e!(R/2¸)< t(0) )MR 2¸ R e!(R/2¸) (t!q) dq'# I##<t(0) )M 2 #I##<t(0) ∀t*0Since <t is the norm of the rotor #ux, then we can "nd that #U #)M#I##2<t(0) (A2) where <t(0)"1 2(t?(0)#t@(0))) 1 2¸(d?#d@) where t?(0))d? t@(0))d@
Then from (A1) and (A2), we can associate the upper bound of the rotor speed with the norm of the stator currents as
" u")k2D# I# (M#I##2<t(0)
)k2M
D # I##k2D
1
¸(d?#d@ #I#
Since the upper bound on the rotor speed is found, the upper bound on the observation error of the rotor speed can be found as
" u!uL")"u"#"uL" )k2M D # I##2 (D)k2 1 ¸(d?#d@ #I##"uL" (A3) APPENDIX B ¹heorem A1 [17]
The second di!erential equation has the form
xR (t)"A(t)x(t)#u2(t)x(t)
(B1)
xR (t)"!u(t)x(t)
where x: [0, R)PRK, x: [0, R)PRL, A(t) and u(t) are, respectively, n;n and n;m matrices of bounded piecewise-continuous functions, and A(t)#A2(t) is uniformly negative de"nite, that is, A(t)#A2(t))!Q(0. Let x"[x2, x2]2. The equibrium point x"0 in Equation (B) is
uniformly asymptotically stable if, and only if, positive constants ¹, d and e exist with a t3[t,
t#¹] such that for any unit vector w3RK,
1 ¹
t#d t
u2 (q)w dq
*e, ∀t*t (B2) When# uR(t) # is uniformly bounded, the condition in Equation (B2) in Theorem 1 can be relaxed as given in Corollary 1.
Corollary A2
If u(t) is smooth, uR (t) is uniformly bounded, and u(t) satis"es the condition 1
¹
t#¹
t # u2 (q)w#dq*e, ∀t*t
for t3R>, and positive constants ¹ and e, and all unit vectors u3RL, then the solution x"0 of Equation (B1) is u.a.s.
Proof: Let#x(t)#(c#x(t)#∀t3[t, t#¹], where c3(0, 1). Since u satis"es the condition
in (B2), there exists a t3[t, t#¹] so that
t#dt #u2 (q) x(q)#dq*e¹#x(t)#!u d supq3[t, t#d]#x(q)#
Integrating the "rst equation over the interval [t, t#d] in (B1) it follows that
# x(t#d)#*[e¹ (1!c!c(u d#A d#1)]#x(t#d)# (B3) since# x(t)#*(1!c#x(t)# and #x(t)#*#x(t#d)#. If
c" ¹e
¹e#(1#b)
b"u d#A d#1
we have from Equation (B3)
# x(t#d)#*c#x(t#d)# (B4)
Integrating <Q (x) over an interval [t, t#¹]L[t, t#¹], we obtain that < (t)!<(t#¹)*j/\
t#¹ t # x(q)# dq *j/\ ¹ t#¹ t # x(q)# dqwhere j/\ is the minimum eigenvalue of Q. Further, by integrating the "rst equation in Equation (B1) over [t, t#¹]
t#¹t #x(q)#dq*¹#x(t)#!¹d(<(t)
where d"A #u , #A(t) #)A and # u(t) #)u . Choosing t"t#d, from Equation (B4), it follows that
<(t#¹))(1!c) <(t) where
c"j/\ ¹(c!¹d) Let ¹"min(t#¹, c/d), we have
<(t#¹))<(t#¹))(1!c)<(t))(1!c)<(t)
Sincej/\ )2, c3(0, 1) and ¹d(c!¹d))4c/27, it follows that c3(0, 8c/27) and hence
x"0 in Equation (B1) is u.a.s.
NOMENCLATURE +<?, <@, stator voltages in the stationary reference frame +I?, I@, stator currents in the stationary reference frame +t?, t@, rotor #uxes in the stationary reference frame u mechanical angular speed of the rotor
R stator resistance R rotor resistance ¸ stator self-inductance ¸P Rotor self-inductance M mutual inductance p number of pole-pairs J rotor inertia D damping coe$cient
¹* load torque k2 torque constant
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