GB2447511A

GB2447511A - Induction motor controller

Info

Publication number: GB2447511A
Application number: GB0706344A
Authority: GB
Inventors: Lin Jiang
Original assignee: UGCS
Current assignee: UGCS
Priority date: 2007-03-14
Filing date: 2007-03-30
Publication date: 2008-09-17
Also published as: GB0706344D0; GB0704983D0

Abstract

An adaptive induction motor controller is described comprising a plurality of high gain perturbation observers, each adapted to estimate a perturbation term. The perturbation terms model nonlinearities of the induction motor. The controller also comprises a feedback arrangement arranged to cancel out nonlinearities in said induction motor using said perturbation terms.

Description

INDUCTION MOTOR

The present invention is directed to induction motors. In particular, the present invention is directed to systems and methods for controlling induction motors.

Induction motors are widely used in many industrial applications due to a simple construction, good reliability and relatively low cost. The control of induction motors is a challenging problem since it is a nonlinear dynamic system, the electrical rotor variables (rotor flux) are not usually measurable, and the physical parameters (rotor resistance and load torque) vary considerably with a significant impact on the system dynamics. Thanks to the development of advanced power electronics and microprocessors, many complex and advanced control schemes have been proposed.

The field-oriented control (FOC) (or vector control) technique is widely used in high performance motion control of induction motors. Because of torque/flux decoupling, FOC tends to control AC induction motors as DC motors to achieve good dynamic response and accurate motion control. By the use of nonlinear state space change of coordinates and nonlinear feedback control, an asymptotic decoupling in the control of speed and rotor flux amplitude is achieved. However, the coupling still exists when flux is not kept in constant value, i.e. when flux is weakened in order to operate at higher speed within input voltage saturation limits or when flux is adjusted to maximize power efficiency.

Nonlinear control theory has attracted much effort for induction motor control in the past several decades.

Input-output linearizing control (IOLC) can achieve the full decoupling of speed and flux dynamics and the connection of FOC to IOLC has been clarified. By adding an integrator to one of its inputs (stator voltage), dynamic feedback linearizing method is investigated and the resulting sixth-order induction motor is input-to-state feedback linearizable. A passivity-based approach which exploits the system energy dissipation property and does not require nonlinearity cancellation is proposed for induction motor control.

The feedback linearizing methods referred to above (including the vector control) rely on the cancellation of nonlinearities and require an accurate induction motor model. They are highly vulnerable to parameter uncertainties and external disturbances. Vector control is very sensitive to parameter variation. Moreover, the methods lead to complicated control algorithms which requires complex nonlinear calculation. Generally speaking, the performance of such control systems depends on the accuracy of the mathematical model of induction motors.

Assuming parameters (load torque and rotor resistance) are unknown constants, nonlinear adaptive control has been proposed for input-output linearizing control of induction motors based on a parameter identification algorithm and Lyapulov redesign methods. Both state feedback and output feedback have been investigated. The augmentation of adaptive algorithms increases the complexity of the total control algorithm and it can only deal with slow varying parameters. Robust control methods, such as sliding mode techniques, have also been investigated for induction motor control which only require the upper bound of uncertainties to be known.

Pdthough many nonlinear control techniques have been proposed to control induction motors, many of these techniques have proved to be extremely difficult and expensive to implement for industrial applications.

Indeed, advanced induction motor control has traditionally been so expensive that it has only been considered feasible for very high power applications in which the cost of implementing such control techniques is negligible compared with the overall cost of the system.

IS Induction motors behave in accordance with the differential equation of motion, which requires, in broad terms, that: =f(x) di In the equation above, f(x) is a vector field and is typically a function of many parameters. Many of those parameters may be determined by the manufacturer of the induction motor, but they are prone to change, for example with age and temperature or with use, and require calibration by experimental works or estimation algorithms.

The traditional approach has been to develop ever more complicated models that seek to model the induction motor more and more accurately. Of course, this approach makes such control systems more complex and does not address the other sources of uncertainty, such as the uncertainty of real-time implementation, external disturbances, unpredictable parameter variations, and unmodeled plant nonlinear dynamics. Consequently, this will deteriorate the dynamic performance of flux and speed significantly.

Thus, there are a number of difficulties with existing induction motor control systems that make advanced control systems difficult to implement.

An alternative approach is discussed in US 5,144,549. In that document, a time delay control system is proposed that can be used in the control of non-linear systems with unknown dynamics.

The time delay algorithm of US 5,144,549 has the advantage that past information can be used to estimate the effects of unknown, nonlinear system dynamics. Its main idea is the time-delayed values of control input and the derivatives of state variables at the previous time steps are used to estimate the unknown nonlinear dynamics and uncertainties. The derivatives of state variables are always calculated by numeric differential methods, such as backwards or forwards difference algorithms. It is well known that the numerical differentiator will magnify the measurement noise.

However, the control algorithm of uS 5,144,549 inherently introduces delay into the control algorithm. Further, as discussed above, noise in the estimation of derivatives reduces the effectiveness of the algorithm.

A further problem with the control algorithm proposed in us 5,144,549 is that it requires that the control gain of the system should not cross zero, i.e. the sign of the control gain should always remain the same. This condition is not satisfied in induction motors.

The present invention seeks to address at least some of the problems outlined above.

The present invention provides an induction motor controller comprising: a plurality of high gain perturbation observers, each adapted to estimate a perturbation term, wherein the perturbation terms model nonlinearities of the induction motor; and a feedback arrangement arranged to cancel out nonlinearities in said induction motor using said perturbation terms. Said perturbation terms may further model parameter uncertainties and said feedback arrangement may be further arranged to cancel out said parameter uncertainties. In some forms of the invention, a plurality of feedback arrangements is provided.

The present invention also provides a method of controlling an induction motor comprising the steps of: using a plurality of high gain perturbation observers to estimate a plurality of perturbation terms, wherein the perturbation terms model nonlinearities of the induction motor; and using a feedback arrangement arranged to cancel out rionlinearities in said induction motor using said perturbation terms. The perturbation terms may further model parameter uncertainties and the method may further comprise the step of using said feedback arrangement to cancel out said parameter uncertainties.

The present invention further provides an induction motor controller comprising: a plurality of high gain perturbation observers, each adapted to estimate a perturbation term, wherein the perturbation terms model parameter uncertainties of the induction motor; and a feedback arrangement arranged to cancel out parameter uncertainties in said induction motor using said perturbation terms. In some forms of the invention, a plurality of feedback arrangements is provided The present invention yet further provides a method of controlling an induction motor comprising the steps of: using a plurality of high gain perturbation observers to estimate a plurality of perturbation terms, wherein the perturbation terms model parameter uncertainties of the induction motor; and using a feedback arrangement to cancel out parameter uncertainties in said induction motor using said perturbation terms.

The control algorithms of the present invention have at least the following advantages: 1. Decoupling between speed and flux dynamics is achieved; 2. Time-varying uncertainties and un-modelled dynamics can be handled; 3. The algorithms do not require an accurate model of the induction model to be provided and are robust to uncertain parameters; and 4. The control algorithm is relatively simple, since nonlinearities can be estimated within perturbation.

In one form of the invention, two perturbation observers and two feedback arrangements are provided. In particular, the high gain perturbation observers of the present invention may comprise: a first perturbation observer having inputs for receiving rotor speed and

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stator voltage signals and outputs providing an estimate of rotor speed, an estimate of the derivative of rotor speed with respect to time and a first perturbation term; and a second perturbation observer having inputs for receiving rotor flux magnitude and stator voltage signals and outputs providing an estimate of rotor flux, an estimate of the derivative of rotor flux with respect to time and a second perturbation term.

In one form of the invention, the feedback arrangements provide input/output linearization.

In a first embodiment of the invention, the perturbation observers are given by =f(x), wherein: di each perturbation observer has an input receiving the signal x arid said perturbation observer provides a first output that is an estimate of x and a second output that is an estimate of f(x); the derivative of said first output of each perturbation observer is equal to the second output plus the product of a first value (h1) and a first error term (e1); and the derivative of said second output of each perturbation observer is equal to the product of a second value (h2) and the first error term (e1) The said first error term (e1) may be the difference between the signal x and the estimate of the signal x.

In a second embodiment of the invention, the functionality of the induction motor is given by the system =f(x)+p1f1(x)+p2f2(x)+gu +gu, wherein f, g, g, * f1 and f2 are vector fields. Further, each perturbation observer may have an input receiving the signal z12 arid said perturbation observer may comprise two high gain perturbation observers designed as follows: d2 = z3 + h,1 (z,2 -z,2) + + = h,2(z,2 -2,2).

Each perturbation observer may have an input receiving the signal z11 and said perturbation observer may comprise two high gain perturbation observers designed as follows: = 2 + h,11 = z,3 + h2z,1 cif d'V z di iii di In one form of the invention, the or each differential is calculated by integration. This is advantageous since estimating differentials can result in large errors due to noise -integration is inherently less prone to such errors.

Embodiments of the invention will now be described, by way of example only, with reference to the following schematic drawings, in which: FIGURE 1 is a block diagram showing the functionality of a perturbation observer in accordance with an aspect of the present invention;

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FIGURE 2 shows time-varying input parameters used in simulating a first exemplary embodiment of the present invention; FIGURE 3 shows first outputs of a simulation of a first exemplary embodiment of the present invention; FIGURE 4 shows perturbation estimates and errors in accordance with the first embodiment of the present invention; FIGURE 5 shows outputs of a simulation of a second exemplary embodiment of the present invention;

FIGURE 6 shows outputs of a simulation a prior art

method; FIGURE 7 shows outputs of a simulation of a further embodiment of the present invention; and FIGURE 8 is a block diagram of an embodiment of the present invention.

The present invention enables control of an induction motor by making use of perturbation estimation.

As discussed above, induction motors are highly non-linear systems. The present invention addresses that problem by lumping all of the nonlinearities of the induction motor system into so-called "perturbation terms". By representing the perturbation terms as a fictitious state in state equations, it is possible to generate estimates of the perturbation terms and thereby compensate for those terms in a control algorithm.

In the present invention, estimates of the perturbation terms are generated using perturbation observers. Figure 1 shows a perturbation observer 2 that receives an input x1 and provides outputs and 2' where is an estimate of x1 and 2 is an estimate of x2.

A first order system is given by where x=x and x2=f(x).

As noted above, the perturbation observer 2 receives an input x1 and provides outputs and i2, where is an estimate of x1 and i2 is an estimate of x2. The perturbation observer 2 is designed as x2 +h1e1 di' d2 e di where h1 and h2 are parameters of the observer, and e1 and e2 are error terms, e1=x1-i1 and e7=x2-7.

The inventor has realised that from the observer above, it is possible to determine the estimated values and i2 by integrating, rather than relying on differentiation.

The observer above is significant since i2 gives an estimate for the perturbation term f(x).

In the control of an induction motor in accordance with the principles of the present invention, all of the nonlinearities in the induction motor system are included in the perturbation term. This results in a control algorithm that does not need a detailed model of the induction motor. This also results in much simpler and therefore much cheaper control algorithms to be implemented.

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The present invention provides a simpler induction motor controller since the perturbation term can be estimated without requiring a complicated and detailed model of the S induction motor system.

A model of induction motor behaviour is given by the following equation: =f(x)+p1J(x)+p,f2(x)+gu +gu Where f, g, g, f and f2 are vector fields defined as follows: P(/'I'h aNd/a -i' +aNMia f(x) -aNyib +flWyJ0 +aNMih aN/id/a +flP/kLfl/Ib)N1a aN/id/h *flp/3WY!a YN1b / g=I0 0 0 0 crL5 / gb=I0 0 0 0 --aL (-1

H j= 0 +Mi

f2= W+Mi, f3yi -M/3i fl1'b /b The parameters are rotor and stator winding resistance (Rn R) and auto-inductances (Lr, La), the mutual inductance M, the moment of inertia of the rotor J, the load torque TL, pole pairs n.

Further, cT=1-M2/LSLR; a=R,/Lr; /JrzM/(crLrLs); y=R/(aL)+Ma/3; /J=(npM)/(JLr); aN =RrNILr; iN -_RSI(crLV)+MaN/3.

When controlling an induction motor, the outputs to be controlled are the rotor speed (y1= 1(x)=) and the square of the rotor flux modulus (y2 = 2(x)=yJ2+çII2) Letz11=y1, z2 = y1, z21=y2 and z22=y2 be state variables of two interconnected system. Considering the induction motor with time-varying parameters TL(t) and a(t) (i. e.

p1!=O and p2!=O, the input/output decoupling of induction motor is obtained as follows: = Lfj+pILf1 1 = Ll+p2Lf2LfI+tLf1 I +LgLf i a + LgLPI 11b = Lf +p2Lf, Z22 = L + p2L j.,L 1çt2 + +f)L2 + (Lg0LpP2 +P2Lg,Lf2)Ua +p2L1Lq + (LgLf +P2LgLf,2)Ub

S where

Lji -P('Ya1bWb'a)j Lp1 = _p13npw(IJI + w,) -pnpco( Yam + Yblb) -p(aN+v)('Yazb-V1b'a) = -2(Xw(IJ/+ t1/i)+2cxiVM(V/aza+ Yb'b) L-ip, = (4a,,+ 2a1,/3M)('Y + Y)+ 2a2,M2(m +i) -(6a1,M+ 2awvM)('YaIa + I/bib) +2aNMnco( 11'a'b -IIb'a) Lj i = -Lj,L1t = -p(1+I1i)('IIIb-I'bl) Lj, = 2M('Yala+'Ybih)2(IIJ+'Y) L fLf, q = 2 aN(2 + MJ3) ( + 2aNM2 (i + i) -2M(y1v + 3av)( + I/bib) +2NMw( Wa1b -V'b'a) Lqi = 2M(3+MP)(tIJaIa+ (I/bib) Lj,Lj RrNLP = -2cxj..rM(3 +M13)('Yala+ /b'b) LgaLf2 = 2t3LrV/a Lgj, Lj 2/3L, Yb Rewrite above equations as follows: (2) = ) + D(x) (un) + ( At) / / p2Lf,Lf I-Lf1I \ / ) = f pLj,Lj +p2L1Lj, I +AD(x) \ 4-pL,4 + Lj4) tmb D(x) = Lg0Lji LgLf I Lg0Lj4 LgLfPi 0 0 AD()=p2 LgLj, LgLf, where D() and AD(.) are the constant and uncertain control gain matrix, respectively, A, and A are combined uncer-tainties which include both the parameters denvation (Pt,P2) and their time dynamics (L, ) D(x) is nonsmgular for + > 0 In the present invention, all system nonlinearities are assumed to be unknown and are grouped together into a single perturbation term. In this way, the following perturbation terms for an induction motor are defined: F,(x,p1,p2,t)=t, +L21 , The control algorithm in accordance with the principles of the present invention, as described above, introduces the following fictitious state: z13 = This leads to the following extended-order model of an induction motor speed and flux dynamics systems: dz,1 2 = + DitUa + D,2uh dz,3 dPi di' di' where i=1,2, D11 is the known part of the control gain [ga 10' LghLJOI [LgaLjO2 LgbLf 02

Examples

To illustrate the concept of perturbation observation, a second-order perturbation observer is described below under the assumption that all sub-system states are available. Taking the second state z12 as input, two high gain perturbation observers (HGPOs) are designed as: d2 +h,,(z,2 -z1,)+D,,ufl+D,2ub =h,2(z,2-,2) Wherein i=1,2 for two HGPOs, i,j=1,2 are gains of high gain observers.

(It should be noted that throughout this document, =z,-1 refers to the estimate error of z1, whereas, refers to the estimate of z1.) From the equations above, the error dynamics of the high gain observers become:

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= z,3 -h,1z, -h di:2:2 di a a Let h,1 = -1-,h,2 = --be observer gain, and i7, = = z,3 be the scaled estimation errors, the equation above can then be represented in a singularly perturbed form as: = A1i, +,B,'-P, where: , [ 771T and r-a, ii ro A=l I,B,=I [-a,2 0] L1 The positive constraints c and a12 are chosen such that A is a Hurwitzian matrix, and, 0 < = 1, is a small positive parameter to be specified.

By using the estimate of perturbation 13 and 23 to cancel the nonlinearities and uncertainties, robust adaptive control law is obtained as: H= D(x)[H13+1v0 Ub) [-Z23) Yb v = -k11z1 -k12z12 + aref Vb = -k21z21 -k22z2, + Vbf Vf = kiiYirej -k12 Yiref+Yiref ref --k1 Y2rcf k,2 Y2rj + Y2ref where k11, i,j=l,2, are chosen to make matrix [0 ii A =1 Ii=12 iO [-Ic, -/c,J' Be Hurwitzian matrix.

Let e,1 = -,ref e12 = -.Yire' and e, = [c e12]T be state variables of the track error system. Substituting the control law into the system model gives the following track error dynamic: è, = A,0e + B,0,3 The control algorithm described above has been simulated in accordance with the following assumptions.

The time varying uncertainties have been simulated based on a number of assumptions, as described below with reference to Figure 2.

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Figure 2(a) is a plot of rotor resistance (Rr) against time and shows the rotor resistance including the time varying term 0.007t + 0.01 + 0.Olsin(2t) added at t=5s.

Figure 2(b) is a plot of load torque (TL) against time and shows that a load torque of 4ONm is added at t = 3.5 seconds and a time varying term 5 + 5sin(2.5t) is added at t=lO seconds.

Figure 3 shows the results of applying the simulation of Figure 2 to the exemplary system described above. Figure 3(a) show plots of rotor speed (ca) and desired rotor speed (Wret) . Figure 3(b) plots the difference 0) -0)ref* Figure 3(c) shows plots of rotor flux amplitude (Y2) and desired flux (Y2ref) . Figure 3 (d) plots the difference Y2-Y2ref The desired unloaded induction motor speed (0)ref) iS 200 rad/second at t=3.0 seconds and the desired rotor flux amplitude =1.3Wb at t=1 second; then at t=5 seconds, the desired motor speed reference is increased to 300 rad/second and returns to 200 tad/second at t=8 seconds.

The flux desired reference is weakened to 0.8Wb at t= 5 seconds and returns back to 1.3Wb at t=8 seconds. Those reference signals are shown in dotted form in Figure 3.

The parameters of liner control were chosen as k11=-900, k12=r-60, i=l,2 to assign the poles at -30.

The parameters of HGPO were assigned as follows: i1= 2500, and c12=-100 to assign poles of A1 at -50, =0.01, i=l, 2. *

The track response of the control is shown in Figure 3 for speed response (Figure 3a) and flux amplitude response (Figure 3c) . It is noted that it is difficult to see the difference between the desired levels (shown in dotted form) and the actual level according to the model (shown in complete form) . However, as noted above, the differences between the desired and actual levels of motor speed and rotor flux amplitude are plotted in Figure 3(b) and 3(d) respectively.

Figure 3 shows that the dynamic of flux and speed are totally decoupled and with fast track dynamic when reference signals change (at t=5 seconds and t=8 seconds), excellent robustness with parameter uncertainties (i.e. unknown load torque at t=3.5 seconds and time varying load torque after t=l0 seconds and up to 50% of time-varying rotor resistance after t=5 seconds) Figure 4 shows the calculation of the perturbation terms 1' (Fig. 4 (a) ) and W. (Fig 4 (C) ) and also the errors in those terms (Figs. 4(b) and 4(d) respectively. Figure 4 shows the fast and accurate estimation of the perturbation terms.

By way of a further example, consider the following scenario in which only rotor speed y1=zi and rotor flux modulus y2=z21 are available as measurements. Two third-order high-gain observers, called high gain state and perturbation observers, can be designed to estimate system states and perturbation as follows:

S dz1 = + dz2

= + h,2z,1 + D11,, + dz3 -z di From equations given above, its error dynamic follows that: = z, + h,1z11 _L=z3+h z1 dt 2; di dP --=--h z di di After defining the scaled observer gain and estimation errors, the observer error equation can be represented in a singularly perturbed form as: , =A,1, +B,1P,(.) El where r, A11, B11 have similar structures to the previous example, but with different dimensions.

Using the estimate of perturbation to cancel the system perturbation, and the estimate of system states and 1;3 to replace the real states the robust and adaptive control law is obtained as: 1u1 = D (X)[1 Z13 +Iva J Ub) [-Z23) Vb where va and Vb are same as in previous description.

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Substituting the control law into the system model gives the following track error dynamic: = A,e, -f-B,K,T,(e,),i, where K, =[k,1,k,2,I1, and T,(,)=diagc,2,g,,13 The control algorithm described above has been simulated in accordance with the assumptions described above with reference to Figures 2 and 3 and further assuming that the parameters of the high gain state and perturbation observer (HGSPO) are as follows: c11=-30; a12=-300; Q3=- 1000, setting poles of A, at -10 and =0.0l. 1=1,2.

Figure 5 shows the results of applying the simulation of Figure 2 to the exemplary system described above. Figure 5(a) show plots of rotor speed () and desired rotor speed k'ref) . Figure 5(b) plots the difference C) -Figure 5(c) shows plots of rotor flux amplitude (y2) and desired flux (Y2ref) . Figure 5 (d) plots the difference {2 Y2ref.

The track response of the proposed control algorithm for speed response (Figure 5(c)) and flux amplitude response (Figure 5(d)) are comparable with state feedback responses with a bigger track error caused by the estimation error of state variables.

Comparison with conventional input/output linearization control (IOLC) is provided. Results of IOLC control using same linear control as the proposed invention are shown in Figure 6. From those figures, we can note there are constant steady errors for both flux and speed dynamics after unknown load torque are applied, with noticeable track error under time-varying uncertainties. This is caused by the incomplete cancellation of system nonlinearities as time-varying nonlinearities are not included in both IOLC and E'OC designs.

The proposed output feedback control has also been simulated using estimate of flux v2=çii+çif obtained from a four-order sliding mode flux observer. Speed and flux responses are shown in Figure 7 and they are very close to those produced with the flux measurements with bigger flux track error. This is because the proposed control treats the introduced estimate error y2-y2 as part of system perturbation. There is a small noticeable track error of flux response after t=lOs, when both Rr(t) and TL(t) are time-varying.

Figure 8 is a block diagram of an induction motor and control system, indicated generally by the reference numeral 10, in accordance with the teaching of the present invention.

The system 10 comprising an induction motor 12 and a controller 13 that is being used to control the induction motor 12. The controller 13 receives inputs from first 14 and second 16 state and perturbation observers and provides outputs to a pulse width modulation circuit 18.

Pulse width modulation circuit 18 provides control signals to the motor 12 via an inverter 20.

The performance of the motor 12 is monitored by monitor 22 and flux observer 24.

The monitor 22 provides a measured rotor speed signal w to first state and perturbation observer 14 and provides measured stator voltage signals (ua, Ub) to both first 14 and second 16 state and perturbation observers. Flux observer 24 provides a rotor flux measurement (iyi) to the second state and perturbation observer 16.

The controller 13 receives control signals indicating desired values for rotor speed (()ref) and rotor flux (Yret) . The controller 13 also receives outputs from the first 14 and second 16 state and perturbation observers as follows.

The first state and perturbation observer 14 provides outputs (,th) and P1. The second state and perturbation observer 16 provides outputs (y,çi) and 2* On the basis of the outputs of the first and second state and perturbation observers, the controller 13 provides control signals for the induction motor in the form of stator voltage signals Ua and ub.

As described above, the present invention describes a robust adaptive controller for an induction motor, based on perturbation estimation and input/output linearization. The perturbation terms, which include parameter uncertainties and all system nonlinearities, are estimated by designing high-gain perturbation observers. Estimates of perturbation are employed to achieve robust and adaptive feedback linearization control. Simulation studies have verified the design.

Claims

CLAIMS: 1. An adaptive induction motor controller comprising: a

plurality of high gain perturbation observers, each adapted to estimate a perturbation term, wherein the perturbation terms model nonlinearities of the induction motor; and a feedback arrangement arranged to cancel out nonhinearities in said induction motor using said perturbation terms.
2. An adaptive induction motor controller as claimed in claim 1, wherein said perturbation terms further model parameter uncertainties and said feedback arrangement is further arranged to cancel out said parameter uncertainties.
3. An adaptive induction motor controller comprising: a plurality of high gain perturbation observers, each adapted to estimate a perturbation term, wherein the perturbation terms model parameter uncertainties of the induction motor; and a feedback arrangement arranged to cancel out parameter uncertainties in said induction motor using said perturbation terms.
4. An adaptive induction motor controller as claimed in any one of claims 1 to 3, wherein said feedback arrangement comprises a plurality of feedback arrangements.
5. An adaptive induction motor controller as claimed in any preceding claim, wherein said feedback arrangements provide input/output linearization. *
6. An adaptive induction motor controller as claimed in any preceding claim, wherein the perturbation observers are given by =f(x), wherein: each perturbation observer has an input receiving the signal x and said perturbation observer provides a first output that is an estimate of x and a second output that is an estimate of f(x); the derivative of said first output of each perturbation observer is equal to the second output plus the product of a first value (h1) and a first error term (e1); and the derivative of said second output of each perturbation observer is equal to the product of a second value (h2) and the first error term (e1)
7. An adaptive induction motor controller as claimed in claim 6, wherein said first error term (ei) is the difference between the signal x and the estimate of the signal x.
8. An adaptive induction motor controller as claimed in any preceding claim, wherein the functionality of the induction motor is given by the system wherein f, g, g, f1 and

f2 are vector fields.
9. An adaptive induction motor controller as claimed in claim 8, wherein each perturbation observer has an input receiving the signal z12 and said perturbation observer comprises two high gain perturbation observers designed as follows: * = z,3 + h,1(z,2 z12) + D,11 + D1211h h,2(z,2 -
10. An adaptive induction motor controller as claimed in claim 9, wherein each perturbation observer has an input receiving the signal z11 and said perturbation observer comprises two high gain perturbation observers designed as follows: _!_ = z + hz,1 __!_ = + h,2z1 d3 = -h z + di di'
ii. An adaptive induction motor controller as claimed in any preceding claim, wherein said plurality of high gain perturbaLion observers comprises: a first perturbation observer having inputs for receiving rotor speed and stator voltage signals and outputs providing an estimate of rotor speed, an estimate of the derivative of rotor speed with respect to time and a first perturbation term; and a second perturbation observer having inputs for receiving rotor flux magnitude and stator voltage signals and outputs providing an estimate of rotor flux, an estimate of the derivative of rotor flux with respect to time and a second perturbation term.
12. An adaptive induction motor controller as claimed in any one of claims 6 to 11, wherein the or each differential is calculated by integration.

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13. An adaptive induction motor controller as claimed in any preceding claim, wherein said induction motor controller is a robust induction motor controller.
14. A method of controlling an induction motor comprising the steps of: using a plurality of high gain perturbation observers to estimate a plurality of perturbation terms, wherein the perturbation terms model nonlinearities of the induction motor; and using a feedback arrangement to cancel out nonlinearities in said induction motor using said perturbation terms.
15. A method as claimed in claim 14, wherein said perturbation terms further model parameter uncertainties, wherein the method further comprises using said feedback arrangement to cancel out said parameter uncertainties.
16. A method of controlling an induction motor comprising the steps of: using a plurality of high gain perturbation observers to estimate a plurality of perturbation terms, wherein the perturbation terms model parameter uncertainties of the induction motor; and using a feedback arrangement to cancel out parameter uncertainties in said induction motor using said perturbation terms.