Stator flux and torque decoupling control for

Stator flux and torque decoupling control for induction motors with resistances adaptation H. Wang, W. Xu, T. Shen and G. Yang Abstract: An adaptive state feedback control has been designed for direct torque and stator flux regulation with a fifth-order model of an induction motor. The control design is based on exact input–output decoupling linearisation via non-linear state feedback. To achieve decoupling control, it is shown that the tracking dynamics with respect to the outputs (stator flux amplitude and electrical torque) is asymptotically stable at an admissible operating equilibrium. In the adaptive control, the uncertainty in the resistances of the stator and the rotor is considered, and the adaptation law is design to guarantee the boundedness of parameter estimation and output regulation performance. Finally, simulation results are presented to demonstrate the availability of the proposed controller. 1 Introduction Induction motors have been widely used in industry because of their advantages of simple construction, ruggedness, reliability, low cost, and minimum maintenance. However, due to the highly coupled non-linear structure, the high- performance control of induction motors is still a challen- ging problem. In the past two decades, several feedback control approaches based on input–output decoupling and linearisation have been proposed for induction motors (see [1] and references therein). A classical control technique for induction motors, field oriented control (FOC) [2], has been improved by exploiting the exact input–output decoupling method [3]. Recently, using non- linear adaptive control theory, input–output decoupling control has been extended with rotor speed and stator current measurements only, to the case where load torque and rotor resistance are unknown [4]. Attempts to utilise adaptive control theory have also been reported [5–7] for rotor speed and rotor flux tracking control problems of induction motors. Attention has also been focused on the direct torque control (DTC) problem of induction motors [8, 9]. Different from FOC, the control objective of DTC is expressed in terms of torque and stator flux regulation. In [10], mathematical analysis of the stabilisation mechanism of DTC and a modified strategy was proposed. A decoupling property for the regulation of torque and stator flux based on output regulation subspaces (ORS) was established, which is similar to the decoupling of torque and rotor flux exploited in FOC. However, decoupling performance using the ORS-based strategy was achieved approximately because control signals were directly generated by a lookup table in discrete form. An input–output decoupling controller with switching function was proposed in [11] for DTC of induction motors. In this paper, we propose an adaptive control approach to the stator flux and torque regulation problem of the induction motor. First, a fifth-order model of an induction motor in the fixed stator reference frame is transformed into polar coordinates. With these new coordinates, we show that the control problems of tracking dynamics with respect to stator flux and torque can be solved by the input–output decoupling control method. Exact decoupling and linearisa- tion regulation of stator flux amplitude and torque via non- linear state feedback is presented for the case when the physical parameters are exactly known. Further, the feed- back controller is extended to cope with parameter uncertainty in the resistances of the stator and the rotor. Simulation results are given to demonstrate the suitability of the proposed controllers. 2 Model and problem formulation 2.1 Induction motors As is well known [1, 6, 10], the model of a three-phase induction motor consists of electrical and mechanical dynamics, which are coupled via the generated torque. The electrical dynamics of the induction motor with one pole pair in a fixed stator reference frame ða;bÞ is described by the following voltage balance equation: _ffs ¼ �Rsis þ us _ffr ¼ �Rrir þ orJfr � ð1Þ where the subscripts s and r stand for stator and rotor quantities, fj ¼ ½fja;fjb�T and ij ¼ ½ija; ijb�T ð j ¼ s; rÞ denote the fluxes and currents respectively. or is the rotor speed and the applied stator voltages us ¼ ½usa; usb�T are the control inputs. R denotes the resistance and the matrix J is defined as J ¼ 0 �1 1 0 � � Under the assumptions of linearity of the magnetic circuits and of equal mutual inductances and neglecting iron losses, we have q IEE, 2005 IEE Proceedings online no. 20045031 doi: 10.1049/ip-cta:20045031 H. Wang, W. Xu, and G. Yang are with the Department of Automation, Tsinghua University, Beijing, 100084, China T. Shen is with the Department of Mechanical Engineering, Sophia University, Tokyo, 102-8554, Japan E-mail: wanghuangang99@mails.tsinghua.edu.cn Paper first received 9th May and in revised form 22nd August 2004. Originally published online 4th March 2005 IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005 363 fs ¼ Lsis þMir fr ¼ Lrir þMis � ð2Þ where L and M represent the self and mutual inductance, respectively. The mechanical dynamics is given by _oor ¼ 1 Dm ðtc � tLÞ ð3Þ where Dm denotes the rotor inertia, tL is the load torque and the electrical torque te is given by te ¼ iTs Jfs ð4Þ When the control objective of induction motors is expressed in terms of stator flux and torque, it has been reported in [10] that the output regulation is coupled with the stator flux orientation, unless is ¼ 0 and or ¼ 0: And in [10], a decoupling property was established accordingly by trans- formation of the system into polar coordinates. However, the input–output maps are still coupled by the dynamics of the rotor flux in polar coordinates. In order to propose an exact input–output decoupling control scheme, we adopt a similar state space change of coordinates to that in [10], and define fs ¼ cse jys ; fr ¼ cre jyr ð5Þ where ys; yr denote the angle of stator and rotor flux, and cs; cr denote the flux amplitude respectively. Then the electrical dynamics (1) is transformed into the following form: _ccs ¼ �aRsLrcs þ aRsMcr cos gþ uc cs _yys ¼ �aRsMcr sin gþ ut _ccr ¼ �aRrLscr þ aRrMcs cos g cr _yyr ¼ aRrMcs sin gþ cror 8>>< >>: ð6Þ where g ¼ ys � yr; and the parameter a is defined with the electrical parameter by a ¼ ðLsLr �M2Þ�1 ð7Þ The relationship between the stage voltage components ðuc; urÞ and ðusa; usbÞ is usa usb � � ¼ cos ys � sin ys sin ys cos ys � � uc ut � � ð8Þ Moreover, the torque te in the new polar coordinates is given by te ¼ aMcscr sin g ð9Þ 2.2 Control problem formulation The stator flux and torque decoupling control problem is formulated as follows: for the given motor dynamics (6) and (3) with outputs y1 ¼ cs; y2 ¼ te ð10Þ find a state feedback uc ¼ acðx;or; y�Þ ut ¼ atðx;or; y�Þ � ð11Þ such that for any admissible initial condition xð0Þ 2 D; we have exact decoupling and linearisation of the stator flux and torque regulation, and lim t!1ðy� y �Þ ¼ 0 ð12Þ where y ¼ ½ y1 y2 �T ; y� ¼ ½ y�1 y�2 �T ; y�1 ¼ c�s ; y�2 ¼ t�e are constant desired values, x ¼ ½cs;cr; ys; yr�T andD � R4 is a given region. The adaptive decoupling control problem is as follows: assume that the rotor and stator winding resistances Rr and Rs are unknown but constant. For the given motor dynamics (6) and (3) and desired values y�1 and y � 2; design a dynamical feedback controller uc ¼ acðx;or; y�; p^pÞ ut ¼ atðx;or; y�; p^pÞ _^pp^pp ¼ Pðx;or; y� Þ 8< : ð13Þ such that for any xð0Þ 2 D and p^pð0Þ; p^p is bounded and (12) holds, where p^p is a parameter estimate vector. 3 Decoupling via state feedback With the model (6) it is easily to check that the induction motor with outputs cs and te has relative degree {1, 1} in all regions of the state space where cs 6¼ 0 and cr 6¼ 0: Notice that cs 6¼ 0 and cr 6¼ 0 is a physical singularity of the induction motor in the starting conditions, and it is common practice to first magnetise the motor before closing the control loops. Therefore, throughout the rest of this paper, we will assume that the system is located in the open set O ¼ fx 2 R4 : cs; cr 6¼ 0g: Then, we can obtain _yy1 ¼ �aRsLry1 þ aRsMcr cos gþ uc _yy2 ¼ y2y1 _yy1 � aRrLsy2 þ aMcry1 cos gu 0 t _ccr ¼ �aRrLscr þ aRrMy1 cos g _gg ¼ �aRrM y1c2 sin gþ u 0 t 8>>< >>: ð14Þ where u0t ¼ 1 y1 ut � Rs y21 y2 � or ð15Þ The state feedback control uc ¼ �aRsMcr cos gþ aRsLry1 þ vc ut ¼ Rs y2y1 þ y1or þ �y2vc=y1þaRrLsy2þvt aMcr cos g ( ð16Þ results in the exact input–output decoupling and linearisa- tion system _yy1 ¼ vc _yy2 ¼ vt _ccr ¼ �aRrLscr þ aRrMy1 cos g _gg ¼ �aRrM y1cr sin gþ u 0 t 8>>< >>: ð17Þ 3.1 Tracking dynamics For the decoupled and linearisable system (17), we want design a state feedback control law such that the outputs y asymptotically track the reference signals y�: The desired feedback controller that solves tracking problem could be found if the tracking dynamics, which describe the motor’s behaviour when y ¼ y�; is stable and input-to-state stable (ISS) with respect to its inputs. Toward this end, the first question that we should tackle is the tracking dynamics. We now show that the tracking dynamics cr and g is locally asymptotically stable and ISS. In order to track the given references y�1 and y � 2 via the control inputs uc and ut respectively, from (17) we choose, for instance vc and vt as vc ¼ �l1 ~yy1 vt ¼ �l2 ~yy2 � ð18Þ IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005364 where l1>0 and l2>0 are given constants, ~yy1 ¼ y1 � y�1; ~yy2 ¼ y2 � y�2: Suppose that the tracking dynamics is achieved by the control inputs u0t ¼ u0�t ¼ atðx; y�Þ ð19Þ then u0�t necessarily satisfies aMy�1cr cos gu 0� t � aRrLsy�2 ¼ 0 ð20Þ which is obtained by setting _yy2 ¼ 0 with y1 ¼ y�1 and y2 ¼ y�2; i.e. u0�t ¼ RrLs Mcr cos g y�2 y�1 ð21Þ and the tracking dynamics is described by _ccr ¼ �aRrLscr þ aRrMy�1 cos g _gg ¼ � Rr c2r y�2 þ aRrLs tan g ( ð22Þ Proposition 3.1: For the tracking dynamics (22), we have the following results: (a) if the given references y�1 and y�2 satisfy y�2 y�21 � M 2 2ðLsLr �M2ÞLs ð23Þ then the tracking dynamics has multiple equilibrium points at g� ¼ 2kpþ 1 2 g0 c�r ¼ MLs y � 1 cos 1 2 g0 � and g� ¼ 2kpþ 1 2 g00 c�r ¼ MLs y � 1 sin 1 2 jg0j � ð24Þ where k ¼ 0;�1;�2; . . . ; g0 ¼ arcsin 2ðLsLr �M2ÞLsy�2 ðMy�1Þ2 and g00 ¼ p� g0; when g0>0�p� g0; when g0 < 0 � (b) The tracking dynamics is attracted to one of the equilibriums, if the tracking dynamics is initialised in the manifold with c2r ð0Þ > M2 2L2s y�21 � LsLr �M2 Ls ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2m � y�22 q tm ¼ M2 2ðLsLr �M2ÞLs y�21 ð25Þ For the proof of this proposition, see the Appendix, Section 9. From the representation (16) and (17), we can easily design a feedback control for driving the system into the manifold where the behaviour of the system is described by the tracking dynamics (22). Thus, our goal can be achieved with input–output decoupling control. However, toward this end, the tracking dynamics must be not only stable but also input-to-state stable with respect to y1 and u 0 r when y1 6¼ y�1 and u0r 6¼ u0�r : Indeed, when y1 6¼ y�1 and u0r 6¼ u0�r ; the dynamics coordinated by cr and g becomes _ccr ¼ �aRrLscr þ aRrMy1 cos g _gg ¼ �aRrM y1cr sin gþ u 0 t � ð26Þ Notice that the angle g is periodic in 2p; therefore, we can establish a new coordinate z1 ¼ cr sin g; z2 ¼ cr cos g ð27Þ which is one-to-one in the region�p � g � p: Then, we get _zz1 ¼ �aRrLsz1 þ z2u0t _zz2 ¼ �aRrLsz2 � z1u0t þ aRrMy1 � ð28Þ Proposition 3.2: Consider the dynamics (28), let Uðz1; z2Þ ¼ r 2 ðz21 þ z22Þ ð29Þ where r>0 is a given constant. Then, there exist positive numbers c1>0; c2>0 and c3>0 such that _UU � �c1ðz21 þ z22Þ; 8kzk � c3 c2 y1; 8u0T ð30Þ holds along the trajectories of the (28), where kzk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi z21 þ z22 p : Remark 3.1: The inequality (30) implies that the dynamics (28) is ISS with respect to y1 and u 0 t: 3.2 Input–output decoupling controller Now we present the exact input–output decoupling controller for direct torque control with stator flux regulation as follows. Proposition 3.3: Consider the system (14) with the desired reference values for the stator flux and the torque that satisfies (23). Let the control inputs uc and ut be given by uc ¼ RsfTsuis � l1 ~yy1 ut ¼ y1bðy; y�; �Þ þ Rs y2y1 þ y1or � ð31Þ where fsu ¼ fs=cs; bðy; y�; �Þ is given with any positive number l2>0 by b ¼ 1 � l1 y2 y2 ~yy1 þ aRrLsy2 � l2 ~yy2 � � ð32Þ where � ¼ aLry21 � fTs is: If the system is initiated by ðcrð0Þ; gð0Þ; y1ð0Þ; y2ð0ÞÞ which closed to the equilibrium described by Proposition 3.1 and crð0Þ satisfies the condition (25), then the control of stator flux and torque is exact decoupled and y1 ! y�1; y2 ! y�2: ð33Þ Proof: From the magnetic equations (2), we have aMcry1 cos g ¼ aMfTs fr ¼ afTs ðLrfs � a�1isÞ ¼ aLry21 � fTs is ð34Þ Observing the derivation above, it is clearly found that substituting the control law uc into the system gives _~yy~yy1 ¼ �aRsLry1 þ Rs y1 ðaLry21 � fTs isÞ þ RsfTsuis � l1 ~yy1 ¼ �l1 ~yy1 ð35Þ and IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005 365 _~yy~yy2 ¼� l1 y2 y1 ~yy1 � aRrLsy2 þ aMcry1 cos g 1 y1 ut � Rs y2 y21 � or � � ¼� l2 ~yy2 ð36Þ The proof thus follows. Remark 3.2: The proposed exact decoupling controllers (31) require the measurement of stator flux amplitude and torque, which can be calculated using the stator flux and stator current by cs ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f2sa þ f2sb q te ¼ iTs Jfs ( ð37Þ Although the stator flux is also a difficult quantity to measure, it is dynamically related to measurable and=or known quantities, as illustrated by the first dynamical equation of (1), hence, it can be calculated by online integration in the open-loop fashion c^cs ¼ Z us � Rsis ð38Þ Due to the dependence on stator resistance Rs; roundoff errors in numerical integration routines and the open-loop nature of (38), the calculated stator flux fs may be subject to drift effects during implementation. Currently, the authors are not aware of a procedure that completely eliminates the problem of integration drift. However, in an attempt to combat the effect of numerical drift, we refer the reader to [7] where the authors utilised a modified trapezoidal integration routine for the calcu- lation of stator flux during experimental trials. Remark 3.3: We have assumed that cs 6¼ 0 and cr 6¼ 0 at the beginning of this section and that the motor has been magnetised before any control loop is closed. Actually, the decoupling control (31) can be improved to magnetise the motor. In the staring conditions, if we set fsuð0Þ ¼ 0 then the derivative of stator flux amplitude will satisfy _ccsð0Þ ¼ l1y � 1>0: The motor is thus magnetised. 4 Adaptive decoupling controller When the values of the rotor and the stator resistances Rr and Rs are not known exactly, the controller presented in Section 3 will lose the input–output decoupling property, and consequently steady-state tracking errors will occur. In this section, an adaptive controller is designed by introdu- cing a dynamical parameter estimation mechanism into the controller (31). Let R^Rs and R^Rr be estimates of the resistances Rs and Rr; respectively, and define ~RRs ¼ R^Rs � Rs; ~RRr ¼ R^Rr � Rr ð39Þ First, replacing Rs by R^Rs in the control law uc; i.e. uc ¼ R^RsfTsuis � l1 ~yy1 ð40Þ we have _yy1 ¼ �l1 ~yy1 þ ~RRsfTsuis ð41Þ Furthermore, in this case, we obtain _yy2 ¼� l1 y2 y1 ~yy1 � aRrLsy2 þ � 1 y1 ut � Rs y2 y21 � or � � þ y2 y1 ~RRsf T suis ð42Þ Since Rr and Rs are unknown, we are not able to use the exact compensation ut for the decoupling control. However, motivated by proposition 3.3, let ut ¼ y1b^bðy; y�; �Þ þ R^Rs y2 y1 þ y1or ð43Þ with the estimate b^bðy; y�; �Þ ¼ 1 � l1 y2 y1 ~yy1 þ aR^RrLsy2 � l2 ~yy2 � � ð44Þ Then, the derivative of y2 becomes _yy2 ¼ �l2 ~yy2 þ ~RRs fTsuis y2 y1 þ � y2 y21 � � þ a ~RRrLsy2 ð45Þ With (41) and (45), we can find a parameter adaptation law for the input–output decoupling controller (40) and (43) as follows: Proposition 4.1: Consider the system (14) with controller (40) and (43). Let the parameter adaptation law be given by _^RR^RRs ¼ �k�11 fTsuis ~yy1 þ fTsuis y2y1 ~yy2 þ � y2 y2 1 ~yy2 � � _^RR^RRr ¼ �k�12 aLsy2 ~yy2 ( ð46Þ where k1>0 and k2>0 are adaptation gains. Then, for any given initial values R^Rsð0Þ and R^Rrð0Þ; the state ðy1; y2;cr; gÞ and the parameter estimation R^Rs; R^Rr are bounded, and y! y�: Proof: Obviously, the dynamics of the closed loop system with the dynamical parameter estimates is represented by (41), (45) and (46). For this system, we now construct a positive definite function Vð~yy1; ~yy2; ~RRs; ~RRrÞ ¼ 1 2 ~yy21 þ ~yy22 þ k1 2 ~RR 2 s þ k2 2 ~RR 2 r ð47Þ Note that, the unknown parameters Rs and Rr are constant. Then, a straightforward calculation obtains _VV ¼� l1 ~yy21 � l2 ~yy22 þ ~RRs fTsuis ~yy1 þ fTsuis y2 y1 ~yy2 þ � y2 y21 ~yy2 � � þ ~RRraLsy2 ~yy2 þ k1 ~RRs _^RR^RRs þ k2 ~RRr _^RR^RRr Thus, instituting the parameter adaptation law, gives _VV ¼ �l1 ~yy21 � l2 ~yy22 < 0; 8y1 6¼ y�1; y2 6¼ y�2 ð48Þ so that the proof follows by this inequality with the LeSalle invariant set theorem. The system block diagram of the adaptive decoupling control for induction motors is showed in Fig. 1. In the figure, ec ¼ c�s � c^cs and et ¼ t�e � t^te; which denote �~yy1 and �~yy2 in the controllers respectively. The estimated values of c^cs; t^te and y^ys can be calculated using (37) and (38). Moreover, the transformation form ðusa; usbÞ to ðusA; usB; usCÞ is better suited to induction motor control. Remark 4.1: Observing the design process, it is clear that adaptive control relies on the linearity of the unknown parameters in the state feedback law obtained in Section 3. In fact, this adaptive controller can be extended to cope with the uncertainty in other physical parameters, even for those uncertain parameters with non-linearity. However, we only IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005366 investigate, for the sake of simplicity, the resistances of the stator and the rotor. 5 Simulation results In order to verify the performance of the proposed control scheme, simulation was carried out using the software package SIMULINK, MATLAB with the ODE 45 integration method and variable step size. In the simulation a squirrel-cage induction motor was employed whose nominal parameters are: 4 kW, 220V, 50Hz, np ¼ 2; orN ¼ 1415 rpm; Rs ¼ 1:55O; Rr ¼ 1:25O; Ls ¼ 0:172H; Lr¼ 0:172H; M ¼ 0:166H; J ¼ 0:065 kgm2: In simulations we calculated the stator flux and torque by (37) and (38). The stator voltage components ðuc; utÞ were transformed into ðusa; usbÞ; which were applied to the induction motor. Moreover, the magnitude of the stator voltage was saturated jusj � 220 ffiffiffi 2 p V: We adopted an ideal voltage source in the simulations, which does not model the real switched voltages. In fact, in the implementation the stator voltages were applied to the motor as pulse width modulated patterns, which may affect the control perform- ance, e.g. by increasing the current harmonic. In the first simulation test, using the true values of motor parameters, the following operating sequence was tested. The stator flux amplitude reference is c�s ¼ 0:9Wb and the motor was unloaded before 0.1 s. At t ¼ 0:1 s a desired torque t�e ¼ 20Nm was applied to the system and decreased to 10Nm at t ¼ 0:5 s: From 1 to 1.5 s, the stator flux amplitude reference was weakened from 0.9Wb to 0.7Wb, and the torque reference set to 15Nm at the same time. A 10Nm load torque was applied to the motor after 0.l s. The decoupling controller given by (31) was used with control parameters l1 ¼ 80 and l2 ¼ 100: Figure 2 shows simulation results of flux amplitude and torque from 0 s to 0.7 s. The motor was magnetised first, and