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Research article
PI controller design for indirect vector controlled induction motor:
A decoupling approach
Jitendra Kr. J...
low load conditions for which the coupling is itself weak. Also, it
requires low-pass filters to remove the chattering effe...
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The sensitivity expression is determined following procedure in
[32]. For the purpose, first, the linearized closed-loop IV...
perturbations may affect the IVCIM performance. The effect of
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PI controller design for indirect vector controlled induction motor

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Decoupling of the stator currents is important for smoother torque response of indirect vector controlled induction motors. Typically, feedforward decoupling is used to take care of current coupling that requires exact knowledge of motor parameters, additional circuitry and signal processing. In this paper, a method is proposed to design the regulating proportional-integral gains that minimize coupling without any requirement of the additional decoupler. The variation of the coupling terms for change in load torque is considered as the performance measure. An iterative linear matrix inequality based H∞ control design approach is used to obtain the controller gains. A comparison between the feedforward and the proposed decoupling schemes is presented through simulation and experimental results. The results show that the proposed scheme is simple yet effective even without additional block or burden on signal processing.

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PI controller design for indirect vector controlled induction motor

  1. 1. Research article PI controller design for indirect vector controlled induction motor: A decoupling approach Jitendra Kr. Jain a,n , Sandip Ghosh b , Somnath Maity a , Pawel Dworak c a Department of Electrical Engineering, National Institute of Technology, Rourkela, Orissa 769008, India b Department of Electrical Engineering, Indian Institute of Technology (BHU), Varanasi 221005, India c Department of Control Engineering and Robotics, West Pomeranian University of Technology, Szczecin, Poland a r t i c l e i n f o Article history: Received 23 August 2016 Received in revised form 3 April 2017 Accepted 25 May 2017 Available online 16 June 2017 Keywords: Indirect vector controlled induction motor Coupling Iterative linear matrix inequality H1 controller Proportional-integral design a b s t r a c t Decoupling of the stator currents is important for smoother torque response of indirect vector controlled induction motors. Typically, feedforward decoupling is used to take care of current coupling that requires exact knowledge of motor parameters, additional circuitry and signal processing. In this paper, a method is proposed to design the regulating proportional-integral gains that minimize coupling without any requirement of the additional decoupler. The variation of the coupling terms for change in load torque is considered as the performance measure. An iterative linear matrix inequality based ∞H control design approach is used to obtain the controller gains. A comparison between the feedforward and the proposed decoupling schemes is presented through simulation and experimental results. The results show that the proposed scheme is simple yet effective even without additional block or burden on signal processing. & 2017 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Vector control in induction motor (IM) is used to make it behave like a separately excited direct current (DC) motor. The vector control of IM is realized in either of two ways: direct and indirect. Because of its simplicity and high performance, indirect one is often preferred over the direct one. To generate the instantaneous torque using indirect vector control (IVC) scheme [1], two current com- ponents (the direct and quadrature ones) are to be controlled in- dependently. The available current-loop control schemes are based on synchronous reference frame control (SRFC), stationary re- ference control (SRC), and hysteresis control [2]. Although SRC is simpler than the SRFC one, the latter one is preferred since it op- erates on DC quantities and zero steady-state error. These benefits can be achieved if the regulator gains are chosen efficiently [2]. For such regulation, the reference quadrature current is given by the outer speed-loop, which is realized by linear, intelligent control techniques and nonlinear methods [3–8]. However, these compo- nents are inherently coupled through the motor dynamics which affects the IM torque response. The coupled flux and torque dynamics behavior along with rotor resistance variation effect have been studied in [9]. The severity of this coupling on motoring operation depends on the control application. During the low-speed IM operation, the coupling effect can be neglected. Erstwhile, for high-speed operation, fast torque response is difficult to achieve without proper decoupling [10]. For overcoming the coupling effect, many decoupling techniques have been discussed. A complex vector synchronous frame proportional- integral (PI) controller design has been proposed in [11] to enhance the performance of synchronous PI controller. Specifically, para- meter variation causes errors in the estimated flux that may lead to the reduction of the torque performance. To overcome this, a modified decoupling control has been proposed in [12]. Additional PI controller has been used to perform the decoupling in [13]. An internal-model-control (IMC) based decoupling has also been re- ported in [14], where additional integrator dynamics has been used to minimize the coupling. Further, a dynamic controller structure for the current controllers have been used in [10] to achieve de- coupling. The rotor time-constant estimation based decoupling has been suggested in [15]. A new adaptive observer based speed esti- mation technique has been proposed in [16] that may be used for decoupling. Multivariable PI based decoupling scheme has been discussed in [17]. Here, similar to the IMC based decoupling [14], inclusion of integrators have been used for mitigating the coupling. Despite these modifications, the feedforward decoupling method [18] is still preferred due to its straight-forwardness and faster de- coupling ability. Apart from the above linear decoupling schemes, a sliding mode control based decoupling has been developed in [19,20]. Due to the lack of antiwindup capability in [19], the proposed controller may saturate at high speed limiting its application to Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions http://dx.doi.org/10.1016/j.isatra.2017.05.016 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved. n Corresponding author. E-mail addresses: jitendra1610@gmail.com (J.Kr. Jain), sghosh.eee@itbhu.ac.in (S. Ghosh), somnathm@nitrkl.ac.in (S. Maity), pawel.dworak@zut.edu.pl (P. Dworak). ISA Transactions 70 (2017) 378–388
  2. 2. low load conditions for which the coupling is itself weak. Also, it requires low-pass filters to remove the chattering effect. The de- coupling is also been attempted in sensorless control scheme [21]. A nonlinear observer is proposed to estimate the motor speed, which is used to compensate the speed dependent cross-coupled terms. A modified IVC scheme based decoupling for current source IM drive has been given in [22]. Besides these decoupling meth- ods, energy efficient model based control (MBC) has been dis- cussed in [23] to improve the torque performance. In this work, the MBC dynamical behavior for load transition has been analyzed in detail and new control strategies have been proposed. In a re- cent work [24], an SRC based cascaded combination of a PI con- troller and a compensator has been designed for satisfying speci- fied decoupling performance. In this scheme, no feedforward terms are employed, and the design criterion is simple based on gain margin (GM) and phase margin (PM) criterion. However, this robust stability based design involves a comparatively complicated structure than the conventional SRFC. In addition, since GM and PM of a multi-input multi-output (MIMO) system is involved in this design, the trade-offs of different performances, e.g., decou- pling and robustness, are hard to obtain. Modified PI controller based decoupling methods have been developed in [11,13,14,17]. However, all these methods in- corporate additional dynamics and/or induce complexity in the conventional indirect vector controlled induction motor (IVCIM) system. Moreover, the tuning of the PI controller, otherwise used for current regulations, also has an impact on the coupling [19]. Despite the fact that such tuning has limited scope to minimize coupling. However, due to its simplicity and classical performance, it is considered as a better method for decoupling rather than introducing additional dynamics. Typically, for designing a PI controller, frequency response (Bode plot) method is considered as simple. However, the ro- bustness of the system is difficult to assess using such method [25]. On the other hand, for the controller design of finite di- mensional linear time invariant systems, static output feedback (SOF) technique has been used in [26,27]. This technique has the advantages of simple closed-loop controller structure and flex- ibility of the design combined with an ∞H control. The present work is on SOF based PI design of the speed as well as the current controllers to achieve decouple transient performance. By this method, the same classical IVCIM structure is retained. The fol- lowing contributions are made in this work: (i) Sensitivity of stator currents on inner-loop controller gains is determined. (ii) The decoupling effect is formulated as a performance criterion for speed and current-loop controller design. (iii) An MIMO SOF based closed-loop structure is formulated for the linearized model of the IVCIM. (iv) An iterative linear matrix inequality (ILMI) based controller design technique is applied to design the PI gains. (v) Simulation and experimental results are presented that cor- roborates the effectiveness of the proposed controller design technique as compared to the conventional feedforward decou- pling one. The remaining of the paper is organized as follows: Section 2 presents the IVCIM model. The classical feedforward decoupler and sensitivity of stator currents are discussed in Section 3. Section 4 shows the formulation of the coupling minimization problem. The PI controller design algorithm is presented in Section 5. Comparison of the proposed method with feedforward one through simulation and experimental results is given in Section 6. Finally, conclusions are pointed out in Section 7. 2. State-space model of IVCIM The IVCIM drive system [28,29] for speed regulation, is shown in Fig. 1. In this system, the rotor speed is regulated by an outer loop PI controller. The output of this controller is used to generate the torque and the flux reference components of the current. These reference currents are compared with the actual motor currents and applied to respective PI current controllers in order to regulate the input voltage to the induction motor. The IM model is structured with the underneath assumptions: 1. Both the stator and rotor windings are arranged symmetrically. 2. The following are neglected: the dynamics associated with nonlinear magnetic circuits; the harmonic content of the mag- neto motive force (MMF) wave; variation in rotor resistance due to the changes in temperature and frequency. 3. The load torque is considered as constant. 4. The reference flux is taken as the rated flux of the motor. The dynamic induction motor model in synchronous reference frame is given by [28]: Fig. 1. Indirect vector control of induction motor. J.Kr. Jain et al. / ISA Transactions 70 (2017) 378–388 379
  3. 3. ⎫ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ( ) ω σψ ω ψ ω ω ψ σψ ψ σ σψ ω ω ψ ψ σ ω ω ψ σψ ω ω = − + + + + = − − − + + = − + ( − ) = − ( − ) − = ( − ) − 1 di dt ai i b b cv di dt i ai b b cv d dt L i d dt L i J d dt K T T B ds ds e qs dr r qr ds qs e ds qs r dr qr qs dr m ds dr e r qr qr m qs e r dr qr r m e L m r where ( ) ψ ψ= ( − ) = = = + − − T K i i K K a, , , ,e t qs dr ds qr t PL L m P R L L L L L R L L L L 3 4 2 m r s r r s m m r r s m r 2 2 2 = = − − b c, L L L L L L L L m r s m r r s m 2 2 , ids and iqs are the direct and quadrature components of stator current; ψdr and ψqr are the direct and quadrature components of rotor fluxes; ωr is angular electrical speed of the rotor; Te is the motor electromechanical torque; ωe is the speed of the synchronously rotating reference frame; Ls, Lr and Lm are the stator, rotor, and mutual inductance, respectively; Rs and Rr are the stator and rotor resistances, respectively; σ = γ 1 is the inverse of rotor time constant where γ = L R r r ; P is the number of poles; J is the moment of inertia; TL is the external load torque; Bm is the viscous friction coefficient. The output of the current controller are the stator voltages, which can be written from Fig. 1 as: ( ) ( )∫ ∫= + ( − ) = + ( − ) ( ) v K K dt i i v K K dt i i, 2ds pd id dsref ds qs pq iq qsref qs where Kpd and Kpq are the proportional constants for the direct and quadrature current components, respectively; Kid and Kiq are the integral constants for the direct and quadrature current controllers, respectively. Now, the IVC scheme for induction motor is governed by the following equation [28]: ⎫ ⎬ ⎪ ⎪ ⎭ ⎪ ⎪( )∫ ( ) ψ γψ ψ ω σ ψ ω ω ω ω ω = + ̇ = = = + ( − ) = +ω ω 3 i L i T K L i T K K dt , , , , dsref drref drref m qsref re t drref sl m qsref drref re p i ref r e r sl where idsref is the reference flux component of current; iqsref is the reference torque component of current; ψdrref is the reference flux; Tre is the torque output from the speed loop controller; ωref is the re- ference speed command; ωKp and ωKi are the speed controller gains. By defining =x ids1 , =x iqs2 , ψ=x dr3 , ψ=x qr4 , ω ω= ( − )x ref r5 , =x Tre6 , ∫= ( − )x i i dtdsref ds7 , ∫= ( − )x i i dtqsref qs8 , the state space model can be written as: ⎡ ⎣ ⎤ ⎦ ⎫ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ( ) [ ] [ ] ( ) ω σ ψ σ ω ψ ω σ ψ ω σ ψ σ σ σ ψ σ σ σ ψ ω ω γψ ψ ψ ̇ = − + [ − + ( ) ] + + ( − ) + − + ̇ = −[ − + ( ) ] − − ( − ) + + ( ) − + ̇ = − + ( ) ̇ = − − ( ) ̇ = − ( − ) − + ( − ) ̇ = − [ ( − ) − − ( − )] + ̇ = ̇ + − ̇ = ( ) − ω ω − − − − − − − − − 4 x ax x L K x x b x b x x cK L x cK x x x L K x x ax b x x b x cK K x x cK x x L x x L K x x x L x x L K x x Jx K K x x x x T B x Jx K K K x x x x T B x JK x x L L x x K x x . ref m t drref ref pd m drref id ref m t drref ref pq t drref iq m m t drref m m t drref m t L m ref p m t L m ref i m drref m drref t drref 1 1 5 2 1 6 2 3 5 4 1 1 7 2 5 2 1 6 1 2 5 3 4 1 6 2 8 3 1 3 2 1 6 4 4 2 4 2 1 6 3 5 2 3 1 4 5 6 2 3 1 4 5 5 7 1 1 1 8 1 6 2 3. Inner-loop feedforward decoupler In order to elaborate the coupling problem, consider rewriting the stator current dynamics following (4) as: ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ω σ ψ σ ω ω σ ψ ω σ ̇ = − + − + ( ) + + ( − ) + ̇ = − − + ( ) − − ( − ) + + ( ) − − x ax x L K x x b x b x x cv x x L K x x ax b x x b x cv 5 ref m t drref ref ds ref m t drref ref qs 1 1 5 2 1 6 2 3 5 4 2 5 2 1 6 1 2 5 3 4 In the above, both ids and iqs are dependent on each other, ad- ditionally on ψdr, ψqr and ωr. These are coupled through the re- maining system dynamics. However, for obtaining a torque re- sponse similar to a DC motor, the flux, and torque components of currents are required to be decoupled while regulating the re- spective currents. The torque performance can be improved by compensating the coupled terms. However, the perfect compensation of all the connected components requires an accurate update knowledge of the motor parameters. This is almost impossible in practice since induction motor parameters are known to have variations during operation, e.g., the rotor resistance variation [30]. In [10], it has been observed that (i) ω ie ds and ω ie qs are the dominating compo- nents in the respective currents, and (ii) the coupling effect in- creases as motor speed increases. So, either by removing or minimizing these dominating components, the effect of coupling phenomenon can be reduced. One way to achieve such reduction in coupling is, by introducing a supplementary feedforward con- troller to the current controller output as shown in Fig. 2(a) [31]. In this, a direct feedforward compensation vds comp and vqs comp cor- responding to the dominating components are used. The compensation voltage signals are: ⎡ ⎣⎢ ⎤ ⎦⎥( )ω ω σ ψ = − = − − + ( ) − v i c x x L K x c 6ds comp e qs ref m t drref2 5 2 1 6 ⎡ ⎣⎢ ⎤ ⎦⎥( )ω ω σ ψ = = − + ( ) − v i c x x L K x c . 7qs comp e ds ref m t drref1 5 2 1 6 The effect of such decoupling method for the IM (Parameters are given in Section 6) is shown in Fig. 2(b). This can be seen that for a change in TL (6.5 N Á m applied at 10 s), the effect in ids is con- siderably reduced by the decoupler. Note that, the above result is obtained while considering same speed and current-loop controller gains. However, for a different set of these controller gains, the performance of the decoupling scheme varies. Moreover, it may be possible to improve the de- coupling by suitable choice of the regulatory PI gains. Since feed- forward decoupling scheme introduces complexity in IVCIM and it is more susceptible to parameter variations [10], an alternate method is proposed in this work. 3.1. Decoupler sensitivity 3.1.1. Current controller tuning effect For IVCIM, the coupling is primarily affected by inner-loop controller tuning. Since cross-couplings exist between the two current loops, controller tuning for the d-axis stator current may affect the q-axis one and vice versa. The sensitivity of the currents with respect to the cross-controller gains are studied here. J.Kr. Jain et al. / ISA Transactions 70 (2017) 378–388380
  4. 4. The sensitivity expression is determined following procedure in [32]. For the purpose, first, the linearized closed-loop IVCIM model is obtained as follows. Let xe (xi e , = …i 1, , 8) be the equilibrium state of (4). It can be determined following the vector control principle [28,33] as: ψ ω ψ ψ ψ = = + = = = = ( ) x L x B K T K K x x x x K x , , , 0, , 8 e drref m e m ref m L m t drref e drref e e e t drref e 1 2 3 4 5 6 2 x7 e and x8 e depends on inner-loop control parameters and can also be determined from (4). The linearized model is obtained by re- placing = + Δx x xe in (4). Neglecting the current-loop integrators and outer speed-loop dynamics, the linearized closed-loop IVCIM model in SISO form (only ids as the output) can be written as: Δ ̇ = Δ Δ = Δ ( )− − −x G x x C x, 91 4 1 4 1 1 4 where ⎡⎣ ⎤⎦Δ = Δ Δ Δ Δ−x x x x x , T 1 4 1 2 3 4 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = ⋯ ⋮ ⋮ ⋮ ⋯ G g g g g 11 14 41 44 , = − −g a cKpd11 , ω=g e12 , σ=g b13 , ω=g b ref14 , ω= −g e21 , = − −g a cKpq22 , ω= −g b ref23 , σ=g b24 , σ=g Lm31 , =g 032 , σ= −g33 , ω=g sl34 , =g 041 , σ=g Lm42 , ω= −g sl43 , σ= −g44 , ⎡⎣ ⎤⎦=C 1 0 0 0 . With a perturbation ΔKpq in Kpq, the above can be written as: Δ ̇ = Δ + Δ Δ = Δ ( )− − − −x G x Gx x C x, 10 e 1 4 1 4 1 4 1 1 4 where ΔG represents perturbed system matrix as ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Δ = Δ Δ × × × G M K K 0 0 0 ,pq pq 2 2 2 2 2 2 is the variation in K ,pq ⎡⎣ ⎤⎦=−x x x x xe e e e e T 1 4 1 2 3 4 , ⎡ ⎣⎢ ⎤ ⎦⎥= − M c 0 0 0 . Now, the sensitivity function for Δids can be derived from (10) as: Δ Δ = − ( ) i K ch i Q 11 ds pq qs e 12 where s represents complex frequency, =h g g g12 12 34 43 + + − − −g g g g g g g g g g g g g g g13 32 44 14 33 42 12 33 44 13 34 42 14 32 43 , = −Q sI G. Similarly, considering the dynamics involving only iqs as the output, the sensitivity function for Δiqs can be determined as: Δ Δ = − ( ) i K ch i Q 12 qs pd ds e 21 where ΔKpd is the variation in Kpd, = +h g g g g g g21 21 34 43 23 31 44 + − − −g g g g g g g g g g g g .24 33 41 21 33 44 23 34 41 24 31 43 The above sensitivities are studied here for an IM with the parameters given in Section 6 later. For evaluating (11), the equi- librium point is determined at TL ¼ 5 N Á m (chosen arbitrarily) and ω ω≃e r ¼ 80 rad/s. These sensitivity functions are analyzed with the magnitude plot as shown in Fig. 3(a). It is observed that the current perturbations can be minimized by tuning the controller with high Kpd and Kpq. However, due to non-ideality, noises and delay in physical implementation, higher gains may actuate os- cillations of large magnitude or instability, hence a systematic design is required. The sensitivity of ids with feedforward decou- pler is shown in Fig. 3(b). For this, the equilibrium point is de- termined at TL ¼ 5 N Á m and with ωe ¼ 0 (perfect compensation is considered). It is observed that high controller gains reduce the current perturbations, however, sensitivity is slightly lower with this decoupling as compared to the without decoupler case. The sensitivity of ids (without decoupling) with variation in frequency is shown in Fig. 4(a). It is observed that sensitivity is high at low frequency (ω ¼ 0.1 rad/s). As frequency reaches 20 rad/s, a small peak is observed due to the cancellation of motor dynamics [14]. Furthermore, it is reduced with increasing frequency and Kpd, Kpq. Note that, h12 and Q depend on Rr and Lm. Since an accurate information of these parameters is impossible to get in practice, Fig. 2. (a) Feedforward decoupling of inner-loop. (b) Behavior of ids for feedforward decoupling scheme. Fig. 3. Sensitivity of ids on controller gains: (a) Without decoupler. (b) With decoupler. J.Kr. Jain et al. / ISA Transactions 70 (2017) 378–388 381
  5. 5. perturbations may affect the IVCIM performance. The effect of improper parameter estimation on current controller gain sensi- tivity is also shown in Fig. 4(b). It is observed that, in feedforward decoupling, overestimation in compensation signals increase the magnitude. However, it is less affected when decoupler is not used. The variation effect of ωr on sensitivity is depicted in Fig. 5. It is observed that higher ωr increases sensitivity. 3.1.2. Parameter variation effect In feedforwad decoupling scheme, the compensation terms vds comp and vqs comp are function of ωsl and c, which depends on Rr and Lm, respectively. Both of these IM parameters may vary with temperature, ageing and other environmental reasons [30]. For determining Rr variation effect on feedforward decoupling, ΔG in (10) is written as: ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ωΔ = − − Δ ( ) × × G E E 0 0 13 sl 2 2 2 2 where ⎡ ⎣⎢ ⎤ ⎦⎥= − E 0 1 1 0 , ωΔ = ψ Δ sl R L i L r m qs e r drref , ΔRr represents perturbation in Rr. The corresponding sensitivity is obtained as ω ψ ψΔ Δ = − + + − ( ) i h i h i h h Q 14 ds sl qs e ds e qr e dr e 11 12 13 14 where = + + − − −h g g g g g g g g g g g g g g g g g g12 12 34 43 13 32 44 14 33 42 12 33 44 13 34 42 14 32 43 , = + + − − −h g g g g g g g g g g g g g g g g g g13 12 23 44 13 24 42 14 22 43 12 24 43 13 22 44 14 23 42 , = + + − − −h g g g g g g g g g g g g g g g g g g14 12 24 33 13 22 34 14 23 32 12 23 34 13 24 32 14 22 33 . The stator current sensitivity magnitude for Rr variation ( )20% is shown in Fig. 6(a). It is observed that sensitivity is comparatively larger with decoupler (ωe ¼ 0). The corresponding sensitivity magnitude varia- tion with change in Kpd and Kpq is shown in Fig. 6(b). It is observed that, initially, the magnitude with decoupler is higher for low fre- quency (up to 20 rad/s). However, it is reduced with higher Kpd and Kpq, e.g., = =K K 100pd pq and frequency. A further increment in Kpd and Kpq reduces the sensitivity magnitude with a marginal value. This shows that the system is quite sensitive to parameter variations. 4. Coupling minimization problem Considering the disturbance input to the system through the variations in load torque TL that can be modeled as disturbance ⎡⎣ ⎤⎦Δ = Δw TL , the linearized dynamics of (4) can be represented as: ⎫ ⎬ ⎪ ⎭ ⎪ Δ ̇ = Δ + Δ + Δ Δ = Δ Δ = Δ ( ) x A x B w B u z C x y C x 15 1 2 1 2 where ⎡⎣ ⎤⎦Δ = Δ Δ Δ Δ Δ Δ Δ Δx x x x x x x x x T 1 2 3 4 5 6 7 8 represents the linearized states, Δy and Δz are the measured and desired outputs, respectively, Δw is the disturbance and Δu represents the control input. The corresponding plant matrix A is given by: Fig. 4. Sensitivity of ids with variation in frequency at ωe ¼ 80 rad/s: (a) Varying Kpd, Kpq. (b) =K Kpd pq ¼ 100 and 35% overestimation of Rr and Lm. ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ( ) ( ) ω σ ω ω ω σ σ σ σ σ= − − + − − − − − − − − + − − − − − − − − − − σ ψ σ ψ σ ψ σ ψ σ ψ σ ψ σ ψ σ ψ ψ A a x b b x x bx x a b x b x bx L L 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 . ref e L x K ref e e e L x K e ref L x K ref e e e L x K m L x K L x K m L x K L x K K K x J K K x J K K x J K K x J B J K 11 5 5 2 4 5 5 1 3 1 m e t drref m e t drref m e t drref m e t drref m e t drref m e t drref m e t drref m e t drref m t e m t e m t e m t e m t drref 6 2 2 2 6 2 1 2 6 2 4 2 6 2 3 2 4 3 2 1 J.Kr. Jain et al. / ISA Transactions 70 (2017) 378–388382
  6. 6. ⎡⎣ ⎤⎦= ( )×A A 0 1611 8 2 where The input matrices are as: ⎡ ⎣ ⎢ ⎤ ⎦ ⎥= ( ) B K J 0 0 0 0 0 0 0 17 m T 1 and ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥= ( ) B c c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 . 18 T 2 The PI controller is defined as: Δ = Δ ( )u F y 19 where F contains PI gains of all the loops as: ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = ω ω F K K K K K K 0 0 0 0 0 0 0 0 0 0 0 0 pd id pq iq p i For minimizing the effect of iqs variation, the dominating nonlinear terms ω ie ds and ω ie qs are linearized and chosen as the performance variables. Correspondingly, output performance matrix C1 is ob- tained as ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ( ) ω σ ψ ω σ ψ σ ψ σ ψ = − + − + 20 C x L x K x L x K x L x K x L x K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ref e m e t drref ref e m e t drref e m e t drref e m e t drref 1 5 6 2 5 6 2 2 2 2 1 1 2 and output feedback matrix is: ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ( ) ψ = − − − − − 21 C K K K x J K K x J K K x J K K x J B J 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 . t drref m t e m t e m t e m t e m 2 4 3 2 1 In the coupling minimization problem, one need to design F (static feedback controller matrix). Here, F is in a block-diagonal structure which is treated in decentralized controller design framework. In this work, the decoupling objective is defined as: to tune F for minimizing the effect of load perturbation Δw on performance variables Δz. 5. PI controller design For the present problem, SOF based PI controller design algo- rithm [27] is used. Referred to the system dynamics (15), the effect of the disturbance Δw on Δz can be quantified by the ∞ norm of the closed-loop system H(F) as: ρ∥ ( )∥ = Δ ( ) Δ ( ) ≤ ( ) ∞ Δ ( )≠ H F z t w t max 22w t 0 2 2 where . 2 represents the L2-norm. In respect to the above, minimizing ρ yields a controller with good disturbance rejection behavior. For this, the below result is well known [26]. Theorem 1. System (15) along with feedback gain (19) satisfies ∞H performance of ρ in (22) if there exists a = >L L 0T satisfying Fig. 5. Sensitivity of ids with variation in frequency at = =K K 100pd pq for different choices of ωr. Fig. 6. Sensitivity of ids on change in Rr with variation in frequency for feedforward decoupler: (a) = =K K 10pd pq . (b) Varying Kpd, Kpq. J.Kr. Jain et al. / ISA Transactions 70 (2017) 378–388 383
  7. 7. ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ρ ρ + − − < ( ) A L LA LB C B L I D C D I 0 23 c T c c c T c T c T c c where = +A A B FCc 2 2, = +B B B FDc 1 2 21, =C Cc 1 and =D 0c . The above criterion satisfies ∞H performance (22) for a chosen ρ and is a nonlinear matrix inequality for synthesis of F due to multi- plication of unknown L and F matrices. In order to design the PI controllers embedded in F, procedures utilizing the benefits of solving LMIs is adopted. It may be noted that LMIs are convex optimization problems and computational tools are available for solving them [34]. Note that, (23) can be written as: ¯ ¯ + ¯ ¯ + (¯ ¯ ¯ ) + ¯ ¯ ¯ < ( )A L LA LBFC LBFC 0 24 T T where ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ρ ρ ¯ = − − ¯ = ¯ = ¯ =A A B I C I L L I I B B C C D 0 0 /2 0 0 /2 , 0 0 0 0 0 0 , 0 0 , 0 . T 1 1 2 2 21 The inequality (24) is satisfied if one can achieve ¯ ¯ + ¯ ¯ − ¯ ¯ ¯ ¯ + ( ¯ ¯ + ¯ ) ( ¯ ¯ + ¯ ) < ( )A L LA LBB L B L FC B L FC 0 25 T T T T T In (25), the ¯ ¯ ¯ ¯LBB L T term is nonlinear since it involves multiplication of the unknown variable ¯L with itself. To solve (25) from here, one adhere to an iterative process introducing an updating variable ¯X in the place of ¯L. For the purpose, note that, for any ¯X the following is satisfied. ¯ ¯ ¯ ¯ ≥ ¯ ¯ ¯ ¯ + ¯ ¯ ¯ ¯ − ¯ ¯ ¯ ¯ ( )L BB L X BB L L BB X X BB X 26 T T T T T T T T Replacing (26) into (25), one gets ( )¯ ¯ + ¯ ¯ − ¯ ¯ ¯ ¯ − ¯ ¯ ¯ ¯ + ¯ ¯ ¯ ¯ + ( ¯ ¯ + ¯ ) ( ¯ ¯ + ¯ ) < 27A L LA XBB L LBB X XBB X B L FC B L FC 0 T T T T T T T The above is an LMI on the variables ¯L and F that satisfies (25) and thereby (23) for any ¯X. Moreover, ¯ = ¯X L yield the least con- servative solution of (27) with respect to the bounding (26). Fur- ther, it may be considered that there exists a <r 0 for which (26) is equivalent to ¯ ¯ + ¯ ¯ − ¯ ¯ ¯ ¯ − ¯ ¯ ¯ ¯ + ¯ ¯ ¯ ¯ + ( ¯ ¯ + ¯ ) ( ¯ ¯ + ¯ ) − ¯ < ( ) A L LA XBB L LBB X XBB X B L FC B L FC rL 0 28 T T T T T T T The LMI (28) is solved iteratively starting with an initial ¯X and then updating ¯X with the new ¯L in each iteration with the re- laxation introduced by the use of r. Once r reaches negative value, the feasibility of (27) is attained. Using Schur complement, (28) can be written in LMI form as: ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ( ) ¯ ¯ + ¯ ¯ − ¯ ¯ ¯ ¯ − ¯ ¯ ¯ ¯ + ¯ ¯ ¯ ¯ − ¯ ( ¯ ¯ + ¯) * − < 29 A L L A X BB L L BB X X BB X rL B L FC I 0 T i i i T i i T i i T i i i T i T Finally, to start with an initial ¯X following (26), one considers solving for ¯L satisfying ¯ ¯ + ¯ ¯ − ¯ ¯ ¯ ¯ < ( )A L LA LBB L 0. 30T T T Using Schur complement, (30) can be written in LMI form as: ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ρ ρ ^ + ^ − ^ * − * * − < ( ) LA AL B B B LC I I 0 0 31 T T T 2 2 1 1 Table 1 Experimental IM parameters. Notation Value Notation Value P 4 Ωb (rated speed) 1410 rpm Rs 5.12 Ω Rr 2.26 Ω Ls 0.332 H Lr 0.338 H Lm 0.320 H J 0.085 kgm2 Fig. 7. Bode plot for inner-loop PI controllers: (a) d current component. (b) q current component. Table 2 Initial parameters for Algorithm 1 Notation Value Notation Value ψdrref 0.65 Wb ωref 80 rad/s Bm 0 N Á m/ rad/s TL 0 N Á m Table 3 Controller gains. Description Notation and value Existing design Current controller [19] Kpd ¼ 8.97, Kid ¼ 1536, Kpq ¼ 8.97, Kiq ¼ 2142 Current controller [2] Kpd ¼ 0.53, Kid ¼ 247, Kpq ¼ 0.53, Kiq ¼ 247 Proposed design Current controller Kpd ¼ 108, Kid ¼ 4636, Kpq ¼ 2.2, Kiq ¼ 123 Speed controller ωKp ¼ 0.4, ωKi ¼ 1.2 J.Kr. Jain et al. / ISA Transactions 70 (2017) 378–388384
  8. 8. Based on the above development, the below ILMI algorithm is used to design the controller F. Algorithm 1. Obtain the values A, B1, B2, C1, C2 of the IVCIM model as per (15). 1: Set i¼0. Specify a ρ > 0 and solve for >L 0 satisfying (31). 2: Update = +i i 1 and =X L1 . 3: Solve for = >L L 0T and F while minimizing r using gevp program in MATLAB satisfying (28). 4: If <r 0 or the iteration i reached a limit then go to next step, else go to step 3. 5: If <r 0 then assign =F Ffinal , ρ ρ=final , reduce ρ and go to step 2. Else stop, Ffinal is the required controller gain matrix. Remark 1. Since the final controller design depends on the con- vergence of the above Algorithm 1, the proposed method may be hindered by possible convergence failure in online implementa- tion, particularly with variation in ψdrref and ωref. This can be overcome by offline tuning of the controller parameters and then using look-up tables for online implementation. 6. Validation and discussions 6.1. Simulation results An IVCIM (Fig. 1) having the machine parameters in Table 1 is used for validating the proposed design. First, the conventional feedforward design is tested. The inner-loop PI design techniques have been discussed in [19,2,35]. However, corresponding to higher bandwidth requirement, current controller gains are Fig. 8. Comparison between proposed and feedforward decoupling for a step =T 6.5L N Á m applied at 10 s: (a) Stator current ids. (b) Stator current iqs. Fig. 9. ωr for change in ωref in step ( → )100 150 . Fig. 10. Parameter variation effect on behavior of ids. Fig. 11. Measurement noise affect on ids: (a) Feedforward decoupling. (b) Proposed decoupling. J.Kr. Jain et al. / ISA Transactions 70 (2017) 378–388 385
  9. 9. selected according to the placement of the closed-loop poles and zeros of the IVCIM [19], not the sliding mode controller. This de- sign considers ids and iqs are decoupled. Here, the poles are chosen as 300. Next, zeros are selected to cancel the plant IM poles. The obtained gains are given in Table 3. Fig. 7 shows the Bode plot of the transfer function ( ) ( )C s H s , where C(s) is the current-loop PI controller and H(s) is the corresponding open-loop transfer func- tion considering single-input single-output model. The bandwidth of the current loops for the designs of [19] and [2] are 380 rad/s, 400 rad/s and 45 rad/s, 45 rad/s for the d and q current compo- nents, respectively. Whereas, the proposed approach provides a bandwidth of 5000 rad/s for the d current component, which is Fig. 12. Experimental setup. Fig. 13. Experimental result for variation in TL: (a) Without decoupling [19]. (b) Without decoupling [2]. (c) Feedforward decoupling. (d) Proposed decoupling. J.Kr. Jain et al. / ISA Transactions 70 (2017) 378–388386
  10. 10. higher than both the existing designs and represents faster inner- loop response and 38 rad/s for the q current component. For the proposed design, first, the parameter ρ in Algorithm 1 is chosen as 12. The reference inputs and the obtained gains are shown in Table 2 and 3, respectively. It may be noted that the proposed one yields higher Kpd, Kid but lower Kpq, Kiq. However, without systematic design procedure, it is difficult to arrive at such gains. Note that, the same speed-loop gains are used to compare the decoupling performance of the current controller design methods [19,2]. First, the above two designs are assessed through simulation. The load variation effect on coupling is shown in Fig. 8. It is assessed that as TL is increased from 0 to 6.5 N Á m (chosen arbitrarily), ids of the feedforward decoupling scheme produces a larger shoot. Whereas, in the proposed decoupling, a smaller overshoot occurs in ids. The outer speed-loop robustness is tested with a step change in ωref from →100 150 rad/sec. The corresponding ωr response in Fig. 9 shows robustness to the sudden variation in ωref. Since ωsl and parameter c are susceptible to variations, the feedforward scheme may lead to mismatch in compensation [10]. By considering total 50% variation (35% in Rr, 15% in Lm and Ls), the effect of feedforward decoupling on ids is shown in Fig. 10. It is observed that as TL is changed at 10 s, such parameter variation deteriorates ids current of the feedforward decoupling method. This behavior corroborates the result in Section 3. Whereas, the proposed decoupling method yields a smoother ids current. Moreover, measurement noise also affects the dynamic motoring response. The effect of such noise on ids behavior for a change in TL is shown in Fig. 11. It is observed that, measurement noise affects ids current more in the feedforward scheme than with the pro- posed decoupling design. Here, the measurement noise is con- sidered in the feedback speed signal. The noise parameters are taken as: mean value of 0, variance 0.5 and initial seed as 50,000. Remark 2. The objective of the proposed decoupling technique is on the design of controller gains while minimizing the coupled terms ω ie ds and ω ie qs effect on the system dynamics. As seen, this technique yields higher Kpd, Kid, while lesser values for Kpq, Kiq are obtained. These gains corroborate the result in Section 3. The current controller sensitivity expression in Eqs. (11) and (12) shows that higher pro- portional gains may reduce the perturbation in respective currents. Since the impact of coupled dynamics is more on ids current, the corresponding controller gains is obtained as higher than that with the iqs. Therefore, when TL is changed, due to high bandwidth control for ids, less perturbation occurs in the corresponding current. 6.2. Experimental results The developments are next validated on an experimental bench having KIRLOSKAR make 2.2 kW, 1410 rpm IM with a coupled dc generator load. The experimental prototype is shown in Fig. 12. The controller is implemented through dSPACE 1103 development board. The sampling frequency for the control process is taken as 10 kHz. An SPWM3 block is used to generate the sinusoidal pulse width mod- ulation (PWM) pulses. The obtained PWM pulses are amplified from 5 to 15 volts using optocoupler 6N137 and CD4504 ICs corresponding to driver circuit (SKYPER 32 R) requirement of SEMIKRON make three phase rectifier- inverter stack. The rectifier circuit consists of an SKD160/18 diode bridge and two dc link capacitor of 4700μF. The switching frequency of the IGBT (SKM75GB12T4) based inverter is taken as 5 kHz. Three LA 25-NP sensors are used to feedback the stator currents to dSPACE ADCs. An infrared sensor is placed nearby an 18 slotted disk to sense the motor speed. The motor parameters are the same as the one used for simulation purpose (Table 1). Since inner-loop gains primarily affect the coupling between ids and iqs, therefore, first, the effect of load variation is tested with the pro- posed decoupling speed-loop gains and inner-loop gains in [19,2]. When the motor speed reaches to applied reference speed (130 rad/s), a step load torque of about 6.5 NÁ m is exerted. The corresponding experimental result for ωr, ids and iqs are shown in Fig. 13(a)–(b). It is observed that as TL is changed, a large overshoot of about 10 A occurs in ids. Next, the feedforward compensation is implemented with the inner-loop PI gains in [19]. For this, the IM dynamics behavior with a change in TL is shown in Fig. 13(c). Further, same load variation is Fig. 14. Experimental result for change in ωref in step ( → → )100 150 100 . Fig. 15. Experimental result of current tracking performance: (a) For change in TL. (b) For change in ωref. J.Kr. Jain et al. / ISA Transactions 70 (2017) 378–388 387
  11. 11. applied to the proposed decoupling gains as shown in Fig. 13(d). It is seen that ids with the feedforward and proposed design shows a small overshoot of about 6A (zoomed portion), which is much lesser than the ids in Fig. 13(a)–(b). Such behavior confirms the simulation results in Fig. 8. Here, ids response is almost similar in both the feedforward and proposed decoupling. Hence, without additional arrangements (as required in feedforward decoupling), similar decoupling performance is achieved using the proposed method with the same regulating PI controllers. The experimental result for outer speed-loop control is shown in Fig. 14. It is seen that ωr tracks smoothly even at large step change in ωref. The current tracking performance of iqs for step change in TL (6.5 NÁ m) and ωref ( → → )100 150 100 is also shown in Fig. 15, which shows smooth current tracking is achieved using proposed design. 7. Conclusions In this work, controller parameters (that are conventionally used for regulation) are designed for minimizing coupling effect of the multi-loop IVCIM system. The traditional approaches rely on addi- tional circuitry or modified controller which induce complexity in conventional IVC approach. The proposed method can improve the torque response yet inheriting other features of the IVC scheme. Moreover, the sensitivity of current controller gains are also de- termined which shows larger gains may reduce coupling. However, it may introduce oscillations or even instability. To avoid this, a design procedure has been given where the coupling terms are treated as disturbance in order to achieve decoupled performance. The benefits of the proposed method are: (i) the same current controllers can be used for decoupling (no additional controller or signal processing re- quired), (ii) all the loops are designed simultaneously that embeds interactions of the individual loops in a single step design and (iii) the design method offers incorporating the additional design criterion, such as, robustness with respect to structured uncertainties since the design is based on Lyapunov analysis and solving LMI criterion. Comparison of the proposed design with the feedforward one shows that feedforward decoupling scheme, even though it requires addi- tional circuitry or signal processing, can only produce similar decou- pling performance. The simulation and experimental results verify that the proposed decoupling scheme provides an alternative for de- coupled performance than the conventional one, yet maintaining regulating performance. 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