DESIGN OF ROBUST CONTROL SYSTEM FOR THE PMS MOTOR

Journal of ELECTRICAL ENGINEERING, VOL 58, NO 6, 2007, 326 333 DESIGN OF ROBUST CONTROL SYSTEM FOR THE PMS MOTOR Ahmed Azaiz Youcef Ramdani Abdelkader Meroufel The field orientation control (FOC) consists in regulating the flux by a component of the current and the torque by the other component So, it is necessary to choose a system of axes d and q and a law of control ensuring the decoupling of the torque and flux In the case of the permanent magnets synchronous machine, if current i d is maintained null, the reaction flux of the stator reaction flux is in quadrature with the flux produced by the permanent magnets Then the torque becomes directly proportional to the current i q, and the model of the machine becomes equivalent to that of the DC machine As a result the model of the PMS machine is divided into two independent SISO systems In this article, we present the design of H suboptimal controllers for the PMS motor K e y w o r d s: stabilizable, detectable, structured singular value, uncertainty, robust stability, robust performance 1 INTRODUCTION The variation of stator resistances and cyclic inductances of the PMS motor induces a difference between the process and the model used for the synthesis of the controllers We consider the design of robust control systems for the PMS motor, which takes into account the structured (parametric) perturbations in the plant coefficients Robust stability and robust performance analysis of the closed-loop systems are presented To design the H suboptimal controllers, we use Glover s and Doyle s method [2, 3,4,6 for a system P, where P is the interconnection matrix for control design The system P is partitioned as follows: P A B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22 The following assumptions are made: (i) (A, B 2, C 2 ) is stabilizable and detectable, (ii) D 12 and D 21 have full rank The controller K is designed such that the H norm of the closed-loop transfer function matrix is less than a given positive number γ, ie F l (P, K) < γ (1) where γ > λ 0 : min K stabilizing F l (P, K) Fig 1 A feedback system Two H suboptimal controllers are designed, one controls the current i q and the other controls the current i d The feedback configuration is shown in Fig 1 2 MODELS AND DECOUPLING OF THE PMSM MACHINE The model of the PMSM machine can be described by the following equations expressed in the rotor reference frame (qd frame): v d r s i d + L d di d dt ωi q (2) v q r s i q + di q dt + ωl di d + ωφ f (3) T e 3 2 pφ fi q + (L d )i d i q (4) ω pω r The mechanical equation is given by J dω r dt T e Fω r T m (5) ω r dθ dt (6) The decoupling by compensation is performed by regulating the currents i d and i q, neglecting the coupling terms in expressions (2) and (3) The model of the machine becomes: v d0 r s i d + L d di d dt v q0 r s i q + di q dt (7) (8) The coupling terms are introduced again at the outputs of the current controllers K d (s) and K q (s) in order to Department of Electronics, Faculty of Sciences of the engineer, Djilali Liabes University Sidi Bel Abbes Algeria; azaiz ah@yahoofr; Department of Electrical Engineering, Faculty of Sciences of the engineer, Djilali Liabes University Sidi Bel Abbes Algeria; ramdaniy@yahoofr ISSN 1335-3632 c 2007 FEI STU

Journal of ELECTRICAL ENGINEERING VOL 58, NO 6, 2007 327 Fig 2 Reference voltages generation v dr Fig 3 Reference voltage generation v qr Fig 4 Field orientation control of the PMSM motor obtain the reference voltages v dr and v qr as shown in Figs 2 and 3 The entire block diagram representing the field orientation control (FOC) of the PMSM machine with decoupling by compensation is shown in Fig 4 3 H DESIGN OF CONTROLLERS 31 Design of the controller K q In a realistic system, the physical parameters r s, L d, and are not known exactly Then their values are assumed to be within certain known intervals That is: r s r s (1 + p r δ r ) (9) L d L d (1 + p d δ d ) (10) (1 + p q δ q ) (11) Where r s, L d and are the nominal values of r s, L d and, p r, p d, and p q, and δ r, δ d, and δ q are the possible relative perturbations on these parameters Fig 5 The q-axis block diagram of the PMS motor To design the controller K q for the q-axis model, we derive, first, from equation (8) the following expression: i q di q dt 1 (v q0 r s i q ) (12) The block diagram of the PMS motor in the q-axis is shown in Fig 5 The expressions of r s and are given by equations (9) and (11) respectively Let us take: p r p q p d 07, 1 δ r 1, 1 δ q 1 This represents to up 70 % uncertainty in the resistance r s and the inductances L d and The quantity 1 can be represented as a linear fractional transformation (upper LFT) in δ q 1 1 (1 + p q δ q ) 1 P q δ q (1+p q δ q ) 1 ( ) F u Mq, δ q with M q [ 1 pq 1 p q [ 07 3571429 07 3571429 Similarly, the parameter r s r s (1 + p r δ r ) can be represented as an upper LFT in δ r [ ( ) 0 rs r s F u Mr, δ r with Mr p r r s [ 0 06 07 06 To represent the q-axis model as an LFT of the unknown real perturbations δ r, δ q, we represent the above LFTs as block diagrams, and denote the inputs and outputs of δ r, δ q as y r, y q and u r, u q respectively as shown in Fig 6

328 A Azaiz Y Ramdani A Meroufel: DESIGN OF ROBUST CONTROL SYSTEM FOR THE PMS MOTOR Fig 6 Block diagram of the q-axis model with uncertain parameters [ rs r s [ 2142857 C 1, 06000 [ p q [ pq D 11 07000 2500000 0 0 0 0 [ 1 [ D 12 3571429, C 0 0 2 1 D 21 [ 0 0, D 22 0 The uncertain behaviour of the original system can be described by an upper LFT representation: y F u ( Gq, ) v q0 with diagonal uncertainty matrix: [ δq 0 diag(δ q, δ r ) 0 δ r, Fig 7 Closed-loop system structure [ [ 1 i pq [ [ [ [ q L q u q yr 0 rs ur y 1, q p q v q0 v r v r p r r s i q Let us choose the state variable x 1 and the output y as follows: x 1 i q and y i q The equations governing the system dynamic behaviour are as follows: ẋ 1 y q y r y [ uq u r r s p p q 1 q r s p p q 1 q r s 0 0 0 1 0 0 0 [ [ δq 0 yq 0 δ r y r x 1 u q u r v q0, Let G q denote the input/output dynamics of the PMS motor in the q-axis system, which takes into account the uncertainty of parameters The state space representation of G q is G q A B 1 B 2 C 1 D 11 D 12, C 2 D 21 D 22 B 1 [ p q A r s 2142587, p q [ 07000 2500000, B 2 1 3571429, Fig 8 Structure of open-loop system The block diagram of the closed-loop system showing the feedback structure and including the elements reflecting model uncertainty and performance requirements is given in Fig 7 The mechanical torque T m is considered as a disturbance This disturbance is reflected on the output of the plant Then T m is divided by the quantity 3 2 pφ f in order to obtain the disturbance d The performance weighting function is a scalar function and is chosen as W p (s) s2 + 200s + 100 s 2 + 05s + 006 which ensures good disturbance attenuation, good transient response ( settling time less than 002 second, and an overshoot less than 1 % for he nominal system) The control weighting function is chosen as the scalar W u 001 The structure of the open-loop system is represented in Fig 8 The open-loop system sys iq consists of four inputs, five outputs and three states (one state (x 1 ) of the plant and two states (x 2, x 3 ) of the weighting function W p ) [ sys iq ẋ 1 ẋ 2 ẋ 3 y q y r e p e u y c 21428 0 0 070 25000 0 35714 1182 013 010 0 0 1182 0 448 010 036 0 0 448 0 21428 0 0 21428 25000 0 35714 060 0 0 0 0 0 0 060 1182 448 0 0 0600 0 0 0 0 0 0 0 001 100 0 0 0 0 100 0 x 1 x 2 x 3 u q u r d v q0

Journal of ELECTRICAL ENGINEERING VOL 58, NO 6, 2007 329 The controller to be designed for the connection sys iq of type SYSTEM, is an H suboptimal controller This controller minimizes the infinite-norm of F L (P, K) over all stabilizing controllers K F L (P, K) is the transfer function matrix of the nominal closed-loop system from the disturbance d to the errors (e), where e [ e p e u The corresponding transfer function matrix P is extracted from the variable sys iq It consists of two inputs d, and v q0, three outputs e p, e u, and y c, and three states x 1, x 2, and x 3 P 2142857 0 0 0 3571429 118250 01326 01062 118250 0 44867 01062 03674 44867 0 06000 118250 44867 06000 0 0 0 0 0 00100 10000 0 0 10000 0 In order to compute the controller K, and find the positive number γ, the Glover s and Doyle s algorithm [2, 3, 4, 6 is used Based on the matrix P, the number of measurements, the number of controls, the lower bound on gamma (γ min 01), and the upper bound on gamma (γ max 1), this algorithm computes the controller K, which is given as a transfer function matrix; it consists of one output v q0, one input (e k i qr i q ), and three states x k1, x k2, and x k3 K q K 00514 10 7 10129 10 7 03843 10 7 0 0 01326 01061 02573 0 01061 03673 00976 00066 10 7 01303 10 7 00494 10 7 0 The final value of gamma found by the algorithm is γ 06008 The Redheffer star product of the two matrices P and K gives the transfer function matrix of the closed-loop system from d to e, clp: 2142857 0 0 118249 01326 01061 44867 01061 03673 0 0 0 clp 02573 0 0 00976 0 0 06000 118249 44867 0 0 00 02361 10 8 46548 10 8 17661 10 8 0 0 0 0 118249 0 0 0 44867 514157 10 4 01012 10 8 00384 10 8 0 01326 010061 002573 0 01061 03673 00976 0 0 0 06000 6613035 130336461 49452944 0 The transfer function matrix clp consists of six states, two outputs e p and e u, and one input d clp < 06008 Fig 9 Robust stability analysis of K q 32 ROBUST STABILITY AND ROBUST PERFORMANCE The Redheffer star product of the open-loop system matrix sys iq and the controller matrix K gives the closed-loop transfer matrix clp iq It consists of three inputs (u q, u r, d), four outputs (y q, y r, e p, e u ), and six states clp iq 2142857 0 0 0236 10 8 118249 01326 01061 0 44867 01061 03673 0 0 0 0 514157 10 4 02573 0 0 0 00976 0 0 0 2142857 0 0 0236 10 8 06000 0 0 0 06000 118249 44867 0 0 0 0 6613035 46548 10 8 1766 10 8 0700 250 0 0 0 0 0 118249 0 0 0 0 44867 01012 10 8 0384 10 8 0 0 0 01326 01061 0 0 02573 01061 03673 0 0 00976 46548 10 8 17666 10 8 0700 250 0 0 0 0 0 0 0 0 0 0 06000 13033 10 4 04945 10 4 0 0 0 The test for stability is conducted on the leading 2 2 diagonal block (rob stab) of the transfer function matrix clp iq Then rob stab consists of the six states of clp iq, the first two inputs u q and u r, and the first two outputs y q, y r of the variable clp iq Since the uncertainty considered is structured, verification of the robust stability needs the frequency response of the transfer function matrix rob stab in terms of structured singular values or µ values, the block structure is set by the matrix BLK [ 1 1 BLK 1 1 The frequency responses of the upper and lower bounds of µ are shown in Fig 9, which indicates that the closedloop system with K q K achieves robust stability

330 A Azaiz Y Ramdani A Meroufel: DESIGN OF ROBUST CONTROL SYSTEM FOR THE PMS MOTOR Fig 10 Nominal and Robust performance of K q Fig 11 Sensitivity function with K q Nominal performance is indicated by the frequency response of the transfer function matrix which consists of the third output (e p ), third input (d), and the six states of the variable clp iq Robust performance is indicated by the frequency response of clp iq in terms of µ values, the block structure is given by BLK 1 1 1 1 1 2 The frequency responses of the upper and lower bounds of µ are shown in Fig 10, which indicates that the system with K q K fails to satisfy robust performance criterion but achieves nominal performance Fig 12 Structure of the closed-loop system The simulation of closed-loop systems with designed controllers is based on the structure shown in Fig 12 The current i q is the q-axis stator current, i qr is the reference current The closed-loop transfer function matrix of the system of Fig 12 is given by clp q: clp q 2142857 02361 10 8 0 00051 10 8 02573 0 00976 0 2142857 02361 10 8 06000 0 10000 0 46548 10 8 1766 10 8 07 250 0 0 0102 10 8 00384 10 8 0 0 0 0 01326 01061 0 0 02573 02573 01061 03673 0 0 00976 00976 46548 10 8 17661 10 8 07 250 0 0 0 0 0 0 0 0 0 0 0 0 0 10000 The closed-loop transfer function matrix clp q consists of three outputs (y q, y r, i q ), four inputs (u q, u r, i qr, d) and four states The transient responses to the reference input (i qr ) and to the disturbance input (d) are shown in Fig 13 and Fig 14 respectively From the transient response to the reference input we deduce the rise time t r and the settling time t s as t r 0011 sec, t s 0015 sec 33 DESIGN OF THE CONTROLLER K d The d-axis model of the PMS given by equation (7) is similar in form to the q-axis model given by equation (8) The d-axis model can represented by equation (12), where i q, i q, v q0 and are replaced respectively by i d, i d, v d0 and L d, respectively The above procedure is used to design the controller K d, with the parameters of the q- axis model (i q, i q, v q0,, σ q ) replaced by the parameters of the d-axis model (i d, i d, v d0, L d, σ d ) in all the above equations The block diagram of the closed-loop system showing the feedback structure and including the elements reflecting model uncertainty and performance requirements is given in Fig 15 W p 06s2 + 120s + 60 s 2 + 05s + 006, W u 001 The open-loop system sys id consists of four inputs, five outputs and three states sys id 4285714 0 0 07 500 0 7142857 118250 01326 01062 0 0 01062 0 44867 01026 03674 0 0 44867 0 4285714 0 0 07 500 0 7142857 06000 0 0 0 0 0 0 06000 118250 44867 0 0 06000 0 0 0 0 0 0 0 00100 10000 0 0 0 0 10000 0

Journal of ELECTRICAL ENGINEERING VOL 58, NO 6, 2007 331 Fig 13 Transient response to the reference Fig 14 Transient response to the disturbance The interconnection matrix P for control design is 4285714 0 0 0 7142857 118250 01326 01062 118250 0 P 44867 01062 03674 44867 0 06000 118250 44867 06000 0 0 0 0 0 00100 10000 0 0 10000 0 In order to compute the controller K, and find the positive number γ, the Glover s and Doyle s algorithm [2, 3, 4, 6 is used Based on the matrix P, the number of measurements, the number of controls, the lower bound on gamma (γ min 01), and the upper bound on gamma (γ max 1), this algorithm computes the controller K, which is given as a transfer function matrix: 00925 10 7 18228 10 7 06916 10 7 0 00000 01326 01061 01819 K d K 0 01061 03673 00690 00084 10 7 01658 10 7 00629 10 7 0 The final value of gamma found by the algorithm is γ 06008 The Redheffer star product of the two matrices P and K gives the transfer function matrix of the closed-loop system from i dr to e, clp: 4285714 0 0 118250 01326 01061 44867 01061 03673 0 0 0 clp 01819 0 0 00690 0 0 06000 118250 44867 0 0 0 06009 10 8 118467 10 8 4499 10 8 0 0 0 0 118250 0 0 0 44867 00092 10 8 01822 10 8 00691 10 8 0, 00000 01326 01061 01819 0 01061 03673 00690 0 0 0 06000 8413025 16584176 62929297 0 clp < 06008 Fig 15 Closed-loop system structure The analysis of closed-loop system with K d is conducted with the same manner as for K q Tests of robust stability and robust performance are performed based on the closed-loop transfer function matrix clp id ( clp id sys id K d ) Figure 16 indicates that the closed-loop system with K d K achieves nominal performance but fails to satisfy robust performance criterion Figure 17 indicates that the closed-loop system with K d K achieves robust stability Fig 16 Nominal and Robust performance of K d The simulation of the closed-loop system with the designed controller K d is based on the structure shown in Fig 15 with the weighting filters not included The current i d is the d-axis stator current, i dr is the reference current The closed-loop transfer function matrix of this system is given by clp d:

332 A Azaiz Y Ramdani A Meroufel: DESIGN OF ROBUST CONTROL SYSTEM FOR THE PMS MOTOR Fig 17 Robust stability analysis of K d Fig 18 Transient response to the reference (i dr ) Fig 19 Stator currents, electromagnetic torque and rotor speed The closed-loop transfer function matrix clp d consists of three outputs (y d, y r, i d ), three inputs (u d, u r, i dr ) and four states clp d 4285714 06009 10 8 0 00092 10 8 01819 00000 00690 0 4285714 06009 10 8 06000 00 10000 0 118467 10 8 44949 10 8 07 500 0 01822 10 8 00691 10 8 0 0 0 01326 01061 0 0 01819 01061 03673 0 0 00690 118467 10 8 44949 10 8 07 500 0 0 0 0 0 0 0 0 0 0 0 The transient responses to the reference input (i dr ) is shown in Fig 18 From the transient response to the reference input we deduce the rise time t r and the settling time t s as t r 0011 sec, t s 0015 sec 4 SIMULATION In order to perform the simulation using the system of Fig 4, the speed controller is a PID controller with the following coefficients: K P 05043, K D 00049, K I 0495 The global system shown in Fig 4 is tested by numerical simulation The input ω ref is a step function with magnitude 175 rad/s The mechanical torque applied to the motor s shaft is originally 0 Nm Figure 19 shows that the torque climbs to nearly 135 Nm when the motor starts and stabilizes rapidly when the motor reaches the reference value The mechanical torque steps to 3 Nm at instant t 008 second At this instant, the stator current i q and the electromagnetic torque increase to maintain the speed at its reference value

Journal of ELECTRICAL ENGINEERING VOL 58, NO 6, 2007 333 5 CONCLUSION The closed-loop system with K q (K d ) achieves robust stability The maximum value of µ is 07099 (07105) that shows that the structured perturbations with norm less than 1/07099 ( 1/07105) are allowable, ie the stability maintains for < 1 07099 ( < 1 07105) The system with K q (K d ) achieves nominal performance but fails to satisfy the robust performance criterion This follows from the fact that the frequency response of the nominal performance has a maximum of 06002 (06001), while the µ curves for the robust performance have a maximum of 10191 (10184) With respect to the robust performance, this means in the present case that the size of the perturbation matrix must be limited ( < 1 10184) to ensure the per- to < 1 10191 formance function satisfying W p(1 + F u (G q, q )K q ) 1 W u K q (1 + F u (G q, q )K q ) 1 1 ( ) W p (1 + F u (G d, d )K d ) 1 W u K d (1 + F u (G d, d )K d ) 1 1 The current i q tracks the reference i qr and i d tracks i dr with an error of 03752 10 5 as specified by the filters w p (s) 5 PARAMETERS OF THE PMSM MACHINE The parameters of the PMSM machine are: L d 14 mh, 28 mh, r s 06 Ω, I q max 20 A, P 4, φ f 012 Wb, ω rn 230 rad/s, T m 85 Nm, J 111 10 3 Kgm 2, F 14 10 3 Nm/rds 1 References [1 BOSE, B K : Power Electronics and AC Drives, Prentice Hall, New-York, 1986 [2 DOYLE, J GLOVER, K KHARGONEKAR, P FRAN- CIS, B: State-Space Solutions to Standard H 2 and H Control Problems, IEEE Trans Automat Contr AC-34 No 8 (Aug 1989), 831 847 [3 DUC, G FONT, S : Commande H infty et µ-analyse, Des outils pour la robustesse, HERMES Science Publications, Paris, 1999 [4 GLOVER, K DOYLE, J C : State Space Formulae for All Stabilizing Controllers that Satisfy an H -Norm Bound and Relations to Risk Sensitivity, Systems and Control Letters (1988) [5 KHARGONEKAR, P P PETERSEN, I R ROTEA, M A: H Optimal Control with State Feedback, IEEE Trans Automat Contr AC-33 (1988), 783 786 [6 PETKOV, D W GU, P H KONSTANTINOV, M M: Robust Control Design with MATLAB, Springer-Verlag, London, 2005 [7 STOORVOGEL, A A: The H Control Problem: A state Space Approach, Prentice Hall, Englewood Cliffs, NJ, 1992 [8 STURTZER, G SMIGIEL, E : Modlisation et commande des moteurs triphasés, Ellipses, 2000 [9 ZHOU, K DOYLE, J C GLOVER, K : Robust and Optimal Control, Prentice Hall, Upper Saddle River, NJ, 1995 Rreceived 15 December 2006 Ahmed Azaiz was born in 1956 in Mascara Algeria He received his BS degree in electrical engineering, computer option from the Institute of Electricity and Electronics (IN- ELEC) Boumerdes in 1988 and the MS degree in Electronics from the University of Sidi Bel Abbes in 1993 He is currently professor of Electronics at the University of Sidi Bel Abbes His research interests are on microprocessor systems and robust control Youcef Ramdani was born in 1952 in Sidi Bel Abbes Algeria He received his BS degree in electrical engineering from USTO University in 1978, DEA in Electronics from Bordeaux, France in 1986, PhD in electrical engineering from the Faculty of sciences of the University of Bordeaux, France, in 1989 He is currently Professor of electrical engineering at the University of Sidi Bel Abbes His research interests are on control of electrical machines and drives Abdelkader Meroufel was born in 1954 in Sidi Bel Abbes Algeria He received his BS degree in electrical engineering from USTO University in 1979, the MS degree from USTO University (Algeria) in 1990, and the PhD degree from the University of Sidi Bel Abbes in 2004 He is currently Professor of electrical engineering at the University of Sidi Bel Abbes His research interests are on robust control of electrical machines and drives