Anti-Windup Design for Synchronous Machines. Journée GT CSE (MACS/SEEDS) Paris, May 17th, 2018

Transcription

1 1 / 39 Anti-Windup Design for Synchronous Machines Andreea Beciu Giorgio Valmorbida Journée GT CSE (MACS/SEEDS) Paris, May 17th, 218

2 2 / 39 Outline 1 Motivation: Use of electric machines, PI control 2 Problem statement: Improve transients of speed control of synchronous machines. 3 Proposed solution: Static Anti-windup. 4 Example: Application to a PMSM.

3 2 / 39 Outline 1 Motivation: Use of electric machines, PI control 2 Problem statement: Improve transients of speed control of synchronous machines. 3 Proposed solution: Static Anti-windup. 4 Example: Application to a PMSM.

4 2 / 39 Outline 1 Motivation: Use of electric machines, PI control 2 Problem statement: Improve transients of speed control of synchronous machines. 3 Proposed solution: Static Anti-windup. 4 Example: Application to a PMSM.

5 2 / 39 Outline 1 Motivation: Use of electric machines, PI control 2 Problem statement: Improve transients of speed control of synchronous machines. 3 Proposed solution: Static Anti-windup. 4 Example: Application to a PMSM.

6 Electric Motors Use of electric motors Anti-windup for electric motors: Saqib, Rehan, Iqbal, and Hong. Static Antiwindup Design for Nonlinear Parameter Varying Systems With Application to DC Motor Speed Control Under Nonlinearities and Load Variations, IEEECST 18; March and Turner. Anti-Windup Compensator Designs for Nonsalient Permanent-Magnet Synchronous Motor Speed Regulators, IEEETIA 9; Sepulchre, Devos, Jadot, and Malrait. Antiwindup Design for Induction Motor Control in the Field Weakening Domain, IEEECST 13; 3 / 39

7 4 / 39 Permanent Magnet Synchronous Motors Internal vs Surface Salient vs Non-Salient

8 5 / 39 Permanent Magnet Synchronous Motors Choice of coordinates: Park and Clarke transformation

9 6 / 39 Permanent Magnet Synchronous Motors Architecture for speed control

10 7 / 39 Park Transformation Choice of coordinates, in the rotor we have di d dt di q dt dω e dt = Rs i L d + Lq ω d L d ei q+ 1 u L 1 d = Rs L q i q L d L q ω ei d ψ f L q ω e+ 1 L q u 2 = Np γ(i J d, i q) f J ] [ u1 u 2 ωe Np J γ L U γ(i d, i q) = 3 2 Np(ψ f +(L d L q)i d )i q i q direct current, i q quadrature current, ω e electrical speed. ψ f, rotor flux, R s, stator resistance, L d, L q direct and quadrature inductances. J the moment of inertia of the rotor, N p the number of pairs of poles, f viscous friction coefficient. Vas. Sensorless Vector and Direct Torque Control, Oxford 98; Grellet and Clerc. Actionneurs Electriques, Principes, Modèles, Commande, Eyrolles 97;

11 7 / 39 Park Transformation Choice of coordinates, in the rotor we have di d dt di q dt dω e dt = Rs i L d + Lq ω d L d ei q+ 1 u L 1 d = Rs L q i q L d L q ω ei d ψ f L q ω e+ 1 L q u 2 = Np γ(i J d, i q) f J ] [ u1 u 2 ωe Np J γ L R 2 γ(i d, i q) = 3 2 Np(ψ f +(L d L q)i d )i q i q direct current, i q quadrature current, ω e electrical speed. ψ f, rotor flux, R s, stator resistance, L d, L q direct and quadrature inductances. J the moment of inertia of the rotor, N p the number of pairs of poles, f viscous friction coefficient. Vas. Sensorless Vector and Direct Torque Control, Oxford 98; Grellet and Clerc. Actionneurs Electriques, Principes, Modèles, Commande, Eyrolles 97;

12 8 / 39 Speed control For speed control, a strategy consists in taking γ(i d, i q) = 3 2 Np(ψ f +(L d L q)i d )i q as the input for the speed control. A reference torque γ r is obtained with where K ψ = ( 3 2 Npψ f) 1. (i dr, i qr) = (, K ψ γ r).

13 9 / 39 PID for speed control The control inputs(v d, v q) generated (using (i dr, i qr)) ė d ė q ė ωe γ r v dr v qr = i d +i dr = i q+i qr = ω e+ω er = K i e T ωe K i ω ii e+k i ω er = K d T id e d K d i d +K d i dr = Kq T iq e q K qi q+k qi qr.

14 9 / 39 PID for speed control The control inputs(v d, v q) generated (using (i dr, i qr)) ė d ė q ė ωe γ r v dr v qr = i d = i q+k ψ γ r = ω e+ω er = K i e T ωe K i ω ii e+k i ω er = K d T id e d K d i d +K d i dr = Kq T iq e q K qi q+k qi qr.

15 1 / 39 Non-linear compensation Remove the nonlinear terms with [ Lqω ei q v c = L d ω ei d +ψ f ω e and applying in v = v r + v c. ]

16 11 / 39 Non-saturated system The closed-loop with ξ = [i d i q ω e e d e q e ωe ] T ξ = Aξ + f(ξ)+b ωω er + B γγ L Rs K d K d L d T id L d Rs Kq ψ f Kq K ψ K i Kq Kq K ψ K i Lq Lq Lq T iq Lq T ii N A = 3 p 2 2 J ψ f f J 1 K 1 K ψ K i ψ K i T ii 1 N 3 p 2 ; f(ξ) = 2 J (L d Lq)i d iq Kq K ψ K i Lq B ω = K ψ K i 1 ; B γ = Np J.

17 12 / 39 Input Constraints The actual constraint set is U = { u R 2 u 2 u max } v q U v d Define a mapping σ : R 2 U such that u = σ(v), v = [ vd v q ].

18 13 / 39 Standard saturation Let us denote σ A : R 2 { u R 2 u } 2 2 umax U the standard decentralized saturation with a fixed saturation level. [ sign(v σ A (v) = d ) max( v d, ] 2 umax) 2 sign(v q) max( v 2 q, umax) 2 v q v v d

19 14 / 39 Alternative saturation Let us denoteσ B : R 2 U a mapping that gives priority to the first input [ ] sign(v d ) max( v d, u max) σ B (v) = sign(v q) max( v q, umax 2 σb1 2 (v)) v q v σ B1 (v) v d Introducing a variable saturation level for the second input

20 15 / 39 Directionality preserving saturation Let us denoteσ C : R 2 U the directionality preserving map σ C (v) = u max max(u max, v 2 ) v v q v v d

21 16 / 39 Sector conditions for directionality preserving For the same value v different inputs v q v σ C σ A σ B v d How to incorporate these functions for the analysis/synthesis? Sector conditions

22 16 / 39 Sector conditions for directionality preserving For the same value v different inputs v q v σ C σ A σ B v d How to incorporate these functions for the analysis/synthesis? Sector conditions

23 17 / 39 Sector condition Consider the directionality preserving non-linearityσ(v) = σ C (v) and define σ(v) := v σ(v). We have σ(v) = v σ(v) u max = v ( max(u max, v 2 ) v ) u max = 1 v max(u max, v 2 ) }{{} [,1] v q σ(v) v d

24 18 / 39 State constraints, current limitation Also need to prevent high peak currents, that is total peak current i peak 2 total steady state current i ss 2 These are state constraints involving the total current i 2 2 = i 2 d + i 2 q. Possible solution: bound the reference signal i dr and i qr.

25 18 / 39 State constraints, current limitation Also need to prevent high peak currents, that is total peak current i peak 2 total steady state current i ss 2 These are state constraints involving the total current i 2 2 = i 2 d + i 2 q. Possible solution: bound the reference signal i dr and i qr.

26 19 / 39 Speed Control Architecture A block diagram of the control loop of the linearized system including the saturation is given below u AW γ L! v r v C + σ σ(v) r (i PMSM d ;i q ;! e ) + v

27 19 / 39 Speed Control Architecture A block diagram of the control loop of the linearized system including the saturation is given below u AW γ L! v r v C + σ σ(v) r (i PMSM d ;i q ;! e ) + v uaw! iq uawq!e!er 1 s Ki Ti γr K iqr 1 s Kq Tq vqr Ki Kq Kd C 1 s Kd Td vdr id uawd

28 2 / 39 Speed Control Architecture And the control inputs(v d, v q) are obtained from ė d ė q ė ωe γ r v d v q = i d +v AWd = i q+k ψ γ r+v AWq = ω e+ω er+v AWω = K i e T ωe K i ω ii e+k i ω er = K d T id e d K d i d +K d i dr = Kq T iq e q K qi q+k qi qr.

29 21 / 39 Speed control with anti-windup input Using σ(v) := v σ(v) the closed-loop system becomes ξ = Aξ B σ(v)+b AW v AW + f(ξ)+b ωω er + B γγ L 1 L d 1 L q B = ; B AW = v AWd 1 ; v AW = v AWq 1 v AWω 1 Compute a static Anti-Windup u AW = K AW σ(v). The LMI based method used here can be found in Zaccarian and Teel. Modern Anti-Windup Synthesis - Control Augmentation for Actuator Saturation, Princetown Series in Applied Mathematics 211(Section 4.3.1); Tarbouriech, Garcia, Gomes da Silva and Queinnec. Stability and Stabilization of Linear Systems with Saturating Actuators, 211 (Section 7.3.2);.

30 21 / 39 Speed control with anti-windup input Using σ(v) := v σ(v) the closed-loop system becomes ξ = Aξ +( B + B AW K AW ) σ(v)+f(ξ)+b ωω er + B γγ L 1 L d 1 L q B = ; B AW = v AWd 1 ; v AW = v AWq 1 v AWω 1 Compute a static Anti-Windup u AW = K AW σ(v). The LMI based method used here can be found in Zaccarian and Teel. Modern Anti-Windup Synthesis - Control Augmentation for Actuator Saturation, Princetown Series in Applied Mathematics 211(Section 4.3.1); Tarbouriech, Garcia, Gomes da Silva and Queinnec. Stability and Stabilization of Linear Systems with Saturating Actuators, 211 (Section 7.3.2);.

31 21 / 39 Speed control with anti-windup input Using σ(v) := v σ(v) the closed-loop system becomes ξ = Aξ +( B + B AW K AW ) σ(v)+f(ξ)+b ωω er + B γγ L 1 L d 1 L q B = ; B AW = v AWd 1 ; v AW = v AWq 1 v AWω 1 Compute a static Anti-Windup u AW = K AW σ(v). The LMI based method used here can be found in Zaccarian and Teel. Modern Anti-Windup Synthesis - Control Augmentation for Actuator Saturation, Princetown Series in Applied Mathematics 211(Section 4.3.1); Tarbouriech, Garcia, Gomes da Silva and Queinnec. Stability and Stabilization of Linear Systems with Saturating Actuators, 211 (Section 7.3.2);.

32 22 / 39 Static Anti-Windup Consider the system ξ = A cl ξ +(B cl,q + B cl,aw K AW ) σ(v)+b cl,w w v = C cl ξ + D cl,u σ(v)+d cl,w w z = C cl,z ξ + D cl,z σ(v)+d cl,z w Theorem 2.1 If there exist Q S n >, S R m n, T D m >, and a scalar γ > such that A cl Q B cl,q T + B cl,aw S B cl,w n p He C cl,u Q (D cl,u I m)t D cl,uw m p mw n mw m 1 2 γimw mw p < C cl,z Q D cl,z T D cl,zw 1 2 γip Then K AW = ST 1 guarantees w 2 z 2 < γ for the closed-loop system.

33 23 / 39 Experimental results We applied the above to a non-salient machine with R s =.95Ω, L d = L q = 13.6mH, φ f =.284Wb, and u max = 34V. N p = 4, J = kgm 2, f =.1Nms 1. PI gains K q = K d = 34 T iq = T id =.143 and K i =.211 T ii =.796. With and we obtain K AW = w = γ L z = ω e ω er,

34 24 / 39 Experimental results Experimental setup

35 Experimental results 14 Angular speed ω e [rad/s] ω er With AW Without AW Time [s] 25 / 39

36 26 / 39 Experimental results 5 Saturated inputs σ(v d ), σ(v q ) [V] σ(v d ) σ(v q ) Time [s]

37 27 / 39 Experimental results 3 Deadzone of v q 25 σ(v q) σ(vq)[v] Time [s]

38 28 / 39 Experimental results 12 Currents, no Anti-Windup i d no AW 1 i q no AW 8 i d, i q [A] Time [s]

39 29 / 39 Experimental results Currents with Anti-Windup id AW iq AW i d, i q [A] Time [s]

40 3 / 39 Experimental results 12 Current response i q no AW i q AW 1 8 i q [A] Time [s]

41 31 / 39 Steps for non-linear analysis Due to the nonlinear terms, the compensation is lost when the system saturates. For the non-salient case we have ξ = A cl ξ +(B cl,q + B cl,aw K AW ) σ(v)+b cl,w w v = C cl ξ+v c(ξ)+d cl,u σ(v)+d cl,w w z = C cl,z ξ + D cl,z σ(v)+d cl,z w not ξ = A cl ξ +(B cl,q + B cl,aw K AW ) σ(v)+b cl,w w v = C cl ξ + D cl,u σ(v)+d cl,w w z = C cl,z ξ + D cl,z σ(v)+d cl,z w [ ] Lqω ei q v c(ξ) = L d ω ei d

42 31 / 39 Steps for non-linear analysis Due to the nonlinear terms, the compensation is lost when the system saturates. For the non-salient case we have ξ = A cl ξ +(B cl,q + B cl,aw K AW ) σ(v)+b cl,w w v = C cl ξ+v c(ξ)+d cl,u σ(v)+d cl,w w z = C cl,z ξ + D cl,z σ(v)+d cl,z w not ξ = A cl ξ +(B cl,q + B cl,aw K AW ) σ(v)+b cl,w w v = C cl ξ + D cl,u σ(v)+d cl,w w z = C cl,z ξ + D cl,z σ(v)+d cl,z w [ ] Lqω ei q v c(ξ) = L d ω ei d

43 Steps for non-linear analysis Due to the nonlinear terms, the compensation is lost when the system saturates. We have 15 Speed 1 rad/s 1 ω e [rad/s] 5 ω er Without AW Linear With AW Linear With AW Without AW Time [s] 32 / 39

44 Steps for non-linear analysis Due to the nonlinear terms, the compensation is lost when the system saturates. The Anti-windup helps recover a similar behavior. 45 Speed 29 rad/s ω e [rad/s] ω er Without AW With AW Without AW Linear With AW Linear Time [s] 33 / 39

45 Steps for non-linear analysis Due to the nonlinear terms, the compensation is lost when the system saturates. The Anti-windup helps recover a similar behavior. 45 Speed 29 rad/s ω e [rad/s] ω er Without AW With AW Without AW Linear With AW Linear Time [s] 33 / 39

46 34 / 39 Steps for non-linear analysis Consider the quadratic function with P (p 1 ) = P (p 1 ) > provided < p 1 < V (i d, i q,ω e) = 1 2 ( Lq L d i d i q ω e T P (p 1 ) 1 p 1 ( 3 Np 2 J d L q) ) 2, if L d > L q or p 1 > ) 1( Lq ( Lq L d L d i d i q ω e L d L q p 1 ) ) 2, if L d < L q.

47 34 / 39 Steps for non-linear analysis Consider the quadratic function with P (p 1 ) = P (p 1 ) > provided < p 1 < V (i d, i q,ω e) = 1 2 ( Lq L d i d i q ω e T P (p 1 ) 1 p 1 ( 3 Np 2 J d L q) ) 2, if L d > L q or p 1 > ) 1( Lq ( Lq L d L d i d i q ω e L d L q p 1 ) ) 2, if L d < L q.

48 35 / 39 Energy-preserving nonlinearities We have V, f(ξ) L q ω L d ei q V, L d L q ω ei d 3 Np 2 (L 2 J d L q)i d i q Take P c > and consider = i d i q ω e T V(x) = V (i d, i 1 q,ω e)+ 2 P (p 1 ) e d e q e ωe T L q ω L d ei q L d L q ω ei d 3 Np 2 (L 2 J d L q)i d i q P c e d e q e ωe = which, from the above gives V as a quadratic function... However with the above V.

49 35 / 39 Energy-preserving nonlinearities We have V, f(ξ) L q ω L d ei q V, L d L q ω ei d 3 Np 2 (L 2 J d L q)i d i q Take P c > and consider = i d i q ω e T V(x) = V (i d, i 1 q,ω e)+ 2 P (p 1 ) e d e q e ωe T L q ω L d ei q L d L q ω ei d 3 Np 2 (L 2 J d L q)i d i q P c e d e q e ωe = which, from the above gives V as a quadratic function... However with the above V.

50 36 / 39 Concluding Remarks Applied static Anti-Windup synthesis to a PMSM model. Defined non-linearities to exploit the input set. Obtained first experimental results. Next steps include: saturation of the reference current i qr, non-linear analysis, feed-forward compensation, dynamic anti-windup.

51 36 / 39 Concluding Remarks Applied static Anti-Windup synthesis to a PMSM model. Defined non-linearities to exploit the input set. Obtained first experimental results. Next steps include: saturation of the reference current i qr, non-linear analysis, feed-forward compensation, dynamic anti-windup.

52 36 / 39 Concluding Remarks Applied static Anti-Windup synthesis to a PMSM model. Defined non-linearities to exploit the input set. Obtained first experimental results. Next steps include: saturation of the reference current i qr, non-linear analysis, feed-forward compensation, dynamic anti-windup.

53 36 / 39 Concluding Remarks Applied static Anti-Windup synthesis to a PMSM model. Defined non-linearities to exploit the input set. Obtained first experimental results. Next steps include: saturation of the reference current i qr, non-linear analysis, feed-forward compensation, dynamic anti-windup.

54 37 / 39 Review Thank you!