2012 IEEE International Conference on Vehicular Electronics and Safety July 24-27, 2012. Istanbul, Turkey
Linear Parameter Varying Control of Permanent Magnet Synchronous Motor via Parameter-Dependent Lyapunov Function for Electrical Vehicles Yusuf ALTUN
Kayhan GULEZ
Department of Control and Automation Engineering Yildiz Technical University Istanbul, Turkey
[email protected]
Department of Control and Automation Engineering Yildiz Technical University Istanbul, Turkey
[email protected]
Abstract— This paper presents linear parameter varying
stator currents. In this paper, the LPV controller is designed instead of PI current controllers; because PI controllers do not take into account the noises of sensors and disturbances. Besides it needs online tuning during the operation. The PI controllers used for PMSM vector control may not provide desired performance for EV that acquires the high speed and torque control performance.
(LPV) control of permanent magnet synchronous motor (PMSM), which have nonlinear model in d-q rotating frame, using Linear Matrix Inequality (LMI). PMSM model is rewritten in the LPV form depending on the angular speed of rotor which can be measured or estimated. Parameterdependent Lyapunov function is used for the LPV control design such that the robust stability is assured for all varying parameters using multi-convexity functions. The proposed LPV structure of speed tracking problem is based on the field oriented control (FOC) in the AC motor control methods. Thus, the proposed LPV controller acquires the control of stator currents in d-q rotating frames for the speed control of PMSM in Electrical Vehicle (EV) presented in the simulation results. Keywords-component; LPV Control, Permanent Magnet Synchronous Motor, Vector Control, LMI, FOC.
I. INTRODUCTION Recently, Permanent Magnet Synchronous Motor (PMSM) among the electric motors has been preferred for Electrical Vehicle (EV) excitation applications particularly during city traffic, and so it has particular interest in the literature. It has higher power density, efficiency and smaller size when compared to the other used electrical motors for EV. Although PMSM mathematical model has multivariable nonlinear coupled system equations, the classical control methods of PMSM are based on linear control design for dynamical control parts. It has also multi input- multi-output (MIMO) structure which has disturbances and two sinusoidal control inputs. Hence, the control of PMSM is a difficult engineering problem. Many researchers in the literature have studied the based on vector control methods in the ac motor control systems in the last decade [1]. Vector control techniques have made possible the application of PMSM for high performance applications where traditionally only dc drives were applied. PMSM torque and speed control has traditionally been achieved using vector control methods which are used mostly Field Oriented Control (FOC) and Direct Torque Control (DTC). In this paper, the controller based on FOC is designed for LPV control of PMSM. The classical vector and scalar control methods use PI controllers for the control of 978-1-4673-0993-6/12/$31.00 ©2012 IEEE
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Recently, linear parameter varying (LPV) control design has been presented as an alternative control method for nonlinear systems. In general, the LPV control approach is based on H∞ control theory for the LPV systems. It takes into account parameter varying, disturbances, sensor noises the whole parameter changing of operating circumstances, and so this enables the designed controllers to guarantee of the stability, performance and robustness for the whole changing of operating conditions. The LPV control methods have already been widely used in various dynamical systems requiring high-performance such as aircraft, active magnetic bearing system, robotic manipulators and vehicles [2-7]. In the literature, there are many nonlinear control studies for PMSM control system. The most of presented control papers are based on sliding mode control (SMC) [8, 9]. The classical SMC have the chattering problem in the systems that involve disturbances and uncertainties, but it can be developed different approaches for the problem [10]. It is also designed independently while it is designed instead of two PI current controllers. The LPV control design is developed as an alternative control method for the nonlinear systems. The papers achieving the LPV control of PMSM are only [11, 12]. The polytopic approach was used for LPV control design in the papers. Hence, the LPV control design papers are very few for PSMM control systems in the literature. This paper presents different approach for the LPV control of PMSM in which parameter dependent Lyapunov function is used. The proposed LPV controller depends on rotor speed and takes into account disturbances, sensor noises. II. LPV REPRESENTATION OF PMSM MODEL The nonlinear model of PMSM in the d-q axis field oriented coordinate system is expressed from (1) to (7) [13, 14 and 15].
Vsd = Rs isd +
Vsq = Rs isq +
d ψ sd − pωmψ sq dt
(1)
d ψ sq + pωmψ sd dt
(2)
disd 1 = (Vsd − Rs isd + Lsq pωm isq ) dt Lsd disq dt
=
1 (Vsq − Rs isq + Lsd pωm isd − λ pωm ) Lsq
(
Te = 1,5 p λisq + ( Lsd − Lsq ) isd isq
)
(3)
(4)
(5)
Rs − L sd Ap (ωm ) = Lsd − pωm Lsq
1 L sd Bp = 0
Lsq Lsd R − s Lsq
pωm
0 1 Lsq
0 λ B f ( ωm ) = −ω mL sq
(10)
(11)
(12)
d ωm 1 = (Te − TL − Bωm ) dt J
(6)
1 0 0 0 Cp = 0 1 0 0
(13)
dθ m = ωm dt
(7)
isd isq Vsd x = , u = , ϕrd Vsq ϕ rq
(14)
Ap = Ap 0 + ωm Ap1
(15)
0 A = p0 - p Lsd Lsq
(16)
Where Vsd stator voltage d axis, Vsq stator voltage q axis, isd stator current d axis, isq stator current q axis, ψsd stotor flux d axis, ψsq stator flux q axis, Rs stator resistance, Lsd stator inductance d axis, Lsq stator inductance q axis, λ pole flux, ωm rotor mechanical speed, J moment of inertia, B viscous damping constant, p number of pole pairs. The general representation of the LPV system in state space form is as in Equation (8). The system requires at least one state space matrix of the system, which depends on at least one parameter, so that such a system can be described as an LPV system. x = A ( r ) x + B ( r ) u
y = C (r ) x + D (r ) u
(8)
The state space form of PMSM can be defined by Equation (9) where Ap, Bp and Bf are system matrix, input matrix and disturbance vector respectively. The state space matrices are expressed in Equations (10)-(13) as state variables and control signals are expressed in Equation (14). x = Ap (ωm ) x + B p u + B f (ωm ) y = Cp x
Rs − Lsd A = p1 0
p
Lsq Lsd 0
Rs − Lsq 0
(17)
(9)
Ap matrix of PMSM model in d-q axis affinely depends on the mechanical angular speed of rotor. The model is rewritten in the LPV form in Equation (15), where Ap 0 and
Ap1 are defined by Equations (16) and (17).
Figure 1. The field oriented control scheme without speed controller part (Texas Instruments Europe, February 1998)
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III. LPV CONTROL OF PMSM Two PI controllers are independently designed and for the control of stator currents in classical FOC design scheme as in Figure 1 because PMSM model is multi input-multi output (MIMO). The controllers are designed by using linear controllers such as pole assignment, Ziegler Nichols, classical control design, and alone these methods do not take into account sensor noises, disturbances and nonlinear terms. However, PMSM model is nonlinear and includes the disturbances terms. On the other hand, the designed LPV controllers take into account sensor noises, disturbances and nonlinear terms. The controllers are also dependently designed for the control of stator currents in d and q axis in place of two PI controllers since they are centralized control. The designed LPV control of PMSM is based on FOC where the stator currents of d and q axis are controlled. The designed LPV controller depends on rotor speed and is designed by taking into account internal disturbances and sensor noises. The LPV control system scheme is presented in Figure 2. The generalized LPV form is expressed in Equation (18).
ω
z
A ( r ) B1 ( r ) B2 ( r ) C1 ( r ) D11 ( r ) D12 ( r ) C (r ) D (r ) 0 21 2
Ac ( r ) Bc ( r ) Cc ( r ) Dc ( r )
z ( t ) = C1 ( r ( t ) ) x ( t ) + D11 ( r ( t ) ) ω ( t ) + D12 ( r ( t ) ) u ( t )
(18)
y ( t ) = C2 ( r ( t ) ) x ( t ) + D21 ( r ( t ) ) ω ( t ) + D22 ( r ( t ) ) u ( t )
xc = Ac ( r ) xc + Bc ( r ) y
ref disturbance
--
xs = As xs + Bs y
z1 = CT xT + DT u
z2 = Cs xs + Ds y
(20)
(21)
(22)
+
xcl = Acl ( r ) x + Bcl ( r ) ω
(23)
A ( r ) + B2 ( r ) Dc ( r ) C2 ( r ) B2 ( r ) Cc ( r ) Acl ( r ) = Ac ( r ) Bc ( r ) C2 ( r )
(24)
Ccl ( r ) = ( C1 ( r ) + D12 ( r ) Dc ( r ) C2 ( r ) D12 ( r ) Cc ( r ) )
Ws
WT + +
xT = AT xT + BT u
B ( r ) + B2 ( r ) Dc ( r ) D21 ( r ) Bcl ( r ) = 1 Bc ( r ) D21 ( r )
z2
+
0 WT
z = Ccl ( r ) x + Dcl ( r ) ω
(19)
u = Cc ( r ) xc + Dc ( r ) y
PMSM MODEL
83 0, 002 , WT = s +0,17 s +0, 002
By using Equations (18), (19) and the structure of the control system showed in Figures 2 and 3; the closed loop control system is showed as in Equation (23) and the system matrices are obtained as in Equation (24).
x ( t ) = A ( r ( t ) ) x ( t ) + B1 ( r ( t ) ) ω ( t ) + B2 ( r ( t ) ) u ( t )
u
Ws =
The matrices of generalized system in (18) are obtained using open loop control structure in Figures 3a and 3b.
Figure 2. The LPV control scheme
z1
The designed LPV structure includes WT, and Ws weight functions. The weighting functions are selected as in Equation (20). The functions are rewritten in the diagonal form by (21) because the system has MIMO structure. The representation of the functions in state space form is as in Equation (22).
0 W WT Ws = s , WT = 0 W 0 s
y
u
Figure 3b. Detailed open loop control system
Dcl ( r ) = D11 ( r ) + D12 ( r ) Dc ( r ) D21 ( r )
error
n
Lemma 1 [16]: r ∈ ℜ the quadratic form which depends on r ( t ) can be written. The form is known multi-convex
WN
function as in (25). sensor noises
f ( r1 ,…, rn ) = α 0 + α i ri + βij ri rj + γ i ri 2
Figure 3a. Basic open loop control system
i
342
i< j
(25)
In this case, in order to provide the stability,
f ( .)
function, which is negative, guarantees for all values of r ( t ) if following multi-convexity condition is obtained as in (26).
by (32). Hence, LMIs in (34) and (35) and its multiconvexity conditions hold for all pairs ( r , r ) in r × r . In such a case, the LPV controller is obtained as in (33).
∂ f ( r ) ≥ 0, i = 1,..., n ∂ri 2
(26)
For the control synthesis, affine parameter-dependent Lyapunov function is selected as in Equation (27), its time derivative is expressed as in Equation (28), n V ( x) = xT P0 + ri Pi x i =1
(
)
d (V ( x, r ) ) = xT P ( r ) A ( r ) + A ( r )T P ( r ) + P ( r ) x ≤ 0 dt
c
(28)
c
{ E = {r : r = r veya r = r , ∀
} = 1,..., q}
D = ri : ri = ri veya ri = r i , ∀i = 1,..., q i
i
(30)
c
X I
IV. SIMULATION RESULTS TM
Simulink is used to simulate the LPV control system of PMSM as shown in Figure 4. The sample of simulation is 100µsn and the parameters of PMSM are shown in Table I. For LPV control synthesis, LMIs of the forms (34) and (35) with their multi-convexity conditions and given idea [10] are solved minimizing γ, and so the state space matrices of the parameter-dependent controller (Ac, Bc, Cc, Dc) are obtained by (33). In this paper, YALMIP [19] parser and SEDUMI solver [20] was used for LMI solutions.
Aˆ + AT + C2T Dˆ T B2T AY + YAT + B2Cˆ + Cˆ T B2T − Y
ˆ XB1 + BD 21 B + B Dˆ D
*
−γ I
*
*
I 0 Y
(33)
2
Bc = X −1 Bˆ − B2 Dc ˆ + ( A + B C )YZ −1 − B C + X −1 XX −1 Z −1 Ac = − X −1 AY 2 c c 2
Theorem 2 [18]: Consider the LPV plant of the form (18) and its parameter trajectories is constrained by (30). The closed loop system has L2 gain, and there exists the LPV output-feedback controller of the form (33) which guarantees internal stability, if there exist symmetric positive definite matrices X and Y defined by (31) with compatible a parameter-dependent of state space data Aˆ , Bˆ , Cˆ , Dˆ defined
ˆ + C T Bˆ T + X AT X + XA + BC 2 2 * * *
(31)
(32)
Cˆ Dc C2Y − Ck Z Bˆ X ( Bc + B2 Dc )
Dc = Dˆ ˆ −1 + D C YZ −1 C = −CZ
(29)
i
i =1
−1 Aˆ X ( A + Bc C2 ) Y − X ( Ac + B2 Cc ) Z + XX
Acl T ( r ) P ( r ) + P ( r ) Acl ( r ) + P ( r ) P ( r ) Bcl ( r ) Ccl T ( r ) Bcl T ( r ) P ( r ) −γ I Dcl T ( r ) 0 Ccl Dcl −γ I
i
i =1
(27)
system is stable and assured by multi-convexity conditions in Lemma 1 for all r ( t ) ∈ D and r ( t ) ∈ E in (30)
i
n
Y Z + X −1 Dˆ D
Theorem 1 [17]: If there exists positive definite symmetric matrix P ( r ) for LMI form in (29), the closed loop control
i
n
X (r ) = X 0 + ri X i ≥ 0 and Y (r ) = Y0 + rY i i ≥ 0
2
1
2
21
TABLE I.
MOTOR PARAMETERS
Rs Lsd=Lsq J B λ p
0.8 ohm 3 mH 0.000235 k.g.m2 0 N.m.s 85.45x10-3 Wb 4
C1T + C2T Dˆ T D12T YC1T + Cˆ T D12T 0 D11T + D21T Dˆ T D12T −γ I
(34)
(35)
343
t
C ontinuous 0 U
isdref
control
Load v sd
wm
Uref
v sq v sabc
LPV controller
iabc
Clock
powergui
Vdq
theta
IGBT
v dq
theta
v abc
Idq TL
TL
Te
IGBT selector
PWM
Vdq
v abc rotor angle
vabc
PMSM IGBT inverter
-K-
wm
i-abc
wm
6 signals
dq-abc wmref
Iabc
PID(s)
rpm-rad
-K-
PID
rad-rpm Rotor speed
Figure 4 The simulink model for the LPV control of PMSM
As a result of the simulation, as shown in Figure 6 the rotor speed of PMSM has good tracking reference under a sample road load in Figure 5. During simulation, applied control signals in d-q axis are shown in Figure 7 and the applied voltages of a, b, c phases are shown in Figure 8. The motor phase currents are also shown as in Figure 9. As shown in simulation results, LPV controller has fast speed and current responses for PMSM control.
It is clear that since the controllers such as PI and pole assignment are linear controller, they do not provide adequate performance for nonlinear systems. Furthermore, designed linear controller could be efficient for only designed phase margins. Yet, the designed LPV controller works rotor speed-dependent for all phase margins. Thus, the motor control system has high performance for EV. 300
1.6
250
1.4
200 150 voltage(volt)
amplitude(N.m)
1.2
1
0.8
100 50 0 -50
0.6
-100 0.4
0.2
-150 -200 0
0.1
0.2
0.3
0.4
0.5 t(sn)
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5 t(sn)
0.6
0.7
0.8
0.9
1
Figure 7 The applied voltages of d-q axis
Figure 5 The sample road load 250 1400
200 1200
150 100 voltage(volt)
rotor speed (rpm)
1000
800
600
50 0 -50 -100
400
-150 200
0
-200 -250 0
0.1
0.2
0.3
0.4
0.5 t(sn)
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5 t(sn)
0.6
0.7
0.8
Figure 8 The voltages of a, b and c phases
Figure 6 The mechanical rotor speed
344
0.9
1
[7]
Witte, J.; Balini, H.M.N.K.; Scherer, C.W.; , "Robust and LPV control of an AMB system," American Control Conference (ACC), 2010 , vol., no., pp.2194-2199, June 30 2010-July 2 2010. [8] Ningzhi Jin; Xudong Wang; Hanying Gao; Jinfeng Liu; , "Sliding mode based speed regulating of PMSM MTPA control system for electrical vehicles,"Electronic and Mechanical Engineering and Information Technology (EMEIT), 2011 International Conference on , vol.2, no., pp.987-992, 12-14 Aug. 2011. [9] Wang Zhengjun; Wang Junzheng; Zhao Jiangbo; Liu Zhigang; , "Switching gain adaptive sliding mode model-following speed control of PMSM," Control Conference (CCC), 2010 29th Chinese , vol., no., pp.3238-3243, 29-31 July 2010. [10] Yan Li; Ju-Beom Son; Jang-Myung Lee; , "PMSM speed controller using switching algorithm of PD and Sliding mode control," ICCASSICE, 2009 , vol., no., pp.1260-1266, 18-21 Aug. 2009. [11] Pohl, L.; Blaha, P.; , "Linear parameter varying approach to robust control of a permanent magnet synchronous motor," Intelligent Engineering Systems (INES), 2011 15th IEEE International Conference on , vol., no., pp.287-291, 23-25 June 2011.
20 15 10
current (A)
5 0 -5 -10 -15 -20
0
0.1
0.2
0.3
0.4
0.5 t(sn)
0.6
0.7
0.8
0.9
1
Figure 9 The currents of a, b and c phases
V. CONCLUSION In this paper, LPV controller design was carried out for nonlinear PMSM model in which the controller is based on field oriented control. In place of two PI current controllers in FOC, LPV controller is designed using LMI. High performance was achieved for PMSM control. The controller guarantees the stability for all rotor speed parameters which highly affects nonlinearity during the operation circumstances. The controller has good performance for system requiring high performance EV and including road load which changes with the road conditions. Hence, the designed LPV controller eliminates some disadvantages of PI control such as requiring online tuning, designing problem for nonlinear systems and taking no account of disturbances and noises in classical FOC design. The simulation results of LPV control of PMSM show quite fast currents and speed response, and so the controller has adequate performance for Electrical Vehicle. REFERENCES [1]
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