Resonant control

reaches the reference after 4-5 sampling periods and has overshot.

Figure 5.25.

2 2

2 1 0

2 2

( ) 2

r r res r res

res

res res

k s k s k

C s

s d s

ω ω

ω ω

+ +

= + + (49)

The FEM simulation delivers a pulsating thrust force which is used as feedback in the control loop of the machine. This eliminates the sag in the force when the vehicle travels between two consecutive segments. Figure 5.26 shows a standard field oriented control structure in the rotor d-q coordinate system for the segmented linear drive with Fx* as input, one current loop for the d-axis with a resonant element, and one force loop for the q-axis with a resonant element too, all for odd and even segments. In fact, a resonant element in the d-axis is not necessary, but it helps to suppress harmonics from the EMF.

The resonant controller Cres(s) is in parallel to a PI controller. The parameters of the PI and resonant controller must be tuned for the entire closed loop transfer function.

Figure 5.27 shows a simplified linear model of the controller, considering the inverter

Hardware level

Space vector modulator Switching times

dqαβ dq

abc Space vector

modulator dqαβ dq

abc

(odd segments)

(even segments)

Switching times

π Even current and

force control

Odd current and force

control

P_lm=52 Relative position

βmech

C_res(s) C_res(s)

F_x^FEM(i_q,x)

i_d^*=0 i_d^*=0

F_x^FEM(i_q,x) i_q

i_q

i_d i_d

-x_even x_odd

Inverter address calculation

F_x^*

S1 S1

C_res(s) S1

C_res(s) S1

ωres ωres ωres ωres

Figure 5.26: control block diagram for the segmented linear drive with resonant controller.

dynamics and the motor. TD and kc are the control time delay and inverter gain respec-tively. L and R denote the inductance and the resistance of the motor segment respec-tively.

To obtain the open loop transfer function it is necessary to simplify the force model to an equivalent gain kf. The ripple introduced by the force model must be rejected by the resonant controller and it is not necessary to consider it in tuning. Equation (50) shows the transfer function Hc(s) of the entire controller (PI+Cres(s)) and (51) the plant Hp(s) (Inverter + Motor + Force model).

2 2

2 1 0

2 2

( ) ( ) 1 1

2 ( )

( )

r r res r res c

c p

i res res c

res

k s k s k N s

H s k

T s s d s D s

C s PI

ω ω

ω ω

⎛ ⎞ + +

= ⎜⎝ + ⎟⎠+ + + = (50)

( )=( )( 1)

+ +

c f p

D

H s k k

Ls R T s (51)

At first the electric pole of the plant can be eliminated with one zero of the controller, i.e. the numerator of Hc(s) must be factorized by (Ls+R). Using Ncd(s) (52) as the de-sired polynomial for the numerator of the controller Hc(s), it is possible to rewrite it as (53).

( ) (

^{2 2 1}

)

( ) 1

Ncd s = Ls R b s+ +b s+ (52)

( ) ( )

( )

2

2 1

2 2

( ) 1

c 2

i res res

Ls R b s b s H s T s s dω s ω

+ + +

= + + (53)

The benefit of using (52) is the order reduction in the closed loop transfer function Hclose(s). By matching the numerator of (49) with polynomial (52), kr1, kr2, kr3 and kp are defined now as function of b2, b1, Ti, d, and ω^res (54)(55)(56)(57).

F_x(i_d=0,i_q,x)

Inverter Motor

PI C_res(s) Resonant

Element F_x^*

-u_q* u_q

u_qEMK

-i_q

x E

Force model H_c(s)

ωres

c D

k

T s+1 Ls R+ 1

Figure 5.27: Simplification of the resonant controller and the plant for the q-axis.

2 2 2 r

i res

b L R

k ⁼ T ⁻ω ⁽⁵⁴⁾

( )

2 2

1 2

1 3

2 1

res res i res

r

i res

b L R b dT

k T

ω ω ω

ω

+ − −

= (55)

( )

(

¹

)

0 3

res i 2

r

i res

L R b T dR

k T

ω

ω

+ − −

= (56)

p 2 res

k R

=ω (57)

The pole assignment technique aims, by designing the controller, to place the poles of the closed loop system at desired locations. The poles of the closed loop transfer function Hclose(s) (58) are now described by a fourth order polynomial. A desired char-acteristic polynomial Dclose(s) (59) is obtained using the principle of the “double ratios”

[13]. In (59) τ is a parameter for the position of the poles (Appendix A.6).

( ) ( ) ( ) 1 ( ) ( )

c p

close

c p

H s H s

H s

H s H s

= + (58)

4 3 2

4 3 2

( ) 1

64 8 2

close

D s =τ s +τ s +τ s + +τs (59)

The solution of the equation system for the coefficients of the denominator of (58) with the coefficients of (59) delivers the parameters b2 (60), b1 (61), Ti (62) and τ ^(63).

( )

( )

( )

2 2 2 2

2 4

32 1 2 2 1

1 2

D D res

D res

T T d

b dT

ω ω

+ −

= + (60)

( )

( )

2 2 2 2 2 3 3 3

1 4

8 1 6 8 12 8

1 2

D D res D res D res D res

D res

T dT T d T d T

b dT

ω ω ω ω

ω

+ − + +

= + (61)

( )

3 4

64 1 2

c f D i

D res

k k T

T ⁼ + dT ω (62)

8 1 2

D D res

T τ dT

= ω

+ (63)

The damping factor d must be smaller than 1 2 in order to achieve resonance. For example, in Figure 5.25 the damping factor d is 0.01, i.e. τ is limited to (64). Then, the dynamic of the closed loop is less dependent on the resonant frequency ωres, and it is limited to the poles described by (65). The Root Locus of the open loop transfer func-tion Hc(s)Hp(s) in Figure 5.28 confirm the above mentioned.

800 0,0012 100 2

D D res

T τ T

= ω ≈

+ (64)

1,2,3,4

2 2

1666 1666

s j j

τ

= − ± ≈ − ± (65)

To increase the degrees of freedom in the closed loop transfer function, it is neces-sary to add a derivative part in the PI controller, i.e. a PID. The new transfer function of the controller Hc1(s) is described by (66), and the desired numerator function Ncd1(s) is given by (67). The resulting controller transfer function using Ncd1(s) is described by (68).

2

2 1 0 1

1 2 2

1

( ) ( ) 1 1

2 ( )

( )

r r res r c

c p d

i res res c

res

k s k s k N s

H s k T s

T s s d s D s

C s PID

ω

ω ω

⎛ ⎞ + +

= ⎜⎝ + + ⎟⎠+ + + = (66)

( ) (

^{3 2}

)

1( ) 3 2 1 1

Ncd s = Ls R b s+ +b s +b s+ (67)

( ) ( )

( )

3 2

3 2 1

1 2 2

( ) 1

c 2

i res res

Ls R b s b s b s H s T s s dω s ω

+ + + +

= + + (68)

The PID controller adds a ZERO in the numerator of the open loop transfer function of the system, adding a degree of freedom for the poles in the closed loop. Again kr1, kr2, kr3 Td, and kp (69)(70)(71)(72)(73) are obtained from (66) and (68).

-6000 -4000 -2000 0

-6000 -4000 -2000 0 2000 4000

6000 0.72 0.58 0.44 0.32 0.2 0.1

0.84

0.96

0.1 0.2 0.32 0.44

0.58 0.72

0.84 0.96

1e+003 2e+003

3e+003 4e+003

5e+003 6e+003

Real Axis

Imag Axis

Figure 5.28: Root-locus of the open loop transfer function H_c(s)H_p(s) for ωres=156Hz.

( )

2 3

2 2

2 _res

r

i res

b L b R dL R

k T

ω

ω

+ −

= − (69)

(

^{2 2}

)

²

(

^{1 3 2}

)

1 3

2 1

res i res res res

r

i res

R b dT L b b

k T

ω ω ω ω

ω

− − + −

= (70)

( )

(

¹

)

0 3

res i 2

r

i res

L R b T dR

k T

ω

ω

+ − −

= (71)

2

3 res

d

i

T b L RT

= ω (72)

p 2 res

k R

=ω (73)

The desired characteristic polynomial for the closed loop Dclose(s) (59) does not change, because the order for the denominator is the same as in (58) using Hc1(s) instead of Hc(s). The parameters b3 (74), b2 (75), b1 (76), and Ti (77) are obtained by solving the equations for the coefficients.

( )

3 3

8 2

64

D D res

D

T d T

b T

τ τ − + τ ω

= − (74)

102 103 104 105

-270 -225 -180 -135 -90 -45

P.M.: 59.7 deg

Frequency (rad/sec) -20

0 20 40 60 80 100

G.M.: -12 dB

-15000 -10000 -5000 0 -1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

1 0.86 0.76 0.64 0.5 0.16

0.94

0.985

0.16 0.5

0.64 0.76 0.86

0.94 0.985

2e+003 1e+004

Real Axis

Imag Axis

x10⁴ a) b)

L = 12,5[mH]

R = 1,9[Ω] T_D = 150[µs]

k_c = 0,1 k_f = 20 [N/A]

d = 0.01 ωres= 156[Hz]

Figure 5.29: Root Locus (a) and Bode diagram (b) of the open loop for a PID+Res con-troller.

( )

2 4

2

32 2

64

D res D res

D

T d T

b T

τ −τ ω + ω

= (75)

4 2

1 64

res D

b T

τ τ ω

= − (76)

4

64

c f i

D

T k k T

= τ (77)

Now, the closed loop poles (65) can be chosen at a desired position. To achieve the dynamics of a second order system tuned with the amplitude optimum criteria, the real part of the poles must be placed in -1/2TD. This condition determines τ (78) and the poles of the closed loop (79).

2T_D 0,0003

τ = = (78)

1,2,3,4

2 2

6666,7 6666,7

s j j

τ

= − ± = − ± (79)

Figure 5.29a shows the Root Locus for the system with the four poles in the closed loop in the desired position (79). In Figure 5.29b, the resonance peak in the Bode

dia-0 2 4

0 1 2 3

0.035 0.036 0.037

0 1 2

-5 0 5

400 500 600 700 x10^-4

x10² x10³ x10⁴

x10⁸

x10² x10³ x10⁴

x10² x10³ x10⁴

x10⁶

x10² x10³ x10⁴

x10⁶

x10² x10³ x10⁴

x10² x10³ x10⁴

k_p

K_p/T_i

K_pT_d

k_r0

k_r1

k_r2

ω_res ^[rad/s]

ω_res ^[rad/s]

ω_res ^[rad/s] ω_res ^[rad/s]

ω_res ^[rad/s]

ω_res ^[rad/s]

Figure 5.30: Parameter for the PID+Res controller for frequencies from 100 to 10000 rad/s.

gram of the open loop transfer function can be clearly appreciated.

In order to guaranty stability at low speed, for frequencies under 100 rad/s (i.e.

16 Hz) the resonant controller and the derivative part is disconnected and the PI take the parameter given by the amplitude optimum criteria (32)(33).

Ti is the only parameter that does not depend on ω^res and has a constant value deter-mined by τ. Under the condition of (78), Ti takes the value 1,75x10^-12. With a damping factor of 0.01 and replacing (74), (75) and (76) into (69), (70) and (71), the controller parameters are obtained as function of the resonance frequency ω^res. Figure 5.30 shows the parameters of the controller for frequencies from 100 to 10000rad/s. However, the large variation for the integral component kp/Ti and the resonant parameter kr1 and kr0, destabilize the closed loop.

A practical solution is to preserve the PI controller tuned with the Amplitude Opti-mum (AO) Criteria and add only a sine resonant element Csin(s) ((46) page 74) or a co-sine resonant element Ccos(s) ((47) page 74).

-6000 -4000 -2000 0

-4000 -3000 -2000 -1000 0 1000 2000 3000

4000 0.84 0.72 0.6 0.46 0.30.14

0.92

0.98

0.14 0.3 0.46 0.6 0.72 0.84

0.92 0.98

1e+003 3e+003

5e+003

Real Axis

Imag Axis

965 970 975 980 985

-10 -5 0

965 970 975 980 985

Real Axis

PI+C_sin(s)

PI+C_cos(s) a)

b)

c)

Figure 5.31: Root Locus diagram of the open loop for a PI+Csin(s) (a)(b) and a PI+Ccos(s) (C) controller. The PI controller is tuned with the AO criteria. Kr is set to 1. the

reso-nance frequency is set to 156 Hz.

From the literature it is known that the phase margin of Ccos(s) is 90° higher than that of Csin(s). Therefore Csin(s) has a poorer phase margin. Poor phase margin can cause a highly under-damped behavior [61] of the system. In case of a PI+Csin(s) or a PI+Ccos(s), the number of poles and zeros of the controller does not change, i.e. there is no effect concerning the phase of the system. Figure 5.31 shows the Root Locus dia-gram for the open loop Hc(s)Hp(s) for both resonant controllers, tuning the PI according to the AO criteria. Roughly, there are no differences between both controllers (Figure 5.31a) but looking in detail around the resonant pole, the sine resonant element shows a zero on the right-half s-plane (Figure 5.31b), i.e. the system has non-minimum-phase zeros. Even so, both controllers will be analyzed and applied to the force control loop.

To demonstrate the behavior of the PI+Csin(s) (and PI+Ccos(s)) controller, two ex-periments are carried out. At first, the linear machine is moved at constant speed v = 1.56 m/s. The force reference Fx* is set to 200 N. At the beginning, the switch S1 is open (Figure 5.26), i.e. only the PI controller is active. Then, at position x = 240 mm, the switch S1 is closed. Figure 5.32 shows the poor behavior of the Csin(s) controller, which needs more than five cycles of the force ripple to begin to suppress it. On the other hand, the controller Ccos(s) needs only one cycle of the detent force to begin the suppression of the ripple. For an extended traveling distance, Figure 5.33 shows the force FxFEM along 3 segments. The resonant controller Csin(s) is more affected in the transition area than the Ccos(s). In general, for all positions the cosine variant of the resonant element shows a better behavior and ripple rejection.

240 260 280 300 320

180 190 200 210 220

240 260 280 300 320

180 190 200 210 220

F xFEM [N]

x [mm]

FxFEM [N]

x [mm]

F_x^FEM

F_x^*

F_x^FEM

F_x^*

6[N]

a) b)

Figure 5.32: Comparison of PI+Csin(s) (a) and PI+Ccos(s) (b) activated at position x = 240 mm for the force control. Force reference Fx* = 200 N. Experiment carried out at constant speed

v = 1.56 m/s. Time windows: ≈ 64 ms for 100 mm.

In order to investigate the dynamic effect of the resonant element added to the loop, a second experiment was carried out. With the same conditions as the previous experi-ment, the cosine resonant element was always activated. Then, at position x = 240 mm a step in the command force Fx* from 200 N to 240 N is applied. Figure 5.34 shows the command force Fx* and the calculated force FxFEM. There are nearly no differences comparing the dynamic behavior of the superimposed force-flux control loop as shown in Figure 5.24 and the behavior of the force+resonant controller in Figure 5.34, but in the steady state the force ripple is considerably reduced.

30 150 270 390 510 630 750 870 990 1.110 1.230 180

190 200 210 220

30 150 270 390 510 630 750 870 990 1.110 1.230 180

190 200 210 220

Segment 1 fed

Segment 2 fed

Segment 3 fed

x [mm] = Vehicle position F xFEM [N]F xFEM [N]

x [mm]

a)

b)

6 [N]

Figure 5.33: Extension of the force measurement F_x^FEM for 3 segments. The reso-nant element Csin(s) (a) and C_cos(s) (b) is activated at x = 240 mm respectively.

Experiment carried out at constant speed v = 1.56 m/s and constant reference force Fx*.

230 235 240 245 250 150

200 250 300

x [mm]

F xFEM [N] F_x^*

F_x^FEM

Figure 5.34: Force step response of a PI+Ccos(s) controller. Command step from 200 N to 240 N at

position x = 240 mm. Experiment carried out at constant speed v = 1.56 m/s.

Im Dokument Investigation of Control Methods for Segmented Long Stator Linear Drives (Seite 90-101)