Dual robust controller design for high power AC servo drive

Dual Robust Controller Design for High Power AC

Servo drive

Stone Cheng

†

_{, Yuan-Yong Huang, Hsin-Hung Chou}

†_{Department of Mechanical Engineering, National Chiao-Tung University, Hsinchu, Taiwan, ROC}

Mechatronics Control Dept., Intelligent Machinery Technology Div., MSL/ITRI, Hsinchu, Taiwan, ROC

Abstract- High power AC motors have a highly interacting

multivariable control structure, and it is difficult to design high dynamic performance AC drive with traditional PID-like controller for high power AC servo motor. This paper presents analysis, design and simulation of velocity loop dual robust controller for 11kw permanent magnetic synchronous motor (PMSM) in the AC servo system. By combining PDFF-MA and Hf control algorithms with its own capability of achieving good performance criteria such as dynamic reference tracking and load torque disturbance rejection,

The PDFF-MA controller is designed and analyzed in the forward loop to provide low frequency stiffness and overcome low-frequency disturbances like friction. To compensate the system response, moving average(MA) error filter is added. While in the feedback loop, Hf controller is designed to meet system robust stability with the existence of external disturbance and model perturbations. The proposed PDFF-MA and Hf controllers are designed based on the transfer function of the poly-phase synchronous machine in the synchronous reference frame at field orientation control (FOC). The parameter variations, load changes, and set-point variations of synchronous machine are taking into consideration to study the dynamic performance.

Keywords: AC servo motor, PDFF-MA controller, Hf feedback control,

I. INTRODUCTION

The AC servo drive plays an important role in industrial motion control applications including machine tools, factory automation and robotics in the low-to-medium power range. Several situations encountered in these applications: 1) Load inertia and friction variation during operation as the payload changes. 2) The presence of a tensional resonance of the mechanical system limited the System bandwidth. 3) In AC servo motors, higher torque ripple and coupled dynamics with magnetic flux caused the nonlinearities in torque response and torque transients. 4) During these applications, the set-point tracking capability in both dynamic and steady-state conditions and the load torque disturbance rejection capability are varying. Several control techniques [1-7] have been developed to overcome these issues. Derived from generalized PID controller, the PDFF controller is allowing the user to eliminate overshoot and provide much more DC stiffness than PI by properly choosing the controller parameters and is less sensitive to plant parameter variations, and its disturbance rejection characteristics are much better than that of the PI controller. Along with PDFF controller, Hf control theory is one of the successful algorithms for robust control problem in AC servo drive to provide better tolerance to disturbance and

modeling uncertainties. In this paper, the Hf design procedure[5,9,10] is proposed and consists three main stages: 1) using weighting matrices W1 and W2 to shape the singular values of the nominal plant follows the elementary open-loop shaping principles; 2) the normalized coprime factor Hf problem is used to find a robust central controller stabilizing this shaped plant, and the observer is obtained from the left coprimeness of the central controller; 3) the H parameter in the controller is used as a tradeoff between robust stability and performance.

II. MATHEMATICAL MODEL OF THE PMSM

The field orientation of the AC servo motor is defined as d-axis, and q-axis that leads the d-axis 90 electric degrees. In the d-q coordinates, the AC servo motor voltage-current and flux equations are shown as follows:

d d d r q

v

Ri



O Z O





(1) q q q r d

v

Ri

 

O Z O



(2) d

L i

d d PM

O



O

(3) q

L i

q q

O

(4)

Where vd and vq are voltages of the d, q axis; R is the stator resistance; id and iq are the d, q axis stator currents; Zr is the rotor speed; Od and Oq are the d, q axis flux induced by the currents of the d, q axis inductance; Ld and Lq are the q, d axis inductances with the same value, and OPM the constant mutual flux of the permanent magnet.

When the stator current vector is oriented perpendicular to the rotor magnetic field, the field-oriented control for Ac servo motor yields id =0. In the case, the electromagnetic torque is in strict positive proportion to iq:

3

4

e PM q T q

P

T

˜

O

˜

i

K

˜

i

(5)

where P is the number of poles of motor, and KT is the motor torque constant.

The mechanic motion equation is:

r e T q d r

d

T

K i

T

B

J

dt

Z

Z





(6)

where J is the moment of inertia; B is the viscous friction, and Td is the torque disturbance such as the load resistance, the torque ripple and the resistance caused by nonlinear factors.

III. DESIGN OF THE CONTROL SYSTEM

A. Control Scheme

The proposed control scheme is presented in Fig. 1 where the nominal plant is G(s) = 1/(Js+B); K(s) is the velocity feedback controller designed by the loop shaping design procedure (LSDP) and the algebraic method, and the velocity lop controller is a servo controller. K(s) is used for attenuating the disturbance TL, and plant uncertainty, and the PDFF controller is used as velocity loop adjuster to improve the low-frequency stiffness.

Fig. 1 Dual robust control scheme B. Velocity Feedback Controller

In this paper, a continuous time control design approach based on HЌ

-optimization control design is performed for a model of the AC servo system as seen from the digital computer control design approach. Consequently, performance is specified at the controller disturbance instants.

Minimum phase W1 and W2 are proper stable, real rational

function denoted by RHf.The left and right coprime factorizations of W1GW2 are

M N

 

_S1 _S and

N M

_{S S}1 ,

respectively. Moreover, a doubly coprime factorization exists as follows: r r S l S l r r S S S l S l S S X Y M Y M Y X Y I N M N X N X N M   ª º ª º ª º ª º «_ » « » « » «_ » ¬   ¼ ¬ ¼ ¬ ¼ ¬   ¼ (7) where N _S, MS, N S, MS, Xr, Yr, Xl, and Yl are over RH

f_.

Then, the velocity controller K(s) is defined as follows:

1 2

( )

( )

_v

( )

( )

K s

W s K s W s

(8) Where _{( ) [} 1 _{] [}1 1 _] v r l r l K s X H Y N   Y H Y M  and H is a unit over RHf. With K(s) of (8), the velocity feedback loop is internally stable. Moreover, Xr and Yr of Kv(s) in (8) play the similar role as central controller although H in Kv(s) cannot be 0. According to this property, Xr and Yr can be designed using the LSDP and H will be used to reject step and sinusoidal disturbance, as follows.

C.Design of Velocity Controller Using the LSDP and the Algebraic Method

The first stage in the LSDP uses a pre-matrix W1 and/or a

post-matrix W2 to shape the singular values of the nominal

plant G as a desired open-loop shape GS = W2GW1. Constant or

dynamic W1 and W2 are selected such that GS has no hidden modes. Constant weighting matrices can improve the performance at low frequencies and increases the crossover frequency. Moreover, the dynamic W1 or W2 is used as the

integral action with the phase-advance term for rejecting the input and output step disturbances. W1 orW2 is selected as the

diagonal matrix and each principal element is (s+I)/s where I>0 is lower than the crossover frequency. The integral action improves the performance at low frequencies, and the phase-advance term s+I avoids the slope of the open-loop shaping at the crossover frequency more than 2, and adjusts

the robustness in the feedback system. If I is closer to the imaginary axis, the robustness is larger. The stage is the same as the velocity controller herein.

[11-14] advocate an expression of coprime factor uncertainty in terms of additive stable perturbations to coprime factors of the nominal plant. Such a class of perturbations has advantages over additive or multiplicative unstructured uncertainty model. For example, the number of unstable zeros and poles may change as the plant is perturbed. The perturbed plant [See Fig. 2.] is written

1

(

_{S N}

) (

_{S M}

)

G

_'

N

 ' ˜

M

 '

 (9)

where the pair (MS, NS) is a normalized right coprime

factorization of GS, and 'M and 'N are stable, unknown transfer functions representing the uncertainty and satisfying

N M

H

f

'

ª

º



«

_'

»

¬

¼

, where

H

( 0)

!

presents the stability margin.

In the second stage of the LSDP, the robust stabilization Hf problem is applied to the normalized right coprime factorization of GS, and obtains a robust controller Kf

satisfying

Fig. 2 Right coprime factor robust stabilization problem

>

@

1

₍

₎

1 1 S S

M



I

K G

_f 

K

_f

I

H

 f



d

(10)

Suppose the shaped plant of GS has the minimal realization

(A, B, C, D). A central controller satisfying (10) is obtained as follows [15]: 2₍ T₎1 T_{( )} 2₍ T₎1 T T T A BF W ZC C DF W ZC K B X D J  J  f ª     º « » ¬ ¼ (11) where _{F S}1_{(D C}T _{B X}T _);

_W

_

_I

₍

_XZ

_

J

2

_I

₎

, and X and Z are the solutions to the two algebraic Riccati equations as follows: 1 1 1 1 ( T )T ( T ) T T 0 A BS D C  XX A BS D C  XBS B X C R C (12) 1 1 1 1 (_{A BS D C Z}  T ) _{Z A BS D C}( _  T )T_{ZC R CZ}T  _{BS B} T 0 (13) where R = I+DDT_{, and S = I+D}T_D.

If the plant is assumed to be strictly proper, i.e. D = 0, the realizations for the doubly coprime factorization can be presented as follows.

0

S S

A BF

B

M

F

I

N

C



ª

º

ª

_{º «}

_»

«

_{» «}

_»

¬

_{¼ «}

_»

¬

¼

(14) 1 S

M



N

S M

'

'

N 6 6



K

_f



 r Z 1 JsB ( ) G s r   PDFF controller H f _feedback controller K s( ) I K s KT KFB KFR  + + + + L T

0

S S

A QLC

B QL

N

M

C

I



ª

º

ª

_{º «}

_»

¬





_{¼ ¬}

_¼

(15)

>

r r

@

₀

A QLC

B

QL

X

Y

F

I





ª

º

«

_

»

¬

¼

(16)

>

l l

@

₀

A BF

B

QL

X

Y

C

I





ª

º

«

»

¬

¼

(17)

The pair (NS, MS) in (15) is the left coprime factorization of GS, but not the normalized left coprime factorization.

Moreover, the pair (Xr ,Yr) are the left coprime factorization of Kf when D = 0. That is, Kf = Xr1Yr. The result presents for the second stage of the velocity controller that the pair (Xr,Yr) in Kv(s) of (8) can be obtained from the left coprime factorization of Kf when D = 0.

In Fig. 1, the transfer function from TL to Zr is (18).

1 1

1

(

)

1

r

W N X

S r

H Y N W

l

T

L

Z

_

_







_˜

(18) For a step in TL, Zr with the zero steady state must satisfy the following equation, according to the final value theorem.

1 1 0 0

(

_{r l S}

)

(

_{S r}

)

0

s s

X



H Y N





H



M



X

 (19) For rejecting a sinusoidal disturbance with known frequency V in TL, the following equation must be satisfied obviously.

1 1

(

_{r l S}

)

(

_{S r}

)

0

s j s j

X

H Y N

H

M

X

V V  









(20)

Hence, for rejecting a step and/or sinusoidal disturbance in TL, H can be designed algebraically. For example, if only the step disturbance exists in TL, H is designed to be constant as follows. 1 0

(

_{S r}

)

s

H

M



X

 (21)

If only a sinusoidal disturbance with known frequency

V

₁ exists in TL, H needs two unknown coefficients and is designed as follows: 1 1

( )

s k

H s

h

s

p





(22) where H of (22) satisfies 1 1 1

( )

_{s j}

(

_{S r}

)

s j

H s

_V

M

X

V 



(23)

p(>0) is given, and h1 and k1 can be solved according to (23).

Analogously, if a number of n sinusoidal disturbances with n known frequencies V1~Vn, H needs 2n coefficients to be solved as follows. 3 2 2 1 2 2 1

( )

(

)

(

)

n n

h

h

h

H s

h

s

p

s

p

s

p







 





"



(24)

Hence, since the pair (Xr ,Yr) in Kv is the left coprime factorization of Kf in the LSDP, the completed velocity

controller has several properties of the LSDP, including consideration of plant input and output performance, limited deteriorations at plant input and output, and bounded closed-loop objective functions. The three major properties of the LSDP are listed in [16]. Moreover, the velocity controller can use the H parameter to reject step and/or sinusoidal disturbances.

The velocity feedback loop also has robustness with coprime factor uncertainty, and satisfies the following robust inequality:

1 1 1 r l S r l S v

Y

H Y M



X

H Y N



H

 f

ª





º

d

¬





¼

(25)

where Hv is the stability margin in the velocity feedback loop. Eq. (25) presents that the H parameter can affect the value of the stability margin Hv.Herein, H is selected according to the control requirements and then the value of Hv can be checked. H may need several redesigns to obtain a satisfactory value of Hv. Moreover, for the sake of the numerical realization, Kv also can be written as Kv =(1+Cv Xr)-1CvYr where Cv=HMS.

D.PDFF Velocity Control Method

In digital control systems of AC servo drive, most of applications are using its velocity and torque control mode. The position loop of AC servo drive is taken control by outside multi-axis controller such as CNC controller. Many controllers design use PI velocity loops, eliminating the derivative term. Tuning PI loop is easy and is ideal for maximum responsiveness applications such as pick-and-place machines. But PI control has a weakness—because of its integral gain must remain relatively small to avoid excessive overshoot provides that it does not have good low frequency "stiffness". PDFF velocity control was developed to combat this problem. Fig. 3 shows the block diagram in frequency domain of a plant with a PDFF controller of the form:

 

 

 

( )

I

( )

FR FB

K

u s

d s

K

r s

e s

K

y s

s



˜



˜



˜

(26)

Fig. 3 Plant and disturbance with PDFF Controller

The transfer function of disturbance to output with the plant is simplified as a first order model is derived by





 

 

2

( )

d FB I

y s

s J

G s

s



B J



K

J s K J



d s

(27)

One of the most important specifications in many motion control applications is the load-torque disturbance rejection capability. The disturbance response can be tuned by moving closed poles more to the left side in the complex plane, and tracking response can be further optimized by adding zeros to the system via feedforward, as shown in (28).









 

 

2

( )

I FR c FB I

K

K s J

y s

G s

s

B J

K

J s K J

r s









(28)

The PDFF controller which locates the zero at an optimal place that shortens the step response rise time without overshoot.

IV. RESULTS OF SIMULATION RESEARCH

An high power AC servo motor model is included in the simulation, its mechanical parameters are: J = 6.37 and B = 0.1. According to the method discussed in part C of Section III, W1, W2, Xr, Yr, H and Cv are given as follows.

1 JsB ( ) G s r PDFF controller I K s KFB KFR y d + _ _ + e + + + u

3 1

5 10 (

s

2500)

W

s

u



, W2 = 1, 2 4 7 2 4 7

1.395 10

2.348 10

1.181 10

1.216 10

r

s

s

X

s

s



u



u



u



u

, 4 7 2 4 7

2.016 10

1.216 10

1.181 10

1.216 10

r

s

Y

s

s

u



u



u



u

3

0.393(

2.713 10 )

1

s

H

s





u



3 3 2 6 9 3 3 2 6 6 1.393 1.903 10 3.040 10 2.090 10 2.132 10 1.965 10 1.962 10 v s s s C s s s   u  u  u  u  u  u

The Simulink model of the velocity control loop with PDFF and Hf feedback controller is shown is Fig. 3. The comparison on the simulation results of the velocity control loop with PI and PDFF plus Hf feedback controller is indicated in Fig. 4(d) shown the response of the two types of controller when the step and sine disturbance is added to the system.

The design yields that GS has the crossover frequency about 300Hz as shown in Fig. 4(a), and the velocity feedback loop have the stability margin 19.36%. Moreover, it yields that the velocity feedback loop can reject the 250Nm step at 0.02 sec and 300Hz sinusoidal at 0 sec disturbances in TL as shown in Fig. 4(b), and the input sensitivity, W1MSXrW1-1 is presented in

Fig. 4(c). The effect of PDFF controller also has contribution on the disturbance rejection, as shown in Fig. 4(b).

PDFF

controller H-inf feedback

controller T ransport Delay nK/1(s) dK(s) Transfer Fcn t To Workspace3 y To Workspace2 u To Workspace1 Step x' = Ax+Bu y = Cx+Du State-Space Sine Wave PID Ki/s -K-KFR -K-KFB 1 Gain2 12:34 Digital Clock 0 Constant

Fig. 3. Simulink model of PDFF plus Hf feedback controller

Frequency (rad/sec) P h as e ( d eg ); M agni tude ( d B ) 0 100 200 10-1 100 101 102 103 10 -180 -160 -140 -120 -100

(a) Bode plot of Gs

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 -0.01 -0.005 0 0.005 0.01 0.015 0.02 sec without PDFF with PDFF (b) -150 -100 -50 0 50 Mag ni tu d e ( d B ) 101 102 103 104 105 -180 0 180 360 540 P h as e ( d eg ) Bode Diagram Frequency (rad/sec) w ithout PDFF w ith PDFF (c) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 sec

(d) Disturbance responses comparison of PI and dual robust controller Fig. 4 (a) GS shape (b) disturbance responses with 250Nm step (at 0.02sec) and

sin600St (at 0 sec) (c) input sensitivity (d) Disturbance responses comparison of PI and dual robust controller with 250Nm step (at 0.05sec) and sin600St (at 0.02 sec)

In digital systems, many manufacturers use PI velocity loops, eliminating the derivative term. PI loops are easy to tuneand is ideal for applications that demand the maximum responsiveness such as pick-and-place machines. But PI

PI controller with gain increasing

PDFF+Hf feedback controller

control has a weakness—it does not provide good low frequency "stiffness". The key difference between PI and PDFF is that PDFF forces the entire error signal through integration. This makes PDFF less responsive to the velocity command than PI. Although the feed-forward term injects the command ahead of the integral making the system more responsive to commands, moving average (MA) filter of error signal is considered to improve the system responsiveness. Fig. 6 shows the step response of a AC servo motor and drive system with MA filter compensation in the velocity loop. Figure 6(a)(b) shown that MA compensation can improve the system response while in the heavy load driving period.

Fig. 5 Block diagram of PDFF controller with MA filter.

1

2

3

1

2

3

(a) No load

1

2

3

1

2

3

(b) With iron round plate load

Velocity feedback Current feedback Command input 2 1 Velocity feedback Current feedback Command input 2 1

(c) Velocity feedback and Current feedback response

velocity velocity feedback feedback current current feedback feedback w/o MA compensation w/o MA compensation with MA compensation with MA compensation velocity velocity feedback feedback current current feedback feedback w/o MA compensation w/o MA compensation with MA compensation with MA compensation

(d) Velocity feedback and Current feedback response Fig. 6 AC servo drive step response: (a) no load (b) with load. (c)(d)These

comparisons are using the same set of parameters, new program has better response and less overshoot. nw/o compensation, owith compensation, pMA compensation signal.

V. CONCLUSIONS

This paper proposes a dual robust controller design for the velocity loop of a high performance AC servo motor speed servo using PDFF-MA and Hf feedback control to meet the requirements of robust stability, exterior load disturbances rejection, low-frequency stiffness and responsiveness. The simulation and experimental results demonstrate the good control performance of the proposed control scheme.

ACKNOWLEDGMENT

Research supported by MSL project, ITRI, Taiwan, ROC. Corresponding author E-mail: [email protected]

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₊

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