Small Signal Modeling and Analysis of Control Speed for Two Mass Resonant System
Mohammad Reza Yousefi, Gahzanfar Shahgholian, Afshin Etesami, Pegah Shafaghi
Department of Electrical Engineering, Islamic Azad University Najaf Abad BranchEsfahan, Iran
[email protected], [email protected], [email protected], [email protected]
Abstract—This paper presents a speed control strategy for the torsional vibration systems. A state feedback control law using an integrator is designed. The pole-placement controller and integr- al of time multiplied by absolute error (ITAE) is used to assign closed-loop poles of the system characteristic equation. The simu- lation results are show to verify good performance obtained using the proposed controller.
Keywords-two-mass system; stability analysis; speed control;
state feedback control; pole placement technique.
I. INTRODUCTION
A mechanical system composed of some masses connected with flexible shafts such as flexible arms, rolling mills is calle- d multi-mass resonant system. In motor drive system, if link up motors and load by a flexible shaft, the motor drives system becomes a mechanical resonance system called two-mass torsional system [1].
The tasks of speed controller for two-mass resonant system are suppression of shaft torsional vibration, faster tracking the load speed of the speed reference without overshoot, rejection the effect of the load disturbance torque and robust controller.
Several papers have discussed the configurations and control strategies for two mass systems, such as H∞ control [2], sliding mode control [3, 4] and PID control [5]. In [6] cheap control theory for single-input two-output systems is used to quantify performance fundamental limitations on the class of all the stabilizing controllers for two-mass systems. The control of a two mass system with uncertain parameter values, encompass- ing friction and hysteretic effects, modeled as a functional differential equation in [7]. The neural estimators, based opti- mal brain damage method, of the state variable of the torsional torque and the load machine speed for drive system with elastic joints presented in [8]. In [9], an optimal controller to achieve better reference tracking and disturbance rejection performance is introduced by pole assignment using ITAE criterion for the two-inertia stabilization system is proposed.
The paper is structured as follows: in section II a two-mass model is derived. In section III a state feedback control law us- ing an integrator is designed. The stability analysis of the syste- m is show in section IV. Simulation results are illustrated in se- ction V to verify good performance obtained using the propose- d controller. Conclusion is given in section VI.
II. TWO MASS SYSTEM MODEL
The two mass system resonant mechanical system which has a motor and a load connected with a flexible shaft is shown in Fig. 1.
Figure 1. Two mass resonant system
Assuming that JL is load inertia, BL is load damping coeffi- cient, KS is the torsional stiffness of the shaft, BS is shaft dam- ping coefficient, JM is motor inertia and BM is motor damping coefficient, with choosing state variables as motor speed (ωM), load speed (ωL) and shaft torsional torque (TS) and input varia- bles motor torque (TM) and load disturbance torque (TL) the st- ate equation of two-mass resonant system is as follows [10]:
M M S M M M
M
M T
J T 1 J
1 J
B dt
d ω =− ω − + (1)
L L
S L S M M
S M S
S )
J B K B
( J )
B K B
( dtT
d = − ω − − ω
L
L S M M
S S L M
S T
J T B J T B J )
1 J ( 1
B + + +
− (2)
L L S L L L
L
L T
J T 1 J
1 J
B dt
d ω =− ω + − (3)
Output variable which can be measured is the motor speed.
Controlled variable is the load speed and disturbance is injected into the load. With the state equations the circuit can be easily modeled by using the functional blocks. The mechanical reson- ance frequency of the open-loop system is given by:
) 1 K J ( K
J L
R = S +
ω (4)
Increasing the inertia ratio KJ=JL/JM and shaft stiffness will decrease the mechanical vibration of the system. Generally,
978-1-4244-7398-4/10/$26.00 ©2010 IEEE 1000 IPEC 2010
the speed ωM and position θM of the motor shaft differ from
the respective variables ωL and θL, on the load side. During tr- ansients, speeds of motor and load differ, and torsional torque is given by:
)]
s ( ) s ( [ K )]
s ( ) s ( [ B ) s (
TS = S ωM −ωL + S θM −θL (5) The simplified block diagram of the two-mass system is shown in Fig. 2.
Figure 2. Block diagram of two-mass resonant system
Assume that the model of two-mass is defined by U
B X A
X = + andY=CX , where X=
[
ωM TS ωL]
T is state vector, U=[TM]is control signal and Y=Xis output signal. The load speed and motor speed are:) s ( )T s ( G ) s ( G ) s ( G ) s ( G 1
) s ( G ) s ( G ) s ( ) G
s
( M
) s LM( H
S L S M
S M L L
+ +
= ω
) s ( )T s ( G ) s ( G ) s ( G ) s ( G 1
)]
s ( G ) s ( G 1 [ ) s ( G
L )
s LL( H
S L S M
S M L
+ +
− + (6)
) s ( )T s ( G ) s ( G ) s ( G ) s ( G 1
)]
s ( G ) s ( G 1 [ ) s ( ) G
s
( M
) s MM( H
S L S M
L S M
M
+ +
= + ω
) s ( )T s ( G ) s ( G ) s ( G ) s ( G 1
) s ( G ) s ( G ) s ( G
L )
s ML( H
S L S M
S M L
+ +
− (7)
where the transfer functions of the motor, load and shaft are GM(s)=1/(sJM+BM), GL(s)=1/(sJL+BL) and GS(s)=BS+KS/s, resp- ectively. A two-input single-output process be represented by the block diagram of the system shown in Fig. 3.
Figure 3. Two-input and single-output of the system
III. STATE FEEDBACK CONTROLLER
Fig. 4 shows a block diagram of the speed control system using a state feedback controller. The system is to insert an int- egrator in the feed forward path between the error comparator and the model. The control signal is:
V S H L L M M
M K K K T E
T =− ω − ω − + (8) where EV is output of the integrator. The system dynamic in close-loop can be described by X AX BU
+
= , where
[
M TS L EV]
TX = ω ω
, KA =A−B
and B =BKEωC . The ωC is reference input speed and K is gain vector:
[
KM KH KL]
TK= (9)
If the desired eigenvalues of matrix A
are specified, then the state feedback gains vector and the integral gain constant KE can be determined by the pole placement technique.
Figure 4. Speed control system using a state feedback controller with an integrator
IV. DESIGN CONTROLLER
The gains controller is evaluated using the integral of time multiplied by the absolute error (ITAE) criterion for a step reference input. The ITAE performance index provides the best selectivity by minimizing overshoot and settling time for a given undershoot. As for the ITAE criterion, the standard form coefficients for a step input is given by:
4 n 3 n 2
2 n 3
n
4 2.1 s 3.4 s 2.7 s
s + ω + ω + ω +ω (10) where ωn represents the 3dB bandwidth [11]. The gains of the controller obtained from ITAE criterion are given by:
4n S
L E MK
J
K = J ω (11)
1001
n M S
3n L
L M 2.1J
K J J 7 .
K =2 ω − ω (12)
1 J ) K K
4 J . 3 K ( K J
L 4 S n S 2 L n S
H = M ω − ω − − (13)
n M M 2.1J
K = ω (14) V. STABILITY ANALYSIS AND SIMULATION RESULTS Small signal stability is a fundamental condition for a safe and reliable operation of electric power systems. The small signal stability of electric power systems is given by the eigen- values of the state matrix. Eigenvalues may be real or complex.
A real eigenvalue represents a non-oscillatory mode, so if it is positive it corresponds to a periodic instability. The two-mass system parameters used in this paper are given in Table I.
TABLE I. NOMINAL PARAMETERS OF ATWO-MASS PLANT
Component Quantity Rating value
KS shaft stiffness 242 Nm/rad
BS shaft damping coefficient 15×10-2 Nm/rad/s JM motor inertia 641×10-4 kg.m2 BM motor viscosity coefficient 2.1×10-3 Nm/rad/s
JL load inertia 523×10-4 kg.m2 BL load viscosity coefficient 5.3×10-2 Nm/rad/s ωR natural frequency 91.7 rad/s
η damping ratio 0.0284
The transfer function from TM to ωM, which is most impo- rtant in the closed loop design, is given by:
) s ( s
s J ) 1 s (
H 2
R 2
2 A 2
M
MM +ω
ω
= + (15)
This transfer function has two particular points: the anti- resonant (ωA) and resonant (ωR) frequencies. At these frequen- cies, the phase characteristics change drastically. Fig. 5 show the frequency response of motor torque to motor speed (real line), shaft torque (dot line), and load speed (break line) in open-loop system. Fig. 6 show the step response of motor speed to motor torque and load torque. The dominant eige- nvalues for open-loop system are shown in Table II. The resonant frequency is ωR= 91.7 rad/s and anti-resonant freque- ncy is ωA=68.0 rad/s. The damping factor of the original plant without the controller is η=0.0315. Since all eigenvalues of the system are on the left hand of the plane, the system is stable but highly damped.
Parameters of state feedback controller for change in ωn
have been shown in Table III with ITAE criterion. Step respo- nses of all gains controller are plotted in the same figure in order to compare different cases (Figures 7, 8 and 9). It is clear that the damping increases, and there is no oscillation in the system response. However, there is still a large overshoot and steady-state error in the shaft torque and response is even worse.
Figure 5. The response frequency of transfer function relative to motor torque without controller
Figure 6. The step response of motor speed without controller TABLE II. OPEN LOOP SYSTEM DOMINANT EIGENVALUES
Mode
number Eigenvalue Frequency Damping
ratio Variable 1, 2 -2.8905±j91.6183 14.5888 0.0315 TS
3 -0.4734 - - ωM
TABLE III. GAINS CONTROLLER WITH ITAE CRITERION
Case ωn KE KL KM KH Eigenvalue A 0.8819 ωA 179.4 0 8.1 0.3 -25.4±j75.8 -37.6±j24.8
B ωA 296.6 2.6 9.2 0.3 -28.8±j85.9
-42.6±j28.2 C 1.2 ωA 615.0 9.4 11.0 1.2 -34.6±j103.1
-51.1±j33.8
1002
Figure 7. The step response of motor torque with controller
Figure 8. The step response of shaft torque with controller
Figure 9. The step response of motor speed with controller
VI. CONCLUSION
A space-state mathematical analysis and design methods for a two-mass system is develop. To eliminate the steady state error, an
integral feedback is added to the motor speed feedback. The simul- ation results of the speed control of the two-mass using the propos- ed controller are investigated to verify the effectiveness of the proposed method.
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