IJRRAS 4 (2) ● August 2010
Ramana & al. ● Design Of The H-∞ Controller For Vsi Fed Pmsm Drive
DESIGN OF THE H-∞ CONTROLLER FOR VSI FED PMSM DRIVE Pilla Ramana 1, K. Alice Mary2 & M. Surya Kalavathi3 1
GMR Institute Of Techonology, Rajam, Srikakulam, AP-532127, India Viznan Institute Of Information Technology,Visakhapatnam, AP-520G4G,India 3 JNTU College Of Engineering, Kukatpally, Hyderabad, AP-500072, India E-mail:
[email protected],
[email protected],
[email protected] 2
ABSTRACT This paper presents a design method of H- ∞ controller for a permanent magnet motor drive included in the current loop in addition to the PI controller included in the speed loop for improving the system dynamics. A simulation has been performed using MATLAB software and the performance figures for different cases with and without H-∞ controller are obtained. From the results it is clear that performance of the system have increased for the different cases. The software realization for the Design of H -∞ controller for a nonlinear motor drive is performed and as an extension to this a hardware implementation may be expected. Keywords:
Permanent magnet synchronous motor (PMSM);
Exact feedback linearization; field oriented
control
1.
INTRODUCTION
This H-∞ techniques have the advantage over classical control techniques is that they are readily applicable to problems involving multivariable systems with cross-coupling between channels; disadvantages of H-∞ techniques include the level of mathematical understanding needed to apply them successfully and the need for a reasonably good model of the system to be controlled. With the development of control theory, more and more new control methods have being used to design controllers which will control the operation of synchronous electrical machine. The H-∞ control theory which use the H-∞ -norm as the performance evaluation is one of the most mature and wellused theory in the field of robust control, and the H-∞ control theory is already become one of the most popular subject which researched by many people in both control theory and engineering application. In recent years, the H∞ control problem have been simplified in both concept and algorithm, and the correlative software package have been developed and well used, so the H-∞ control theory gradually become a practical engineering theory. In this research, H-∞ control theory is used in the design of the motor controller to make the regulating process of the output terminal voltage become smoother. H-∞ control has developed its own terminology, notation, and paradigm, the basic object of interest in H-∞ control is the transfer function. In fact, we will be optimizing over the pace of the transfer functions. Optimization presupposes the cos (or objective) function, because we want to compare different transfer functions and choose the best one in the space. In H-∞ control [2], we compare transfer functions according to their H-∞ norm (a mathematical term for the concept of size). The permanent magnet motors are similar to the salient pole motors, except that there is no field winding excitation and the field is provided instead by mounting permanent magnets in the rotor. The excitation voltage can not be varied. The elimination of field coil, dc supply and slip rings reduces the motor loss and complexity. These motors also known as ‘Brushless motors’ are finding increasing applications in robots and machine tools. The model of a PMSM drive with damper windings on the rotor is derived with the reference frame fixed on rotor (Park’s equations). 2.
MODELLING OF PMSM
The voltage equations for the permanent magnet motor in rotor reference frame are
vqs raiqs lqs piqs laq piqr r ldsids r ladidr r
(1)
vds raids lds pids lad pidr r lqsiqs r laqiqr
(2)
vqr rqriqr lqr piqr laq piqs
(3)
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IJRRAS 4 (2) ● August 2010
Ramana & al. ● Design Of The H-∞ Controller For Vsi Fed Pmsm Drive
vdr rdridr ldr pidr lad pids
(4)
Where, ψ- air gap flux linkage The eqn. (1) can be rewritten as ' vqs vqs r ra iqs lqs piqs l aq piqr r ldsids r l ad idr
(5)
The electrical torque developed is
Te
3P lad laq iqsids ladiqsidr laqiqrids iqs 22
(6)
The torque balance equation is 2 2 (7) jpr Te Tl r P P Where all voltages (v) and currents (i) refer to the rotor reference frame. The subscripts qs, ds, qr and dr correspond to q and d axis quantities for the stator(s) and rotor(r) in all combinations, r a denotes the armature resistance and lfr the field inductance, while other inductances are denoted by l qs, lds etc. and Te is the developed torque. The rotor speed is given by ωr and the load torque by T l. J is moment of inertia, P is the number of poles and β is the coefficient of viscous friction. The derivative operator is represented by the symbol p. Representing the voltage eqns.(1)-(4) a state space representation and all matrices connected with it is of the form
r lds 0 r lad iqs lqs 0 laq 0 piqs v'qs ra v l ds r qs ra r laq 0 .ids 0 lds 0 lad . pids vqr 0 0 rqr 0 iqr laq 0 lqr 0 piqr 0 0 rdr idr 0 lad 0 ldr pidr vdr 0
(8)
It may be noted that vqr= vdr= 0. As these relate to damper windings, in d-q axes ralqr 1 wr lqsldr 2 A r alaq 1 l w ad r lqs 2
l qr 1 0 B l aq 1 0
wr ldslqr
rqrlaq
1 raldr 2 wr laqlds
1 wr laqldr
1 ralad 2
1 laqlad wr
2 rqrlqs
2
wr ladlqr 1 rdrlad 2 ladlaqwr 1 rdrlds 2
0 l dr 2 0 l ad 2
(9)
(10)
Partitioning A into linear and nonlinear terms and expressing them in matrices, we have
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IJRRAS 4 (2) ● August 2010
ra lqr 1 0 A' ra laq 1 0
0 l A' ' qs 0 0
rqr laq
0
1
ra ldr 2
0 rqr lqs
0
1
ra lad 2
0
l ds 0
0 l aq
0 0
0 0
Ramana & al. ● Design Of The H-∞ Controller For Vsi Fed Pmsm Drive
rdr lad 2 0 rdr lds 2 0
(11)
l ad 0 0 0
(12)
and
u ' [v 'qs
vds ]T
(13)
where
1 lqs lqr laq2 2 2 ldsldr lad
3. DESCRIPTION OF THE H-∞ CONTROL PROBLEM The above equations are described as a generalized system, which is shown in Fig.1. The diagram shown contains two main blocks, the plant and the controller. The plant section has two inputs and two outputs. The plant inputs are classified as control inputs and exogenous inputs. The control inputs u is the output of the controller, which becomes the inputs to the actuators driving the plant. The exogenous input w is actually, a collection of inputs (a vector). The main distinction between w and u is that the controller cannot manipulate these exogenous inputs. Typical inputs that are lumped into w are external disturbances, noise from the sensors, and tracking or command signals. The plant outputs are also categorized into two groups. The first group y is signals that are measured and feedback. These become the inputs to the controller. The second group z, are the regulated outputs. These are all the signals we are interested to control or regulate. They could be states, error signals, and control signals. In Fig.1, K(s) is the controller which is to be designed; P(s) is a linear time-invariant system, or called the generalized control object, described as the following state-space equations. .
x Ax B1w B2u
(14)
z = C1x+D11w +D12 u y= C2x+D21w +D22 u
(15) (16)
where x Rnq- the state variable; w Rm - the external disturbance signal which include reference signals, disturbances and noises. u Rr - the control input signal; z R -the output signal which include tracking signals, regulation signals and actuator’s outputs p y R - the measurement signal, as the sensor’s output signals. A transfer function representation of the system is given by
z Pzw w Pzu u, y Pyw w Pyuu, u Ky
(17)
The closed-loop transfer function between the regulated outputs and the exogenous inputs is obtained as follows. First, we substitute for u in the equation for
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IJRRAS 4 (2) ● August 2010
Ramana & al. ● Design Of The H-∞ Controller For Vsi Fed Pmsm Drive
y Pyw w Pyu ku
(18)
Solving for y
( I Pyu K ) y Pyw w
(19)
y ( I Pyu K )1 Pyw w
(20)
u Ky K ( I Pyu K )1 Pyw w
(21)
Therefore, u becomes
Hence the system can be represented in state space form as
x Ax B1w B2u
(22)
The equation (14-16) can be represented as follows A B1 B2 P( s ) C1 D11 D12 C D D22 21 2 A B P( s ) C D The transfer function of a system with state-space matrices (A, B,C, D) is given by G(s) = C(sI − A)-1 B+ D The relationship between the inputs and outputs can be described as z P11 P y 21
P12 w P22 u
(23)
(24)
(25)
u Ky
(26)
Figure 1: Description of the generalized system Taking the equation (26) into (25) to eliminate the measurement signal y, and the closed loop transfer function which from w to z can be expressed as Twz= F(P, K) = (P11+ P12K(I − KP22)−1P21) (27) Based on the above deduction, the standard H∞ control problem can be described as H∞ optimal control problem. 3.1Advantages of H-∞ controller It effectively controls the speed of the nonlinear motor drive considered. It makes the system fast and robust. Thus the time taken to reach the steady state condition will be reduced.
4. CONTROLLER DESIGN The control problem considered in this paper is to regulate output values to a specified value. Basically, damper windings have no influience on the regulation of output value, so the damping windings are ignored. The phase currents ids and iqs, being state variables, cannot be the input variables at the same time. There fore, a fictitious terminal resistance Ra was added to the model. This resistance has a relatively high value and does not effect
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IJRRAS 4 (2) ● August 2010
Ramana & al. ● Design Of The H-∞ Controller For Vsi Fed Pmsm Drive
other variables. So the equivalent circuit of the synchronous motor can be represented in Fig. 2. The equations corresponding to Fig. 2 can be described as a linear system in state-space form. The parameters of synchronous motor were obtained from literature [1]. The state variables x, input variables u, reference signals , and the measurement signals z are defined respectively as follows
iqs iq vqs x ids , u id , z , w V f Nld vds i fr i f
(28)
In the system, the measurement signals z are the output variables y. A PA PBR1 B P Q 0 *
-1
P (A +A − BR or, PX+Q=0
(29)
*
B ) +Q = 0
(30) (31)
post multiplying with X-1 on both sides of eqn.(31) we get, PXX-1 + Q X-1=0, i.e PI=-Q X-1 i.e P = -Q X-1 and K=R-1B *P Thus the H-∞ controller gain matrix is obtained.
(32) (33)
Figure 2: Equivalent circuit of the PM synchronous motor Table: Performance figures for a step change in speed corresponding to a frequency of 4 Hz to 6 Hz Specified figures
Upf
11
11
0
Lead pf
-50
11
-39
Lag pf
34
69
103
Field oriented control
0
40
40
Achieved values for
With H- controller
Without H controller
iqs ids iqs ids iqs ids iqs
2.15 -5.4 2.65 -10 1.7 -0.27 2.07
1.72 -0.3 1.92 -2.25 1.5 3.8 1.7
ids
-4.85
0.05
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IJRRAS 4 (2) ● August 2010
Ramana & al. ● Design Of The H-∞ Controller For Vsi Fed Pmsm Drive
Figure 3: Field oriented case -Simulation results for step change in speed with respect tort to a frequency of 2 to 6 Hz a)without H−∞ controller b) with H − ∞ controller
Figure 4:Field oriented case-Simulation results for step change in load torque from 1 to 3 N-m at a constant frequency of 6 Hz a) without H − ∞ controller b) with H − ∞ controller 5.
RESULTS AND DISCUSSIONS
The simulation results (dynamic responses) are presented for the permanent magnet synchronous motor (a) For a change in speed corresponding to a frequency of 2 Hz to 6 Hz at a constant load torque of 3 Nm for field oriented case. (b) For a change in load torque from 1 to 3 Nm at a constant frequency of 6 Hz for field oriented case. A
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IJRRAS 4 (2) ● August 2010
Ramana & al. ● Design Of The H-∞ Controller For Vsi Fed Pmsm Drive
laboratory scale Variable reluctance synchronous motor with a damper winding is taken up as a case study. The rating of the machine as well as the values of the parameters are given in the Appendix. For the motor considered here, for the value of if r to be 1.0 Amp and for different values of Te*, we can obtain the characteristics of i*qsandi*ds as a function of δ and ψ as shown in fig.3. From these currents, we can compute the currents per phase (rms) as well as the power factor for given values of δ and T e*. The performance figures from table clearly shows the results of steady state values with and without H-∞ controller. It is clear that in case of the permanent magnet motor, one can achieve Lagging, leading or unity power factor. It is seen that 1) The transient response has improved and steady state is obtained fast with H- controller compared to without H-∞ controller. 2) The initial overshoots in the currents have reduced with the use of the H-∞ controller. 3) The settling time is reduced with H-∞ controller compared to without H-∞ controller in all the cases. Field oriented control is achieved for ψ = 0◦. The currents are, however, not very sensitive to variation in ψ. It has been shown in Fig 4 that a desired power factor can be achieved at any load by specifying δ and ψ independently. The armature current is minimum for unity power factor leading to an optimum value of torque/ampere. By virtue of a controllable field excitation, the system can be controlled at any desired power factor allowing high efficiency for various loads. It should be noted here that many more simulations have been performed, but all the results are not included here. Only typical results have been shown in order to maintain a compact presentation. A practical real-time implementation of the above scheme has also been made and reported elsewhere. 6.
MACHINE RATINGS AND PARAMETERS
Machine rating and parameters of the Variable reluctance synchronous motor Stator : 400 V, 2.17 A, 3-phase, 50 Hz, 1500 r/min,4-pole. Power rating : 1.2/1.5 kW, 0.8/1.0 pf. ra = 5.5 Ω , rdr= 16.0 Ω, rqr= 4.2 Ω ,rf r= 0.31 Ω, ll= 0.016H, lad= 0.082H, laq= 0.048H, lf r= 0.092H, 2 ldr= 0.14 H, lqr= 0.14 H, J= 0.048 kg· m , D=0.0049 N·m/sec/rad.
7. REFERENCES [1] Alice Mary, K., ‘Nonlinear Modeling, Control System Design and Implementation for an Inverter-fed Synchronous Motor Drive’, Ph.D. Thesis, Department of Electrical Engineering, I.I.T. Kharagpur, 1998. [2] Jianying Gong, Rong Xie, Weiguo Liu., et.al., ’H − ∞ controller design of the synchronous generator’IEEE Computer Society, International conference on Intelligent Computation Technology and Automation. 978-0-7695-3357-5/08, 2008, pp-370-374. [3] Wang S. G, Yeh. H. Y and Roschke,’Robust control for Structural systems with parametric and unstructured uncertainties’ Journal of Vibration and control,vol- 7, 2001, pp753-712. [4] C Charbonnel. H and LMI attitude control design: towards performances and robustness enhancement. Acta Astronautica,2004,54(5):307-314. [5] D.Xie, L.Wang. ”Robust stability analysis and control syn-thesis for discrete-time uncertain switched systems”. Proceedings of the 42nd IEEE conference on decision and control, Maui, Hawaii USA, December 2003,pp. 4812-4817.
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