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Proc. of Int. Conf. on Advances in Electrical & Electronics 2010

An Online Tuning of a PMSM for Improved Transient Response Using Ziegler-Nichol’s Method 1

B.Jaganathan, 2 R.Brindha, 3 Pallavi Murthy, 4Nagulapati Kiran,5Swetha.S 1,2,3,4

EEE Department, SRM University, Kattankulathur, Kanchipuram (Dt), TamilNadu, INDIA. jagana78@gmail.com, brindha_apr16@yahoo.co.in pllv.murthy@gmail.com n.kiran234@gmail.com catchsw@gmail.com Abstract: Tuning of PID controllers is one of the important ways to achieve desired performance of a system. PMSM drives are the upcoming parts in the field of Hybrid Electric Vehicles, etc., Many methods are available for the tuning of PID controllers. In this paper an online tuning of PID controllers using Zeigler-Nichol’s method for PMSM drives is presented to improve the transient response of the drive. A conventional PID controller is also used to control the machine. These two methods are compared with respect to their transient response, i.e., with respect to their settling times. It is found that the Zeigler-Nichol’s method gives much improved transient response than the other. The transfer function of the PMSM is obtained from its state model defined in d-q-o model which is made use in the simulation. MATLAB/Simulink is used for the simulation and the results are also shown. Keywords: Ziegler-Nichol’s tuning, PMSM, PID controllers, Tuning, Hybrid Electric Vehicles

I.INTRODUCTION In recent years, permanent magnet synchronous motor drives have been widely used in many industrial applications such as robots, rolling mills and machine tools. The inherent advantages of these machines include high power density, low inertia, and high speed capabilities. However, the control performance of the PMSM is greatly affected by the uncertainties of the plant which usually are mismatched motor parameters, external load disturbance, and unmodelled and nonlinear dynamics [1].Advanced control techniques such as nonlinear control [2], adaptive control [3], robust control [4], variable structure control [5], and intelligent control [6, 7] have been developed to deal with plant uncertainties under various operating conditions. In these control schemes, the speed or position signal is necessary for establishing the outer speed loop feedback and also in the flux and torque control algorithms. From the viewpoints of reliability, robustness, and cost, several approaches have been proposed that address the elimination of the mechanical sensors. Some approaches are based on the motor equations in order to express rotor positions and speed as functions of terminal quantities [8, 9]. However, the sensitivity to motor parameters is a major

drawback of this method. In other approach, sensor less PMSM drives have been developed on the basis of state observers [2,10,11]. However, the overall stability may not be guaranteed in these schemes due to certain assumptions introduced, complicated controller design, and feedback linearization. In a third approach, the estimation of the rotor position and speed have been proposed using the extended Kalman filter technique [12-15]. When used in Hybrid electric vehicles these require perfect transient response i.e., lesser settling time. For this many ideas have been proposed. Ziegler Nichol’s tuning is one the best method. Many procedures have been adopted for the tuning of PID controllers. These procedures are now accepted as standard in control systems practice. Both techniques make a priori assumptions on the system model, but do not require that these models be specifically known. ZieglerNichols formulae for specifying the controllers are based on plant step responses. However, this method has some inherent disadvantages such as the effect of noise characteristic, the computational burden, parameter sensitivity, and the absence of design and tuning criteria. II. THE PMSM PMSM drives are nowadays replacing the induction motor drives for they have many advantages such as high efficiency, high T/I ratio, higher speed, etc.,. Permanent magnet machines are, due to their high efficiency, power density, and torque to inertia ratio a common choice in EV and HEV concepts although other machine types, such as induction and switched reluctance machines, also have been adopted [4],[6]. Permanent magnet machines are, depending on the supply voltage waveform, divided into Brushless DC machines (BLDCs) which are fed with trapezoidal voltage waveforms and Permanent Magnet Synchronous Machines (PMSMs) which are fed with sinusoidal waveforms [7]. Both types are found in EVs and HEVs. However, in the present work the scope is limited and only PMSMs are considered. As pointed out before, adopting PMSM drives in EVs and HEVs can contribute significantly to improve the overall efficiency of the 189

© 2010 ACEEE DOI: 02.AEE.2010.01.181


Proc. of Int. Conf. on Advances in Electrical & Electronics 2010

vehicle. Thereby, the operating range can be increased and for HEVs the fuel consumption is reduced.

III. MATHEMATICALMODEL OF PMSM The dynamic model of the PMSM can be described in the d-q rotor frame as follows [14]: Vd = Rs i + pλ d - ωe λq (1) V q = Rs i + pλ q + ωe λd (2) Where: λd = L d i d + φ (3) λq = Lq iq (4) ωe = Pωr (5) The mechanical motion of the PMSM can be expressed as: Te = Jp ω r + D ω r + TL (6) Where Te is the electromagnetic torque developed by the machine which is given by: T e = (3/ 2) P [ λd i q + ( Ld - Lq)iq id] (7) IV.METHODS OF TUNING Currently, more than half of the controllers used in industry are PID controllers. In the past, many of these controllers were analog; however, many of today's controllers use digital signals and computers. When a mathematical model of a system is available, the parameters of the controller can be explicitly determined. However, when a mathematical model is unavailable, the parameters must be determined experimentally. Controller tuning is the process of determining the controller parameters which produce the desired output. Controller tuning allows for optimization of a process and minimizes the error between the variable of the process and its set point.There are several methods for tuning a PID loop Manual tuning, Cohen-Coons, Ziegler-Nichols are some of them. The choice of method will depend largely on whether or not the loop can be taken "offline" for tuning, and the response time of the system. If the system can be taken offline, the best tuning method often involves subjecting the system to a step change in input, measuring the output as a function of time, and using this response to determine the control parameters. Types of controller tuning methods include the trial and error method, and process reaction curve methods. The most common classical controller tuning methods are the Ziegler-Nichols and Cohen-Coon methods. These methods are often used when the mathematical model of the system is not available. The Ziegler-Nichols method can be used for both closed and open loop systems, while Cohen-Coon is typically used for open loop systems. A closed-loop control system is a system which uses feedback control. In an open loop system, the output is not compared to the input. The equation below shows the equation governing a PID controller:

(8) where, u is the control signal å is the difference between the current value and the set point, Kc is the gain for a proportional controller, ôi is the parameter that scales the integral controller. ôd is the parameter that scales the derivative controller. t is the time taken for error measurement. b is the set point value of the signal, also known as bias or offset. The experimentally obtained controller gain which gives stable and consistent oscillations for closed loop systems, or the ultimate gain, is defined as Ku . Kc is the controller gain which has been corrected by the Ziegler-Nichols or Cohen-Coon methods, and can be input into the above equation. Ku is found experimentally by starting from a small value of Kc and adjusting upwards until consistent oscillations are obtained, as shown below. If the gain is too low, the output signal will be damped and attain equilibrium eventually after the disturbance occurs as shown below.

V. Ziegler–Nichols method In this method, the Ki and Kd gains are first set to zero. The P gain is increased until it reaches the critical gain, Kc, at which the output of the loop starts to oscillate. Kc and the oscillation period Pc are used to set the gains as shown: Ziegler–Nichols method Control type

P

kp

0.50Kc

ki

Kd

-

-

-

PI

0.45Kc

1.2Kp / Pc

PID

0.60Kc

2Kp / Pc

KpPc / 8

Table .1 The parameters in Z-N method

Tuning rules simplify or perhaps over-simplify the PID loop tuning problem to the point that it can be solved with slide-rule technology. The Ziegler-Nichols rule is a heuristic PID tuning rule that attempts to produce good values for the three PID gain parameters namely, Kp – the controller path gain, Ti - the controller's integrator time constant, Td - the controller's derivative time constant given two measured feedback loop parameters derived from measurements: the period Tu of the oscillation frequency at the stability limit the gain margin Ku for loop stability with the goal of achieving good regulation (disturbance rejection). V. STATE MODEL AND TRANSFER FUNCTION OF PERMANENT MAGNET SYNCHRONOUS MOTOR The general form of the state model of a system is : of the machine. 190

© 2010 ACEEE DOI: 02.AEE.2010.01.181


Proc. of Int. Conf. on Advances in Electrical & Electronics 2010

dx=Ax+Bu (9) y=Cx+Du (10) where, A-System Matrix, B-Input matrix, C-output matrix and D-Transmission matrix. With the specified parameters in the table 1.,the state model of the motor used for simulation is:

VI. SIMULINK CIRCUITS Two circuits are simulated, one without the Z-N method, i.e., the conventional method of tuning and the other is the Z-N method of tuning. The transfer function of the PMSM is obtained and the same is used in both the Simulink circuits. Figure 2 shows the Simulink circuit with conventional method.

(11) The transfer function of the PMSM can be obtained from its state model by using the following formula:

T(S) = C [SI-A] B + D

(12)

and the transfer function of the PMSM considered is : (13)

Fig.2 Simulink Circuit without Ziegler-Nichol’s Tuning

The Simulink circuit without the implementation of Zmethod is shown in figure 2. The transfer. The Simulink circuit without the implementation of Z-N method is shown in figure 2. The transfer function of the PMSM model is the plant. The tuning parameters are shown in the following section.

This transfer function of the PMSM is made use in simulation and also for stability analysis discussed in the later sections. PARAMETERS Stator Resistance/phase(Ω)

VALUES 2.8750

Ld /phase (mH)

8.5

Lq/phase (mH)

8.5

Flux produced by the

0.175

magnets(V-s) Voltage Constant(V/K rp m) Torque constant (N-m/A)

127

Fig. 3 Simulink Circuit with Ziegler-Nichol’s tuning

The Simulink circuit with the implementation of Z-N method is shown in figure 2. The transfer function of the PMSM model is the plant. VII.WAVEFORMS AND OBSERVATIONS The response of the system figure 2 , i.e., without the Z-N control tuning is shown in figure 4. It can be observed that the transient response of the system gets settled down at about 12 seconds. In fact this is very much undesirable in practical systems.

1.05 2

Moment of inertia (Kg -m )

0.0008

Viscous Friction Coefficient

0.001

(N-m-s) Pole Pairs

4

Table 1. Parameters of the considered PMSM model

Fig. 4 Response c(t) with conventional tuning Method

The response of the system shown in figure 3 , i.e., with the Z-N control tuning is shown in figure 5. 191

© 2010 ACEEE DOI: 02.AEE.2010.01.181


Proc. of Int. Conf. on Advances in Electrical & Electronics 2010

Fig. 5 Response c(t) with Ziegler Nichol’s method

The root locus of the system with Z-N control method is shown in figure 6 It can be observed that all the poles and zeros lie on the left hand S-plane, though the locus crosses the imaginary axis and a small segment of the locus is on the RHP.

Fig. 6 Root locus of the overall system with the Z-N Control

It can thus be concluded that the overall system is stable. CONCLUSIONS Tuning of PID controllers is one of the important ways to achieve desired performance of a system. PMSM drives are the upcoming drives in the field of Hybrid Electric Vehicles, etc., Many methods are available for the tuning of PID controllers. In this paper an online tuning of PID controllers using Zeigler-Nichol’s method for PMSM drives is presented to improve the transient response of the drive. A conventional PID controller, i.e., without Z-N method is also used to control the drive. These two methods are compared with respect to their transient response, i.e., with respect to their settling times. It is found that the Zeigler-Nichol’s method gives much improved transient response than the other. The transfer function of the PMSM is obtained from its state model defined in d-q-o model which is made use in the simulation. MATLAB/Simulink is used for the simulation and the results are also shown. From the simulation results, it can be observed that with the conventional method of tuning of PID controllers, the settling time is about 12 seconds whereas with that of Zeigler-Nichol’s tuning method the settling time of the system is only 1.8 seconds. Hence it can be concluded that the Zeigler-Nichol’s method of tuning of PID controller gives much improved transient response. REFERENCES [1] Westlake A. J. G., Bumby J.R., Spooner E., “Damping the power-angle oscillations of a permanent magnet

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