IMPLEMENTATION OF A ROBUST CONTROLLER FOR BLDC MOTOR DRIVE
ABSTRACT
PID control has the advantages of simple structure, high reliability and
easy to engineering applications, so it still widely used. Traditional
speed control system for brushless DC motor is usually used PID control.
Despite it is simple, easy to regulate the parameters and has a certain
control precision, the PID control algorithm do has limitations. Today
controlled objects are increasingly complicated; the traditional PID can
no longer meet the requirements of rapid, stability and robustness for
speed control system of brushless DC motor. Three PID control parameters
of proportional, integral and differential, while coordinating with
each other, are nonetheless checking on, which is not a simple linear
combination. The desired control effect will be obtained as long as by
maintaining the best relationship between changing the non-linear
relationships.
The PID control based neural network show good self adaptability and
robustness for the uncertain system and the controlled system with
parameter variation time-delay. It overcomes the problem that the
traditional PID controller is difficult to determine the parameters on
line moment and effectively control nonlinear time varying system, and
it has a high value of practical application in the present neural
network control.
This project presents a robust controller with two degree of freedom to
track the input reference signal. Speed is considered as a reference
signal. The mathematical model of BLDC motor is presented. The proposed
approach has superior features, including easy implementation, stable
convergence characteristic and good computational efficiency. Comparing
with PID controller, the proposed method is more efficient in improving
the step response characteristics such as, reducing the steady-states
error; rise time, settling time and maximum overshoot in speed control
of a linear brushless DC motor. The brushless DC motor with proposed
method is modeled in Simulink.
INTRODUCTION Brushless DC Motor A new design concept of permanent magnet brushless motors has been developed in 1980. The permanent magnet brushless motor can be classified upon to the back-EMF waveform, where it can be operated in either brushless AC (BLAC) or brushless DC (BLDC) modes. Usually the Brushless AC motors have a sinusoidal back-EMF waveform and Brushless DC motors have a trapezoidal back-EMF. In modern electrical machines industry productions the brushless direct current (Brushless DC) motors are rapidly gaining popularity. Brushless DC motors are used in industries such as Appliances, HVAC industry, medical, electric traction, road vehicles, aircrafts, military equipment, hard disk drive, etc. Comparing Brushless DC motors with DC motors, the DC motor have high starting torque capability, smooth speed control and the ability to control their torque and flux easily and independently. In the DC motor, the power losses occur mainly in the rotor which limits the heat transfer and consequently the armature winding current density, while in Brushless DC motor the power losses are practically all in the stator where heat can be easily transferred through the frame, or cooling systems can be used specially in large machines. In general the induction motor has many advantages as: their simplest construction, simple maintenance, low price and reliability. Furthermore, the disadvantages of induction machines make the Brushless DC motors more efficient to use and become more attractive option than induction motors. Some of the disadvantages of induction machines are poor dynamic characteristics, lower torque at lower speeds and lower efficiency. PID Controller Fig.: Plant with PID controller Three-term controller The transfer function of the PID controller Kp= Proportional'gain KI= Integral gain Kd = Derivative gain Characteristics of P, I, and D controllers CLOSED LOOP RESPONSE RISE TIME OVER SHOOT SETTLIN GTIME STEADY STATE ERROR KP Decrease Increase Small Change Decrease Kt Decrease Increase Increase Eliminate Kd Small Change Decrease Decrease Small Change Robust Controller In order to gain a perspective for robust control, it is useful to examine some basic concepts from control theory. Control theory can be broken down historically into two main areas: conventional control and modern control. Conventional control covers the concepts and techniques developed up to 1950. Modern control covers the techniques from 1950 to the present. Each of these is examined in this introduction "Robust control refers to the control of unknown plants with unknown dynamics subject to unknown disturbances". Clearly, the key issue with robust control systems is uncertainty and how the control system can deal with this problem. Figure shows an expanded view of the simple control loop. Uncertainty is shown entering the system in three places. There is uncertainty in the model of the plant. There are disturbances that occur in the plant system. Also there is noise which is read on the sensor inputs. Each of these uncertainties can have an additive or multiplicative component. Fig.: Plant control loop with uncertainty The figure also shows the separation of the computer control system with that of the plant. It is important to understand that the control system designer has little control of the uncertainty in the plant. The designer creates a control system that is based on a model of the plant. However, the implemented control system must interact with the actual plant, not the model of the plant [13]. It is the property of a controller to overcome small perturbations while reference signal tracking without loosing stability. Robust control refers to the control of unknown plants with unknown dynamics subject to unknown disturbances. Clearly, the key issue with robust control systems is uncertainty and how the control system can deal with this problem Mathematical Model of BLDC Motor: The three phase star connected BLDC motor can be described by following four equations [11]. Vab= R(ia- ib) + L (ia - ib) + ea – eb Vbc= K(i&-ic) + L (ib-ic) + eb-ec Vca- R(ic- to) + L Where i = phase currents e=back emf (phase) V=phase to phase voltage R=resistance per phase L=inductance per phase Te=electric torque TI=load torque Kf=friction constant wm = rotor speed J = rotor inertia Where Ke=back – emf constant Kt=torque constant =electrical angle m=rotor angle P=pole pairs F= function gives the trapezoidal wave form of the back emf The voltage equation must be written in state space form, the current relationship is given as The voltage equation becomes The complete model is Dynamic Model of BLDCM It is assumed that the BLDC motor is connected to the output of the inverter while the input terminals are connected to constant supply voltage as shown in figure Fig. Brushless DC Motor model For symmetrical winding & balanced system the voltage equations are The back emf wave forms eaebec are functions of angular velocity of the rotor shaft, Where back emf constant The BLDC Motor mathematical model can be represented as The stator self inductance are independent of the rotor position hence The mutual inductance will be Assuming balanced system, all the phases resistance are equal Rearranging the equation 5 yields The electromechanical torque is expressed as The instantaneous electromagnetic torque can be represented as Integral of Time multiplied by Absolute Error: The design of control systems to minimize the integral of time multiplied by absolute error (ITAE) is discussed in this section. For the quadratic overall system [9]. The ITAE, the integral of absolute error (IAE0 and the integral of square error (ISE) as a function of the damping ratio are shown in fig. Fig.: Comparison of various design criteria The IATE has largest changes as varies and therefore has the best selectivity. The ITAE also yields a system with a faster response than other criteria. The system that has sense of the ITAE or the ITAE optimal system. Consider the overall transfer function Fig.: Optimal pole location. Fig.: Step response of ITAE optimal systems with zero position error. Table: List of integral of time multiplied by absolute error method equations. CONTROLLER DESIGN CALCULATIONS: PID Controller The plant transfer function is Where Substation of the above values in the transfer function we get Where Ts=Setting Time = 0.01 Sec. %Mp=Percentage overshoot = 2%.0.02 By doing calculations we get the values of and Therefore and Where By combining the plant transfer function and PID controller transfer function we get the controller values the equation is as follows. Where By doing the calculations we get the third degree characteristic equation. Where Table: Parameters of PID Controller Parameters Values & Units Kp 66.26 0.082 21032.88 0.02 0.01 sec R 11.05Ohms L 000215H J 0.0001Kgm2 0.1433VS/rad 0.1433kg-m/A D 0.0001 kg-ms/rad RST Controller Table: Parameters of RST Controller Parameters Values R 14675.85s+600.89 S s+394.21 T 17506.46s+5610749.165 K 7.165 0.02 0.01sec R 11.05 Ohms L 0.0215H J 0.0001Kgm2 0.1433VS/rad 0.1433kg-m/A D 0.0001 kg-ms/rad SIMULATION MODEL: Simulation Results: PID Controller Fig.: Speed waveform of Brushless Fig: Torque waveform of Brushless DC Motor with PID Controller DC Motor with PID Controller RST Controller Fig.: Speed waveform of Brushless Fig.: Torque waveform of Brushless DC Motor with RST Controller. DC Motor with RST Controller. CONCLUSION The PID controller has over shoot, rise time and has low settling time. The RST controller has fast settling time and does not have over shoot and rise time which is observed in the simulated waveforms. Table 9.1 Parameters of PID Controller and RST Controller Parameters PID controller RST Controller Rise time 0.5 sec 0 Over shoot 52% 0 Setting time 0.08se 0.002sec FUTURE SCOPE The proposed RST controller can be applied to DC motors, Induction motors, Servo motors, Synchronous motors, this controller can also be used for signal tracking, load frequency control, the simulation study can extended for neuro fuzzy controller. REFERENCES 1. P.Pillay and R.Krishnan, Modeling, Simulation and Analysis of Permanent –Magnet Motor Drives, Part- 1: The Permanent – Magnet Synchronous Motor Drive,” IEEE Trans, Ind. Appl, Vol.25, pp-265-273, 1989. 2. P. Pillay and R. Krishnan, “Modeling, Simulation and Analysis of Permanent – Magnet Motor Drives, Part 2: The Brushless DC Motor Drive, “ IEEE Trans, Ind. Appl, Vol. 25, pp 274-279, 1989. 3. Gwo-Ruey Yu and Rey-Cheu Hwang, “optimal PID Speed Control of Brushless Dc Motors using LQR Approach,” IEEE International Conference on system, pp-473-478, 2004. 4. www.controltheorypro.com. 5. www.engin.umich.edu
INTRODUCTION Brushless DC Motor A new design concept of permanent magnet brushless motors has been developed in 1980. The permanent magnet brushless motor can be classified upon to the back-EMF waveform, where it can be operated in either brushless AC (BLAC) or brushless DC (BLDC) modes. Usually the Brushless AC motors have a sinusoidal back-EMF waveform and Brushless DC motors have a trapezoidal back-EMF. In modern electrical machines industry productions the brushless direct current (Brushless DC) motors are rapidly gaining popularity. Brushless DC motors are used in industries such as Appliances, HVAC industry, medical, electric traction, road vehicles, aircrafts, military equipment, hard disk drive, etc. Comparing Brushless DC motors with DC motors, the DC motor have high starting torque capability, smooth speed control and the ability to control their torque and flux easily and independently. In the DC motor, the power losses occur mainly in the rotor which limits the heat transfer and consequently the armature winding current density, while in Brushless DC motor the power losses are practically all in the stator where heat can be easily transferred through the frame, or cooling systems can be used specially in large machines. In general the induction motor has many advantages as: their simplest construction, simple maintenance, low price and reliability. Furthermore, the disadvantages of induction machines make the Brushless DC motors more efficient to use and become more attractive option than induction motors. Some of the disadvantages of induction machines are poor dynamic characteristics, lower torque at lower speeds and lower efficiency. PID Controller Fig.: Plant with PID controller Three-term controller The transfer function of the PID controller Kp= Proportional'gain KI= Integral gain Kd = Derivative gain Characteristics of P, I, and D controllers CLOSED LOOP RESPONSE RISE TIME OVER SHOOT SETTLIN GTIME STEADY STATE ERROR KP Decrease Increase Small Change Decrease Kt Decrease Increase Increase Eliminate Kd Small Change Decrease Decrease Small Change Robust Controller In order to gain a perspective for robust control, it is useful to examine some basic concepts from control theory. Control theory can be broken down historically into two main areas: conventional control and modern control. Conventional control covers the concepts and techniques developed up to 1950. Modern control covers the techniques from 1950 to the present. Each of these is examined in this introduction "Robust control refers to the control of unknown plants with unknown dynamics subject to unknown disturbances". Clearly, the key issue with robust control systems is uncertainty and how the control system can deal with this problem. Figure shows an expanded view of the simple control loop. Uncertainty is shown entering the system in three places. There is uncertainty in the model of the plant. There are disturbances that occur in the plant system. Also there is noise which is read on the sensor inputs. Each of these uncertainties can have an additive or multiplicative component. Fig.: Plant control loop with uncertainty The figure also shows the separation of the computer control system with that of the plant. It is important to understand that the control system designer has little control of the uncertainty in the plant. The designer creates a control system that is based on a model of the plant. However, the implemented control system must interact with the actual plant, not the model of the plant [13]. It is the property of a controller to overcome small perturbations while reference signal tracking without loosing stability. Robust control refers to the control of unknown plants with unknown dynamics subject to unknown disturbances. Clearly, the key issue with robust control systems is uncertainty and how the control system can deal with this problem Mathematical Model of BLDC Motor: The three phase star connected BLDC motor can be described by following four equations [11]. Vab= R(ia- ib) + L (ia - ib) + ea – eb Vbc= K(i&-ic) + L (ib-ic) + eb-ec Vca- R(ic- to) + L Where i = phase currents e=back emf (phase) V=phase to phase voltage R=resistance per phase L=inductance per phase Te=electric torque TI=load torque Kf=friction constant wm = rotor speed J = rotor inertia Where Ke=back – emf constant Kt=torque constant =electrical angle m=rotor angle P=pole pairs F= function gives the trapezoidal wave form of the back emf The voltage equation must be written in state space form, the current relationship is given as The voltage equation becomes The complete model is Dynamic Model of BLDCM It is assumed that the BLDC motor is connected to the output of the inverter while the input terminals are connected to constant supply voltage as shown in figure Fig. Brushless DC Motor model For symmetrical winding & balanced system the voltage equations are The back emf wave forms eaebec are functions of angular velocity of the rotor shaft, Where back emf constant The BLDC Motor mathematical model can be represented as The stator self inductance are independent of the rotor position hence The mutual inductance will be Assuming balanced system, all the phases resistance are equal Rearranging the equation 5 yields The electromechanical torque is expressed as The instantaneous electromagnetic torque can be represented as Integral of Time multiplied by Absolute Error: The design of control systems to minimize the integral of time multiplied by absolute error (ITAE) is discussed in this section. For the quadratic overall system [9]. The ITAE, the integral of absolute error (IAE0 and the integral of square error (ISE) as a function of the damping ratio are shown in fig. Fig.: Comparison of various design criteria The IATE has largest changes as varies and therefore has the best selectivity. The ITAE also yields a system with a faster response than other criteria. The system that has sense of the ITAE or the ITAE optimal system. Consider the overall transfer function Fig.: Optimal pole location. Fig.: Step response of ITAE optimal systems with zero position error. Table: List of integral of time multiplied by absolute error method equations. CONTROLLER DESIGN CALCULATIONS: PID Controller The plant transfer function is Where Substation of the above values in the transfer function we get Where Ts=Setting Time = 0.01 Sec. %Mp=Percentage overshoot = 2%.0.02 By doing calculations we get the values of and Therefore and Where By combining the plant transfer function and PID controller transfer function we get the controller values the equation is as follows. Where By doing the calculations we get the third degree characteristic equation. Where Table: Parameters of PID Controller Parameters Values & Units Kp 66.26 0.082 21032.88 0.02 0.01 sec R 11.05Ohms L 000215H J 0.0001Kgm2 0.1433VS/rad 0.1433kg-m/A D 0.0001 kg-ms/rad RST Controller Table: Parameters of RST Controller Parameters Values R 14675.85s+600.89 S s+394.21 T 17506.46s+5610749.165 K 7.165 0.02 0.01sec R 11.05 Ohms L 0.0215H J 0.0001Kgm2 0.1433VS/rad 0.1433kg-m/A D 0.0001 kg-ms/rad SIMULATION MODEL: Simulation Results: PID Controller Fig.: Speed waveform of Brushless Fig: Torque waveform of Brushless DC Motor with PID Controller DC Motor with PID Controller RST Controller Fig.: Speed waveform of Brushless Fig.: Torque waveform of Brushless DC Motor with RST Controller. DC Motor with RST Controller. CONCLUSION The PID controller has over shoot, rise time and has low settling time. The RST controller has fast settling time and does not have over shoot and rise time which is observed in the simulated waveforms. Table 9.1 Parameters of PID Controller and RST Controller Parameters PID controller RST Controller Rise time 0.5 sec 0 Over shoot 52% 0 Setting time 0.08se 0.002sec FUTURE SCOPE The proposed RST controller can be applied to DC motors, Induction motors, Servo motors, Synchronous motors, this controller can also be used for signal tracking, load frequency control, the simulation study can extended for neuro fuzzy controller. REFERENCES 1. P.Pillay and R.Krishnan, Modeling, Simulation and Analysis of Permanent –Magnet Motor Drives, Part- 1: The Permanent – Magnet Synchronous Motor Drive,” IEEE Trans, Ind. Appl, Vol.25, pp-265-273, 1989. 2. P. Pillay and R. Krishnan, “Modeling, Simulation and Analysis of Permanent – Magnet Motor Drives, Part 2: The Brushless DC Motor Drive, “ IEEE Trans, Ind. Appl, Vol. 25, pp 274-279, 1989. 3. Gwo-Ruey Yu and Rey-Cheu Hwang, “optimal PID Speed Control of Brushless Dc Motors using LQR Approach,” IEEE International Conference on system, pp-473-478, 2004. 4. www.controltheorypro.com. 5. www.engin.umich.edu
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