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The application of BLDC motor drives in industries is becoming more popular nowadays. An error will occur in the drive that is originated by some disturbances which are the major problems to reduce the stability of the system. To obtain the minimum performance index, the optimal control signal is formulated, which is the main objective of this paper. Based on quadratic performance index, the optimal control system of BLDC motor drive is a design which spotlights in this paper. The complexity of the mathematical expressions has been reduced by using state space approach to the BLDC system. The burden to the control engineers has reduced based on tedious computation by using thus optimal design. To provide the desired operating performance, this optimal design helps to realize the BLDC system with practical components.

Presently, consumers demand for lower energy costs, higher efficiency, better performance, reduced acoustic noise and more convenient features. The traditional technologies like DC motor and AC motor cannot meet these demands. Hence, motor manufactures are gravitating towards Brushless Direct Current (BLDC) motor. The use of the BLDC motor in these applications is becoming very common due to the features of high efficiency, high flux density per unit volume and high power density due to the absence of field winding. In addition, the absence of brushes leads to high reliability, low maintenance requirements and low electromagnetic interference problems [

Due to change in load variations, BLDC motor exhibits oscillations in speed response and hence the stability of the system gets affected. The stability issues can be recovered by two ways, namely reducing the gain of speed controller and regulating output method [

The optimal control design is aimed to obtain a best possible system of a particular type with respect to a certain performance index. In the optimal control design, the performance index replaces the conventional design criteria. A transfer function for the BLDC drive is derived in this paper. In this paper, for digitally PWM controlled BLDC motor drive, the optimal control system is designed. To achieve this optimization, the state and control variables are formulated in this paper. Based on optimal control theory, the BLDC performance characteristics close to the optimal control system are synthesized.

By changing the applied voltage across the motor phases, the speed of BDC motor can be controlled. This is achieved by utilizing pulse amplitude modulation, PWM or hysteresis control. Another method of speed control involves sensorless techniques.

The torque equation is given by,

where T_{em}, ω(t), B, J and T_{L} denote electromagnetic torque, rotor angular velocity, viscous friction constant, rotor moment of inertia and load torque respectively.

where K_{t} = torque constant and I = average current. For the purpose of analysis, the digital controller was considered equivalent to a proportional controller with high gain and saturation.

From the above equations it is possible to derive the transfer function

The state variable equation for this BLDC drive is given by

Arranging in matrix form, the following equations are obtained

The output equation is given by,

The design of a state feedback BLDC motor control system is based on a suitable selection of a feedback system structure. If the state variables are known, then they can be utilized to design a feedback controller so that the input becomes u = kx. It is necessary to measure and utilize the state variables of the system in order to control the speed of the BLDC motor. This design approach of state variable feedback control gives sufficient information about the stability of the BLDC drive system. Feedback control system will be chosen so that,

Then Equations (6) & (7) become,

Arranging in matrix form the following state equation is obtained

This is in the form

where

Let k_{1}=1 and determine a suitable value for k_{2} so that the performance index is minimized.

Completing matrix multiplication, addition and solving the following are obtained

To minimize as a function of k_{2}

Therefore,

The system matrix H obtained for the compensated system is,

The feedback control signal is obtained as,

This optimal controlled BLDC drive system results in a minimum value for the performance index. Also, the control law given by Equation (25) yields optimal result for any initial state under the given performance index. This design based on the quadratic performance index yields a stable control system for the BLDC drive system.

This section considers the expenditure of control signal energy for the BLDC drive system. To account for the expenditure of the optimal energy of the performance index

Therefore the matrix Q is

Let,

To simplify the algebra without any loss, let

S. No | Parameter | Symbol | Unit | Value |
---|---|---|---|---|

1 | Stator Winding Resistance | Ra | Ω | 1.4 |

2 | Stator Winding Inductance | L_{a} | H | 0.0066 |

3 | Rotor Inertia | J | Kg-m^{2} | 0.00176 |

4 | Motor Viscous Friction Coefficient | B | Nm/rad/sec | 0.00038818 |

5 | Torque Constant | k_{t} | Nm/Amp | 0.03 |

6 | Velocity Constant | k_{e} | Volts/rad | 0.0000181 |

To obtain minimum performance index, set

where

When

Applying Lin’s Method the above equation becomes

This section discusses about the design of a stable control system for BLDC drive based on quadratic performance indexes. The system design will be stable by using the quadratic optimal control scheme has the main advantage except in the case where the system is not controllable. The matrix “P” is determined from the solution of the matrix Riccatti equation. This optimal control is called the Linear Quadratic Regulator (LQR) [

Let,

Solving we obtain the following three equations

Solving these three equations we get,

The optimal feedback gain matrix

This control signal yields an optimal result for any initial state under the given performance index.

To achieve the desired system response, the BLDC drive system had designed with state variable feedback. Also the performance index value is minimum for BLDC drive which was designed by optimal control. In accordance with the expenditure of energy and resources, the control signal is often included in the performance integral. From the foregoing analysis, the value of K_{2} is obtained as 1.0149 so that the performance index is minimized. The minimum value of performance index is obtained as 1.47. This optimal controlled BLDC drive system results in a minimum value for the performance index. Also, the control law given by Equation (54) yields optimal

results for any initial state under the given performance index. This design based on the quadratic performance index yields a stable control system for the BLDC drive system.

Murugan Marimuthu,Jeyabharath Rajaiah, (2016) An Optimal Control Theory Based Analysis of Brushless DC Motor Drive. Circuits and Systems,07,3384-3391. doi: 10.4236/cs.2016.710288