A feasibility study report is carried out in order to introduce us to the aims, the objectives and the analysis of the impact of the designed project. First of all, a description of the model is given and the main idea is to fully understand how the Brushless DC (BLDC) motor works by analyzing its principles of operation. Secondly, an approach to control either the speed or the torque of the DC motor is demonstrated. Finally, the on-line optimisation of the speed control system is designed and an algorithm for on line resolution of the control system is proposed.
Before analysing the objectives of the project, it is important to refer to the previous work done in optimization of speed control of DC brushless motor. This project is based on the off-line optimization of speed control for a brushless DC motor, last year's dissertation carried out by Martin Maya Gonzalez [3]. The previous project is going to be the base in order to design an algorithm for real time optimisation of speed control. The design of the algorithm for real time optimisation of speed control, is based on previous work done by Lyantsev, Kulikov, Breikin and Arkov on on-line performance optimization of aero engine control system [1].
Objectives and outcome of the project
The main objective of the project is to increase the efficiency of the motor by increasing the power output of the motor. This is achieved if we eliminate the mechanical power of the motor (power input) by setting up a motor controller and choosing a fixed switching angle (phase between Back-EMF and current on the windings of the motor). At the same time, the motor controller that is going to be implemented has to achieve speed reference tracking. Increasing the efficiency of the motor implies controlling the flow of energy to the motor [10]. The energy is supplied to the motor via the motor shaft and the physical quantities that describe the shaft are:
speed
torque
These variables describe the main characteristics of a typical DC motor. In order to control the flow of energy it is necessary to focus on these two controllable variables, so we are going to refer to torque control and speed control.
In the following figures, the relationship between speed torque and output power of the motor is illustrated. It is observed that the torque is linearly and inversely proportional to the speed and that there are two important points on the graph which represent the maximum torque when the shaft is not rotating (- stall torque) and the maximum output speed of the motor (no load speed) [7]. It is also illustrates the point where maximum optimal power output is achieved.
Figure1. Speed-torque curve [7]
Either speed control or torque control can be used to maximize the efficiency of the motor because they are linearly and inversely related. By using these two control approaches, and obtaining a steady state model with speed reference tracking and maximum efficiency, we are going to shift to the next important part of the final project. In this part, we are going to introduce the real time optimisation of the control system. This means that if operating conditions of the motor change, the system will re-evaluate and calculate the optimal set-points of either speed or torque, in order to achieve maximization of energy efficiency of the motor which implies minimization of fuel consumption. This is achieved by using Model Predictive Control, a strategy which is currently the most widely implemented advanced process control technology for process plants. Consequently, the outcome of the project will be to improve the power motor performance by changing set-points of the system. The MPC strategy will be described briefly in section 3.
Problem Domain
In this section, the principle of operation of the Brushless DC motor will be described briefly. Furthermore, the controlled variables and the manipulated variables of the motor are analysed in order to understand the variables we want to control and in what way we are going to manipulate them to achieve our objectives.
In the following figure, the layers of the hierarchical control system architecture are illustrated.
Top Layer: Process Optimization
Middle Layer: Model Predictive Control
Bottom Layer: Control System Structure-PID Regulators
Figure2. Hierarchical control system architecture
The problem that is going to be solved lies on the bottom and middle layer. The bottom layer, receives set-points from Multivariable Control and outputs the control action to the motor. In this layer lies the torque and speed control.
In the top layer, lies the Process Optimization. In this layer, there are all the objectives of the project that have to be fulfilled. For example, process optimization provides the set-points for the middle layer. In this case, it might be an increase of the percentage of efficiency of the motor or a decrease of the percentage in the fuel consumption of the motor. The optimization layer doesn't care about dynamics of the model. The process optimization problem is not solved as often as MPC layer. The MPC is responsible for implementing the optimal solution to the problem and achieving the objectives of the top layer by taking into account the set points and solutions given for the bottom layer. In other words, there is a strong interaction between the layers in order the objectives to be achieved.
Before discussing the control approach that is going to be implemented on the bottom and the middle layer, the principle of operation of the motor and the understand the characteristics that describe the motor will be described.
Operation of motor
In the following figure, the structure of a BLDC motor is illustrated.
Figure3. Brushless Permanent Magnet DC Motor Basics [11]
The construction of a BLCD motor is very similar to AC motors and it is known as the permanent magnet synchronous motor.
The BLDC motor consists of the stator (in our model the induced part of the machine, 3-phase windings) and the rotor (inductor of the machine) which is a permanent magnet (Figure3).
Commutation refers to the process which converts the input direct current to alternating current and properly distributes it to each winding in the armature [12]. In BLDC motors, this process is done by semiconductor devices such as transistors as there are no brushes and commutator.
The BLDC motor has two main characteristics [12]:
the stator flux is synchronized with the flux of the permanent magnet.
an electromotive force which is proportional to the speed of the motor is created when the permanent magnet rotates. This force is called Back-EMF and it is opposite of the direction of the current on the windings of the motor.
As referred in the previous section in order to maximize efficiency, we have to eliminate the power input of the motor.
Efficiency, n is described by the following equation:
(1)
(2)
(3)
w = rotational speed
= Load torque
E= Back-EMF
I=current in the windings
The back-EMF will be at its maximum when the variation of the flux is at maximum. So when the rotor passes from a North pole to a South pole for example this means that the back-EMF is maximum when the rotor flux is perpendicular to the phase [12].
Maximum efficiency is obtained if we keep the back-EMF voltage and the current on the windings in phase or in other words, if we minimize the difference of phase angle between the Back-EMF and the current. This is the control principle that puts the base in controlling the motor's speed.
Hence, the Back-EMF force is the key to know the position of the rotor and the relationship between the angular speed is w= (θ=motor position)
In the next section a brief description is given on which way are we going to use the above characteristics of the motor in order to control the speed and the torque and achieve our objectives.
2.2 Control Theory
In figure 4, the motor control is based on a six-step principle with a standard triple half bridge [12]. We see that the transistors T1, T2, T3, T4, T5, T6 are connected with the windings of the motor and A, B, C are the phases of the motor. It is observed that in each step of the six-step cycle, there are always two or three windings that are biased [12]. These have opposite directions. Hence, there will be one phase that will have zero current. For example, in step 1, phase A is positive biased (positive current) and phase B is negative biased (negative current) and phase C has no current. When controlling the motor, we will read information on the phase which is not energised. This information will allow us to determine the real position of the rotor so we can modify the PWM outputs accordingly and drive the motor with maximum efficiency [12].
In figure 5, it is illustrated what is described previously about the difference of the phase angle α between the back-EMF and the current.
Figure4a. Ideal Current on the windings of the motor [12]
Figure4b. Six-step principle with a standard triple half bridge [12]
Figure5. Back-EMF and current in phase [12]
The following figure, illustrates the effect of α in the efficiency of the motor. It is observed that the efficiency of the motor is increased if the angle alpha is minimised. In the first graph, the rotational speed of the motor is kept constant and we obtain different graphs in different operational conditions with variance in the torque load.
In the second graph, the load torque is kept constant and we obtain different graphs of efficiency for different operational conditions with variance in the rotational speed (figure 6).
Figure6. Efficiency vs. alpha. Qinetiq motor [3]
Methods and approaches
Control approach for speed reference tracking by PID regulators
In this section, a brief description of the methods that are going to be used in the project is given. First of all, the control structure of the system consists of PID regulators which achieve speed reference tracking. Figure 7, illustrates the two different driving modes of the motor.
The maximum efficiency loop consists of the current regulation loop, which means that by changing the reference current of the motor we can achieve torque control. This loop keeps the back-EMF and the current on the windings at the same phase. The position of the rotor is taken into account and calculated by the back-EMF on the winding of the motor that is not energised [12].
Speed control is achieved by changing the voltage reference of the motor. The optimum speed loop maintains the motor at its target speed. The above system structure represents cascade control, with current regulation loop being the slave controller and the speed regulation loop being the master.
Figure 7 Control approach of the system [12]
The process of the driving of the motor takes place in the bottom level of the hierarchical control system architecture.
Model Predictive Control
The control approach described so far is going to be the base before constructing the MPC. Assuming that the system represents steady state operating conditions maintained by closed loop control system, the MPC is going to be built based on the control approach described by Lyanstev [1] for engine control of gas turbines.
The MPC strategy is selected in order to solve the optimization of speed control problem. In this project, linear MPC is going to be implemented, so the development of a linear dynamic simulation motor model is required. The limitation of linear models is that they only consider a small neighbourhood around steady state operation conditions (on our case speed and torque load) [1]. Therefore, because we want transient performance over the operating range of the motor, another type of model is going to be built which consists of a set of linear models connected with the nonlinear static line. This is the idea of constructing a Piecewise Linear Dynamic Model and allows good performance of the model in different operational points and describes the motor's dynamics around steady state conditions.
MPC structure
MPC controller is a discrete time controller and uses the measurements taken by a model of the process in order to predict the outputs [5]. Based on the prediction of the outputs, at each sampling instant in time, an objective function is optimised on line, taking into account some limitations about the inputs and the outputs (constraints). The MPC controller will implement a control strategy that is going to minimise a cost function.
This constrained optimisation problem is solved by mathematical programming. In this project the optimisation problem is going to be solved using Quadratic Programming and the objective function, the process model and the constraints are in linear form.
Figure 8 illustrates the structure of MPC. The main components of the MPC are the process model which predicts the outputs from past inputs and outputs and by future inputs taken by the optimizer where quadratic programming is implemented. In figure 8, MPC seems to be a feedback control but what distinguishes from that standard form of feedback control is the use of the finite prediction horizon.
Figure 8 Basic Structure of MPC [4]
The receding prediction horizon is a key ingredient that characterises the MPC and is going to be described briefly.
We consider the single input single output (SISO) system in the linear form:
Where x is the state vector, u is the control vector, y is the output vector and A, B, C, and D are the matrices that contain the dynamics of the motor. In this case of the BLDC motor, the manipulated input of the system is the amplitude of voltage and the output is the speed w.
The vector of states is defined as:
where and are the currents on the two of the three phases of the motor (the third phase is not energised as described in section 2.2 so it is zero ) and w is the speed.
In figure 9, the concept of MPC and its receding prediction horizon is presented.
Figure 9 Receding Prediction Horizon of MPC [9]
At the present time k, the response of the output is predicted over the prediction horizon of M samples. The manipulated input varies over the control horizon. The main idea is that changes of the manipulated input are computed such that future deviations between the predicted output and the desired set-point are minimised. This is an advantage of MPC because the prediction horizon allows to the MPC to take control action at the current time step if an error between the set-point and the output occurs and after taking control action the algorithm repeats the same procedure again for the next time interval enabling the MPC to correct the error continually.
In order to minimise the deviations from the set-point the following form of quadratic objective function is used [4]
min J= (4)
where and are the weights which describe the importance of output to follow its set-point r compared to the action of the manipulated input u, Δu is the change of the input, denotes the estimate of at sample k.
Note that Δu input moves, are optimised starting from sampling k+1 (l=1) to minimise the predicted output deviations from sample k+2 (l=2).
The minimisation of the above objective function has to be done subjected to linear constraints.
The constraints in this case are [4]:
Constraints on the states of the motor (for example maximum or minimum operational speed of the motor or amplitude of current).
Constraints on the input signals u (for example maximum amplitude of voltage).
A linearly constrained optimization problem with a quadratic objective function is called Quadratic Program (QP). [8]
The general quadratic program has the general form:
min , (5)
where G is an n-dimensional row vector describing the coefficients of the linear terms in the objective function, and H is an (n ï‚´ï€ n) symmetric matrix describing the coefficients of the quadratic terms [8]
When the objective function f(x) is strictly convex for all feasible points the problem has a unique local minimum which is also the global minimum. A sufficient condition to guarantee strict convexity is for H to be positive definite [8].
In the standard QP constraints take the following form:
(6)
The following figure presents the trajectory of search for minimum of objective function for the engine control of gas turbines [1]. The same figure is used for the minimisation of the objective function in our case because it deals with the same idea of optimisation. In this case the control inputs are fuel feed Wf and nozzle area An. In our case the control inputs are the amplitude of voltage (function of alpha) on two of the three windings of the motor.
The current operating point of the system (current value of speed) is inside the boundaries.
The unconditional minimum of the objective function J=0 is the point . This point is the intersection of two straight lines. These straight lines are obtained if we equate to zero the two sum parts of (4). If we connect the two points with a straight line we obtain the , which lies on the edge of the boundaries and determines optimal control at the current step.[1]
Figure 10.Trajectory of search for minimum of objective function [1]
Project Plan
Figure 11 Gantt chart
Conclusion
The final thesis will be divided in to two parts. The first part is about model simulation and analysis of its dynamics. An engine model is will be built using Simulink and data provided. It is very important to analyse the techniques used in controlling the BLDC motor as well as the control objectives (controlled outputs, manipulated inputs, disturbances), because this information is going to be used to meet the objectives. In the second part of the project, the optimisation method will be analyzed and the algorithm and implementations used in this study are will be discussed.
In this feasibility study, the problem domain and the solution of the problem are briefly described and the main part of this project is the optimisation of the control problem via MPC. The MPC implementation has been discussed briefly in this report, detailed information on the implementation of MPC like tuning methods of MPC, robustness, stability and fragility of constrained MPC is given in [4].
