Project 2 - Speed Control of a Direct Current (DC) motor
Q) In this project, a Simulink model for the DC motor will be first developed per the schematic shown in Figure 1.
Use the Proportional, Integral, and Derivative (PID) control design knowledge gained in the training to design, develop and test a controller to maintain a desired speed for the motor.
Solution:
Modeling of a DC Machine:
What is a DC Motor?
A DC motor is an electrical machine that converts electrical energy into mechanical energy. In a DC motor, the input electrical energy is the direct current which is transformed into mechanical rotation.
Modeling of a DC Motor:
Modeling of a dc motor by using the below circuit model and taking the parameters as per.
J = Moment of Inertia of Rotor.
B = Damping Coefficient.
R = Armature resistance
L = Armature Inductance
Vb = Back emf constant
KT = Motor Torque Constant
V = Supply Voltage
Motor Torque T = KT * I
Back EMF Vb = Kb * ω = Kb * dθ/dt
Finding equations, from the below figure:
As per Kirchhoff’s Voltage Law
V = IR + L (di/dt) + Vb
L (di/dt) + IR = V - Kb (dθ/dt) ----equ-(a)
As per Newtons Law
J (d2θ/dt2) + B (dθ/dt) = KT * I ----equ-(b)
Converting the above equations to the S domain (Laplace Transform)
Laplace of equ – (a)
LsI(s) + RI(s) = V(s) - Kbsθ(s) ----equ-(c)
I(s) = V(s) - Kbsθ(s)/Ls + R ---- equ-(d)
Laplace of equ – (b)
Js2 θ(s) + Bs θ(s) = KT * I(s) ---equ-(e)
ωs = s θ(s) = KT * I(s)/Js+B ----equ-(f)
Substitute I(s) value in equ-(e)
Js2 θ(s) + Bs θ(s) = KT * I(s) ---equ-(e)
Js2 θ(s) + Bs θ(s) = KT * (V(s) – Kbsθ(s))/ (Ls + R) ---equ-(g)
s θ(s)[(Js+B) * (Ls+R)] = KTV(s) – KT * Kbsθ(s)
s θ(s)[(Js+B) * (Ls+R)] + KT * Kbsθ(s)= KTV(s)
s θ(s)[(Js+B) * (Ls+R) + (KT * Kb)]= KT V(s)
The transfer function of output angle θ(s) to the input angle V(s) is given by
Ga(s) = θ(s)/V(s) = KT / s[(Ls+R) * (Js+B) + (KT * Kb)] ----equ-(h)
The transfer function of the output angle ω(s) to the input angle V(s) is given by
Ga(s) = (s)/V(s) = KT / [(Ls+R) * (Js+B) + (KT * Kb)] ----equ-(i)
Simulink Model of DC Motor:
Model building:
Firstly, we are taking a sum block with two input terminals one is positive for taking input from the constant or step block input as reference input. And another port is a negative port this port is connected to a feedback gain block output port. Which is the feedback signal taken from the speed (output of the second transfer function). Then the output of the sum block is given to a transfer function block which is calculating the La and Ra. Then the output of the transfer function block is multiplied with a gain block named Kt and here we can get the torque from the output of the gain block it is calculated with the second transfer function block and the output is speed. Then the speed is integrated with an integrator here the displacement is the output and it is given to a workspace block. By plotting the graph between the time vs speed, current, and voltage we can observe the system response.
The DC MOTOR PARAMETERS in m-script:
% This file contains the design paramters to model a Dc motor
j = 0.01; % momemt of inertia
b = 0.1; % motor friction
kb = 0.01; % back emf
kt = 0.01; % Motor torque constant
Ra = 1.0; % aramture resistance
La = 0.5; % aramture inductance
Result:
What is PID Controller?
A proportional-integral-derivative controller (PID controller or three-term controller) is a control loop mechanism employing feedback that is widely used in industrial control systems and a variety of other applications requiring continuously modulated control. The proportional integral derivative controller produces an output, which is the combination of the outputs of proportional, integral, and derivative controllers. The feedback from the plant whichever is a controllable parameter is given as input to the PID controller and the processed error is minimized to get the desired output if tuning of the parameter is done optimally. The mathematically PID controller can be defined by the following equation where e(t) is the error signal with respect to time, Kp, Ki and KD are PID constants.
DC motor with PID controller simulation model shown below:
Blocks used in model:
Model explanation:
For building the DC motor with the PID controller simulation model we use the Simulink primitive blocks. Firstly, we took a clock block and a to workspace block that is connected to each other. Then for a reference speed, we took a step input block with a step size of 5 and a final value of 10. Then it was connected to a sum block which is having two input ports positive and negative. The positive input port has a step input and the negative port is connected between the integrator and to workspace named ‘w’. then the output of the sum block is given to a controller gain block named Kp. For the Integral Ki a gain block is used with initial conditions as zero. For the derivative gain, we are taking a derivative block connected to a gain block named Kd, and all the PID gains output is connected to a sum block with positive ports. Then its output is given to another sum block positive port and the negative port is connected to the back emf gain block. Ant its input is taken from the second transfer function output as a feedback signal. The second sum block output is given to a transfer function having impedance and resistance values and its output is multiplied with a gain block which is torque constant Kt. By multiplying the kt and transfer function we get Torque as output and it is given to another transfer function then the output is a speed and it is given to the workspace block. this simulation is run for 10 seconds and the system response is verified by plotting the graph between time vs speed and ref_speed.
M-script Explanation:
To run the model, we use a script file. In this script, we are defining the model parameters as taken in the lecture. For the PID controller design, we are taking the equations which are solved to develop the Simulink model. And here we are defining the matrices A and B. for the pole placement we take the fn as natural frequency. And wn is calculated for natural frequency in rad/sec. the zeta value is taken as 0.5. to calculate the poles, we use the below formulas: p1=-zeta*wn+wn*sqrt(zeta^2-1);
p2=-zeta*wn-wn*sqrt(zeta^2-1);
the by using the “place” command we place the poles in the plot. We define the PID values as Kp, Kd, and Ki. The transfer function is defined and poles are mapped for the given G1 transfer function.
After running the script successfully without errors for 10 seconds. We can observe the result in the workspace window and the poles are plotted in a new window.
% This file contains the design paramters to model a Dc motor
j = 0.01; % momemt of inertia
b = 0.1; % motor friction
kb = 0.01; % back emf
kt = 0.01; % Motor torque constant
Ra = 1.0; % aramture resistance
La = 0.5; % aramture inductance
A=[-b/j kt/j; -kb/La -Ra/La];
B=[0; 1/La];
fn=1;
wn=2*pi*fn;
zeta=0.5;
p1=-zeta*wn+wn*sqrt(zeta^2-1);
p2=-zeta*wn-wn*sqrt(zeta^2-1);
p=[p1;p2];
k=place(A,B,p);
kp=k(1);
kd=k(2);
ki=0.5;
Test-1: kp=38.3133, kd= -2.8584, Ki= 0.5000
The plot response with the above gain’s values is shown below.
Test-2: Kp=30, Kd=-1.5, Ki= 30
Conclusion:
The dc motor without control and with the controller is designed and its plot response is observed. The controller is tested under various values of controller gains