Project - Speed Control of a Direct Current (DC) motor
Ans:
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.
Key Highlights:
- Mathematical modeling from the first principles
- Modeling an Electrical Engineering system
- Simulink based modeling
- Control law design and test
- Stability of a dynamic system
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 the lecture.
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)
Case-1: DC motor without a controller.
By using the above transfer function equations, the DC motor is developed.
Case-2: DC Motor with PID controller
In case 1, we developed a DC Motor without a controlling system.
Here we use a PID controller to control the motor.
Modeling of a DC Motor with PID controller
State Space equations for the DC Motor
State equations will be derived from equations (a) and (b)
dI/dt = - (R/L) I + (1/L) V – (Kb/L) d/dt
dI/dt = - (R/L) I + (1/L) V – (Kb/L) ω dI/dt
dω/dt= - (B/J) ω + (1/J) KTI
Y(t) = ω = d/dt
A state space representation of the DC motor from the above equations
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.
Apply Laplace transform on both sides.
The parameter which is under control must have some properties or be free from some properties such as rise time, settling time, overshoot, and steady-state error. The impact of PID parameters on such properties can be found in the below table.
Rise Time:
Rise time is defined as the time taken for a signal to cross from a specified low value to a specified high value. In analog and digital electronics, the specified lower value and specified higher value are 10% and 90% of the final or steady-state value. So, the rise time is typically defined as how long it takes for a signal to go from 10% to 90% of its final value.
Settling Time
The settling time of a dynamic system is defined as the time required for the output to reach and steady within a given tolerance band. It is denoted as Ts. Settling time comprises propagation delay and time required to reach the region of its final value. It includes the time to recover the overload condition incorporated with slew and steady near to the tolerance band.
The tolerance band is the maximum allowable range in which the output can be settled. Generally, the tolerance bands are 2% or 5%.
Peak Over Shoot:
(MP) is defined as the deviation of the response at the peak time from the final value of the response. it is also called the maximum overshoot. We can write it as
Steady State Error:
Steady-state error is defined as the difference between the desired value and the actual value of a system output in the limit as time goes to infinity (i.e., when the response of the control system has reached steady-state).
Steady-state error is a property of the input/output response for a linear system. In general, a good control system will be one that has a low steady-state error.
Stability of the System:
A system is stable if every bounded input yields a bounded output. We call this statement the bounded-input, bounded-output (BIBO) definition of stability.
The stability of a linear closed loop system can be determined from the locations of the closed loop poles in the s-plane.
The location of the poles of the transfer function decides the stability of the system. This is shown in the above figure for critically stable, unstable, and stable systems.
For our case, the transfer function is determined from the constants of motor and state space equations.
Modeling of DC Motor with PID Controller:
