ENGR 3350 control systems
Design a controller for an electric vehicle using PID / PI controllers
Electric motors are widely used in industries, such as traction motors installed in electric vehicles. In this project an electric vehicle, being propelled by a DC motor, is considered. The model of this electric vehicle is shown in the block diagram of Figure 1:
Figure 1: Block diagram of the electric vehicle powertrain
The driver applies the input R_{v}(s) to the vehicle by pushing the gas pedal. The block G_{sc}(s) represents the electric powertrain controller. The numerical values of all the parameters involved are tabulated below:
Parameter |
Value |
Parameter |
Value | ||
Wheel radius |
r |
0.25 |
Armature gain constant |
R_{a} |
1 |
Vehicle mass |
M_{tot} |
950 |
Armature time constant |
T_{a} |
5 |
Total inertia moment |
Jtot |
8.6 |
Actuator gain constant |
K_{A} |
10 |
Friction torque gain |
K_{f} |
0.1 |
Actuator time constant |
T_{A} |
5 |
Armature gain |
K_{t} |
2 |
Back emf gain |
K_{b} |
0.1 |
Efficiency |
ηtot |
0.9 |
In order to design a fast and reliable control system for the attitude control, the following tasks should be performed:
- Find the closed-loop transfer function without the controller, i.e., G_{sc}(s) =1.
- Perform the stability analysis of the closed-loop system by using the Routh-Hurwitz stability criterion. In order to achieve a stable system, what would be the relation between the current sensitivity gain, K_{cs} and the speed sensitivity gain K_{ss }?
- For the value of K_{cs} = 0.087, determine the gain K_{ss} such that the system has a steady-state error of 2% for the unit-step input.
- Plot the root locus of the two systems and discuss about the stability of the system from the plots. Also, for the case of marginal stability determine the system gain.
- Plot the unit-step response of the system and find the attributes of the response.
- Plot the Bode diagram of the system and find the Phase Margin (PM), Gain Margin (GM) and the Band Width (BW). Determine their corresponding frequencies.
- According to Figure 1, find the transfer function G(s) of the plant:
Figure 2: The cascade controller G_{sc}(s)
- Select an appropriate controller for G_{sc}(s) as either PI or PID, and then design the controller to obtain M_{p} < 3% and t_{s }< 35 seconds for the unit-step input. Explain your design procedure fully and justify your chosen controller type.
- Find the closed-loop transfer function of the compensated system. Determine the type of the system and calculate the steady-state error to an appropriate input.
- Plot the root locus of the compensated system. Discus the stability of the system.
- Plot the unit-step response of the compensated system. Find the attributes of the unit-step response and compare them with the uncompensated system.
- Plot the Bode diagram of the compensated system and determine the PM, GM and BW. Compare with the uncompensated system frequency response attributes.