Assignment

1. You are to design a driveline speed controller for a four cylinder SI engine. Assume that there are flexibilities in the drive shaft (axle) and the clutch. Using the state space model equations for such a driveline model (equations 7.84 - 7.86), derive the transfer function between crankshaft speed (output) and engine torque (input). Assume that J₁ = 0.13; J₂ = 0.07; J₃ = 0.25; k_c = 5; k_d = 2.0; d_c = 1, d_d = 0.9; d₂ = 1.0; d₃ = 0.7, l = 0.02; i_t = 1.5; and i_f = 3. The units of all parameters can be assumed to be in SI units. You may use the ss2tf function in MATLAB to obtain the transfer function.

For the open loop transfer function obtained above, plot the root locus using MATLAB function. What are the stability limits for the transfer function? Note your observations.

2. For problem 1, design a PID controller for crankshaft speed such that the following requirements are met: Maximum Overshoot < 8% and Settling Time (2% definition) = 2.2 seconds. Assume any additional desired pole locations that are less dominant than the pole pair obtained from the above specifications, if necessary. Note that the control variable is the engine torque.

Using SIMULINK, model and simulate the controller performance (open loop vs. closed loop) for a step command for the crankshaft output speed.

3. Consider the transmission torque control of an automobile driveline. Assume that the driveline torque model for the transmission torque control problem (with only axle flexibility) can be represented by the state space model in equations (7.175 in conjunction with 7.79 - 7.83) of your textbook. It is desired to design a full state feedback controller with proportional component, i.e.

u = -K_p x

Derive the control law for minimizing the transmission torque with no constraints on the control (u) itself. The design requirements are the same as in problem 2. You may need to assume additional non-dominant desired pole locations. Assume that the driveline parameters have the following values: J₁ = 0.01, J_t1 = 0.015, J_e = 0.08, d_t1= 1.1, d₁ = 1.03, i_t = 2, i_f = 3, d = 1.25, k = 2.5. Using MATLAB, plot the step response for both the open and closed loop systems. Assume that J₂ = 0.25; d₂ = 0.75, l = 0. [Hint: Need to represent the state space model in equations 7.79-7.83 in block diagram form; then use the state feedback].

4. Derive the transfer function of the closed-loop system in problem 3. Find the roots of the characteristic equation. State your observation on the relative stability of the closed loop system.