Me 50400 And ECE 59500 Automotive Control Homework Set
Me 50400 And Ece 59500 Automotive Controlhome Work Set 41 You Are To
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Me 50400 And Ece 59500 Automotive Controlhome Work Set
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; kc = 5; kd = 2.0; dc = 1, dd = 0.9; d2 = 1.0; d3 = 0.7, l = 0.02; it = 1.5; and if = 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
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. xKu pï€ï€½ 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, Jt1 = 0.015, Je = 0.08, dt1= 1.1, d1 = 1.03, it = 2, if = 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.
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Paper For Above instruction
The design and analysis of driveline control systems in automotive engineering involve complex mathematical modeling and control theory applications. This paper addresses the process of deriving transfer functions, designing controllers, and assessing system stability for a four-cylinder spark-ignition engine's driveline, incorporating model parameters, MATLAB simulations, and control strategies, including PID and state feedback controllers.
Introduction
Automotive control systems ensure optimal engine performance, vehicle stability, and drivability. Accurate modeling of the driveline, incorporating engine dynamics and flexibility aspects, is fundamental to controller design. The system’s mathematical representation often uses state-space models which facilitate the derivation of transfer functions, stability analysis via root locus, and controller synthesis using classical control and modern state feedback approaches.
Modeling and Transfer Function Derivation
The first step is establishing the state-space representation based on equations (7.84 - 7.86) from the course material, involving parameters such as moments of inertia (J1, J2, J3), stiffness coefficients (kc), damping factors (kd, dc, dd, d2, d3), and other system specifics. Using MATLAB’s ss2tf function simplifies converting the state-space model to the transfer function from engine torque to crankshaft speed.
Given the parameters (e.g., J1=0.13, J2=0.07, J3=0.25, kc=5, kd=2.0, dc=1, dd=0.9, d2=1.0, d3=0.7, l=0.02, it=1.5, if=3), the transfer function can be computed and analyzed. The root locus plot indicates the system's stability margins and how pole locations move as gain varies, guiding controller design.
Controller Design: PID Tuning
The specification of a maximum overshoot of 8% and a settling time of 2.2 seconds allows for pole placement via PID controller tuning. The PID parameters are adjusted to ensure transient response criteria are met, employing MATLAB and SIMULINK for simulation. The open-loop response depicts how the system inherently behaves, while the closed-loop response reveals improved stability and transient characteristics after control intervention.
Transmission Torque Control via State Feedback
For transmission torque control, a state-space model encapsulates system dynamics with parameters like Jt1, Je, dt1, d1, etc. A full-state feedback control law (xKu) is derived to minimize transmission torque deviations. This involves pole placement or optimal control methods. MATLAB implementation includes defining the augmented system, selecting desired pole locations that ensure stability and transient performance, and evaluating the closed-loop response.
Characteristic Equation and Stability Analysis
The roots of the characteristic polynomial, obtained from the determinant of (sI - A + BK), determine the system's stability. If all roots have negative real parts, the system is stable. Variations in pole locations influence convergence rates and damping; hence, careful selection ensures robust and responsive control performance.
Conclusion
The integration of system modeling, transfer function derivation, control design, and stability assessment provides a comprehensive framework for automotive driveline control. MATLAB tools facilitate this process, enabling simulation and analysis, which are critical in practical controller implementation. Effective control enhances vehicle performance, safety, and driver comfort, underscoring the importance of these methodologies in modern automotive engineering.
References
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