Reference no: EM13858974
Part 1 The nonlinear disturbance free cruise control system for automobiles can be approximately given by
Dv/ dt = (cα/m)u - (cf|v|v)/m
Where the speed is given in m/s and the control throttle is given in m where ca is the proportionality constant and u is the throttle position. cf is the friction/wind resistance coefficient and m is the mass of the automobile. The linearized system (around operating point v0 = 20 m/s is given by
Dv/dt = (cα/m)u - (σcf/ m) v(t)
where σ = 2* v0 = 40.
(A)Prove that the above linearization is correct. Also, show that the above linearized system (with initial values x1(0) = 0, x2(0) = v0) can be written as
![757_1.png](https://secure.expertsmind.com/CMSImages/757_1.png)
The transfer function of the linearized system is
given by
V(s)/ U(s) =ca/(MS+σCF)
The above expression can be written as
V(s)/ U(s) = ac /( Tcs + 1)
(B) Find the expressions of the cruise control constant a0 and cruise time constant Tc. If an unit step is applied to the system, then the output speed is given by
V(s) = (αc/ Τcs + 1)(1/s)
The inverse Laplace of Eq. 7 gives the time domain expression for the speed as
v(t) = αc (1 - exp(- t/Τe ))
(C) At what value of t will v(t) = 0.63α0?
(D) If αc was found to be αc, = 75, find Τc using SIMULINK®. Hint: Use the min command.
(E) Again find the Τc, this time using the step command for time grid t1 = o : 0.5 : 500. Compare it with the Τc found using SIMULINK®, which one is more accurate? and why?
Part 2
(A) Develop a closed-loop proportional controller using gain kp for the above cruise system to control the speed of the automobile. The proportionality constant that converts the output speed to throttle displacement is kt and kr converts the reference speed to throttle displacement. Draw the block diagram and find error E(s) for a given disturbance D(s). Also find steady state error e,,.
(B) Practically speaking, can the the steady state error go to zero?
(C) Find the expression for kr for the steady state error to be zero in the absence of disturbance D(s).
(D) Let kt = 4/300, obtain kr from theisbove expression. Apply a disturbance d(t) = 2u(t - 200) and the reference speed is a step vr(t) = 9u(t). Obtain a plot for 3 values of kp = 1, kp = 5 and kp = 10. Plot them in the same figure
(E) Calculate the theoretical and experimental steady-state error. Is the steady-state error zero? what happens to the steady state error if kp is increased? At what value of kp will the steady-state error go to zero?
(F) It is desired to reduce steady-state error to zero. Introduce an integral controller and make the block diagram.
(G) Let ki = 1, kp = 1, kr = kt and d(t) = 2u(t - 200). Plot the output speed. Is the response oscillatory? what is the steady-state error?
If it is oscillatory, this is dangerous as it can lead the automobile to get out of control. Therefore we are going to use the ITAE index to obtain a value of /4 that gives us a less oscillatory response.
(H) Can you explain why we are taking the ITAE index?
(I) Obtain the expression for damping ratio ς and natural frequency ωn for the PI control system we have developed so far with disturbance d = 0.
(J) Vary ς with kp, kr, kt fixed as above and disturbance d = 0 such that ςvec = 0.01 : 0.01 : 2. Obtain kt for each ς value and obtain the JITAE. Plot JITAE against ki. Choose the kt that gives the least J1TAE (use min command).
(I) From the plot, comment on its selectivity.
(J) Now again plot the output speed with d = 2u(t-200) and use the optimum kt.
(K) Comment on the steady-state and transient-state behavior in the plot. Also, how does it reacts to the disturbance?
Part 3
(A) At the end we are going to implement the PI controller on the nonlinear system given in Equation 1. The system was linearized around v0 = 20m/s. Implement the PI controller on the nonlinear model in SIMULINK®. Plot the output speed against time for both the linear and nonlinear model on the same figure. Also plot the control signal u against time for both the linear and nonlinear systems. Are the control signals different? Can you explain?