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

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) = (α_c/ Τ_cs + 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 α_c was found to be α_c, = 75, find Τ_c using SIMULINK®. Hint: Use the min command.

(E) Again find the Τ_c, this time using the step command for time grid t₁ = o : 0.5 : 500. Compare it with the Τ_c found using SIMULINK®, which one is more accurate? and why?

Part 2

(A) Develop a closed-loop proportional controller using gain k_p for the above cruise system to control the speed of the automobile. The proportionality constant that converts the output speed to throttle displacement is k_t and k_r 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 k_t = 4/300, obtain kr from theisbove expression. Apply a disturbance d(t) = 2u(t - 200) and the reference speed is a step v_r(t) = 9u(t). Obtain a plot for 3 values of k_p = 1, k_p = 5 and k_p = 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 k_p is increased? At what value of k_p 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 k_i = 1, k_p = 1, k_r = k_t 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, k_r, kt fixed as above and disturbance d = 0 such that ς_vec = 0.01 : 0.01 : 2. Obtain kt for each ς value and obtain the J_ITAE. Plot J_ITAE against k_i. Choose the k_t 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?