Question 4: Again consider the standard (FINITE-horizon) LQR problem for the double integrator system:

with time optimal performance index (whirl must be minimize I):

the control is bounded lu(t)I < 1. The goal is move any arbitrary state to the final state (0,0) in minimmn time tf

(i)  Determine and plot the switching curve on phase plot. 7 = 7₊ U-y_. Give the equations for both branches.

(ii)   Assume that the starting state is (-0.5, -1). determine the appropriate phase curve and its im ervrtion with the switching curve. Compute the time take on each interval. Determine the (shortest) time ti to reach the final target state (0,0).

(iii) Hand sketch (or plot) the state trajectory from starting state to origin (0,0) on the phase plane. (You may use the unmarked  phase Platte plot on the last page of this exam. You can print it. out and
hand sketch the state trajectory neatly and scan it with the rest of your exam submission. Remember to label it. If you attempt. Question 5, sketch the trajectory on a separate sheet, scan and upload. )

(iv) Plot the states xt (t). x2(t) and u(t) between t = 0 and t = ti. (Note for the two phase segments, each has own time starting from 0 to miptvtive end time t₀. Thus segment 1 may have traverse time I_I and segment; 2 may have traverse time t₂.When computing the state x_i, x₂and u on segment. 1 use time sequence 0, T₁, 271, t_i. Similarly for segment 2 use 0, T2, 2T2, t₂. When plotting wrt time stack the intervals with offsets. So for time me
0, T1, 2T1, ...... ti (t₁ + T₂)  (t₁ + 2T₂)  (t₁+ t₂)
                         II added tot h6 hegetnent

and stack the states: eg. xi(t)
x₁(0),x₁ (T₁), .....- x₁ (it ), x_i (T₂), xi (²2) ..... ' X₁ (t₂) •

       segment I                                              segment 2

Similarly the state x2(t) and u(t).

Hint for Questions 4 & 5: Recall :in the intervals where u'is constant, denote u* = U = ±1. x₂(t) = x₂(0) + Ut and x_l(t) = x₁(0) + x₂(0)t + 4Ut²

where initial states are x₁₀ = si(0), x₂₀ = x2(0), U = ±1 = b.
The phase plane representations:
x_l(t)= x_1o - 4Ux1_:, + 4-₂Ux3(t) and t = U (x₂(t) - x₂,₃).

if U = +1,  then       ti = (x₂ - x_2o)
xi = xu, - Pio + Pi = al. + 44 if U . -1, then 1 ^t = - ^(x₂ - ^x₂₂)

Question 5 (Optional Question for Bonus 20 pts) : Again consider the standard (infinite-horizon) LQR problem for the double integrator system:

with time optimal performance index (whirl must be minimized):

the control is bounded lu(01 < 1. The goal is move any arbitrary state to the final state (-0.5, -1). This is opposite of the previous problem.

(i)  Determine and plot the switching curve on phase plot 7 = 7₊ O7_.Give the equations for both branches and corresponding regions of validity (ie. limits on values of x₂).

(ii)   Assume that the starting state is origin (0,0), determine the appropriate phase curve and its intersection with the switching curve. Compute the time take on each interval. Determine the (shortest) time ti to reach the final target state (-0.5, -1).

(iii) Hand sketch (or plot) the state trajectory from starting state (0,0) to (-0.5, -1) on the phase plane.

(iv)  Plot the states xi (t), x2(t) and u(!) between t =  0 and t = t_f

Question 2 : Consider the LQR minimization problem

J = r (mx²(t) + u²(0) dt; m > 0

and constraint:

1(t) = ax(t) + u(t); y(t) = x(t) Is the system controllable? Observable?

(ii)  Find the optimal state feedback. Solve the ARE for two possible solutions. Which is the positive definite solution?

(iii) Verify that selecting the positive definite solution of ARE in (ii) gives a stable re:vottse by  analyzing the optimal closed-loop g coefficient for optimal state equation r(t) = gx• (t)

(iv)  Explain how the time constant of the optimal closed-loop system varies with parameter m.

In your answer concentrate on the speed with which the state x(t) decays to zero starting from some arbitrary initial condition x(0) = x_o (e.g. the time constant of the response) and on the peak value of the optimal control signal u(t) i.e. MX0{1lu(t)I. (Note this is not related. to the Bang-luny cot,. It merely refers to the maximum value of absolute value of u(t).) Compare the two parameters (time constant and the max_t>0 kW') in relation to the real physical limitation on the maximum magnitude of the optimal control.