Provide the correctanswers only. Question 1 (50%): In this question, we deal with a very important concept:

 Provide the correctanswers only.

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Question 1 (50%): In this question, we deal with a very important concept: reachable sets. Given a system the reachable set with unit energy is defined as i(t) Ax(t) + Bw(t) R = = {x(T) : (T): x, w satisfy the dynamics above, x(0) = 0 and To w w (t)w(t)dt < a} This is the set that one can reach with perturbations of unit energy. It is important in terms of establishing safety; i.e., can perturbation w take the system to undesirable places or not. a) Show that if P satisfies the following LMIS then the ellipsoid P> 0 and contains the reachable set; i.e., b) Show that the volume of an ellipsoid ε = {x = x² Px < 1} [ATP+ PA PB BTP PB subject to EQR. E = {x: "Ma < 1} for some M > 0 is proportional to det(M) where the constant depends only on the dimension of x; i.e., if dimension of a is n, then the volume of the ellipsoid is of the form volume(E) = C(n) det (M). x = Hence, we can determine the minimum volume ellipsoid that contains the reachable set. R by solving the following optimization problem min log [det (P-¹)] [ATP+ PA PB BIP -I P>0 c) It can be shown that log [det (P-1)] is a convex function of P if P is positive definite. Take T 5s. Use CVX (or any other toolbox that can solve convex optimization prob- lems) and the formulation in the previous question to determine the ellipsoid of smallest volume that contains the reachable set R of the system 1 -10- 30 I + 201 30 Formulate the problem of state feedback design d) Generate 100 inputs w at random of energy less than or equal to 1 in the interval [0, 5] and simulate the response of the system above to these inputs. Verify that the obtained ellipsoid contains all trajectories. e) Now assume that we have a control input u; i.e., x(t) Ax(t) + Bw(t) + Buu(t) ( u = Ka that leads to the ellipsoid of smallest volume that contains the reachable set R. Can you formulate it as a convex problem? Why or why not? Question 1 (50%): In this question, we deal with a very important concept: reachable sets. Given a system the reachable set with unit energy is defined as i(t) Ax(t) + Bw(t) R = = {x(T) : (T): x, w satisfy the dynamics above, x(0) = 0 and To w w (t)w(t)dt < a} This is the set that one can reach with perturbations of unit energy. It is important in terms of establishing safety; i.e., can perturbation w take the system to undesirable places or not. a) Show that if P satisfies the following LMIS then the ellipsoid P> 0 and contains the reachable set; i.e., b) Show that the volume of an ellipsoid ε = {x = x² Px < 1} [ATP+ PA PB BTP PB subject to EQR. E = {x: "Ma < 1} for some M > 0 is proportional to det(M) where the constant depends only on the dimension of x; i.e., if dimension of a is n, then the volume of the ellipsoid is of the form volume(E) = C(n) det (M). x = Hence, we can determine the minimum volume ellipsoid that contains the reachable set. R by solving the following optimization problem min log [det (P-¹)] [ATP+ PA PB BIP -I P>0 c) It can be shown that log [det (P-1)] is a convex function of P if P is positive definite. Take T 5s. Use CVX (or any other toolbox that can solve convex optimization prob- lems) and the formulation in the previous question to determine the ellipsoid of smallest volume that contains the reachable set R of the system 1 -10- 30 I + 201 30 Formulate the problem of state feedback design d) Generate 100 inputs w at random of energy less than or equal to 1 in the interval [0, 5] and simulate the response of the system above to these inputs. Verify that the obtained ellipsoid contains all trajectories. e) Now assume that we have a control input u; i.e., x(t) Ax(t) + Bw(t) + Buu(t) ( u = Ka that leads to the ellipsoid of smallest volume that contains the reachable set R. Can you formulate it as a convex problem? Why or why not?