Question 1:  In this question, we deal with a very important concept: reachable sets. Given a system

X^·(t) = Ax(t) + B_ww(t)

the reachable set with unit energy is defined as

R = {x(T) : x, w satisfy the dynamics above, x(0) = 0 and ₀∫^Tw^T(t)w(t)dt ≤ 1

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

P > 0 and

then the ellipsoid

ε =^· {x : x^T Px ≤ 1}

contains the reachable set; i.e.,

ε ⊇ R.

b) Show that the volume of an ellipsoid

ε^~ =^· {x: x^T M^-2x ≤ 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 x is n, then the volume of the ellipsoid is of the form

volume(ε^~) = C(n) det(M).

Hence, we can determine the minimum volume ellipsoid that contains the reachable set R. by solving the following optimization problem

min_P log [det (P^-1)]

subject to

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

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) + B_ww(t) + B_uu(t)

Formulate the problem of state feedback design

u = Kx

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 2: In this question we show and use the fact that many system specifications can be formulated as a convex problem on the Youla parameter. First, recall that a set A is convex if for any x ∈ A and any y ∈ A,

λx + (1 - λ)y ∈ A for all λ ∈ [0, 1]

Also, a function f is said to be convex if for any x and y

f [λx + (1 - λ)y] ≤ λf(x) + (1 - λ) f(y)  for all λ λ [0, 1]

Now, consider a system with performance input w and performance output z and controller input u and output to the controller y. Assume that all these are scalars. As mentioned in class, one can parameterize all achievable closed loop transfer functions by using the Youla parameter Q. In this problem, given Q, we denote by

T^Q_zw

the closed loop transfer function from w to z and (with some abuse of notation)

z(t) = [T^Q_zw, w](t)

the response of the system at time t to the input w.

a) Show that, for any system norm ||.||

f(Q) = ||T^Q_zw||

is a convex function of Q.

b) We now look at time domain constraints on the response of the system. It turns out that this indeed leads to a "nice" design problem. Let the input w be given and fixed and consider upper and lower bounds on the response to this input; i.e., we want

zl(t) < [T^Q_zw](t) ≤ z_u(t) for all t

where z₁ and z_u are given upper and lower bounds.

Show that the set

Q = {Q : zl(t) < [T^Q_zww](t) ≤ z_u(t) for all t}

is convex.

c) As discussed in class, it is hard to find Q directly since it "lives" in an infinite dimensional space. So, a possible finite dimensional approximation is to express Q as

Q (α) = α_o + ∑^N_i=1α_i(s-γ)/(s+γ)ⁱ,

for a given γ > 0. Moreover, imposing time restrictions like the ones in the previous question might be hard to handle, so one usually tries to enforce them at specific time instants t₁, t₂, ... , t_M and then test to see it is satisfied for all t. Show that the set

A = {α : z_l/(t_i) < [T^Q(α)_zww,2,')w](t_i) ≤ z_u(t_i) for all i = 1,2, ... , M}

is a polytope; i.e., there exist matrix A and vector b such that

A= {α : Aα ≤ b}

d) Consider now the nominal model of the ACC benchmark problem; i.e., take k = 1.25 and assume that there is no uncertainty. Using the results above and a linear program solver, minimize the peak of the control input when the disturbance is an impulse with the additional restriction that the settling time should be below 15 sec.