Part -1:

1. Consider a SISO LTI system whose state-matrix is given as

        -1   2   0

A =    0  -1   0

        0   0   -2

(a) Cite an example of the input matrix B and output matrix C such that the system is

i. neither controllable nor observable

ii. controllable but not observable

iii. not controllable but observable

iv. both controllable and observable

(b) Compute the state transition matrix eAt for this system.

(c) Let E₂ be the second column of e^At computed above. What is the L₂-norm of E₂?

2. Let B be a 4 × 4 matrix to which we apply the following operations:

(a) double column 1
(b) halve row 3
(c) add row 3 to row 1
(d) interchange columns 1 and 4
(e) subtract row 2 from each of the other rows (f) replace column 4 by column 3
(g) delete column 1

Write the result as a product of eight matrices. Illustrate by taking B as the identity matrix.

3. Determine the variation of the following functional (Kirk, 4-4b, pg. 179):

J(x) = _to∫^tf [x₁²(t) + x₁(t)x₂(t) + x²₂(t) + 2x'1(t)x'₂(t)]dt (2)

Assume that the endpoints are fixed and specified.

4. Draw the surface of the nonlinear function using mesh(.) in Matlab:

f (x) = (x₁ - x₂)(x₁ - 3x²₂) (3)

Calculate the Hessian and show that it is singular at the origin.

Part -2:

1. Find Euler's equation for the following variational problem:

min ₀∫^T g x(t), x?(t), x¨(t), ...,dⁿx(t)/dtⁿ, t) t

where x(t) is a scalar function of time. Assume that T , x(0) and x(T ) are known.

2. Find the optimal trajectory x∗(t) for:

J = ₀∫^T + 3x(t)x' (t) + 2x2(t) + 4x(t))dt (2)

with boundary conditions x(0) = 1 and x(1) = 4.

3. Find the optimal trajectory x∗(t) for:

J = ₀∫² (x'(t)²/2 + x(t)x' (t) + x' (t) + x(t) dt (3)

with boundary conditions x(0) = 1 and x(2) free. Find all constants in your solution.

Part - 3:

1. Find the optimal trajectory x∗(t) for:

J = ₀∫^tf (x'₁²(t) + x'₂²(t) + 2x₁(t)x₂(t)) dt (1)

x₁(0) = x₂(0) = 0, tf is free, and x(t_f ) lies on the curve θ(t) = [5t + 3 t³/2 ]T .

2. Consider two distinct planes

S₁ : a₁x + b₁y + c₁z = k₁ (2)
S₂ : a₂x + b₂y + c₂z = k₂ (3)

where the coefficients a, b, c and k's are all constant real numbers. Consider a point P = (l, m, n) that lies on S1. Find the equation of the straight line passing through P and cutting S2 such that the length of the line segment between the two planes is minimum. Pose the problem first as a constrained optimization problem, and then solve (symbolically) using the method of Lagrange multipliers.

3. It is desired to heat a room using least possible amount of energy. Let θ(t) be the temperature of the room, θ_a= ambient temperature of air outside the room which is considered to be fixed at 60?, and u(t) = rate of heat supply in the room.

The temperature dynamics is:

θ'(t) = -0.1(θ(t) - θa) + 2u(t) (4)

where the coefficients follow from room insulation data. Define x(t) = θ(t) - θ_a so that x' (t) = -0.1x(t) + 2u(t). Our objective is to minimize

J = 1/2 ₀∫²⁰ u2(t)dt

Assume initial room temperature θ(0) = θ_a = 60?, and the control should drive the ?nal room temperature to θ(20) = 70?. Find u∗(t).

Part -4 :

1. Consider the scalar LTI system: x' (t) = -x(t) + u(t)

The objective is to bring the system state from a fixed initial value x₀ > 0 to the origin in T seconds while minimizing

J = 1/2 ₀∫^T u²(t)dt (1)

Prove that if T is free then it will take infinite amount of time to do so (i.e., T → ∞).

2. Consider the minimization of

J = 1/2(x(t_f ) - 1)² + 1/2 ₀∫^tf u²(t)dt (2)

subject to x? (t) = -x(t) + u(t), with both t_f and x(t_f) free. The initial state is x(0) = x₀. Also, consider the two sets S1 = [0, 1 ) and S2 = (1, ∞). Prove that if x0 does not lie in any of these two sets then the only way to reach a final state will be to reach it in infinite time - i.e., if x₀ ∈/ (S1 ∪ S2) then t_f → ∞.