Exam 2018: https://mega.nz/#!z1AljAQT!qAzDR69lO9XzoA9eXQheBIMssRSHVIXZAYVtp86Px5o

The lecturer sent a list of issues. On exam he chose 2 of them for each group. During the exam, he did not pay attention to anyone is cheeting.

1. Write the definition of exp(At). Present two selected methods for computing exp(AT).

Definition:

$$ e^{At} = \sum_{i=0}^\infty \frac{A^i t^i}{i!} $$

Result of this equation is matrix with the same size as A, but consisting of a function. There at least four methods to computing matrix exponential:

• from definition:

  • If matrix A is nilpotent (raised to the power of a finite number of times is equal to 0) then it is possible to calculate the sum directly (because the sum consists of a finite number of elements).
  • If matrix A is not nilpotent, but it is possible to calculate the explicit formula for every element of any matrix power then using the Taylor series it's possible to find an solution.

• using Laplace transform,

  • dx/dt = A x

    s X(s) - x(0) = A X(s)

    (I s - A) X(s) = x(0)

    X(s) = (I s - A)^(-1) * x(0)

    x(t) = L^(-1){ (I s - A)^(-1) } * x(0)

  • https://wolfweb.unr.edu/~fadali/ee472/SolState.pdf

    [ I s - A]^(-1) = 1/s [ I - 1/s A ] ^(-1) = 1/s [ I + 1/s A^1]

• using modal transformations (diagonalization of matrix)

• using Cayley-Hamillton theorem

2. Write an Input/State/Output representation of a linear dynamical system. Present a general solution of this differential equation and justify your answer.

u - input, x - state, y - output

$$ \dot{x} = Ax + Bu \\ y = Cx + Du $$

Solution of this equation is explicit formula x(t).

dx(t)/dt = Ax(t) + Bu(t)

S dx(t) = S Ax(t) dt + S Bu(t) dt

x(t) = S Ax(t) dt + S Bu(t) dt

3. Write the definition of controllability. Present the Kalman's controllability rank condition and the Hautus test. Prove one of the mentioned controllability tests.

The system is controllable if for any initial state x_0 and any desired state x_d there exist a control signal u(t) such that in time T ≥ 0: