Question

Problem 1. (Transformation to controllable canonical form). Consider the following third-order SISO LTI system $\dot{x} = Ax + Bu,$\quad $x \in \mathbb{R}^3, u \in \mathbb{R}^1$ (AB-CLTI) Assume that the characteristic polynomial of A is given by $det(sI - A) = s^3 + a_1s^2 + a_2s + a_3$ and consider the 3 $\times$ 3 matrix $\(T := C \begin{bmatrix} 1 & \alpha_1 & \alpha_2\\ 0 & 1 & \alpha_1\\ 0 & 0 & 1 \end{bmatrix}$\ where C is the system's controllability matrix. (a) Show that the following equality holds: $\(B = T \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}$\ (b) Show that the following equality holds: $\(AT = T \begin{bmatrix} -\alpha_1 & -\alpha_2 & -\alpha_3\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{bmatrix}$\

          Problem 1. (Transformation to controllable canonical form). Consider the following third-order SISO LTI system
\(\dot{x} = Ax + Bu,\)\quad \(x \in \mathbb{R}^3, u \in \mathbb{R}^1\)
(AB-CLTI)
Assume that the characteristic polynomial of A is given by
\(det(sI - A) = s^3 + a_1s^2 + a_2s + a_3\)
and consider the 3 \(\times\) 3 matrix
$\(T := C \begin{bmatrix} 1 & \alpha_1 & \alpha_2\\ 0 & 1 & \alpha_1\\ 0 & 0 & 1 \end{bmatrix}$\
where C is the system's controllability matrix.
(a) Show that the following equality holds:
$\(B = T \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}$\
(b) Show that the following equality holds:
$\(AT = T \begin{bmatrix} -\alpha_1 & -\alpha_2 & -\alpha_3\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{bmatrix}$\

Problem 1. (Transformation to controllable canonical form). Consider the following third-order SISO LTI system
ẋ = Ax + Bu, x ∈ℝ^3, u ∈ℝ^1
(AB-CLTI)
Assume that the characteristic polynomial of A is given by
det(sI - A) = s^3 + a1s^2 + a2s + a3
and consider the 3 × 3 matrix
T := C
< b m a t r i x >
where C is the system's controllability matrix.
(a) Show that the following equality holds:B = T
< b m a t r i x >
(b) Show that the following equality holds:AT = T
< b m a t r i x >

Added by Glen J.

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