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Problem 5: (20 points) Consider the second-order LTI system $\dot{x}(t) = Ax(t)$. Assume that $M$ is a non-singular matrix such that $M^{-1}AM = J_r$, and $J_r$ takes the form of $\begin{bmatrix} \alpha & -\beta\\ \beta & \alpha \end{bmatrix}$ where $\alpha$ and $\beta$ are real numbers. (P1) What are the eigenvalues of $A$? (P2) Define $z \triangleq M^{-1}x$. Derive the state-space equation where the state is $z$. (P3) Define $r \triangleq \sqrt{z_1^2 + z_2^2}$ and $\theta \triangleq \tan^{-1}(\frac{z_2}{z_1})$. Derive the state-space equation where the states are $r$ and $\theta$. Find the solution $r(t)$ and $\theta(t)$ with initial state $r_0$ and $\theta_0$. (P4) Use (P3) to show that (i) $x(t) \to 0$ as $t \to \infty$ when $\alpha < 0$; (ii) $x(t)$ goes unbounded as $t \to \infty$ when $\alpha > 0$.

          Problem 5: (20 points) Consider the second-order LTI system $\dot{x}(t) = Ax(t)$. Assume that $M$ is a non-singular
matrix such that $M^{-1}AM = J_r$, and $J_r$ takes the form of
$\begin{bmatrix} \alpha & -\beta\\ \beta & \alpha \end{bmatrix}$ where $\alpha$ and $\beta$ are real numbers.
(P1) What are the eigenvalues of $A$?
(P2) Define $z \triangleq M^{-1}x$. Derive the state-space equation where the state is $z$.
(P3) Define $r \triangleq \sqrt{z_1^2 + z_2^2}$ and $\theta \triangleq \tan^{-1}(\frac{z_2}{z_1})$. Derive the state-space equation where the states are $r$ and $\theta$. Find
the solution $r(t)$ and $\theta(t)$ with initial state $r_0$ and $\theta_0$.
(P4) Use (P3) to show that (i) $x(t) \to 0$ as $t \to \infty$ when $\alpha < 0$; (ii) $x(t)$ goes unbounded as $t \to \infty$ when $\alpha > 0$.

Problem 5: (20 points) Consider the second-order LTI system ẋ(t) = Ax(t). Assume that M is a non-singular
matrix such that M^-1AM = Jr, and Jr takes the form of
< b m a t r i x > where α and β are real numbers.
(P1) What are the eigenvalues of A?
(P2) Define z ≜ M^-1x. Derive the state-space equation where the state is z.
(P3) Define r ≜√(z1^2 + z2^2) and θ≜tan^-1((z2)/(z1)). Derive the state-space equation where the states are r and θ. Find
the solution r(t) and θ(t) with initial state r0 and θ0.
(P4) Use (P3) to show that (i) x(t) → 0 as t →∞ when α < 0; (ii) x(t) goes unbounded as t →∞ when α > 0.

Added by Iker P.

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