Exercise 1. Cayley-Hamilton Theorem

Given

Find A¹⁰ and e^At uisng C-H.

Exercise 2. Linear dynamics solution

Let x₁(t) be the water level in Tank 1 and x₂(t) be the water level in Tank 2 . Let be the rate of outflow from Tank 1 and be rate of outflow from Tank 2 . Let u be the supply of water to the system. The system can be modelled into the following differential equations:

Figure 1: Tank Problem

Given = 0.1, = 0.2, u = 1, x₁(0) = 2, x₂(0) = 1, find the water level in both tanks after 5s. Solve with C-H theorem. You may use calculator but do not directly use programming.

Exercise 3. Jordan form, decomposition (20 points, 5 for each A) Derive the Jordan-form of the following matrices manually.

A A A A

(Write the Jordan form such that eigenvalues should be in ascending order of their absolute

values. The absolute value of a complex number is defined as |a + bi| = a² + b²)

Exercise 4. CT and DT Dynamics

Given the following system. Assume x(0) = 0 and u is a unit step input. Answer the following questions with both manual derivation and Python programming. Include all your derivation. Solutions without derivation will receive zero points. Submit the code to Gradescope.

1. i) Find y(5) for CT system (5 points) ii) Find the discretized state space representation using sample time T = 1s (5 points) iii) Find y(5) of Discrete Time system. (5 points) Also plot signals y(t) for both CT and DT systems in the same figure. (5 points)

Exercise 5. Diagonalization

In the Fibonacci sequence, a Fibonacci number is the sum of the two previous Fs, starting from 0, 1

Fibonacci numbers: 0,1,1,2,3,5,8,13,.

Fibonacci equation: F_k+2 = F_k+1 + F_k

How could we find the 20th Fibonacci number, without starting at F₀ = 0 and F₁ = 1, and add the numbers one by one all the way out to F₂₀? Hint: construct a discrete linear time invariant system.

You are allowed to use programming in the final matrix multiplication, but do not use the for loop to directly computer Fibonacci sequence