Exercise 1. Cayley-Hamilton Theorem (20 points)
Given
Find A10 and eAt uisng C-H.
Exercise 2. Linear dynamics solution (20 points)
Let x1(t) be the water level in Tank 1 and x2(t) be the water level in Tank 2 . Let α be the rate of outflow from Tank 1 and β 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 α = 0.1, β = 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 (20 points)
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.
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 (20 points)
In the Fibonacci sequence, a Fibonacci number is the sum of the two previous F’s, starting from 0, 1
Fibonacci numbers: 0,1,1,2,3,5,8,13,....
Fibonacci equation: Fk+2 = Fk+1 + Fk
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.
