Question

The A matrix of a state space system takes the following form, TO 1 07 A= 0 0 1. L-2 1 2] It is also known that one eigenvalue of A is 1. Find the initial condition of the system with which the states go to origin over time (assume zero input).

Answer 1

Answer:

λ₁ = 1, λ₂ = 2, λ₃ = -1

This is the origin condition value

Step-by-step explanation:

Given that

A = $\left[\begin{array}{ccc}0&1&0\\0&0&1\\-2&1&2\end{array}\right]$

given one Elegen value is 1

Let λ₁ = 1

its known that

λ₁ + λ₂ + λ₃ = trace

1 + λ₁ + λ₃ = 0 + 0 + 2

λ₂ + λ₃ = 1   (eqn 1)

λ₁ . λ₂ . λ₃ = /A/

where

/A/ = -1(2)

/A/ = -2

λ₂ . λ₃ = -2 (eqn 2)

solve eqn 1 and eqn 2

λ₂ = 2, λ₃ = -2

therefore

λ₁ = 1, λ₂ = 2, λ₃ = -1

this is the origin condition value