Question 1. Consider a mechanical system shown in the figure below, where F₁ (t) and F₂(t) are the inputs. k₁ m m1 mm The equations of motion are derived as follows, → Fi(t) b m₂ x2 F₂(t) m₁x₁ + k₁x₁ − k₂(x2 − x₁) – b(x₂ − x₁) = F₁(t) m₂x₂ + k₂(x₂ − x₁) + b( x2 − x₁) = F2(t) a) Obtain the state-space form to represent this system. For the output equations, assume that the variables of interest (outputs) are x₁ and x₂. b) Re-write the state-space model obtained in Part a in the matrix-vector form, that is x Ax+ Bu, y = Cx + Du Clearly show the state vector, input vector, output vector, and the state, input, output and transition matrices.

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Question 1. Consider a mechanical system shown in the figure below, where F₁ (t) and F₂(t)
are the inputs.
k₁
m
m1
mm
The equations of motion are derived as follows,
→ Fi(t)
b
m₂
x2
F₂(t)
m₁x₁ + k₁x₁ − k₂(x2 − x₁) – b(x₂ − x₁) = F₁(t)
m₂x₂ + k₂(x₂ − x₁) + b( x2 − x₁) = F2(t)
a) Obtain the state-space form to represent this system. For the output equations, assume
that the variables of interest (outputs) are x₁ and x₂.
b) Re-write the state-space model obtained in Part a in the matrix-vector form, that is
x Ax+ Bu,
y = Cx + Du
Clearly show the state vector, input vector, output vector, and the state, input, output and
transition matrices.
Transcribed Image Text:Question 1. Consider a mechanical system shown in the figure below, where F₁ (t) and F₂(t) are the inputs. k₁ m m1 mm The equations of motion are derived as follows, → Fi(t) b m₂ x2 F₂(t) m₁x₁ + k₁x₁ − k₂(x2 − x₁) – b(x₂ − x₁) = F₁(t) m₂x₂ + k₂(x₂ − x₁) + b( x2 − x₁) = F2(t) a) Obtain the state-space form to represent this system. For the output equations, assume that the variables of interest (outputs) are x₁ and x₂. b) Re-write the state-space model obtained in Part a in the matrix-vector form, that is x Ax+ Bu, y = Cx + Du Clearly show the state vector, input vector, output vector, and the state, input, output and transition matrices.
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