# Linear Systems and Linearized State Space

1. a) A system has the following model:

x_1 = x_2

x_2 = (-x_2)92+cos(x_1))-3sin(x_1 - u)

y = tan(x_1 - u)

Find the linearized state space model about the steady state with x = (0,0)^T.

b) Draw a Simulink implementation diagram of the system in question a. The input comes from a function generator and the output and simulation time must be sent to the workspace.

2. b) Find the natural response of the system (see attached), with input u(t) = 0 and initial condition x(0) = (-1,1)^T. Sketch your solution.

bii) Use the Laplace transform to find the output of the system (see attached), y = (0 1)x,

with x_0 = (1 0)^T and ut) = e^-3t.

bii) Sketch the solution and confirm that it makes sense.

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#### Solution Summary

This solution includes formated calculations and explanation in the attached Word document. The Simulink diagram for 1(b) is given and the .mdl model file is attached. 287 words.