Using Simulink, simulate both the nonlinear model (1) and the linearized model (2), for the case when all the initial states are zero except a (0) 0.1 [rad], and with no input. Plot the outputs (t) and a(t), and compare the output responses between (1) and (2). Add your Matlab code(s) (m-file and Simulink block) in your report. = The equations of motion can be written (no derivation is required here) as (Jr + J, sin a) + m,rl cos a +2.J, sin a cos ad - m,rl sin ad = 7-6,0, Jp+mprl cos af - Jp sin a cos a0 +mpgl sin a = -b, (1) where the notations are indicated in the figure, and := Lp/2. If we approximate the system around 0 = 0 and a = 0, using sinaa, cos a 1 and sin a 0, we can simplify these equations as J, + mrl = T - b,0, From these two equations, we can derive where mrl + Jp = -ba-mpgla. a }, {.Jp(7 b0) + mrl(b+m,gla)}} {J, (b + m,gla) mrl(7 b0)} - JJrJp - (mprl). By introducing the state variables as = 0, I = 0, 1:= I3 = 0, I4 = , and the input and outputs as we can get the state-space model as i = where A:= 1 -Jpbr/Jt 0 0 0 mrlb,/J 0 u= r, y = 0, y = 0, Notation m., T br mp Lp bp 9 [ 8]. The parameter values are given in the table below. J, := Ar+ Bu, Cx, 0 0 (mpl)rg/J mprlbp/J 0 1 J.mpgl/J Jbp/Jt C:= 1000 0010 1 1 =\m,r, Jp = {ml, l= Meaning rotary arm mass rotary arm length viscous friction coefficient pendulum mass pendulum length viscous friction coefficient gravitational acceleration B: 4/2 1 J Value and unit 0.095 kg 0.085 m Jp -mprl 0.001 Nms/rad 0.024 kg 0.129 m 5 105 Nms/rad 9.81 m/s (2) Hint: You may want to realize the nonlinear model (1) in Simulink in the following way. S Matlab function 3 [:] [B] a