Consider the following mechanical system: y m +f dy(t) d’y(t) +b dt dt + ky(t) = f (t) m Obtain the state space model of the system with input f(t) and output y(t) . Calculate the system matrices for m = 1, k = 1 and b = 2. Check the stability by using the second method of Lvapunov. 3.

A mechanical system is described by the following transfer function -4s H(s) = sª-2s² + s-17 If u is the input, and y the output, Cast this system into the state variable format. (Do not find the solution for the system) Y

1. Reduce the following differential equation to the state space equation form. a. y" (t) + 3y' (t) + 2y (t) = u(t) b. y" (t) = u(t) — b₁y' (t) — boy (t) c. 4y" (t) cos (t) y' (t) + sin (t) y (t) = u(t)

Derive the state-space model (state equation and output equation) in vector form for the following system. The system outputs are the displacements of each spring. Assume that the connection between the spring and rope is massless and that the rope is inextensible. Assume that gravity is an input as well as the applied forces Fi(t) and F2(t). Neglect friction forces on mass m₂. If q₁ is the state variable for the bottom spring connected to m₁, q2 is the state variable for the mass m₁, q3 is the the top spring connected to the pulley, and q4 is the state variable connected to the mass m2, then you should expect to get the following state-space representation: 92 43 y 0 LaLa (L+L) 0 4₂ kL₂ mi Li 0 m2 0 13 (L+L) 0 CA = - [8] 92 93 + 92 93 + L94 m₂ F₂(t) m₂ TOL F₂(t) Figure 2: Diagram for problem 2 X5 00 X2 [000] 00 U₁ U12 U13 m₂. 21 142 143

Using masons rule find the transfer function G(s) = C(s)/R(s)

Part A and B  SOLVE CAREFULLY!! Please Write Clearly and Box the final Answer for both

For each transfer function write the corresponding differential equation. For problem c simplify the answer by explicitly computing all derivatives

For the mechanical system shown in Fig-1, u₁, u₂ are the inputs and y₁, y2 are the outputs YI Y Y2Y k₁ mi UI U2 + m2 Fig-1 a) Write the differential equations that model this system b) Find a state space representation of the system.

Use MATLAB to obtain a state model for the following equations; obtain the expressions for the matrices A, B, C, and D. In both cases, the input is f(t); the output: is y. a. 5d³yd²y +7. b. dy +3 dt³ dt² dt Y(s) 5 = F(s) s² +7s+4 - +6y=f(t)

b m dy(t) +b. m d² y(t) dt² (5 points) Obtain the state space model of the system with input ƒ(t) and output y(t). Calculate the system matrices for m = 1, k = 4 and b = 1. dt ►y Hon - + ky(t) = f(t) f