Problem 5: [25 point(s)] In this problem, we will compute the pulse transfer function in two different ways. Consider the simplified quarter car model depicted in the Fig. 2. Here v is the longitudinal speed of the car. SPRUNG MASS SUSPENSION WHEEL r1 NEGLECTED Figure 2: Simplified quarter car model The mass m represents a fourth of the mass of the car. The stiffness k and damping b represent the spring and shock. The term r(t) is the road excitation, and serves as an input to the system. Using the states p=z-r and w=, the dynamics can be written as 1 |-|-x/m m2+5m50 k/m -b/m The velocity of the car body is considered as the output [10]= For this problem consider the values m=300[kg] k=18000[N/m] b = 1200[N-s/m] Suppose we want to know the response of the system given the input u = P(r). In the Fig. 3, D(z) is the digital filter, and G(s) is the transfer function of the system. For the sake of simplicity, let's assume D(z) = 1, and sampling period 7 = 1s. Answer the following questions: 5.1 [6 point(s)] Find the (analog) transfer function Go(s) = GZOH G(s). 5.2 [6 point(s)] Find the Z transform of Go(s). 6 u(t) y(t) D(z) ZOH G(s) T 1 1 Digital filter Quarter car system Figure 3: Signal flow of system 5.3 [7 point(s)] Find the discrete-time, state space representation of the system. Remember in homework 1, we proved: Ade B = = (S e^ dv) B. C = C 5.4 [6 point(s)] Find the transfer function of the discretized system.