Problems 2.2, 2.9, 2.16 about control system.

Please give a brief explanation about your answer.

2.2. Ednsider the med1anieal system shown in Figure 2.42. Here a} is an external fnree applied to the mass M, yt] is the displacement of the mass 1with respect tn the pnsitinn 1when the spring is related. The spling erce and frietinn force are g'wen respecti'rnrelf,r by felt} = H1 + yzitllyiiL felt} = bait} Figure 2.42: A mam and spring system fer Prehlsm 3.2- 1. Write the differential equation mndel of this system. 2. Write a. state space description of the system. 3. Is the system linear':III [Fit is net linear, Iinearise it arnund the operating point 1with up. = . 4. Find the transfer funetien of the linearized system. 2.16. Use MATLAB to strive this problem. A exible beam system is described by the following state spaoe equation 1.3333 13.3334 1.3333 13.1433 3.3313 it] _ 1.1T'F3 15.2ti4r1 l.354$ l4.5115 it] + 1.31111 {t} I ' 3.3433 3.3333 3.4313 3.3333 3 3.1331 \" 1.3313 13.3333 1.1313 13.4333 1.1334 sftl= [ 33.3333 33.3433 33.4333 13.3133 he}. 54 Chapter 2 Modeling and Simulation . Form a system 1variable representing this system. I[Use as.) . Find the transfer function of the system. {Use til} . Find the pollen, zeros. and the DC gain of the system. [Use spin} . Write a state space equation of the system from its transfer metion using the formulas in Section 23?. 5. Obtain a state space equation of the system from its transfer metion by using 55. Compare the result with the original state spaoe equation and the result of part :1. Are they the same? Explain why. amen- 2.9. Consider Figure 2.43. HEIJIIIE that a state space made] of Pia} is :I':I[t} = Az} + bruit] Hit} = it} and K is a pure gain. htain a state space model of the closed-[nap system 1with. input rt} and nutput git}. Figure 2.45: Ujmy feedback system with a pmparlianal feedback far Pruhlem 3-9