(c)What is the phase margin PM for the system shown in Figure 3? 

(Notes: (b) will counted only when (a) is correct! (c) will counted only when (b) is correct!)

(a) Derive the state equation for the system in Fig. 4(a). In this figure, C represents a linear damper which generates a reaction force proportional to its velocity (with C being the proportional constant), M is a mass, and F is the applied force input. You should use the velocity of mass, v, as the state variable.(3pts) (b) Derive the state equation for the system in Fig. 4 (b). In this figure, in addition to the damper (C) and the force input F, K represents a linear spring (with K being the spring constant). You should use the deformation of the spring, x, as the state variable. 

(c) Derive the state equations and the output equation for the system in Fig. 4 (c) using x and v as the state variables and the endpoint velocity No as the output. 

(d) Find the condition relating M, K, C1, and C2 that renders the system in Fig. 4 (c) uncontrollable. Also find the condition that results in an unobservable system. 

(a) Derive the state equation and the output equation of the plant. 

(b) For the system you derived in (a), design a state feedback control law to place the closed-loop pole at -2. 

(c) For the system you derived in (a), design an observer that has the observer pole located at -10.

(d) Combine the results in (b) and (c) and derive a dynamic compensator for the system. What is the control transfer function

(e) Consider the control system in Fig.5 with the C(s) you derived in (d). Roughly sketch the root locus of the system as 0 ＜K＜ㆀ. Where does the root locus intersect with the imaginary axis and what is the corresponding K?

1.請就你預計進行的論文研究,設計一個研究計畫,簡要的說明(1)研究主題; (2)相關理論框架和分析取徑(或是你想回應的人類學識題與既存研究);(3) 田野地點、對象、與資料收集方式；(4)可能的研究限制和倫理考量·請注意, 你的研究設計必須要能對應你的問題意識,而且必須具體可行。

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