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The Bode straight line asymptote plot of the magnitude of transfer function, G(s). is shown below: 10 10 10 The Bode straight line asymptote plots of some compensated open loop transfer functions, D(s)G(s), are shown on the page Identify the type of compensator used in each case. The possible compensators for D(s) are: (a) PD (Proportional plus Derivative) (b) PI (Proportional plus Integral) (c) PID (Proportional plus Integral plus Derivative) (d) Lead (e) Lag (f) Lead-Lag The thicker and darker lines in each figure are the plots of whereas the original plot of IG(jw)l is shown with thinner and lighter lines as reference Match the type of compensator with each Bode plot on page using the table on the next page (or reproduction thereof in your homework). giving brief comment on your reasoning in each case Bode Plot Comments Reasons (b) (f) DG), (DG), G G 10 (i) (ii) 10 10° 10¹ 10 10° Frequency (rad/sec) Frequency (rad/sec) DG), (DG|, G G 10 10° (iii) (iv) 10° Frequency (rad/sec) 10,18 10 Frequency (rad/sec) (DG|. |DG|, 10 G G 10° (v) (vi) Frequency (rad/sec) 10,10 Frequency (rad/sec) Consider the plant x = Ax+ Bu y Cx where -2 B and C=[1-2| (a) What is the transfer function G(s) from to y? (b) Compute the state feedback gain vector K such that when Kx the closed loop poles of the system are at and -3. (c) Design an estimator with poles at -8 and -9, That is. what is the estimator gain vector L that will piace the estimator poles at -8 and -97 (d) What is the transfer function D(s) of the state-space based compensator that includes both the estimator and state-feedback control law designed in the previous two parts? (e) How should reference input r be introduced such that the estimator error dynamics are indepen dent of and such that the output becomes equal to constant reference input in steady state (i.e., no steady state error)? Provide all the necessary equations, as well as the particular values of all matrices, gains, and constants needed. (f) Extra credit (worth up to points for 5138 students, and up to points for 4138 students). Given the state-space based controller designed in all of previous parts of this problem, including how an external reference input i s introduced, what is the overall transfer function from to y? Hint: The following block matrix identity may be useful. When A and B exist, then A 1BB B-¹ where A. B. D. and 0 are each square matrices of equal size.

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