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Stability of Linear Systems

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Design a proportional and integral controller, G= k_p ((a+1/T_1)/s) for the plant,

P = 1/((X+1)(S+3)) to have dominant poled with (see attachment)

i) Use rough sketches of the root locus to determine the location of the controller zero (i.e. find T_1)

ii) Calculate the required controller gain Kp, (An accurate root locus is not required)

Question 1: Nyquist Analysis

Consider the following open loop system: L= (s+3)/(s^2+2)

a) Make a rough sketch of the Bode plot
b) Sketch the full Nyquist diagram
c) Determine if the closed loop system is stable or not
d) Sketch the positive gain root locus of the system with open loop transfer function, kL(s) and comment on your answer in corresponding to A - 1

Plot the root locus for the following system: L= k (s^2+2s+2)/(s(s+1)(s+2)) as the feedback gain is varied from -infinity to positive infinity. 1/L(s) has turning points at s= -1.49 and s=-0.51

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Stability and subspaces of a linear system

Please use a step-by-step method to solve the two following exercises:

Exercise 1:

Consider the linear system x' = Ax, where A is a 2 x 2 matrix with lamda in the diagonal as follows;

A = [lamda, -2]
[1 , lamda],

and lamda is real. Determine if the system has a saddle, node, focus, or center at the origin and determine the stability of each node or focus.

Exercise 2:

Consider the linear system x' = Ax, where A is a 3x3 matrix as follows:

A = [ 1, 0 , 0 ]
[ 0, -1, -1 ]
[ 0 , 1, -1 ]

a) Find the general solution of the linear system.

b) Find the stable, unstable, and the center subspaces E^s, E^u and E^c.

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