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# Stable Control Systems

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1. a. Draw the Bode diagram for the attached system.
b. Is the system stable?

2. Robots can be used in manufacturing and assembly operations where accurate, fast and versatile manipulation is required. The open-loop transfer function of a direct-drive arm may be approximated by:

G H (s) = K(s+2) / s(s+5)(s^2+2s+5)

a. Determine the value of gain K when the system oscillates.
b. Calculate the roots of the closed-loop system for the K determined in part a)..

3. For the system attached, determine the range of K for which the system is stable.

https://brainmass.com/engineering/robotics/stable-control-systems-505671

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1. a. To derive the Bode plots for amplitude and phase we need to find the poles and zero's of the closed loop transfer function.

b. In order to check the stability of the system we need to examine the characteristic equation of the denominator of the closed loop transfer function, namely the denominator of:

T(s) = 20,000(0.1s + 1) / (s + 10.258)(s^2 + 14.472s + 1950)

This as mentioned in part a. has poles at s = -10.258 and s = -7.38 +/- 43.58j

As both poles lie in the left hand plane of the s plane (negative real parts for each of the poles) we can say this system is stable.

2. a. The system oscillates in the limit that K >= 28.12 or when K = 28.12.
b. The roots occur when s = +/- j2.755

3. We can manipulate the inner loop as this is a negative feedback loop and can therefore be represented by a single block, where the single block transfer function is given by (1/s) / (1 + 3(1/s)) = 1/(s+3).

The system is stable only if the coefficients in the first column of the Routh Array are the same sign. Clearly the first 2 coefficients are >0 so requiring K > - 0.443.

Also from the last coefficient in the column there is a requirement that this coefficient is >0 so meaning K>0. This is an over riding requirement so meaning for the system to be stable we should have K>0.

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