ME 176Control Systems Engineering

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Mechanical Engineering

Root Locus Technique

Introduction : Root Locus

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Mechanical Engineering

"... a graphical representation of closed loop poles as a system parameter is varied, is a powerful method of analysis and design for stability and transient response."

"...real powere lies in its ability to provide for solutions for systems of order higher than 2."

Refining Sketch: Root Locus Rules:1. Branches equal to closed-loop poles2. Symentrical about real axis. 3. Left of odd number of real-axis finite open-loop poles and/or zeros.4. Begins on finite/infinite poles, ends on finite/infinite zeros of G(s)H(s).5. Approaches asymptotes as the

locus approaches infinity: Refinements: 1. Break-away and break-in: 2. jw-Axis crossing using Routh-Hourwitz criterion.

Department of

Mechanical Engineering

Refining Sketch: Root Locus 3. Angles of Departure and Arrival:

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Mechanical Engineering

Root locus departs at complex open-looppoles and arrives at open-loop zeros at angels given by: assuming points close to such poles and zeros, summing all angles drawn from all poles and zeros will equal(2k+1)180. Zeros - postivivePoles - negative

Refining Sketch: Root Locus 4. Plotting and Calibrating the Root Locus:

Department of

Mechanical Engineering

Evaluate the root locus at a point on the s-plane by first solving if that point yields a summation of angles (zero angles - pole angles) equal to an odd multiple of 180. Then calculate the gain by multiplying the pole lengths drawn to that point divided by product of zero lengths.

Refining Sketch: Root Locus Approximation to 2nd order systems

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Mechanical Engineering

1. Higher-order poles are much farther into the left half of the s-plane than the dominant second order pair of poles. The response from a higher order pole does not appreciably change the transient response expected from the dominant second order pole.

Refining Sketch: Root Locus Approximation to 2nd order systems

Department of

Mechanical Engineering

2. Closed-loop zeros nea the closed loop second-order pole pair are nearly canceled by the close proximity of higher-order closed-loop poles.

Refining Sketch: Root Locus 4. Plotting and Calibrating the Root Locus:

Department of

Mechanical Engineering

Evaluate the root locus at a point on the s-plane by first solving if that point yields a summation of angles (zero angles - pole angles) equal to an odd multiple of 180. Then calculate the gain by multiplying the pole lengths drawn to that point divided by product of zero lengths.

Refining Sketch: Root Locus Example:

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Mechanical Engineering

Do the following: 1. Sketch the root locus .2. Find the imaginary-axis crossing.3. Find the gain, K, at the jw-crossing.4. Find the angles of departure.

Refining Sketch: Root Locus Example:

Department of

Mechanical Engineering

Do the following: 1. Sketch the root locus .2. Find the imaginary-axis crossing.3. Find the gain, K, at the jw-crossing.4. Find the angles of departure.

Refining Sketch: Root Locus Example: Do the following:

1. Sketch the root locus .2. Find the imaginary-axis crossing.3. Find the gain, K, at the jw-crossing.4. Find the angles of departure.

Department of

Mechanical Engineering

Refining Sketch: Root Locus Lab

Find F(s) at point s= -7+j9 Find G(s) at point s=-3+j0

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Mechanical Engineering

Refining Sketch: Root Locus Lab a. Sketch the root locus.b. Find Imaginary-axis crossing.c. Find K at jw-axis crossing.d. Find the break-in point.e. Find the point where the locus crosses the 0.5 damping ration line.f. Find the gain at the point where the locus crosses the 0.5 damping ration line.g. Find the range of gain, K, for which the system is stable.

Department of

Mechanical Engineering

Refining Sketch: Root Locus Lab

Department of

Mechanical Engineering

Sketching Rules: 1. Branches: 22. Symmetric about real axis3. Real-axis segment: Left of -2 pole4. Start at poles, ends at zeros5. Behavior at infinity:

Sigma = 0Theta = infinity

Use Matlab: 6. Real-axis breakaway points7. jw - crossing

Refining Sketch: Root Locus Lab

Department of

Mechanical Engineering