Stability

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Most Slides from the Routh-Hurwitz Criterion by Brian Douglas and Control by Prof. Richard Hill

Stability of Open Loop System

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Stability of Open Loop System

• When a pole is negative
• This root exists in the left half plane
• Transfer function will ultimately die out
• The system will eventually be at rest (stable)
• When a pole is positive
• This root exists in the right half plane
• Transfer function will blow up into infinity
• The system is unstable

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• Transfer function of multiple poles
• The last one blows up to infinity to make the whole transfer function unstable
• Conclusion: a single root in the right half plane makes the whole system unstable

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Routh-Hurwitz Criterion

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Routh-Hurwitz Criterion

• Calculating the roots of the system for larger than the second-order polynomial becomes time-consuming and possibly even impossible in a closed-form

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• How can we determine the stability of a higher order polynomial without solving for the roots directly?
• The great thing about the Routh-Hurwitz criterion is that you do not have to solve for the roots of the characteristic equation

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• If all of the signs are not the same, the system is unstable
• If you build up a transfer function with a series of poles, then the only way to get a negative coefficient is to have at least one pole exists in right-half plane

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Routh-Hurwitz Criterion

• However, we cannot claim that all positive coefficients are still either stable or unstable

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Normal Case (1/2)

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Normal Case (2/2)

• Determine the number of roots in RHP by counting the number of sign changes
• We can determine the number of roots in the right-half plane by looking at this first column
• It changes sign twice which means that there are two roots in the right half plane

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Special Case 1 (1/2)

• A zero in a row with at least one non-zero appearing later in that same row
• If you are attempting to access stability of the system, you do not need to complete the rest of the table at this point
• The system is always unstable because completing Routh array will always result in a sign change of the first column

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Special Case 1 (2/2)

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Special Case 2 (1/2)

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Special Case 2 (2/2)

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Stability with State Space Representation

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Stability with State Space Representation

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Stability with State Space Representation

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Stability of Closed Loop System

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Root Locus (Stability in Time)

• We are interested in the stability of a closed loop system from an open loop system.

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• The closed-loop system is

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• A pole exists when the characteristic polynomial in the denominator becomes zero.

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Root Locus (Stability in Time)

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Relative Stability (Stability in Frequency)

• Suppose the Bode plot of the open-loop transfer function is given.

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• Question:
• tell the stability of a closed-loop system from the open-loop frequency response

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Relative Stability (Stability in Frequency)

• At 180^o of phase lag of the loop, the reference and feedback signal are added.
• If the magnitude of the loop is greater than 1 the error grows exponentially (unstable)

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Relative Stability

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Relative Stability

• In order to be stable, both gain and phase margin must be positive

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• Gain and phase margins tell how stable the system would be in closed-loop
• These quantities can be read from the open-loop data

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Relative Stability (Stability in Frequency)

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Relative Stability in MATLAB

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Relative Stability in MATLAB

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• Is stable the closed-loop system with a unity negative feedback?

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Gain Margin

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Gain Margin

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Phase Margin

• Add more delay

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Phase Margin

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Phase Margin

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Stability in Nyquist Plot

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Stability in Nyquist Plot

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Stability in Nyquist Plot

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Nyquist Stability in MATLAB

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Nyquist Stability in MATLAB

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Nyquist Stability in MATLAB

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Nyquist Stability in MATLAB

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Nyquist Stability in MATLAB

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