Root locus

reference: slides, and awesome calculator

final value theorem

If a system is stable and has a final constant value, then one can find steady state value without solving for system’s response. Formally:
$\lim_{t \to \infty} x(t) = \lim_{s \to 0} sX(s)$

sketching root locus

Rule	Description
Number of Branches	Number of closed-loop poles, or the number of finite open-loop poles = number of finite open-loop zeros
Symmetry	About the real axis
Start and End Points	Starts at poles of open loop transfer function and ends at finite and infinite open loop zeros
Behaviour at $\infty$	Real axis: $\sigma_a = \frac{\Sigma{\text{finite poles}} - \Sigma{\text{finite zeros}}}{\text{\# finite poles - \# finite zeros}}$ Angle: $\theta_a = \frac{(2k+1)\pi}{\text{\# finite poles - \# finite zeros}}$ where $k = 0, \pm 1, \pm 2, \pm 3$
Breakaway/Break-in Points	Located at roots where $\frac{d[G(s)H(s)]}{ds} = 0$

reference: slides, and awesome calculator

excerpt from real-time system slides

final value theorem

If a system is stable and has a final constant value, then one can find steady state value without solving for system’s response. Formally:
$\lim_{t \to \infty} x(t) = \lim_{s \to 0} sX(s)$

sketching root locus

Rule	Description
Number of Branches	Number of closed-loop poles, or the number of finite open-loop poles = number of finite open-loop zeros
Symmetry	About the real axis
Start and End Points	Starts at poles of open loop transfer function and ends at finite and infinite open loop zeros
Behaviour at $\infty$	Real axis: $\sigma_a = \frac{\Sigma{\text{finite poles}} - \Sigma{\text{finite zeros}}}{\text{\# finite poles - \# finite zeros}}$ Angle: $\theta_a = \frac{(2k+1)\pi}{\text{\# finite poles - \# finite zeros}}$ where $k = 0, \pm 1, \pm 2, \pm 3$
Breakaway/Break-in Points	Located at roots where $\frac{d[G(s)H(s)]}{ds} = 0$

Root locus

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sketching root locus

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