Root Locus Method

Root Locus Method

Root Locus Method

Root Locus Method

Roots of the characteristic equation Depends on K c (tuning) of the loop.

1- This control loop will never go unstable. 2- When Kc=0, the root loci originates from The OLTF poles:-1/3, and The number of root loci/branches=number Of OLTF poles=2 4- As Kc increases, the root loci approaches infinity

1- This control loop can go unstable. 2- When Kc=0, the root loci originates from The OLTF poles:-1/3, -1, The number of root loci/branches=number Of OLTF poles=3 4- As Kc increases, the root loci approaches infinity

1- This control loop can never go unstable. As Kc increases the root loci move away from I-axis, and D- mode adds a lead to the loop makes it more stable. Addition of lag reduces stability 2- When Kc=0, the root loci originates from the OLTF poles:-1, -1/3 3- The number of root loci/branches=number Of OLTF poles=2 4- As Kc increases, one rout locus approaches – infinity and the other -5, the zero of the OLTF

The rout locus must satisfy the MAGNITUDE and the ANGLE conditions

The rout locus must satisfy the MAGNITUDE and the ANGLE conditions

MAGNITUDE CONDITION ANGLE CONDITION

MAGNITUDE CONDITION ANGLE CONDITION

Example:

Matlab comands: rlocus Evans root locus Syntax rlocus(sys) rlocus(sys,k) rlocus(sys1,sys2,...) [r,k] = rlocus(sys) r = rlocus(sys,k)

Matlab comands: Find and plot the root-locus of the following system. h = tf([2 5 1],[1 2 3]); Rlocus(h, k)

Frequency Response Technique Process Identification: A- Step Test Open-Loop Response B- Frequency Response.

Frequency Response Technique B- Frequency Response.

Frequency Response Technique Recording from sinusoidal testing

Frequency Response Technique Mathematical Interpretation:

Frequency Response Technique Mathematical Interpretation (Continued): Amplitude of the response radian degrees

Frequency Response Technique Mathematical Interpretation (Continued): Amplitude of the response Amplitude Ratio Magnitude Ratio

Frequency Response Technique Mathematical Interpretation (Continued): All these terms (AR, MR, and Phase angle) are functions of Frequency response is the study of how AR(MR) and phase angle of different components change as frequency changes. Methods of Generating Frequency Response: A- Experimental Method B- Transforming the OLTF after a sinusoidal input

Frequency Response Technique Methods of Generating Frequency Response: B- Transforming the OLTF after a sinusoidal input

Frequency Response Technique Methods of Generating Frequency Response: B- Transforming the OLTF after a sinusoidal input

Frequency Response Technique Methods of Generating Frequency Response: B- Transforming the OLTF after a sinusoidal input

Frequency Response Technique Methods of Generating Frequency Response: B- Transforming the OLTF after a sinusoidal input Long time

Frequency Response Technique Methods of Generating Frequency Response: B- Transforming the OLTF after a sinusoidal input

Frequency Response Technique Methods of Generating Frequency Response: B- Transforming the OLTF after a sinusoidal input

Frequency Response Technique Example:

Frequency Response Technique Example:

Frequency Response Technique Example:

Frequency Response Technique Generalization

Frequency Response Technique Generalization

Frequency Response Technique 1- Bode Plots, 2-Nyquist Plots, and 3- Nichols Plots 1- Bode Plots

Frequency Response Technique 1- Bode Plots

Frequency Response Technique 1- Bode Plots

Frequency Response Technique 1- Bode Plots

Frequency Response Technique 1- Bode Plots

Frequency Response Technique 1- Bode Plots

Bode Plots

Frequency Response Technique 1- Bode Plots

Frequency Response Technique 1- Bode Plots EXAMPLE:

Frequency Response Technique 1- Bode Plots EXAMPLE:

Frequency Response Technique 1- Bode Plots EXAMPLE:

Frequency Response Technique 1- Bode Plots Frequency Response Stability Criterion

Frequency Response Technique 1- Bode Plots Frequency Response Stability Criterion

Frequency Response Technique 1- Bode Plots Frequency Response Stability Criterion

Frequency Response Technique 1- Bode Plots Frequency Response Stability Criterion

Frequency Response Technique 1- Bode Plots Frequency Response Stability Criterion

Frequency Response Technique 1- Bode Plots Frequency Response Stability Criterion EXAMPLE:

Frequency Response Technique 1- Bode Plots Frequency Response Stability Criterion EXAMPLE: Without dead time With dead time ω u =0.160 rad/s It is easier for the process with dead time to go unstable

Frequency Response Technique 1- Bode Plots Frequency Response Stability Criterion EXAMPLE: Without dead time With dead time ω u =0.160 rad/s It is easier for the process with dead time to go unstable

MATLAB CONTROL TOOL BOX

Bode(num, den)