EE 3CL4, §5 1 / 65 Tim Davidson Preliminary examples Principles EE3CL4: Sketching the Root Locus, Introduction to Linear Control Systems Steps 1–4 Steps 1 and 2 Section 5: Root Locus Procedure Review of Principles Review of Steps 1, 2 Step 3 Step 4 Compensator Tim Davidson design for VTOL aircraft Sketching the McMaster University Root Locus, Steps 5–7 Review of Steps 1–4 Winter 2020 Step 5 (approx’d) Step 6 Step 7 Example Parameter Design “Negative” Root Locus
EE 3CL4, §5 2 / 65 Outline Tim Davidson Preliminary examples 1 Preliminary Principles 2 examples Sketching the Root Locus, Steps 1–4 3 Principles Steps 1 and 2 Sketching the Root Locus, Review of Principles Steps 1–4 Steps 1 and 2 Review of Steps 1, 2 Review of Principles Review of Steps 1, 2 Step 3 Step 3 Step 4 Step 4 Compensator Compensator design for VTOL aircraft design for 4 VTOL aircraft Sketching the Root Locus, Steps 5–7 5 Sketching the Root Locus, Review of Steps 1–4 Steps 5–7 Step 5 (approx’d) Review of Steps 1–4 Step 5 (approx’d) Step 6 Step 6 Step 7 Step 7 Example Parameter Example Design Parameter Design 6 “Negative” Root Locus 7 “Negative” Root Locus
EE 3CL4, §5 4 / 65 Simple example Tim Davidson Preliminary examples Principles Sketching the Root Locus, Steps 1–4 Steps 1 and 2 Review of Principles Review of Steps 1, 2 Step 3 Open loop transfer function: K amp G ( s ) Step 4 Compensator K amp G ( s ) design for Closed loop transfer function T ( s ) = VTOL aircraft 1 + K amp G ( s ) Sketching the Char. eqn: s 2 + 2 s + K amp = 0 Root Locus, Steps 5–7 Review of Steps 1–4 � Closed-loop poles: s 1 , s 2 = − 1 ± 1 − K amp Step 5 (approx’d) Step 6 Step 7 What paths do these closed-loop poles take as K amp goes Example from 0 to + ∞ ? Parameter Design “Negative” Root Locus
EE 3CL4, §5 5 / 65 Simple example Tim Davidson Preliminary examples Principles Sketching the Root Locus, Steps 1–4 Steps 1 and 2 Review of Principles Review of Steps 1, 2 Step 3 Step 4 Compensator design for VTOL aircraft Sketching the Root Locus, Steps 5–7 Review of Steps 1–4 Step 5 (approx’d) Step 6 Step 7 Example Parameter Design “Negative” Root Locus
EE 3CL4, §5 6 / 65 Another example Tim Davidson Preliminary examples Principles Sketching the Root Locus, Steps 1–4 Steps 1 and 2 Review of Principles Review of Steps 1, 2 K amp G ( s ) Closed loop transfer function T ( s ) = Step 3 1 + K amp G ( s ) Step 4 Compensator Consider K amp to be fixed design for VTOL aircraft Char. eqn: s 2 + as + K amp = 0 Sketching the Root Locus, Steps 5–7 � � � a 2 − 4 K amp Closed-loop poles: s 1 , s 2 = − a ± / 2 Review of Steps 1–4 Step 5 (approx’d) Step 6 What paths do these closed-loop poles take as a goes from Step 7 Example 0 to + ∞ ? Parameter Design “Negative” Root Locus
EE 3CL4, §5 7 / 65 Another example Tim Davidson Preliminary examples Principles Sketching the Root Locus, Steps 1–4 Steps 1 and 2 Review of Principles Review of Steps 1, 2 Step 3 Step 4 Compensator design for VTOL aircraft Sketching the Root Locus, Steps 5–7 Review of Steps 1–4 Step 5 (approx’d) Step 6 Step 7 Example Parameter Design “Negative” Root Locus
EE 3CL4, §5 8 / 65 What to do in the general case? Tim Davidson Preliminary examples Principles Sketching the Root Locus, Steps 1–4 In the previous examples we exploited the simple Steps 1 and 2 Review of Principles factorization of second order polynomials Review of Steps 1, 2 Step 3 Step 4 Compensator However, it would be very useful to be able to draw the design for paths that the closed-loop poles take as K amp increases for VTOL aircraft Sketching the more general open-loop systems Root Locus, Steps 5–7 Review of Steps 1–4 Step 5 (approx’d) Step 6 Step 7 Example Parameter Design “Negative” Root Locus
EE 3CL4, §5 10 / 65 Principles of general procedure Tim Davidson Preliminary examples Principles Sketching the Root Locus, 1 + K amp G ( s ) = p ( s ) K amp G ( s ) Steps 1–4 Closed loop transfer function T ( s ) = q ( s ) Steps 1 and 2 Review of Principles Closed loop poles are solutions to q ( s ) = 0 Review of Steps 1, 2 Step 3 These are also sol’ns to 1 + K amp G ( s ) = 0; i.e., K amp G ( s ) = − 1 + j 0 Step 4 In polar form, | K amp G ( s ) | ∠ K amp G ( s ) = 1 ∠ ( 180 ◦ + ℓ 360 ◦ ) Compensator design for VTOL aircraft Therefore, for an arbitrary point on the complex plane s 0 to be a Sketching the closed-loop pole for a given value of K amp the following equations must Root Locus, be satisfied Steps 5–7 Review of Steps 1–4 ∠ K amp G ( s 0 ) = ( 180 ◦ + ℓ 360 ◦ ) | K amp G ( s 0 ) | = 1 and Step 5 (approx’d) Step 6 Step 7 where ℓ is any integer. (Note: book uses k , but we will use ℓ to avoid Example confusion with K ) Parameter Design We will also keep in mind that R ( s ) and Y ( s ) correspond to real signals. “Negative” Hence, closed-loop poles are either real or occur in complex-conjugate Root Locus pairs
EE 3CL4, §5 11 / 65 In terms of poles and zeros Tim Davidson For s 0 to be a closed-loop pole, we must have Preliminary examples ∠ K amp G ( s 0 ) = ∠ ( 180 ◦ + ℓ 360 ◦ ) | K amp G ( s 0 ) | = 1 and Principles Sketching the Root Locus, � M Write G ( s ) = K G i = 1 ( s + z i ) Steps 1–4 j = 1 ( s + p j ) , which means that the � n Steps 1 and 2 Review of Principles open loop zeros are − z i ’s; open loop poles are − p j ’s Review of Steps 1, 2 Step 3 For s 0 to be a closed-loop pole Step 4 Compensator design for | K amp K G | � M i = 1 | s 0 + z i | VTOL aircraft = 1 � n Sketching the j = 1 | s 0 + p j | Root Locus, Steps 5–7 M n Review of Steps 1–4 ∠ ( s 0 + p j ) = 180 ◦ + ℓ 360 ◦ � � ∠ K amp + ∠ K G + ∠ ( s 0 + z i ) − Step 5 (approx’d) Step 6 i = 1 j = 1 Step 7 Example (From the definition of the factorization of G ( s ) , when M = 0 the Parameter Design terms related to the zeros “disappear” a natural way) “Negative” Root Locus Can we interpret these expressions in a geometric way?
EE 3CL4, §5 12 / 65 Vector difference Tim Davidson • Let u and v be complex numbers. Preliminary examples • Can you describe v − u in geometric terms? Principles • Use the fact that v = u + ( v − u ) . Sketching the Root Locus, Steps 1–4 • That means that v − u is the vector from u to v Steps 1 and 2 Review of Principles Review of Steps 1, 2 Step 3 Step 4 Compensator design for VTOL aircraft Sketching the Root Locus, Steps 5–7 Review of Steps 1–4 Step 5 (approx’d) • v − u = ℓ e j θ . That is, Step 6 Step 7 • | v − u | is the length of the vector from u to v . Example • ∠ ( v − u ) is the angle of the vector from u to v Parameter Design • In our expressions we have terms of the form “Negative” Root Locus s 0 + z i = s 0 − ( − z i ) and s 0 + p j = s 0 − ( − p j )
EE 3CL4, §5 13 / 65 Geometric interpretation Tim Davidson Magnitude criterion: Preliminary | K amp K G | � M examples i = 1 | s 0 + z i | = 1 Principles � n j = 1 | s 0 + p j | Sketching the | K amp K G | � M Root Locus, i = 1 distances from zeros (if any) of G ( s ) to s 0 Steps 1–4 = 1 � n Steps 1 and 2 j = 1 distances from poles of G ( s ) to s 0 Review of Principles Review of Steps 1, 2 Step 3 Phase criterion: Step 4 Compensator M n ∠ ( s 0 + p j ) = 180 ◦ + ℓ 360 ◦ design for � � ∠ K amp + ∠ K G + ∠ ( s 0 + z i ) − VTOL aircraft i = 1 j = 1 Sketching the Root Locus, Steps 5–7 M Review of Steps 1–4 Step 5 (approx’d) � ∠ K amp + ∠ K G + angles from zeros (if any) of G ( s ) to s 0 Step 6 Step 7 i = 1 Example n Parameter � − angles from poles of G ( s ) to s 0 Design “Negative” j = 1 Root Locus = 180 ◦ + ℓ 360 ◦
EE 3CL4, §5 14 / 65 Now for the challenge Tim Davidson Preliminary examples Principles • Can we build on these geometric interpretations of the Sketching the equations in the simple case of amplifier gains to Root Locus, Steps 1–4 develop a broadly applicable approach to control Steps 1 and 2 Review of Principles system design? Review of Steps 1, 2 Step 3 Step 4 • The first step will be to develop a formal procedure for Compensator design for sketching the paths that the closed-loop poles take as a VTOL aircraft design parameter (often an amplifier gain) changes. Sketching the Root Locus, These are called the root loci. Steps 5–7 Review of Steps 1–4 Step 5 (approx’d) • We will develop the formal procedure in a slightly more Step 6 Step 7 general setting than what we have seen so far Example Parameter Design “Negative” Root Locus
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