Lecture Outline Regeltechniek Previous lecture: Root locus, - PowerPoint PPT Presentation

Lecture Outline Regeltechniek Previous lecture: Root locus, frequency response derivation. Lecture 7 – Frequency Response, Bode Plots Robert Babuˇ ska Today: • Handout for the remaining computer sessions. Delft Center for Systems and Control Faculty of Mechanical Engineering • Bode plots. Delft University of Technology • Non-minimum-phase systems. The Netherlands • System type in Bode plots. e-mail: r.babuska@dcsc.tudelft.nl www.dcsc.tudelft.nl/ ˜ babuska tel: 015-27 85117 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 1 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 2 Frequency Response Magnitude and Phase Periodic input: Magnitude: � { Re[ G ( jω )] } 2 + { Im[ G ( jω )] } 2 u ( t ) = M sin ωt | G ( jω ) | = Steady-state output: Phase: � Im[ G ( jω )] � ∠ G ( jω ) = tan − 1 � � y ( t ) = | G ( jω ) | · M sin ωt + ∠ G ( jω ) Re[ G ( jω )] Both the magnitude and phase are generally functions of ω ! | G ( jω ) | . . . magnitude (gain) Fully describe G ( s ) , can also can be measured experimentally. ∠ G ( jω ) . . . phase Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 3 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 4

Magnitude and Phase: Example Magnitude and Phase Plot 2 U(s) Y(s) G(s) 1.5 G(s) Magnitude 1 0.5 0 0 5 10 15 20 25 30 ω [rad/s] 0 −20 Phase −40 −60 −80 −100 0 5 10 15 20 25 30 ω [rad/s] Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 5 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 6 Bode Plot Logarithmic Scale: Decibels Plotting on a linear scale is not so useful – plots are hard to inter- The 10-base logarithm of a power gain is called a Bell (B): pret and cannot be easily drawn by hand. P out x B = log 10 P in If logarithmic scales are introduced, drawing becomes easier. This unit appeared too large ( x was usually small) the decibel (dB) was introduced: P out x dB = 10 log 10 Such a logarithmic plot is called the Bode plot: P in – frequency is plotted on a logarithmic scale (log 10 ) – amplitude is plotted using logarithmic units (decibels) – phase is plotted on a linear scale (degrees) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 7 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 8

Logarithmic Scale: Decibels Decomposing Transfer Functions Furthermore, power is proportional to the square of voltage: Decompose a transfer function into: αV 2 αV out out x dB = 10 log 10 = 20 log 10 G ( s ) = G 1 ( s ) G 2 ( s ) · · · G n ( s ) βV 2 βV in in Letting s = jω we have: Therefore for a gain K the corresponding value in dB is: G ( jω ) = | G ( jω ) | e j ∠ G ( jω ) x = 20 log 10 ( K ) with | G ( jω ) | = | G 1 ( jω ) | · · · | G n ( jω ) | That is: x is the value of K expressed in dB . ∠ G ( jω ) = ∠ G 1 ( jω ) + · · · + ∠ G n ( jω ) In words: magnitudes multiply, phases add. Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 9 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 10 Expressing Magnitude in Decibels Bode Form of Transfer Function ω n,i ) 2 + 2 ζ ( s i [( s � i ( τ i s + 1) · � ω n,i ) + 1] G ( s ) = K · s k · dB ( | G ( jω ) | ) = 20 log 10 | G ( jω ) | j [( s ω n,j ) 2 + 2 ζ ( s � j ( τ j s + 1) · � ω n,j ) + 1] which implies: Example: 2000( s + 0 . 5) 2( s/ 0 . 5 + 1) dB ( | G ( jω ) | ) = dB ( | G 1 ( jω ) | ) + · · · + dB ( | G n ( jω ) | ) G ( s ) = s ( s + 10)( s + 50) = s ( s/ 10 + 1)( s/ 50 + 1) In words: in dB, we can add magnitudes too. 2( jω/ 0 . 5 + 1) G ( jω ) = jω ( jω/ 10 + 1)( jω/ 50 + 1) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 11 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 12

Bode Plots: Basic Transfer Functions Bode Plots: G ( jω ) = K, K > 0 Any transfer function G ( s ) can be represented as a product of (some of) the following terms: 20 log 10 K 0 • K • ( s ) ± 1 Frequency�(rad/sec) • ( τs + 1) ± 1 ω n ) 2 + 2 ζ ( s • [( s ω n ) + 1] ± 1 ∠ K 0 We can draw the magnitudes and phases of these basic terms and add them up graphically. Frequency�(rad/sec) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 13 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 14 Bode Plots: G ( jω ) = jω Bode Plots: G ( jω ) = 1 jω 20 20 20 log 10 | jω | 0 20 log 10 | 1 / ( jω ) | 0 -20 -20 0 . 1 1 10 0 . 1 1 10 log( ω ) log( ω ) ∠ ( jω ) 90 ∠ 1 / ( jω ) -90 0 . 1 1 10 0 . 1 1 10 log( ω ) log( ω ) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 15 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 16

Bode Plots: G ( jω ) = jωτ + 1 Bode Plots: G ( jω ) = 1 jωτ +1 30 0 20 -10 20 log 10 | ( jωτ + 1) | 20 log 10 | 1 / ( jωτ + 1) | 10 -20 0 0 . 1 /τ 1 /τ 10 /τ 0 . 1 /τ 1 /τ 10 /τ log( ω ) log( ω ) 90 0 60 -30 ∠ ( jωτ + 1) ∠ 1 / ( jωτ + 1) 30 -60 0 -90 0 . 1 /τ 1 /τ 10 /τ 0 . 1 /τ 1 /τ 10 /τ log( ω ) log( ω ) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 17 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 18 Bode Plots: G ( jω ) = [( s ω n ) 2 + 2 ζ ( s Non-Minimum-Phase Systems ω n ) + 1] − 1 A system with zeros z i such that Re { z i } > 0 is called non-minimum ζ = 0 . 1 .. 1 steps of 0 . 1 20 phase system. 20 log 10 | G ( jω ) | 0 Example: -20 G 1 ( s ) = s + 1 G 2 ( s ) = − s + 1 -40 ω n 0 . 1 ω n 10 ω n s + 10 s + 10 log( ω ) 0 ζ = 0 . 1 .. 1 steps of 0 . 1 System G 2 ( s ) undergoes a larger net change in phase than G 1 ( s ) , ∠ G ( jω ) i.e., G 2 ( s ) is called non-minimum phase. -90 -180 ω n 0 . 1 ω n 10 ω n log( ω ) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 19 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 20

MP vs. NMP System: Bode Plots Type of System 0 From the bode diagram of the open loop system G ( s ) it is possi- Magnitude [dB] −5 ble to see of what type the loop transfer L ( s ) is, if proportional −10 controller is used. −15 −20 −1 0 1 2 3 10 10 10 10 10 • If the slope of the magnitude on the extreme left of the Bode Frequency [rad/s] plot is 0 ⇒ no pure integrator ⇒ Type 0. 100 50 Phase [deg] 0 • If the slope of the magnitude on the extreme left of the Bode −50 plot is − 20 n dB/decade ⇒ n integrators ⇒ Type n . −100 −150 −200 −1 0 1 2 3 10 10 10 10 10 Frequency [rad/s] Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 21 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 22 Type 0 System: Example Type 1 System: Example 1 2 20 10 40 10 1 Magnitude [dB] Magnitude [dB] 20 10 0 Magnitude Magnitude 0 10 0 0 10 −1 −20 10 −1 −20 10 −2 −40 10 −2 −3 −40 10 −60 10 −1 −1 0 0 1 1 −1 −1 0 0 1 1 10 10 10 10 10 10 10 10 10 10 10 10 Frequency [rad/s] Frequency [rad/s] 0 −50 −100 −50 Phase [deg] Phase [deg] −150 −100 −200 −150 −250 −200 −300 −1 0 1 −1 0 1 10 10 10 10 10 10 Frequency [rad/s] Frequency [rad/s] Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 23 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 24