Lecture Outline Systeem- en Regeltechniek II Previous lecture: Bode plots, non-minimum-phase systems. Lecture 8 – Frequency Domain Design Robert Babuˇ ska Today: • Bode’s gain-phase relation. Delft Center for Systems and Control Faculty of Mechanical Engineering • Neutral stability. Delft University of Technology • Gain and phase margin, performance specs. The Netherlands • Controller design. 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 Domain Methods Bode’s Gain-Phase Relation For any stable minimum-phase system, phase ∠ G ( jω ) is uniquely Bode Diagrams related to magnitude | G ( jω ) | : 0 -10 Phase (deg); Magnitude (dB) -20 Bode plot -30 transfer -40 -5 0 � ∞ -1 0 0 function -1 5 0 ∠ G ( jω 0 ) = 1 dM 10 -1 1 0 0 1 0 1 Frequency (rad/sec) frequency du W ( u ) du π ∞ response Nyquist plot data where M = ln | G ( jω ) | , u = ln ω/ω 0 , W ( u ) = ctanh | u/ 2 | . (experiment) For a constant slope, we can approximate the above by: ∠ G ( jω 0 ) ≃ nπ • Now we now how to sketch and plot Bode diagrams. 2 • The next step is analysis of system properties and design. where n is the slope ( 1 for 20 dB /dec, 2 for 40 dB /dec, etc). Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 3 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 4
Consequence of the Gain-Phase Relation Bode Plot: Closed-Loop Stability For open loop stable minimum-phase system, it is sometimes suf- ficient to look at the magnitude only. R s ( ) E s ( ) U s ( ) Y s ( ) + C s ( ) G s ( ) - This property can be used to derive a simple design rule for control. But first, we must be able determine, from the Bode plot, whether the system is stable! L ( s ) = G ( s ) C ( s ) Can we infer closed-loop stability from a Bode plot of the loop transfer function L ( 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 Proportional Controller: Loop Transfer Proportional Controller: Loop Transfer K > 1 L ( s ) = Y ( s ) E ( s ) = K G ( s ) e For the Bode plot, the following holds: d K < 1 u t i n g a ∠ G ( jω ) = ∠ ( KG ( jω )) ( K is a real number) M -2 0 2 4 10 10 10 10 (multiplication by a gain) | G ( jω ) | = | K | · | G ( jω ) | shift the magnitude response of G ( jω ) by 20 log( K ) phase does not change | G ( jω ) | dB = | K | dB + | G ( jω ) | dB Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 7 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 8
Example: DC Motor DC Motor: Open-Loop Bode Plot G ( s ) = θ ( s ) 2 V ( s ) = Transfer function: s ( s + 10)( s + 2) G ( s ) = θ ( s ) K t 50 V ( s ) = s [( Ls + R )( Js + b ) + K 2 t ] magnitude (dB) 0 −50 = 0 . 01 kg · m 2 inertia of the rotor J −100 −150 damping (friction) b = 0 . 1 Nms −2 −1 0 1 2 10 10 10 10 10 0 back emf K t = 0 . 01 Nm / A phase (deg) −100 resistance R = 1 Ω −200 inductance L = 0 . 5 H −300 −2 −1 0 1 2 10 10 10 10 10 frequency (rad/sec) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 9 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 10 Influence of Proportional Gain Let’s See Whether Root Locus Helps . . . 15 K · 2 L ( s ) = KG ( s ) = s ( s + 10)( s + 2) 10 5 Imag Axis 0 Use Matlab: sisotool(’bode’,G) −5 −10 OK, the magnitude moves up and down with the gain and the −15 −12 −10 −8 −6 −4 −2 0 2 Real Axis phase does not change . . . ∠ G ( s ) = − 180 ◦ Basic properties of RL: and | KG ( s ) | = 1 . . . but, is there anything on the Bode plot that would hint on the stability of the closed loop? Neutral stability: | K max G ( jω ) | = 0 dB and ∠ G ( jω ) = − 180 ◦ Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 11 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 12
Back to the Bode Plot Point of Neutral Stability 100 K max |G(j ω )| System is stable if: | KG ( jω ) | < 0 dB at ∠ G ( jω ) = − 180 ◦ magnitude (dB) 0 100 magnitude (dB) −100 0 −100 −200 −1 0 1 2 3 10 10 10 10 10 −200 −1 0 1 2 3 10 10 10 10 10 0 0 phase (deg) −100 phase (deg) −100 −200 −200 −300 −1 0 1 2 3 −300 10 10 10 10 10 −1 0 1 2 3 10 10 10 10 10 frequency (rad/sec) 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 Crossover Frequency and Stability Margins Crossover Frequency and Stability Margins 50 • The crossover frequency ω c is the frequency for which the loop 0 Gain margin TF has gain 0 dB. -50 -100 • The gain margin (GM) is the factor (or amount dB) by which -150 the loop gain can be raised before instability occurs. 0 -100 • The phase margin (PM) is the amount (in degrees) by which Phase margin the phase exceeds 180 ◦ at ω c . -200 -300 -2 -1 0 1 2 10 10 10 10 10 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 15 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 16
Importance of Stability Margins Robustness: Example The margins tell us how far the closed-loop system is from the Suppose our model is: point of neutral stability. This indicates the robustness w.r.t. un- 10 ˆ certainty in the plant model: L ( s ) = s 2 + 0 . 4 s + 1 • Gain margin: by what factor the total process gain can increase. while the true plant is: • Phase margin: by how much the phase can decrease. 10 L ( s ) = ( s 2 + 0 . 4 s + 1)(0 . 1 s + 1) and performance: • Phase margin: related to closed loop damping (overshoot). Relatively small mismatch in terms of step-response behavior, ma- • Crossover frequency: related to response speed jor difference in terms of closed-loop stability! (bandwidth). 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 Plot of Model and True Plant Closed-Loop Bandwidth Bode Diagram Bandwidth = frequency up to which the input is “well reproduced” 50 at the output of the closed-loop system. 0 Defined as frequency ω bw at which the magnitude has an Magnitude (dB) −50 attenuation of 0.707 (3dB) – corresponds to 0.5 power gain. −100 0 −150 -3 dB 0 Magnitude −90 System: G1 Phase (deg) Phase Margin (deg): 7.59 Delay Margin (sec): 0.0401 At frequency (rad/sec): 3.3 Closed Loop Stable? Yes −180 System: G2 Phase Margin (deg): −10.1 Delay Margin (sec): 1.89 At frequency (rad/sec): 3.23 Closed Loop Stable? No −270 −1 0 1 2 3 10 10 10 10 10 Frequency (rad/sec) Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 19 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 20
Bandwidth and Crossover Frequency Phase Margin and Overshoot The larger PM, the larger damping (less overshoot): Typically: ω c ≤ ω bw ≤ 2 ω c ζ ≈ PM 100 this holds up to PM = 60 ◦ The required speed of response (e.g., the settling time or rise time) can be expressed in terms of ω c . Recall: See the Franklin et al. for a graphical relationship between the overshoot and PM (page 357). t r = 1 . 8 /ω n (for second-order dominant response). Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 21 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 22 Recall Specs for Second-Order Systems More Complex Plants 18 . • System unstable for small K and stable for large K , e.g.,: = t ω r n L ( s ) = K ( s + 2) π s 2 − 1 = t p ω − ζ 2 1 n • Conditionally stable systems (unstable for small and large K , 4 6 . = ± t for 1% stable for some intermediate values), e.g.,: s ζω n πζ K ( s + 2) − L ( s ) = − ζ 2 ( s + 10) 2 ( s + 1)( s − 1) = 1 M e p In the sequel, we consider systems with no poles in RHP. Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 23 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 24
The Basic Idea Bode Plots: Homework Assignments • Read Sections 6.1 through 6.6, except for the Nyquist criterion. • Adjust the proportional gain to get the required crossover fre- quency and/or steady-state tracking error. • Work out examples in these sections and verify the results by using Matlab. • If needed, use the derivative action to add phase in the neigh- borhood of ω c in order to increase the phase margin. • Reproduce the derivation of the frequency response as given on the overhead sheets. • If needed, use the integral action to increase the gain at low fre- quencies in order to guarantee the required steady-state tracking error. • Work out a selection of problems 6.3 through 6.9 and verify your results by using Matlab. Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 25 Robert Babuˇ ska Delft Center for Systems and Control, TU Delft 26
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