# CDS 101/110a: Lecture 8-1 Frequency Domain Design

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1 CDS 11/11a: Lecture 8-1 Frequency Domain Design Richard M. Murray 17 November 28 Goals: Describe canonical control design problem and standard performance measures Show how to use loop shaping to achieve a performance specification Work through a detailed example of a control design problem Reading: Åström and Murray, Feedback Systems, Ch 11 Advanced: Lewis, Chapter 12 CDS 21: DFT, Chapters 4 and 6

2 Frequency Domain Performance Specifications Specify bounds on the loop transfer function to guarantee desired performance d + e C(s) + u P(s) y -1 Steady state error: SS Tracking PC > 1/X BW zero frequency ( DC ) gain Bandwidth: assuming ~9 phase margin sets crossover freq Tracking: X% error up to frequency ω t determines gain bound (1 + PC > 1/X) CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 2

3 Relative Stability Relative stability: how stable is system to disturbances at certain frequencies? System can be stable but still have bad response at certain frequencies Typically occurs if system has low phase margin get resonant peak in closed loop (M r ) + poor step response Solution: specify minimum phase margin. Typically 45 or more Phase (deg); Magnitude (db) Bode Diagrams 1.5 Step Response From: U(1) Amplitude To: Y(1) 1.5 Magnitude (db) M r Time (sec.) Frequency (rad/sec) CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 3

4 Canonical Control Design Problem Noise and disturbances d = process disturbances n = sensor noise Keep track of transfer functions between all possible inputs and outputs η y = u P P C P 1 P C C P CF P CF CF d n r Design represents a tradeoff between the quantities Keep L=PC large for good performance (H er << 1) Keep L=PC small for good noise rejection (H yn << 1) F = 1: Four unique transfer functions define performance ( Gang of Four ) Stability is always determined by 1/(1+PC) assuming stable process & controller Numerator determined by forward path between input and output More generally: 6 primary transfer functions; simultaneous design of each CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 4

5 Two Degree of Freedom Design Typical design procedure Design C to provide good load/noise response Design F to provide good response to reference Sensitivity Function S = T = P S = CS = P C P C 1 + P C P 1 + P C C 1 + P C Sensitivity function Complementary sensitivity Load sensitivity Noise sensitivity CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 5

6 Algebraic Constraints on Performance r + - e d C(s) + u P(s) y Sensitivity function Goal: keep S & T small S small low tracking error T small good noise rejection (and robustness [CDS 11b]) Problem: S + T = 1 Can t make both S & T small at the same frequency Solution: keep S small at low frequency and T small at high frequency Loop gain interpretation: keep L large at low frequency, and small at high frequency + n Magnitude (db) L(s) 1 Complementary sensitivity function L(s) < 1 Transition between large gain and small gain complicated by stability (phase margin) CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 6

7 Loop Shaping Revisited Disturbance rejection H ed = P 1 + L Would like Hed to be small make large L(s) Typically require this in low frequency range High frequency measurement noise H un = L P (1 + L) Want to make sure that Hun is small (avoid amplifying noise) small L(s) Typically generates constraints in high frequency range Robustness: gain and phase margin Focus on gain crossover region: make sure the slope is gentle at gain crossover Fundamental tradeoff: transition from high gain to low gain through crossover CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 7

8 Lead compensation Use to increase phase in frequency band Effect: lifts phase by increasing gain at high frequency Very useful controller; increases PM Bode: add phase between zero and pole Nyquist: increase phase margin r + - e u P(s) y CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 8

9 Process Inversion Simple trick: invert out process Write all performance specs in terms of the desired loop transfer function Choose L(s) that satisfies specfiications Choose controller by inverting P(s) Pros Very easy design process L(s) = 1/s often works very well Can be used as a first cut, with additional shaping to tune design Cons High order controllers (at least same order as the process you are controlling) Requires perfect model of your process (since you are inverting it) Does not work if you have right half plane poles or zeros (get internal instability) S = P C C(s) = L(s)/P (s) T = P C 1 + P C P S = P 1 + P C CS = C 1 + P C CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 9

10 Example: Control of Vectored Thrust Aircraft System description Vector thrust engine attached to wing Inputs: fan thrust, thrust angle (vectored) Outputs: position and orientation States: x, y, θ + derivatives Dynamics: flight aerodynamics Control approach Design inner loop control law to regulate pitch (θ ) using thrust vectoring Second outer loop controller regulates the position and altitude by commanding the pitch and thrust Basically the same approach as aircraft control laws CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 1

11 Performance Specification and Design Approach Phase (deg); Magnitude (db) Performance Specification 1% steady state error Zero frequency gain > 1 1% tracking error up to 1 rad/sec Gain > 1 from -1 rad/sec 45 phase margin Gives good relative stability Provides robustness to uncertainty Frequency (rad/sec) Design approach Open loop plant has poor phase margin Add phase lead in 5-5 rad/sec range Increase the gain to achieve steady state and tracking performance specs Avoid integrator to minimize phase CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 11

12 Control Design and Analysis Select parameters to satisfy specs Place phase lead in desired crossover region (given by desired BW) Phase lead peaks at 1X of zero location Place pole sufficiently far out to insure that phase does not decrease too soon Set gain as needed for tracking + BW Verify controller using Nyquist plot, etc bode Imaginary Axis To: Y(1) Nyquist Diagrams From: U(1) Real Axis nyquist bode step CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 12

13 Control Verification: Gang of 4 Remarks Check each transfer function to look for peaks, large magnitude, etc Example: Noise sensitivity function (CS) has very high gain; step response verifies poor step response Implication: controller amplifies noise at high frequency will generate lots of motion of control actuators (flaps) Fix: roll off the loop transfer function faster (high frequency pole) CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 13

14 Summary: Loop Shaping Loop Shaping for Stability & Performance Steady state error, bandwidth, tracking 5-5 Main ideas Performance specs give bounds on loop transfer function Use controller to shape response Gain/phase relationships constrain design approach Standard compensators: proportional, lead, PI CDS 11/11, 17 Nov 8 Richard M. Murray, Caltech CDS 14

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