Lecture 15: State Feedback Control: Part I Pole Placement for SISO Systems Illustrative Examples.
A: SISO Feedback Control A.1 Internal Stability and Youla ...Controllability analysis with SISO...
Transcript of A: SISO Feedback Control A.1 Internal Stability and Youla ...Controllability analysis with SISO...
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A: SISO Feedback Control
A.2 Sensitivity and Feedback Performance
Reference:
A.3 Loop Shaping
[SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design,Second Edition, Wiley, 2005.
[SP05, Sec. 2.2, 5.2]
[SP05, Sec. 2.4, 2.6]
A.1 Internal Stability and Youla Parameterization[SP05, Sec. 3.2, 4.1.5, 4.7, 4.8]
Robust and Optimal Control, Spring 2015Instructor: Prof. Masayuki Fujita (S5-303B)
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Complementary SensitivitySensitivity
Internal Stability
Noise SensitivityLoad Sensitivity
[AM08] K.J. Astrom and R.M. Murray, Feedback Systems, Princeton University Press, 2008
Gang of Four [AM08, p. 317]
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Internal Stability[SP05, Ex. 4.16]
Sensitivity Comp. Sensitivity
Load Sensitivity Noise Sensitivity
Time [s] Time [s] Time [s] Time [s]
Step Response
Unstable
(p. 144)
Stable?
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Internal Stability
C.A. Desoer and W.S. Chan, Journal of the Franklin Institute, 300 (5-6) 335-351, 1975
C. Desoer
[SP05, Theorem 4.6] (p. 145)
The feedback system in the above figure is internally stable if and only if all “Gang of Four ( ) ” are stable
Well-posedness:(Gang of Four: well-defined and proper)
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Youla Parameterization ( Parameterization)
−
Stable Plant
−
−
Gang of Four: Proper Stable Transfer Function-parameter
All Stabilizing Controllers:
Internal Model Control (IMC) Structure
Case 1: [SP05, p. 148]
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: Integer[Ex.]
Bezout Identity: Proper Stable Transfer Functions
: Coprime
Youla Parameterization
Coprime Factorization
: Proper Stable Transfer Functions
[SP05, Ex. 4.1]
: Integer
Unstable PlantCase 2:
Coprime: No common right-half plane(RHP) zeros
(*)
[SP05, p. 149]
[SP05, p. 122]
[SP05, Ex.] : (*)
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Youla ParameterizationUnstable PlantsCase 2:
A Stabilizing Controller
[SP05, Ex.]
Gang of Four
Affine Functions of
All Stabilizing Controllers
[SP05, p. 149]
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Disturbance AttenuationOpen-loop Closed-loop
small: good Feedback Performance
: Sensitivity
Sensitivity and Feedback Performance
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Insensitivity to Plant Variations
small : good Feedback Performance
[SP05, p. 23]
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• Insensitivity to Plant Variations
Benefits of Feedback
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• Disturbance Attenuation
• Reference Tracking
• Stabilization (Unstable Plant)
: small
• Linearizing Effects
Two-degrees-of-freedom Control Feedback + Feedforward
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Waterbed Effects
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There exists a frequency range over which the magnitude of the sensitivity function exceeds 1 if it is to be kept below 1 at the other frequency range.
[SP05, p. 167]
100 101
10
10−
0
]dB[
Frequency [rad/s]
[SP05, Ex., p. 170](unstable)
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Maximum Peaks of and
: Maximum Peak Magnitude of
: Bandwidth Frequency of
: Maximum Peak Magnitude of
: Bandwidth Frequency of
Complementary SensitivitySensitivity
[SP05, p. 36]
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Loop ShapingLoop Transfer Function
+
large small
Sensitivity:
small small
Comp. Sensitivity:
Constraint
Loop ShapingClosed-loop
Open LoopStability, Performance, Robustness 13
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Loop Transfer Function
[SP05, Ex. 2.4] (p. 34)
Gain Crossover Frequency
Gain MarginPhase MarginTime Delay MarginStability Margin
Stability Margins [SP05, p. 32]
[SP05, Ex. 2.4] (p. 34) 14
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[Ex.]
[Ex.]
Maximum Peak Criteria [SP05, p. 36]
[SP05, Ex. 2.4] (p. 34)
Frequency Domain Performance
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Bode Gain-phase Relationship
Slope of the Gain Curve at
Steep Slope: Small Phase Margin
(minimum phase systems)
[SP05, p. 18]
[SP05, Ex., p. 20]0
-2-1 -2
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Fundamental LimitationsBound on the Crossover Frequency RHP (Right half-plane) Zero
Time Delay
Im
Re0 z
betterworse
Step Response
Time [s]
Tight RestrictionsSlow RHP Zeros ( small):Fast RHP Zeros ( large): Loose Restrictions
Frequency [rad/s]
[SP05, pp. 183]
Unstable zero
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Fundamental LimitationsBound on the Crossover Frequency
Im
Re0 p×××××××
worsebetter
RHP (Right half-plane) Pole
Fast RHP Poles ( large): Tight RestrictionsLoose RestrictionsSlow RHP Poles ( small):
Frequency [rad/s]
[SP05, pp. 192, 194]
Unstable pole
Poles on imaginary axis
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SISO Loop Shaping
Robust Stability(+ Roll-off)
Performance
Loop Shaping Specifications• Gain Crossover Frequency• Shape of• System Type, Defined as the Number of Pure Integrators in• Roll-off at Higher Frequencies
[SP05, pp. 41, 42, 343]
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Step response analysis/Performance criteria
[QZ07] L. Qiu and K. Zhou (2007) Introduction to Feedback Control, Prentice Hall.
First-order System
Settling time
Rise time
Error tolerance
Peak timeSettling timeRise time
Overshoot0
0.1
0.91
Second-order System
Overshoot
Settling time
Rise time
OvershootPeak Time
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Design Relations
Phase Margin
[FPN09] G.F. Franklin, J.D. Powell and A. E.-Naeini (2009) Feedback Control of Dynamic Systems, Sixth Edition, Prentice Hall.
: Maximum Peak Magnitude of
: Bandwidth Frequency of
Complementary Sensitivity
Maximum Peak Magnitude of
Bandwidth
ifif
if
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Controllability analysis with SISO feedback control[SP05, pp. 206-209]
Typically, the closed-loop bandwidth of the spacecraft is an order of magnitude less than the lowest mode frequency, and as long as the controller does not excite any of the flexible modes, the sampling period may be selected solely based on the closed-loop bandwidth.
[Le10] W.S. Levine (Eds.) (2010) The Control Handbook, Second Edition: Control System Fundamentals, Second Edition, CRC Press.
Margin to stay within constraintsMargin for performanceMargin because of RHP-pole
Margin because of RHP-zero
Margin because of frequency where plant has phase lag
Margin because of delay
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For systems with a RHP pole and RHP zero (or a time delay ), any stabilizing controller gives sensitivity functions with the property
RHP Poles/Zeros, Time Delays and Sensitivityp z
τ
zpzpjSM S −
+≥= )(sup ω
ω
τ
ωω p
T ejTM ≥= )(sup
SMTM
)( ωjT)( ωjS
RHP pole and zero and time delay significantly limit the achievable performance of a system
zpzp
−+ τpe
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RHP Poles/Zeros, Time Delays and Sensitivity
or pz /6 <6/1/ <pz 3.0<τp
allowable phase lag of at : °= 90lϕapP gcω
All-pass system( , , )1=p bz =
1)(
−−
=s
sbsPap 1)(
−=
−
sesP
s
ap
τ
RHP pole/zero pair RHP pole and time delay
The zero and the pole must be sufficiently far apart
The product of RHP pole and time delay must be sufficiently small
τ
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