Basics of H ∞ Control - PowerPoint Presentation

Slide1

Basics of H

∞ Control PART #1 the Loop-Shaping/Mixed Sensitivity Approach (to Robust & Optimal Control)

Leonidas Dritsas PhD ldri@otenet.gr

Version of 08-April-2012

Timestamp08-Apr-12

THANKS TO…

Prof. Kostas Kyriakopoulos (NTUA CSL)

Panos Marantos (PhD candidate)Slide2

Note

An H∞ “Crash Course” divides naturally into (at least) 3 parts: Intro /

Nominal Stability + Perform. / Freq. Domain (mainly) - Loopshaping

– Mixed Sensitivity – Riccati Approach (DGKF’89)

ROBUST

Stability + ROBUST Performance +

“

μ

”

-SSV

Linear Matrix Inequality (LMI), Bounded Real Lemma

(BRL),

Dissipativity

,

LMIreg

, LPV + Gain Scheduling, Nonlinearities

THIS PRESENTATION IS AN ATTEMPT FOR PART #1 !Slide3

Contents

Intro – Classical Feedback Control(internal) stability + performance + robustnessSVD – Norms – {H∞

, H2 } spaces & concepts in Control – the Engineering Interpratation

(Nominal) Controller Design

3.1 Loopshaping

3.2 Mixed Sensitivity (1-DOF architecture)

3.3 Generalized Plant + LFT

3.4 WHAT do the 3 filters represent ?

3.5 HOW do we select them? Fundamental Limitations in Control

MATLAB commands + examples (

mixsyn/hinfsyn)What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities}

Αλέξης Παπαχαραλαμπόπουλος – Διπλωματική Εργασία

3

/60Slide4

http://www.nt.ntnu.no/users/skoge/book/Slide5

The Classical (1-DOF) Control LoopSlide6

The “Gang of Four”

The “Gang of Four”: For internal stability must “check-stab” Four TFsSlide7

Two Degrees of Freedom

Use a “prefilter” to meet both regulator and tracking performanceThe Gang of Six

For internal stability must “check-stab” SIX TFsSlide8

Two Degrees of Freedom

Alternative config.Slide9
Slide10

1-DOF feedback control system

Closed-loop response

Control error

:

Plant input:Slide11

Design TRADE OFFs

= “CANNOT HAVE IT ALL”Slide12

Closed-loop

PerformanceClosed-loop RobustnessBode / Nyquist plots of L(jω)Slide13

Max. (Peak) Sensitivity

MsSlide14
Slide15
Slide16
Slide17

Controller design =

TRADE OFFs = “CANNOT HAVE IT ALL”Slide18
Slide19
Slide20

Contents

Intro – Classical Feedback Control - (internal) stability + performance + robustnessSVD – Norms – {H∞, H

2 } spaces & concepts in Control – the Engineering Interpratation

(Nominal) Controller Design3.1

Loopshaping

3.2 Mixed Sensitivity (1-DOF architecture)

3.3 Generalized Plant + LFT

3.4 WHAT do the 3 filters represent ?

3.5 HOW do we select them? Fundamental Limitations in Control

MATLAB commands + examples (

mixsyn/hinfsyn)What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities}

Αλέξης Παπαχαραλαμπόπουλος – Διπλωματική Εργασία

20

/60Slide21
Slide22

Signal & System

NormsSlide23

Vector & Matrix Norm(s)Slide24

(Induced)

Matrix NormsVector & Matrix Norm(s)Slide25
Slide26
Slide27

Vector & Matrix Norm(s)Slide28

Signal & System

NormsSlide29

Signal & System

NormsSlide30
Slide31

Signal & System

NormsSlide32

A paradigm shift: Generic control configuration (John Doyle)

The concept of Generalized Plant Will be explained shortly Slide33
Slide34

Contents

Intro – Classical Feedback Control - (internal) stability + performance + robustnessSVD – Norms – {H∞, H

2 } spaces & concepts in Control – the Engineering Interpratation

(Nominal) Controller Design3.1

Loopshaping

3.2 Mixed Sensitivity (1-DOF architecture)

3.3 Generalized Plant + LFT

3.4 WHAT do the 3 filters represent ?

3.5 HOW do we select them? Fundamental Limitations in Control

MATLAB commands + examples (

mixsyn/hinfsyn)What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities}

34/60Slide35

Recall Classical Loop ShapingSlide36

Recall Classical Loop ShapingSlide37

Recall Classical Loop ShapingSlide38

Recall Classical Loop Shaping - Relation to stability marginsSlide39
Slide40
Slide41
Slide42
Slide43
Slide44
Slide45

Contents

Intro – Classical Feedback Control - (internal) stability + performance + robustnessSVD – Norms – {H∞, H

2 } spaces & concepts in Control – the Engineering Interpratation

(Nominal) Controller Design3.1

Loopshaping

3.2

Mixed Sensitivity (1-DOF architecture)

3.3 Generalized Plant + LFT

3.4 WHAT do the 3 filters represent ?

3.5 HOW do we select them? Fundamental Limitations in Control

MATLAB commands + examples (mixsyn/hinfsyn)What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities}

45

/60Slide46
Slide47
Slide48
Slide49
Slide50
Slide51
Slide52
Slide53
Slide54
Slide55
Slide56
Slide57

Loop gain specifications

S as a function of LSlide58

T as a function of L

Relation to stability marginsSlide59

Probably the most important slide

(for Design purposes)Same as slide #44Slide60

Probably the most important slide (for Design purposes)Slide61
Slide62
Slide63

Contents

Intro – Classical Feedback Control(internal) stability + performance + robustness

SVD – Norms – H∞ - H2 - Engineering Interpr.

(Nominal) Controller Design

Loopshaping

– Mixed Sensitivity (1-DOF architecture)

Generalized Plant + LFT

WHAT do the 3 filters represent ?

HOW do we select the 3 filters ? Fundamental Limitations

MATLAB commands + examples (

mixsyn/hinfsyn)Slide64
Slide65

Separate Lecture on

RobustnessSlide66

The SMALL GAIN TheoremSlide67

Robustness Interpretation of W1, W2, W3 filtersSlide68

Robustness Interpretation of W1, W2, W3 filtersSlide69

Generalized Plant

+ LFT Slide70

Generalized Plant

+ LFT Slide71

Generalized Plant

+ LFT Slide72

Generalized Plant

+ LFT Slide73

Generalized Plant for the Classical Control LoopSlide74

Standard Problem: P-K-Structure +

LFT = Linear Fractional Transformations (Upper & Lower)Slide75

LFT = Linear Fractional Transformations (Upper & Lower)Slide76

Contents

Intro – Classical Feedback Control - (internal) stability + performance + robustnessSVD – Norms – {H∞, H

2 } spaces & concepts in Control – the Engineering Interpratation

(Nominal) Controller Design3.1

Loopshaping

3.2 Mixed Sensitivity (1-DOF architecture)

3.3 Generalized Plant + LFT

3.4 WHAT do the 3 filters represent ?

3.5 HOW do we select them? Fundamental Limitations in Control

MATLAB commands + examples (

mixsyn/hinfsyn)What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities}

76/60Slide77

Fundamental Performance Limitations reflected in filter limitations

1. Perfect control & plant inversionSlide78

Fundamental Performance Limitations reflected in filter limitations

2 Constraints on S and T3 The waterbed effects (BODE sensitivity integrals 1945)

4 Interpolation constraints from internal stabilitySlide79

Fundamental Performance Limitations reflected in filter limitations

5 Sensitivity peaks - Maximum modulus Principle.Slide80

Fundamental Performance Limitations reflected in filter limitationsSlide81

Fundamental Performance Limitations reflected in filter limitationsSlide82

Fundamental Performance Limitations reflected in filter limitationsSlide83

Fundamental Performance Limitations reflected in filter limitations

Limitations imposed by RHP-polesSlide84

Limiting factorsSlide85

Contents

Intro – Classical Feedback Control - (internal) stability + performance + robustnessSVD – Norms – {H∞, H

2 } spaces & concepts in Control – the Engineering Interpratation

(Nominal) Controller Design3.1

Loopshaping

3.2 Mixed Sensitivity (1-DOF architecture)

3.3 Generalized Plant + LFT

3.4 WHAT do the 3 filters represent ?

3.5 HOW do we select them? Fundamental Limitations in Control

MATLAB commands + examples (

mixsyn/hinfsyn)What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities}

85/60Slide86
Slide87

hinfsynSlide88

hinfsynSlide89

mixsynSlide90

mixsynSlide91
Slide92

Contents

Intro – Classical Feedback Control - (internal) stability + performance + robustnessSVD – Norms – {H∞, H

2 } spaces & concepts in Control – the Engineering Interpratation

(Nominal) Controller Design3.1

Loopshaping

3.2 Mixed Sensitivity (1-DOF architecture)

3.3 Generalized Plant + LFT

3.4 WHAT do the 3 filters represent ?

3.5 HOW do we select them? Fundamental Limitations in Control

MATLAB commands + examples (

mixsyn/hinfsyn)What’s next ? “μ”-SSV, LMIs, BRL, dissipativity, “LMIreg”, LPV + Gain Scheduling, Handling Nonlinearities

92/60Slide93

THE END THANK YOU FOR YOUR ATTENTION

THANKS TO… Prof. Kostas Kyriakopoulos (NTUA CSL)Panos Marantos (PhD candidate)Slide94
Slide95
Slide96

Contents

Intro – Classical Feedback Control - (internal) stability + performance + robustnessSVD – Norms – {H∞, H

2 } spaces & concepts in Control – the Engineering Interpratation

(Nominal) Controller Design3.1

Loopshaping

3.2 Mixed Sensitivity (1-DOF architecture)

3.3 Generalized Plant + LFT

3.4 WHAT do the 3 filters represent ?

3.5 HOW do we select them? Fundamental Limitations in Control

MATLAB commands + examples (

mixsyn/hinfsyn)What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities}

96/60Slide97

Contents

Intro – Classical Feedback Control(internal) stability + performance + robustness

SVD – Norms – H∞ - H2 - Engineering Interpr.

(Nominal) Controller Design

Loopshaping

– Mixed Sensitivity (1-DOF architecture)

WHAT do the 3 filters represent ?

HOW do we select the 3 filters ? Fundamental Limitations

MATLAB commands + examples (

mixsyn

/hinfsyn)What’s next ? “μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, NonlinearitiesSlide98
Slide99
Slide100

Modeling for “small” (

τk < h) and “Long” delays