Basics of H ∞ Control PART #1 the Loop-Shaping/Mixed Sensitivity Approach (to Robust & Optimal Control) Leonidas Dritsas PhD Version of 08-April-2012 Timestamp 08-Apr-12 THANKS TO… Prof. Kostas Kyriakopoulos (NTUA CSL) Prof. Kostas Kyriakopoulos (NTUA CSL) Panos Marantos (PhD candidate) Panos Marantos (PhD candidate)

Note An H∞ “Crash Course” divides naturally into (at least) 3 parts: 1. Intro / Nominal Stability + Perform. / Freq. Domain (mainly) - Loopshaping – Mixed Sensitivity – Riccati Approach (DGKF’89) 2. ROBUST Stability + ROBUST Performance + “μ”-SSV 3. Linear Matrix Inequality (LMI), Bounded Real Lemma (BRL), Dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities THIS PRESENTATION IS AN ATTEMPT FOR PART #1 !

Contents 1. Intro – Classical Feedback Control (internal) stability + performance + robustness (internal) stability + performance + robustness 2. SVD – Norms – {H ∞, H 2 } spaces & concepts in Control – the Engineering Interpratation 3. (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 4. MATLAB commands + examples (mixsyn/hinfsyn) 5. What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities} Αλέξης Παπαχαραλαμπόπουλος – Διπλωματική Εργασία 3/60

The Classical (1-DOF) Control Loop

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

Two Degrees of Freedom Use a “prefilter” to meet both regulator and tracking performance The Gang of Six For internal stability must “check- stab” SIX TFs

Two Degrees of Freedom Alternative config.

1-DOF feedback control system Closed-loop response Control error : Plant input:

Design TRADE OFFs = “CANNOT HAVE IT ALL”

Closed-loop Performance Closed-loop Robustness Bode / Nyquist plots of L(jω)

Max. (Peak) Sensitivity Ms

Controller design = TRADE OFFs = “CANNOT HAVE IT ALL”

Contents 1. Intro – Classical Feedback Control - (internal) stability + performance + robustness 2. SVD – Norms – {H ∞, H 2 } spaces & concepts in Control – the Engineering Interpratation 3. (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 4. MATLAB commands + examples (mixsyn/hinfsyn) 5. What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities} Αλέξης Παπαχαραλαμπόπουλος – Διπλωματική Εργασία 20/60

Signal & System Norms

Vector & Matrix Norm(s)

(Induced) Matrix Norms Vector & Matrix Norm(s)

Signal & System Norms

A paradigm shift: Generic control configuration (John Doyle) The concept of Generalized Plant Will be explained shortly

Contents 1. Intro – Classical Feedback Control - (internal) stability + performance + robustness 2. SVD – Norms – {H ∞, H 2 } spaces & concepts in Control – the Engineering Interpratation 3. (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 4. MATLAB commands + examples (mixsyn/hinfsyn) 5. What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities} 34/60

Recall Classical Loop Shaping

Recall Classical Loop Shaping - Relation to stability margins

Contents 1. Intro – Classical Feedback Control - (internal) stability + performance + robustness 2. SVD – Norms – {H ∞, H 2 } spaces & concepts in Control – the Engineering Interpratation 3. (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 4. MATLAB commands + examples (mixsyn/hinfsyn) 5. What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities} 45/60

Loop gain specifications S as a function of L

T as a function of L Relation to stability margins

Probably the most important slide (for Design purposes) Same as slide #44

Probably the most important slide (for Design purposes)

Contents 1. Intro – Classical Feedback Control (internal) stability + performance + robustness (internal) stability + performance + robustness 2. SVD – Norms – H∞ - H 2 - Engineering Interpr. 3. (Nominal) Controller Design Loopshaping – Mixed Sensitivity (1-DOF architecture) Loopshaping – Mixed Sensitivity (1-DOF architecture) Generalized Plant + LFT Generalized Plant + LFT WHAT do the 3 filters represent ? WHAT do the 3 filters represent ? HOW do we select the 3 filters ? Fundamental Limitations HOW do we select the 3 filters ? Fundamental Limitations 4. MATLAB commands + examples (mixsyn/hinfsyn)

Separate Lecture on Robustness

The SMALL GAIN Theorem

Robustness Interpretation of W1, W2, W3 filters

Generalized Plant + LFT

Generalized Plant for the Classical Control Loop

Standard Problem: P-K-Structure + LFT = Linear Fractional Transformations (Upper & Lower)

LFT = Linear Fractional Transformations (Upper & Lower)

Contents 1. Intro – Classical Feedback Control - (internal) stability + performance + robustness 2. SVD – Norms – {H ∞, H 2 } spaces & concepts in Control – the Engineering Interpratation 3. (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 4. MATLAB commands + examples (mixsyn/hinfsyn) 5. What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities} 76/60

Fundamental Performance Limitations reflected in filter limitations 1. Perfect control & plant inversion

Fundamental Performance Limitations reflected in filter limitations 2 Constraints on S and T 3 The waterbed effects (BODE sensitivity integrals 1945) 4 Interpolation constraints from internal stability

Fundamental Performance Limitations reflected in filter limitations 5 Sensitivity peaks - Maximum modulus Principle.

Fundamental Performance Limitations reflected in filter limitations

Fundamental Performance Limitations reflected in filter limitations

Fundamental Performance Limitations reflected in filter limitations

Fundamental Performance Limitations reflected in filter limitations Limitations imposed by RHP- poles

Limiting factors

Contents 1. Intro – Classical Feedback Control - (internal) stability + performance + robustness 2. SVD – Norms – {H ∞, H 2 } spaces & concepts in Control – the Engineering Interpratation 3. (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 4. MATLAB commands + examples (mixsyn/hinfsyn) 5. What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities} 85/60

hinfsyn

mixsyn

Contents 1. Intro – Classical Feedback Control - (internal) stability + performance + robustness 2. SVD – Norms – {H ∞, H 2 } spaces & concepts in Control – the Engineering Interpratation 3. (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 4. MATLAB commands + examples (mixsyn/hinfsyn) 5. What’s next ? “μ”-SSV, LMIs, BRL, dissipativity, “LMIreg”, LPV + Gain Scheduling, Handling Nonlinearities 92/60

THANK YOU FOR YOUR ATTENTION THE END THANK YOU FOR YOUR ATTENTION THANKS TO… Prof. Kostas Kyriakopoulos (NTUA CSL) Prof. Kostas Kyriakopoulos (NTUA CSL) Panos Marantos (PhD candidate) Panos Marantos (PhD candidate)

Contents 1. Intro – Classical Feedback Control - (internal) stability + performance + robustness 2. SVD – Norms – {H ∞, H 2 } spaces & concepts in Control – the Engineering Interpratation 3. (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 4. MATLAB commands + examples (mixsyn/hinfsyn) 5. What’s next ? {“μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities} 96/60

Contents 1. Intro – Classical Feedback Control (internal) stability + performance + robustness (internal) stability + performance + robustness 2. SVD – Norms – H∞ - H 2 - Engineering Interpr. 3. (Nominal) Controller Design Loopshaping – Mixed Sensitivity (1-DOF architecture) Loopshaping – Mixed Sensitivity (1-DOF architecture) WHAT do the 3 filters represent ? WHAT do the 3 filters represent ? HOW do we select the 3 filters ? Fundamental Limitations HOW do we select the 3 filters ? Fundamental Limitations 4. MATLAB commands + examples (mixsyn/hinfsyn) 5. What’s next ? “μ”-SSV, LMIs, BRL, dissipativity, LMIreg, LPV + Gain Scheduling, Nonlinearities

Modeling for “small” ( k < h) and “Long” delays Modeling for “small” ( τ k < h) and “Long” delays