1 © 2013 The MathWorks, Inc.

Robust Control

Meets

Nonsmooth Optimization

Pascal Gahinet Pierre Apkarian

MathWorks, USA ONERA, France

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Outline

1. Motivation (Set Robust Control free!)

2. What is fixed-structure synthesis?

3. Why does it work?

4. Cool applications

5. Remaining challenges

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The World As We Like to See It

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Tune this!

The World As Engineers See It

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Applying Robust Control theory to real-world problems

remains challenging:

• Perceived as “complex, PhD required!”

(weighting functions, high-order black-box controllers,

MIMO frequency-domain formulation,…)

• Does not fit familiar control structures and design

workflows

• Hard to mix with other requirements

(time response, pole damping,…)

• Difficult to fine tune on-site, gain schedule,…

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How can we increase the adoption of Robust Control ideas

in industry?

• Better education

Caveat: Still mostly limited to MS and PhDs

• More bridges

𝑇𝑤𝑧 < 𝛾

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• LMI formulations

Elegant but patchy results

Computationally demanding

Pays for convexity with conservative relaxations

Can’t handle fixed controller structure

• Brute-force optimization (generic NLP, global search,…)

Long running times

Requires good starting points

Many pitfalls (nonlinear, nonconvex, nonsmooth,…)

Controller Synthesis Beyond DGKF

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Flexibility

Tra

cta

bili

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Generic

NLP

DGKF

LMIs

SYSTUNE

0

0

1

1

mu

Flexibility vs. Tractability of Synthesis

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Outline

1. Motivation

2. What is fixed-structure synthesis?

3. Why does it work?

4. Cool applications

5. Remaining challenges

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Optimization-based tuning of free parameters in a given

control architecture

No restriction on control structure, number of feedback loops,

compensator types, …

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SYSTUNE maps all control architectures to standard form:

Efficient management of

tunable blocks and I/O maps

Cheap gradient computation

Structural analysis and

more…

The LFT rules!

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𝐻∞ requirements

Gain bounds Disk margins Loop shape

Requirement-Driven Tuning

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𝐻2 requirements

Stochastic disturbance

attenuation

Step response matching

LQR objective

(𝒙𝑻𝑸𝒙 + 𝒖𝑻𝑸𝒖)𝒅𝒕∞

𝟎

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Constraints on closed-loop dynamics

Decay rate, damping, natural frequency

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SYSTUNE takes any combination of such requirements

Each requirement can apply to any I/O transfer or loop

transfer, including open-loop configurations

Closed-loop system Requirements

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SYSTUNE turns these requirements into normalized

functions 𝑓 𝑥 of the tuned parameters 𝑥 …

… and solves the resulting min-max program:

min𝑥max𝑖𝑓𝑖 𝑥 subject tomax

𝑗𝑔𝑗 𝑥 < 1

𝑓 𝑥 = (𝑇 𝑠, 𝑥 − 𝑇𝑟𝑒𝑓(𝑠))/𝑠 2

𝛿 (1 − 𝑇𝑟𝑒𝑓(𝑠))/𝑠 2

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Demo: Helicopter Flight Control

8+6 states, 21 tunable parameters

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Outline

1. Motivation

2. What is fixed-structure synthesis?

3. Why does it work?

4. Cool applications

5. Remaining challenges

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Too good to be true, you can’t be serious!

Nonlinear, nonconvex, nonsmooth program!

Why is it any better than brute-force optimization?

Objection!

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Multiple active frequencies

Nearly active

frequency

Waterbed effect causes loss of differentiability when

minimizing 𝑇 𝑠, 𝑥 ∞

Not Just Any Optimizer

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Why does this matter?

𝑥

𝑑

−𝑔1

−𝑔2

One active frequency Two active frequencies

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SYSTUNE uses a dedicated nonsmooth optimizer

Remedy

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Yes, but

• Operates on low-dimensional parameter space

(no huge Lyapunov matrices)

• Uses well-behaved objectives (𝐻∞ and 𝐻2 norms)

• Well-posed synthesis problems tend to have few

undesirable local minima

• Runs in seconds, not hours or days

Wait, This Is Not Convex!

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Outline

1. Motivation

2. What is fixed-structure synthesis?

3. Why does it work?

4. Cool applications

5. Remaining challenges

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Makes Hard Problems Look Easy

• Static output feedback

• Strong stabilization

• Mixed 𝐻2 / 𝐻∞

• Fixed-order synthesis

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Tune controller against multiple plant models

Another approach to robustness…

Multi-Model Synthesis

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Three-loop autopilot, scheduled in 𝛼, 𝑉

Gain Scheduling

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1. Parameterize the gains

as functions of α, V

2. Tune the coefficients

K0, K1, K2 , … to enforce

requirements over a

grid of design points

𝐾𝑝 𝛼, 𝑉 = 𝐾0 + 𝐾1𝛼 + 𝐾2𝑉 + 𝐾3𝛼𝑉

Local sub-model LTI (α, V)

Tune Entire Gain Surfaces!

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Conclusion

• SYSTUNE helps bring Robust Control to the masses

• Good compromise between flexibility and tractability

• What else can we do if we give up Riccati’s and LMIs?

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Remaining Challenges

Control engineers are a demanding kind!

• “I’m not comfortable with the frequency domain”

• “My system is very nonlinear”

• “What’s wrong with hand tuning?”

• “Can you tune my controller directly on the hardware?”

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The quest continues!

Flexibility, usability

Tra

cta

bili

ty

DGKF

LMIs

SYSTUNE

0 0

1

1

mu

Generic

NLP