Robust Control meets nonsmooth optimization

1 © 2013 The MathWorks, Inc.

Robust Control

Meets

Nonsmooth Optimization

Pascal Gahinet Pierre Apkarian

MathWorks, USA ONERA, France

2

Outline

1. Motivation (Set Robust Control free!)

2. What is fixed-structure synthesis?

3. Why does it work?

4. Cool applications

5. Remaining challenges

3

The World As We Like to See It

4

Tune this!

The World As Engineers See It

5

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,…

6

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

7

• 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

8

Flexibility

Tra

cta

bili

ty

Generic

NLP

DGKF

LMIs

SYSTUNE

0

0

1

1

mu

Flexibility vs. Tractability of Synthesis

9

Outline

1. Motivation

2. What is fixed-structure synthesis?

3. Why does it work?

4. Cool applications

5. Remaining challenges

10

Optimization-based tuning of free parameters in a given

control architecture

No restriction on control structure, number of feedback loops,

compensator types, …

11

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!

12

𝐻∞ requirements

Gain bounds Disk margins Loop shape

Requirement-Driven Tuning

13

𝐻2 requirements

Stochastic disturbance

attenuation

Step response matching

LQR objective

(𝒙𝑻𝑸𝒙 + 𝒖𝑻𝑸𝒖)𝒅𝒕∞

𝟎

14

Constraints on closed-loop dynamics

Decay rate, damping, natural frequency

15

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

16

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

17

Demo: Helicopter Flight Control

8+6 states, 21 tunable parameters

18

19

Outline

1. Motivation

2. What is fixed-structure synthesis?

3. Why does it work?

4. Cool applications

5. Remaining challenges

20

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

Nonlinear, nonconvex, nonsmooth program!

Why is it any better than brute-force optimization?

Objection!

21

Multiple active frequencies

Nearly active

frequency

Waterbed effect causes loss of differentiability when

minimizing 𝑇 𝑠, 𝑥 ∞

Not Just Any Optimizer

22

Why does this matter?

𝑥

𝑑

−𝑔1

−𝑔2

One active frequency Two active frequencies

23

SYSTUNE uses a dedicated nonsmooth optimizer

Remedy

24

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!

25

Outline

1. Motivation

2. What is fixed-structure synthesis?

3. Why does it work?

4. Cool applications

5. Remaining challenges

26

Makes Hard Problems Look Easy

• Static output feedback

• Strong stabilization

• Mixed 𝐻2 / 𝐻∞

• Fixed-order synthesis

27

Tune controller against multiple plant models

Another approach to robustness…

Multi-Model Synthesis

28

Three-loop autopilot, scheduled in 𝛼, 𝑉

Gain Scheduling

29

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!

30

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?

31

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?”

32

The quest continues!

Flexibility, usability

Tra

cta

bili

ty

DGKF

LMIs

SYSTUNE

0 0

1

1

mu

Generic

NLP