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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