SLIDE 1
CNRS - RAS cooperation Seminar - 18th June, 2004 - IPME, St. - - PowerPoint PPT Presentation
CNRS - RAS cooperation Seminar - 18th June, 2004 - IPME, St. - - PowerPoint PPT Presentation
CNRS - RAS cooperation Seminar - 18th June, 2004 - IPME, St. Petersburg Quadratic Separation for Robustness and Design Dimitri Peaucelle What is LAAS-CNRS? 1 French National Center for Scientific Research. Public basic-research org.
SLIDE 2
SLIDE 3
Methods and Algorithms in Control group
2
MAC group http://www.laas.fr/MAC
❏ Research fields : Robust control & Non-linear control ❏ Application fields : Aeronautics & Space industry & Environment
Research in robust control
❏ MIMO LTI systems with parametric uncertainty ❏ State-space modeling and Lyapunov theory ❏ Stability and performance (H∞, H2, pole location, impulse to peak) ❏ Analysis & Control design (state-feedback, full-order and static output-feedback) ❏ LMI based results & Semi-definite programming ❏ Robust MULti-Objective Control toolbox (V1 in September)
http://www.laas.fr/OLOCEP
Quadratic Separation for Robustness and Design
SLIDE 4
Outline
3
Quadratic separation for LTI systems Examples of results for robustness and design
➙ Preliminaries and notations ➙ Robust analysis and losslessness of quadratic separators ➙ Quadratic separation and control design Quadratic Separation for Robustness and Design
SLIDE 5
Methodology and notations
4
Uncertain model
❏ Engineering problem modeled as uncertain differential equations with constraints ❏ State-space LTI systems / parametric uncertainty / pole and induced norm constraints
- Optimization problem
❏ At best: lossless formulation with a global polynomial-time algorithm ❏ Conveniently: Conservative formulation with a global polynomial-time algorithm ❏ Usually: Conservative with sub-optimal heuristic algorithm ❏ LMI formulated results ➾ convex SDP &
✁n6
✂5
✄algorithms ❏ YALMIP interface in Matlab & Solvers: SeDuMi, SDPT3, CSDP,... v
☎C
✁∆
✄x
✆D
✁∆
✄v ˙ x
☎A
✁∆
✄x
✆B
✁∆
✄v ∞ g v
✝min γ :
✁P
✞Θ
✞γ
✄✠✟ ✡Quadratic Separation for Robustness and Design
SLIDE 6
Topological Separation
5
Graph of Σ1 and inverse graph of Σ2:
✁Σ1
✄ ☎ ☛ ☞ ✌x
☎z w : Σ1
✁z
✞w
✄ ☎ ✍ ✎ ✏I
✁Σ2
✄ ☎ ☛ ☞ ✌x
☎z w : Σ2
✁w
✞z
✄ ☎ ✍ ✎ ✏Stability result [Safonov]:
The interconnected system
z w
2
Σ
1
Σ
is stable ❏ iff
✁Σ1
✄ ✑I
✁Σ2
✄ ☎ ✒ ✓❏ iff
✔d :
☛ ☞ ✌d
✁x
✄✠✕ ✞✗✖x
✘ ✁Σ1
✄d
✁x
✄ ✙ ✞✗✖x
✘I
✁Σ2
✄d : topological separator (see also “supply rate” in dissipative theory [Willems]) Quadratic Separation for Robustness and Design
SLIDE 7
Quadratic Separation
6
Quadratic function for the topological separator:
d
✁x
✄ ☎x
✚Θx Θ
☎Θ
✚ ✘ ✛ ✜m
✢p
✣✠✤ ✜m
✢p
✣Lossless for linear systems
E.g. for matrix gains
The interconnected system
w z
Σ1
✥ ✦p
✧m
Σ2
✥ ✦m
✧p
is well-posed ❏ iff det
✁✩★✫✪Σ1Σ2
✄ ✬ ☎❏ iff
✔Θ :
☛ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ☞ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✌w
✚ ✮Σ
✚1
★ ✯Θ
✰ ✱Σ1
★ ✲ ✳w
✕ ✞✗✖w
✬ ☎z
✚ ✮ ★Σ
✚2
✯Θ
✰ ✱ ★Σ2
✲ ✳z
✙ ✞✗✖z
✬ ☎Quadratic Separation for Robustness and Design
SLIDE 8
One interconnected operator is uncertain
7
Robust analysis
w z
Σ1
✥ ✦p
✧m
Σ2
is robustly well-posed for all Σ2
✘ ✴ ✵ ✛m
✤p
❏ iff det
✁✩★✫✪Σ1Σ2
✄ ✬ ☎ ✞✗✖Σ2
✘ ✴❏ iff
✔Θ :
☛ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ☞ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✌ ✮Σ
✚1
★ ✯Θ
✰ ✱Σ1
★ ✲ ✳ ✕ ✡ ✮ ★Σ2
✚ ✯Θ
✰ ✱ ★Σ2
✲ ✳ ✙ ✡ ✞ ✖Σ2
✘ ✴➥ Quadratic separation results are LMI-based. ➥ Is it possible to handle the infinite-dimensional constraint without conservatism? Quadratic Separation for Robustness and Design
SLIDE 9
One interconnected operator is uncertain
8
Example: Lyapunov matrix
A
✘ ✶n
✤n
s
✷1
★z
✸˙ x w
✸x
is stable (interconnection well-posed for all s
✷1
✘ ✛ ✢) ❏ iff
✔Θ :
☛ ✭ ✭ ✭ ✭ ✭ ✭ ☞ ✭ ✭ ✭ ✭ ✭ ✭ ✌ ✮AT
★ ✯Θ
✰ ✱A
★ ✲ ✳ ✕ ✡ ✮ ★s
✷ ✚ ★ ✯Θ
✰ ✱ ★s
✷1
★ ✲ ✳ ✙ ✡ ✞ ✖s
✷1
✘ ✛ ✢❏ iff
✔Θ
☎ ✰ ✱ ✡ ✪P
✪P
✡ ✲ ✳: P
✕ ✡ ✞ ✮AT
★ ✯Θ
✰ ✱A
★ ✲ ✳ ☎ ✪ATP
✪PA
✕ ✡➥ Lossless quadratic separator. ➥ Other sets than
✛ ✢➾ pole location. Quadratic Separation for Robustness and Design
SLIDE 10
One interconnected operator is uncertain
9
Example: bounded real lemma
w z
✰ ✱A B C D
✲ ✳ ✰ ✱s
✷1
★ ✡ ✡∆
✲ ✳is robustly stable ( s
✷1
✘ ✛ ✢ ✞∆T∆
✙ ★) ❏ iff there exists a separator such as: Θ
☎ ✰ ✹ ✹ ✹ ✹ ✹ ✱ ✡ ✡ ✪P
✡ ✡ ✪τ
★ ✡ ✡ ✪P
✡ ✡ ✡ ✡ ✡ ✡τ
★ ✲ ✺ ✺ ✺ ✺ ✺ ✳, P
✕ ✡τ
✕that satisfies the LMI constraint:
✮Σ
✚1
★ ✯Θ
✰ ✱Σ1
★ ✲ ✳ ☎ ✪ ✰ ✱ATP
✆PA
✆τCTC PB
✆τCTD BTP
✆τDTC
✪τ
★ ✆τDTD
✲ ✳ ✕ ✡➥ Lossless quadratic separator. Quadratic Separation for Robustness and Design
SLIDE 11
Conservative and lossless separators
10
Lossless quadratic separators
❏ Full-block dissipative ( -procedure)
✮ ★∆
✚D
✯ ✰ ✱X Y Y
✚Z
✲ ✳ ✰ ✱ ★∆D
✲ ✳ ✙ ✡ ✻Θ
☎τ
✰ ✱X Y Y
✚Z
✲ ✳ ✞τ
✕❏ Disk located, repeated, complex valued scalar ∆
☎δc
★α
✆δcβ
✆δ
✚cβ
✚ ✆δcδ
✚cγ
✙ ✻Θ
☎ ✰ ✱αP βP β
✚P γP
✲ ✳ ✞P
✕ ✡❏ Bounded, repeated, real valued scalar ∆
☎δr
★α
✆2δrβ
✆δ2
rγ
✙ ✻Θ
☎ ✰ ✱αP βP
✆Q βP
✆Q
✚γP
✲ ✳ ✞P
✕ ✡Q
☎ ✪Q
✚Quadratic Separation for Robustness and Design
SLIDE 12
Conservative and lossless separators
11
Conservative quadratic separators for block diagonal uncertainty
❏ Repeated full-block dissipative ∆
☎ ★r
✼∆D
✮ ★∆
✚D
✯ ✰ ✱X Y Y
✚Z
✲ ✳ ✰ ✱ ★∆D
✲ ✳ ✙ ✡ ✝Θ
☎ ✰ ✱T
✼X T
✼Y T
✼Y
✚T
✼Z
✲ ✳T
✕ ✡❏ Block diagonal polytopic ∆
☎diag
✁∆1
✞✠✽ ✽ ✽ ✞∆r
✄∆
☎∑ζi∆
✾i
✿: ∑ζi
☎1
✞ζi
❀ ✝ ✮ ★∆
✾i
✿ ✯Θ
✰ ✱ ★∆
✾i
✿ ✲ ✳ ✙ ✡Θ22
ii
❀ ✡❏ Any block diagonal structure of previous types (lossless if 2
✁mr
✆mc
✄ ✆mD
✙3) ∆
☎diag
✁∆1
✞ ✽ ✽ ✽ ✄ ✝Θ
☎ ✰ ✱diag
✁Θ11
1
✞ ✽ ✽ ✽ ✄diag
✁Θ12
1
✞ ✽ ✽ ✽ ✄diag
✁Θ21
1
✞ ✽ ✽ ✽ ✄diag
✁Θ22
1
✞ ✽ ✽ ✽ ✄ ✲ ✳Quadratic Separation for Robustness and Design
SLIDE 13
Both interconnected operators are uncertain
12
Robust analysis: parameter-dependent separators
w z
Σ1
❁∆1
❂∆2
is robustly well-posed for all
✖∆1
✘ ✴1
✞✗✖∆2
✘ ✴2
❏ iff
✖∆1
✘ ✴1
✔Θ
✁∆1
✄:
☛ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ☞ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✌ ✮Σ
✚1
✁∆1
✄ ★ ✯Θ
✁∆1
✄ ✰ ✱Σ1
✁∆1
✄ ★ ✲ ✳ ✕ ✡ ✮ ★∆2
✚ ✯Θ
✁∆1
✄ ✰ ✱ ★∆2
✲ ✳ ✙ ✡ ✞✗✖∆2
✘ ✴2
➥ Infinite number of LMI variables & infinite number of constraints Quadratic Separation for Robustness and Design
SLIDE 14
Both interconnected operators are uncertain
13
Example: µ-analysis
w z
∆ Σ
❁jω
❂is robustly stable ( ω
✘ ✶ ✢ ✞∆
✘ ✴) ❏ iff
✖ω
✘ ✶ ✢ ✔Θ
✁jω
✄:
☛ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ☞ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✌ ✮Σ
✚ ✁jω
✄ ★ ✯Θ
✁jω
✄ ✰ ✱Σ
✁jω
✄ ★ ✲ ✳ ✕ ✡ ✮ ★∆
✚ ✯Θ
✁jω
✄ ✰ ✱ ★∆
✲ ✳ ✙ ✡ ✞ ✖∆
✘ ✴➥ An optimistic bound on µ can then be obtained by gridding
✒ω1
✞ ✽ ✽ ✽ ✞ωN
✓ ✵ ✶ ✢. ➥ For each ωi, build finite dimensional LMIs. Quadratic Separation for Robustness and Design
SLIDE 15
Both interconnected operators are uncertain
14
Example: Parameter-dependent Lyapunov Function
w z
A
❁∆
❂s
❃1
❄is robustly stable ( s
✷1
✘ ✛ ✢ ✞∆
✘ ✴) ❏ iff
✖∆
✘ ✴ ✔P
✁∆
✄:
☛ ☞ ✌AT
✁∆
✄P
✁∆
✄ ✆P
✁∆
✄A
✁∆
✄ ✟ ✡P
✁∆
✄ ✕ ✡Question: When A
✁∆
✄ ☎A
✆B∆
✁✩★✫✪D∆
✄ ✷1C
how is this result related to the robust stability of w z
✰ ✱A B C D
✲ ✳ ✰ ✱s
✷1
★ ✡ ✡∆
✲ ✳? Quadratic Separation for Robustness and Design
SLIDE 16
Towards lossless robust analysis
15
Answer: Consider the implicit augmented system
diag
✁s
✷1
★ ✞s
✷1
★ ✞∆
✞∆
✄Σ Σ :
☛ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ☞ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✌˙ x
☎Ax
✆Bw ˙ w
☎˙ w z
☎Cx
✆Dw ˙ z
☎CAx
✆CBw
✆D ˙ w
☎ ✪w
✆w Quadratic separation
✝parameter-dependent Lyapunov matrix P
✁∆
✄ ☎ ✮ ★CT
✁✩★✫✪D∆
✄ ✷T∆T
✯P
✰ ✱ ★∆
✁ ★✫✪D∆
✄ ✷1C
✲ ✳Prospective work:
❏ In relation with [Bliman], build asymptotically lossless P-D Lyapunov functions ❏ Take into account information on the derivatives of ∆. Quadratic Separation for Robustness and Design
SLIDE 17
Quadratic separation v.s. control law
16 There exists a matrix K such that
w z
K Σ
is stable ❏ iff
✔K
✞ ✔Θ :
✮Σ
✚ ★ ✯Θ
✰ ✱Σ
★ ✲ ✳ ✕ ✡ ✞ ✮ ★K
✚ ✯Θ
✰ ✱ ★K
✲ ✳ ✙ ✡❏ iff
✔Θ
☎ ✰ ✱X Y Y
✚Z
✲ ✳:
☛ ✭ ✭ ☞ ✭ ✭ ✌ ✮Σ
✚ ★ ✯Θ
✰ ✱Σ
★ ✲ ✳ ✕ ✡X
✙YZ
✷1Y
✚Z
✕ ✡➾ All matrices K such that
✮ ★K
✚ ✯Θ
✰ ✱ ★K
✲ ✳ ✙ ✡, stabilize the interconnection. ➥ The non-linear constraint X
✙YZ
✷1Y
✚is not convex Quadratic Separation for Robustness and Design
SLIDE 18
Improvements and challenge
17
Quadratic separation for design, features:
∆K Σ K ∆
❏ Design of sets of controllers: handles fragility ❏ LMI formulations for state-feedback and full-order output-feedback ❏ All robust multi-objective problems can be recast as LMIs + X
✙YZ
✷1Y
✚Challenge:
❏ Algorithms to handle the non-linear matrix inequality. ❏ Successful results of a gradient-type algorithm - to be improved. Quadratic Separation for Robustness and Design
SLIDE 19