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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
Nonsmooth H∞ Synthesis P. Pellanda, P. Apkarian & D. Noll IME, ONERA & Universit´ e Paul Sabatier/Math.
WCSMO6 Conference, 2005
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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nonsmooth formulations
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nonsmooth descent algorithms
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Outline ♦ context ♦ nonsmooth formulations ♦ nonsmooth descent algorithms ♦ applications ♦ concluding remarks
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
hard non-LMI problems many synthesis problems do not reduce to LMI/SDP !
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
hard non-LMI problems many synthesis problems do not reduce to LMI/SDP !
à à à à à à
reduced- and fixed-order synthesis (PID H∞ , etc.) structured and decentralized synthesis problems general robust control with uncertain and/or nonlinear components simultaneous model/controller design, multimodel control unrelaxed LTI and LPV multi-objective combinations of the above, etc
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
hard non-LMI problems many synthesis problems do not reduce to LMI/SDP !
à problems are nonconvex and/or nonsmooth à even large size LMIs/SDPs still difficult to solve
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
hard non-LMI problems many synthesis problems do not reduce to LMI/SDP !
⇒ need new algorithms for hard problems
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
current potential techniques à coordinate descent schemes (D-K iterations,...) ⇒ slow and lack of convergence certificate à solutions based on relaxations ⇒ unknown conservatism
• other proposals à TR method by Leibfritz à nonsmooth gradient sampling by Burke, Lewis & Overton à non-quadratic penalty method for SDP casts by Kocvara, Zibulesky, à ...
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
applications
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concluding remarks
fundamental limitation of SDP casts benchmark : Boeing 767 at flutter condition
• static stabilization pb. x˙ A = y C A = 55 × 55
B 0
B = 55 × 2
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
x u
,
u = Ky
C = 2 × 55
K =2×2
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
applications
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concluding remarks
fundamental limitation of SDP casts benchmark : Boeing 767 at flutter condition
• static stabilization pb. x˙ A = y C A = 55 × 55
B 0
B = 55 × 2
x u
,
u = Ky
C = 2 × 55
K =2×2
à BMI ⇒ n = 1540 variables ! à minK α(A + BKC ) ⇒ 4 variables where α(.) := maxi Re λi (.) is spectral abscissa
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
applications
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concluding remarks
fundamental limitation of SDP casts benchmark : Boeing 767 at flutter condition
• static stabilization pb. x˙ A = y C A = 55 × 55
B 0
B = 55 × 2
x u
,
u = Ky
C = 2 × 55
K =2×2
à BMI ⇒ n = 1540 variables ! à minK α(A + BKC ) ⇒ 4 variables where α(.) := maxi Re λi (.) is spectral abscissa
accept nonsmoothness and design appropriate algorithms
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
nonsmooth formulations
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
applications
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concluding remarks
nonsmooth formulations • stabilization minimize K
α(A + BKC ),
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
α := max Re λi i
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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concluding remarks
nonsmooth formulations • stabilization minimize K
α(A + BKC ),
• BMI minimize λ1 x
A0 +
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
r X i=1
xi Ai +
α := max Re λi i
r X s X
! x` xk B`k
`=1 k=1
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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concluding remarks
nonsmooth formulations • stabilization minimize
α(A + BKC ),
K
• BMI minimize λ1
A0 +
x
r X i=1
xi Ai +
α := max Re λi i
r X s X
! x` xk B`k
`=1 k=1
• H∞ synthesis minimize sup σ (Tw →z (K , jω)) K
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
ω
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
applications
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concluding remarks
nonsmooth formulations • stabilization minimize
α(A + BKC ),
K
• BMI minimize λ1
A0 +
x
r X i=1
xi Ai +
α := max Re λi i
r X s X
! x` xk B`k
`=1 k=1
• H∞ synthesis minimize sup σ (Tw →z (K , jω)) K
ω
• many others
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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concluding remarks
closed-loop H∞ norm w
z P(s)
K (s)
Tw →z (K (s)) := P11 + P12 K (I − P22 K )−1 P21 à composite k · k∞ ◦ Tw →z is Clarke regular à ⇒ exhaustive description of subdifferential ∂(k · k∞ ◦ Tw →z )(K ) à ⇒ variety of algorithms P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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nonsmooth formulations
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nonsmooth descent algorithms
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Clarke subdifferential of H∞ norm with standard notations define closed-loop data A(K ) C(K )
:= A + B2 KC2 , B(K ) := C1 + D12 KC2 , D(K )
:= B1 + B2 KD21 , := D11 + D12 KD21 ,
introduce notation Tw →z (K , s) G12 (K , s) := G21 (K , s) ? C(K ) D(K ) D12 −1 B(K ) B2 + (sI − A(K )) . C2 D21 ?
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
Clarke subdifferential of H∞ norm subdif. is convex set of subgradients ΦY ’s Pp
ν=1
T Re G21 (K , jων ) Tw →z (K , jων )H Qν Yν QνH G12 (K , jων ) kTw →z (K )k−1 ∞
à ων are active (peak) frequencies at K à Y := (Y1 , . . . , Yp ) ranges over (convex) {Y = (Y1 , . . . , Yp ) : Yi = YiH , 0,
p X
Tr (Yν ) = 1}
ν=1
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
convergent nonsmooth descent method à Ω(K ) set of active frequencies
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
convergent nonsmooth descent method à Ω(K ) set of active frequencies à define enriched set Ωe (K ) ⊇ Ω(K )
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
convergent nonsmooth descent method à Ω(K ) set of active frequencies à define enriched set Ωe (K ) ⊇ Ω(K ) à define descent function (1st-order model) θ(K ) :=
inf
sup
sup
H∈R m2 ×p2 ω∈Ωe (K ) Tr Yω =1,Yω 0
{. . .
−kTw →z (K )k∞ + σ(Tw →z (K , jω)) + hΦYω , Hi + 21 δkHk2F }
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
♦
applications
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concluding remarks
convergent nonsmooth descent method à Ω(K ) set of active frequencies à define enriched set Ωe (K ) ⊇ Ω(K ) à define descent function (1st-order model) θ(K ) :=
inf
sup
sup
H∈R m2 ×p2 ω∈Ωe (K ) Tr Yω =1,Yω 0
{. . .
−kTw →z (K )k∞ + σ(Tw →z (K , jω)) + hΦYω , Hi + 21 δkHk2F } à H is (controller) search direction computed explicitly
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
convergent nonsmooth descent method à Fenchel duality =⇒ descent direction at K H(K ) := −
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
1 δ
X
τω Φ Y ω .
ω∈Ωe (K )
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
nonsmooth descent algorithms
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applications
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concluding remarks
convergent nonsmooth descent method à Fenchel duality =⇒ descent direction at K H(K ) := −
1 δ
X
τω Φ Y ω .
ω∈Ωe (K )
à and SDP dual form θ(K )
:=
sup sup {−kTw →z (K )k∞ . . . Tr Yω =1,Yω 0 Ωe (K ) τω =1,τω ≥0 P P 1 k τω ΦYω k2F } τω σ(Tw →z (K , jω)) − 2δ
P
+
Ωe (K )
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
Ωe (K )
IME, ONERA & Universit´ e Paul Sabatier/Math.
♦
context
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nonsmooth formulations
nonsmooth descent algorithms
♦
applications
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concluding remarks
convergent nonsmooth descent method à Fenchel duality =⇒ descent direction at K H(K ) := −
1 δ
X
τω Φ Y ω .
ω∈Ωe (K )
à and SDP dual form θ(K )
:=
sup sup {−kTw →z (K )k∞ . . . Tr Yω =1,Yω 0 Ωe (K ) τω =1,τω ≥0 P P 1 k τω ΦYω k2F } τω σ(Tw →z (K , jω)) − 2δ
P
+
Ωe (K )
Ωe (K )
à θ(K ) ≤ 0, ∀K and θ(K ) = 0 ⇔ 0 ∈ ∂(k · k∞ ◦ Tw →z )(K )
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
♦
context
♦
♦
nonsmooth formulations
nonsmooth descent algorithms
♦
applications
♦
concluding remarks
convergent nonsmooth descent method à Fenchel duality =⇒ descent direction at K H(K ) := −
1 δ
X
τω Φ Y ω .
ω∈Ωe (K )
à and SDP dual form θ(K )
:=
sup sup {−kTw →z (K )k∞ . . . Tr Yω =1,Yω 0 Ωe (K ) τω =1,τω ≥0 P P 1 k τω ΦYω k2F } τω σ(Tw →z (K , jω)) − 2δ
P
+
Ωe (K )
Ωe (K )
à θ(K ) ≤ 0, ∀K and θ(K ) = 0 ⇔ 0 ∈ ∂(k · k∞ ◦ Tw →z )(K ) à algorithm : line search using H(K ) with θ(K ) as stopping test
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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concluding remarks
example of enriched set Ωe (K ) • Ωe (K ) captures extra info. on σ(Tw →z (K , jω)) =⇒ better steps 2.2
2.1
max singular value
2
1.9
1.8
1.7
1.6
1.5 −5 10
−4
10
−3
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
10
freq. (rad./s.)
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
fixed-order output-feedback H∞ synthesis examples from Leibfritz’s collection
problem AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(n, m, p) (9, 1, 5) (4, 2, 1) (4, 2, 2) (55, 2, 2) (55, 2, 2) (82, 4, 4) (130, 1, 2) (240, 1, 2) (480, 1, 2)
order 0 0 0 0 1 0 0 0 0
iter 20 4 31 15 46 44 11 2 2
cpu (sec.) 45 7 51 294 408 1501 1112 3052 4785
nonsmooth H∞ 2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
H∞ AL 2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
FW 2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
H∞ full 1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
H∞ synthesis with nonsmooth algorithm low-level SUNW, Sun-Blade-1500
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
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applications
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concluding remarks
extensions à pure stabilization problems A + B2 KC2 is Hurtwitz iff kC2 (sI − (A + B2 KC2 ))−1 B2 k∞ < ∞ à multidisk problems minimize f (K ) := max kTw i →z i (K )k∞ K
i=1,...,N
à structured feedback design K = K0 + L diag(κ)R à etc
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
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context
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nonsmooth formulations
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nonsmooth descent algorithms
♦
applications
♦
concluding remarks
concluding remarks à nonsmooth approach quite effective on variety of problems
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
♦
context
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nonsmooth formulations
♦
nonsmooth descent algorithms
♦
applications
♦
concluding remarks
concluding remarks à nonsmooth approach quite effective on variety of problems à easily extended to any controller structure
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
♦
context
♦
nonsmooth formulations
♦
nonsmooth descent algorithms
♦
applications
♦
concluding remarks
concluding remarks à nonsmooth approach quite effective on variety of problems à easily extended to any controller structure à convergence certificate
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
♦
context
♦
nonsmooth formulations
♦
nonsmooth descent algorithms
♦
applications
♦
concluding remarks
concluding remarks à à à à
nonsmooth approach quite effective on variety of problems easily extended to any controller structure convergence certificate can handle large state systems
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
♦
context
♦
nonsmooth formulations
♦
nonsmooth descent algorithms
♦
applications
♦
concluding remarks
concluding remarks à à à à à
nonsmooth approach quite effective on variety of problems easily extended to any controller structure convergence certificate can handle large state systems promising for IQC and µ synthesis
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
♦
context
♦
nonsmooth formulations
♦
nonsmooth descent algorithms
♦
applications
♦
concluding remarks
concluding remarks à à à à à à
nonsmooth approach quite effective on variety of problems easily extended to any controller structure convergence certificate can handle large state systems promising for IQC and µ synthesis second-order version in progress
P. Pellanda, P. Apkarian & D. Noll: Nonsmooth H∞ Synthesis
IME, ONERA & Universit´ e Paul Sabatier/Math.
