nonsmooth structured control design P. APKARIAN
(ONERA and UPS/Institut de Mathématiques de Toulouse)
V. BOMPART
(ONERA UPS/Institut de Mathématiques de Toulouse)
D. NOLL
(UPS/Institut de Mathématiques de Toulouse)
ALCOSP 2007 - St. Petersburg, August 29
motivation nonsmooth optimization controller design conclusion key references
outline
motivation nonsmooth optimization controller design conclusion
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 2/22
motivation nonsmooth optimization controller design conclusion key references
context challenging practical design problems: reduced- and fixed-order synthesis structured and fixed-architecture synthesis problems general robust control with uncertain and/or nonlinear components simultaneous model/controller design, multimodel control multi-objective frequency- and time-domain designs combinations of the above, etc
classical design methods (Riccati, LMI) fail on sizeable plants. D-K schemes and heuristic methods (homotopy,. . . ) not satisfactory solutions based on LMI relaxations often very conservative nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 3/22
motivation nonsmooth optimization controller design conclusion key references
context challenging practical design problems: reduced- and fixed-order synthesis structured and fixed-architecture synthesis problems general robust control with uncertain and/or nonlinear components simultaneous model/controller design, multimodel control multi-objective frequency- and time-domain designs combinations of the above, etc
classical design methods (Riccati, LMI) fail on sizeable plants. D-K schemes and heuristic methods (homotopy,. . . ) not satisfactory solutions based on LMI relaxations often very conservative nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 3/22
motivation nonsmooth optimization controller design conclusion key references
context challenging practical design problems: reduced- and fixed-order synthesis structured and fixed-architecture synthesis problems general robust control with uncertain and/or nonlinear components simultaneous model/controller design, multimodel control multi-objective frequency- and time-domain designs combinations of the above, etc
classical design methods (Riccati, LMI) fail on sizeable plants. D-K schemes and heuristic methods (homotopy,. . . ) not satisfactory solutions based on LMI relaxations often very conservative nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 3/22
motivation nonsmooth optimization controller design conclusion key references
context challenging practical design problems: reduced- and fixed-order synthesis structured and fixed-architecture synthesis problems general robust control with uncertain and/or nonlinear components simultaneous model/controller design, multimodel control multi-objective frequency- and time-domain designs combinations of the above, etc
classical design methods (Riccati, LMI) fail on sizeable plants. D-K schemes and heuristic methods (homotopy,. . . ) not satisfactory solutions based on LMI relaxations often very conservative nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 3/22
motivation nonsmooth optimization controller design conclusion key references
an illustrative example Boeing 767 at flutter condition stabilization with static output feedback u = Ky : x˙ A B x = y C 0 u A = 55 × 55
B = 55 × 2
C = 2 × 55
K =2×2
BMI: (A + BKC )T P + P(A + BKC ) ≺ 0 et P = P T 0 ⇒ 1544 variables most of them (1540) for Lyapunov matrix P min α(A + BKC ) where α , max Re λi is spectral abscissa K
i
⇒ 4 variables : controller parameters
⇒ recast controller design as a nonsmooth program nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 4/22
motivation nonsmooth optimization controller design conclusion key references
an illustrative example Boeing 767 at flutter condition stabilization with static output feedback u = Ky : x˙ A B x = y C 0 u A = 55 × 55
B = 55 × 2
C = 2 × 55
K =2×2
BMI: (A + BKC )T P + P(A + BKC ) ≺ 0 et P = P T 0 ⇒ 1544 variables most of them (1540) for Lyapunov matrix P min α(A + BKC ) where α , max Re λi is spectral abscissa K
i
⇒ 4 variables : controller parameters
⇒ recast controller design as a nonsmooth program nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 4/22
motivation nonsmooth optimization controller design conclusion key references
an illustrative example Boeing 767 at flutter condition stabilization with static output feedback u = Ky : x˙ A B x = y C 0 u A = 55 × 55
B = 55 × 2
C = 2 × 55
K =2×2
BMI: (A + BKC )T P + P(A + BKC ) ≺ 0 et P = P T 0 ⇒ 1544 variables most of them (1540) for Lyapunov matrix P min α(A + BKC ) where α , max Re λi is spectral abscissa K
i
⇒ 4 variables : controller parameters
⇒ recast controller design as a nonsmooth program nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 4/22
motivation nonsmooth optimization controller design conclusion key references
controller design as nonsmooth (minimax) optimization stabilization min max Re λi (A + BKC ) K
i
H∞ synthesis min K
max ω∈[0,+∞]
max σi Tw →z (K , jω) i
time-domain design n + o + min max max z(K , t)−zmax (t) , zmin (t)−z(K , t) K
t∈[0,+∞]
generic BMI problem X XX min max λi A0 + xk Ak + xk xl Bkl x
i
k
k
l
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 5/22
motivation nonsmooth optimization controller design conclusion key references
controller design as nonsmooth (minimax) optimization stabilization min max Re λi (A + BKC ) K
i
H∞ synthesis min K
max ω∈[0,+∞]
max σi Tw →z (K , jω) i
time-domain design n + o + min max max z(K , t)−zmax (t) , zmin (t)−z(K , t) K
t∈[0,+∞]
generic BMI problem X XX min max λi A0 + xk Ak + xk xl Bkl x
i
k
k
l
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 5/22
motivation nonsmooth optimization controller design conclusion key references
controller design as nonsmooth (minimax) optimization stabilization min max Re λi (A + BKC ) K
i
H∞ synthesis min K
max ω∈[0,+∞]
max σi Tw →z (K , jω) i
time-domain design n + o + min max max z(K , t)−zmax (t) , zmin (t)−z(K , t) K
t∈[0,+∞]
generic BMI problem X XX min max λi A0 + xk Ak + xk xl Bkl x
i
k
k
l
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 5/22
motivation nonsmooth optimization controller design conclusion key references
controller design as nonsmooth (minimax) optimization stabilization min max Re λi (A + BKC ) K
i
H∞ synthesis min K
max ω∈[0,+∞]
max σi Tw →z (K , jω) i
time-domain design n + o + min max max z(K , t)−zmax (t) , zmin (t)−z(K , t) K
t∈[0,+∞]
generic BMI problem X XX min max λi A0 + xk Ak + xk xl Bkl x
i
k
k
l
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 5/22
motivation nonsmooth optimization controller design conclusion key references
controller design as nonsmooth optimization • these optimization problems are nonsmooth and/or nonconvex • special composite structure max λ1 ◦ T (K , x) x∈X
where maxx∈X λ1 is convex and T (K , x) differentiable wrt. K , ∀x Clarke subdifferential Clarke regularity ⇒ specialized and efficient techniques can be developed • directly formulated in controller parameter space (no Lyapunov variables) • flexible: to handle controller structural constraints • efficient: to handle large control problems nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 6/22
motivation nonsmooth optimization controller design conclusion key references
controller design as nonsmooth optimization • these optimization problems are nonsmooth and/or nonconvex • special composite structure max λ1 ◦ T (K , x) x∈X
where maxx∈X λ1 is convex and T (K , x) differentiable wrt. K , ∀x Clarke subdifferential Clarke regularity ⇒ specialized and efficient techniques can be developed • directly formulated in controller parameter space (no Lyapunov variables) • flexible: to handle controller structural constraints • efficient: to handle large control problems nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 6/22
motivation nonsmooth optimization controller design conclusion key references
controller design as nonsmooth optimization • these optimization problems are nonsmooth and/or nonconvex • special composite structure max λ1 ◦ T (K , x) x∈X
where maxx∈X λ1 is convex and T (K , x) differentiable wrt. K , ∀x Clarke subdifferential Clarke regularity ⇒ specialized and efficient techniques can be developed • directly formulated in controller parameter space (no Lyapunov variables) • flexible: to handle controller structural constraints • efficient: to handle large control problems nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 6/22
motivation nonsmooth optimization controller design conclusion key references
subdifferential and subgradients nonsmooth minimax optimization problems of the form min max f (K , x) K x∈X | {z } ,f∞ (K )
X stands for: a finite set of indices : X = {1, · · · , q} min max Re λi (A + BKC ) K
i
a time- or frequency-domain closed interval : X = [0, +∞], X = [0, tmax ], X = [ω1 , ω2 ], · · · min K
double max :
max
t∈[0,+∞]
z(K , t) − zmax (t)
minK maxω∈[0,+∞] maxi σi Tw →z (K , jω)
.
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 7/22
motivation nonsmooth optimization controller design conclusion key references
subdifferential and subgradients nonsmooth minimax optimization problems of the form min max f (K , x) K x∈X | {z } ,f∞ (K )
X stands for: a finite set of indices : X = {1, · · · , q} min max Re λi (A + BKC ) K
i
a time- or frequency-domain closed interval : X = [0, +∞], X = [0, tmax ], X = [ω1 , ω2 ], · · · min K
double max :
max
t∈[0,+∞]
z(K , t) − zmax (t)
minK maxω∈[0,+∞] maxi σi Tw →z (K , jω)
.
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 7/22
motivation nonsmooth optimization controller design conclusion key references
subdifferential and subgradients nonsmooth minimax optimization problems of the form min max f (K , x) K x∈X | {z } ,f∞ (K )
X stands for: a finite set of indices : X = {1, · · · , q} min max Re λi (A + BKC ) K
i
a time- or frequency-domain closed interval : X = [0, +∞], X = [0, tmax ], X = [ω1 , ω2 ], · · · min K
double max :
max
t∈[0,+∞]
z(K , t) − zmax (t)
minK maxω∈[0,+∞] maxi σi Tw →z (K , jω)
.
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 7/22
motivation nonsmooth optimization controller design conclusion key references
subdifferential and subgradients nonsmooth minimax optimization problems of the form min max f (K , x) K x∈X | {z } ,f∞ (K )
X stands for: a finite set of indices : X = {1, · · · , q} min max Re λi (A + BKC ) K
i
a time- or frequency-domain closed interval : X = [0, +∞], X = [0, tmax ], X = [ω1 , ω2 ], · · · min K
double max :
max
t∈[0,+∞]
z(K , t) − zmax (t)
minK maxω∈[0,+∞] maxi σi Tw →z (K , jω)
.
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 7/22
motivation nonsmooth optimization controller design conclusion key references
subdifferential and subgradients nonsmooth minimax optimization problems of the form min max f (K , x) K x∈X | {z } ,f∞ (K )
assume (for simplicity) that f is continuous and K 7→ f (K , x) is C 1 ∀x ∈ X . then f∞ is loc. Lipschitz and admits a Clarke subdifferential ∀K : ∂f∞ (K ) =
co
ˆ (K ) x∈X
∇K f (K , x)
ˆ (K ) , {x ∈ X : f∞ (K ) = f (x, K )} active set. where X vectors φ ∈ ∂f∞ (K ) are Clarke subgradients of f∞ at K . nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 7/22
motivation nonsmooth optimization controller design conclusion key references
nonsmooth optimality condition Theorem ([Clarke, 1983]) K ? is a local minimum of f∞ =⇒ 0 ∈ ∂f∞ (K ? ) • define optimality function θ to check whether 0 ∈ ∂f∞ (K ) satisfied at K . • if not, find a descent step H(K ) for f∞ (K ), i.e. ∃tK > 0 such that, for all t ∈ (0, tK ] f∞ (K + tH(K )) < f∞ (K ) • steepest descent:
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 8/22
motivation nonsmooth optimization controller design conclusion key references
nonsmooth optimality condition Theorem ([Clarke, 1983]) K ? is a local minimum of f∞ =⇒ 0 ∈ ∂f∞ (K ? ) • define optimality function θ to check whether 0 ∈ ∂f∞ (K ) satisfied at K . • if not, find a descent step H(K ) for f∞ (K ), i.e. ∃tK > 0 such that, for all t ∈ (0, tK ] f∞ (K + tH(K )) < f∞ (K ) • steepest descent:
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 8/22
motivation nonsmooth optimization controller design conclusion key references
nonsmooth optimality condition Theorem ([Clarke, 1983]) K ? is a local minimum of f∞ =⇒ 0 ∈ ∂f∞ (K ? ) • define optimality function θ to check whether 0 ∈ ∂f∞ (K ) satisfied at K . • if not, find a descent step H(K ) for f∞ (K ), i.e. ∃tK > 0 such that, for all t ∈ (0, tK ] f∞ (K + tH(K )) < f∞ (K ) • steepest descent: H(K ) , −arg min H |
max φ∈∂f∞ (K )
hφ, Hi + 12 kHk2 {z }
,θ1 (K ) nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 8/22
motivation nonsmooth optimization controller design conclusion key references
nonsmooth optimality condition Theorem ([Clarke, 1983]) K ? is a local minimum of f∞ =⇒ 0 ∈ ∂f∞ (K ? ) • define optimality function θ to check whether 0 ∈ ∂f∞ (K ) satisfied at K . • if not, find a descent step H(K ) for f∞ (K ), i.e. ∃tK > 0 such that, for all t ∈ (0, tK ] f∞ (K + tH(K )) < f∞ (K ) • steepest descent: H(K ) = −arg
min φ∈∂f∞ (K )
1 2
kφk2
⇒ K 7→ H(K ) not continuous ! convergence problem. nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 8/22
motivation nonsmooth optimization controller design conclusion key references
nonsmooth descent direction H(K ) , arg min max f (K , x) + h∇K f (K , x), Hi + 12 δkHk2 −f∞ (K ) H x∈X ˆe (K ) | {z } 1st-order quadratic local model of f (·,x) at K
ˆe (K ) ⊃ X ˆ (K ), finite extension of active set δ>0,X
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 9/22
motivation nonsmooth optimization controller design conclusion key references
nonsmooth descent direction H(K ) , arg min max f (K , x) + h∇K f (K , x), Hi + 12 δkHk2 −f∞ (K ) H x∈X ˆe (K )
|
{z
}
1st-order piecewise quadratic local model of f∞ (·) at K
ˆe (K ) ⊃ X ˆ (K ), finite extension of active set δ>0,X
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 9/22
motivation nonsmooth optimization controller design conclusion key references
nonsmooth descent direction H(K ) , arg min max f (K , x) + h∇K f (K , x), Hi + 12 δkHk2 − f∞ (K ) H x∈X ˆe (K )
|
{z
optimality function θ(K )
ˆe (K ) ⊃ X ˆ (K ), finite extension of active set δ>0,X Proposition ([Apkarian and Noll, 2006-I]) θ(K ) ≤ 0 and θ(K ) = 0 ⇐⇒ 0 ∈ ∂f∞ (K ) θ(K ) criticality measure of K for f∞ θ(K ) termination test and (local) optimality certificate K 7→ θ(K ) continuous 0 (K ; H(K )) ≤ θ(K ) − 1 kH(K )k2 < 0 f∞ 2δ H(K ) descent direction for f∞ at K
K 7→ H(K ) continuous nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 9/22
}
motivation nonsmooth optimization controller design conclusion key references
nonsmooth descent direction H(K ) , arg min max f (K , x) + h∇K f (K , x), Hi + 12 δkHk2 − f∞ (K ) H x∈X ˆe (K )
|
{z
optimality function θ(K )
ˆe (K ) ⊃ X ˆ (K ), finite extension of active set δ>0,X Proposition ([Apkarian and Noll, 2006-I]) θ(K ) ≤ 0 and θ(K ) = 0 ⇐⇒ 0 ∈ ∂f∞ (K ) θ(K ) criticality measure of K for f∞ θ(K ) termination test and (local) optimality certificate K 7→ θ(K ) continuous 0 (K ; H(K )) ≤ θ(K ) − 1 kH(K )k2 < 0 f∞ 2δ H(K ) descent direction for f∞ at K
K 7→ H(K ) continuous nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 9/22
}
motivation nonsmooth optimization controller design conclusion key references
nonsmooth optimality condition • primal QP problem min max f (K , x) + h∇K f (K , x), Hi + 12 δkHk2 − f∞ (K ) H x∈X ˆ e (K )
|
{z
}
optimality function θ(K )
• dual QP problem (equivalent) θ(K ) = − min
τx ≥0 P τx =1
X ˆ e (K ) x∈X
2
1 X
τx f∞ (K )−f (K , x) + τx ∇K f (K , x)
2δ
with H(K ) = −
ˆ e (K ) x∈X
1 δ
X
τx? ∇K f (K , x)
ˆ e (K ) x∈X
• select primal or dual with smaller dimension nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 10/22
motivation nonsmooth optimization controller design conclusion key references
nonsmooth optimality condition • primal QP problem min max f (K , x) + h∇K f (K , x), Hi + 12 δkHk2 − f∞ (K ) H x∈X ˆ e (K )
|
{z
}
optimality function θ(K )
• dual QP problem (equivalent) θ(K ) = − min
τx ≥0 P τx =1
X ˆ e (K ) x∈X
2
1 X
τx f∞ (K )−f (K , x) + τx ∇K f (K , x)
2δ
with H(K ) = −
ˆ e (K ) x∈X
1 δ
X
τx? ∇K f (K , x)
ˆ e (K ) x∈X
• select primal or dual with smaller dimension nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 10/22
motivation nonsmooth optimization controller design conclusion key references
nonsmooth optimality condition • primal QP problem min max f (K , x) + h∇K f (K , x), Hi + 12 δkHk2 − f∞ (K ) H x∈X ˆ e (K )
|
{z
}
optimality function θ(K )
• dual QP problem (equivalent) θ(K ) = − min
τx ≥0 P τx =1
X ˆ e (K ) x∈X
2
1 X
τx f∞ (K )−f (K , x) + τx ∇K f (K , x)
2δ
with H(K ) = −
ˆ e (K ) x∈X
1 δ
X
τx? ∇K f (K , x)
ˆ e (K ) x∈X
• select primal or dual with smaller dimension nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 10/22
motivation nonsmooth optimization controller design conclusion key references
nonsmooth descent algorithm Algorithm ([Apkarian and Noll, 2006-I, Polak, 1997]) set β ∈ (0, 1), δ > 0, εθ > 0. Initialize with controller K0 .
2
set counter l ← 0 ˆ (Kl ). find active set X
3
ˆe (Kl ). build a finite extension X
4
compute the optimality function value (convex QP) θ(Kl ) and the search direction H(Kl ).
5
if |θ(Kl )| < εθ (criticality test), stop. else compute the step-size tl such that
1
f∞ Kl + tl H(Kl ) − f∞ (Kl ) ≤ tl βθ(Kl ) 6
set Kl +1 ← Kl + tl H(Kl ), l ← l + 1 and go to step 2.
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 11/22
motivation nonsmooth optimization controller design conclusion key references
synthesis formulation z
w A x˙ P(s) z = C1 y C2
B1 D11 D21
P
B2 x D12 w 0 u
y
u K(κ)
• K(κ) defines controller structure • find κ free parameters to achieve stability, frequency- or time-domain constraints
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 12/22
motivation nonsmooth optimization controller design conclusion key references
structured controller design controller structural constraints K(κ) with κ ∈ Rq vector of free controller parameters
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 13/22
motivation nonsmooth optimization controller design conclusion key references
structured controller design controller structural constraints K(κ) with κ ∈ Rq vector of free controller parameters decentralized control (N subcontrollers)
A1K
K(κ) = C1 K
BK1
A2K ..
.
CK2
DK1
...
(vec DKN )T
. BKN
DK2 ..
. CKN
with κ = (vec A1K )T
BK2 ..
AN K ..
.
DKN T
.
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 13/22
motivation nonsmooth optimization controller design conclusion key references
structured controller design controller structural constraints K(κ) with κ ∈ Rq vector of free controller parameters PID control (m2 × m2 )
0 K(κ) = 0 Im2 Rd such that K (s) = DK + Rsi + s+τ . T here κ = τ (vec Ri ) (vec Rd )T
0 −τ Im2 Im 2
Ri Rd DK
(vec DK )T
T
.
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 13/22
motivation nonsmooth optimization controller design conclusion key references
structured controller design controller structural constraints K(κ) with κ ∈ Rq vector of free controller parameters observer-based control (LQG) K(κ) =
A − B2 Kc − Kf C2 −Kc
Kf 0
Kf state estimator gain, Kc state feedback gain. T κ = (vec Kf )T (vec Kc )T
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 13/22
motivation nonsmooth optimization controller design conclusion key references
structured controller design controller structural constraints K(κ) with κ ∈ Rq vector of free controller parameters fractional representations K (s, κ) = (Nm s m + . . . + N0 )(Dn s n + . . . + D0 )−1 Ni numerator coefficients Dj denominator coefficients T κ = . . . (vec Ni )T . . . (vec Dj )T . . .
etc any differentiable possibly nonlinear K(κ)
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 13/22
motivation nonsmooth optimization controller design conclusion key references
structured controller design controller structural constraints K(κ) with κ ∈ Rq vector of free controller parameters the nonsmooth problem becomes min max f (K(κ), x) | {z }
κ∈Rq x∈X
f∞ (K(κ))
chain rule for subdifferentials with K ∈ C 1 (Rq ): ∂(f∞ ◦ K)(κ) = K0 (κ)? [∂f∞ (K(κ))] subgradients ψ = JK (κ)T φ where JK Jacobian matrix of K, and φ ∈ ∂f∞ (K(κ)) nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 13/22
motivation nonsmooth optimization controller design conclusion key references
stabilization min max Re λi (A + B2 KC2 ) K i | {z } f∞ (K )
minimize spectral abscissa α , maxi Re λi . active set of closed-loop eigenvalues indices : ˆ (K ) = {i = 1 . . . n : Re λi (A + B2 KC2 ) = α(A + B2 KC2 )} X ˆe (K ) ⊃ X ˆ (K ) with neighboring eigenvalues. extended set X ˆe (K ), are simple. working assumption: all the closed-loop eigenvalues λi , i ∈ X subgradients φ(K ) ∈ ∂f∞ (K ) : X φ(K ) = τi Re (C2 vi uiH B2 )T ˆ (K ) i ∈X
vi right eigenvector associated with λi uiH left eigenvector associated with λi P τi ≥ 0 and i ∈Xˆ (K ) τi = 1 stop before convergence if α(A + B2 KC2 ) < 0 (K stabilizing). nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 14/22
motivation nonsmooth optimization controller design conclusion key references
stabilization min max Re λi (A + B2 KC2 ) K i | {z } f∞ (K )
minimize spectral abscissa α , maxi Re λi . active set of closed-loop eigenvalues indices : ˆ (K ) = {i = 1 . . . n : Re λi (A + B2 KC2 ) = α(A + B2 KC2 )} X ˆe (K ) ⊃ X ˆ (K ) with neighboring eigenvalues. extended set X ˆe (K ), are simple. working assumption: all the closed-loop eigenvalues λi , i ∈ X subgradients φ(K ) ∈ ∂f∞ (K ) : X φ(K ) = τi Re (C2 vi uiH B2 )T ˆ (K ) i ∈X
vi right eigenvector associated with λi uiH left eigenvector associated with λi P τi ≥ 0 and i ∈Xˆ (K ) τi = 1 stop before convergence if α(A + B2 KC2 ) < 0 (K stabilizing). nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 14/22
motivation nonsmooth optimization controller design conclusion key references
stabilization min max Re λi (A + B2 KC2 ) K i | {z } f∞ (K )
minimize spectral abscissa α , maxi Re λi . active set of closed-loop eigenvalues indices : ˆ (K ) = {i = 1 . . . n : Re λi (A + B2 KC2 ) = α(A + B2 KC2 )} X ˆe (K ) ⊃ X ˆ (K ) with neighboring eigenvalues. extended set X ˆe (K ), are simple. working assumption: all the closed-loop eigenvalues λi , i ∈ X subgradients φ(K ) ∈ ∂f∞ (K ) : X φ(K ) = τi Re (C2 vi uiH B2 )T ˆ (K ) i ∈X
vi right eigenvector associated with λi uiH left eigenvector associated with λi P τi ≥ 0 and i ∈Xˆ (K ) τi = 1 stop before convergence if α(A + B2 KC2 ) < 0 (K stabilizing). nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 14/22
motivation nonsmooth optimization controller design conclusion key references
stabilization min max Re λi (A + B2 KC2 ) K i | {z } f∞ (K )
minimize spectral abscissa α , maxi Re λi . active set of closed-loop eigenvalues indices : ˆ (K ) = {i = 1 . . . n : Re λi (A + B2 KC2 ) = α(A + B2 KC2 )} X ˆe (K ) ⊃ X ˆ (K ) with neighboring eigenvalues. extended set X ˆe (K ), are simple. working assumption: all the closed-loop eigenvalues λi , i ∈ X subgradients φ(K ) ∈ ∂f∞ (K ) : X φ(K ) = τi Re (C2 vi uiH B2 )T ˆ (K ) i ∈X
vi right eigenvector associated with λi uiH left eigenvector associated with λi P τi ≥ 0 and i ∈Xˆ (K ) τi = 1 stop before convergence if α(A + B2 KC2 ) < 0 (K stabilizing). nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 14/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m2 , p2 )
iter. stab.
cpu (s)
iter. conv.
cpu (s)
α
θ
AC8 (SOF) AC8 (nK = 1)
(9, 1, 5) (9, 1, 5)
1 5
< 0.1 0.2
9 17
0.2 0.4
−4.45 · 10−1 −4.45 · 10−1
−5.60 · 10−17 −5.01 · 10−27
HE2 (SOF) HE2 (nK = 1)
(4, 2, 1) (4, 2, 1)
1 1
< 0.1 < 0.1
216 (28)
2.3 0.7
−2.39 · 10−1 −2.31 · 10−1
−9.77 · 10−6 −1.53
REA2 (SOF) REA2 (PID)
(4, 2, 2) (4, 2, 2)
1 1
< 0.1 < 0.1
(49) (19)
0.83 0.6
−2.46 −1.27
−1.1 · 10−2 −1.4 · 10−1
AC10 (SOF)
(55, 4, 4)
1
0.5
(99)
27.9
−7.99 · 10−2
−8.00 · 10−4
AC8 (SOF) : modes en boucle fermée 0.5 0.4 0.3
Im[λ(A+B2KC2)]
0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −1
−0.9
−0.8
−0.7
−0.6
−0.5 −0.4 −0.3 Re[λ(A+B2KC2)]
−0.2
−0.1
0
0.1
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 15/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m2 , p2 )
iter. stab.
cpu (s)
iter. conv.
cpu (s)
α
θ
AC8 (SOF) AC8 (nK = 1)
(9, 1, 5) (9, 1, 5)
1 5
< 0.1 0.2
9 17
0.2 0.4
−4.45 · 10−1 −4.45 · 10−1
−5.60 · 10−17 −5.01 · 10−27
HE2 (SOF) HE2 (nK = 1)
(4, 2, 1) (4, 2, 1)
1 1
< 0.1 < 0.1
216 (28)
2.3 0.7
−2.39 · 10−1 −2.31 · 10−1
−9.77 · 10−6 −1.53
REA2 (SOF) REA2 (PID)
(4, 2, 2) (4, 2, 2)
1 1
< 0.1 < 0.1
(49) (19)
0.83 0.6
−2.46 −1.27
−1.1 · 10−2 −1.4 · 10−1
AC10 (SOF)
(55, 4, 4)
1
0.5
(99)
27.9
−7.99 · 10−2
−8.00 · 10−4
AC8 (SOF) : modes en boucle fermée 0.5 0.4 0.3
Im[λ(A+B2KC2)]
0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −1
−0.9
−0.8
−0.7
−0.6
−0.5 −0.4 −0.3 Re[λ(A+B2KC2)]
−0.2
−0.1
0
0.1
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 15/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m2 , p2 )
iter. stab.
cpu (s)
iter. conv.
cpu (s)
α
θ
AC8 (SOF) AC8 (nK = 1)
(9, 1, 5) (9, 1, 5)
1 5
< 0.1 0.2
9 17
0.2 0.4
−4.45 · 10−1 −4.45 · 10−1
−5.60 · 10−17 −5.01 · 10−27
HE2 (SOF) HE2 (nK = 1)
(4, 2, 1) (4, 2, 1)
1 1
< 0.1 < 0.1
216 (28)
2.3 0.7
−2.39 · 10−1 −2.31 · 10−1
−9.77 · 10−6 −1.53
REA2 (SOF) REA2 (PID)
(4, 2, 2) (4, 2, 2)
1 1
< 0.1 < 0.1
(49) (19)
0.83 0.6
−2.46 −1.27
−1.1 · 10−2 −1.4 · 10−1
AC10 (SOF)
(55, 4, 4)
1
0.5
(99)
27.9
−7.99 · 10−2
−8.00 · 10−4
AC8 (SOF) : modes en boucle fermée 0.5 0.4 0.3
Im[λ(A+B2KC2)]
0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −1
−0.9
−0.8
−0.7
−0.6
−0.5 −0.4 −0.3 Re[λ(A+B2KC2)]
−0.2
−0.1
0
0.1
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 15/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m2 , p2 )
iter. stab.
cpu (s)
iter. conv.
cpu (s)
α
θ
AC8 (SOF) AC8 (nK = 1)
(9, 1, 5) (9, 1, 5)
1 5
< 0.1 0.2
9 17
0.2 0.4
−4.45 · 10−1 −4.45 · 10−1
−5.60 · 10−17 −5.01 · 10−27
HE2 (SOF) HE2 (nK = 1)
(4, 2, 1) (4, 2, 1)
1 1
< 0.1 < 0.1
216 (28)
2.3 0.7
−2.39 · 10−1 −2.31 · 10−1
−9.77 · 10−6 −1.53
REA2 (SOF) REA2 (PID)
(4, 2, 2) (4, 2, 2)
1 1
< 0.1 < 0.1
(49) (19)
0.83 0.6
−2.46 −1.27
−1.1 · 10−2 −1.4 · 10−1
AC10 (SOF)
(55, 4, 4)
1
0.5
(99)
27.9
−7.99 · 10−2
−8.00 · 10−4
AC8 (SOF) : modes en boucle fermée 0.5 0.4 0.3
Im[λ(A+B2KC2)]
0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −1
−0.9
−0.8
−0.7
−0.6
−0.5 −0.4 −0.3 Re[λ(A+B2KC2)]
−0.2
−0.1
0
0.1
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 15/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m2 , p2 )
iter. stab.
cpu (s)
iter. conv.
cpu (s)
α
θ
AC8 (SOF) AC8 (nK = 1)
(9, 1, 5) (9, 1, 5)
1 5
< 0.1 0.2
9 17
0.2 0.4
−4.45 · 10−1 −4.45 · 10−1
−5.60 · 10−17 −5.01 · 10−27
HE2 (SOF) HE2 (nK = 1)
(4, 2, 1) (4, 2, 1)
1 1
< 0.1 < 0.1
216 (28)
2.3 0.7
−2.39 · 10−1 −2.31 · 10−1
−9.77 · 10−6 −1.53
REA2 (SOF) REA2 (PID)
(4, 2, 2) (4, 2, 2)
1 1
< 0.1 < 0.1
(49) (19)
0.83 0.6
−2.46 −1.27
−1.1 · 10−2 −1.4 · 10−1
AC10 (SOF)
(55, 4, 4)
1
0.5
(99)
27.9
−7.99 · 10−2
−8.00 · 10−4
AC8 (SOF) : modes en boucle fermée 0.5 0.4 0.3
Im[λ(A+B2KC2)]
0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −1
−0.9
−0.8
−0.7
−0.6
−0.5 −0.4 −0.3 Re[λ(A+B2KC2)]
−0.2
−0.1
0
0.1
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 15/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m2 , p2 )
iter. stab.
cpu (s)
iter. conv.
cpu (s)
α
θ
AC8 (SOF) AC8 (nK = 1)
(9, 1, 5) (9, 1, 5)
1 5
< 0.1 0.2
9 17
0.2 0.4
−4.45 · 10−1 −4.45 · 10−1
−5.60 · 10−17 −5.01 · 10−27
HE2 (SOF) HE2 (nK = 1)
(4, 2, 1) (4, 2, 1)
1 1
< 0.1 < 0.1
216 (28)
2.3 0.7
−2.39 · 10−1 −2.31 · 10−1
−9.77 · 10−6 −1.53
REA2 (SOF) REA2 (PID)
(4, 2, 2) (4, 2, 2)
1 1
< 0.1 < 0.1
(49) (19)
0.83 0.6
−2.46 −1.27
−1.1 · 10−2 −1.4 · 10−1
AC10 (SOF)
(55, 4, 4)
1
0.5
(99)
27.9
−7.99 · 10−2
−8.00 · 10−4
AC8 (SOF) : modes en boucle fermée 0.5 0.4 0.3
Im[λ(A+B2KC2)]
0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −1
−0.9
−0.8
−0.7
−0.6
−0.5 −0.4 −0.3 Re[λ(A+B2KC2)]
−0.2
−0.1
0
0.1
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 15/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m2 , p2 )
iter. stab.
cpu (s)
iter. conv.
cpu (s)
α
θ
AC8 (SOF) AC8 (nK = 1)
(9, 1, 5) (9, 1, 5)
1 5
< 0.1 0.2
9 17
0.2 0.4
−4.45 · 10−1 −4.45 · 10−1
−5.60 · 10−17 −5.01 · 10−27
HE2 (SOF) HE2 (nK = 1)
(4, 2, 1) (4, 2, 1)
1 1
< 0.1 < 0.1
216 (28)
2.3 0.7
−2.39 · 10−1 −2.31 · 10−1
−9.77 · 10−6 −1.53
REA2 (SOF) REA2 (PID)
(4, 2, 2) (4, 2, 2)
1 1
< 0.1 < 0.1
(49) (19)
0.83 0.6
−2.46 −1.27
−1.1 · 10−2 −1.4 · 10−1
AC10 (SOF)
(55, 4, 4)
1
0.5
(99)
27.9
−7.99 · 10−2
−8.00 · 10−4
AC8 (SOF) : modes en boucle fermée 0.5 0.4 0.3 3 modes non commandables
Im[λ(A+B2KC2)]
0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −1
−0.9
−0.8
−0.7
−0.6
−0.5 −0.4 −0.3 Re[λ(A+B2KC2)]
−0.2
−0.1
0
0.1
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 15/22
motivation nonsmooth optimization controller design conclusion key references
H∞ synthesis
Tw →z (K (s)) = w
z min K
max ω∈[0,+∞]
|
max σi Tw →z (K , jω) i {z }
=kTw →z (K ,.)k∞ =f∞ (K )
P(s) y
u K (s)
Tw →z (K (s)) := P11 + P12 K (I − P22 K )−1 P21 composite k · k∞ ◦ Tw →z is Clarke regular as a composite of convex and differentiable maps ⇒ exhaustive description of subdifferential ∂(k · k∞ ◦ Tw →z )(K ) nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 16/22
motivation nonsmooth optimization controller design conclusion key references
Clarke subdifferential of H∞ norm with standard notations define closed-loop data A(K ) := A + B2 KC2 , B(K ) := B1 + B2 KD21 , C(K ) := C1 + D12 KC2 , D(K ) := 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 ? • use appropriate plant augmentation if dynamic controller nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 17/22
motivation nonsmooth optimization controller design conclusion key references
Clarke subdifferential of H∞ norm Apkarian & Noll, 2006, [Apkarian and Noll, 2006-I] subdif. is convex-compact set of subgradients ΦY ’s Pp
ν=1 Re
T G21 (K , jων ) Tw →z (K , jων )H Qν Yν QνH G12 (K , jων ) kTw →z (K )k∞
ων are active (peak) frequencies at K Y := (Y1 , . . . , Yp ) ranges over spectraplex (convex) {Y = (Y1 , . . . , Yp ) : Yi = YiH , 0,
p X
Tr (Yν ) = 1}
ν=1
use chain rule K(κ)0∗ ∂(k · k∞ ◦ Tw →z )(K(κ)) if structured controllers nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 17/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m, p)
order
iter
cpu (sec.)
nonsmooth H∞
H∞ AL
FW
H∞ full
AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(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)
0 0 0 0 1 0 0 0 0
20 4 31 15 46 44 11 2 2
45 7 51 294 408 1501 1112 3052 4785
2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
Synthèse H∞ (AC8) 5.5 5 4.5 4
σ1(T(K,jω))
3.5 3 2.5 2 1.5 1 0.5 −4 10
−3
10
−2
10
−1
10
0
ω (rad/s)
10
1
10
2
10
3
10
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 18/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m, p)
order
iter
cpu (sec.)
nonsmooth H∞
H∞ AL
FW
H∞ full
AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(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)
0 0 0 0 1 0 0 0 0
20 4 31 15 46 44 11 2 2
45 7 51 294 408 1501 1112 3052 4785
2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
Synthèse H∞ (AC8) 5.5 5 4.5 4
σ1(T(K,jω))
3.5 3 2.5 2 1.5 1 0.5 −4 10
−3
10
−2
10
−1
10
0
ω (rad/s)
10
1
10
2
10
3
10
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 18/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m, p)
order
iter
cpu (sec.)
nonsmooth H∞
H∞ AL
FW
H∞ full
AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(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)
0 0 0 0 1 0 0 0 0
20 4 31 15 46 44 11 2 2
45 7 51 294 408 1501 1112 3052 4785
2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
Synthèse H∞ (AC8) 5.5 5 4.5 4
σ1(T(K,jω))
3.5 3 2.5 2 1.5 1 0.5 −4 10
−3
10
−2
10
−1
10
0
ω (rad/s)
10
1
10
2
10
3
10
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 18/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m, p)
order
iter
cpu (sec.)
nonsmooth H∞
H∞ AL
FW
H∞ full
AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(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)
0 0 0 0 1 0 0 0 0
20 4 31 15 46 44 11 2 2
45 7 51 294 408 1501 1112 3052 4785
2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
Synthèse H∞ (AC8) 5.5 5 4.5 4
σ1(T(K,jω))
3.5 3 2.5 2 1.5 1 0.5 −4 10
−3
10
−2
10
−1
10
0
ω (rad/s)
10
1
10
2
10
3
10
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 18/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m, p)
order
iter
cpu (sec.)
nonsmooth H∞
H∞ AL
FW
H∞ full
AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(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)
0 0 0 0 1 0 0 0 0
20 4 31 15 46 44 11 2 2
45 7 51 294 408 1501 1112 3052 4785
2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
Synthèse H∞ (AC8) 5.5 5 4.5 4
σ1(T(K,jω))
3.5 3 2.5 2 1.5 1 0.5 −4 10
−3
10
−2
10
−1
10
0
ω (rad/s)
10
1
10
2
10
3
10
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 18/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m, p)
order
iter
cpu (sec.)
nonsmooth H∞
H∞ AL
FW
H∞ full
AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(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)
0 0 0 0 1 0 0 0 0
20 4 31 15 46 44 11 2 2
45 7 51 294 408 1501 1112 3052 4785
2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
Synthèse H∞ (AC8) 5.5 5 4.5 4
σ1(T(K,jω))
3.5 3 2.5 2 1.5 1 0.5 −4 10
−3
10
−2
10
−1
10
0
ω (rad/s)
10
1
10
2
10
3
10
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 18/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m, p)
order
iter
cpu (sec.)
nonsmooth H∞
H∞ AL
FW
H∞ full
AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(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)
0 0 0 0 1 0 0 0 0
20 4 31 15 46 44 11 2 2
45 7 51 294 408 1501 1112 3052 4785
2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
Synthèse H∞ (AC8) 5.5 5 4.5 4
σ1(T(K,jω))
3.5 3 2.5 2 1.5 1 0.5 −4 10
−3
10
−2
10
−1
10
0
ω (rad/s)
10
1
10
2
10
3
10
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 18/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m, p)
order
iter
cpu (sec.)
nonsmooth H∞
H∞ AL
FW
H∞ full
AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(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)
0 0 0 0 1 0 0 0 0
20 4 31 15 46 44 11 2 2
45 7 51 294 408 1501 1112 3052 4785
2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
Synthèse H∞ (AC8) 5.5 5 4.5 4
σ1(T(K,jω))
3.5 3 2.5 2 1.5 1 0.5 −4 10
−3
10
−2
10
−1
10
0
ω (rad/s)
10
1
10
2
10
3
10
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 18/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m, p)
order
iter
cpu (sec.)
nonsmooth H∞
H∞ AL
FW
H∞ full
AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(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)
0 0 0 0 1 0 0 0 0
20 4 31 15 46 44 11 2 2
45 7 51 294 408 1501 1112 3052 4785
2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
Synthèse H∞ (AC8) 5.5 5 4.5 4
σ1(T(K,jω))
3.5 3 2.5 2 1.5 1 0.5 −4 10
−3
10
−2
10
−1
10
0
ω (rad/s)
10
1
10
2
10
3
10
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 18/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m, p)
order
iter
cpu (sec.)
nonsmooth H∞
H∞ AL
FW
H∞ full
AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(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)
0 0 0 0 1 0 0 0 0
20 4 31 15 46 44 11 2 2
45 7 51 294 408 1501 1112 3052 4785
2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
Synthèse H∞ (AC8) 5.5 5 4.5 4
σ1(T(K,jω))
3.5 3 2.5 2 1.5 1 0.5 −4 10
−3
10
−2
10
−1
10
0
ω (rad/s)
10
1
10
2
10
3
10
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 18/22
motivation nonsmooth optimization controller design conclusion key references
examples from Leibfritz’s collection problem
(n, m, p)
order
iter
cpu (sec.)
nonsmooth H∞
H∞ AL
FW
H∞ full
AC8 HE1 REA2 AC10 AC10 BDT2 HF1 CM4 CM5
(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)
0 0 0 0 1 0 0 0 0
20 4 31 15 46 44 11 2 2
45 7 51 294 408 1501 1112 3052 4785
2.005 0.154 1.192 13.11 10.21 0.8364 0.447 0.816 0.816
2.02 0.157 1.155 ∗ ∗ ∗ ∗ ∗ ∗
2.612 0.215 1.263 ∗ ∗ ∗ ∗ ∗ ∗
1.62 0.073 1.141 3.23 3.23 0.2340 0.447 ∗ ∗
Synthèse H∞ (AC8) 5.5 5 4.5 4
σ1(T(K,jω))
3.5 3 2.5 2 1.5 1 0.5 −4 10
−3
10
−2
10
−1
10
0
ω (rad/s)
10
1
10
2
10
3
10
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 18/22
motivation nonsmooth optimization controller design conclusion key references
time-domain design min max K ∈ t∈[0,+∞] |
n
+ + o z(K , t) − zmax (t) , zmin (t) − z(K , t) {z } =f∞ (K )
minimize maximum violation of SISO step response envelope specification zmin (t) ≤ z(K , t) ≤ zmax (t) stabilizing initial controller K0 . simulate step response for tl ∈ [0, T ]. active times (finite set) : ˆ (K ) = X
n o tl : z(K , tl ) − zmax (t) = f∞ (K ) or zmin (t) − z(K , tl ) = f∞ (K )
ˆe (K ) ⊃ X ˆ (K ) with neighboring times. extended set X subgradients φ(K ) ∈ ∂f∞ (K ) from time-domain simulations (Iterative Feedback Tuning, [Hjalmarsson et al., 94]). nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 19/22
motivation nonsmooth optimization controller design conclusion key references
time-domain design min max K ∈ t∈[0,+∞] |
n
+ + o z(K , t) − zmax (t) , zmin (t) − z(K , t) {z } =f∞ (K )
minimize maximum violation of SISO step response envelope specification zmin (t) ≤ z(K , t) ≤ zmax (t) stabilizing initial controller K0 . simulate step response for tl ∈ [0, T ]. active times (finite set) : ˆ (K ) = X
n o tl : z(K , tl ) − zmax (t) = f∞ (K ) or zmin (t) − z(K , tl ) = f∞ (K )
ˆe (K ) ⊃ X ˆ (K ) with neighboring times. extended set X subgradients φ(K ) ∈ ∂f∞ (K ) from time-domain simulations (Iterative Feedback Tuning, [Hjalmarsson et al., 94]). nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 19/22
motivation nonsmooth optimization controller design conclusion key references
time-domain design min max K ∈ t∈[0,+∞] |
n
+ + o z(K , t) − zmax (t) , zmin (t) − z(K , t) {z } =f∞ (K )
minimize maximum violation of SISO step response envelope specification zmin (t) ≤ z(K , t) ≤ zmax (t) stabilizing initial controller K0 . simulate step response for tl ∈ [0, T ]. active times (finite set) : ˆ (K ) = X
n o tl : z(K , tl ) − zmax (t) = f∞ (K ) or zmin (t) − z(K , tl ) = f∞ (K )
ˆe (K ) ⊃ X ˆ (K ) with neighboring times. extended set X subgradients φ(K ) ∈ ∂f∞ (K ) from time-domain simulations (Iterative Feedback Tuning, [Hjalmarsson et al., 94]). nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 19/22
motivation nonsmooth optimization controller design conclusion key references
time-domain design min max K ∈ t∈[0,+∞] |
n
+ + o z(K , t) − zmax (t) , zmin (t) − z(K , t) {z } =f∞ (K )
minimize maximum violation of SISO step response envelope specification zmin (t) ≤ z(K , t) ≤ zmax (t) stabilizing initial controller K0 . simulate step response for tl ∈ [0, T ]. active times (finite set) : ˆ (K ) = X
n o tl : z(K , tl ) − zmax (t) = f∞ (K ) or zmin (t) − z(K , tl ) = f∞ (K )
ˆe (K ) ⊃ X ˆ (K ) with neighboring times. extended set X subgradients φ(K ) ∈ ∂f∞ (K ) from time-domain simulations (Iterative Feedback Tuning, [Hjalmarsson et al., 94]). nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 19/22
motivation nonsmooth optimization controller design conclusion key references
time-domain design min max K ∈ t∈[0,+∞] |
n
+ + o z(K , t) − zmax (t) , zmin (t) − z(K , t) {z } =f∞ (K )
minimize maximum violation of SISO step response envelope specification zmin (t) ≤ z(K , t) ≤ zmax (t) simulated subgradients φ(K ) =
X
±τl
ˆ (K ) tl ∈ X
with τl ≥ 0 and
P
∂z (K , tl ) ∂Kij
ˆ (K ) τl tl ∈X
=1
0
P(s)
i ,j
∂z ∂Kij ∂y ∂Kij
∂u ∂Kij + +
K
yj ei nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 19/22
motivation nonsmooth optimization controller design conclusion key references
numerical results 2-DOF PID control of a SISO plant (4th-order) ts settling time at ±2% zos overshoot plant G1 G1 G1
contr. ZN∗ IFT∗ NS
ts (s) 46 .22 21 .34 20 .94
zos (%) 46 .92 5 .37 1 .95 time−domain design (G1 with PID) 1.6
[Lequin et al., 03]
1.4
1.2 1
0.8 z(K,t)
∗
0.6
0.4 0.2 step response z(K,t) zmax(t)
0
zmin(t) −0.2 0
10
20
30
40
50 t
60
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 20/22
70
80
90
100
motivation nonsmooth optimization controller design conclusion key references
numerical results 2-DOF PID control of a SISO plant (4th-order) ts settling time at ±2% zos overshoot plant G1 G1 G1
contr. ZN∗ IFT∗ NS
ts (s) 46 .22 21 .34 20 .94
zos (%) 46 .92 5 .37 1 .95 time−domain design (G1 with PID) 1.6
[Lequin et al., 03]
1.4
1.2 1
0.8 z(K,t)
∗
0.6
0.4 0.2 step response z(K,t) zmax(t)
0
zmin(t) −0.2 0
10
20
30
40
50 t
60
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 20/22
70
80
90
100
motivation nonsmooth optimization controller design conclusion key references
numerical results 2-DOF PID control of a SISO plant (4th-order) ts settling time at ±2% zos overshoot plant G1 G1 G1
contr. ZN∗ IFT∗ NS
ts (s) 46 .22 21 .34 20 .94
zos (%) 46 .92 5 .37 1 .95 time−domain design (G1 with PID) 1.6
[Lequin et al., 03]
1.4
1.2 1
0.8 z(K,t)
∗
0.6
0.4 0.2 step response z(K,t) zmax(t)
0
zmin(t) −0.2 0
10
20
30
40
50 t
60
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 20/22
70
80
90
100
motivation nonsmooth optimization controller design conclusion key references
numerical results 2-DOF PID control of a SISO plant (4th-order) ts settling time at ±2% zos overshoot plant G1 G1 G1
contr. ZN∗ IFT∗ NS
ts (s) 46 .22 21 .34 20 .94
zos (%) 46 .92 5 .37 1 .95 time−domain design (G1 with PID) 1.6
[Lequin et al., 03]
1.4
1.2 1
0.8 z(K,t)
∗
0.6
0.4 0.2 step response z(K,t) zmax(t)
0
zmin(t) −0.2 0
10
20
30
40
50 t
60
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 20/22
70
80
90
100
motivation nonsmooth optimization controller design conclusion key references
numerical results 2-DOF PID control of a SISO plant (4th-order) ts settling time at ±2% zos overshoot plant G1 G1 G1
contr. ZN∗ IFT∗ NS
ts (s) 46 .22 21 .34 20 .94
zos (%) 46 .92 5 .37 1 .95 time−domain design (G1 with PID) 1.6
[Lequin et al., 03]
1.4
1.2 1
0.8 z(K,t)
∗
0.6
0.4 0.2 step response z(K,t) zmax(t)
0
zmin(t) −0.2 0
10
20
30
40
50 t
60
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 20/22
70
80
90
100
motivation nonsmooth optimization controller design conclusion key references
numerical results 2-DOF PID control of a SISO plant (4th-order) ts settling time at ±2% zos overshoot plant G1 G1 G1
contr. ZN∗ IFT∗ NS
ts (s) 46 .22 21 .34 20 .94
zos (%) 46 .92 5 .37 1 .95 time−domain design (G1 with PID) 1.6
[Lequin et al., 03]
1.4
1.2 1
0.8 z(K,t)
∗
0.6
0.4 0.2 step response z(K,t) zmax(t)
0
zmin(t) −0.2 0
10
20
30
40
50 t
60
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 20/22
70
80
90
100
motivation nonsmooth optimization controller design conclusion key references
numerical results 2-DOF PID control of a SISO plant (4th-order) ts settling time at ±2% zos overshoot plant G1 G1 G1
contr. ZN∗ IFT∗ NS
ts (s) 46 .22 21 .34 20 .94
zos (%) 46 .92 5 .37 1 .95 time−domain design (G1 with PID) 1.6
[Lequin et al., 03]
1.4
1.2 1
0.8 z(K,t)
∗
0.6
0.4 0.2 step response z(K,t) zmax(t)
0
zmin(t) −0.2 0
10
20
30
40
50 t
60
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 20/22
70
80
90
100
motivation nonsmooth optimization controller design conclusion key references
numerical results 2-DOF PID control of a SISO plant (4th-order) ts settling time at ±2% zos overshoot plant G1 G1 G1
contr. ZN∗ IFT∗ NS
ts (s) 46 .22 21 .34 20 .94
zos (%) 46 .92 5 .37 1 .95 time−domain design (G1 with PID) 1.6
[Lequin et al., 03]
1.4
1.2 1
0.8 z(K,t)
∗
0.6
0.4 0.2 step response z(K,t) zmax(t)
0
zmin(t) −0.2 0
10
20
30
40
50 t
60
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 20/22
70
80
90
100
motivation nonsmooth optimization controller design conclusion key references
numerical results 2-DOF PID control of a SISO plant (4th-order) ts settling time at ±2% zos overshoot plant G1 G1 G1
contr. ZN∗ IFT∗ NS
ts (s) 46 .22 21 .34 20 .94
zos (%) 46 .92 5 .37 1 .95 time−domain design (G1 with PID) 1.6
[Lequin et al., 03]
1.4
1.2 1
0.8 z(K,t)
∗
0.6
0.4 0.2 step response z(K,t) zmax(t)
0
zmin(t) −0.2 0
10
20
30
40
50 t
60
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 20/22
70
80
90
100
motivation nonsmooth optimization controller design conclusion key references
numerical results 2-DOF PID control of a SISO plant (4th-order) ts settling time at ±2% zos overshoot plant G1 G1 G1
contr. ZN∗ IFT∗ NS
ts (s) 46 .22 21 .34 20 .94
zos (%) 46 .92 5 .37 1 .95 time−domain design (G1 with PID) 1.6
[Lequin et al., 03]
1.4
1.2 1
0.8 z(K,t)
∗
0.6
0.4 0.2 step response z(K,t) zmax(t)
0
zmin(t) −0.2 0
10
20
30
40
50 t
60
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 20/22
70
80
90
100
motivation nonsmooth optimization controller design conclusion key references
numerical results 2-DOF PID control of a SISO plant (4th-order) ts settling time at ±2% zos overshoot plant G1 G1 G1
contr. ZN∗ IFT∗ NS
ts (s) 46 .22 21 .34 20 .94
zos (%) 46 .92 5 .37 1 .95 time−domain design (G1 with PID) 1.6
[Lequin et al., 03]
1.4
1.2 1
0.8 z(K,t)
∗
0.6
0.4 0.2 step response z(K,t) zmax(t)
0
zmin(t) −0.2 0
10
20
30
40
50 t
60
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 20/22
70
80
90
100
motivation nonsmooth optimization controller design conclusion key references
conclusion benefits of nonsmooth approaches in control design: general enough to apply to a variety of problems numerically efficient for complex plants flexible for structured practical controllers provide local termination test and optimality certificates
faster convergence with 2nd order elements: by improving local model θ with BFGS matrix, or by recasting as a smooth local constrained problem and solving with SQP [Bompart et al., 2007]
Extensions to: multidisk and multiband H∞ synthesis [Apkarian and Noll, 2006-II] constrained design problems (mixed H2 /H∞ synthesis) IQC analysis and synthesis [Apkarian and Noll, 2006-III]
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 21/22
motivation nonsmooth optimization controller design conclusion key references
conclusion benefits of nonsmooth approaches in control design: general enough to apply to a variety of problems numerically efficient for complex plants flexible for structured practical controllers provide local termination test and optimality certificates
faster convergence with 2nd order elements: by improving local model θ with BFGS matrix, or by recasting as a smooth local constrained problem and solving with SQP [Bompart et al., 2007]
Extensions to: multidisk and multiband H∞ synthesis [Apkarian and Noll, 2006-II] constrained design problems (mixed H2 /H∞ synthesis) IQC analysis and synthesis [Apkarian and Noll, 2006-III]
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 21/22
motivation nonsmooth optimization controller design conclusion key references
conclusion benefits of nonsmooth approaches in control design: general enough to apply to a variety of problems numerically efficient for complex plants flexible for structured practical controllers provide local termination test and optimality certificates
faster convergence with 2nd order elements: by improving local model θ with BFGS matrix, or by recasting as a smooth local constrained problem and solving with SQP [Bompart et al., 2007]
Extensions to: multidisk and multiband H∞ synthesis [Apkarian and Noll, 2006-II] constrained design problems (mixed H2 /H∞ synthesis) IQC analysis and synthesis [Apkarian and Noll, 2006-III]
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 21/22
motivation nonsmooth optimization controller design conclusion key references
conclusion benefits of nonsmooth approaches in control design: general enough to apply to a variety of problems numerically efficient for complex plants flexible for structured practical controllers provide local termination test and optimality certificates
faster convergence with 2nd order elements: by improving local model θ with BFGS matrix, or by recasting as a smooth local constrained problem and solving with SQP [Bompart et al., 2007]
Extensions to: multidisk and multiband H∞ synthesis [Apkarian and Noll, 2006-II] constrained design problems (mixed H2 /H∞ synthesis) IQC analysis and synthesis [Apkarian and Noll, 2006-III]
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 21/22
motivation nonsmooth optimization controller design conclusion key references
conclusion benefits of nonsmooth approaches in control design: general enough to apply to a variety of problems numerically efficient for complex plants flexible for structured practical controllers provide local termination test and optimality certificates
faster convergence with 2nd order elements: by improving local model θ with BFGS matrix, or by recasting as a smooth local constrained problem and solving with SQP [Bompart et al., 2007]
Extensions to: multidisk and multiband H∞ synthesis [Apkarian and Noll, 2006-II] constrained design problems (mixed H2 /H∞ synthesis) IQC analysis and synthesis [Apkarian and Noll, 2006-III]
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 21/22
motivation nonsmooth optimization controller design conclusion key references
conclusion benefits of nonsmooth approaches in control design: general enough to apply to a variety of problems numerically efficient for complex plants flexible for structured practical controllers provide local termination test and optimality certificates
faster convergence with 2nd order elements: by improving local model θ with BFGS matrix, or by recasting as a smooth local constrained problem and solving with SQP [Bompart et al., 2007]
Extensions to: multidisk and multiband H∞ synthesis [Apkarian and Noll, 2006-II] constrained design problems (mixed H2 /H∞ synthesis) IQC analysis and synthesis [Apkarian and Noll, 2006-III]
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 21/22
motivation nonsmooth optimization controller design conclusion key references
conclusion benefits of nonsmooth approaches in control design: general enough to apply to a variety of problems numerically efficient for complex plants flexible for structured practical controllers provide local termination test and optimality certificates
faster convergence with 2nd order elements: by improving local model θ with BFGS matrix, or by recasting as a smooth local constrained problem and solving with SQP [Bompart et al., 2007]
Extensions to: multidisk and multiband H∞ synthesis [Apkarian and Noll, 2006-II] constrained design problems (mixed H2 /H∞ synthesis) IQC analysis and synthesis [Apkarian and Noll, 2006-III]
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 21/22
motivation nonsmooth optimization controller design conclusion key references
key references F.H. Clarke. Optimization and Nonsmooth Analysis. John Wiley & Sons, 1983. E. Polak. On the mathematical foundations of nondifferentiable optimization in engineering design. SIAM Review, 29(1):21-89, 1987. E. Polak. Optimization: Algorithms and Consistent Approximations. Springer, Applied Mathematical Sciences, 1997. J.V. Burke, D. Henrion, A.S. Lewis and M.L. Overton. Stabilization via Nonsmooth, Nonconvex Optimization, IEEE Trans. on Automatic Control, 51(11):1760-1769, 2006.
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 22/22
motivation nonsmooth optimization controller design conclusion key references
key references http://www.cert.fr/dcsd/cdin/apkarian/ P. Apkarian and D. Noll. Nonsmooth H∞ synthesis. IEEE Trans. on Automatic Control, 51(1):71-86, 2006. P. Apkarian and D. Noll. Nonsmooth Optimization for Multidisk H∞ Synthesis. Eur. J. of Control, 3(12):229-244, 2006. P. Apkarian and D. Noll. IQC analysis and synthesis via nonsmooth optimization. Systems and Control Letters, 55(12):971-981, 2006. V. Bompart, D. Noll and P. Apkarian. Second-order nonsmooth optimization for H∞ synthesis. Numerische Mathematik, in press, 2007. P. Apkarian, V. Bompart and D. Noll. Nonsmooth Structured Control Design with Application to PID Loop-Shaping of a Process. Int. J. of Robust and Nonlinear Control, in press, 2007.
nonsmooth structured control design - P. APKARIAN - ALCOSP 2007 - 22/22
