seminaire-optimisation-aerodynamique

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An introduction to optimal control of fluid flows
MECAERO - Engineering and design of aerodynamic structures
P. Meliga
CNRS - Center for Material Forming (CEMEF)
February 2022
Plan
Introduction
Selected applications
Guidelines
Introduction
Flow control for aerodynamics
Selected applications
Introduction
Flow control for aerodynamics
Trial and error
The featherie, gutta percha, Haskell and modern golf balls (æ)
Introduction
Flow control for aerodynamics
Physics-based
Vortex generators delaying local flow
separation and stall
Introduction
Flow control for aerodynamics
Physics-based
Vortex generators delaying local flow
separation and stall
(does not preclude trial and error...)
Introduction
Flow control for aerodynamics
Bio-inspired
Kingfisher: master of the dive
Introduction
Flow control for aerodynamics
Bio-inspired
Forefront of a high-speed Shinkansen bullet train mimicking the beak
of a kingfisher to reduce noise
Introduction
Flow control for aerodynamics
Bio-inspired
The humpback whale and bubble net fishing
Introduction
Flow control for aerodynamics
Bio-inspired
Grooved wind turbine blades mimicking the flipper of the humpback-
whale to delay stall (increased drag, decreased lift)
Introduction
Flow control for aerodynamics
Bio-inspired
Polyurethane swimsuits mimicking the denticles of the shark skin to
reduce viscous drag
Guidelines
A flow control digest
Use relevant cost function (know the physics! )
Choose strategy (passive, active, shape optimization...)
Set proper objective (improve? optimize?)
Plan
Introduction
Various approaches to flow control
Modelisation
A “simple” aerodynamics problem
Find the angle of attack such that CL/CD is maximal?
Several approaches available that vary in sophistication and ease of
implementation
• Low dimensional models
• Direct methods
• Optimization methods
Modelisation
Low dimensional models
Use simple analytical expressions of lift and drag
Modelisation
Low dimensional models
Thin airfoil theory
• Inviscid, steady flow
• zero-thickness wings with infinite span (2-D)
Replace the wing by its equivalent thin airfoil (flat plate if no cam-
ber)
Lift coe�cient Drag coe�cient
CL = 2fi– CD =?
Modelisation
Low dimensional models
Lifting line theory
• Inviscid, steady flow
• Finite wings with no sweep and a reasonably large aspect ratio
Lift coe�cient Drag coe�cient
CL = 2fi
3 AR
AR + 2
4
– CD = CD0 +
C2L
fiARe
Modelisation
Low dimensional models
Lifting line theory
CL = 2fi
3 AR
AR + 2
4
–
CD = CD0 +
C2L
fiARe
Objective: lift-to-drag ratio
ˆCL/CD
ˆ–
= 0 =
CD ˆCLˆ– ≠ CL ˆCDˆ–
C2D
… CD
ˆCL
ˆ–
≠ CL
ˆCD
ˆ–
= 0
ˆCL
ˆ–
= 2fi
3 AR
AR + 2
4
ˆCD
ˆ–
= 2CL
fiARe
ˆCL
ˆ–
Modelisation
Low dimensional models
Lifting line theory
CL = CL0 + 2fi
3 AR
AR + 2
4
–
CD = CD0 +
C2L
fiARe
Objective: lift-to-drag ratio
ˆCL/CD
ˆ–
= 0 …
3
CD0 ≠
C2L
fiARe
4
2fi
3 AR
AR + 2
4
= 0
…CL = (fiAReCD0)1/2
…–opt =
AR + 2
2
3eCD0
fiAR
41/2
Modelisation
Low dimensional models
Lifting line theory
CL = CL0 + 2fi
3 AR
AR + 2
4
–
CD = CD0 +
C2L
fiARe
Optimal angle: CD0 = 0.3, e = 0.75
AR = 0.5: ≥ 27¶ AR = 5: ≥ 24¶ AR = 50: ≥ 56¶
Modelisation
Low dimensional models
• Easy to use
• Risk of oversimplification (many underlying assumptions)
Flow physics: instationarities, viscous friction, turbulence...
Potential flow Viscous flow
Baez et al. (2011)
Modelisation
Low dimensional models
• Easy to use
• Risk of oversimplification (many underlying assumptions)
Flow physics: occurrence of stall
https://www.youtube.com/watch?v=SiOiVHUEYao
Modelisation
Low dimensional models
• Easy to use
• Risk of oversimplification (many underlying assumptions)
Flow physics: occurrence of stall
Modelisation
Low dimensional models
• Easy to use
• Risk of oversimplification (many underlying assumptions)
Flow physics: shape of the airfoil
Modelisation
Direct methods
Use exact values of lift and drag (no model)
Modelisation
Direct methods
Wind tunnel testing, numerical simulation
Sinn & Barrett (2010) Munz (2017)
Modelisation
Direct methods
• Rigorous (few assumptions)
• Time consuming
• Hard to generalize
• Inherently suboptimal (limited parameter space)
Modelisation
Optimization methods
Smartly use exact values of lift and drag
Modelisation
Optimization methods
Smartly use exact values of lift and drag
æ
Modelisation
Optimization methods
Smartly use exact values of lift and drag
æ
• Gradient methods: rely on evaluations of the gradient of the target
function at the current point
• Adjoint method
• Machine learning
• Gradient-free methods: rely on evaluations of the target function at
the current point (genetic algorithms, particle swarm optimization...)
Modelisation
Optimization methods
• Rigorous (few assumptions)
• Systematical (no trial and error)
• Large parameter spaces
• Built for optimality
• Implementation can be intricate
• Optimality can be local
Modelisation
Optimization methods
• Rigorous (few assumptions)
• Systematical (no trial and error)
• Large parameter spaces
• Built for optimality
• Implementation can be intricate
• Optimality can be local
Some pitfalls of gradient methods: e�ect of descent step
Modelisation
Optimization methods
• Rigorous (few assumptions)
• Systematical (no trial and error)
• Large parameter spaces
• Built for optimality
• Implementation can be intricate
• Optimality can be local
Some pitfalls of gradient methods: e�ect of initial condition
Plan
Introduction
Various approaches to flow control
A general framework for optimal control
System dynamics
Lagrange multipliers
Optimal control
General framework
State equation
ˆtq + N(q, „, Â) = 0 with q(t = 0) = q i
with
• q: state vector (prescribed initial state q i)
• „: control
• Â: noise
• N: time-varying function of (q, „, Â) representing a smooth ODE
Optimal control
General framework
State equation
ˆtq + N(q, „, Â) = 0 with q(t = 0) = q i
Objective: find „ minimizing a relevant cost function J(q, „, Â)
(drag related, energy, pressure drop...)
" Governing equation (controlled linear equation) must be satisfied
∆ Constrained optimization problem
min J(q, „, Â) subject to ˆtq + N(q, „, Â) = 0
Optimal control
General framework
Constrained optimization problem
min J(q, „, Â) subject to ˆtq + N(q, „, Â) = 0
• Uncontrolled … „ = 0
• Ideal (no noise) … Â = 0
• is a control elaborated for a specific set of parameters still
e�cient in the presence of model uncertainties ? (e.g., noise,
variations of model parameters, external parameters...)
• if the optimization is not robust, the control can yield excellent
results for the nominal model, but catastrophic ones for the
modified model.
Optimal control
General framework
Constrained optimization problem
min J(q, „, Â) subject to ˆtq + N(q, „, Â) = 0
Applies to low-dimensional ODEs and high-dimensional discretiza-
tions of classical PDEs
• Poisson equation
• Navier–Stokes equations
• Stokes equations
• Heat equation
• Maxwell equations...
Optimal control
General framework
Constrained optimization problem
min J(q, „, Â) subject to ˆtq + N(q, „, Â) = 0
Tractable if low state d.o.f. and relations between variables can be
made explicit
min/max f (x , y) subject to g(x , y) = c̀
˘
Constraint parameter
Optimal control
General framework
Constrained optimization problem
min J(q, „, Â) subject to ˆtq + N(q, „, Â) = 0
Tractable if low state d.o.f. and relations between variables can be
made explicit
min/max f (x , y) subject to g(x , y) = c … y = h(x)
min/max F (x) = f (x , h(x)) … (x0, h(x0)) |
dF
dx
----
x0
= 0
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
• Make relation between variablesexplicit
y = ≠83x = h(x) ∆ min/max F (x) = f (x , h(x)) = ≠
8
3x
3 ≠ ln(x)
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
• Make relation between variables explicit
y = ≠83x = h(x) ∆ min/max F (x) = f (x , h(x)) = ≠
8
3x
3 ≠ ln(x)
• Solve
dF
dx = ≠8x
2 ≠ 1x = 0 … x0 = ≠
1
2 ∆ y0 = h(x0) =
4
3
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
• Make relation between variables explicit
y = ≠83x = h(x) ∆ min/max F (x) = f (x , h(x)) = ≠
8
3x
3 ≠ ln(x)
• Solve
(x0, y0) =
3
≠12 ,
4
3
4
Optimal control
Lagrange multiplier technique
Constrained optimization problem
min/max f (x) subject to g(x) = c
Assume we walk along the contour line with an g = c
• Only when the contour line for g = c intersects contour lines of f
tangentially, do we not increase or decrease the value of f
• This occurs when normal vectors to the contour lines are parallel:
Òf = ⁄Òg
Optimal control
Lagrange multiplier technique
Constrained optimization problem
min/max f (x) subject to g(x) = c
• Form Lagrange function L(x, ⁄) = f (x) ≠ ⁄(g(x) ≠ c)
• Solutions x0 to the constrained problem are saddle points of L
• Unconstrained problem: seek stationary points (x0, ⁄0) of L
ˆL
ˆx0
= 0
ˆL
ˆ⁄0
= 0 = g(x) ≠ c … constraint is satisfied
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
• Form Lagrangian function
L(x , y , ⁄) = x2y ≠ ln(x) ≠ ⁄(8x + 3y)
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
• Form Lagrangian function
L(x , y , ⁄) = x2y ≠ ln(x) ≠ ⁄(8x + 3y)
• Seek stationary points
ˆL
ˆx = 2xy ≠
1
x ≠ 8⁄ = 0
ˆL
ˆy = x
2 ≠ 3⁄ = 0 ∆ ⁄ = 13x
2
ˆL
ˆ⁄
= 8x + 3y = 0¸ ˚˙ ˝
Constraint
∆ y = ≠83x
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
• Form Lagrangian function
L(x , y , ⁄) = x2y ≠ ln(x) ≠ ⁄(8x + 3y)
• Seek stationary points
ˆL
ˆx = ≠
16
3 x
2 ≠ 1x ≠
8
3x
2 = 0
ˆL
ˆy = x
2 ≠ 3⁄ = 0 ∆ ⁄ = 13x
2
ˆL
ˆ⁄
= 8x + 3y = 0 ∆ y = ≠83x
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
• Form Lagrangian function
L(x , y , ⁄) = x2y ≠ ln(x) ≠ ⁄(8x + 3y)
• Seek stationary points
ˆL
ˆx = ≠8x
2 ≠ 1x = 0
ˆL
ˆy = x
2 ≠ 3⁄ = 0 ∆ ⁄ = 13x
2
ˆL
ˆ⁄
= 8x + 3y = 0 ∆ y = ≠83x
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
• Form Lagrangian function
L(x , y , ⁄) = x2y ≠ ln(x) ≠ ⁄(8x + 3y)
• Seek stationary points
ˆL
ˆx = ≠8x
2 ≠ 1x = 0 ∆ x0 = ≠
1
2
ˆL
ˆy = x
2 ≠ 3⁄ = 0 ∆ ⁄ = 13x
2
ˆL
ˆ⁄
= 8x + 3y = 0 ∆ y = ≠83x
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
• Form Lagrangian function
L(x , y , ⁄) = x2y ≠ ln(x) ≠ ⁄(8x + 3y)
• Seek stationary points
ˆL
ˆx = ≠8x
2 ≠ 1x = 0 ∆ x0 = ≠
1
2
ˆL
ˆy = x
2 ≠ 3⁄ = 0 ∆ ⁄ = 13x
2 ∆ ⁄0 =
1
12
ˆL
ˆ⁄
= 8x + 3y = 0 ∆ y = ≠83x ∆ y0 =
4
3
Optimal control
Basic example
Constrained optimization problem
min/max f (x , y) = x2y ≠ ln(x) subject to 8x + 3y = 0 = g(x , y)
• Form Lagrangian function
L(x , y , ⁄) = x2y ≠ ln(x) ≠ ⁄(8x + 3y)
• Seek stationary points
(x0, y0, ⁄0) =
3
≠12 ,
4
3 ,
1
12
4
Optimal control
Another example
Find the discrete probability distribution {p1 . . . pN} maximizing the
Shannon entropy
f (p1 . . . pN) = ≠
Nÿ
1
pj log2 pj subject to g(p1 . . . pN) =
Nÿ
1
pj = 1
Optimal control
Another example
Find the discrete probability distribution {p1, p2, . . . pN} maximizing
the Shannon entropy
f (p1 . . . pN) = ≠
Nÿ
1
pj log2 pj subject to g(p1 . . . pN) =
Nÿ
1
pj = 1
• Form Lagrangian function
L(pj , ⁄) = ≠
Nÿ
j=1
pj log2 pj ≠ ⁄(
Nÿ
j=1
pj ≠ 1)
Optimal control
Another example
Find the discrete probability distribution {p1 . . . pN} maximizing the
Shannon entropy
f (p1 . . . pN) = ≠
Nÿ
1
pj log2 pj subject to g(p1 . . . pN) =
Nÿ
1
pj = 1
• Form Lagrangian function
L(pj , ⁄) = ≠
Nÿ
j=1
pj log2 pj ≠ ⁄(
Nÿ
j=1
pj ≠ 1)
• Seek stationary points
ˆL
ˆpj
= ≠ log2 pj ≠ pj
1
ln 2pj
≠ ⁄ = 0 ’j
ˆL
ˆ⁄
=
Nÿ
j=1
pj ≠ 1 = 0
Optimal control
Another example
Find the discrete probability distribution {p1 . . . pN} maximizing the
Shannon entropy
f (p1 . . . pN) = ≠
Nÿ
1
pj log2 pj subject to g(p1 . . . pN) =
Nÿ
1
pj = 1
• Form Lagrangian function
L(pj , ⁄) = ≠
Nÿ
j=1
pj log2 pj ≠ ⁄(
Nÿ
j=1
pj ≠ 1)
• Seek stationary points
ˆL
ˆpj
= ≠ log2 pj ≠ pj
1
ln 2pj
≠ ⁄ = 0 ∆ log2 pj = ≠
1
ln 2 ≠ ⁄ ’j
ˆL
ˆ⁄
=
Nÿ
j=1
pj ≠ 1 = 0
Optimal control
Another example
Find the discrete probability distribution {p1 . . . pN} maximizing the
Shannon entropy
f (p1 . . . pN) = ≠
Nÿ
1
pj log2 pj subject to g(p1 . . . pN) =
Nÿ
1
pj = 1
• Form Lagrangian function
L(pj , ⁄) = ≠
Nÿ
j=1
pj log2 pj ≠ ⁄(
Nÿ
j=1
pj ≠ 1)
• Seek stationary points
ˆL
ˆpj
= ≠ log2 pj ≠ pj
1
ln 2pj
≠ ⁄ = 0 ∆ pj = p1 ’j
ˆL
ˆ⁄
=
Nÿ
j=1
pj ≠ 1 = Np1 ≠ 1 = 0 … p1 =
1
N = pj ’j
Optimal control
Lagrange multiplier technique
Constrained optimization problem
min/max f (x) subject to g(x) = c
• Form Lagrange function L(x, ⁄) = f (x) ≠ ⁄(g(x) ≠ c)
• Physical interpretation
⁄ = ˆL
ˆc
Lagrange multiplier © rate of change of the quantity being optimized
with respect to the constraint parameter
Optimal control
Lagrange multiplier technique
Constrained optimization problem
min/max f (x) subject to gj(x) = cj
• Lagrange function L(x, ⁄) = f (x) ≠ qj ⁄j(g(x) ≠ cj) = ⁄t · (g ≠ c)
• Physical interpretation
⁄j =
ˆL
ˆcj
Lagrange multiplier © rate of change of the quantity being optimized
with respect to the constraint vector
Plan
Introduction
Various approaches to flow control
A general framework for optimal control
Time-invariant systems
Linear systems
Optimal control / Time-invariant, linear systems
General framework
State equation
ˆtq + N(q, „, Â) = 0 with q(t = 0) = q i
with
• q: state vector (prescribed initial state q i)
• „: control
• Â: noise
• N: time-varying function of (q, „, Â) representing a smooth ODE
Optimal control / Time-invariant, linear systems
General framework
State equation (ideal … no noise)
L̀
˘
q + B̀
˘
„ = 0
constant matrices © generally discretized di�erential op. (gradient,
divergence, laplacian) ∆ depends on a space discretization scheme
Optimal control / Time-invariant, linear systems
General framework
State equation
Lq + B„ = 0
Objective: minimize some measure of the energy
J(q) = qt · q = Èq; qÍ
with Èa; bÍ = at · b (canonical inner product)
Optimal control / Time-invariant, linear systems
General framework
State equation
Lq + B„ = 0
Objective: minimize some measure of the energy
J(q) = Èq; qÍ
Constrained optimization problem
min J(q) subject to state equation
Optimal control / Time-invariant, linear systems
General framework
State equation
Lq + B„ = 0
Objective: minimize some measure of the energy
J(q) = Èq; qÍ
Constrained optimization problem
min J(q) subject to state equation
Lagrangian function
L(q†`̆ , q, „) = Èq; qÍ ≠ Èq
†; Lq + B„Í
Co-state/adjoint variable = Lagrange multiplier (dim.q† =dim.q)
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
How to impose stationarity? æ Gateau di�erential
ˆL
ˆx x
Õ = lim
‘æ0
L(x + ‘x Õ) ≠ L(x)
‘
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = lim
‘æ0L(q† + ‘q†Õ, q, „) ≠ L(q†, q, „)
‘
= 0 ’ admissible q†Õ
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = lim
‘æ0
L(q† + ‘q†Õ, q, „) ≠ L(q†, q, „)
‘
= 0 ’ admissible q†Õ
L(q† + ‘q†Õ, q, „) = Èq; qÍ ≠ Èq† + ‘q†Õ; Lq + B„Í
L(q† + ‘q†Õ, q, „) = L(q†, q, „) ≠ ‘Èq†Õ; Lq + B„Í
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = lim
‘æ0
L(q† + ‘q†Õ, q, „) ≠ L(q†, q, „)
‘
= 0 ’ admissible q†Õ
L(q† + ‘q†Õ, q, „) ≠ L(q†, q, „)
‘
= ≠Èq†Õ; Lq + B„Í
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = ≠Èq†Õ; Lq + B„Í = 0 ’ admissible q†Õ
∆ (q, „) must satisfy the state equation
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
= 0 ’ admissible qÕ
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
= 0 ’ admissible qÕ
L(q†, q + ‘qÕ, „) = Èq + ‘qÕ; q + ‘qÕÍ ≠ Èq†; L(q + ‘qÕ) + B„Í
L(q†, q + ‘qÕ, „) = L(q†, q, „) + 2‘Èq; qÕÍ ≠ ‘Èq†; LqÕÍ + ‘2ÈqÕ; qÕÍ
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
= 0 ’ admissible qÕ
L(q†, q + ‘qÕ, „) = Èq + ‘qÕ; q + ‘qÕÍ ≠ Èq†; L(q + ‘qÕ) + B„Í
L(q†, q + ‘qÕ, „) = L(q†, q, „) + 2‘Èq; qÕÍ ≠ ‘Èq†; LqÕÍ + ‘2ÈqÕ; qÕÍ
L(q†, q + ‘qÕ, „) = L(q†, q, „) + 2‘Èq; qÕÍ ≠ ‘ÈLtq†; qÕÍ + ‘2ÈqÕ; qÕÍ
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
= 0 ’ admissible qÕ
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
= È2q ≠ Ltq†; qÕÍ + ‘ÈqÕ; qÕÍ
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = È2q ≠ Ltq†; qÕÍ = 0 ’ admissible qÕ
∆ q† must satisfy the adjoint equation
Ltq† ≠ 2q = 0
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
= 0 ’ admissible „Õ
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
= 0 ’ admissible „Õ
L(q†, q, „ + ‘„Õ) = Èq; qÍ ≠ Èq†; Lq + B(„ + ‘„Õ)Í
L(q†, q, „ + ‘„Õ) = L(q†, q, „) ≠ ‘Èq†; B„ÕÍ
L(q†, q, „ + ‘„Õ) = L(q†, q, „) ≠ ‘ÈBtq†; „ÕÍ
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
= 0 ’ admissible „Õ
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
= ≠ÈBtq†; „ÕÍ
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = ≠ÈBtq†; „ÕÍ = 0 ’ admissible „Õ
If optimality
Btq† = 0
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = ≠ÈBtq†; „ÕÍ = 0 ’ admissible „Õ
Otherwise
dJ = dL =
�
��
ˆL
ˆq q
Õ +
⇢
⇢
⇢⇢ˆL
ˆq† q
†Õ + ˆL
ˆ„
„Õ
(… state/adjoint equations are satisfied)
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = ≠ÈBtq†; „ÕÍ = 0 ’ admissible „Õ
Otherwise
dJ = È≠Btq†; „ÕÍ = ÈÒ„J ; „ÕÍ ∆ Ò„J = ≠Btq†dJ = È≠Btq†; „ÕÍ Ò„J`̆dJ = È≠B
tq†; „ÕÍ =
gradient of cost function with respect to control variable
Plan
Introduction
Various approaches to flow control
A general framework for optimal control
Time-invariant systems
Linear systems
Nonlinear systems
Optimal control / Time-invariant, nonlinear systems
General framework
State equation
ˆtq + N(q, „, Â) = 0 with q(t = 0) = q i
with
• q: state vector (prescribed initial state q i)
• „: control
• Â: noise
• N: time-varying function of (q, „, Â) representing a smooth ODE
Optimal control / Time-invariant, nonlinear systems
General framework
State equation
N(q, „) = 0
Objective: minimize some measure of the energy
J(q) = Èq; qÍ
Optimal control / Time-invariant, nonlinear systems
General framework
State equation
N(q, „) = 0
Objective: minimize some measure of the energy
J(q) = Èq; qÍ
Constrained optimization problem
min J(q) subject to state equation
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; N(q, „)Í
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = lim
‘æ0
L(q† + ‘q†Õ, q, „) ≠ L(q†, q, Â)
‘
ˆL
ˆq† q
†Õ = ≠Èq†Õ; Lq + B„Í = 0 ’ admissible q†Õ
∆ (q, „) must satisfy the state equation
Optimal control / Time-invariant, nonlinear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; N(q, „)Í
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = lim
‘æ0
L(q† + ‘q†Õ, q, „) ≠ L(q†, q, Â)
‘
ˆL
ˆq† q
†Õ = ≠Èq†Õ; N(q, „)Í = 0 ’ admissible q†Õ
∆ (q, „) must satisfy the state equation
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
= 0 ’ admissible qÕ
ˆL
ˆq q
Õ = 2Èq; qÕÍ ≠ Èq†; LqÕÍ
Optimal control / Time-invariant, nonlinear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; N(q, „)Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
= 0 ’ admissible qÕ
ˆL
ˆq q
Õ = 2Èq; qÕÍ ≠ Èq†; LqÕÍ
with
LqÕ © L(q)qÕ = lim
‘æ0
N(q + ‘qÕ, „) ≠ N(q, „)
‘
Optimal control / Time-invariant, nonlinear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; N(q, „)Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
ˆL
ˆq q
Õ = È2q ≠ Ltq†; qÕÍ = 0 ’ admissible qÕ
∆ q† must satisfy the adjoint equation
Ltq† ≠ 2q = 0
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
= 0 ’ admissible „Õ
ˆL
ˆ„
„Õ = ≠Èq†; B„ÕÍ
Optimal control / Time-invariant, nonlinear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; N(q, „)Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
= 0 ’ admissible „Õ
ˆL
ˆ„
„Õ = ≠Èq†; B„ÕÍ
with
B„Õ © B(q)„Õ = lim
‘æ0
N(q, „ + ‘„Õ) ≠ N(q, „)
‘
Optimal control / Time-invariant, nonlinear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; N(q, „)Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = ≠ÈBtq†; „ÕÍ = 0 ’ admissible „Õ
If optimality
Btq† = 0
Optimal control / Time-invariant,nonlinear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; N(q, „)Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = ≠ÈBtq†; „ÕÍ = 0 ’ admissible „Õ
Otherwise
dJ = dL =
�
��
ˆL
ˆq q
Õ +
⇢
⇢
⇢⇢ˆL
ˆq† q
†Õ + ˆL
ˆ„
„Õ
Optimal control / Time-invariant, nonlinear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; N(q, „)Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = ≠ÈBtq†; „ÕÍ = 0 ’ admissible „Õ
Otherwise
dJ = È≠Btq†; „ÕÍ = ÈÒ„J ; „ÕÍ ∆ Ò„J = ≠Btq†
Plan
Introduction
Various approaches to flow control
A general framework for optimal control
Time-invariant systems
Linear systems
Nonlinear systems
Weighing the cost of control
Weighed optimal control / Time-invariant, linear systems
General framework
State equation (no noise)
Lq + B„ = 0
Objective: minimize some measure of the energy
J(q) = qt · q = Èq; qÍ
Weighed optimal control / Time-invariant, linear systems
General framework
State equation
Lq + B„ = 0
Objective: minimize some penalized measure of the energy
J(q, „) = qt · q+—2`̆ „
t · „ = Èq; qÍ+—2È„; „Í
penalty parameter weighing the cost of the control
(© upper bound on the control amplitude)
Weighed optimal control / Time-invariant, linear systems
General framework
State equation
Lq + B„ = 0
Objective: minimize some penalized measure of the energy
J(q, „) = qt · q+—2„t · „ = Èq; qÍ+—2È„; „Í
Constrained optimization problem
min J(q, „) subject to state equations
Lagrangian function
L(q†, q, „) = Èq; qÍ+—2È„; „Í ≠ Èq†; Lq + B„Í
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = ≠Èq†Õ; Lq + B„Í = 0 ’ admissible q†Õ
∆ (q, „) must satisfy the state equation
Weighed optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ+—2È„; „Í ≠ Èq†; Lq + B„Í
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = ≠Èq†Õ; Lq + B„Í = 0 ’ admissible q†Õ
∆ (q, „) must satisfy the state equation
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = È2q ≠ Ltq†; qÕÍ = 0 ’ admissible qÕ
∆ q† must satisfy the adjoint equation
Ltq† ≠ 2q = 0
Weighed optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ+—2È„; „Í ≠ Èq†; Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = È2q ≠ Ltq†; qÕÍ = 0 ’ admissible qÕ
∆ q† must satisfy the adjoint equation
Ltq† ≠ 2q = 0
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
= 0 ’ admissible „Õ
L(q†, q, „ + ‘„Õ) = L(q†, q, „) ≠ ‘ÈBtq†; „ÕÍ
Weighed optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ+—2È„; „Í ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
= 0 ’ admissible „Õ
L(q†, q, „ + ‘„Õ) = L(q†, q, „)+2‘—2È„; „ÕÍ≠‘Èq†; B„ÕÍ+‘2—2È„Õ; „ÕÍ
Weighed optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ+—2È„; „Í ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
= 0 ’ admissible „Õ
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
= È2—2„ ≠ Btq†; „ÕÍ+‘—2È„Õ; „ÕÍ
Weighed optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ+—2È„; „Í ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = È2—2„ ≠ Btq†; „ÕÍ = 0 ’ admissible „Õ
If optimality
Btq†≠2—2„ = 0
Weighed optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ+—2È„; „Í ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = È2—2„ ≠ Btq†; „ÕÍ = 0 ’ admissible „Õ
Otherwise
dJ = dL =
�
��
ˆL
ˆq q
Õ +
⇢
⇢
⇢⇢ˆL
ˆq† q
†Õ + ˆL
ˆ„
„Õ
Weighed optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ+—2È„; „Í ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = È2—2„ ≠ Btq†; „ÕÍ = 0 ’ admissible „Õ
Otherwise
dJ = È≠Btq†; „ÕÍ = ÈÒ„J ; „ÕÍ ∆ Ò„J = 2—2„ ≠ Btq†
Plan
Introduction
Various approaches to flow control
A general framework for optimal control
Time-invariant systems
Time-variant systems
Optimal control / Time-variant, linear systems
General framework
State equation
ˆtq + N(q, „, Â) = 0 with q(t = 0) = q i
with
• q: state vector (prescribed initial state q i)
• „: control
• Â: noise
• N: time-varying function of (q, „, Â) representing a smooth ODE
Optimal control / Time-variant, linear systems
General framework
State equation (ideal … no noise)
ˆtq + L̀˘
q + B̀
˘
„ = 0
constant matrices © generally discretized di�erential op. (gradient,
divergence, laplacian) ∆ depends on a space discretization scheme
Control strategy?
• Regulation control: evolution over [0, T ]
• Terminal control: final state at t = T
Optimal control / Time-variant, linear systems
General framework
State equation
ˆtq + Lq + B„ = 0
Objective: minimize some measure of the energy
J(q, „) =
⁄ T
0
qt · q dt + —2
⁄ T
0
„t · „ dt = Èq; qÍ + —2È„; „Í
with Èa; bÍ =
⁄ T
0
at · b dt (canonical time-space inner product)
Optimal control / Time-variant, linear systems
General framework
State equation
ˆtq + Lq + B„ = 0
Objective: minimize some measure of the energy
J(q, „) = Èq; qÍ + —2È„; „Í
⁄ T
0
qt · q dt
Constrained optimization problem
min J(q, „) subject to state equations
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; ˆtq + Lq + B„Í
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; Lq + B„Í
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = lim
‘æ0
L(q† + ‘q†Õ, q, „) ≠ L(q†, q, Â)
‘
ˆL
ˆq† q
†Õ = ≠Èq†Õ; Lq + B„Í = 0 ’ admissible q†Õ
∆ (q, „) must satisfy the state equation
Optimal control / Time-variant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; ˆtq + Lq + B„Í
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = lim
‘æ0
L(q† + ‘q†Õ, q, „) ≠ L(q†, q, „)
‘
ˆL
ˆq† q
†Õ = ≠Èq†Õ; ˆtq + Lq + B„Í = 0 ’ admissible q†
Õ
∆ (q, „) must satisfy the state equation
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
= 0 ’ admissible qÕ
ˆL
ˆq q
Õ = È2q ≠ Ltq†; qÕÍ
Optimal control / Time-variant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; ˆtq + Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
ˆL
ˆq q
Õ = ≠Èq†; ˆtqÕÍ + È2q ≠ Ltq†; qÕÍ = 0 ’ admissible qÕ
Optimal control / Time-variant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; ˆtq + Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
ˆL
ˆq q
Õ = ≠Èq†; ˆtqÕÍ + È2q ≠ Ltq†; qÕÍ = 0 ’ admissible qÕ
Integrating by parts
≠Èq†; ˆtqÕÍ = Ȉtq†; qÕÍ ≠ q†(T )t ·qÕ(T ) +⇠⇠⇠⇠
⇠⇠q†(0)t ·qÕ(0)
Initial state is prescribed
Optimal control / Time-variant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; ˆtq + Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
ˆL
ˆq q
Õ = Ȉtq† + 2q ≠ Ltq†; qÕÍ≠q†(T )t·qÕ(T ) = 0 ’ admissible qÕ
∆ q† must satisfy the adjoint equation
≠ˆtq† + Ltq† ≠ 2q = 0 with q†(T ) = 0
to be solved backwards in time (from t = T to 0).
Optimal control / Time-invariant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
ˆL
ˆ„
„Õ = È2—2„ ≠ Btq†; „ÕÍ = 0 ’ admissible „Õ
Optimal control / Time-variant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; ˆtq + Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
ˆL
ˆ„
„Õ = È2—2„ ≠ Btq†; „ÕÍ = 0 ’ admissible „Õ
If optimality
Btq† ≠ 2—2„ = 0
Optimal control / Time-variant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; ˆtq + Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
ˆL
ˆ„
„Õ = È2—2„ ≠ Btq†; „ÕÍ = 0 ’ admissible „Õ
Otherwise
dJ = dL =
�
��
ˆL
ˆq q
Õ +
⇢
⇢
⇢⇢ˆL
ˆq† q
†Õ + ˆL
ˆ„
„Õ
Optimal control / Time-variant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ ≠ Èq†; ˆtq + Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
ˆL
ˆ„
„Õ = È2—2„ ≠ Btq†; „ÕÍ = 0 ’ admissible „Õ
Otherwise
dJ = È2—2„ ≠ Btq†; „ÕÍ = ÈÒ„J ; „ÕÍ ∆ Ò„J = 2—2„ ≠ Btq†
Plan
Introduction
Various approaches to flow control
A general framework for optimal control
Time-invariant systems
Time-variant systems
Extension to optimal and robust control
Towards optimal and robust control
General framework
State equation
ˆtq + N(q, „, Â) = 0 with q(t = 0) = q i
with
• q: state vector (prescribed initial state q i)
• „: control
• Â: noise
• N: time-varying function of (q, „, Â) representing a smooth ODE
Towards optimal and robust control
General framework
State equation
ˆtq + L̀˘
q + B̀
˘
„+C̀
˘
 = 0
constant matrices © generally discretized di�erential op. (gradient,
divergence, laplacian) ∆ depends on a space discretization scheme
Towards optimal and robust control
General framework
State equation
ˆtq + Lq + B„ + CÂ = 0
Objective: minimize some perturbed measure of the energy
J(q, „, , Â) =
⁄ T
0
qt · q dt + —2
⁄ T
0
„t · „ dt≠“2
⁄ T
0
Ât · Â dt
J(q, „, Â) = Èq; qÍ + —2È„; „Í≠“2`̆ ÈÂ; ÂÍ
penalty parameter weighing the importance of noise
Towards optimal and robust control
General framework
State equation
ˆtq + Lq + B„ + CÂ = 0
Objective: minimize some weighed measure of the energy
J(q, „, Â) = Èq; qÍ + —2È„; „Í≠“2ÈÂ; ÂÍ
Constrained optimization problem
min J(q, „, Â) subject to state equations
Lagrangian function
L(q†, q, „, Â)=Èq; qÍ+—2È„; „Í≠“2ÈÂ; ÂÍ≠Èq†; ˆtq + Lq + B„ + CÂÍ
Optimal control / Time-variant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; ˆtq + Lq + B„Í
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = lim
‘æ0
L(q† + ‘q†Õ, q, „) ≠ L(q†, q, „)
‘
ˆL
ˆq† q
†Õ = ≠Èq†Õ; ˆtq + Lq + B„Í = 0 ’ admissible q†
Õ
∆ (q, „) must satisfy the state equation
Towards optimal and robust control
General framework
Lagrangian function
L(q†, q, „, Â)=Èq; qÍ+—2È„; „Í≠“2ÈÂ; ÂÍ≠Èq†; ˆtq + Lq + B„ + CÂÍ
Stationarity with respect to adjoint variable
ˆL
ˆq† q
†Õ = lim
‘æ0
L(q† + ‘q†Õ, q, „, Â) ≠ L(q†, q, „, Â)
‘
ˆL
ˆq† q
†Õ = ≠Èq†Õ; ˆtq + Lq + B„+CÂÍ = 0 ’ admissible q†
Õ
∆ (q, „, Â) must satisfy the state equation
Optimal control / Time-variant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; ˆtq + Lq + B„Í
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „) ≠ L(q†, q, „)
‘
ˆL
ˆq q
Õ = Ȉtq† + 2q ≠ Ltq†; qÕÍ≠q†(T )t ·qÕ(T ) = 0 ’ admissible qÕ
∆ q† must satisfy the adjoint equation
≠ˆtq† + Ltq† ≠ 2q = 0 with q†(T ) = 0
to be solved backwards in time (from t = T to 0).
Towards optimal and robust control
General framework
Lagrangian function
L(q†, q, „, Â)=Èq; qÍ+—2È„; „Í≠“2ÈÂ; ÂÍ≠Èq†; ˆtq + Lq + B„ + CÂÍ
Stationarity with respect to state variable
ˆL
ˆq q
Õ = lim
‘æ0
L(q†, q + ‘qÕ, „, Â) ≠ L(q†, q, „, Â)
‘
ˆL
ˆq q
Õ = Ȉtq† + 2q ≠ Ltq†; qÕÍ ≠ q†(T )t ·qÕ(T ) = 0 ’ admissible qÕ
∆ q† must satisfy the adjoint equation
≠ˆtq† + Ltq† ≠ 2q = 0 with q†(T ) = 0
to be solved backwards in time (from t = T to 0).
Optimal control / Time-variant, linear systems
General framework
Lagrangian function
L(q†, q, „) = Èq; qÍ + —2È„; „Í ≠ Èq†; ˆtq + Lq + B„Í
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ) ≠ L(q†, q, „)
‘
ˆL
ˆ„
„Õ = È2—2„ ≠ Btq†; „ÕÍ = 0 ’ admissible „Õ
Towards optimal and robust control
General framework
Lagrangian function
L(q†, q, „, Â)=Èq; qÍ+—2È„; „Í≠“2ÈÂ; ÂÍ≠Èq†; ˆtq + Lq + B„ + CÂÍ
Stationarity with respect to control variable
ˆL
ˆ„
„Õ = lim
‘æ0
L(q†, q, „ + ‘„Õ, Â) ≠ L(q†, q, „, Â)
‘
ˆL
ˆ„
„Õ = È2—2„ ≠ Btq†; „ÕÍ = 0 ’ admissible „Õ
Towards optimal and robust control
General framework
Lagrangian function
L(q†, q, „, Â)=Èq; qÍ+—2È„; „Í≠“2ÈÂ; ÂÍ≠Èq†; ˆtq + Lq + B„ + CÂÍ
Stationarity with respect to noise variable
ˆL
ˆÂ
ÂÕ = lim
‘æ0
L(q†, q, „, Â + ‘ÂÕ) ≠ L(q†, q, „, Â)
‘
ˆL
ˆ„
„Õ = ≠È2“2 + Ctq†; ÂÕÍ = 0 ’ admissible ÂÕ
Towards optimal and robust control
General framework
Lagrangian function
L(q†, q, „, Â)=Èq; qÍ+—2È„; „Í≠“2ÈÂ; ÂÍ≠Èq†; ˆtq + Lq + B„ + CÂÍ
Stationarity with respect to control/noise variable
ˆL
ˆ„
„Õ = È2—2„ ≠ Btq†; „ÕÍ = 0 ’ admissible „Õ
ˆL
ˆÂ
ÂÕ = ≠È2“2 + Ctq†; ÂÕÍ = 0 ’ admissible ÂÕ
Gradient evaluation
dJ = dL =
�
��
ˆL
ˆq q
Õ +
⇢
⇢
⇢⇢ˆL
ˆq† q
†Õ + ˆL
ˆ„
„Õ+ ˆL
ˆÂ
ÂÕ
Towards optimal and robust control
General framework
Lagrangian function
L(q†, q, „, Â)=Èq; qÍ+—2È„; „Í≠“2ÈÂ; ÂÍ≠Èq†; ˆtq + Lq + B„ + CÂÍ
Stationarity with respect to control/noise variable
ˆL
ˆ„
„Õ = È2—2„ ≠ Btq†; „ÕÍ = 0 ’ admissible „Õ
ˆL
ˆÂ
ÂÕ = ≠È2“2 + Ctq†; ÂÕÍ = 0 ’ admissible ÂÕ
Gradient evaluation
dJ = È2—2„ ≠ Btq†; „ÕÍ+È≠2“2 ≠ Ctq†; ÂÕÍ = ÈÒ„J ; „ÕÍ+ÈÒÂJ ; ÂÕÍ
∆ Ò„J = 2—2„ ≠ Btq† and ÒÂJ = ≠2“2Â ≠ Ctq†
Plan
Introduction
Various approaches to flow control
A general framework for optimal control
Time-invariant systems
Time-variant systems
Extension to optimal and robust control
Application to flow systems
Application to flow systems
Adjoint-based control
Remarks
• Tractable even for high-dimensional discretization of PDEs with
d.o.f.> 106
• Linear and nonlinear systems
Navier–Stokes
Ò · u = 0`̆ (ˆtu) + u · Òu + Òp ≠ ‹Ò2u = 0`̆
Continuity eq.
(constraint #1)
Momentum eq.
(constraint #2)
State vector q = (p, u) ∆ adjoint vector q† = (p†, u†)
Application to flow systems
Adjoint-based control
Remarks
• Tractable even for high-dimensional discretization of PDEs with
d.o.f.> 106
• Linear and nonlinear systems
Navier–Stokes
Ò · u = 0 (ˆtu) + u · Òu + Òp ≠ ‹Ò2u = 0
Core adjoint Navier–Stokes
Ò · u† = 0 (≠ˆtu†) ≠ u · Òu† + u† · Òut + Òp† ≠ ‹Ò2u† = 0`̆
© ≠ˆtq† + Ltq† = 0
Application to flow systems
Adjoint-based control
Remarks
• Tractable even for high-dimensional discretization of PDEs with
d.o.f.> 106
• Linear and nonlinear systems
• Quadratic / nonquadratic cost functions
• Energy J(u) =
s
� u
t · udV
• Drag J(u, p) =
s
� ≠pn + µÒu · ndS
• Pressure drop (duct flows) J(u, p) =
s
�1fi�2
p(u · n)dS
• Thermal power J(u, �) =
s
�1fi�2
�(u · n)dS
Application to flow systems
Adjoint-based control
Remarks
• Tractable even for high-dimensional discretization of PDEs with
d.o.f.> 106
• Linear and nonlinear systems
• Quadratic / nonquadratic cost functions
• Energy J(u) =
s
� u
t · udV
• Drag J(u, p) =
s
� ≠pn + µÒu · ndS
• Pressure drop (duct flows) J(u, p) =
s
�1fi�2
p(u · n)dS
• Thermal power J(u, �) =
s
�1fi�2
�(u · n)dS
Application to flow systems
Adjoint-based control
Remarks
• Tractable even for high-dimensional discretizationof PDEs with
d.o.f.> 106
• Linear and nonlinear systems
• Quadratic / nonquadratic cost functions
• Energy J(u) =
s T
0
s
� u
t · udV dt
• Drag J(u, p) =
s T
0
s
� ≠pn + µÒu · ndS dt
• Pressure drop (duct flows) J(u, p) =
s T
0
s
�1fi�2
p(u · n)dS dt
• Thermal power J(u, �) =
s T
0
s
�1fi�2
�(u · n)dS dt
Application to flow systems
Adjoint-based control
Remarks
• Tractable even for high-dimensional discretization of PDEs with
d.o.f.> 106
• Linear and nonlinear systems
• Quadratic / nonquadratic cost functions
• Energy J(u) =
s
� u
t(T ) · u(T )dV
• Drag J(u, p) =
s
� ≠p(T )n + µÒu(T ) · ndS
• Pressure drop (duct flows) J(u, p) =
s
�1fi�2
p(T )(u(T ) · n)dS
• Thermal power J(u, �) =
s
�1fi�2
�(T )(u(T ) · n)dS
Application to flow systems
Adjoint-based control
Remarks
• Tractable even for high-dimensional discretization of PDEs with
d.o.f.> 106
• Linear and nonlinear systems
• Quadratic / nonquadratic cost functions
" Adjoint problem depends on cost function
• Volume cost function © sources terms + homogeneous boundary
conditions
• Surface cost function © inhomogeneous boundary conditions
Application to flow systems
Adjoint-based control
Remarks
• Tractable even for high-dimensional discretization of PDEs with
d.o.f.> 106
• Linear and nonlinear systems
• Quadratic / nonquadratic cost functions
" Adjoint problem depends on cost function
• Energy J(u) =
s T
0
s
� u
t · udV dt
≠ˆtu†≠u · Òu†+u† · Òut +Òp†≠‹Ò2u†≠2u = 0`̆ with
;
q†(T ) = 0
q†|ˆ� = 0
© ≠ˆtq† + Ltq† ≠ 2q = 0
Application to flow systems
Adjoint-based control
Remarks
• Tractable even for high-dimensional discretization of PDEs with
d.o.f.> 106
• Linear and nonlinear systems
• Quadratic / nonquadratic cost functions
" Adjoint problem depends on cost function
• Drag J(u, p) =
s T
0
s
� ≠p(T )n + µÒu(T ) · ndS dt
≠ˆtu†≠u · Òu†+u† · Òut +Òp†≠‹Ò2u† = 0`̆ with
Y
]
[
q†(T ) = 0
q†|� = (1, 0, 0)
q†|ˆ�\� = 0
© ≠ˆtq† + Ltq† ≠⇢⇢2q = 0
Algorithms
Single-step adjoint-based control (no noise)
Start from uncontrolled („ = 0)
• Solve state equation
• Solve adjoint equation
• Compute gradient
• Use knowledge to get
æ q
æ q†
Ò„J =���2—2„ ≠ Btq†
dJ = ÈÒ„J ; „Í < 0
Cost= one nonlinear NS + one linear adjoint NS solutions
Algorithms
Single-step control for unsteady Navier–Stokes
Start from uncontrolled („ = 0)
• Solve state equation
• Solve adjoint equation
• Compute gradient
• Use knowledge to get
ˆtq + N(q,◆◆„) = 0 | q(0) = q i
≠ˆtq† + Ltq† ≠ 2q = 0 | q†(T ) = 0
Ò„J =���2—2„ ≠ Btq†
dJ = ÈÒ„J ; „Í < 0
Cost= one nonlinear NS + one linear adjoint NS unsteady solutions
Algorithms
Single-step control for unsteady Navier–Stokes)
Start from uncontrolled („ = 0)
• Solve state equation
• Solve adjoint equation
• Compute gradient
• Use knowledge to get
ˆtq + N(q,◆◆„) = 0 | q(0) = q i
≠ˆtq† + Ltq† ≠ 2q = 0 | q†(T ) = 0
Ò„J =���2—2„ ≠ Btq†
dJ = ÈÒ„J ; „Í < 0
" Adjoint problem requires full knowledge of state time history
State
Adjoint
0 T
∆ storage, checkpointing algorithms, receding strategies...
Algorithms
Illustrations
Drag minimization with passive device
Algorithms
Illustrations
Drag minimization with passive device
• Assume knowledge of Ò„=0J (gradient of uncontrolled solution)
• Assume prescribed control „ © „(x) modelling the e�ect of a small
device
• Find e�cient positions from maps of ”J(x) = ÈÒ„J ; „Í
Algorithms
Illustrations
Drag minimization with passive device
Meliga (2017)
• Presence of the device modeled by a body force „(x , y) tuned to
numerical data
• Map of ÈÒ„J ; „Í
Algorithms
Illustrations
Drag minimization with passive device
Meliga (2017)
• Presence of the device modeled by a body force „(x , y) tuned to
numerical data
• Map of ÈÒ„J ; „Í
Feasible to similarly optimize the placement of vortex generators
Rouillon et al. (2011)
Algorithms
Illustrations
Drag minimization with blowing and suction
?
Algorithms
Illustrations
Drag minimization with blowing and suction
• Assume knowledge of Ò„=0J (gradient of uncontrolled solution)
• Set „ = ≠–2Ò„J ∆ dJ = ≠–2ÈÒ„J ; Ò„JÍ < 0
Algorithms
Illustrations
Drag minimization with blowing and suction
• Velocity distribution set in the descent direction
• Optimality guaranteed if linear assumption holds (small amplitudes)
Meliga et al. (2018)
Algorithms
Iterative adjoint-based control (no noise)
Start from uncontrolled („0 = 0)
• Solve state equation
• Solve adjoint equation
• Compute gradient
• Update via user-defined minimization algorithm
• Start over
æ q0
æ q†0
҄0J =��
�2—2„0 ≠ Btq†0
„0 æ „1
Algorithms
Iterative adjoint-based control (no noise)
Start from current control („1)
• Solve state equation
• Solve adjoint equation
• Compute gradient
• Update via user-defined minimization algorithm
• Start over
æ q1
æ q†1
Ò„1J = 2—2„1 ≠ B
tq†1
„1 æ „2
Algorithms
Iterative adjoint-based control (no noise)
Start from current control („n)
• Solve state equation
• Solve adjoint equation
• Compute gradient
• Update via user-defined minimization algorithm
• Repeat until stationarity
æ qn
æ q†n
Ò„n J = 2—2„n ≠ B
tq†n
„n æ „n+1
Cost= N nonlinear NS + N linear adjoint NS solutions
Algorithms
Iterative adjoint-based control (with noise)
Start from current control and noise („n, Ân)
• Solve state equation
• Solve adjoint equation
• Compute gradients
• Update and via user-defined minimization
algorithm
• Repeat until stationarity
" Tractable but much more complex (requires finding a saddle
point, not just a minimum
æ qn
æ q†n
Ò„n J and ÒÂn J
„n æ „n+1 Ân æ Ân+1
Algorithms
Iterative adjoint-based control
Vanilla minimization algorithm for gradient update
„n+1 = „n ≠ –ndn
with
• dn: descent direction
• Steepest descent: dn = Ò„n J
• Conjug. gradient: dn = Ò„n J + ’ǹ˘
dn≠1
’(Ò„n J , Ò„n≠1J)
Polak-Ribiere, Fletcher-Reeves...
Algorithms
Iterative adjoint-based control
Vanilla minimization algorithm for gradient update
„n+1 = „n ≠ –ndn
with
• dn: descent direction
• Steepest descent: dn = Ò„n J
• Conjug. gradient: dn = Ò„n J + ’ndn≠1
• –n: descent parameter
• Fixed
• Determined by line search: min–n J(qn, „n ≠ –ndn) w/ root
finding algorithm (Brent method)
Algorithms
Illustrations
Approach well suited to shape optimization
æ control variable © position of a solid/fluid interface
Drag minimization
• Stokes limit (creeping flow)
• Navier–Stokes
Pironneau (1973)
Kondoh et al. (2011)
Algorithms
Illustrations
Approach well suited to shape optimization
æ control variable © position of a solid/fluid interface
Total pressure drop minimization
Abdel Nour et al.
Algorithms
Illustrations
Approach well suited to shape optimization
æ control variable © position of a solid/fluid interface
Thermal power minimization (balance with total pressure drop)
Subramaniam et al. (2019)