optimal control of fractional systems: numerics under …fa/cdps/talks/matignon.pdffractional...
Post on 26-Oct-2020
6 Views
Preview:
TRANSCRIPT
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Optimal control of fractional systems:numerics under diffusive formulation
Denis MATIGNON & Nicolas THERME
ISAE, DMIA department & Universite de Toulouse.
CDPS’11, Wuppertal, Germany.— Tuesday, July 19th, 2011 —
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
1 Introduction
2 Fractional operators and adjoints under diffusiverepresentation
State-space representationsAdjoints of Fractional Operators
3 Models under study
4 Optimal control of the toy model
5 Optimal diffusive representationsl2 criteria and closed-form solutionsl1 criteria, Linear Programming formulation and simplexalgorithm
6 Conclusion and Future works
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Outline
1 Introduction2 Fractional operators and adjoints under diffusive
representationState-space representationsAdjoints of Fractional Operators
3 Models under study
4 Optimal control of the toy model
5 Optimal diffusive representationsl2 criteria and closed-form solutionsl1 criteria, Linear Programming formulation and simplexalgorithm
6 Conclusion and Future works
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Fractional guys ? almost everywhere !
Fractional differential systems have become quite popular inthe recent decades, giving rise to a wide literature, both on thetheoretical and on the applied sides :
monogaphs,international journals : Fractional Calculus and AppliedAnalysis,Fractional Dynamics and Applications,special issues of international journals,
and alsointernational conferences : Fractional Differentiation and itsApplicationsworkshops of international conferences,
are now devoted to this active research field !
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
What about optimal control, then ?
However, even if different scientific communities seem to havebeen involved in these questions, still very few papers areconcerned with the question of optimal control of fractionaldifferential systems.
In e.g. [Tricaud & Chen (2010)] or [Defterli (2010)],1 ad hoc finite-dimensional approximations of fractional
derivatives are used in the first place,2 classical optimal control methods are being applied in the
second place ;But no proof of convergence of the process is provided.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Why is it so ?
Possible answers :1 optimal control of infinite-dimensional systems is a quite
involved and technical field,2 the very nature of fractional operators itself : causal, but
highly non-local in time ; hence their adjoint becomesnecessarily anti-causal and still... non-local in time.
Thus, we will be left with coupled forward and backwardfractional dynamics in order to solve the optimal controlproblem for fractional differential systems.=⇒ at first glance, it seems very unlikely that Riccati equationscould be either analysed or even solved (not to speak ofadequate numerical schemes for these) in such a complicatedsetting !
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
So, what ?
In order to overcome this intrinsic difficulty, we propose to usethe equivalent diffusive representations of fractional systems,and to work on it, as for infinite dimensional systems of integerorder !
Let us recall diffusive representations of fractional operatorsand their adjoints and see how these can be useful for optimalcontrol problems, on a series of models of decreasingcomplexity, namely :
1 Webster-Lokshin Wave equation,2 A Fractionally Damped Oscillator,3 An Oscillator Damped by Memory Variables.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Outline
1 Introduction2 Fractional operators and adjoints under diffusive
representationState-space representationsAdjoints of Fractional Operators
3 Models under study
4 Optimal control of the toy model
5 Optimal diffusive representationsl2 criteria and closed-form solutionsl1 criteria, Linear Programming formulation and simplexalgorithm
6 Conclusion and Future works
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Introduction
Some useful identities
Let β ∈ (0, 1), in the frequency domain, we have :
Hβ : C \ R− → C
s 7→∫ ∞
0µβ(ξ)
1s + ξ
dξ =1sβ
,with µβ(ξ) ∝ ξ−β .
So to speak, fractional transfer functions Hβ are nothing but asuperposition of first-order systems, with appropriate weight µβ .Equivalently, in the time domain, this reads :
hβ : R+ → R
t 7→∫ ∞
0µβ(ξ) e−ξ t dξ =
1Γ(β)
tβ−1 .
So to speak, fractional kernels hβ are nothing but asuperposition of decaying exponential, with weight µβ .=⇒ Input-output & State-space representations can be derivedfor fractional integrals Iβ and derivatives Dα.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
State-space representations
Input-output representation
Let u and y = Iβu be the input and output of the causalfractional integral of order β, defined by the Riemann-Liouvilleformula y = hβ ? u =
∫ t0 hβ(t − τ) u(τ) dτ in the time domain,
which reads Y (s) = Hβ(s) U(s) in the Laplace domain :
y(t) =
∫ ∞
0µβ(ξ) [eξ ? u](t) dξ ,
with eξ(t) := e−ξ t 1t≥0, and [eξ ? u](t) =∫ t
0 e−ξ (t−τ) u(τ) dτ .
Now for fractional derivative of order α ∈ (0,1) in the sense ofdistributions of Schwartz, we have y = Dαu = D[I1−αu], and acareful computation shows that :
y(t) =
∫ ∞
0µ1−α(ξ) [u − ξ eξ ? u] (t) dξ .
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
State-space representations
State-space representations
Let ϕ(ξ, .) := [eξ ? u](t) be the state, parametrized by ξ.
∂tϕ(ξ, t) = −ξ ϕ(ξ, t) + u(t), ϕ(ξ,0) = 0 , (1)
y(t) =
∫ ∞
0µβ(ξ)ϕ(ξ, t) dξ ; (2)
and
∂t ϕ(ξ, t) = −ξ ϕ(ξ, t) + u(t), ϕ(ξ, 0) = 0 , (3)
y(t) =
∫ ∞
0µ1−α(ξ) [u(t)− ξ ϕ(ξ, t)] dξ . (4)
are state-space representations for Iβ and Dα, respectively.
Note : functional spaces must be specified for theserepresentations to make sense ; more precisely :
ϕ belongs to Hβ := {ϕ s.t .∫∞
0 µβ(ξ)|ϕ|2 dξ <∞},ϕ belongs to Hα := {ϕ s.t .
∫∞0 µ1−α(ξ)|ϕ|2 ξ dξ <∞} ;
see e.g. [Haddar and M. (2008)].
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Adjoints of Fractional Operators
Adjoints of fractional integrals
On H = L2(0,T ), the adjoint of the causal fractional integratorIβ0+ : u 7→ hβ ? u, defined by
y(t) := Iβ0+u(t) =1
Γ(β)
∫ t
0(t − τ)β−1 u(τ) dτ ,
is IβT− : z 7→ v , the anti-causal fractional integral, defined by
v(τ) := IβT−z(τ) =1
Γ(β)
∫ T
τ(t − τ)β−1 z(t) dt .
=⇒ quite difficult to handle, especially in coupled situations ofoptimal control !=⇒ Need to make it easier.=⇒ Extend diffusive representation to anti-causal context !(see e.g. [M. (2009)] for a first definition of those).
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Adjoints of Fractional Operators
The backward diffusive realization (1)
Let the backward dynamical system :
∂tψ(ξ, τ) = +ξ ψ(ξ, τ)− z(τ), for 0 < τ < T , (5)with ψ(ξ,T ) = 0 as final condition ; (6)
together with the output, defined by :
v(τ) =
∫ ∞
0µβ(ξ)ψ(ξ, τ) dξ ;
they provide a realization for v = IβT−z.
Moreover, the fundamental equality holds :
(Iβ0+u, z)L2(0,T ) = (u, IβT−z)L2(0,T ) . (7)
Proof : straightforward, simply relies on properties ofreal-valued exponentials.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Adjoints of Fractional Operators
Adjoints of fractional derivatives
On H = L2(0,T ), the adjoint of the causal fractional derivativeDα
0+ : u 7→ ddt (h1−α ? u), defined on its domain by
y(t) := Dα0+u(t) =
ddt
[1
Γ(1− α)
∫ t
0(t − τ)−α u(τ) dτ
],
is DαT− : z 7→ v , the anti-causal fractional derivative, defined on
its domain by
v(τ) := DαT−z(τ) = − d
dτ
[1
Γ(1− α)
∫ T
τ(t − τ)−α z(t) dt
].
Note : the derivatives are to be understood in the sense ofdistributions of Schwartz.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Adjoints of Fractional Operators
The backward diffusive realization (2)
Let the backward dynamical system :
∂t ψ(ξ, τ) = +ξ ψ(ξ, τ)− z(τ), for 0 < τ < T , (8)with ψ(ξ,T ) = 0 as final condition ; (9)
together with the extended output, defined by :
v(τ) =
∫ ∞
0µ1−α(ξ)
[z(τ)− ξ ψ(ξ, τ)
]dξ ;
they provide a realization for v = DαT−z.
Moreover, the fundamental equality holds :
(Dα0+u, z)L2(0,T ) = (u,Dα
T−z)L2(0,T ) , (10)
Proof : less straightforward, but still relies on properties ofreal-valued exponentials.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Outline
1 Introduction2 Fractional operators and adjoints under diffusive
representationState-space representationsAdjoints of Fractional Operators
3 Models under study
4 Optimal control of the toy model
5 Optimal diffusive representationsl2 criteria and closed-form solutionsl1 criteria, Linear Programming formulation and simplexalgorithm
6 Conclusion and Future works
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Webster-Lokshin Wave equation
1 Webster-Lokshin Wave equation :
∂2t w +ε(x)D1/2
0+ [∂tw ]+d(x) ∂tw +η(x) I1/20+ [∂tw ]−∂2
x w = 0 ,
for 0 < x < L and t > 0, with boundary control ue(t) atx = 0, and initial conditions.Provided ε(x) > 0, d(x) ≥ 0 and η(x) > 0, once thediffusive reformulation has been used, we can prove :
existence and uniqueness, see e.g. [Haddar & M. (2008)],asymptotic stability, see [M. (2006)], [M. & Prieur (2011)],consistent and accurate numerical schemes, see e.g.[Haddar, Li & M. (2009)], also [Li (2010)].
A finite horizon optimal control problem, reformulated in thenew framework presented above, will become tractablewith the theory of optimal control for linear PDEs, becausethe system is now no more than the coupling of a 1D waveequation with two 1D diffusion equations... still to be done !
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
A Fractionally Damped Oscillator
2 A Fractionally Damped Oscillator : together with dynamicboundary conditions of Robin type, the Lokshin model hasa Riesz basis of eigenvectors (studied in [M. (1996)], seealso [Kergomard, Debut & M. (2006)]) : the projection ofthe PDE onto one mode gives rise to a fractionally dampedoscillator, studied in [M. & Prieur (2005)], and for whichelementary propreties and numerical simulations havebeen presented in e.g. [Deu & M. (2010)].
x + Dα0+[x ] + x + Iβ0+[x ] + ω2 x = ue ,
The above framework is well suited to the formulation of anoptimal control problem of this system in a classical setting,with no more fractional operators and no more heredity :only the diffusive subsystems are infinite dimensional.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Oscillator Damped by Memory Variables
3 An Oscillator Damped by Memory Variables (toy model) :Discretizing the diffusive representations of y = Iβ0+u on Kpoints, and y = Dα
0+u on L points, in a consistent way, weget :
x + y + x + y + ω2 x = ue ,
with three types of damping :x = u, instantaneous w.r.t u ;y(u), with memory and low-pass behaviour : measure µconsists of finitely many (K ) Dirac measures located atsome ξk with positive weights µk ;y(u) with memory and high-pass behaviour : measure νconsists of finitely many (L) Dirac measures located atsome ξl with positive weights νl .
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Outline
1 Introduction2 Fractional operators and adjoints under diffusive
representationState-space representationsAdjoints of Fractional Operators
3 Models under study
4 Optimal control of the toy model
5 Optimal diffusive representationsl2 criteria and closed-form solutionsl1 criteria, Linear Programming formulation and simplexalgorithm
6 Conclusion and Future works
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Methodology
The objective is to minimize the energy functional
J(ue) =12
∫ T
0X t(τ) Q X (τ) + ue(τ)2 dτ +
12
X t(T ) DT X (T )
with an external input ue on the toy model, a controlleddynamical systems.Why ? Because the diffusive components with small ξk (big τk )display a long-memory behaviour that is typical for fractionalsystems ! Thus, the objective is to make the convergence toequilibrium much faster.=⇒ solve Dynamic Riccati Equation on S(t), of dimension(2 + K + L)× (2 + K + L), thanks to a Runge-Kutta method,then apply the time-varying feedback on the state X .
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Time domain simulations
Parameters : K = L = 3, and T = 20, DT = 1.
Left : Open Loop, Right : Closed Loop (feedback from DRE).
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
A plot of the SVD of the Riccati matrix
Parameters : K = L = 3, and T = ∞.
Left : SVD plot, Right : Closed Loop (feedback from ARE).
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
An interesting idea ?
An interesting idea follows from the plot the singular values ofthe Riccati matrix versus time : why not apply the infinite-timefeedback, solution of the Algebraic Riccati equation, then ?(much easier, allows greater values of K and L).
But... this heuristics cannot be used to prove any convergenceresults, since the diffusive approximations converge on finitehorizons only.
=⇒ there is indeed a need for low dimensional representationof complexity, by :
interpolation methods,optimization methods.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Outline
1 Introduction2 Fractional operators and adjoints under diffusive
representationState-space representationsAdjoints of Fractional Operators
3 Models under study
4 Optimal control of the toy model
5 Optimal diffusive representationsl2 criteria and closed-form solutionsl1 criteria, Linear Programming formulation and simplexalgorithm
6 Conclusion and Future works
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Re-interpreting Sobolev spaces
Optimization in the frequency domain, stemming from
h(f ) = limε→0+
H(ε+ 2iπf )
Norms in L2, or Sobolev spaces Hs, are defined as :
‖h‖2Hs(Rt )
=
∫Rf
ws(f ) |H(2iπf )|2 df , with ws(f ) = (1+4π2f 2)s .
where s ∈ R tunes the balance between low and highfrequencies.For specific applications, more general frequencydependent weights can be used : bounded frequencyrange, logarithmic scale, relative error measurement,bounded dynamics ... see e.g. [Helie & M. (2006)].
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Re-interpreting Sobolev spaces
Optimization in the frequency domain, stemming from
h(f ) = limε→0+
H(ε+ 2iπf )
Norms in L2, or Sobolev spaces Hs, are defined as :
‖h‖2Hs(Rt )
=
∫Rf
ws(f ) |H(2iπf )|2 df , with ws(f ) = (1+4π2f 2)s .
where s ∈ R tunes the balance between low and highfrequencies.
For specific applications, more general frequencydependent weights can be used : bounded frequencyrange, logarithmic scale, relative error measurement,bounded dynamics ... see e.g. [Helie & M. (2006)].
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Re-interpreting Sobolev spaces
Optimization in the frequency domain, stemming from
h(f ) = limε→0+
H(ε+ 2iπf )
Norms in L2, or Sobolev spaces Hs, are defined as :
‖h‖2Hs(Rt )
=
∫Rf
ws(f ) |H(2iπf )|2 df , with ws(f ) = (1+4π2f 2)s .
where s ∈ R tunes the balance between low and highfrequencies.For specific applications, more general frequencydependent weights can be used : bounded frequencyrange, logarithmic scale, relative error measurement,bounded dynamics ... see e.g. [Helie & M. (2006)].
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Building up specific weights for audio applications
For audio applications, w(f ) can be adapted and modifiedaccording to the following requirements :
1 a bounded frequency range f ∈ [f−, f +] : w(f ) 1[f−,f +](f ) ;2 a frequency log-scale : w(f )/f ;3 a relative error measurement : w(f )/|H(2iπf )|24 a relative error on a bounded dynamics :
w(f )/(SatH,Θ(f )
)2 where the saturation function SatH,Θ
with threshold Θ is defined by
SatH,Θ(f ) =
{|H(2iπf )| if |H(2iπf )| ≥ ΘHΘH otherwise
Note : normalization of the samples is desirable in mostaudio applications, before the sequence is sent to DACaudio converters.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Regularized criterion with equality constraints
Let Hµ(s) =∑K
k=1 µk (s + ξk )−1 ; based on Bode diagrams,a heuristic choice for the {ξk}1≤k≤K leads to a geometricsequence on a frequency range of interest.
The regularized criterion reads :
CR (µ) =
∫R+
∣∣∣Hµ(2iπf )−H(2iπf )∣∣∣2w(f )df +
K∑k=1
εk |µk |2,
Equality constraints for Hµ(dj ) at prescribed frequency
points ηj , 1 ≤ j ≤ J are taken into account thanks to aLagrangian CR,L by adding to CR :
<e
`∗
H(d1)(2iπη1)− Hµ
(d1)(2iπη1)
...
H(dJ)(2iπηJ)− Hµ(dJ)
(2iπηJ)
,
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Regularized criterion with equality constraints
Let Hµ(s) =∑K
k=1 µk (s + ξk )−1 ; based on Bode diagrams,a heuristic choice for the {ξk}1≤k≤K leads to a geometricsequence on a frequency range of interest.The regularized criterion reads :
CR (µ) =
∫R+
∣∣∣Hµ(2iπf )−H(2iπf )∣∣∣2w(f )df +
K∑k=1
εk |µk |2,
Equality constraints for Hµ(dj ) at prescribed frequency
points ηj , 1 ≤ j ≤ J are taken into account thanks to aLagrangian CR,L by adding to CR :
<e
`∗
H(d1)(2iπη1)− Hµ
(d1)(2iπη1)
...
H(dJ)(2iπηJ)− Hµ(dJ)
(2iπηJ)
,
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Regularized criterion with equality constraints
Let Hµ(s) =∑K
k=1 µk (s + ξk )−1 ; based on Bode diagrams,a heuristic choice for the {ξk}1≤k≤K leads to a geometricsequence on a frequency range of interest.The regularized criterion reads :
CR (µ) =
∫R+
∣∣∣Hµ(2iπf )−H(2iπf )∣∣∣2w(f )df +
K∑k=1
εk |µk |2,
Equality constraints for Hµ(dj ) at prescribed frequency
points ηj , 1 ≤ j ≤ J are taken into account thanks to aLagrangian CR,L by adding to CR :
<e
`∗
H(d1)(2iπη1)− Hµ
(d1)(2iπη1)
...
H(dJ)(2iπηJ)− Hµ(dJ)
(2iπηJ)
,
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Discrete criterion
Discrete version of the criterion for frequencies increasingfrom f1 = f− to fN+1 = f+ is, with sn = 2iπfn :
C(µ) ≈N∑
n=1
wn
∣∣∣Hµ(sn)− H(sn)∣∣∣2 with wn =
∫ fn+1
fnw(f )df .
In matrix notations, this rewrites
CR,L(µ) =(Mµ−h
)∗W (Mµ−h
)+µtEµ+<e
(`∗ [k − Nµ]
),
with
M : model N × KN : constraint model J × KE : regularization K × KW : weights N × Nh : data N × 1k : constraints J × 1
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Discrete criterion
Discrete version of the criterion for frequencies increasingfrom f1 = f− to fN+1 = f+ is, with sn = 2iπfn :
C(µ) ≈N∑
n=1
wn
∣∣∣Hµ(sn)− H(sn)∣∣∣2 with wn =
∫ fn+1
fnw(f )df .
In matrix notations, this rewrites
CR,L(µ) =(Mµ−h
)∗W (Mµ−h
)+µtEµ+<e
(`∗ [k − Nµ]
),
with
M : model N × KN : constraint model J × KE : regularization K × KW : weights N × Nh : data N × 1k : constraints J × 1
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Closed-form solutions !
If J = 0 (no constraint), the solution reduces to
µ=M−1H ,
where M=<e(M∗WM+E
)and H=<e
(M∗Wh
).
For J ≥ 1, the solution reads :
µ = M−1[H+ N tN−1
(k − NM−1H
)],
where N = NM−1N t is invertible for non-redundant
constraints, and{
N t denotes [<e(N t),=m(N t)]
k t denotes [<e(k t),=m(k t)].
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Closed-form solutions !
If J = 0 (no constraint), the solution reduces to
µ=M−1H ,
where M=<e(M∗WM+E
)and H=<e
(M∗Wh
).
For J ≥ 1, the solution reads :
µ = M−1[H+ N tN−1
(k − NM−1H
)],
where N = NM−1N t is invertible for non-redundant
constraints, and{
N t denotes [<e(N t),=m(N t)]
k t denotes [<e(k t),=m(k t)].
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l2 criteria and closed-form solutions
Our example : Hβ(s) = s−β, µβ(−ξ) ∝ ξ−β
Top : Interpolation, K = 16. Bottom : Optimization, K = 10 !
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
l1 criteria, Linear Programming formulation and simplex algorithm
Identification in l1 setting
Suppose we want to identify K values of µk with N prescribedmeasurements, and N >> K . Following [Boyd et al. (2004)],
1 Consider the following free optimization problem :
minµ∈RK
‖M µ− h‖l1(RN) , i.e. minµ∈RK
N∑n=1
|(M µ)n − hn| .
It can be rewritten in an equivalent Linear Programmingproblem, as follows, where ≤ means componentwise :
min{µ ∈ RK , t ∈ RN
+, −t ≤ M µ− h ≤ t}1t t
2 Using the simplex algorithm, the LP problem can be solvedefficiently. Moreover, the algorithm searches for vertices(corners of the polytope) as particular solutions : manyequalities are fulfilled !
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Outline
1 Introduction2 Fractional operators and adjoints under diffusive
representationState-space representationsAdjoints of Fractional Operators
3 Models under study
4 Optimal control of the toy model
5 Optimal diffusive representationsl2 criteria and closed-form solutionsl1 criteria, Linear Programming formulation and simplexalgorithm
6 Conclusion and Future works
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Many things are... still to be done !
1 Optimal weights ? Refine constrained l2 methods,thourough study of l1 methods, comparison of the results.Frequency domain versus time-domain formulation ?
2 Optimal control ? Solve dynamic Riccati equation throughthe Hamiltonian matrix, using symplectic numericalmethods on an invariant manifold.
3 Top-down methodology instead of bottom-up strategy ?Derive the infinite-dimensional optimal control system inthe first place, discretize the equations second place.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
S. BOYD AND L. VANDENBERGHE, Convex Optimization,Cambridge Univ. Press, 2004.
O. DEFTERLI, A numerical scheme for two-dimensionaloptimal control problems with memory effect, Computersand Math. with Applications, 59 (2010), pp. 1630–1636.
J.-F. DEU AND D. MATIGNON, Simulation of fractionallydamped mechanical systems by means of aNewmark-diffusive scheme, Computers and Math. withApplications, 59 (2010), pp. 1745–1753.
G. GARCIA AND J. BERNUSSOU, Identification of thedynamics of a lead acid battery by a diffusive model,ESAIM : Proc., 5 (1998), pp. 87–98.
G. GRIPENBERG, S.-O. LONDEN, AND O. STAFFANS,Volterra integral and functional equations, vol. 34 of
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
Encyclopedia of Mathematics and its Applications,Cambridge University Press, 1990.
H. HADDAR, J.-R. LI, AND D. MATIGNON, Efficient solutionof a wave equation with fractional order dissipative terms, J.Computational and Applied Mathematics, 234 (2010),pp. 2003–2010.
H. HADDAR AND D. MATIGNON, Theoretical and numericalanalysis of the Webster-Lokshin model, Res. Rep. RR6558,Institut National de la Recherche en Informatique etAutomatique (INRIA), jun. 2008.
T. HELIE AND D. MATIGNON, Diffusive representations forthe analysis and simulation of flared acoustic pipes withvisco-thermal losses, Mathematical Models and Methods inApplied Sciences, 16 (2006), pp. 503–536.
, Representations with poles and cuts for thetime-domain simulation of fractional systems and irrational
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
transfer functions, Signal Processing, 86 (2006),pp. 2516–2528.
J. KERGOMARD, V. DEBUT, AND D. MATIGNON, Resonancemodes in a 1-D medium with two purely resistiveboundaries : calculation methods, orthogonality andcompleteness, J. Acoust. Soc. Amer., 119 (2006),pp. 1356–1367.
J.-R. LI, A fast time stepping method for evaluatingfractional integrals, SIAM J. Sci. Comput., 31 (2010),pp. 4696–4714.
D. MATIGNON, Fractional modal decomposition of aboundary-controlled-and-observed infinite-dimensionallinear system, in Mathematical Theory of Networks andSystems (MTNS), Saint Louis, Missouri, june 1996, 5 p.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
, Asymptotic stability of the Webster-Lokshin model, inMathematical Theory of Networks and Systems (MTNS),Kyoto, Japan, jul 2006, 11 p. CD–Rom. (invited session).
, Diffusive representations for fractional Laplacian :systems theory framework and numerical issues, Phys.Scr., 136 (2009).
, An introduction to fractional calculus, ch. 7 in Scaling,Fractals and Wavelets, by P. ABRY, P. GONDCALVES, AND
J. L. VEHEL, eds., Digital Signal and Image Processingseries, ISTE - Wiley, 2009, pp. 237–278.
D. MATIGNON AND G. MONTSENY, eds., FractionalDifferential Systems : models, methods and applications,vol. 5 of ESAIM : Proceedings, SMAI, December 1998.
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
D. MATIGNON AND C. PRIEUR, Asymptotic stability of linearconservative systems when coupled with diffusive systems,ESAIM Control Optim. Calc. Var., 11 (2005), pp. 487–507.
, Asymptotic stability of Webster-Lokshin equation, tobe submitted (2011).
D. MATIGNON AND H. ZWART, Standard diffusive systemsas well-posed linear systems, (2009), submitted.
J.-P. RAYMOND, Optimal control of partial differentialequations, French Indian Cyber-University in Science,2007. Lecture Notes.http ://www.math.univ-toulouse.fr/∼raymond/book-ficus.pdf.
S. G. SAMKO, A. A. KILBAS, AND O. I. MARICHEV,Fractional integrals and derivatives : theory andapplications, Gordon & Breach, 1987. (transl. fromRussian, 1993).
Outline Introduction Diffusive Rep. Models under study Optimal control of the toy model Optimal diffusive representations Conclusion and Future works Some references
O. J. STAFFANS, Well-posedness and stabilizability of aviscoelastic equation in energy space, Trans. Amer. Math.Soc., 345 (1994), pp. 527–575.
R. F. STENGEL, Optimal control and estimation, Dover,1994.
N. THERME, Controle Optimal d’un systeme couple d’EDPhyperbolique-parabolique, Internship Report, ISAE, July2011.
C. TRICAUD AND Y. Q. CHEN, Solution of fractional orderoptimal control problems using SVD-based rationalapproximations, in Proc. American Control Conference, St.Louis, Mo, USA, June 2009, ACC’09, pp. 1430–1435.
, Time-optimal control of systems with fractionaldynamics, Int. J. Differential Equations, 2010 (2010).
top related