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Optimal Control Techniques for Sampled-Data Control Systems with Medical Applications
Bernard Bonnard, Toufik Bakir, Piernicola Bettiol, Loïc Bourdin, Jérémy Rouot, Sandrine Gayrard
To cite this version:
Bernard Bonnard, Toufik Bakir, Piernicola Bettiol, Loïc Bourdin, Jérémy Rouot, et al.. Optimal Control Techniques for Sampled-Data Control Systems with Medical Applications. PGMO Days 2020, Dec 2020, Paris, France. �hal-03021332�
Optimal Control Techniques for Sampled-Data Control Systems with Medical Applications
PI: Bernard Bonnard (UBFC, INRIA)
Participants:T. Bakir, P. Bettiol, L. Bourdin, J. Rouot, S. Gayrard1
PGMO Days December 2020
1. Thèse CIFRE :The Mathematical Frame and the Application to functional Muscular Control, 2020–
Control systemof the form
˙
x(t)=f(x(t),u(t)), x(0)=x0, x(T)=xT
and an optimal control of the Mayer type
minu(·) ϕ(x(T))
withϕ:Rn→R.
Permanent Control :u: [0,T]7→U,u∈U whereU are the absolutely continuous maps valued inU.
Sampled Data Caseu∈Usampl ed: fixn∈N, nsampling times
0<t1< · · · <tn<T.
(n+1)-amplitudes
η=(η0, . . . ,ηn)∈[0, 1]n+1
The control is constant over[ti,ti+1].
Numerical schemes
In thepermanent case, the optimal control can be computed using Direct scheme :the problem is transformed into a finite dimensional optimization problem using
discretization scheme for the dynamics discretization scheme for the control
Indirect scheme :the problem can be analyzed using Pontryagin Maximum Principle which leads to Hamiltonian equations
˙ x=∂H
∂p, p˙= −∂H
∂x H(x,p,u)=max
v∈U H(x,p,v)
withH(x,p,v)=p·f(x,v),pbeing the adjoint vector satisfying
p(T)=p0∂ϕ
∂x(x(T)) (transversality condition).
This necessary optimality condition can be handled using ashooting method
Shooting method
x(0) x
p
t p(T)
p(t1) p(t2) p(0)
0 x(t1) x(t2)
x(T)
t1 t2 T
Force-Fatigue muscular model
2FES inputi.Dirac impulsesδat timest=0,t1,t2, . . . ,tN. i(t)=
N
X
i=0
Riηiδ(t−ti), ηi∈[0, 1]
where
Ri:=
1, fori=0,
1+( ¯R−1) exp µ
−ti−ti−1 τc
¶
, fori=1, . . . ,N,
takes into account thetetaniccontraction.
FES signalEs.
Es(t)= 1 τc
N
X
i=0
RiηiH(t−ti) exp µ
−t−ti
τc
¶
H: Heaviside
2. based on the Ding et al. / Hill-Huxley model
The FES signal drives the evolution of the dynamics :
C˙N(t)= −CN(t)
τc +Es(t;ti,ηi), F˙(t)= −F(t)γ(t)+Aβ(t).
where the Hill functions are given by
β(t):= CN(t)
Km+CN(t), and γ(t):= 1 τ1+τ2β(t). (A,Km,τ1,τ2)are the fatigue parameters.
The problem fits in the sampled data control frame with : u0=η0e−t/τc/τcon[0,T]
u1=u0(t1)+η1R1e−(t−t1)/τc/τcon[t1,T] ...
Note that in this form each control splits into Head: restricting to[ti,ti+1]
Tail: restricting to[ti,T]
Optimal control problems considered.
(OCP1)maxti,ηiF(T) (OCP2)min
ti,ηi
Z T
0 |F(t)−Fr e f|2dt(Fr e f : reference force).
Main theoretical results
cN(T)= 1 τc
n
X
i=0
e−(t−ti)/τc(T−ti)Ri
F(T)is a piecewiseC∞mapping with respect toηi,ti and the problem fits in a finite dimensional optimization problem.
The non-fatigue modelfor a sequence of train, fatigue parametersP satisfy a dynamics of the form
P˙(t)=P(t)−Pr est
τp +αpF(t).
First order necessary optimality conditions can be obtained adapting [5] and are described in [2]. They were numerically implemented and compared with a direct optimization scheme in [3].
Numerical results using direct method
0.000 0.088 0.157 0.215 0.263 0.305 0.341 0.375 0.404 0.432 0.451 0.500
0.0000 0.1852 0.3704
0.0000 0.4636 0.9272
2.962 2.986 3.009
0.1030 0.1041 0.1052
0.0510 0.0522 0.0534
0.000 0.088 0.157 0.215 0.263 0.305 0.341 0.375 0.404 0.432 0.451 0.500
0.6093 40.2200 79.8200
Direct method :max
ti F(T),N=10,10ms≤ti+1−ti,i=0, . . . ,N.
Theorem (see [2])
If(η∗0,η∗1, . . . ,η∗N,t1∗, . . . ,tN∗)is optimal, then there existspsatisfying the co-state equation and the transversality condition.
Moreover, the necessary conditions are : (i) the inequality
ÃZT t∗i
p1(s)b(s) ds
! η˜i
≤
0,for alli=0, . . . ,nand all admissible perturbationη˜i ofη∗i ; (ii) and the inequality
NCi:= µ
−p1(ti∗)b(ti∗)G(ti∗−1,ti∗)η∗i +b(−ti∗)η∗i
Z T ti∗
p1(s)b(s) ds
+b(−ti∗)( ¯R−1)η∗i+1 ZT
t∗i +1
p1(s)b(s) ds
¶ t˜i
≤
0,for alli=1, . . . ,nand all admissible perturbationt˜i ofti∗.
Work in progress
We need efficient algorithms (real-time application) : Explicit expression of
(t1, . . . ,tn,η0, . . . ,ηn)→F(T) to apply a direct optimization scheme
LQmethods MPC methods
Online parameter estimation coupled with optimization methods [1]
Extension to the non-isometric case withjoint angle variableto produce a motion [4]
References
[1] T. Bakir, B. Bonnard B. and J. Rouot, A case study of optimal input-output system with sampled-data control : Ding et al. force and fatigue muscular control model.Netw. Hetero.
Media,14no.1 (2019), 79–100
[2] T. Bakir, B. Bonnard B., L. Bourdin and J. Rouot, Pontryagin-type conditions for optimal muscular force response to functional electrical stimulations.J. Optim. Theory Appl.,184 (2020), 581–602.
[3] T. Bakir, B. Bonnard B., L. Bourdin and J. Rouot, Direct and Indirect Methods to Optimize the Muscular Force Response to a Pulse Train of Electrical Stimulation, preprint, hal-02053566.
[4] B. Bonnard B. and J. Rouot, Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model.
Accepted inJournal of Geometric Mechanics(2020)
[5] L. Bourdin and E. Trélat, Optimal sampled-data control, and generalizations on time scales, Math. Cont. Related Fields,6(2016), 53–94
