OPTIMAL CONTROL APPLIED TO OPTIMIZATION
OF MOBILE SWITCHING SURFACES
PART I: ALGORITHM
Jean-Claude Jolly, Céline Quémard, Jean-Louis Ferrier
LISA – FRE 2656 CNRS, Université d’Angers, 62, avenue Notre-Dame du Lac, 49000 Angers, France
Keywords: Hybrid Dynamical System, Optimization, Optimal Control, Mobile Switching Surface.
Abstract: Following (Boccadoro et al., 2004) and (Wardi et al., 2004), we consider hybrid dynamical systems with
parameterized switching surfaces. The goal is to optimize the choice of parameters in relation with a
criterion. In an optimal control framework we deepen and generalize results of these authors. We get that
thanks a known algorithm, usually not totally explicit, that can be here specified up to obtain an efficient
one. Ideas of some new or classical applications are given. They will be developped in a second paper,
enforcing the theoric results expanded here.
1 INTRODUCTION
Authors of (Boccadoro et al., 2004, 2005) and
(Wardi et al., 2004), have pointed out and studied an
optimization problem of switching surfaces in
hybrid dynamical systems (h.d.s. – for general
notions see, for example, (Bensoussan et al., 1997),
(Van Der Schaft and Schumacher, 1999), (Zaytoon
et al., 2001)). Here, drawing our inspiration from the
classical reference book (Bryson and Ho, 1969), we
get the results of (Boccadoro et al., 2004) and
(Wardi et al., 2004), by another method. This one
uses the variational calculus with an augmented
criterion. It readily gives the searched relations. As
opposed to the more technical method used by the
previous authors, here the meaning of the costate in
the framework of optimal control becomes clear.
Moreover, our results are more general, including
mobility of switching surfaces and specific terms in
the criterion at switching instants. An important
result is the determination of the optimal switching
instants.
First, the problem is stated. Varitional calculus is
then applied to an augmented criterion. This supplies
a method for the criterion gradient calculus which
reduces the optimization problem to the use of a
classical steepest descent algorithm. In our
conclusion we give ideas of classical or new
applications. They are developped in a second paper
(Quémard et al., 2005d), enforcing the theoric
results expanded here.
2 PRESENTATION OF THE
OPTIMIZATION PROBLEM
Let ,
0
t
(
)
00
n
xxt=∈
be a given initial instant and
a given initial state. At the beginning the considered
h.d.s. follows a given (classical) dynamical system
(
)
1
,
x
fxt=

up to a switching instant . This one
corresponds to the first instant at which the
trajectory hits a given mobile (or fixed) switching
surface of equation
1
t
(
)
1111
,, 0xta ψ =
for state
(
)
11
x
xt=
. This surface depends on a parameter
. Then the h.d.s. follows
1
1
r
a ∈
(
)
2
,
x
fxt=

up to
such that
2
t
(
)
2222
,, 0xta ψ =
for
()
22
x
xt=
and
. By induction, this defines
2
2
r
a ∈
11
,,,
NN
ttt
+
…
an
increasing sequence of switching instants linked to
some given switching surfaces of equations
(
)
,, 0, 1, , 1,
iiii
xta i N ψ
=
=… +
(1)
for states
(
)
ii
x
xt=
and some parameters .
In
i
r
i
a ∈ 
[
]
01
,
N
tt
+
state
(
)
x
t
is supposed to be continuous,
378
Jolly J., Quémard C. and Ferrier J. (2005).
OPTIMAL CONTROL APPLIED TO OPTIMIZATION OF MOBILE SWITCHING SURFACES PART I: ALGORITHM.
In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and
Control, pages 378-382
DOI: 10.5220/0001184903780382
Copyright
c
 SciTePress

and in
[
]
1
,, 1,, 1
ii
tti N
−
=… +
state
(
)
x
t
complies
with some given dynamical systems
(
)
,.
i
x
fxt=

(2)
Functions
i
ψ
and are from class with values
in
and respectively. Instant
i
f
1
C

n

1N
t
+
is
interpreted as a final instant and
1N
ψ
+
as a final
constraint.
Notations — For a composed function as
(
)
(
)
,uv
α
α
we can note more simply
u
α
. For
example,
()
(
,
i
)
f
xt t
and
()
(
,,
iiii
)
x
tta ψ
can be
noted
i
t
f
and
i
i
t
ψ
.
We make the following assumptions:
Assumption A1 (consistency) — For
if parameters are given,
then
is the smallest instant such that
1, , 1,iN=… +
1
)
1
,,
i
aa
−
…
i
t
1
,
i
tt t
−
>
() ()
(
,
i
x
tfxtt=

and . This
defines
as a function of . Moreover, we
assume that such a sequence
exists for all
()
()
,, 0
ii
xt ta ψ =
i
t
1
,,
i
a… a
11
,,
N
tt
+
…
(
)
1
,,
N
aa
+
…
1
+
1
belonging to an open set of
.
1
1
N
r
r
+
××
Assumption A2 (transversality) — For
if parameters are given,
then
as partial function of is one from
class obtained by application of the implicit theorem
to constraints
1, , 1,iN=…
1
,,
i
aa
−
…
i
t
i
a
1
C
() ()
(
)
,,
iiii
x
tfxtt=

()
(
,,
iiii
)
x
tta ψ
. In particular, we have (3)
0=
1
0,
i
i
iiiii
ii
t
iiiii
t
t
ff
i
i
x
ta xta
ψψ ψψψ
−
⎛⎞
∂∂∂∂∂∂
+≠ =− +
⎜⎟
∂∂∂∂∂
⎝⎠
∂
Remark — For A1 as for A2, case where is
specified, i.e.
i
t
ii
tcst
ψ
=−
, is possible.
Relation (3) comes from
0
i
i
i
d
x
ψ
ψ
∂
==
∂
i
dx
,
ii
ii
ii
dt da
ta
ψψ ∂∂
++
∂∂
i
iii
and given
assumptions.
t
dx f dt=
Criterion — Let
0
J
be a criterion, to minimize or
maximize, in the form
1
00
1
N
i
i
J
J
+
=
=
∑
where
() ()
1
0
,, ,
i
i
t
iiiii i
t
J
xta L xtdt φ
−
=+
∫
(4)
with
i
φ
and from class.
i
L
1
C
Optimization problem — Assuming A1, A2, we
consider
as a function of , .
We search values for
which optimize
criterion
i
t
1
,,
i
aa…
1, , 1iN=+…
1
,,
N
aa
+
…
1
0
J
under constraints (1) and (2).
In this paper, we limit ourselves to the search of a
calculus method for the variation
0
1
0
1
N
i
i
i
dJ
dJ da
da
+
=
=
∑
.
Here, notation
0
i
dJ
da
means that the variation of
0
J
according to
is to be considered both directly
through
, namely
i
a
i
a
0
,
i
J
a
∂
∂
and indirectly through all
and
j
t
(
)
,
jj
x
xt=
,, 1ji N
=
+…
. The use of a
classical steepest descent algorithm (Polak, 1997)
permits then pursuing the resolution.
3 VARIATIONAL CALCULUS
For
1, , 1,iN
=
+…
we introduce a costate variable
(
)
ii
t λλ =
from class in
[
1
C
]
1
,
ii
tt
−
and a control
parameter
i
ν
∈

that define an augmented
criterion
i
J
by
0
ii
1
i
J
JJ
=
+
with
0
i
J
like in (4) and
() ()
()
1
1
,, , .
i
i
t
iiiiii ii
t
J
xta f xt xdt νψ λ
−
=+−
∫

We define a global augmented criterion J by
1
1
N
i
i
J
J
+
=
=
∑
. In accordance with paragraph 2, we have
also
1
0
1
N
i
i
0
J
J
+
=
=
∑
. Defining
1
1
1
N
i
i
1
J
J
+
=
=
∑
, we have
therefore
01
J
JJ
=
+ .
Let
(
)
(
)
,, ,
ii
F
xt x f xt x
=
−

. Constraints
(1), (2) give
0, 0, 0, 0
iiii
dFdF
ψ
ψ
=
== =
. We can
deduce that
1
1
i
i
t
T
iii ii
t
J
Fdt νψ λ
−
=+
∫
satisfies:
OPTIMAL CONTROL APPLIED TO OPTIMIZATION OF MOBILE SWITCHING SURFACES PART I: ALGORITHM
379

{}
1
1
1
1
0.
i
i
i
i
T
iiiiiiii
t
t
TTT
ii i i i i i
t
t
dJ d d F dt
Fdt d F dFdt
νψ ν ψ λ
λλλ
−
−
−
=⋅+⋅+
−+⋅−⋅
∫
=
i
Hence, we have
and therefore
. In the sequel we will enforce conditions
on
01
iii
dJ dJ dJ dJ=−=
0
dJ dJ=
(
)
i
t λ
and
i
ν
in order to obtain an expression
for
1
1
N
i
i
i
dJ
dJ da
da
+
=
=
∑
as explicit as possible.
With
iiii
φ
νψ +Φ=
, we have
(
)
1
.
i
i
t
T
ii ii
t
J
Hxd λ
−
=Φ + −
∫

t
i
()
{}
()
1
0
1
1
1
1
11
0
1
10 0 1 0 1 1
1
1
1
1
ii
i
i
i
N
N
N
T
iii
ii ii i
i
iii
tt
N
t
TT
i
iiiiii
t
t
i
TT
N
NN
t
N
t
N
NN
N
t
dx H dt da
xta
H
x
dt t dx H dt
x
t dx H dt dx
x
Hdt
t
λ
λδ λ
λλ
−
+
+
+
=
++
=
+
++
+
+
+
+
⎧
⎛⎞⎛⎞
∂Φ ∂Φ ∂Φ
⎪
=−+++
⎨
⎜⎟⎜⎟
∂∂∂
⎝⎠⎝⎠
⎪
⎩
∂⎫
⎛⎞
++ + −
⎬
⎜⎟
∂
⎝⎠
⎭
⎛⎞
∂Φ
=−+−
⎜⎟
∂
⎝⎠
⎛⎞
∂Φ
++
⎜⎟
∂
⎝⎠
∑
∑
∫

11
1
i
N
TT
i
ii i
i
i
t
dx
x
λλ
++
=
⎧
⎛⎞
∂Φ
⎪
+−+
⎨
⎜⎟
∂
⎝⎠
⎪
⎩
∑
For variations
11
,, , , ,
iii
x
xdx dx dt dt
δ
δ
−−

satisfying

x
x
δ
δ =
i

and
()
11 1ii i1i
x
dx x t dt δ
−− −
=−

−
i
,
()
ii i
x
dx x t dt δ =−

(see Bryson and Ho, 1969, fig.
2.7.1) we calculate variation
under constraints
(1) and (2):
,
dJ
1
1
N
i
i
dJ dJ
+
=
=
∑
1
1
1
.
i
ii
i
iii
iiii
iii
t
T
i
iii i i
tt
t
dJ dx dt da
xta
H
Ldt L dt x xdt
x
δλδ
−
−
−
∂Φ ∂Φ ∂Φ
=++
∂∂∂
∂
⎛⎞
+− + −
⎜⎟
∂
⎝⎠
∫

Integrating by parts yields
1
1
1
,
i
i
i
i
i
i
t
T
i
i
t
t
t
TT
i
ii
t
t
H
xxdt
x
H
xdt x
x
δλδ
λδ λδ
−
−
−
∂
⎛⎞
−
⎜⎟
∂
⎝⎠
∂
⎛⎞
⎡⎤
=+−
⎜⎟
⎣⎦
∂
⎝⎠
∫
∫


and therefore
()
()
1
1
11 1
.
ii
i
i
i
TT
ii
iiiiii
ii
tt
TT
i
iii i iii i
t
i
t
T
i
i
t
dJ dx L f dt
xt
da t dx L f dt
a
H
xdt
x
λλ
λλ
λδ
−
−
−− −
⎛⎞⎛ ⎞
∂Φ ∂Φ
=− +++
⎜⎟⎜ ⎟
∂∂
⎝⎠⎝ ⎠
∂Φ
++ −+
∂
∂
⎛⎞
++
⎜⎟
∂
⎝⎠
∫

i
We can deduce
()
1
1
11
11
11 1
ii
i
i
i
NN
T
ii
iiii
ii
ii
tt
T
i
iii i i i
t
i
t
T
i
i
t
dJ dJ dx H dt
xt
da t dx H dt
a
H
xdt
x
λ
λ
λδ
−
−
++
==
−− −
⎧
⎛⎞⎛⎞
∂Φ ∂Φ
⎪
== − ++
⎨
⎜⎟⎜⎟
∂∂
⎝⎠⎝⎠
⎪
⎩
∂Φ
++ −
∂
∂⎫
⎛⎞
++
⎬
⎜⎟
∂
⎝⎠
⎭
∑∑
∫

i
1
.
i
ii
ii i i
ii
t
H
Hdtd
ta
+
⎫
⎛⎞
∂Φ ∂Φ
⎪
+− + + +
⎬
⎜⎟
∂∂
⎝⎠
⎪
⎭
a
(5)
Let us choose to compel
(
)
,
ii
t
λ
ν
to comply with
1
0 in ,
T
i
ii
H
ttt
x
λ
−
∂
+= ≤≤
∂

i
(6)
() ()
1
,
TT
i
ii i i
i
tt
x
λλ
+
∂Φ
=+
∂
(7)
1
.
ii
i
ii
tt
i
HH
t
+
∂Φ
=−
∂
(8)
Those relations are given for
and with
the definition for notation convenience that
1, ,1iN=+…
()
1
1
21 2 2
0, 0, 0.
N
N
NN N N
t
t
tL H λ
+
+
++ + +
=
==
(9)
One can find jump relations (7), (8) in (Bryson and
Ho, 1969), or (El Bagdouri et al., 2005).
Combination (8) – (7).
gives
i
f
()
11
i
i
TT
ii
iii i ii i
t
ii
t
H
fH f f
tx
λλ
++
⎛⎞
∂Φ ∂Φ
−=−−−
⎜⎟
∂∂
⎝⎠
()
111
.
i
i
i
T
ii
iiiii i
t
ii
t
ii
ii
ii
t
LL ff f
tx
f
tx
φφ
λ
ψψ
ν
+++
⎛⎞
⎛⎞
∂∂
=+ −−+
⎜⎟
⎜⎟
⎜⎟
∂∂
⎝⎠
⎝⎠
⎛⎞
∂∂
−+
⎜⎟
∂∂
⎝⎠
We can deduce
()
111
1
.
i
i
T
ii
iiiiii i
ii
t
ii
i
ii
t
LL f f f
tx
f
tx
φφ
νλ
ψψ
+++
−
⎛⎞
⎛⎞
∂∂
=−+ −−+
⎜⎟
⎜⎟
⎜⎟
∂∂
⎝⎠
⎝⎠
⎛⎞
∂∂
⋅+
⎜⎟
∂∂
⎝⎠
(10)
Thus
i
ν
is explicitly determined as a function of
(
)
1ii
t λ
+
. According to
iiii
φ
νψ
Φ
=+
, substituting
ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL
380

this expression of
i
ν
in (7) gives explicitely
(
)
ii
t λ
as a function of
:
()
1ii
t λ
+
() ()
1
.
TT
i
ii i i i
ii
tt
i
x
x
φ
ψ
λλ ν
+
∂∂
=++
∂∂
(11)
By this way system (6) with limit condition (9) or
(11) can be solved backwards starting from
up to . This resolution is efficient
because condition
1iN=+ 1i =
0
i
ii
i
ii
t
f
tx
ψψ
⎛⎞
∂∂
+≠
⎜⎟
∂∂
⎝⎠
that ensure existence of
i
ν
given by (10) follows
from (3) of our assumption A2.
Let
(
)
00 0
,tx xt=
be fixed initial conditions. We
have
. According to (5), preceding
choices made for
00
0, 0dt dx==
(
)
,
ii
t νλ
give
1
1
.
N
i
i
i
i
dJ da
a
+
=
∂Φ
=
∂
∑
As we have established
, the searched
expression for our gradient calculus is then
0
dJ dJ=
0
.
i
iii
dJ dJ
da da a
∂Φ
==
∂
Using
iii
f
i
νψ
Φ= +
and constraint (1), we can
precise
0
,1,, 1
iii
ii
ii ii
ii
i
ii
d
dJ
da a a da
iN
aa
φψν
νψ
φψ
ν
∂∂
=+ +
∂∂
∂∂
=+ = +
∂∂
…
.
1
(12)
We have obtained the following algorithm:
Algorithm — Our aim is to compute the gradient
linked to the optimization problem set in paragraph
2. Let
be fixed parameters and let
1
,,
N
aa
+
…
(
)
00 0
,tx xt=
be specified initial conditions. A
calculus algorithm for variation
0
1
0
1
N
i
i
i
dJ
dJ da
da
+
=
=
∑
at
is the following one. For given
, the resolution of forward system
(2) with constraint (1) gives
1
,,
N
aa
+
…
1
)(
11 1
,
ii i
tx xt
−− −
=
(
)
,
ii i
tx xt=
. Starting
from
(
)
00 0
,tx xt=
this gives by forward induction
sequences
. For given
, we get
()
,,1,,
ii i
tx xt i N==… 1+
()
1ii
t λ
+ i
ν
and
(
)
ii
t λ
from (10) and (11).
The resolution of backward system (6) with final
condition
(
)
ii
t λ
gives . Starting from
(
1ii
t λ
−
)
(
)
22
0
NN
t λ
++
=
in (9) this gives by backward
induction sequence
,1,
i
iN ,1
ν
=
+ …
. We get then
0
i
dJ
da
from (12). All this is obtained under
assumptions A1, A2.
Remark — A special case is the one where is not
free, that is to say
i
t
ii
tcst
ψ
=
−
. We have then
() ()
1
TT
i
ii i i
i
tt
x
φ
λλ
+
∂
=+
∂
,
0
i
ii
dJ
da a a
i
i
φ
∂Φ ∂
==
∂∂
. The
value of
i
ν
has no effect on the one of
0
i
dJ
da
. Taking
it equal to zero simplifies the calculuses.
This result has been established by authors of
(Boccadoro et al., 2004) and (Wardi, Y., 2004), but
limited to the case where
()
,
i
x
fx=

(
)
,0
iii
xa ψ
=
,
and
()
1
0
i
i
t
ii
t
J
Lxdt
−
=
∫
. Moreover, counter to their
demonstration more technical, here we make clear
the interpretation of costate
(
)
t λ
in terms of control
(they do not consider an augmented criterion J). In
(Boccadoro, 2004) one can find a nice application to
the optimization of switching rules for a fixed
obstacle avoidance problem in robotics. Our more
general algorithm enables us to extend this
application to the case of a mobile obstacle.
4 CONCLUSION
A natural generalization would be to consider the
case where there is an additional continuous control
term
(
)
ut
between switching times. This is studied
in (Bryson and Ho, 1969), or (El Bagdouri et al.,
2005) but whithout taking into account dependency
of switching surfaces on parameters
11
,,
N
aa
+
…
. It
appears that an explicit determination of
(
)
,
iii
t νλ
as done in (10), (11) is not always possible. In
particular, it is subject to some transversality
conditions more difficult to specify (Bryson and Ho,
1969, p. 59, p. 103 and p. 164). We can read p. 102:
“However, finding solutions to such problems is, in
general, quite involved”.
The question of optimization of switching
surfaces for a hybrid dynamical system is due to
authors of (Boccadoro et al., 2004)
and (Wardi et al.,
OPTIMAL CONTROL APPLIED TO OPTIMIZATION OF MOBILE SWITCHING SURFACES PART I: ALGORITHM
381

2004). Our contribution is about deepening,
simplification and generalization of their works. It
uses the idea of an augmented criterion as in (Bryson
and Ho, 1969) or (El Bagdouri et al., 2005). Like in
these references, at cost of more complexity and
more place, it would be easy to generalize our
algorithm to the case of controlled jumps for state
variable
x
at switching instants (Bryson and Ho,
1969, p. 106-107).
i
t
Diversity of possible applications for our
theoretical results motivates a second
communication (Quémard, 2005d). Let us mention
the applications, classical or new, for which a
resolution is performed:
y Optimization of limit cycles. Application to a
thermal device with hysteresis phenomenon
(Quémard et al., 2005a, 2005b, 2005c).
y Optimization of switching instants for a minimum
time problem for a car with two gears
y Optimization of switching rules for a mobile
obstacle avoidance problem in robotics.
REFERENCES
Bensoussan, A. and Menaldi, J.L., 1997. Hybrid Control
and Dynamic Programming. In Discrete and Impulsive
Systems, 3, p. 395-442.
Boccadoro, M., 2004. Optimal Control of Switched
Systems with Applications to Robotics and
Manufacturing. Ph. D Thesis, Perugia, Italia.
Boccadoro, M., Egerstedt, M. and Wardi, Y., 2004.
Optimal Control of Switching Surfaces in Hybrid
Dynamic Systems.
IFAC Workshop on Discrete
Event Systems. Reims, France, Sept. 2004.
Boccadoro, M., Egerstedt, M. and Wardi, Y., 2005.
Obstacle Avoidance for Mobile Robots Using
Switching Surface Optimization. IFAC World
Congress.
Prague, the Czech Republic, July 2005.
Bryson, A.E. and Ho, Y.C., 1969. Applied Optimal
Control, Ginn and Company.
El Bagdouri, M., Cébron, B., Sechilariu, M., Burger, M.,
2005. Variational Formalism Applied to the Control of
Autonomous Switching Systems. In Control &
Cybernetics.
Polak, E., 1997. Optimization. Springer.
Quémard, C., Jolly, J.-C., Ferrier J.-L., 2005a. Search for
cycles in piecewise linear hybrid dynamical systems
with autonomous switchings. Application to a thermal
process. IMACS World Congress. Paris, Jul. 2005.
Quémard, C., Jolly, J.-C., Ferrier, J.-L., 2005b.
Mathematical study of a thermal device as an hybrid
system. IMACS World Congress. Paris, Jul. 2005.
Quémard, C., Jolly, J.-C., Ferrier, J.-L., 2005c.
Optimisation de cycles limites. Application à un
processus thermique. JDMACS. Lyon, Sept. 2005.
Quémard, C., Jolly, J.-C., 2005d. Optimal Control Applied
to Optimization of Mobile Switching Surfaces. Part II:
Applications. ICINCO. Barcelona, Spain, Sept. 2005.
Van Der Schaft, A. and Schumacher, H., 1999. An
Introduction to Hybrid Dynamical Systems. Springer.
Wardi, Y., Egerstedt, M. and Boccadoro, M., Dec. 2004b.
Optimal Control of Switching Surfaces. In Conference
on Decision and Control CDC.
Atlantis, Bahamas,
Dec. 2004.
Zaytoon, J., coordonnateur, 2001. Systèmes dynamiques
hybrides. Hermès.
ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL
382