ASYMPTOTIC THEORY OF THE REACHABLE SETS TO LINEAR
PERIODIC IMPULSIVE CONTROL SYSTEMS
E. V. Goncharova
Institute for System Dynamics and Control Theory, Siberian Branch of Russian Academy of Sciences
134 Lermontov Street, Irkutsk, Russia
A. I. Ovseevich
Institute for Problems in Mechanics, Russian Academy of Sciences
101 Vernadsky Avenue, Moscow, Russia
Keywords:
Linear periodic dynamic systems, impulsive control, reachable sets, shapes of convex bodies.
Abstract:
We study linear periodic control systems with a bounded total impulse of control. The main result is an asymp-
totic formula for the reachable set, which, at the same time, reveals the structure of the attractor — the set of
all limit shapes of the reachable sets. The attractor is shown to be parameterized by a (finite-dimensional) toric
fibre bundle over a circle. The fibre of the bundle can be described via the Floquet multipliers (monodromy
matrix) of the linear system. Moreover, the limit dynamics of shapes of reachable sets can be parametrized by
an explicit curve on the toric bundle.
1 INTRODUCTION
One of the fundamental notions of control theory is
that of reachable sets which provide a visible bound
for control capabilities. In general, these sets have a
complicated shape and dynamics. There is, however,
a kind of problems where the behavior of reachable
sets is well-understood.
Namely, it turns out that the reachable sets of lin-
ear control systems have simple limit properties as
time evolves to infinity provided that a suitable time-
dependent matrix scaling is applied. This kind of re-
sults was found for the first time in (Ovseevich, 1991),
where time-invariant linear control systems with geo-
metric bounds on control were studied. It was shown
that in this setup there is a single limit shape of the
reachable sets, shape being the set regarded up to an
arbitrary nondegenerate linear transform.
At present the scope of this phenomena is not yet
clear cut. It is very likely that there is a natural exten-
sion of these results to general time-dependent linear
systems. Moreover, a similar phenomena was discov-
ered for some nonlinear stochastic dynamic systems
(Dolgopyat et al., 2004).
The purpose of the study is to develop the asymp-
totic theory of the reachable sets to linear impulsive
control systems. A motivation to address impulsive
control systems is also due to the perceived relevance
of the impulsive control theory for hybrid systems
whose state evolution is dictated by the interaction of
conventional time-driven dynamics and event-driven
dynamics (see, e.g., (Aubin, 2000; Branicky et al.,
1998; Miller and Rubinovich, 2003)).
In this paper, we study the periodic linear control
systems with a bounded total impulse of control. The
main result is an asymptotic formula for the reachable
sets (see (3)), that, in particular, reveals the structure
of attractor — the set of all limit shapes of the reach-
able sets.
It would be extremely interesting to understand
the limit behavior of reachable sets for a general lin-
ear system. Unfortunately, the nature of the present
methods is computational and it looks like new ideas
are needed in order to grasp the limit dynamics of the
reachable sets.
2 PROBLEM STATEMENT
Consider a linear control system on the time interval
[0,T]
˙x(t) = A(t)x(t)+ B(t)u(t), x(0) = 0, (1)
131
V. Goncharova E. and I. Ovseevich A. (2008).
ASYMPTOTIC THEORY OF THE REACHABLE SETS TO LINEAR PERIODIC IMPULSIVE CONTROL SYSTEMS.
In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - SPSMC, pages 131-136
DOI: 10.5220/0001490001310136
Copyright
c
SciTePress
under the constraint on the total impulse of control u:
Z
T
0
h f(t),u(t)dti 1, (2)
where x(t) V = R
n
, u(t) W = R
m
, A(t), B(t) are
matrices of appropriate dimensions, U(t) is a given
central symmetric convex body in W, f is an arbitrary
continuous function such that f(t) U
(t), andU
(t)
is the polar of set U(t).
Assume that the Kalman type condition of com-
plete controllability holds, namely, for any vector
u W and a time moment T, function Φ(T,t)B(t)u
does not vanish identically in any interval of time t.
Under these assumptions the reachable sets D (T) to
system (1), (2) are central symmetric convex bodies.
The problem addressed is to study the limit be-
havior of the reachable sets D (T) as T . The
reachable sets are regarded as elements of the metric
space B of central symmetric convex bodies with the
Banach-Mazur distance ρ:
ρ(
1
,
2
) = log(t(
1
,
2
)t(
2
,
1
)),
where t(
1
,
2
) = inf{t 1 : t
1
2
}.
The general linear group GL(V) naturally acts on the
space B by isometries. The factorspace S is called the
space of shapes of central symmetric convex bodies,
where the shape Sh S of a convex body B
is the orbit Sh= {C: detC 6= 0} of the point
with respect to the action of GL(V). The Banach-
Mazur factormetric makes S into a compact metric
space. The convergence of the reachable sets D (T)
and their shapes is understood in the sense of the
Banach-Mazur metric. For two asymptotically equal
functions with values in the space of convex bodies
or the space of their shapes, the following notations
are used:
1
(T)
2
(T), if ρ(
1
(T),
2
(T)) 0
as T , and similarly Sh
1
(T) Sh
2
(T), if
ρ(Sh
1
(T),Sh
2
(T)) 0 as T . The conver-
gence of convex bodies may be also understood in the
sense of convergence of their support functions. Re-
mind that the support function of a convex compact
set is given by formula: H
(ξ) = sup
x
hx,ξi, where
ξ V
, and uniquely defines the set . The equiva-
lence of the two definitions of convergence of convex
bodies — in the terms of convergence of their support
functions and in the sense of the Banach-Mazur met-
ric is established by the following lemma (Figurina
and Ovseevich, 1999):
Lemma 1. A sequence
i
B converges to B
in the sense of the Banach-Mazur metric if and only
if the corresponding sequence of the support func-
tions H
i
(ξ) = H
i
(ξ) converges to the support func-
tion H
(ξ) pointwise and is uniformly bounded on the
unit sphere in the dual space V
.
We address the periodic case, when the con-
stituents A, B, and U of control system (1), (2) are
supposed to be continuous and periodic in t. To fix
ideas, the period is assumed to be 1. It would be in-
teresting to understand the limit behavior of reach-
able sets for a general linear system. This prob-
lem, however, seems rather difficult, since already the
time-invariant case is nontrivial, and, say, for quasi-
periodic systems it is not clear how to prove the cor-
responding natural conjectures.
3 ASYMPTOTIC BEHAVIOR OF
SHAPES OF THE REACHABLE
SETS
We study the limit behavior as T +of the curve
T 7→ ShD (T) under different assumptions on the
spectrum of the monodromy matrix. At the heart of
the considerations below there is an explicit formula
for the support function of the reachable set:
Lemma 2. The support function of the reachable set
D (T) to system (1), (2) is given by
H
D (T)
(ξ) = sup
t[0,T]
H
U(t)
(B(t)
Φ(T,t)
ξ), (3)
where Φ(t,s) is the fundamental matrix of linear sys-
tem ˙x = A(t)x.
Stable Case. Let the system (1) be asymptotically
stable, i.e.
Φ(T,t) = o(1) as T t +,
and o(1) is uniformly small. It is easy to estab-
lish the stability criterion: system (1) is asymptoti-
cally stable iff the spectrum of the monodromy matrix
M = Φ(1,0) is contained in the open unit disk of the
complex plane.
Let us show that the curve T 7→ D (T) is asymptot-
ically periodic as T . In other words, there exists
such a continuous periodic curve f : R/Z B that
D (T) f(T) as T +. Informally speaking, the
curve T 7→ D (T) is reeled on a limit cycle.
The curve f can be given by an explicit formula.
Define function
F (T) = F (T,ξ) = sup
t(,T]
H
U
(B
Φ(T,t)
ξ), (4)
where the argument t of periodic functions B and U
is omitted. Due to the stability condition, F (T) is a
continuous periodic function of T. The periodicity of
F follows from the equality
Φ(T + 1,t + 1) = Φ(T,t)
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132
for the fundamental matrix of a 1-periodic system.
Furthermore, for each T, function ξ 7→ F (T, ξ) is ho-
mogeneous and convex, and therefore it is the support
function of a body f(T). Thus, the curve f is defined.
On the other hand, from (3), (4) it follows that
H
f(T)
(ξ) = H
D (T)
(ξ) for large enough T, (5)
since Φ(T,t) = O
e
β(Tt)
, where β > 0 owing to
the assumed stability property. The asymptotic equal-
ity D (T) f(T) follows from (5).
Unstable Case. Assume that the system (1) is strictly
unstable, i.e.
Φ(T, s) = o(1) as T s ,
where o(1) is uniformly small. System (1) is strictly
unstable iff the spectrum of the monodromy matrix
M = Φ(1, 0) is contained in the complement of the
closed unit disk of the complex plane.
Define the matrix multiplier C(T) = Φ(0,T) and
consider the set
e
D (T)
def
= C(T)D (T).
It is easy to see that
H
e
D (T)
(ξ) = sup
t[0,T]
H
U
(B
Φ(0,t)
ξ),
and from the instability criterion it followsthat Φ(0,t)
decreases exponentially fast as t . Therefore, the
values sup
t[0,T ]
H
U
(B
Φ(0,t)
ξ) converge as T ,
and the convergence of bodies
e
D (T) D
, takes
place, where
H
D
(ξ) = sup
t[0,]
H
U
(B
Φ(0,t)
ξ).
Thus, we have the asymptotical equality:
D (T) Φ(T,0)D
.
In the view of the behavior of shapes of the reach-
able sets, there is an essential difference between sta-
ble and unstable cases. In the unstable case, shapes
ShD (T) convergeas T , while, for stable system,
the curve T ShD (T) is reeled on a limit cycle.
Note that the aboveconsiderations admit an exten-
sion to almost periodic systems.
Yet another choiceC(T) = Φ({T} , T), where {T}
is the fractional part of T, of a normalizing matrix fac-
tor seems also reasonable. It is easy to see at that rate,
that the limit normalized body D
= D
(T) depends
on time periodically. This choice of the matrix factor
fits better the general case, when stable, unstable and
neutral components are present.
Neutral Case. Assume that system (1) is neutral,
meaning that the spectrum of the monodromy matrix
M = Φ(1,0) rests on the unit circle. Consider the Jor-
dan decomposition
M = UD = e
N+D
,
where D is a diagonalizable matrix of the same spec-
trum that the matrix M has, D is such a diagonalizable
real matrix that D = e
D
, N is a nilpotent matrix, and
ND = DN. As is well known, there exists a matrix
F = F(N,T) with the following properties:
FNF
1
= T
1
N, FD = DF,
and F
= F
(N) = lim
T+
F(N,T) is defined.
Put
N(T) = Φ(T, 0)NΦ(0,T), D(T) = Φ(T, 0)DΦ(0,T).
It is easy to see that N(T) and D(T) are periodic func-
tions of T, since matrices N and D commute with
M = Φ(1,0). Matrix function
F
(N(T)) = Φ(T,0)F
(N)Φ(0,T)
is also continuous and periodic. Function φ given by
formula
φ(T,t) = e
(tT)[N(T)+D(T)]
Φ(T,t), (6)
is periodic in T and t. Define the matrix factor
C(T) = F(N(T),T)e
T[N(T )+D(T)]
and consider the normalized set
e
D (T)
def
= C(T)D (T).
It is easy to see that
H
e
D (T)
(ξ) = supH
g
1
T
(t)
= supH
g
2
T
(t)
= supH (g
T
(t)) + o(1),
(7)
where sup is taken over t [0, T], H = H
U
and
g
1
T
(t) = B
φ(T,t)
e
t(N
(T)+D
(T))
F(N(T),T)
ξ,
g
2
T
(t) = B
φ(T,t)
F(N(T),T)
e
t
T
N
(T)+tD
(T)
ξ,
g
T
(t) = B
φ(T,t)
F
(N(T))
e
t
T
N
(T)+tD
(T)
ξ.
The last term o(1) in (7) appears on account of the
difference between F(T) and F
(N(T)). Since other
matrices involved in g
2
T
(t) are uniformly bounded, the
difference of the arguments comes to o(1).
Consider the following function of two arguments
T and L:
I(L,T) = sup
t[0,L]
f
T
(t,t,τ) =
= sup
t[0,L]
H
U
B
φ(T,t)
F
(N(T))
e
t
L
N
(T)+tD
(T)
ξ
.
ASYMPTOTIC THEORY OF THE REACHABLE SETS TO LINEAR PERIODIC IMPULSIVE CONTROL SYSTEMS
133
Function f
T
(t,t,τ), where t = e
tD(T)
, τ = t/L, is pe-
riodic in t and depends on the parameter T periodi-
cally as well. By using the Hermann Weyl averaging
method (Weyl, 1938; Weyl, 1939; Arnold, 1989), like
in (Goncharova and Ovseevich, 2007), we obtain the
asymptotic representation
I(L,T) = sup
T ×J
f
T
(t,t,τ) + o(1) (8)
as L , where o(1) is small uniformly in T. In
formula (8), the interval J = [0,1] and a torus T =
T (T) are involved. The torus is the closure of the
one-parameter subgroup {(e
2πit
,e
tD(T)
)} in the group
S
1
× GL(V). Notice that the torus
T (T) = Φ(T,0)T (0)Φ(0,T)
depends on T continuously and periodically. The
torus can be naturally represented as a fibre bundle
over the circle, at that the fibre over e
2πiT
S
1
is
the closure of the cyclic group generated by matrix
e
D(T)
GL(V).
It is clear that
I(T) = sup
T ×J
f
T
(t,t,τ)
is a periodic function of parameter T. Hence, for large
L,
I(L,T) = I(T) + o(1)
is a periodic function of T up to o(1). In particular,
this is true for L = T and large T. From this we con-
clude that the curve T 7→ ShD (T) is periodic up to
o(1), i.e. it is reeled on a limit cycle.
Stable-neutral Case. Suppose that system (1) is
stable-neutral. This means that fundamental matrix
admits a polynomial estimate:
|Φ(T,t)| = O(1+ |T t|
n
) as T t +.
It is not difficult to obtain the criterion of stable-
neutrality: system (1) is stable-neutral iff the spec-
trum of the monodromy matrix M = Φ(1,0) is con-
tained in the closed unit disk of the complex plane.
Consider the canonical decomposition of the mon-
odromy matrix M = Φ(1,0)
M = M
0
M
into the stable and neutral components (in accordance
with the relations |λ| < 1, |λ| = 1 for eigenvalues),
and the corresponding decomposition of phase space:
V = V
0
V
.
For an arbitrary time moment T, the monodromy ma-
trix
M(T) = Φ(T + 1,T) = Φ(T,0)MΦ(0, T)
depends on T periodically. The corresponding de-
composition
V(T) = V
0
(T) V
(T).
is also periodic in T.
The scaling matric factor C(T) can be taken in the
block-diagonal form
C(T) = C
0
(T) C
(T),
where C
i
(T) : V
i
(T) V
i
(T) are given by formulas
C
0
(T) = F(N(T),T), C
(T) = I.
The support function H
e
D (T)
(ξ) of the normalized
body
e
D (T)
def
= C(T)D (T) is as follows
H
e
D (T)
(ξ) = sup
t[0,T ]
f
T
(t) = sup
t[0,T]
H
U
(g
T
(t)), (9)
g
T
(t) = B
Φ(T,t)
ξ
+
B
φ(T,t)
e
(Tt)(N
(T)+D
(T))
F(N(T),T)
ξ
0
,
ξ
i
= ξ
i
(T) V
i
(T)
, i {−, 0} are componentsof the
canonical decomposition of a vector ξ V
, and the
periodic in both arguments function φ(T,t) is defined
in (6). Notice that in the generic case, matrix F is the
identity one, and therefore, the formula for the sup-
port function is just as simple as (3):
H
e
D (T)
(ξ) = sup
t[0,T]
H
U
(B
Φ(T,t)
ξ).
To study H
e
D (T)
(ξ) as T let us apply the decom-
position method like in the autonomous case (Gon-
charova and Ovseevich, 2007). Owing to the basic
commutativity relations for matrices F, N, and D, we
have
e
(Tt)(N
(T)+D
(T))
F(N(T),T)
ξ
0
=
= F(N(T),T)
e
Tt
T
N
(T)+(Tt)D
(T)
ξ
0
,
and, thus, we have the uniform asymptotic equality
Φ(T,t)
F(N(T),T)
ξ
0
=
φ(T,t)
F
(N(T))
e
Tt
T
N
(T)+(Tt)D
(T)
ξ
0
+ o(1)
as T . Like in (Goncharova and Ovseevich,
2007), we divide the time interval I = [0,T] into the
two subintervals
I = I
0
I
= [0,(1 ε)T] [(1 ε)T,T],
where ε = ε(T) is such that ε(T) = o(1), while
ε(T)T as T . By adopting arguments from
(Goncharova and Ovseevich, 2007), we obtain the
asymptotic equality
H
e
D (T)
(ξ) = max{H
0
(T, ξ),H
0
(T, ξ)} + o(1), (10)
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134
where in accordance with notations (9)
H
0
= sup
tI
f
T
(t) + o(1) = sup
t(,T]
H
U
(g
0T
(t)),
where g
0T
(t) stands for
B
Φ(T,t)
ξ
+ B
φ(T,t)
F
(N(T))
e
(Tt)D
(T)
ξ
0
,
while
H
0
= sup
tI
0
f
T
(t) + o(1) = sup
tR,τ[0,1]
H
U
(g
0T
),
where g
0T
(t,τ) stands for
B
φ(T,t)
F
(N(T))
e
τN
(T)+(Tt)D
(T)
ξ
0
.
Functions H
0
(T, ξ) and H
0
(T, ξ) are periodic in T,
and convex, homogeneous in ξ. From this it fol-
lows that H
i
(T, ξ), i = 0,0, are the support func-
tions of some convex compact sets
0
(T) V(T)
and
0
(T) V
0
(T), which periodically depend on
T. Furthermore, the convex compact
(T) =
0
(T)
0
(T) V(T),
and
0
(T)) is a body with the support function
max{H
0
(T, ξ),H
0
(T, ξ)},
and also periodically depends on T. Here, we use the
join notation:
=
′′
,
meaning that is the convex hull of the union
S
′′
, or what is the same H
= max{H
,H
′′
}.
Thus, the curve T 7→
e
D (T) is reeled on the limit cycle
T 7→ (T).
Unstable-neutral Case. Similarly to stable-neutral
case, fundamental matrix admits a polynomial esti-
mate
|Φ(T,t)| = O(1+ |T t|
n
) as T t .
System (1) is unstable-neutral iff the spectrum of the
monodromy matrix M = Φ(1,0) is contained in the
closed complement of the unit disk of the complex
plane. In this case, the normalizing matrix factor
C(T) can be taken in the block-diagonal form:
C(T) = C
0
(T) C
+
(T),
whereC
i
(T) : V
i
(T) V
i
(T) are defined by formulas
C
0
(T) = F(N(T),T ), C
+
(T) = Φ({T},T).
We note in passing that if we put C
+
(T) = Φ(0,T),
then the block-diagonal structure of the normalizing
matrix would be lost, since, in general, Φ(0,T) do
not map V
+
(T) into itself.
The support function H
e
D (T)
(ξ) of the normalized
body
e
D (T)
def
= C(T)D (T) is similar to (9):
sup
t[0,T ]
H
U
(B
Φ({T},t)
ξ
+
+ B
φ(T,t)
×
×e
(Tt)(N
(T)+D
(T))
F(N(T),T)
ξ
0
),
and like in (10) we have the asymptotics
H
e
D (T)
(ξ) = max{H
+0
(T, ξ),H
0
(T, ξ)} + o(1),
where H
+0
(T, ξ) = sup
t[0,)
H
U
(g
+0T
(t)), and
g
+0T
(t) = B
Φ({T},t)
ξ
+
+
+B
φ(T,t)
F
(N(T))
e
N(T)
e
(Tt)D
(T)
ξ
0
.
The essential difference, in contrast to stable-neutral
situation, consists of that, on this occasion, func-
tion H
+0
(T, ξ) is not periodic in T, however, it be-
comes periodic by the transformation of variable
ξ
0
7→ e
TD
(T)
ξ
0
. Geometrically, this means that the
asymptotic equality
e
D (T) p(t
T
)
+0
(T)
0
(T) as T ,
holds, where t
T
= (e
2πiT
,e
TD(T)
) is an element of the
torus T (T), matrix p(t
T
) = e
TD(T)
is the second com-
ponent of t
T
,
α
(T) are periodically depending on
time convex compacts in spaces V
α
(T), α { +0,0}.
Thus, the limit behavior of the normalized reach-
able sets
e
D (T) is the same as the behavior of the curve
T 7→ t
T
. The closure of the curve might have an ar-
bitrary large dimension so that the curve is reeled on
a multidimensional manifold. Still, the shapes of the
reachable sets ShD (T) have a simpler behavior, since
ShD (T) Sh(
+0
(T)
0
(T)), and, therefore, the
curve T 7→ ShD (T) is reeled on a limit cycle of di-
mension not greater than 1.
General Case. The general result can be obtained
by using the decomposition method (see (Goncharova
and Ovseevich, 2007)) and the considered above
cases. Consider the canonical decomposition of the
monodromy matrix M = Φ(1,0)
M = M
+
M
0
M
into the unstable, neutral, and stable components ( in
accordance with the relations |λ| < 1, |λ| > 1, |λ| = 1
for eigenvalues), and the corresponding decomposi-
tion of phase space
V = V
+
V
0
V
.
For an arbitrary time moment T, the monodromy ma-
trix
M(T) = Φ(T + 1,T) = Φ(T,0)MΦ(0,T)
ASYMPTOTIC THEORY OF THE REACHABLE SETS TO LINEAR PERIODIC IMPULSIVE CONTROL SYSTEMS
135
depends on T periodically, so does the corresponding
decomposition of phase space
V = V
+
(T) V
0
(T) V
(T).
In the general case, the scaling matric factor C(T) can
be chosen in the block-diagonal form
C(T) = C
+
(T) C
0
(T) C
(T),
where C
i
(T) : V
i
(T) V
i
(T) are given by formulas
C
+
(T) = Φ({T},T),
C
0
(T) = F(N(T),T), C
(T) = I.
The normalized body
e
D (T) = C(T)D (T) has the fol-
lowing asymptotics
e
D (T) p(t)
+0
(T)
0
(T)
0
(T) (11)
as T , where t = (e
2πiT
,e
TD(T)
) is an element
of the torus T (T), matrix p(t) = e
TD(T)
is the sec-
ond component of t,
α
(T) are periodically depend-
ing on time convex compacts in V
α
(T), α {+0, 0}.
Asymptotic equality (11) can be naturally interpreted
in the terms of attractors in space S of shapes of con-
vex bodies. Define a fibre bundle over circle P S
1
as follows:
P = {(e
2πiT
,e
TD(T)
z) S
1
× GL(V) : z Z (T)},
(e
2πiT
,e
TD(T)
z) 7→ e
2πiT
,
where Z (T) is the closure in GL(V) of a cyclic group
generated by matrix e
D(T)
. Then, the relation (11) as-
serts that the totality A of all the limit shapes of the
reachable sets (attractor) is parameterized by the set
P : there is a continuous map σ from P onto A . At
that, in the limit, the curve T 7→ ShD (T) is parame-
terized by the curve T 7→ t
T
= (e
2πiT
,e
TD(T)
) in P in
the sense that
ShD (T) σ(t
T
)
as T .
This asymptotic equality is an incarnation of (11)
and the main result of the paper. It says that the limit
dynamics of the shapes ShD (T) can be described via
the “straight winding” T 7→ t
T
in the toric bundle P .
4 CONCLUSIONS
In this paper we determined completely the asymp-
totic behavior of reachable sets to periodic linear dy-
namic systems with impulsive control. This is just a
single step in the long road directed to understanding
the limit behavior of reachable sets for general linear
systems. Still, our results for the periodic case suggest
a reasonable conjectural description of this behavior.
In fact, it is possible to state a precise conjecture per-
taining to the quasi-periodic case.
ACKNOWLEDGEMENTS
The work was partially supported by RFBR (projects
08-01-00156, 08-08-00292).
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