Page 1
Asymptotic Feedback Controllability of Switched Control Systems to the Origin
P. C. Perera,
Department of Mathematics,
University of Texas PanAmerican,
Edinburg, TX 78539
pperera@math.panam.edu
W. P. Dayawansa
Department of Mathematics and Statistics,
Texas Tech University
Lubbock, TX 79409
daya@math.ttu.edu
Abstract—Trajectories of controllable switched systems con
sisting of linear continuoustime timeinvariant subsystems are
arbitrarily closely approximated by those of a controllable
timeinvariant nonswitched polynomial systems. Examples are
obtained to show that the aforementioned switched control sys
tems are not locally asymptotically stabilizable via continuous
switching strategies. Finally, asymptotic feedback controllability
of such switched control systems is established.
I. Introduction
A fundamental requirement for the design of feedback
control systems is the knowledge of the structural prop
erties of the switched control system under consideration.
These properties are closely related to the concepts of con
trollability, observability, stability and stabilizability. There
have been many studies for switched systems primarily on
stability analysis and design in [2], [5], [9]. In the case
of controllability, studies for loworder switched control
systems consisting of linear subsystems have been presented
in [10]. Moreover, some necessary and sufficient conditions
for controllability of switched control systems are presented
in [6] and [15] under the assumption that the switching
strategy is fixed a priori. In [16], necessary and sufficient
condition for the controllability and reachability of switched
control systems consisting of linear continuoustime time
invariant subsystems is presented.
A. The General Form of a Switched Control System
Mathematically, a switched control system can be de
scribed by a differential equation of the form
˙ x(t) = fσ(t)(x(t))
where {fp: p ∈ I} is a family of sufficiently regular vector
fields from Rnto Rnthat is parameterized by some index
set I, and σ : [0,∞) → I is a piecewise constant switching
signal.
The linear continuoustime version has the form
˙ x(t) = Aσ(t)x(t)
(1)
where {Ap: p ∈ I} is a family of n × n matrices with real
entries that is parameterized by some index set I, and σ is
as above. The discretetime counterpart of (1) takes the form
x(k + 1) = Aσ(k)x(k)
*research was partially supported from NSF grants ECS0220314, and
ECS0218245
*research was partially supported from NSF grants ECS0220314, and
ECS0218245
0780383354/04/$17.00 ©2004 AACC
where σ is a function from nonnegative integers to a finite
index set I.
In this context, our focus is on the class of switched control
systems consisting of linear continuoustime timeinvariant
subsystems, which in addition admits a certain algebraic
condition corresponding to controllability. This subclass is
denoted by W and is explicitly described in section 1.2.
B. The Class W of Switched Control Systems
Consider a switched control system consisting of linear
continuoustime timeinvariant subsystems of the form
˙ x = Aix + Bu
(2)
where for each (i ∈ k), Aiis an n × n matrices with real
entries and B is an n × m matrix. To avoid trivialities, it
is assumed that B = (e1...em) where el(l ∈ m) denotes
the lthelement of the standard basis for Rn. Moreover, the
space U of admissible inputs of the switched control system
with subsystems of the form given in (2) is assumed to be
Rm.
The reachability subspace < AiB > of ˙ x = Aix+Bu is
given by
< AiB >= B + AiB + A2
where B is the column space of B. Define the finite sequence
of subspaces {Dl}n
D0
Dl
The necessary and sufficient condition for the controllability
of a switched control system consisting of subsystems of the
form given in (2) is Dn= Rn[16]. To avoid trivial cases, it
is assumed that D0?= Rn.
Remark 1.1: If D0?= Rnand Dn= Rn, then 1 ≤ m ≤
n − 2 and n ≥ 3 where m = dimB.
Definition 1.1: The class W is defined as the set of
switched control systems consisting of subsystems of the
form given in (2) satisfying
D0?= Rnand Dn= Rn.
Generating controls and stabilizing controllers for linear
switched systems has been shown to be a nontrivial problem
[1]. This is an attempt to show that it is, in fact, possi
ble to relate with a given controllable switched system, a
controllable nonswitched timeinvariant polynomial system
with the property that all trajectories of the latter can be
approximated arbitrarily closely by trajectories of the given
switched system.
iB + ··· + An−1
i
B
l=0recursively as
< A1B > +···+ < AkB >
< A1Dl−1> +···+ < AkDl−1> for l ∈ n.
=
=
(3)
Proceeding of the 2004 American Control Conference?
Boston, Massachusetts June 30  July 2, 2004
FrP19.3
5806
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Denote the set of polynomial control systems by P. For
a given w ∈ W, a related nonswitched timeinvariant
controllable polynomial system φ ∈ P of which trajectories
can be arbitrarily closely approximated by those of w, is
constructed. That is, for a given w ∈ W, we aim at defining
a relation S(S ⊂ W × P).
Examples are constructed to demonstrate the fact that, in
general, for w ∈ W, controllability does not imply local
stabilizability via a continuous switching strategy. Since the
trajectories of φ can be approximated arbitrarily closely by
those of w, the asymptotic feedback controllability of w
to the origin is established via the related nonswitched
polynomial system φ.
II. The General Form and the Controllability of φ ∈ P
A. The General Form of φ ∈ P
In this section, we investigate the general form of φ ∈ P
for a given w ∈ W. For a given w ∈ W, consider the non
switched timeinvariant polynomial system φ given by
?
i=1
where Ai (i ∈ k) are n × n matrices, B = (e1...em)
and αi(x) : Rn→ R (i ∈ k) are nonnegative polynomial
functions satisfying
αi(x) > 0 for all x ∈ Rn\{0}. The
functions αi(x) (i ∈ k) are called feedback functions.
˙ x =
k
?
αi(x)Ai
?
x + Bu
(4)
k
?
i=1
B. Controllability of Related φ ∈ P when m = n − 2
Depending on the values of n and m, the related non
switched polynomial system φ for a given w ∈ W becomes
linear or nonlinear. For a given w ∈ W, a sufficient condition
for φ to be linear is established in lemma 2.1.
Lemma 2.1: Suppose w ∈ W with m = n − 2. Then,
there exist nonnegative constants αi (i ∈ k) such that the
related nonswitched system φ with αi(x) = αi(i ∈ k) is
controllable.
Proof: See [12].
Example 2.1: Consider the switched control system with
n > 3 and m < n − 2 consisting of 2 subsystems ˙ x =
Aix + Bu for i = 1,2 where B = e1and A1=
?
a(1)
0
elsewhere
?
a(1)
ij
?
n×n
and A2=
a(2)
ij
?
i − j = 2
n×nare given by
ij=
?
1
a(2)
ij=
?
1
0
(i,j) = (2,1)
elsewhere
(5)
It is left to the reader to verify that D0?= Rnand Dn= Rn.
Thus, w ∈ W.
Letting αi(x) = αifor i = 1,2, in system φ given in (4),
we get
˙ x = (α1A1+ α2A2) + Bu.
(6)
Let the controllability matrix of φ given in (6) be
C. Straightforward calculations yield that for n > 3,
max
2
(α1,α2)∈R2(rank(C)) =?n
?+ 1 < n. Thus, for n > 3, there
exist w ∈ W such that there are no constants αi(i ∈ k) for
which φ with αi(x) = αi(i ∈ k) is controllable.
The above example indicates the fact that for a given w ∈ W
with m < n−2, the related φ is not controllable with constant
feedback functions αi(x) = αi (i ∈ k) in general. This
motivates us to seek some nonnegative nonconstant functions
for αi(x) (i ∈ k) which make φ nonlinear.
C. Suitable Choices for αi(x) When m < n − 2
It is required to choose smooth functions for αi(x) (i ∈ k)
such that
(a)
φ is globally controllable for any w ∈ W and
(b)
αi(x) > 0 for all x ∈ Rn\{0}.
?
µ(i)
l
∈ N for all i ∈ k, l ∈ r. Then, in multiindex notation,
up(i)= (up(i)
1
r
) and p(i) = p(i)
all 1 ≤ i ≤ k. Then, the above requirements can be met by
letting αi(x) = ci+up(i)for all (i ∈ k) with u = (x1,...,xr)
satisfy
p(i)− p(j)t≥ 4 for all i ?= j(i,j ∈ k),
(ii)
p(i)
(iii)
c1> 0 and ci= 0 for all 1 < i ≤ k.
where .tis the taxicab metric in Zr.
D. Controllability of Related φ ∈ P when m < n − 2
Theorem 2.1: If w ∈ W is a multiinput switched control
system (m ≥ 2), then there exist distinct positive semi
definite polynomials αi(x) (i ∈ k) satisfying (7) such that
the related nonswitched polynomial system φ is globally
controllable.
Proof: Since w ∈ W is a multiinput switched control
system, it consists subsystems of the form given in (2) which
satisfy the condition given in (3) with r ≥ 2. Recall that the
related nonswitched system φ given in (4) has the form
?
By choosing αi(x) (i ∈ k) as in (7), straightforward
calculations yield that
k
?
i=1
Let r =
m
2
if m ≥ 2
if m = 1
. Also, let p(i)
l
= 2µ(i)
l
where
1
,...,up(i)
r
1
+ ··· + p(i)
1
for
(i)
r ?= 0 for all i ∈ k,
(7)
˙ x = f(x) +
m
?
l=1
glul=
k
?
i=1
αi(x)Ai
?
x + Bu.
adp(i)
r
grad
p(i)
r−1
gr−1...adp(i)
2
g2adp(i)
1
g1f = Aix+
r
?
l=1
klxlAiblfor all i ∈ k.
(8)
Also note that kl∈ N for all l ∈ r. Letting
hi= adp(i)
r
grad
p(i)
r−1
gr−1...adp(i)
2
g2adp(i)
1
g1f for i ∈ k,
it can be deduced that
adp(i)
l
gl hi= p(i)
for all i ∈ k,l ∈ r. Letting
1!p(i)
2!...p(i)
l−1!(p(i)
l
+ 1)!p(i)
l+1!...p(i)
r!Aibl
(9)
hil= Aibl=
adp(i)
gl hi
l−1!(p(i)
l
p(i)
1!p(i)
2!...p(i)
l
+ 1)!p(i)
l+1!...p(i)
r !
(10)
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for all i ∈ k,l ∈ r, from (8), it yields that
Aix = hi−
r
?
l=1
klxlhilfor all i ∈ k.
(11)
It is obvious that hil, hj −?r
constant vector fields of the strong accessibility Lie algebra
S of φ by computing appropriate Lie brackets using (10)
and (11). Since Dn= Rn, the constant vector fields of strong
accessibility Lie algebra S has full rank. Thus, the system
φ with αi(x) given in (7) is globally controllable [11]. Also
see [4], [7], [8], [12] and [14].
Hitherto, the controllability of the class W of switched
control systems were considered except when m = 1 and
n > 3. To analyze the controllability properties of such
systems, we adhere to a different strategy described as
follows.
Since m = 1, in this case, B = e1. Without loss of
generality, it can be assumed that there exist i ∈ k and j ∈ n
such that
Aib1= γ1e1+ γjej.
l=1klxlhjl ∈ S for all
i,j ∈ k. The basis vectors of Dncan hence be obtained as
(12)
If the system does not inherit this property, by means
of an appropriate coordinate transformation, (12) can be
obtained. Moreover, by means of another coordinate trans
formation, (12) can be obtained as
A1b1= e2.
(13)
Theorem 2.2: If w ∈ W is a switched control system
consisting of singleinput linear subsystems which evolve in
Rn(n > 3) satisfying (13), then there exist distinct positive
semidefinite polynomials αi(x) (i ∈ k) satisfying (7) such
that the related nonswitched polynomial system φ is globally
controllable.
Proof: The lines of this proof are the same as those of
theorem 2.1 with the exception that, in this case, r = 2.
E. Approximation of Trajectories of φ ∈ P by Those of
w ∈ W
In proposition 2.1, it is established that for a given w ∈ W,
the trajectories of the related φ of the form given in (4) can
be arbitrarily closely approximated by those of w.
Proposition 2.1: For any w ∈ W and T < ∞, the
trajectories of a related nonswitched polynomial system φ
in the form of (4) can be approximated arbitrarily closely
by those of w for all t ∈ [0,T].
Moreover, the feedback functions αi(x) (i ∈ k) contain
information of the switching strategy that should be
employed in order to follow the trajectory of φ arbitrary
closely by the switched control system w of our interest.
Proof: Suppose φ is globally controllable for a given
w ∈ W. Thus for given ˆ x ∈ Rn\{0} andˆˆ x ∈ Rn, there
exist ˆ u ∈ Rmand T > 0 such that x(T,0, ˆ x, ˆ u) =ˆˆ x.
Furthermore, K = {x(t,0, ˆ x, ˆ u) : t ∈ [0,T]} is compact.
Thus, for given ? > 0, there exists N ∈ N such that
N?
where {xj}N
(a)
x1= ˆ x, xN=ˆˆ x and
K ⊂
j=1
B(xj,?)
(14)
j=1can be chosen as follows.
xj∈ K\{0} for all j ∈ N − 1.
If x1 = ˆ x, then xj ∈ B(xj−1,?) for all j =
2,...,N.
Suppose tj (j ∈ N) are given by x(tj,0, ˆ x, ˆ u) = xj for all
j ∈ N. By βijwe denote αi(xj). Let γj=
(By definition of αi(x) (i ∈ k), γj (j ∈ N − 1) are well
defined since from (15) it follows that?k
For the sake of notational simplicity, by denote ψt
ψt
(15)
(b)
??k
i=1βij
?−1
.
i=1βij> 0 for all
j ∈ N − 1.)
1,ˆψtand
2, we denote
ψt
1
=
φt
φt
φβkjt
k
?
k
?
Akx+Bγjˆ u◦ ... ◦ φβ1jt
i=1
αi(x)Ai
x+Bˆ u
ˆψt
=
i=1
βijAix
+Bˆ u
ψt
2
=
A1x+Bγjˆ u
Let . be the Euclidean metric in Rn. For given K, defined
as above, there exist a pair ˆ ? > 0 andˆ N(ˆ ?) such that for all
?(0 < ? < ˆ ?) and N(?)[N(?) >ˆ N(ˆ ?)] satisfying (14), the
following are true.
???
2(zj)
???ψt
1(xj) −ˆψt(zj)
???
?????? <(j − 1)?
N
+
?
2N
(16)
and
???ˆψt(zj) − ψt
?????? <
?
2N
(17)
for t ∈ [tj,tj+1] and x ∈ B(xj,?) where zj (j ∈ N) are
given as z1= x1= ˆ x and
zj+1= ψ(tj+1−tj)
2
(zj) = φβkj(tj+1−tj)
Akx+Bγjˆ u◦...◦φβ1j(tj+1−tj)
A1x+Bγjˆ u(zj)
for j ∈ N − 1. (Note that (17) is a direct consequence of
BakerCampbellHausdorff formula.)
From (16) and (17), it follows that
????ψt
1(xj) − ψt
2(zj)????<j?
N
(18)
for t ∈ [tj,tj+1] and x ∈ B(xj,?).
Letting t = tj+1in (18), it yields that xj+1− zj+1 <j?
for j ∈ N − 1. Letting j = N −1, the assertion immediately
follows since zN− xN <(N−1)?
Letting j = 1, (18) yields that
????ψt
N
N
< ?.
1(x1) − ψt
2(z1)????<
?
N.
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for t ∈ [t1,t2] and x ∈ B(x1,?). Note that t1= 0. Thus, it
can be easily understood the fact that steering the state from
x1for a small time t ∈ [0,t2] by means of the system φ given
in (4), is arbitrarily approximately equivalent to steering the
state from x1using the subsystems of w ∈ W given in (2)
sequentially in such a way that the ithsubsystem is employed
for a duration of βi1t with inputs scaled down by a factor γ1.
Since βi1= αi(x1) for i ∈ k, it is obvious that the functions
αi(x) (i ∈ k) contain information of switching strategy.
Remark 2.1: In proposition 2.1, the functions αi(x) (i ∈
k) could be assumed to be nonnegative of arbitrary but
sufficiently smooth functions. Nevertheless, they are assumed
to be nonnegative polynomials since this assumption enables
us to establish the global controllability of φ easily in
theorem 2.1.
Remark 2.2: Note that when t → ∞, the arbitrary close
approximation of trajectories of a given w ∈ W and those
of a related φ ∈ P is not guaranteed by proposition 2.1. If
φ is globally controllable, it will not be an issue since the
state can be steered between any two arbitrary points in finite
time.
Theorem 2.3: Suppose w ∈ W.
(a)If m = n−2, then there exists a related controllable
linear system of which trajectories can be arbitrarily
closely approximated by those of w.
(b)If w ∈ W with m < n − 2, then there ex
ists a related globally controllable nonswitched
polynomial system of which trajectories can be
arbitrarily closely approximated by those of w. In
particular, the nonlinear system can be chosen to
be a homogeneous system.
Proof:
(a)This follows directly from proposition 2.1 and
lemma 2.1.
(b)This is immediate from proposition 2.1, theorem
2.1 and 2.2.
III. Stabilizability and Asymptotic Feedback
Controllability of Linear Switched Control Systems to
the Origin
A. Approximation of trajectories of w ∈ W by those of
stabilizable linear systems when m = n − 2
In theorem 2.3, it was proved that for a given w ∈ W
with m = n − 2, there exists a related linear timeinvariant
controllable system φ given by
?
i=1
of which trajectories can be approximated arbitrarily closely
by those of w where αi(i ∈ k) are nonnegative constants.
Theorem 3.1: Let w ∈ W with m = n − 2. Then, there
exists a related stabilizable linear timeinvariant system of the
form given in (19) of which trajectories can be approximated
arbitrarily closely by those of w.
Proof: This immediately follows from proposition 2.1
and lemma 2.1.
˙ x =
k
?
αiAi
?
x + Bu
(19)
B. NonStabilizability of w ∈ W via Continuous Switch
ing Strategy
A linear timeinvariant controllable system is always sta
bilizable and the system poles can be placed arbitrarily. On
the contrary, in the case of nonlinear systems, controllability
does not even imply local stabilizability, in general.
In example 3.1, it is demonstrated that if m < n − 2,
there exists w ∈ W for which all related nonswitched time
invariant polynomial systems are not stabilizable.
Example 3.1: Consider the switched system w whose sub
systems are of the form
˙ x
˙ x
=
=
A1x
A2x
+
+
Bu
Bu
where
A1=
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
A2=
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
B = [1,0,0,0]T.
It can be easily verified that the above system satisfies (3).
Thus, w ∈ W. Moreover, there exists a related nonswitched
globally controllable polynomial system of the form
˙ x = α1(x)A1x + α2(x)A2x + Bu
(20)
of which trajectories can arbitrarily closely approximated by
those of w where α1(x) and α2(x) are positive semidefinite
polynomial feedback functions.
The system (20) can explicitly be given as
˙ x1
˙ x2
˙ x3
˙ x4
=
=
=
=
u
α2(x)x1
α1(x)x1
α1(x)x2
If the system has the form ˙ x = F(x,u), then for all
nonnegative polynomials α1(x) and α2(x), F(R4×R) does
not contain an open neighborhood of the origin. This implies
that there is no pair of positive semidefinite polynomial
feedback functions α1(x) and α2(x) for which (20) is locally
stabilizable at the origin. (See [3].)
For every n,m ∈ N with m < n − 2, similar examples
can be constructed. Thus, by virtue of example 3.1, it implies
that there exists w ∈ W with m < n − 2 for which there
is no related stabilizable nonswitched polynomial system of
which trajectories can be arbitrarily closely approximated by
those of w.
Definition 3.1: A continuous switching signal is a
switching strategy which is employed in such a way that at
the time of switching between two subsystems, the system
vector field remains continuous.
Observe that any (possibly nonunique) trajectory of a
switched system which is forced with a continuous switching
strategy, is always smooth.
Now, it is shown that controllability does not imply
existence of a continuous switching strategy for an arbitrary
w ∈ W with m < n − 2 via a .
5809
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Example 3.2: Consider the switched system w given in
example 3.1. Consider the system
˙ x = ψ1(t)A1x + ψ2(t)A2x + Bu
(21)
where for any fixed t ∈ [0,∞), ψ1(t)ψ2(t) = 0 and ψ1(t)+
ψ2(t) = 1. If ψ1(t) = 1 and ψ2(t) = 0, then
˙ x1
=
˙ x2
=
˙ x3
=
˙ x4
=
By the same argument used in example 3.1, it follows that the
system given in (22) is not locally asymptotically stabilizable.
If ψ1(t) = 0 and ψ2(t) = 1, then
˙ x1
=
˙ x2
=
˙ x3
=
˙ x4
=
By the same argument as above, it follows that the system
given in (23) is not locally stabilizable. Thus, the system
given in (21) is not locally asymptotically stabilizable via a
continuous switching strategy.
u
0
x1
x2
(22)
u
x1
0
0
(23)
C. Asymptotic Feedback Controllability of w ∈ W when
m < n − 2
Since the condition given in (3) is not sufficient for the
smooth local asymptotic stabilizability of (4) via a continuous
switching strategy, one has to switch to asymptotic feedback
controllability of such systems to the origin which is de
scribed as follows.
A submanifold M of Rnwhich contains the origin, is
constructed in such a way that, on M, the related non
switched polynomial system φ is invariant and is asymp
totically stabilizable to the origin. Moreover, on M, the
systems φ and w are equivalent. Then, the arbitrary close
approximation of trajectories of w ∈ W and those of φ is
utilized to drive the state by means of w from an arbitrary
x0 ∈ Rnto a point y at the vicinity of a point y where
0 ?= y ∈ M. If y ∈ M, then the system w can be employed
to steer the state from y to the origin asymptotically along
M. This phenomenon is called the asymptotic feedback
controllability of switched control systems to the origin.
Let w ∈ W. Without loss of generality, in addition to the
condition given in (3), it can be assumed that A1 has the
form
where the pair (A(1)
submatrix of B consisting its first m1(m1≥ m) rows.
Define the submanifold M of Rnas
A(1)
11
− − −−
0

A(1)
12
− − −−
A(1)
22
−−

(24)
11,B1) is controllable where B1 is the
M = {(x1,...,xn) ∈ Rnxr= xm1+1= ··· = xn= 0}
?
(25)
where r =
m
2
if m ≥ 2
if m = 1
Lemma 3.1: Let w ∈ W with m < n − 2 satisfies the
condition given in (24). Suppose that the feedback functions
αi(x) (i ∈ k) of the related nonswitched polynomial system
φ in the form of (4) satisfy the condition given in (7). Also
suppose M is as in (25). Then for given x0 ∈ Rnand
R > 0, using φ, the state can be steered from x0to y ∈ M
(y = R) with piecewise constant inputs in finite time. M
is invariant for φ. Moreover, φ is asymptotically stabilizable
on M.
Proof: See [13].
Theorem 3.2: Suppose the switched control system w ∈
W. Also suppose that A1has the form given in (24). Then,
w can be used to steer the state satisfying the following. For
a given x0∈ Rn, R > 0 and δ > 0, there exists T > 0 such
that
x(T) − y < δ
where y ∈ M with y = R and x(0) = x0. If x(T) ∈ M,
then for every ? there existsˆT such that
x(t) < ? whenever t >ˆT
whereˆT is given byˆT = T +1
Proof:
Proposition 2.1 implies that for arbitrary
finite times, the trajectories of w can be arbitrarily closely
approximated by those of a related nonswitched globally
controllable polynomial system φ of the form given in (4).
The global controllability of φ implies that for given
x0 ∈ Rn, R > 0 and y ∈ M there exist T < ∞ such
that x(t) = y ∈ M if x(0) = x0. Then, the first assertion
immediately follows from the proposition 2.1.
λln?K
?
?. (λ > 0 and K > R)
For the second assertion, since A1 has the form given
in (24), it can be easily verified that, on M, (4) is a
controllable linear system, namely,
˙ x = c1A(1)
11x + B1u.
Moreover M is invariant for this system if
ur= −c1
n
?
j=1
a(1)
rjxj.
Since (4) is a controllable linear system on M, its poles can
arbitrarily be placed such that
σ(c1A(1)
11+ B1ˆK) < −λ < 0(λ > 0)
for some matrixˆK of order m×m1. Then, from linear system
theory, it follows that
x(t) < ? for t > T +1
λln
?K
?
?
.
IV. REFERENCES
[1] Blondel,
Switched
may
the
http://www.nd.edu/
Dame, August 2002.
V.,Theys
Systems
Unstable,
MTNS
J.,and
are
Vladimirov
Periodically
Proceedings
(available
A.A.,
Stablethat
be
2002
Electronic
Conference,
mtns/talksalph.htm),
of
at
Notre
5810