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STABILITY OF PLANAR SWITCHED SYSTEMS: THE LINEAR
SINGLE INPUT CASE∗
UGO BOSCAIN†
SIAM J. CONTROL OPTIM.
Vol. 41, No. 1, pp. 89–112
c ? 2002 Society for Industrial and Applied Mathematics
Abstract. We study the stability of the origin for the dynamical system ˙ x(t) = u(t)Ax(t)+(1−
u(t))Bx(t), where A and B are two 2×2 real matrices with eigenvalues having strictly negative real
part, x ∈ R2, and u(.) : [0,∞[→ [0,1] is a completely random measurable function. More precisely,
we find a (coordinates invariant) necessary and sufficient condition on A and B for the origin to be
asymptotically stable for each function u(.).
The result is obtained without looking for a common Lyapunov function but studying the locus
in which the two vector fields Ax and Bx are collinear. There are only three relevant parameters:
the first depends only on the eigenvalues of A, the second depends only on the eigenvalues of B,
and the third contains the interrelation among the two systems, and it is the cross ratio of the four
eigenvectors of A and B in the projective line CP1. In the space of these parameters, the shape and
the convexity of the region in which there is stability are studied.
This bidimensional problem assumes particular interest since linear systems of higher dimensions
can be reduced to our situation.
Key words. stability, planar, random switching function, switched systems
AMS subject classifications. 93D20, 37N35
PII. S0363012900382837
1. Introduction. By a switched system we mean a family of continuous-time
dynamical systems and a rule that determines at any time which dynamical system
is responsible for the time evolution. More precisely, let {fu: u ∈ U} be a (finite
or infinite) set of sufficiently regular vector fields on a manifold M, and consider the
family of dynamical systems:
˙ x = fu(x), x ∈ M.(1)
The rule is given assigning the so-called switching function u(.) : [0,∞[→ U. Here we
consider the situation in which the switching function cannot be predicted a priori; it
is given from outside and represents some phenomena (e.g., a disturbance) that it is
not possible to control or include in the dynamical system model.
In the following, we use the notation u ∈ U to label a fixed individual system and
u(.) to indicate the switching function.
Suppose now that all of the fuhave a given property for every u ∈ U. A typical
problem is to study under which conditions this property holds for the system (1) for
arbitrary switching functions. For a discussion of various issues related to switched
systems, we refer the reader to [8].
In [1, 7] the case of switched linear systems was considered:
˙ x = Aux, x ∈ Rn, Au∈ Rn×n, u ∈ U, (2)
and the problem of the asymptotic stability of the origin for arbitrary switching
functions was investigated. Clearly we need the asymptotic stability of each single
∗Received by the editors December 22, 2000; accepted for publication (in revised form) October
17, 2001; published electronically April 2, 2002. This work was supported by a TMR fellowship (Non
Linear Control Network), contract FMRX-CT97-0137 (DG 12-BDCN) (CNRS CON00P140DR04).
http://www.siam.org/journals/sicon/41-1/38283.html
†Universit´ e de Bourgogne, D´ epartement de Math´ ematiques, Analyse Appliqu´ ee et Optimisation,
9, Avenue Alain Savary B.P., 47870-21078 Dijon, France (uboscain@u-bourgogne.fr).
89
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UGO BOSCAIN
subsystem ˙ x = Aux, u ∈ U, in order to have the asymptotic stability of (2) for each
switching function (i.e., the eigenvalues of each matrix Aumust have strictly negative
real part). This will be assumed to be the case throughout the paper.
Notice the important point that in the case of linear systems, the asymptotic sta-
bility for arbitrary switching functions is equivalent to the more often quoted property
of global exponential stability, uniform with respect to switching (GUES); see, for ex-
ample, [2] and references therein.
In [1, 7], it is shown that the structure of the Lie algebra generated by the matrices
Au,
g = {Au: u ∈ U}L.A.,
is crucial for the stability of the system (2) (i.e., the interrelation among the systems).
The main result of [7] is the following theorem.
Theorem 1.1 (Hespanha, Morse, Liberzon). If g is a solvable Lie algebra, then
the switched system (2) is asymptotically stable for each switching function u(.) :
[0,∞[→ U.
In [1] a generalization was given. Let g = r ⊃ + s be the Levi decomposition of
g in its radical (i.e., the maximal solvable ideal of g) and a semisimple subalgebra,
where the symbol ⊃ + indicates the semidirect sum.
Theorem 1.2 (Agrachev, Liberzon). If s is a compact Lie algebra, then the
system (2) is asymptotically stable for every switching function u(.) : [0,∞[→ U.
Theorem 1.2 contains Theorem 1.1 as a special case. Anyway, the converse of
Theorem 1.2 is not true in general: if s is noncompact, the system can be stable or
unstable. This case was also investigated. In particular, if s is noncompact, then it
contains as a subalgebra sl(2,R). Due to that, in the case in which g has dimension at
most 4 as Lie algebra, the authors were able to reduce the problem of the asymptotic
stability of the system (2) to the problem of the asymptotic stability of an auxiliary
bidimensional system. We refer the reader to [1] for details. For this reason, the
bidimensional problem assumes particular interest, and in this paper we give the
complete description of that case for a single input system.
More precisely, we study the stability of the origin for the switched system
˙ x(t) = u(t)Ax(t) + (1 − u(t))Bx(t),(3)
where A and B are two 2 × 2 real matrices with eigenvalues having strictly negative
real part, x ∈ R2, and u(.) : [0,∞[→ [0,1] is an arbitrary measurable switching
function.
It is well known that asymptotic stability for linear switching systems is equivalent
to the existence of a common Lyapunov function. In [11] necessary and sufficient
conditions were obtained for linear bidimensional systems to share a common quadratic
Lyapunov function, but there are linear bidimensional systems for which this function
may fail to be quadratic (see [6]) so that the problem of finding necessary and sufficient
conditions on A and B for the asymptotic stability of the system (3) was open in
general.
In this paper, we give the solution to this problem. Our result is obtained with
a direct method without looking for a common Lyapunov function but analyzing the
locus in which the two vector fields are collinear, to build the “worst trajectory,”
similarly to what people do in optimal synthesis problems on the plane (see [4, 5, 9,
10]). We also use the concept of feedback. The idea of building the worst trajectory
was used also in [6] for analyzing an example.
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STABILITY OF SWITCHED SYSTEMS
91
Three cases are analyzed separately. In the first case, both matrices have complex
eigenvalues (in the following (CC) case). In the second case, one of the two matrices
has real and the other has complex eigenvalues (in the following (RC) case). In the
third case, both the matrices have real eigenvalues (in the following (RR) case).
There are only three relevant parameters: one depends on the eigenvalues of A,
one on the eigenvalues of B (we call them, respectively, ρA and ρB), and the last
contains the interrelation among the two systems, and it is the cross ratio of the four
eigenvectors of A and B in the projective line CP1.
The result can be obtained quite easily except in one case in which the integration
of the vector fields has to be done. In this case, the computations are not difficult but
long, and they are collected in Appendices A and B. In the (CC) and (RR) cases,
we are even able to write the final result in a relatively compact way (see formulas
(5) and (7)).
Fixing the value of the cross ratio, we study the region R in which the system is
asymptotically stable for arbitrary switching functions in the space of the parameters
ρA and ρB. In the (CC) and (RR) cases it is constituted by one or two open
unbounded convex regions, while in the (RC) case it is an open unbounded region
but not always convex.
In section 2 we give the basic definitions, we study the properties of the parameters
describing the problem, and we state the stability theorem giving the main ideas of
the proof. In section 3 we prove the stability theorem separately for the three cases
(CC), (RC), (RR), and we give some examples. In section 4 we study the shape
and the convexity of the region R for fixed values of the cross ratio. In section 5 we
make some final remarks.
2. Basic definitions and statement of the main results. Let A and B be
two diagonalizable 2 × 2 real matrices with eigenvalues having strictly negative real
part. Consider the following property:
(P)
ically stable at the origin for each measurable function u(.) : [0,∞[→ [0,1].
In this section we state the necessary and sufficient conditions on A and B under which
(P) holds. Moreover, we state under which conditions we have at least stability (not
asymptotic) for each function u(.).
Set M(u) := uA + (1 − u)B, u ∈ [0,1]. In the class of constant functions the
asymptotic stability of the origin of the system (3) occurs iff the matrix M(u) has
eigenvalues with strictly negative real part for each u ∈ [0,1]. So this is a necessary
condition. On the other hand, it is known that if [A,B] = 0, then the system (3) is
asymptotically stable for each function u(.). So in the following we will always assume
the following conditions:
H1. Let λ1,λ2 (resp., λ3,λ4) be the eigenvalues of A (resp., B). Then Re(λ1),
Re(λ2), Re(λ3), Re(λ4) < 0.
H2. [A,B] ?= 0. (That implies that neither A nor B are proportional to the
identity.)
For simplicity we will also assume the following.
H3. A and B are diagonalizable. (Notice that if H2 and H3 hold, then λ1?= λ2,
λ3?= λ4.)
H4. Let V1,V2∈ CP1(resp., V3,V4∈ CP1) be the eigenvectors of A (resp.,
B). From H2 and H3 we know that they are uniquely defined, and V1?= V2
and V3?= V4. We assume Vi?= Vjfor i ∈ {1,2}, j ∈ {3,4}.
The dynamical system in R2: ˙ x(t) = u(t)Ax(t)+(1−u(t))Bx(t) is asymptot-
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UGO BOSCAIN
The degenerate cases, in which H1 and H2 hold and H3 or H4 or both do not, are
the following:
• A or B is not diagonalizable. This case (in which (P) can be true or false)
can be treated with techniques entirely similar to the ones of this paper.
• A or B is diagonalizable, but one eigenvector of A coincides with one eigen-
vector of B. In this case, using arguments similar to the ones of the next
section, it is possible to conclude that (P) is true.
Remark 1. One can easily prove that (under the hypotheses H2 and H3), H4
can be violated only in the (RR) case (see also subsection 3.3). Moreover, hypotheses
H2, H3, and H4 imply that Vi?= Vjfor i,j ∈ {1,2,3,4}, i ?= j. This fact permits
us to define the cross ratio without additional hypotheses (see the definition of cross
ratio below).
Theorem 2.3 gives necessary and sufficient conditions for the stability of the sys-
tem (3) in terms of three (coordinates invariant) parameters defined in Definition
2.1. The first (ρA) depends on the eigenvalues of A, the second (ρB) depends on the
eigenvalues of B, and the third (K) depends on Tr(AB), which is a standard scalar
product in the space of 2×2 matrices. Proposition 2.2 gives some properties of these
parameters. Finally, Proposition 2.4 shows the geometrical meaning of K. It is in
one-to-one correspondence with the cross ratio of the four points in the projective line
CP1that corresponds to the four eigenvectors of A and B. This parameter contains
the interrelation among the two systems.
Definition 2.1. Let A and B be two 2 × 2 real matrices, and suppose that H1,
H2, H3, and H4 hold. Moreover, choose the labels (1) and (2) (resp., (3) and (4)) in
such a way that |λ2| > |λ1| (resp., |λ3| > |λ4|) if they are real or Im(λ2) < 0 (resp.,
Im(λ4) < 0) if they are complex. Define
ρA:= −iλ1+ λ2
λ1− λ2;ρB:= −iλ3+ λ4
λ3− λ4;
K := 2Tr(AB) −1
2Tr(A)Tr(B)
(λ1− λ2)(λ3− λ4)
.
Moreover, define the following function of ρA,ρB,K:
D := K2+ 2ρAρBK − (1 + ρ2
A+ ρ2
B).(4)
Notice that ρA∈ R, ρA> 0, iff A has complex eigenvalues and ρA∈ iR, ρA/i > 1,
iff A has real eigenvalues. The same holds for B. Moreover, D ∈ R. The parameter K
contains important information about the matrices A and B. They are stated in the
following proposition, which can be easily proved using the systems of coordinates of
the next section (see also [3]).
Proposition 2.2. Let A and B be as in Definition 2.1. We have the following:
• if A and B have both complex eigenvalues, then K ∈ R and |K| > 1;
• if A and B have both real eigenvalues, then K ∈ R \ {±1};
• A and B have one complex and the other real eigenvalues iff K ∈ iR.
Theorem 2.3. Let A and B be two real matrices such that H1, H2, H3, and
H4 hold, and define ρA,ρB,K,D as in Definition 2.1. We have the following stability
conditions:
Case (CC) If A and B have both complex eigenvalues, then:
Case (CC.1) if D < 0, then (P) is true;
Case (CC.2) if D > 0, then:
Case (CC.2.1) if K < −1, then (P) is false;
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STABILITY OF SWITCHED SYSTEMS
109
θ+
θ−
u=1
u=0
ρ=1
ρ=1
Let ρ(t), θ(t) (resp., x(t),y(t)) be the polar coordinates (resp., Cartesian) of γM(t),
where we fix the initial condition by setting ρ(0) = 1, θ(0) = θ+.
check if at the time a such that θ(a) = θ(0) + 2π we have ρ(a) < 1. Due to the
symmetries of the system, this happens iff at the time¯t such that θ(¯t) = θ++ π we
have ρRC:= ρ(¯t) < 1. Notice that¯t = a/2. The trajectory γM(t) corresponds to the
constant switching function u = +1 up to the time t?in which θ(t?) = θ−+ π. This
time is defined by the equations
We have to
x(t?) = ρ0e−δAt?cos(ωAt?+ θ+
y(t?) = ρ0Ee−δAt?sin(ωAt?+ θ+
?m+
ρ0= (cos2(θ+
y(t?) = m−x(t?).
E),
E),
θ+
E= arctan
E
?
∈
?
[−π/2,π/2[ if θ+∈ [−π/2,π/2[,
]π/2,3π/4[ if θ+∈]π/2,3π/4[,
E) + E2sin2(θ+
E))−1/2,
It follows that tan(ωAt?+θ+
we have t?= (θ−
After time t?, γM(t) corresponds to the constant switching function u = 0 up to
the first time¯t in which θ(¯t) = θ++ π. This time is defined by the equations
E) = m−/E. If we set θ−
E)/ωA.
E= arctan(m−/E) ∈]θ+
E,θ+
E+π[,
E− θ+
x(¯t) = ρ(t?)e−δB(¯ t−t?)cos(ωB(¯t − t?) + θ−+ π),
y(¯t) = ρ(t?)e−δB(¯ t−t?)sin(ωB(¯t − t?) + θ−+ π),
ρ(t?) = ρ0e−δA
ωA(θ−
E−θ+
E)?
cos2(θ−
E) + E2sin2(θ−
E),
y(¯t) = m+x(¯t).
It follows that tan(ωB(¯t−t?)+θ−+π) = tan(ωB(¯t−t?)+θ−) = m+= tan(θ+), and
we have¯t = (θ+− θ−)/ωB+ t?. Finally,
¯ ρ = ρ(¯t) = ρ(t?)e−δB
ωB(¯ t−t?)= e−δA
ωA(θ−
E−θ+
E)−δB
ωB(θ+−θ−)
?
cos2θ−
cos2θ+
E+ E2sin2(θ−
E+ E2sin2(θ+
E)
E).
This formula is not in a good form because it is not explicitly invariant for the exchange
of δA,ωAwith δB,ωBand because the quantity E does not appear only in the form
E + 1/E. Recalling the definition of ρA,ρB,K (see Definition 2.1) and using the
equality
arctana − arctanb = arctan
?ab + 1
b − a
+ π/2
?
,
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UGO BOSCAIN
which holds for a > b, it is possible to obtain the relations
−δA
−δB
ωA(θ−
E− θ+
E) = −ρA
?
?
arctan
?−ρAK + ρB
?ρA− ρBK
√D
√D
?
+ π/2
?
,
ωB(θ+− θ−) = −ρB
arctan
?
+ π/2
?
.
Moreover, with elementary computation we can show that
?
cos2θ−
cos2θ+
E+ E2sin2(θ−
E+ E2sin2(θ+
E)
E)=
?
ρAρB+√D
ρAρB−√D.
Formula (5) is obtained.
Appendix B: Proof of formula (6). To obtain a result that explicitly does
not depend on the choice of the system of coordinates, we need to write the formulas
of section 3.2 in a more invariant way. Set
ψ =
?
E cos2ϕ + 1/E sin2ϕ
E sin2ϕ + 1/E cos2ϕ,
and make the coordinates transformation
x → Ψ(ψ)x, where Ψ(ψ) :=
?
1
0
0
ψ
?
.
In this case (E ≥ 1), the new coordinates A, B, and θ±have the expressions
?
B = Ψ−1(ψ)
cd
−χ ±√−D
2
λ1−λ2
Equivalently, we can use the expressions (15), (16), (18) for A,B,θ±with E ≥ 1 and
ϕ = π/4.
A = Ψ−1(ψ)
λ1
0
0
λ2
?
Ψ(ψ) =
?
+ω√−K2+ 1
λ1
0
0
λ2
?
,
?
ab
?
Ψ(ψ) =
?
−δ − ωK/i
−ω√−K2+ 1
−δ + ωK/i
−χ ±√−D
(−ρA/i − 1)√−K2+ 1.
?
,
θ±= arctanm±,m±=
λ2
√−K2+ 1=
A =
?
λ1
0
0
λ2
?
?
,
B = R−1(π/4)
−δ
ωE
−ω/E
−δ
?
R(π/4),
−χ ±√−D
λ2
θ±= arctanm±,m±:=
λ1−λ2(E + 1/E)= 2
−χ ±√−D
(−ρA/i − 1)(E + 1/E).(29)
The relation between K and E is
K = i1
2(E − 1/E),E = K/i +
?
−K2+ 1.
Moreover, we are considering the case χ < 0 so that θ+,θ−∈]−π/2,0[. From (29) it
follows that θ+< θ−.
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STABILITY OF SWITCHED SYSTEMS
111
In this case, γM(.) corresponds to the feedback (see the following figure):
u(x) =
?
1 if θ ∈]θ+,θ−[ or θ ∈]θ++ π,θ−+ π[,
0 if θ ∈]θ−,θ++ π[ or θ ∈]θ−+ π,θ++ 2π[.
u=+1
u=0
u=+1
θ
θ
u=0
+
−
Make the following coordinates transformation: x → ¯ x = R(π/4)x. We have
?
B →¯B =
ωE
−δ
θ±→¯θ±= θ±− π/4 = arctan ¯ m±∈ [3/4π,π/4[,
A →¯A = R(π/4)
λ1
0
0
λ2
?
?
R−1(π/4),
?
−δ
−ω/E
,
¯ m±:=m±− 1
m±+ 1.
Similarly to Appendix A, we compute γMin polar coordinates with the initial con-
dition ρ(0) = 1, θ(0) =¯θ−. Let t?be the first time such that θ(t?) =¯θ++ π. We
have
t?= (ξ+− ξ−)/ω,
ρ(t?) = e−δ
ω(ξ+−ξ−)
?
cos2ξ++ E2sin2ξ+
cos2ξ−+ E2sin2ξ−,
where ξ±:= arctan(¯ m±/E), ξ+∈]ξ−,ξ−+ π[.
Now we come back to the old coordinates (¯ x → x = R−1(π/4)¯ x), and we integrate
Bx up to the first time¯t such that θ(¯t) = θ−+ π. We have
x(¯t) = ρ(t?)cos(θ++ π)eλ1(¯ t−t?),
y(¯t) = ρ(t?)sin(θ++ π)eλ2(¯ t−t?),
y(¯t) = m−x(¯t).
It follows that
m+e((λ2−λ1)(¯ t−t?))= n−=⇒¯t − t?=
1
λ2− λ1
ln
?m−
m+
?
.
Finally,
ρRC:= ρ(¯t) = ρ(t?)
?
?
cos2θ+e
λ1
λ2−λ1ln(m−/m+)+ sin2θ+e
λ2
λ2−λ1ln(m−/m+)
= e−δ
ω(ξ+−ξ−)
cos2ξ++ E2sin2ξ+
cos2ξ−+ E2sin2ξ−
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UGO BOSCAIN
×
?
?cos2θ+
= e−ρB(ξ+−ξ−)
?
?
?m−
?
m+
?
λ1
λ2−λ1
+ sin2θ+
?m−
m+
?
λ2
λ2−λ1
cos2ξ++ E2sin2ξ+
cos2ξ−+ E2sin2ξ−
×
?
cos2θ+
?m+
m−
?1
2(−ρA/i+1)
+ sin2θ+
?m+
m−
?1
2(−ρA/i−1)
,
which is formula (6). This formula is complicated but acceptable because there are
no further symmetries.
Acknowledgment. The author is grateful to Andrei Agrachev for suggesting the
problem and for helpful discussions that contributed to finding the right invariant.
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