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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 continuoustime
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 socalled 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 FMRXCT970137 (DG 12BDCN) (CNRS CON00P140DR04).
http://www.siam.org/journals/sicon/411/38283.html
†Universit´ e de Bourgogne, D´ epartement de Math´ ematiques, Analyse Appliqu´ ee et Optimisation,
9, Avenue Alain Savary B.P., 4787021078 Dijon, France (uboscain@ubourgogne.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
onetoone 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
93
Case (CC.2.2) if K > 1, then (P) is true iff the following condition
holds:
?
− ρBarctan
?
ρCC:= exp
−ρAarctan
?ρA− ρBK
(ρAρB+ K) +√D
(ρAρB+ K) −√D
?−ρAK + ρB
?
√D
2(ρA+ ρB)
?
(5)
√D
−π
?
×
< 1.
Case (CC.3) If D = 0, then (P) is true or false according, respectively, to
the fact that K > 1 or K < −1.
Case (RC) If A and B have one complex and the other real eigenvalues, define χ :=
ρAK−ρB, where ρAand ρBare chosen in such a way that ρA∈ iR, ρB∈ R.
Then:
Case (RC.1) if D > 0, then (P) is true;
Case (RC.2) if D < 0, then χ ?= 0, and we have:
Case (RC.2.1) if χ > 0, then (P) is false. Moreover, in this case
K/i < 0;
Case (RC.2.2) if χ < 0, then:
Case (RC2.2.A) if K/i ≤ 0, then (P) is true;
Case (RC2.2.B) if K/i > 0, then (P) is true iff the following
condition holds:
ρRC:= e−ρB(ξ+−ξ−)
?
cos2ξ++ E2sin2ξ+
cos2ξ−+ E2sin2ξ−
(6)
×
??m+
m−
?1
2(−ρA/i+1)
cos2θ++
?m+
m−
?1
2(−ρA/i−1)
sin2θ+< 1,
where:E := K/i +
m±:=
?
−K2+ 1,
−χ ±√−D
(−ρA/i − 1)K/i,
θ+:= arctanm+,
ξ±:= arctan
?
m±− 1
E(m±+ 1)
?
, ξ+∈]ξ−,ξ−+ π[.
Case (RC.3) If D = 0, then (P) is true or false according, respectively, to the fact
that χ < 0 or χ > 0.
Case (RR) If A and B have both real eigenvalues, then:
Case (RR.1) if D < 0, then (P) is true. Moreover, we have K > 1;
Case (RR.2) if D > 0, then K ?= −ρAρB(notice that −ρAρB> 1) and:
Case (RR.2.1) if K > −ρAρB, then (P) is false
Case (RR.2.2) if K < −ρAρB, then:
Case (RR.2.2.A) if K > −1, then (P) is true;
Case (RR.2.2.B) if K < −1, then (P) is true iff the following
condition holds:
ρRR:= −fsym(ρA,ρB,K)fasym(ρA,ρB,K)(7)
× fasym(ρB,ρA,K) < 1,
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UGO BOSCAIN
where:
fsym(ρA,ρB,K) :=1 + ρA/i + ρB/i + K −√D
ρB/i − KρA/i −√D
ρB/i − KρA/i +√D
1 + ρA/i + ρB/i + K +√D;
?
faym(ρA,ρB,K) :=
?1
2(ρA/i−1)
.
Case (RR.3) If D = 0, then (P) is true or false according, respectively, to
the fact that K < −ρAρBor K > −ρAρB.
Finally, if (P) is false, then in case (CC.2.2) with ρCC= 1, case (RC.2.2.B)
with ρRC= 1, case (RR.2.2.B) with ρRR= 1, case (CC.3) with K < −1, case
(RC.3) with χ > 0, and case (RR.3) with K > −ρAρB, for every C > 0, there
exists C?≤ C such that if γ(0) < C?, then γ(t) < C for every t ∈ [0,∞[ (i.e., we
have stability of the origin). In the other cases, there exists a trajectory γ(t) such that
limt→∞γ(t) = ∞.
Notice that the expressions (5) and (7) are invariant if we exchange ρAwith ρB.
The last statement says when we have at least stability (not asymptotic) for every
switching function.
Let us study the geometric meaning of K. Let V1,V2,V3,V4 belong to the
complex projective line CP1. Suppose V1 ?= V2 ?= V3, and let (v1,v?
(v3,v?
β(V1,V2,V3,V4) is defined in the following way. Make a Moebius transformation
such that V1,V2become the fundamental points (i.e., of homogeneous coordinates,
respectively, (0,1) and (1,0)) and V3the unity point (i.e., of homogeneous coordinates
(1,1)), and let (¯ v4, ¯ v?
have
1), (v2,v?
The cross ratio
2),
3), (v4,v?
4) be the corresponding homogeneous coordinates.
4) be the new homogeneous coordinates of V4. By definition we
β(V1,V2,V3,V4) := ¯ v?
4/¯ v4=
????
v1
v?
v2
v?
v4
v?
v4
v?
14
????
????
v2
v?
v1
v?
v3
v?
v3
v?
23
????
????
24
????
????
13
????
.(8)
Now the four eigenvectors of A and B are exactly four directions in C2; i.e., they can
be regarded as four points of CP1, and under the conditions H2, H3, H4, it makes
sense to compute their cross ratio (cf. Remark 1).
One can immediately obtain (suggestion: use the systems of coordinates of the
next section) the following proposition.
Proposition 2.4.Let A and B be two 2 × 2 real matrices such that H1,
H2, H3, and H4 hold, and let V1,V2,V3,V4be the four points in the space CP1
corresponding, respectively, to the four eigenvectors of A and B chosen in such a
way that they correspond, respectively, to λ1,λ2,λ3,λ4 (see Definition 2.1).
β(V1,V2,V3,V4) be their cross ratio and K the quantity defined in Definition 2.1.
Then β(V1,V2,V3,V4) and K are in the onetoone relation from C ∪ {∞} to
C ∪ {∞}:
Let
K =β(V1,V2,V3,V4) + 1
β(V1,V2,V3,V4) − 1,β(V1,V2,V3,V4) =K + 1
K − 1.
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95
Notice that K ?= ∞ so that β ?= 1. From Proposition 2.4 and Definition 2.1 we
have the following expression for the cross ratio of the eigenvectors of A and B:
β =Tr(AB) − (λ1λ4+ λ2λ3)
Tr(AB) − (λ1λ3+ λ2λ4).
Theorem 2.3 is proved in the next section. Here we describe the main idea of the
proof.
We build the “worst trajectory,” i.e., the trajectory that at each point has the
velocity forming the angle, with the (exiting) radial direction, having the smallest
absolute value, without taking care of the module of the velocity.
.γ
γ
The main idea is that the system (3) is asymptotically stable iff this trajectory tends
to the origin. The worst trajectory is constructed in the following way. We study the
locus Q−1(0) (the notation is clarified later) in which the two vector fields Ax and
Bx are collinear. We have several cases:
• If Q−1(0) contains only the origin, then, in the (CC) and (RC) cases, one
vector field always points on the same side of the other, and the worst tra
jectory is a trajectory of the vector field Ax or Bx. In this case, the system
is asymptotically stable (cases (CC.1) and (RC.1) of Theorem 2.3).
The situation is similar in case (RR.1). (The worst trajectory tends to the
origin.)
• If Q−1(0) does not contain only the origin, then it is a couple of straight lines
passing from the origin (see the next section). If at each point of Q−1(0) the
two vector fields have opposite versus, then there exists a trajectory going
to infinity corresponding to a constant switching function (see the following
figure).
γ
Q (0)
1
This corresponds to cases (CC.2.1), (RC.2.1), and (RR.2.1) of Theorem
2.3, and it is the situation in which there exists u ∈ [0,1] such that the matrix
M(u) admits an eigenvalue with positive real part. If at each point of Q the
two vector fields have the same versus, then the system is asymptotically
stable iff the worst trajectory turns around the origin and after one turn the
distance from the origin is increasing.
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UGO BOSCAIN
This corresponds to cases (CC.2.2), (RC.2.2), and (RR.2.2) of Theorem
2.3.
• Finally, (CC.3), (RC.3), and (RR.3) are the degenerate cases in which the
two straight lines coincide.
More details are given later.
3. Proof of the stability theorem. In the following, we prove Theorem 2.3
separately for the three cases in which A and B have both complex, one complex and
the other real, and both real eigenvalues.
3.1. The case in which A and B have both complex eigenvalues. Let
−δA± iωA (δA,ωA > 0) be the eigenvalues of A and −δB± iωB (δB,ωB > 0) be
the eigenvalues of B. We have ρA = δA/ωA,
coordinates in which
ρB = δB/ωB. Choose a system of
A =
?
−δA
ωAE
−ωA/E
−δA
?
,B =
?
−δB
ωB
−ωB
−δB
?
,
where E ∈ R\{0}. This corresponds to put B in the normal form in which its integral
curves are “circular spirals” and then, using the invariance of B under rotation, to
rotate the coordinates in such a way that the integral curves of A are elliptical spirals
with axes corresponding to the x1and x2directions (see, for example, Figure 3.1).
We have
[A,B] = ωAωB(E − 1/E)
?
1
0
0
−1
?
,
so we assume E ?= ±1; otherwise, [A,B] = 0. In this case we have K =1
and without loss of generality we may assume E > 1.
The locus in which Ax and Bx are collinear is Q−1(0), where
2(E +1
E),
Q = det(Ax,Bx) = x2
+ x1x2(ωAωB(E − 1/E)) + x2
and x = (x1,x2). Now let DCC be the discriminant of the quadratic form Q. We
have
1(−δAωB+ δBωAE)
2(−δAωB+ δBωA/E)
DCC= ω2
Aω2
Aω2
B(E − 1/E)2− 4(−δAωB+ δBωAE)(−δAωB+ δBωA/E)
BD,
where D is defined in Definition 2.1.
Case 1. If D < 0, then the quadratic form Q has strictly defined sign and Q−1(0) =
{0}. In this case, one vector field always points on the same side of the other.
Making a suitable change of coordinates and possibly exchanging the labels
(A) and (B), we can realize the situation in which Ax always points on the
left of Bx for every x ∈ R2\ {0}. We have two cases.
• Suppose first that E > 1. In this case, Ax always points in the grey
region of the following picture.
(9)
= 4ω2
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STABILITY OF SWITCHED SYSTEMS
97
Fix an arbitrary measurable switching function u(.) : [0,∞[→ [0,1],
and let (x1(t),x2(t)) (resp., (ρ(t),θ(t))) be the Cartesian (resp., polar)
coordinates of the solution of ˙ x(t) = u(t)Ax(t)+(1−u(t))Bx(t), x(0) =
x0∈ R2\{0}. In this case, we have ˙ ρ(t) < 0 for almost every t ∈ [0,+∞[
and (P) is true.
• Suppose now that E < −1. Fix x0∈ R2\{0}, and let γ be a trajectory
of the switched system (3) such that γ(0) = x0. Let γA: [0,tA] → R2
(resp., γB: [0,tB] → R2) be a trajectory of the vector field Ax (resp.,
Bx) such that γA(0) = x0(resp., γB(0) = x0), and define tAand tBin
such a way that γA(tA) = γB(tB) =: ¯ x is the first intersection point of
γAand γBafter x0.
x
x
0
γB
A
γ
Let Ω be the simply connected closed set whose border is
∂Ω = Supp(γA[0,tA]∪ γB[0,tB]).
For every x ∈ ∂Ω we have the following. Define Vu= uAx + (1 − u)Bx.
For each u ∈]0,1[, Vu points inside Ω. Moreover, if x / ∈ {x0, ¯ x}, V1
(resp., V0) points inside Ω or it is tangent to ∂Ω. Fix¯t > max{tA,tB}.
We clearly have ¯ x := γ(¯t) ∈ int(Ω). Using homothety invariance of the
system (3), we may easily conclude that limt→∞γ(t) = 0 and (P) is
true. This proves case (CC.1) of Theorem 2.3 (see Example 1 below).
Case 2. If D > 0, then Q has no definite sign and Q−1(0) is a couple of noncoinciding
straight lines passing from the origin and forming the following angles with
the x1axis:
θ±= arctan(m±), where(10)
m±=−ωAωB(E − 1/E) ±√DCC
2(−δAωB+ δBωA/E)
=−(E − 1/E) ± 2√D
2(−ρA+ ρB/E)
m−= ∞, m+=δAωB− δBωAE
ωAωB(E − 1/E)
=ρA− ρBE
E − 1/E
(11)
if − ρA+ ρB/E ?= 0,
(12)
if − ρA+ ρB/E = 0,
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UGO BOSCAIN
where we assume that θ−∈ [−π
E < −1, then 4(−δAωB+ δBωAE)(−δAωB+ δBωA/E) > 0, which implies
that θ±∈ [−π
Case 2.1. If E < −1 (K < −1), then at each point of Q−1(0) \ {0} the
two vector fields have opposite versus.
components of R2\Q−1(0). In this case, for each point x0belonging to
two of these regions (see the figure below), it is possible to find u0∈ [0,1]
such that u0Ax0+ (1 − u0)Bx0has the exiting radial direction. So the
system is not stable for arbitrary switching functions. This situation
corresponds to the case in which there exists u ∈ [0,1] such that M(u) :=
uA + (1 − u)B admits an eigenvalue with positive real part; i.e., there
exist trajectories γ corresponding to constant switching functions such
that limt→∞γ(t) = ∞. Case (CC.2.1) of Theorem 2.3 is proved (see
Example 4 below).
2,π
2[ and θ+∈]θ−,θ−+ π[. Notice that if
2,0[.
Consider the four connected
Q (0)
1
Case 2.2. If E > 1 (K > 1), then at each point of Q−1(0)\{0} the two vector fields
have the same versus (counterclockwise). Fix x0∈ R2\ {0}, and let γM:
[0,∞[→ R2, γM(0) = x0be the trajectory corresponding to the feedback
?
where θ ∈ [θ−,θ−+ 2π[ is defined by x1= ρcos(θ), x2= ρsin(θ).
uM(x) =
0 if θ ∈ [θ−,θ+[ or θ ∈ [θ−+ π,θ++ π[,
+1 if θ ∈ [θ+,θ−+ π[ or θ ∈ [θ++ π,θ−+ 2π[,
(13)
θ−
u=+1
u=+1
u=0
u=0
θ+
Let (ρM(t),θM(t)) be the polar coordinates of γMand a the time defined by
θM(a) = θM(0) + 2π. If ρM(a) < ρM(0), then let l be the segment joining
the points (ρM(0),θM(0)) with (ρM(a),θM(a)) and Ω the simply connected
region whose border is ∂Ω := Supp(γM[0,a]∪ l).
Page 11
STABILITY OF SWITCHED SYSTEMS
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0.75
0.5
0.25
0.25
0.5
0.75
1
1.5
1
0.50.5
1
1.5
1
0.75
0.5
0.25
0.25
0.5
0.75
1
1.5
1
0.50.5
1
1.5
1
0.75
0.5
0.25
0.25
0.5
0.75
1
1.5
1
0.50.5
1
1.5
1
0.75
0.5
0.25
0.25
0.5
0.75
1
Example 2
Example 1
Example 4
Example 3
Fig. 3.1. Examples in the (CC) case.
For every x ∈ ∂Ω, we have the following. Define Vuas in Case 1, E < −1.
For each u ∈]0,1[, Vupoints inside Ω. Moreover, if x / ∈ {γM(0),γM(a)}, V1
(resp., V0) points inside Ω or is tangent to ∂Ω. Similarly to Case 1 (E < 1),
we can conclude that (P) is true (see Example 3 below). On the other hand if
ρM(a) ≥ ρM(0), then γM(t) does not tend to the origin and (P) is false (see
Example 2 below). The condition ρM(a) < ρM(0) is satisfied iff condition (5)
holds. Formula (5) is obtained in Appendix A . The condition ρM(a) = ρM(0)
(i.e.,ρCC= 1) is the case in which we have at least stability (not asymptotic)
for every switching function. This concludes the proof of case (CC.2.2).
Case 3. If D = 0, then the two straight lines coincide. If E > 1, it is easy to
understand that we are in the same situation as that of Case 1. If E < −1,
then to every x ∈ Q there exists u ∈ [0,1] such that uAx+(1−u)Bx = 0. In
this case, (P) is false, but we have at least stability (not asymptotic). This
proves case (CC.3) of Theorem 2.3.
Examples. In the following, we give some examples of the various situations in
the (CC) case. We refer to Figure 3.1.
Example 1. ρA = 0.05, ρB = 0.06, K = −1.005. In this case, D ∼ −0.002,
Page 12
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UGO BOSCAIN
and (P) is true. In Figure 3.1, two integral curves of the vector fields Ax and Bx
are shown. A similar situation but with the two trajectories rotating with the same
versus can be obtained with the same values of ρAand ρBbut with K = +1.00001.
In this case, D ∼ −0.00008 and (P) is true (see case (CC.1)).
Example 2. ρA= 0.0375, ρB= 0.05, K = 1.67. In this case, D ∼ 1.79, ρCC∼
2.62 and (P) is false. In Figure 3.1, two integral curves of the vector fields Ax and
Bx, D that are the two straight lines (one almost coincides with the x2axis) and a
trajectory γ such that limt→∞γ(t) = ∞ (cf. case (CC.2.2)) are shown.
Example 3.ρA = 0.0375, ρB = 0.0425, K = 1.00455.
0.0091, ρCC∼ 0.96, and (P) is true (cf. case (CC.1)).
Example 4. Suppose ρA= 0.0375, ρB= 0.05, K = −1.67. In this case, D ∼ 1.77
and (P) is false (cf. case (CC.2.1)).
3.2. The case in which A and B have one complex and the other real
eigenvalues. Suppose that A has real eigenvalues λ1, λ2(λ1,λ2< 0, λ2 > λ1)
and B complex eigenvalues λ3 = −δ + iω, λ4 = −δ − iω (δ,ω > 0).
ρA= −i(λ1+ λ2)/(λ1− λ2) and ρB= δ/ω. We recall that ρA/i > 1, ρB> 0. Define
?
and choose a system of coordinates in which
In this case, D ∼
We have
R(ϕ) :=
cos(ϕ)
−sin(ϕ)
sin(ϕ)
cos(ϕ)
?
∈ SO(2),(14)
A =
?
?
?
λ1
0
0
λ2
?
:= R−1(ϕ)
,(15)
B =
a
c
b
d
?
?
−δ
ωE
−ω/E
−δ
?
R(ϕ)(16)
=
−δ − ω(E − 1/E)sin(ϕ)cos(ϕ)
ω(E cos2(ϕ) + 1/E sin2(ϕ))
−ω(E sin2(ϕ) + 1/E cos2(ϕ))
−δ + ω(E − 1/E)sin(ϕ)cos(ϕ)
?
.
We have K = i(E − 1/E)cos(ϕ)sin(ϕ) ∈ iR, and without loss of generality we may
assume that ϕ ∈ [0,π/2[, E ≥ 1. Notice that in this case
?
Similarly to the previous subsection, the locus in which Ax and Bx are collinear is
Q−1(0), where
[A,B] = (λ1− λ2)
0b
0
−c
?
?= 0 for each K ∈ iR.
Q = det(Ax,Bx) = x2
1(λ1c) + x1x2¯ χ + x2
2(−λ2b),
and by definition ¯ χ := λ1d−λ2a = (λ1+λ2)ωK/i−(λ1−λ2)δ = (λ1−λ2)ωχ, where
χ := ρAK − ρB (see Theorem 2.3). In this case, the discriminant of the quadratic
form Q is
DRC= ¯ χ2+ 4λ1λ2bc = ¯ χ2− 4λ1λ2ω2(−K2+ 1) = −ω2(λ1− λ2)2D.(17)
Notice that χ = 0 implies ¯ χ = 0, which implies DRC < 0, i.e., D > 0. Moreover,
χ > 0 implies K/i < 0, which implies E < −1. Similarly to the previous subsection,
we have the following cases.
Case 1. If DRC< 0 (D > 0), then (P) is true (see Example 1 below).
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STABILITY OF SWITCHED SYSTEMS
101
Case 2. If DRC> 0 (D < 0), then Q−1(0) is a couple of noncoinciding straight lines
passing from the origin and forming the following angles with the x1axis:
θ±= arctan(m±),
m±:=−¯ χ ±√DRC
2(−λ2b)
From (17) it follows that DRC< ¯ χ2(i.e., −D < χ2) so that in this case we
have χ ?= 0 and we may assume
?
Case 2.1 If χ > 0, then K/i < 0, which implies E < −1, and we have
θ−θ+∈] − π/2,0[. In this case, at each point of Q−1(0) \ {0} the two
vector fields have opposite versus. The same argument of Case 2.1 of
section 3.1 shows that (P) is false (see Example 4 below).
Case 2.2 If χ < 0, then in both cases where E ≥ 1, E ≤ −1 at each point
of Q−1(0) \ {0} the two vector fields have the same versus.
Case 2.2.A If E ≤ −1 (which implies K/i ≤ 0), then (P) is true be
cause of the following argument.
From χ = ρAK − ρB< 0 we have
−K/i
ρB
(18)
=
−χ ±√−D
2
λ2
λ1−λ2(E sin2(ϕ) + 1/E cos2(ϕ)).
θ−,θ+∈]0,π/2[ if χ and E have the same sign,
θ−,θ+∈] − π/2,0[ if χ and E have opposite sign.
<
1
ρA/i< 1.(19)
Now let γ be an integral of the vector field Bx and (ρ(t),θ(t)) its
polar coordinates. We have
γ(t) = R(ϕ)
?
ρ0e−δtcos(ωt + ϕ0)
ρ0Ee−δtsin(ωt + ϕ0),
?
and ρ(t) = ρ0e−δt?
that the condition (19) implies ˙ ρ(t) ≤ 0 for every t ∈ Dom(γ), which
clearly implies that (P) is true. We have
?
?
− δ
cos2(ωt + ϕ0) + E2sin2(ωt + ϕ0). Now we prove
˙ ρ(t) = ρ0e−δt
(E2− 1)ω sin(ωt + ϕ0)cos(ωt + ϕ0)
cos2(ωt + ϕ0) + E2sin2(ωt + ϕ0)
?
cos2(ωt + ϕ0) + E2sin2(ωt + ϕ0)
?
.
Therefore, ˙ ρ(t) < 0 iff
(E2− 1)ω sin(ωt + ϕ0)cos(ωt + ϕ0)
?
− δ
or, equivalently, iff
cos2(ωt + ϕ0) + E2sin2(ωt + ϕ0)
?
cos2(ωt + ϕ0) + E2sin2(ωt + ϕ0) < 0
cos2(ωt + ϕ0) + E2sin2(ωt + ϕ0)
− (E2− 1)ω
(20)
δsin(ωt + ϕ0)cos(ωt + ϕ0) > 0.
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UGO BOSCAIN
Now if (E2− 1)ω/δ ≤ −2E (that choosing a system of coordinates
in which ϕ = π/4 is equivalent to −K/(iρB) ≤ 1; see Appendix B),
then the condition (20) is satisfied. Hence from (19) we can conclude
that ˙ ρ(t) < 0 for each t ∈ Dom(γ) and (P) is true (see Example 5
below).
Case 2.2.B If E ≥ 1 (which implies K/i ≥ 0), then (P) is true iff
condition (6) is satisfied (see Appendix B). Notice that in the case
where K = 0 we clearly have that ρRC< 1 and (P) is true (see
Examples 2 and 3 below). The case in which ρRC= 1 is the case
in which we have at least stability (but not asymptotic) for every
switching function.
Case 3. If D = 0, then the two straight lines coincide.
understand that we are in the same situation as that of Case 1. If χ > 0,
then to every x ∈ Q−1(0) there exists u ∈ [0,1] such that uAx+(1−u)Bx = 0.
In this case, (P) is false, but we have at least stability (not asymptotic). This
proves case (RC.3) of Theorem 2.3.
This concludes the proof of cases (RC).
Examples. In the following, we give some examples of the various situations in
the (RC) case. We refer to Figure 3.2.
Example 1. ρA/i = 1.11, ρB= 0.045, K/i = 0.095. In this case, χ ∼ −0.15, D ∼
0.2, and (P) is true (cf. case (RC.1)).
Example 2. ρA/i = 1.11, ρB= 0.02, K/i = 1.33. In this case, χ ∼ −1.49, D ∼
−1.62, ρRC∼ 1.4, and (P) is false (cf. case (RC.2.2.B)).
Example 3. ρA/i = 1.11, ρB= 0.03, K/i = 0.75. In this case, χ ∼ −0.85, D ∼
−0.37, ρRC∼ 0.98, and (P) is true (cf. case (RC.2.2.B)).
Example 4. ρA/i = 1.11, ρB= 0.045, K/i = −2.4. In this case, χ ∼ 2.6, D ∼
−5.3, and (P) is false (cf. case (RC.2.1)).
Example 5. ρA/i = 1.14, ρB= 1.67, K/i = −0.42. In this case, χ ∼ −1.19, D ∼
−1.06, and (P) is true (cf. case (RC.2.2.A)).
3.3. The case in which A and B have both real eigenvalues. Let λ1,λ2
(λ1,λ2< 0, λ2 > λ1) be the eigenvalues of A and λ3,λ4(λ3,λ4< 0, λ4 > λ3)
be the eigenvalues of B. Choose a system of coordinates such that
If χ < 0, it is easy to
A =
?
?
λ1
0
0
λ2
?
:= R−1(π/4)
,(21)
B =
a
c
b
d
?
?
λ3
0
α(λ4− λ3)
λ4
?
R(π/4)(22)
=1
2
?
(λ3+ λ4) − α(λ4− λ3)
(λ3− λ4) − α(λ4− λ3)
(λ3− λ4) + α(λ4− λ3)
(λ3+ λ4) + α(λ4− λ3)
?
,
where R(ϕ) is defined as in formula (14) and α ∈ R \ {±1}.
coordinates the eigenvectors of A are proportional to V1= (1,0), V2= (0,1) and the
eigenvectors of B to V3= (1,1), V4= ((α − 1)/(α + 1),1). The geometric meaning
of α is the following. Arctan(α) is the angle between the vector (−1,1) and V4,
measured clockwise. We have K = α. Notice that
[A,B] = −1
In this system of
2(λ1− λ2)(λ3− λ4)
?
0(α − 1)
0(α + 1)
?
,
so [A,B] ?= 0 for every value of α. The case α = ±1 is excluded (otherwise V4 is
parallel to V2or to V1, respectively, and (H4) fails).
Page 15
STABILITY OF SWITCHED SYSTEMS
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2
1.5
1
0.5
0.5
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1.5
2
2
1.5
1
0.50.5
1
1.5
2
2
1.5
1
0.5
0.5
1
1.5
2
21.510.50.51 1.52
2
1.5
1
0.5
0.5
1
1.5
2
2 1.510.5 0.511.52
2
1.5
1
0.5
0.5
1
1.5
2
2
1.5
1
0.50.5
1
1.5
2
2
1.5
1
0.5
0.5
1
1.5
2
Example 5
Example 1
Example 2
Example 3Example 4
Fig. 3.2. Examples in the (RC) case.
The locus in which Ax and Bx are collinear is Q−1(0), where
Q = det(Ax,Bx) = x2
1(λ1c) + x1x2¯ χ + x2
2(−λ2b),
and by definition ¯ χ := λ1d − λ2a =1
−1
2((λ1− λ2)(λ3+ λ4) − K(λ1+ λ2)(λ3− λ4)) =
2i(λ1− λ2)(λ3− λ4)χ, where χ := ρAK − ρB∈ iR. In this case, the discriminant
Page 16
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UGO BOSCAIN
of the quadratic form Q is
DRR= ¯ χ2+ 4λ1λ2bc = ¯ χ2+ λ1λ2(λ3− λ4)2(−K2+ 1)
=1
4(λ1− λ2)2(λ3− λ4)2D.
(23)
Notice that if K < 1, then D > 0. The following lemma states that in the case where
K < 1 (P) is true.
Lemma 3.1. Let A,B be two 2 × 2 real matrices satisfying H1, H2, H3, and
H4 and such that their eigenvalues are real. Fix an arbitrary measurable switching
function u(.) : [0,∞[→ [0,1], and let (x1(t),x2(t)) (resp., (ρ(t),θ(t))) be the Cartesian
(resp., polar) coordinates of the solution of ˙ x(t) = u(t)Ax(t)+(1−u(t))Bx(t), x(0) =
x0∈ R2\ {0}. If K ∈] − 1,1[, we have that ˙ ρ(t) < 0 for almost every t ∈ [0,+∞[.
Proof. In this case, it is possible to choose a system of coordinates such that
A =
?
λ1
0
0
λ2
?
cos2(ϕ)λ3+ sin2(ϕ)λ4
(λ3− λ4)sin(ϕ)cos(ϕ)(λ3− λ4)
?
λ3
0
,
B = R−1(ϕ)
0
λ4
?
R(ϕ)
=
?
(λ3− λ4)sin(ϕ)cos(ϕ)
sin2(ϕ)λ3+ cos2(ϕ)λ4
?
,
where we assume ϕ ∈]0,π/2[. Notice that ϕ = 0 is excluded (otherwise [A,B] = 0).
We have
˙ ρ(t) = ˙ x1(t)cosθ(t) + ˙ x2(t)sinθ(t)
= ρ(t)?u(t)(λ1cos2θ(t) + λ2sin2θ(t))
+ (1 − u(t))(λ3cos2(θ(t) − ϕ) + λ4sin2(θ(t) − ϕ)?.
This means that ρ(t) has the expression
ρ(t) = ρ(0)exp
??t
0
(u(t)f1(t) + (1 − u(t))f2(t))dt
?
,
where f1and f2are analytic functions satisfying f1< λ1, f2< λ3.
If K > 1 we have the following cases.
Case 1. If D < 0, then (P) is true.
Case 2. If D > 0, then Q−1(0) is a couple of noncoinciding straight lines passing
from the origin and forming the following angles with the x1axis:
θ±= arctan(m±), m±:=−¯ χ ±√DRR
(24)
2(−λ2b)
=
−χ/i ±√D
(ρA/i + 1)(1 − K).
From (23) it follows that DRR< ¯ χ2so that in this case we have χ ?= 0 and
we may assume
?
θ−,θ+∈]0,π/2[ if χ/i and K have the same sign,
θ−,θ+∈] − π/2,0[ if χ/i and K have opposite sign.
We have the following lemma.
Lemma 3.2. Let D > 0; then
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STABILITY OF SWITCHED SYSTEMS
105
(a) K > −ρAρB;
(b) at each point of Q−1(0)\{0}, Ax and Bx have the same (resp., opposite)
sign iff K < −ρAρB(resp., K > −ρAρB).
Proof. (a) can be checked directly. Let us prove (b). Define Λ±:= K±√D−
ρAρB. By direct computation it follows that
• Λ±> 0 iff K > −ρAρB;
• (
• (
Ax
?Ax?) = (
Ax
?Ax?) = −(
Bx
?Bx?) for every x = (h,m±h), h ∈ R \ {0}, iff Λ±< 0;
Bx
?Bx?) for every x = (h,m±h), h ∈ R \ {0}, iff Λ±> 0.
This concludes the proof.
From Lemma (3.2) we have the following cases (notice that −ρAρB> 1).
Case 2.1 If K > −ρAρB, then (P) is false.
Case 2.2 If K < −ρAρB, then:
Case 2.2.A If K > 1, one can easily check that the worst trajectory
cannot rotate around the origin and (P) is true.
Case 2.2.B If K < −1, then the worst trajectory rotates around the
origin and (P) is true iff condition (7) is satisfied. Condition (7) can
be obtained with arguments entirely similar to the ones of Appen
dices A and B. The case in which ρRC= 1 is the case in which we
have at least stability (not asymptotic) for every switching function.
Case 3. If D = 0, then the two straight lines coincide. Similarly to the (CC) and
(RC) cases, if K < −ρAρB, then (P) is true. Vice versa, if K > −ρAρB, then
(P) is false but we have stability (not asymptotic).
4. Asymptotic stability in the space of parameters. Fix a value of the cross
ratio, and let R (resp.,¯R) be the region in the (ρA,ρB) plane in which the system
is asymptotically stable (resp., asymptotically stable or only stable) for arbitrary
switching functions. In this section, we study the shape and the convexity of R and
¯R.
4.1. The complexcomplex case. In Figure 4.1 we show R for a fixed value
of K in the case in which both A and B have complex eigenvalues.
In the case K < −1, R is determined by the condition D < 0.
values of ρA and ρB such that D = 0 is the two curved lines of equations ρB =
ρAK ±?(ρ2
?
(ρA,ρB) = (0,(26)
The set of
A+ 1)(K2− 1) of Figure 4.1 (case K < −1). The points of intersection
with the two axes are
(ρA,ρB) = (
K2− 1,0),
?
(25)
K2− 1).
In this case, R is constituted by two connected open convex unbounded regions, while
to get¯R we have to add the points in which D = 0.
In the case where K > 1, R is determined by the condition ρCC< 1. In Figure
4.1 (case K > 1) the locus D = 0 is drawn with dotted lines, while the locus ρCC= 1
is drawn with a solid line. The points in which the two loci intersect each other and
intersect the two axes are given again by formulas (25) and (26). In this case, to study
the convexity of R, we have to check if, expressing the locusρCC= 0 as ρB= fK(ρA),
we find a convex function. In the following, the label (K) is a parameter, and it will be
dropped.Let us indicate the derivative of ρCCwith respect to the first and second
Page 18
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UGO BOSCAIN
ASYMPTOTIC STABILITY
Asymptotic
Stability
Case K<1
Case K>1
ρ
ρ
K
21
A
A
B
B
ρ
ρ
K
21
K
21
K
21
Asymptotic
Stability
Fig. 4.1. R (the grey region) for a fixed value of K, in the complex–complex case.
variable as?ρCC
f?(ρA) = F (ρA,f(ρA)), where F(ρA,ρB) := −
?
1and?ρCC
?
2. We have that
?ρCC
?
?
1
?ρCC
2
= −
arctan(ρB−ρAK
√D
√D
) +π
2
arctan(ρA−ρBK
) +π
2
,
f??(ρA) = G(ρA,f(ρA)), where
G(ρA,ρB) =∂F(ρA,ρB)
∂ρA
+∂F(ρA,ρB)
∂ρB
F(ρA,ρB)
=
2
(1 + ρ2
A) (1 + ρ2
B)
√D
?
π + 2 arctan(ρA−ρBK
√D
)
?3
Aρ2
×
?
?ρ3
AρB+ ρAρ3
B+ 2(1 + ρ2
A+ ρ2
B+ ρAρB+ ρ2
B) + K(2 + ρ2
A+ ρ2
B)?π2
+ 4?1 + ρ2
+ 4?1 + ρ2
+ 4?1 + ρ2
+ 4?1 + ρ2
+ 8?1 + ρ2
B
? ?1 + ρ2
? ?1 + ρ2
?(ρAρB+ K) arctan
?(ρAρB+ K) arctan
? ?1 + ρ2
A+ ρAρB+ K?π arctan
B+ ρBρA+ K?π arctan
?ρA− ρBK
?ρB− ρAK
?2
?2
√D
√D
?
?
A
A
?ρB− ρAK
?ρA− ρBK
?ρB− ρAK
√D
√D
B
AB
?arctan
√D
?
arctan
?ρA− ρBK
√D
??
.
Now the only terms that can be negative are the ones in the third and fourth rows,
but it is easy to check numerically that the sum of these two terms with the one in
the second row is always bigger than zero. The convexity follows. In this case, R is a
convex open unbounded region, while¯R is a convex notopen unbounded region (we
Page 19
STABILITY OF SWITCHED SYSTEMS
107
(K/i)2+1
>0
ρ/i
A
ρ
B
B
ρ
ρ
1
1
Case K/i <0
Case K/i
A/i
ASYMPTOTIC STABILITY
ρ
B=
(K/i)2+1
(K/i)(ρA/i)
(K/i)

Stability
Asymptotic
Fig. 4.2. R (the grey region) for a fixed value of K, in the (RC) case.
have to add the points such that ρCC= 1).
4.2. The realcomplex case. In the case in which A and B have one complex
and the other real eigenvalues, R is drawn in Figure 4.2. We recall that ρA/i > 1,
ρB> 0, K/i ∈ R.
In the case where χ > 0 (which implies K/i < 0 and ρB< (−K/i)(ρA/i)), R is
determined by the condition D > 0. The locus D = 0 is the set of points such that
ρB= −(ρA/i)(K/i)±?(−(ρA/i)2+ 1)(−(K/i)2− 1). The intersection point with the
?
and the intersection with the ρA/i = 1 set is
ρAaxis is
(ρA/i,ρB) = ((K/i)2+ 1,0),(27)
(ρA/i,ρB) = (1,−(K/i)).
In the case when χ < 0 and K/i ≤ 0, we have asymptotic stability. We conclude that
in the case when K/i ≤ 0, R is a convex open unbounded region (see Figure 4.2 (case
K/i ≤ 0)), while to get¯R, we have to add the points in which D = 0.
In the case when χ < 0 and K/i > 0, R is determined by the condition ρRC< 1.
In Figure 4.2 (case K/i > 0), the locus D = 0 is drawn with a dotted line, while the
locus ρCC= 1 is drawn with a solid line. The points in which the two loci intersect
each other are given by formula (27). In this case, R is a nonconvex open unbounded
region. Again the points in which we have at least stability are the points in which
we have asymptotic stability plus the points such that ρRC= 1.
4.3. The realreal case. In the case in which A and B have both real eigen
values, R is drawn in Figure 4.3. We recall that ρA/i,ρB/i > 1, K ∈ R\{±1}.
K < −1, R is determined by ρRR> 0, while, if K > 1, R is determined by D > 0.
Similarly to the (CC) case, we can conclude that R is a convex open unbounded
region, while¯R is a convex notopen unbounded region. (We have to add the points
such that ρCC= 1 and D = 0.)
If
Page 20
108
UGO BOSCAIN
K/ i
i /
A
i /
A
i /
A
i /
A
i /
1
i /
1
i /
1
i /
1
ASYMPTOTIC STABILITY
Case K>1
ρ
ρ
ρ ρ
A
K=
B
Case K<1
ASYMPTOTIC STABILITY
ρ
ρ
Case 1<K<1 : Asymp. Stab.
K/ i
D =0
ρ =0
RR
Fig. 4.3. R (the grey region) for a fixed value of K, in the (RR) case.
5. Final remarks. Using the results of [1, 7] and by Theorem 2.3, we have a
complete algorithm to study the asymptotic stability of a switched linear system in
any dimension at least in the case in which
Au= uA1x + (1 − u)A2x,u ∈ [0,1], A1,A2∈ Rn×n,
where A and B are diagonalizable and dim{A1,A2}L.A.≤ 4. The case in which A or
B is not diagonalizable can be treated with similar techniques.
Generalization can be done for more complex sets U. One is the following minput
system:
˙ x =
m
?
i=1
uiAix,
m
?
i=1
ui= 1,ui≥ 0 (i = 1,...,m),x ∈ Rn,A1,...,Am∈ Rn×n.
With exactly the same techniques used in this paper, one can find a coordinates
invariant necessary and sufficient condition for the stabilizability of a control system
of the kind (2), where all the matrices have eigenvalues with strictly positive real
part. This problem was also studied in [12] but not in terms of a minimum number
of coordinatefree parameters. We refer to [12] for details.
Some results can be obtained for the nonlinear version of the problem treated in
this paper,
˙ x = uF(x) + (1 − u)G(x), (28)
where x ∈ R2, F(.), G(.) are C∞generic functions from R2to R2such that F(0) = 0,
G(0) = 0, and the two dynamical systems ˙ x = F(x), ˙ x = G(x) are globally asymptot
ically stable at the origin. We are interested in studying under which conditions on
F(.) and G(.) the origin of the system (28) is globally asymptotically stable for every
measurable function u(.) : [0,∞[→ [0,1].
Appendix A: Proof of formula (5). We refer to the following figure.
Page 21
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
?
,
Page 22
110
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−.
Page 23
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−
Page 24
112
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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