Page 1

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

Page 2

90

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.

Page 3

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-

Page 4

92

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;

Page 5

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,