Stability Analysis of Second-Order Switched Homogeneous Systems.
We study the stability of second-order switched homogeneous systems. Using the concept of generalized first integrals we explicitly characterize the "most destabilizing" switching-law and construct  a Lyapunov function that yields an easily verifiable, necessary and sufficient condition for asymptotic stability. Using the duality between stability analysis and control synthesis, this also leads to a novel algorithm for designing  a stabilizing switching controller.
Introduction
. We consider the switched homogeneous system:
_
are homogeneous functions (with equal
degree of homogeneity), and Co denotes the convex hull. An important special case
reduces to a switched linear system.
Switched systems appear in many elds of science ranging from economics to
electrical and mechanical engineering [15][18]. In particular, switched linear systems
were studied in the literature under various names, e.g., polytopic linear dieren-
tial inclusions [4], linear polysystems [6], bilinear systems [5], and uncertain linear
systems [20].
is an equilibrium point of (1.1). Analyzing the
stability of this equilibrium point is di-cult because the system admits innitely
many solutions for every initial value 1 .
Stability analysis of switched linear systems can be traced back to the 1940's
since it is closely related to the well-known absolute stability problem [4][19]. Current
approaches to stability analysis include (i) deriving su-cient but not necessary
and su-cient stability conditions, and (ii) deriving necessary and su-cient stability
conditions for the particular case of low-order systems. Popov's criterion, the circle
criterion [19, Chapter 5] and the positive-real lemma [4, Chapter 2] can all be considered
as examples of the rst approach. Many other su-cient conditions exist in
the literature 2 . Nevertheless, these conditions are su-cient but not necessary and
su-cient and are known to be rather conservative conditions.
Far more general results were derived for the second approach, namely, the particular
case of low-order linear switched systems. The basic idea is to single out the
\most unstable" solution ~
x(t) of (1.1), that is, a solution with the following property:
If ~ x(t) converges to the origin then so do all the solutions of (1.1). Then, all that is
left to analyze is the stability of this single solution (see, e.g., [3]).
Department of Theoretical Mathematics, Weizmann Institute of Science, Rehovot, Israel 76100.
(holcman@wisdom.weizmann.ac.il).
y (Corresponding author) Department of Electrical Engineering-Systems, Tel Aviv University, Israel
69978. (michaelm@eng.tau.ac.il).
An analysis of the computational complexity of some closely related problems can be found
in [2]
2 See, for example, the recent survey paper by Liberzon and Morse [11]
D. HOLCMAN AND M. MARGALIOT
Pyatnitskiy and Rapoport [16][17] were the rst to formulate the problem of nd-
ing the \most unstable" solution of (1.1) using a variartional approach. Applying the
maximum principle, they developed a characterization of this solution in terms of a
two-point boundary value problem. Their characterization is not explicit but, never-
theless, using tools from convex analysis they proved the following result. Let be the
collection of all the q-sets of linear functions fA xg for which (1.1) is asymptotically
stable, and denote the boundary 3 of by @. Pyatnitskiy and Rapoport
proved that if fA 1 then the \most unstable" solution of (1.1) is a
closed trajectory. Intuitively, this can be explained as follows. If fAx; Bxg 2 then,
by the denition of , ~
x(t) converges to the origin; if fAx; Bxg
is unbounded. Between these two extremes, that is, when fAx; Bxg 2 @, ~
x(t) is a
closed solution. This leads to a necessary and su-cient stability condition for second
and third-order switched linear systems [16][17], however, the condition is a nonlinear
equation in several unknowns and, since solving this equation turns out to be di-cult,
it cannot be used in practice.
Margaliot and Langholz [14] introduced the novel concept of generalized rst
integrals and used it to provide a dierent characterization of the closed trajectory.
Unlike Pyatnitskiy and Rapoport, the characterization is constructive and leads, for
second-order switched linear systems, to an easily veriable necessary and su-cient
stability condition. Furthermore, their approach yields an explicit Lyapunov function
for switched linear systems.
In the general homogeneous case, the functions f i () are nonlinear functions, and
therefore, the approaches used for switched linear systems cannot be applied. Filippov
derived a necessary and su-cient stability condition for second-order switched
homogeneous systems. However, his proof uses a Lyapunov function that is not constructed
explicitly.
In this paper we combine Fillipov's approach with the approach developed by
Margaliot and Langholz to provide a necessary and su-cient condition for asymptotic
stability of second-order switched homogeneous systems. We construct a suitable
explicit Lyapunov function and derive a condition that is easy to check in practice.
A closely related problem is the stabilization of a several unstable systems using
switching. This problem has recently regained new interest with the discovery that
there are systems that can be stabilized by hybrid controllers whereas they cannot
be stabilized by continuous state-feedback [18, Chapter 6]. To analyze the stability
of (1.1), we synthesize the \most unstable" solution ~
x(t) by switching between several
asymptotically stable systems. Designing a switching controller is equivalent to synthesizing
the \most stable" solution by switching between several unstable systems.
These problems are dual and, therefore, a solution of the rst is also a solution of the
second. Consequently, we use our stability analysis to develop a novel procedure for
designing a stabilizing switching controller for second-order homogeneous systems.
The rest of this paper is organized as follows. Section 2 includes some notations
and assumptions. Section 3 develops the generalized rst integral which will serve as
our main analysis tool. Section 4 analyzes the sets and @. Section 5 provides an
explicit characterization of the \most destabilizing" switching-law. Section 6 presents
an easily veriable necessary and su-cient stability condition. Section 7 describes a
new algorithm for designing a switching controller. The nal section summarizes.
3 The set is open [17]
SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 3
2. Notations and Assumptions. For  > 1, let

that is, the set of homogeneous functions of degree . We denote by E  the set of
Consider the system _
. Transforming to
polar coordinates
we get
_
where R() and A() are homogeneous functions of degree +1 in the variables cos()
and sin().
Following ([9], Chapter III), we analyze the stability of (2.1) by considering two
cases. If A() has no zeros then the origin is a focus and (2.1) yields:
R
is periodic in  with period 2, and h :=2
R 2R(u)
If A has zeros say, then the line  =  is a solution of (2.1) (the origin
is a node) and along this line r(t)
Hence, if ES  := ff 2
asymptotically stableg, then ES
ES N
, where 4
has no zeros and sgn(h) 6= sgn(A)g
ES N
such that

Given f its dierential at x by (Df)(x) :=

The dierential's norm is jj(Df)(x)jj := sup h2R
vector norm on R 2 . The distance between two functions f
is dened by [10]:
(jjf
Note that (E  ; d(; )) is a Banach space and that in the topology induced by d(; )
the set ES  is open.
For simplicity 5 , we consider the dierential inclusion (1.1) with 2:
_
Given an initial condition x 0 , a solution of (2.3) is an absolutely continuous
function x(t), with almost all t. Clearly, there is
4 Here F stands for focus and N for node
5 Our results can be easily generalized to the case q > 2
4 D. HOLCMAN AND M. MARGALIOT
an innite number of solutions for any initial condition. To dierentiate the possible
solutions we use the concept of a switching-law.
Definition 2.1. A switching-law is a piecewise constant function  : [0;
[0; 1]. We refer to the solution of _
as the solution
corresponding to the switching-law .
The solution x(t)  0 is said to be uniformly 6 locally asymptotically stable if:
Given any  > 0, there exists -() > 0 such that every solution of (2.3) with
There exists c > 0 such that every solution of (2.3) satises lim
Since f and g are homogeneous, local asymptotic stability of (2.3) implies global
asymptotic stability. Hence, when the above conditions hold, the system is uniformly
globally asymptotically stable (UGAS).
Definition 2.2. A set P  R 2 is an invariant set of (2.3) if every solution x(t),
with
Definition 2.3. We will say
that
singular if there
exists an invariant set that does not contain an open neighborhood of the origin.
We assume from here on that
Assumption 1. The
set
(x) is not singular.
The role of Assumption 1 will become clear in the proof of Lemma 5.4 below. Note
that it is easy to check if the assumption holds by transforming the two systems _
f (x) and _
to polar coordinates and examining the set of points where _
for each system. For example, if there exists a line l that is an invariant set for both
_
l is an invariant set of (2.3) and Assumption 1 does not
hold.
To make the stability analysis nontrivial, we also assume
Assumption 2. For any xed  2 [0; 1], the origin is a globally asymptotically
stable equilibrium point of _
3. The Generalized First Integral. If the system
_
is Hamiltonian [8] then it admits a classical rst integral, that is, a function H(x)
which satises H(x(t))  H(x(0)) along the trajectories of (3.1). In this case, the
study of (3.1) is greatly simplied since its trajectories are nothing but the contours
const. In particular, it turns out that the rst integral provides a
crucial analysis tool for switched linear systems [14]. The purpose of this section is to
extend this idea to the case where f 2 ES  and, therefore, (3.1) is not Hamiltonian.
dv
dx2
are both homogeneous functions of degree  and, therefore,
the ratio f2 (x1 ;x2 )
is a function of v only which we denote by (v). Hence, along the
trajectories of (3.1): dv
x1 , that is, e
R dv
const. Thus, we dene
6 The term \uniform" is used here to describe uniformity with respect to switching signals
SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 5
the generalized rst integral of (3.1) by
2k
where L(v) :=
R dv
and k is a positive integer. Note that we can write
substituting
Let S be the collection of points where H(x not dened or not continuous
then, by construction, H along the trajectories
of (3.1). If classical rst integral of the system. In general, how-
ever, S 6= ;. Nevertheless, this does not imply that H cannot be used in the analysis
of (3.1). Consider, for example, the case where S is a line and a trajectory x(t) of (3.1)
can cross S but not stay on S. Then, H(x(t)) will remain constant except perhaps
at a crossing time 7 where its value can \jump". Thus, a trajectory of the system is
a concatenation of several contours of H . This motivates the term generalized rst
integral.
To clarify the relationship between the trajectories of _
and the contours
const, we consider an example.
Example 1. Consider the system
_
_
Here (3.2) yields
2k
, and using
In this case 0g. It
is easy to verify that l 1 is an invariant set of (3.3), that is, x(t) \ l
the trivial trajectory that starts and stays on l 1 ). Furthermore, it is easy to see that
a trajectory of (3.3) cannot stay on the line l 2 .
Fig. 3.1 shows the trajectory x(t) of (3.3) for x Fig. 3.2 displays H(x(t))
as a function of time. It may be seen that H(x(t)) is a piecewise constant function
that attains two values. Note that the \jump" in H(x(t)) occurs when x
is, when x(t) 2 S.
4. The Boundary of Stability. Let be the set of all pairs (f ; g) for which (2.3)
is UGAS. In this section we study and its boundary @. Our rst result, whose
proof is given in the appendix, is an inverse Lyapunov theorem.
Lemma 4.1. If (f ; g) 2 , then there exists a C 1 positive-denite function
such that for all x
Furthermore, V (x) is positively homogeneous of degree one 8 .
Lemma 4.2. is an open cone.
Proof. Let (f ; g) 2 . Clearly, (cf ; cg) 2 for all c > 0. Hence, is a cone.
7 That is, a time t 0
such that x(t 0
8 That is,
6 D. HOLCMAN AND M. MARGALIOT
Fig. 3.1. The trajectory of (3.3) for x 0
Fig. 3.2. H(x(t)) as a function of time.
To prove that is open, we use the common Lyapunov function V from Lemma 4.1.
Denote
is a closed curve encircling the origin. Hence, there
exists a < 0 such that for all x 2
rV (x)f(x) < a and rV (x)g(x) < a
If ~ f 2 ES  and ~
are such that
su-ciently small, then for all x 2
It follows from the homogeneity of V , ~ f , and ~
g that ( ~ f ; ~
5. The Worst-Case Switching Law. In this section we provide two explicit
characterizations of the switching-law that yields the \most unstable" solution of (2.3).
SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 7
be the generalized rst integral of _
Definition 5.1. Dene the worst-case switching-law (WCSL) by:
We denote
so the solution corresponding to WCSL satises _
h(x). Note that WCSL is a
state-dependent switching-law and that since
or respectively, that is, the vertices
of
. Furthermore, it is easy to see
that h(x) is homogeneous of degree .
Intuitively, WCSL can be explained as follows. Consider a point x where f (x)
and g(x) are as shown in Fig. 5.1. A solution of _
follows the contour H f
const, whereas, a solution of _
contour going further away from
the origin. In this case, rH f (x)g(x) > 0 so WCSL is which corresponds
to setting _
\pushes" the trajectory away from the origin as
much as possible.
const
Fig. 5.1. Geometrical explanation of WCSL when rH f (x)g(x) > 0
Note that the denition of WCSL using (5.1) is meaningful only for x
since rH f (x) is not dened for x 2 S. However, extending the denition of WCSL to
any x 2 R 2 is immediate since x 2 S implies one of two cases. In the rst case, x 2 l,
where l is a line in R 2 which is an invariant set of _
(with
c < 0 since f is asymptotically stable) so clearly WCSL must use g. In the second
case, the trajectory of _
so the value of the switching-law on the
single point x can be chosen arbitrarily.
We expect WCSL to remain unchanged if we swap the roles of f and g. Indeed,
this is guaranteed by the following lemma whose proof is given in the Appendix.
8 D. HOLCMAN AND M. MARGALIOT
Lemma 5.2. For all x 2 D
where sgn() is the sign function.
We can now state the main result of this section.
Theorem 5.3. (f ; g) 2 @ if and only if the solution corresponding to WCSL is
closed 9 .
Proof. Denote the solution corresponding to WCSL by x(t) and suppose that x(t)
is closed. Let
be the closed curve is the smallest
time such that x(T that using the explicit construction of (x)
(see (5.1)) we can easily dene
explicitly as a concatenation of several contours
of H f (x) and H g (x). Note also that the switching between _
takes place at points x where rH f (see (5.1)), that is, when g(x) and f (x)
are collinear. Hence, the curve
has no corners.
We dene the function V (x) by V
that is, the contours of V are obtained by scaling
(see [1]). The function V (x) is
positively homogeneous (that is, for any c  0: V radially unbounded,
and dierentiable on R 2 n f0g.
Note that both f  (x) and g  (x) belong to E  . We use V (x) to analyze the stability
of the perturbed system _
(x)g. Consider the derivative of V
along the trajectories of _
_
t. If at some x, V (x) corresponds to a contour H f
then rV and, by the denition of WCSL (see (5.1)), rV (x)g(x)  0
so _
corresponds to a contour H g
so rV
any  < 0 we have
_
since this holds for for all x and all (t) 2 [0; 1], we get that for  <
0:
On the other hand, for  > 0 and
_
since this holds for all x, _
x (x) admits an unbounded solution for  > 0. The
derivations above hold for arbitrarily small  and,
therefore,
For the opposite direction, assume that (f ; g) 2 @, and let x(t) be the solution
corresponding to WCSL, that is, x(t) satises _
To prove that x(t) is a closed trajectory we use the following Lemma, whose proof
appears in the Appendix.
9 We omit specifying the initial condition, because the fact that h(x) is homogeneous implies
that, if the solution starting at some x 0
is closed, then all solutions are closed.
SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 9
Lemma 5.4. If (f ; g) 2 @ then the solution corresponding to WCSL rotates
around the origin.
Thus, for a given x 0 6= 0, there exists t 1 > 0 such that x(t), with
since h(x) is homogeneous, we get: x(nt 1
We consider two cases. If c > 1 then x(t) is unbounded and using the
homogeneity of h(x) we conclude that 0 is a (spiral) source. It follows from the
theory of structural stability (see, e.g., [10]) that there exists an  > 0 such that for
all
g) with the origin is a source of the perturbed
dynamical system _
This implies that (f ; g) 62 @ which
is a contradiction.
If c < 1 then x(t) converges to the origin and, by the construction of WCSL, so
does any other solution, so (f ; g) 2 which is again a contradiction. Hence,
that is, x(t) is closed.
The characterization of WCSL using the generalized rst integrals leads to a
simple and constructive proof of Theorem 5.3. However, to actually check whether
the solution corresponding to WCSL is closed, a characterization of WCSL in polar
coordinates is more suitable.
Representing (2.3) in polar coordinates, we get
_
r
_
cos  sin
sin
r
cos
r
If (f ; g) 2 @ then WCSL yields a closed solution. By using the transformation
necessary) we may always assume that this solution rotates around the
origin in a counter clockwise direction, that is, _
that this implies that if at some point x the trajectories of one of the systems are in
the clockwise direction, then WCSL will use the second system. Hence, determining
WCSL is non-trivial only at points where the trajectories of both systems rotate in the
same direction and we assume from here on that both rotate in a clockwise direction
(note that this explains why in Lemma 5.2 it is enough to consider x 2 D).
so F is a parameterization of the set of directions
in
for which _
> 0.
For any (r; ) we dene the switching-law
_
r
_
that is,  is the switching-law that selects, among all the directions which yield _
> 0,
the direction that maximizes d ln r
d . Using (5.4), we get
(cos  sin )j  (r; )
sin  cos )j  (r; )
Let
(cos  sin )j  (r; )
sin  cos )j  (r; )
so that along the trajectory corresponding to : 1
r _
m. Note that since f and g
are homogeneous,
D. HOLCMAN AND M. MARGALIOT
It is easy to verify that the function q(y) := ay+b(1 y)
(where c and
d are such that the denominator is never zero) is monotonic and, therefore, (r; )
in (5.6) is always 0 or 1 and m(r; ) in (5.7) is always one of the two values:
(cos  sin
sin  cos
(cos  sin
sin  cos
respectively.
The next lemma, whose proof is given in the Appendix, shows that the switching-
law  is just the worst-case switching-law .
Lemma 5.5. The switching-law  yields a closed solution if and only if  yields
a closed solution.
Let
I :=
Z 2m()d
d
d
where (r(t); (t)) is the solution corresponding to the switching-law , and T is the
time needed to complete a rotation around the origin. This solution is closed if and
only ln(r(T Combining this with Lemma 5.5 and Theorem 5.3 we
immediately obtain
Theorem 5.6. (f ; g) 2 @ if and only if I = 0.
It is easy to calculate I numerically and, therefore, Theorem 5.6 provides us with
a simple recipe for determining whether (f ; g) 2 @. However, note that we assumed
throughout that the closed solution of the system rotates in a counter clockwise di-
rection. Thus, to use Theorem 5.6 correctly, I has to be compute twice: First, for
the original system, and then for the transformed system r
value by I 0 ). (f ; g) 2 @ if and only if max(I ; I 0 In this way, we nd whether the
system has a closed trajectory, rotating around the origin in a clockwise or counter
clockwise direction.
The following example demonstrates the use of Theorem 5.6.
Example 2. (Detecting the boundary of stability).
Consider the system:
_
where
It is easy to verify that f 2 ES N
3 and since g 0
have
. The
problem is to determine the smallest k  > 0 such that (f(x); g k  (x)) 2 @ .
Transforming to polar coordinates we get:
sin 3  2 cos 3
cos  sin 2
cos
SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 11
Fig. 5.2. I(k) as a function of k
so
sin 3  2 cos 3
cos  sin 2
cos
and
(cos  sin )j  (r; )
sin  cos )j  (r; )
d
where F () includes 0 if ( sin  cos )j 0
Note that although j  is a function of both r and , the integrand in (5.11) is a
function of  (and k) but not of r.
We calculated I(k) numerically for dierent values of k. The results are shown
in Fig. 5.2. The value k  for which I(k
(to four-digit accuracy) and it may be seen that for k <
We repeated the computation for the transformed system
and found that there exists no closed solution rotating around the origin in a clockwise
direction. Hence, the system (5.9) and (5.10) is UGAS for all k 2 [0; k  ) and unstable
for all k > k  .
The WCSL (see (5.6)) for
Fig. 5.3 depicts the solution of the system given by (5.9) and (5.10) with
1:3439, WCSL (5.12), and x It may be seen that the solution is a closed
trajectory, as expected. Note that this trajectory is not convex which implies that the
Lyapunov function used in the proof of Theorem 5.3 (see (5.3)) is not convex. This
12 D. HOLCMAN AND M. MARGALIOT
-0.4
Fig. 5.3. The solution of (5.9) and (5.10) for
is a phenomenon that is unique to nonlinear systems. For switched linear systems the
closed trajectory is convex and, therefore, so is the Lyapunov function V that yields
a su-cient and necessary stability condition [14].
6. Stability Analysis. In this section we transform the original problem of
analyzing the stability of (2.3) to that of detecting the boundary of stability @. The
later problem was solved in section 5.
Given
a new homogeneous function h k (x) with the
following properties: (1) h 0
g. One possible example that satises
the above is:
Consider the switched homogeneous system
_
The absolute stability problem is: nd the smallest k  > 0, when it exists, such
that
Noting
that
and
k1
k2 for all k 1 < k 2 , we immediately obtain the following result.
Lemma 6.1. The system (2.3) is UGAS if and only if k  > 1.
Thus, we can always transform the problem of analyzing the stability of a switched
dynamical system to an absolute stability problem. We already know how to solve
the latter problem for second-order homogeneous systems. To illustrate this consider
the following example.
Example 3. Consider the system (2.3)
It is easy to verify that f (x) and g(x) belong to ES 3 and that both Assumptions 1
and 2 are satised.
SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 13
To analyze the stability of the system we use Lemma 6.1. Dening
we must nd the smallest k  such that (f ; h k  ) 2 @ . We already calculated k  in
Example 2 and found that k 1. Hence, the system (2.3) with f and g
given in (6.2) is UGAS.
7. Designing a Switching Controller. In this section we employ our results to
derive an algorithm for designing a switching controller for stabilizing homogeneous
systems. To be concrete, we focus on linear systems rather than on the general
homogeneous case. Hence, consider the system
_
where K 1 and K 2 are given matrices that represent constraints 10 on the possible
controls. We would like to design a stabilizing state-feedback controller
that satises the constraint u(t) 2 U for all t.
We assume that for any xed matrix K 2 the matrix A
is strictly unstable and, therefore, a linear controller not stabilize the
system. However, it is still possible that a switching controller will stabilize the system
and designing such a controller (if one exists) is the purpose of this section.
Roughly speaking, we are trying to nd a switching-law that yields an asymptotically
stable solution of _
each matrix
in
is strictly unstable. Using the transformation t, we see that such a solution
exists if and only if this switching-law yields an unstable solution of _
Clearly, every matrix
in
is asymptotically stable.
Hence, we obtain the main result of this section.
Theorem 7.1. Let be WCSL for the system _
x be the corresponding solution. There exists a switching controller that
asymptotically stabilizes (7.1) if and only if ~
x is unbounded and, in this case,
stabilizing controller.
Note that Theorem 7.1 provides an algorithm for designing a stabilizing switching
controller whenever such a controller exists. We already solved the problem of
analyzing ~
x for second-order systems.
Example 4. (Designing a stabilizing switching controller).
Consider the system (7.1) with
is a constant. It is easy to verify that for any xed K 2 the
unstable and ,therefore, no linear controller
the system. Therefore, we design a switching controller. By Theorem 7.1 we must
analyze the stability of the switched system (6.1) with
x:
example, by the physical limitations of the actuators
14 D. HOLCMAN AND M. MARGALIOT
Transforming _
to polar coordinates we get:
_
r
_
(cos  sin )r sin
whereas _
becomes:
_
r
_
sin )r sin
Clearly, the solutions of both these systems always rotate in a counter clockwise direction
> 0 for all ) and, therefore, for all , we have
where
(cos  sin ) sin
It is easily veried that m 1 only if tan   0. Hence, WCSL is:
and
Z
Z 3=2
Computing numerically we nd that the value of k for which I = 0 is k
Hence, there exists a switching controller that asymptotically stabilizes (7.1) and (7.2)
if and only if k > 6:98513 and
is a stabilizing controller.
Fig. 7.1 depicts the trajectory of the closed-loop system given by (7.1) and (7.2)
with the switching controller (7.3), and x As we can see, the
system is indeed asymptotically stable.
8.

Summary

. We presented a new approach to stability analysis of second-order
switched homogeneous systems based on the idea of generalized rst integrals. Our
approach leads to an explicit Lyapunov function that provides an easily veriable
necessary and su-cient stability condition.
Using our stability analysis, we designed a novel algorithm for constructing a
switching controller for stabilizing second-order homogeneous systems. The algorithm
determines whether the system can be stabilized using switching, and if the answer
is a-rmative, outputs a suitable controller.
Interesting directions for further research include the complete characterization
of the boundary of stability @ and the study of higher-order switched homogeneous
systems.

Acknowledgments

. We thank the anonymous reviewers for many helpful comments

SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 15
Fig. 7.1. Trajectory of the closed-loop system with the switching controller with x 0


Appendix

.
Proof of Lemma 4.1. The existence of a common Lyapunov function V 0 (x)
follows from Theorem 3.1 in [13] (see also [12]). However, V 0 is not necessarily homo-
geneous. Denote
is a closed curve encircling the origin. We
dene a new function V (x) by V
that is, the contours of V are obtained by scaling
(see [1]). V (x) is dierentiable
on R 2 n f0g, positively homogeneous of order one, and radially unbounded.
For any x 2
we have rV using the homogeneity
of V (x) and f (x) this holds for any x 2 R 2 n f0g. Similarly, rV (x)g(x) < 0 for
all x 2 R 2 n f0g.
Proof of Lemma 5.2. Let
and
. These two
vectors form an orthonormal basis of R 2 and, therefore,
and (rH g (x))
For any x 2 D we have a 1 > 0 and since rH f (x) (rH g (x)) is orthogonal to f (x)
(g(x)), we also have b 2 > 0. Substituting in (8.1) yields sgn(a 2 which is
just (5.2).
Proof of Lemma 5.4. The system _
homogeneous and we can represent
it in polar coordinates as in Eq. (2.1). If
the solution corresponding to WCSL follows the line l := then the
solution follows the line l to the origin. However, by the denition of WCSL this is
only possible if both the solutions of _
coincide with the line l.
D. HOLCMAN AND M. MARGALIOT
Thus, the line l is an invariant set of the system which is a contradiction to Assumption
1. If R()  0, then we get a contradiction to Assumption 2. Hence, A() 6= 0
for all  2 [0; 2] and, therefore, there exists c > 0 such that A() > c or A() < c
for all  2 [0; 2]. Thus, the solution rotates around the origin.
Proof of Lemma 5.5. Suppose that WCSL yields a closed trajectory ~ x(t) that
rotates around the origin in a counter clockwise direction ( _
> 0). Assume that at
some point x along this trajectory,
Note that by the denition of the generalized rst integral: rH f
any x 2 R 2 n S. This implies that rH f
so (8.2) yields
Let  be the polar coordinates of x. Since ~
rotates around the origin in a
counter clockwise direction, and satises _
at x, we have ( sin  cos
on the other hand, ( sin  cos then by the denition of  (see
Eq. (5.6)), only if:
(cos  sin
sin  cos
(cos  sin
sin  cos
Simplifying, we see that (8.4) is equivalent to f 1
which is just Eq. (8.3), hence, we proved that
and only if



--R

Set invariance in control
A survey of computational complexity results in systems and control
Stability of planar switched systems: The linear single input case
Linear Matrix Inequalities in System and Control Theory

A converse Lyapunov theorem for a class of dynamical systems which undergo switching
Stability conditions in homogeneous systems with arbitrary regime switching

Stability of Motion

Basic problems in stability and design of switched systems
A smooth converse Lyapunov theorem for robust stability
A converse Lyapunov theorem for nonlinear switched systems
Necessary and su-cient conditions for absolute stability: The case of second-order systems
Control Using Logic-Based Switching
Criteria of asymptotic stability of di

An Introduction to Hybrid Dynamical Systems
Nonlinear Systems Analysis
Nonquadratic Lyapunov functions for robust stability analysis of linear uncertain systems
--TR
absolute stability;robust stability;hybrid control;hybrid systems;switched linear systems