LMI-based computation of optimal quadratic Lyapunov
functions for odd polynomial systems
G. Chesi1 , A. Garulli1 , A. Tesi2 , A. Vicino1
1
Dipartimento di Ingegneria dell’Informazione, Università di Siena
Via Roma 56, 53100 Siena, Italy
E-mail: chesi,garulli,[email protected]
2
Dipartimento di Sistemi e Informatica, Università di Firenze
Via di S.Marta 3, 50139 Firenze, Italy
E-mail: [email protected]
Abstract
The problem of estimating the domain of attraction (DA) of equilibria is
considered for odd polynomial systems. Specifically, the computation of the
Optimal Quadratic Lyapunov Function (OQLF), i.e. the Quadratic Lyapunov
Function (QLF) which maximizes the volume of the Largest Estimate of the DA
(LEDA), is addressed. In order to tackle this double non-convex optimization
problem, a relaxation approach based on homogeneous polynomial forms is
proposed. The first contribution of the paper shows that a lower bound of the
LEDA for a fixed QLF can be obtained via Linear Matrix Inequalities (LMIs)
based procedures. Also, condition for checking tightness of the lower bound are
provided. The second contribution is a strategy for selecting a good starting
point for the OQLF search, which is based on the volume maximization of the
region where the time derivative of the QLFs is negative and is given in terms
of LMIs. Several application examples are presented to illustrate the numerical
behavior of the proposed approach.
Keywords: Lyapunov functions; domain of attraction; Linear Matrix Inequalities;
homogeneous forms.
1
1
Introduction
The domain of attraction (DA) of an equilibrium point, i.e. the set of initial states
from which the system converges to the equilibrium point, plays a key role in the
analysis and control of nonlinear systems [1]. Unfortunately, it is well known that the
DA is a very complicated set and, in most cases, it does not admit an exact analytic
representation [2]. On the other hand, gridding-based techniques for approximating
the set are almost always untractable from the computational burden viewpoint.
For this reason, the approximation of the DA via an estimate of a simpler shape has
become a fundamental issue since long time (see [2]). In particular, the ellipsoidal
shape, which is associated to a Quadratic Lyapunov Function (QLF), is definitely
the most popular one.
Within this context, a problem of primary importance is the selection of the Optimal
Quadratic Lyapunov Function (OQLF), i.e. the QLF which maximizes the volume
of the estimate. Unfortunately, the computation of the OQLF amounts to solving a
double non-convex optimization problem [3, 4]. Specifically, for any given QLF one
has to compute the Largest Estimate of the DA (LEDA). Then, the OQLF must be
looked for.
In this paper, the problem of computing the OQLF for odd polynomial systems
is addressed via a relaxation approach based on homogeneous polynomial forms.
The key point of the approach is the Complete Square Matricial Representation
(CMSR) of homogeneous forms (Section 2.2), which has already been exploited to
devise Linear Matrix Inequalities (LMIs) procedures to deal with non-convex distance problems [5, 6, 7]. It is worth remarking that the importance of homogeneous
forms in the analysis of control systems has been recognized since long time (see e.g.,
[8, 9]). More recently, homogeneous forms are deserving a renewed attention due
to their strong connection with semidefinite programming and convex optimization
techniques [10], making it possible to tackle several problems of interest.
The first contribution of the paper concerns the computation of the LEDA (Section
3). It is shown that a lower bound of the LEDA can be obtained via a one-parameter
sequence of LMI feasibility tests in the general case, and via one EigenValue Problem
(EVP) if the nonlinear term of the system is a homogeneous form. Both LMI
feasibility test and EVP can be tackled by convex optimization techniques [11]. The
tightness of this lower bound can be tested a posteriori, or guaranteed a priori for
some classes of systems.
The second contribution of the paper is a procedure for selecting a good starting
point for the OQLF search (Section 4). A relaxed criterion is proposed, based on
2
the volume maximization of the region where the time derivative of the QLF is
negative. The criterion involves a one-parameter sequence of LMI feasibility tests in
the general case, and one Generalized EVP (GEVP), which is a quasi-convex LMI
optimization [11], if the nonlinear term of the system is a homogeneous form.
Finally, several application examples are presented to illustrate the behavior of the
proposed approach to compute the OQLF and the related computational burden
(Section 5).
Notation. 0n : origin of Rn ; Rn0 : Rn \ {0n }; In : n × n identity matrix; A′ : transpose
of A; A > 0 (A ≥ 0): symmetric positive definite (semidefinite) matrix; A ⊗ B:
Kronecker’s product of matrices A and B; A[m] : A
| ⊗A⊗
{z. . . ⊗ A}.
m
2
2.1
Problem formulation and preliminaries
Problem formulation
Let us consider the odd polynomial system defined as
ẋ = Ax +
m−1
X
f2i+1 (x),
(1)
i=1
where x ∈ Rn , fi (x) is a vector of n homogeneous forms of degree i, and A ∈ Rn×n
is a Hurwitz matrix. This implies that the origin is a locally asymptotically stable
equilibrium point.
Let us consider the QLF V (x; P ) = x′ P x, where P > 0 is such that the time
derivative
′
"
V̇ (x; P ) = 2x P Ax +
m−1
X
#
f2i+1 (x)
i=1
is locally negative definite. We refer to such a matrix P as feasible P and define the
set of feasible matrices as
P = {P ∈ Rn×n : P = P ′ , P > 0, P A + A′ P < 0}.
(2)
Let us define the ellipsoidal set associated to the QLF
V(P, c) = x ∈ Rn : x′ P x ≤ c ,
and the negative time derivative region,
n
o
D(P ) = x ∈ Rn : V̇ (x; P ) < 0 ∪ {0}.
(3)
(4)
Then, V(P, c) is an ellipsoidal estimate of the DA of the origin if V(P, c) ⊆ D(P ).
3
Definition 1 Given a feasible QLF V (x; P ) for system (1), the corresponding Largest
Estimate of the Domain of Attraction (LEDA) of the origin is given by
S(P ) = V (P, γ(P )) ,
(5)
γ(P ) = sup {c ∈ R : V(P, c) ⊆ D(P )} .
(6)
where
Let us observe that the computation of γ(P ) requires the solution of a non-convex
distance problem.
Proposition 1 The quantity γ(P ) in (6) can be computed as
γ(P ) =
inf x′ P x
x∈Rn
0
s.t.
(7)
V̇ (x; P ) = 0 .
The Optimal QLF (OQLF) is defined as the QLF which maximizes the volume of
the DA.
Definition 2 The OQLF for system (1) is given by V (x; P ∗ ) = x′ P ∗ x, where
P ∗ = argmax δ(P )
(8)
P ∈P
and
δ(P ) =
s
[γ(P )]n
.
det(P )
Notice that, since the volume of V(P, 1) is proportional to
(9)
p
det (P −1 ) up to a scale
factor depending on n, δ(P ) is a measure of the volume of S(P ) . It turns out that
the computation of the OQLF amounts to solve a double non-convex optimization
problem. In fact, the volume function δ(P ) can show local maxima in addition to
the global one δ(P ∗ ). Moreover, each evaluation of δ(P ) requires the computation
of γ(P ), that is the solution of the non-convex distance problem (7).
In order to address (8) let us observe that the set P in (2) can be written as
P = P ∈ Rn×n : P = P ′ , P > 0, P A + A′ P = −Q, Q = Q′ > 0
(10)
and, hence, linearly parameterized through the function
P = F (Q) : F (Q)A + A′ F (Q) = −Q
(11)
where Q is any symmetric positive definite matrix. This means that problem (8)
can be formulated as
P ∗ = F (Q∗ ) , Q∗ = argmax δ(F (Q))
Q∈Q
4
(12)
where
Q = Q ∈ Rn×n : Q = Q′ , Q > 0 .
(13)
Notice that, since δ(P ) is not affected by a positive scale factor on P , and P depends
linearly on Q, matrices in Q can be arbitrarily scaled.
In Section 4, a procedure for the selection of a good starting point in the optimization
problem (12) will be presented. The basic step of the iterative OQLF search is the
computation of the LEDA for a fixed QLF, which will be addressed in Section 3.
Since homogeneous forms play a key role in the computation of the LEDA, some
basic material on homogeneous forms is introduced next.
2.2
Square Matricial Representation of homogeneous forms
A function gm (x) is a homogeneous form of degree m in x ∈ Rn if
X
gm (x) =
ci1 ,i2 ,...,in xi11 xi22 . . . xinn ,
i1 +i2 +...+in =m
where i1 , i2 , . . . , in are nonnegative integers, and ci1 ,i2 ,...,in ∈ R are weighting coefficients. The Square Matricial Representation (SMR) of a homogeneous form of even
degree g2m (x) is defined as
′
g2m (x) = x{m} Gx{m} ,
(14)
where x{m} ∈ Rd is a vector containing all monomials of degree m in x, and G =
G′ ∈ Rd×d is a coefficient matrix. An important property of the SMR is that matrix
G is not unique. Indeed, all the matrices G satisfying (14) can be parameterized as
G + L, with L ∈ L, where
n
o
′
L = L = L′ ∈ Rd×d : x{m} Lx{m} = 0 ∀x ∈ Rn .
Then, the family of matrices G describing g2m (x) can be parameterized affinely as
G(α) = G + L(α), where α ∈ RdL is a vector of free parameters and L : RdL → L
is a linear parameterization of L. Hence, the Complete SMR (CSMR) of g2m (x) is
defined as
′
g2m (x) = x{m} G(α)x{m} .
and G(α) is called the CSMR matrix of g2m (x). The numbers d and dL are given
respectively by
d=
(n + m − 1)!
(n − 1)!m!
1
(n + 2m − 1)!
dL = d(d + 1) −
.
2
(n − 1)!(2m)!
Values of d and dL for different n and m are reported in Table 1.
A procedure for computing the CSMR of a generic form of even degree is reported
in [7].
5
d
n=2
3
4
5
dL
n=2
3
4
5
m=1
2
3
4
5
m=1
0
0
0
0
2
3
6
10
15
2
1
6
20
50
3
4
10
20
35
3
3
27
126
420
4
5
15
35
70
4
6
75
465
1990
5
6
21
56
126
5
10
165
1310
7000
(a)
(b)
Table 1: Values of d (a) and dL (b), for some n and m.
3
Computation of the LEDA for a fixed QLF
The aim of this section is to formulate an LMI-based procedure for the computation
of the LEDA for a fixed QLF. In particular, a lower bound for γ(P ) in (7) will be
derived and a simple test for checking its tightness will be provided. The main idea is
to transform problem (7) into a sequence of positivity tests for suitable homogeneous
forms, and then relax these tests by checking if such forms are sum of squares.
3.1
Computation of the LEDA: generic odd nonlinearity
For any given feasible P ∈ P, let us introduce the polynomial w(x; P ) such that
V̇ (x; P ) = −w(x; P ) = −
m
X
w2i (x; P )
(15)
i=1
where w2i (x; P ) are suitable homogeneous forms of degree 2i. Observe that w(x; P )
is locally positive definite because w2 (x; P ) = x′ Qx.
Let us now introduce the homogeneous form of degree 2m
′
m
X
x P x m−i
h(x; P, c) =
w2i (x; P )
.
c
(16)
i=1
Notice that h(x; P, c) is a polynomial form in the entries of P . If P is fixed, h(x; P, c)
is a homogeneous form in x with one only additional parameter c. The characterization of the LEDA proposed in the following is based on the properties of this
homogeneous form, for different values of c.
Let H(P, c, α) ∈ Rd×d be the CSMR matrix of h(x; P, c) and define the quantity
n
o
(17)
γ(P ) = sup c̃ : H(P, c, α) > 0 for some α ∈ RdL , ∀c ∈ (0, c̃] .
Note that γ(P ) can be computed via a one-parameter sequence of the LMI feasibility
tests
∃α ∈ RdL : H(P, c, α) > 0
6
(18)
which are convex optimizations in dL free parameters.
The following result pertains to γ(P ).
Theorem 1 Let γ(P ) and γ(P ) be given by (7) and (17), respectively. Then,
γ(P ) ≤ γ(P ).
Proof. Consider the boundary of V(P, c), i.e. ∂V(P, c) = {x ∈ Rn : x′ P x = c}. It
turns out that
γ(P ) = sup {c̃ > 0 : w(x; P ) > 0 ∀x ∈ ∂V(P, c), ∀c ∈ (0, c̃]} .
(19)
Then, Lemma 3 in [7] guarantees that for any c ∈ (0, +∞) we have that
w(x; P ) > 0 ∀x ∈ ∂V(P, c) ⇐⇒ h(x; P, c) > 0 ∀x ∈ Rn0 .
(20)
From (19) and (20) it follows that γ(P ) can be computed via a sequence of positivity
tests on homogeneous forms, that is
γ(P ) = sup {c̃ > 0 : h(x; P, c) > 0 ∀x ∈ Rn0 , ∀c ∈ (0, c̃]} .
The proof is completed by observing that if H(P, c, α) > 0 for some α ∈ RdL , then
h(x; P, c) > 0.
Remark 1. Tightness of the lower bound γ(P ) is strictly related to the property of
positive homogeneous forms to be represented as the sum of squares of homogeneous
forms [12, 10]. Indeed, it has been proved that γ(P ) is tight if and only if the
homogeneous form h(x; P, c) satisfies this property for all c ∈ (0, γ(P )) [7]. For the
cases n = 2, ∀m and n = 3, m = 2 such property is guaranteed a priori. Moreover,
tightness of γ(P ) can be easily checked a posteriori by exploiting the fact that
γ(P ) = γ(P ) ⇐⇒ ∃x ∈ Rn : x{m} ∈ ker H(P, γ(P ), α̂)
where α̂ is the optimizing vector in (18), i.e. the vector α for which the supremum
in (17) is achieved (see also Theorem 2 in [7]). In this respect, it has been experienced that the lower bound is almost always tight, except for ad hoc examples [7].
The proposed procedure for the computation of the LEDA is summarized in Table
2.
7
Table 2: Procedure for computing the LEDA
0. Let P be given.
1. For each c > 0:
− Form the CSMR matrix H(P, c, α) of the homogeneous form h(x; P, c) in (16);
− Run an LMI feasibility test to check condition (18).
2. Perform a bisection search on c, to find γ(P ) in (17).
s
[γ(P )]n
.
3. Set V(P, γ(P )) as the sought LEDA, and δ(P ) =
det(P )
3.2
A special case: homogeneous nonlinearity
If the nonlinear term in system (1) is a homogeneous form, then the computation of
the lower bound γ(P ) can be simplified. Let system (1) be described as
ẋ = Ax + f2m−1 (x)
(21)
where f2m−1 (x) is a homogeneous form of degree 2m − 1. Let us define w(x; P ) as
in (15). Then, the homogeneous form h(x; P, c) in (16) boils down to
h(x; P, c) = c1−m h1 (x; P ) + h2 (x; P ),
where
h1 (x; P ) = −2x′ P Ax(x′ P x)m−1 ,
h2 (x; P ) = −2x′ P f2m−1 (x).
Let us introduce the SMR of h1 (x; P ) and h2 (x; P )
′
hi (x; P ) = x{m} Hi (P )x{m} ∀x ∈ Rn , i = 1, 2.
Then, the CSMR matrix of h(x; P, c) is given by
H(P, c, α) = c1−m H1 (P ) + H2 (P ) + L(α).
Proposition 2 Let H̄1 (P ) ∈ Rd×d be such that H1 (P ) = H̄1′ (P )H̄1 (P ), and define
E(P, α) = [H̄1′ (P )]−1 [H2 (P ) + L(α)] [H̄1 (P )]−1 .
(22)
Then, the lower bound γ(P ) in (17) is given by
γ(P ) =
(
1
[b(P )] 1−m
if b(P ) > 0,
∞
otherwise
8
(23)
where
b(P ) = −
max
t∈R,α∈RdL
s.t.
t
(24)
E(P, α) − tId > 0.
Proof. First, observe that matrix H1 (P ) can always be selected positive definite.
Indeed, let us rewrite h1 (x; P ) as follows
h1 (x; P ) = −2x′ P Ax(x′ P x)m−1 ,
′
= x[m] (−P A − A′ P ) ⊗ P [m−1] x[m] ,
[m] ′
= x
(25)
J(P )x[m] .
Notice that J(P ) is positive definite since it is the Kronecker’s product of positive
definite matrices. Let K ∈ Rn
m ×d
be the matrix satisfying
x[m] = Kx{m} ,
∀x ∈ Rn .
(26)
Since K has full column rank, being nm ≥ d for all n, m, the matrix H1 (P ) =
K ′ J(P )K is positive definite.
Then, H(P, c, α) can be rewritten as
H(P, c, α) = H̄1′ (P ) c1−m Id + E(P, α) H̄1 (P )
and the results immediately follows.
According to Proposition 2, the computation of the lower bound γ(P ) requires the
solution of the EVP (24), with dL + 1 free parameters. This allows one to avoid the
sweep on the parameter c required in the general case (see eq. (18)).
4
Computation of the optimal QLF via LMI optimization
In this section a strategy for selecting a good starting point to initialize the search for
the OQLF (12) is proposed. The basic idea is to find the matrix P that maximizes
the volume of a fixed shape ellipsoid, which is included in the negative time derivative
region D(P ) in (4). The rationale is that, in order to maximize the size of the LEDA,
a good strategy is to enlarge the set D(P ) which contains the LEDA itself.
4.1
Computation of optimal QLF: generic odd nonlinearity
Let V(U, c) be a given ellipsoid, where U ∈ Rn×n is a fixed symmetric positive
definite matrix. Then, the criterion described above can be formulated as
sup ρ(U, P )
P ∈P
9
(27)
where
ρ(U, P ) = sup {c : V(U, c) ⊆ D(P )} .
The volume of the ellipsoid V(U, ρ(U, P )) is proportional to
(28)
p
[ρ(U, P )]n / det(U );
hence, the optimization problem (27) looks for the largest ellipsoid of fixed shape U
contained in the region D(P ), for some feasible QLF matrix P . Figure 1 illustrates
the rationale behind (27)-(28) for an example with n = 2: three ellipsoidal estimates
of the DA with the same shape U are depicted (dashed ellispoids), corresponding to
three different QLF matrices Pi , i = 1, 2, 3, with the corresponding regions D(Pi ).
6
D(P1 )
D(P2 )
D(P3 )
4
x2
2
0
−2
−4
−6
−6
−4
−2
0
2
4
6
x1
Figure 1: Three ellipsoidal estimates of the DA with the same shape (dashed),
corresponding to different QLF matrices P1 , P2 , P3 , and the corresponding regions
D(P1 ) (solid thin), D(P2 ) (dash-dotted), D(P3 ) (solid thick).
In order to check the inclusion of V(U, c) in D(P ), the same relaxation technique
adopted in Section 3 for the computation of the LEDA can be employed. Define the
homogeneous form of degree 2m
g(x; U, P, c) =
m
X
w2i (x; P )
i=1
x′ U x
c
m−i
.
(29)
It is worth observing that, from (6) and (28), one has ρ(P, P ) = γ(P ). Analogously,
from (16) and (29) it turns out that g(x; P, P, c) = h(x; P, c). The main idea here
is that, being U and P different matrices, the form g is linear in P (while the
form h in (16) is polynomial in P ). Hence, for fixed U , it is possible to pursue the
maximization of ρ(U, P ) in (27) by solving LMI feasibility tests for different values
of the parameter c, as explained next.
10
Let G(U, P, c, α) ∈ Rd×d be the CSMR matrix of g(x; U, P, c) and define the quantity
ρ(U ) = sup
n
o
c̃ : G(U, F (Q), c, α) > 0 for some α ∈ RdL and Q ∈ Q, ∀c ∈ (0, c̃]
(30)
where F (Q) and Q are given by (11) and (13), respectively.
Theorem 2 Let ρ(U, P ) and ρ(U ) be given by (28) and (30), respectively. Then,
ρ(U ) ≤ sup ρ(U, P ).
P ∈P
Proof. Following the same reasoning of the proof of Theorem 1, one has
ρ(U, P ) = sup {c̃ > 0 : w(x; P ) > 0 ∀x ∈ ∂V(U, c), ∀c ∈ (0, c̃]} .
(31)
Then, Lemma 3 in [7] guarantees that for any c ∈ (0, +∞) we have that
w(x; P ) > 0 ∀x ∈ ∂V(U, c) ⇐⇒ g(x; U, P, c) > 0 ∀x ∈ Rn0 .
(32)
From (31) and (32), one gets
ρ(U, P ) = sup {c̃ > 0 : g(x; U, P, c) > 0 ∀x ∈ Rn0 , ∀c ∈ (0, c̃]} .
Clearly, a sufficient condition for g(x; U, P, c) > 0 is that the CSMR matrix G(U, P, c, α)
is positive definite, for some α ∈ RdL . Now, let P = F (Q) and let Q vary in the set
of symmetric positive definite matrices Q. If G(U, F (Q), c, α) is positive definite for
some α ∈ RdL and for some Q ∈ Q, then c ≤ supP ∈P ρ(U, P ), and the result follows
by maximizing over c.
Now, let Q̂ be the matrix Q for which the supremum in (30) is achieved. According
to the above reasoning, Q̂ is the starting point we choose for the OQLF search (12),
and P̂ = F (Q̂) the corresponding QLF matrix. Observe that Q̂ can be computed
via a one-parameter sequence of LMI feasibility tests in dµ = dL + (n2 + n − 2)/2
parameters (being (n2 + n − 2)/2 the degrees of freedom in the set Q).
In order to improve the choice of the starting point of the OQLF search, one can
iterate the procedure proposed above by setting U at each iteration equal to the
matrix P̂ obtained at the previous iteration. In this way, one obtains a sequence
P̂ (i) of feasible QLF matrices, for which an inner approximation of the DA is given
by V(P̂ (i−1) , γ(P̂ (i−1) )). Examples of the application of this iterative procedure will
be provided in Section 5. The overall procedure is summarized in Table 3.
11
Table 3: Iterative U -P procedure
0. Let U = P̂ (0) be given. Set i = 1.
1. For each c > 0:
− Form the CSMR matrix G(U, F (Q), c, α) of the homogeneous form g(x; U, P, c)
in (29);
− Run an LMI feasibility test to check if ∃α ∈ RdL and Q ∈ Q such that
G(U, F (Q), c, α) > 0.
2. Perform a bisection search on c, to find ρ(U ) in (30).
3. Let Q̂ such that G(U, F (Q̂), ρ(U ), α) ≥ 0 and set P̂ (i) = F (Q̂).
4. Set U = P̂ (i) and i = i + 1.
5. If the stopping criterion is not satisfied go to 1. Otherwise, let i∗ = arg max γ(P̂ (i) )
∗
i
∗
and return V(P̂ (i ) , γ(P̂ (i ) )) as the best inner approximation of the DA obtained so
far.
4.2
Computation of optimal QLF: homogeneous nonlinearity
In the case of homogeneous nonlinearity, treated in Section 3.2, the computation
of the starting point Q̂ can be simplified. Following a reasoning similar to that
in Section 3.2 for the computation of the LEDA, it will be shown that Q̂ can be
obtained by solving one single Generalized EVP (see [11]).
Consider again system (21). The homogeneous form g(x; U, P, c) in (29) boils down
to
g(x; U, P, c) = c1−m g1 (x; U, P ) + g2 (x; P ),
where
g1 (x; U, P ) = −2x′ P Ax(x′ U x)m−1 ,
g2 (x; P ) = −2x′ P f2m−1 (x).
Let us introduce the SMR of g1 (x; U, P ) and g2 (x; P ):
′
g1 (x; U, P ) = x{m} G1 (U, P )x{m} ∀x ∈ Rn ,
′
g2 (x; P ) = x{m} G2 (P )x{m} ∀x ∈ Rn .
Then, the CSMR matrix of g(x; U, P, c) is given by
G(U, P, c, α) = c1−m G1 (U, P ) + G2 (P ) + L(α).
Proposition 3 The lower bound ρ(U ) in (30) is given by
(
1
a(U ) 1−m
if a(U ) < 0,
ρ(U ) =
∞
otherwise
12
(33)
(34)
where
a(U ) =
min
d
Q∈Rn×n
,t∈R,α∈R L
s.t.
t


 tG1 (U, F (Q)) > −G2 (P ) − L(α)
G1 (U, F (Q)) > 0


 Q∈Q
(35)
Proof. It follows immediately by solving problem (30), with G(U, P, c, α) given by
(33) and P = F (Q).
The QLF matrix returned by Proposition 3 is given by P̂ = F (Q̂), where Q̂ is the
optimizing matrix Q of problem (35). Observe that (35) is a GEVP with dµ + 1 free
parameters. Regarding the set of constraints in (35), it is worth noting that it is
always possible to find a positive definite G1 (U, F (Q)) for any Q ∈ Q. Specifically,
such G1 (U, F (Q)) is given by
G1 (U, F (Q)) = K ′ Q ⊗ U [m−1] K
where K is defined as in (26).
5
Examples
In this section some numerical examples are presented to illustrate the proposed
technique. The
(
ẋ1 =
(S1)
ẋ2 =
(
ẋ1 =
(S2)
ẋ2 =
(
ẋ1 =
(S3)
ẋ2 =
(
ẋ1 =
(S4)
ẋ2 =



 ẋ1 =
(S5)
ẋ2 =


 ẋ =
3



 ẋ1 =
(S6)
ẋ2 =


 ẋ =
3
following systems have been considered.
x2 ,
−2x1 − 3x2 + x21 x2
−5.2x1 − 0.1x2 − 0.5x31 + 1.9x21 x2 − x1 x22 − 1.7x32 ,
2x1 − x2 + 0.4x31 − 0.3x21 x2 + 2.9x1 x22 + 1.6x32
−1.3x1 − 1.4x2 − x51 − 1.8x41 x2 + 1.7x31 x22 + 3.2x21 x32 − 0.4x1 x42 + 0.9x52 ,
2x1 − 0.8x2 − 4x51 + 3.5x41 x2 − 2.8x31 x22 − 2.2x21 x32 − 0.1x1 x42 − 0.8x52
x2 ,
−2x1 − x2 + x1 x22 − x51 + x1 x42 + x52
x2 ,
x3 ,
−4x1 − 3x2 − 3x3 + x21 x2 + x21 x3
−x1 + x3 + x31 + x21 x2 + x21 x3 ,
−2x2 + x3 ,
x1 − 2x3
13
For each system, the OQLF matrix P ∗ is estimated by solving (12) via the routine
fminsearch of Matlab; the function δ(P ) in (9) is evaluated using the procedure
outlined in Table 2. Since the obtained result depends on the matrix Pinit used to
initialize the search, we denote the computed OQLF matrix as P ∗ |Pinit .
In order to pick a good starting point Pinit for the OQLF search, the U -P iteration
summarized in Table 3 has been adopted. We denote by P̂ (i) the matrix computed
at the i-th step of the U -P iteration, being P̂ (0) = F (In ). The iteration has been
stopped after 5 steps. In Table 4, the volume of the LEDA corresponding to matrix
P ∗ |F (In ) is compared to that of P ∗ |P̂ ⋆ , where P̂ ⋆ is the matrix P̂ (i) that achieves
the largest volume δ(P̂ (i) ), for 1 ≤ i ≤ 5.
system
P̂ ⋆
δ (P ∗ |F (In ))
δ(P ∗ |P̂ ⋆ )
(S1)
P̂ (3)
10.21
10.21
(S2)
P̂ (4)
13.38
27.10
(S3)
P̂ (3)
1.76
9.51
(S4)
P̂ (5)
0.63
0.85
(S5)
P̂ (5)
13.77
23.47
(S6)
P̂ (4)
7.48
10.90
Table 4: Comparison between volumes of the LEDA for P ∗ |F (In ) and P ∗ |P̂ ⋆ .
Table 4 shows that for all the considered systems except (S1), the search initialized in
F (In ) ends up in a local maximum. In several cases, the size of the LEDA obtained
by initializing the OQLF search in P̂ ⋆ is significantly larger than the one obtained
by starting the search in F (In ): this motivates the heuristic procedure used to select
the starting point of the OQLF search. Figure 2 shows the LEDA relative to the
OQLF P ∗ |P̂ ⋆ (dashed ellipses), for systems (S1)-(S4). The solid lines represent the
boundary of the negative derivative region D(P ∗ |P̂ ⋆ ). It can be observed that the
computed OQLF always corresponds to the case in which the ellipsoid of the LEDA
has 4 tangency points to the region D(P ∗ |P̂ ⋆ ).
For the systems in the form (21), i.e. all except (S4), the size γ(P ) of the LEDA
has been approximated through the lower bound γ(P ), computed via one EVP as in
(23)-(24). This EVP involves only dL + 1 parameters, which means that the LEDA
evaluation requires a very low computational burden. For the system (S4), the lower
bound γ(P ) has been computed via a one-parameter sequence of LMI feasibility tests
with dL parameters according to (17)–(18).
If the computational burden is an issue, an appealing alternative to solving prob-
14
8
6
10
4
5
2
0
x
x
2
2
0
−2
−5
−4
−10
−6
−8
−3
−2
−1
0
x1
1
2
3
−8
−6
−4
−2
(a) S1
0
x1
2
4
6
8
(b) S2
5
4
1
3
2
0.5
2
0
x
x
2
1
0
−1
−2
−0.5
−3
−4
−1
−5
−4
−3
−2
−1
0
x
1
2
3
−1
4
−0.5
0
x
0.5
1
1
1
(c) S3
(d) S4
Figure 2: Systems (S1)-(S4): LEDA provided by P ∗ |P̂ ⋆ (dashed ellipses) and boundaries of D(P ∗ |P̂ ⋆ ).
lem (12) via a general purpose optimization tool, is to employ the U -P iteration.
Consider for example system (S5). In order to compute P ∗ |P̂ ⋆ via fminsearch, 559
evaluations of the function δ(P ) have been performed. For system (S5), each of
these evaluations requires the solution of an EVP with dL + 1 = 7 free parameters.
From Table 4, the size of the obtained LEDA is δ(P ∗ |P̂ ⋆ ) = 23.47. Otherwise, one
can iterate the U -P procedure of Table 3, for which each step has approximately the
same computational burden as one computation of δ(P ). Indeed, for a system like
(S5) it requires the solution of a GEVP with dµ + 1 = 12 free parameters. After 5
iterations one gets δ(P̂ (5) ) = 21.25, while after 15 iterations δ(P̂ (15) ) = 22.98: these
are good approximations of the size of the LEDA corresponding to the estimated
OQLF P ∗ |P̂ ⋆ .
15
Based on the above discussion, it can be concluded that the U -P iteration in Table
3, although it is a heuristic strategy for which there is no guarantee of convergence,
can be successfully employed either as a “cheap” algorithm for the approximation of
the DA, or as a way to compute a starting point for a more accurate OQLF search.
6
Conclusions
The problem of computing the OQLF for odd nonlinear systems has been considered.
This is a double non-convex optimization problem which requires at each step the
computation of the LEDA. The OQLF is the one providing the LEDA of maximum
volume.
A relaxation approach based on the CMSR of homogeneous forms has been pursued
to deal with this problem. A lower bound of the LEDA is provided via a procedure
based on LMIs. Tightness of this lower bound can be guaranteed a priori for some
system dimensions and it can be always checked a posterori via a simple test. In
this respect, it is worth mentioning that extensive simulations have shown that the
non-tight cases occur rarely. Also, a criterion for selecting a good starting point for
the OQLF search is given in term of LMI feasibility tests.
Extensive simulation results have shown the effectiveness of the proposed approach
to compute the OQLF. In particular, the computational burden required for the
LEDA turns out to be low, especially if compared with existing methods. Also, the
criterion provides quite satisfactory candidates for the OQLF, preventing in several
cases the search procedure to stop in local maxima.
References
[1] H. Khalil, Nonlinear Systems. New York: McMillan Publishing Company, 1992.
[2] R. Genesio, M. Tartaglia, and A. Vicino, “On the estimation of asymptotic stability regions: State of the art and new proposals,” IEEE Trans. on Automatic
Control, vol. 30, pp. 747–755, 1985.
[3] E. Davison and E. Kurak, “A computational method for determining quadratic
Lyapunov functions for nonlinear systems,” Automatica, vol. 7, pp. 627–636,
1971.
[4] A. Michel, N. Sarabudla, and R. Miller, “Stability analysis of complex dynamical systems. some computational methods,” Circ. Syst. Signal Processing, vol. 1,
pp. 561–573, 1982.
16
[5] G. Chesi, R. Genesio, and A. Tesi, “Optimal ellipsoidal stability domain estimates for odd polynomial systems,” in Proc. of 36-th IEEE CDC, (San Diego),
pp. 3528–3529, December 1997.
[6] G. Chesi, New convexification techniques and their applications to the analysis
of dynamic systems and vision problems. Università di Bologna: Ph.D. Thesis,
2001.
[7] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “Solving quadratic distance problems: an LMI-based approach,” IEEE Trans. on Automatic Control, vol. 48,
pp. 200-212, 2003.
[8] R. W. Brockett. Lie algebra and Lie groups in control theory. In D.Q. Mayne
and R.W. Brockett, editors, Geometric Methods in Systems Theory, pages 43–
82. Dordrecht, Reidel, 1973.
[9] N. K. Bose. Applied Multidimensional Systems Theory. Van Rostrand Reinhold,
New York, 1982.
[10] P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry
Methods in Robustness and Optimization. California Institute of Technology:
Ph.D. Thesis, 2000.
[11] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia: SIAM, 1994.
[12] G. Hardy, J. Littlewood, and G. Pólya, Inequalities: Second edition. Cambridge:
Cambridge University Press, 1988.
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