2012 IEEE International Conference on Control Applications (CCA) Part of 2012 IEEE Multi-Conference on Systems and Control October 3-5, 2012. Dubrovnik, Croatia
A Nagumo Type Result for Linear Dynamical Systems Octavian Pastravanu, Member, IEEE, and Mihaela-Hanako Matcovschi*, Member, IEEE
linear systems obtained in Section II in comparison with some existing results on set invariance, proved by different strategies. Section III also shows that, for time-invariant systems and some particular cases of sets, Nagumo’s condition is equivalent to well-known algebraic properties satisfied by the system matrix. Section IV illustrates our theoretical developments by relevant examples. Section V comments on the possible exploitation of the key ideas presented by this paper in the advanced training programs offered to Control Engineering students.
Abstract— Nagumo’s condition, also known as the subtangency condition, characterizes the set invariance with respect to the trajectories of dynamical systems. The general formulation of Nagumo’s condition considers a comprehensive framework, referring to time-variant nonlinear systems and generic sets with arbitrary shapes. For time-variant linear dynamical systems and sets defined by arbitrary norms, we prove that Nagumo’s condition has an equivalent form expressing the nonpositiveness of the system-matrix measure. This equivalent form is more attractive for application than the general formulation, since it points out the role of the timedependent model coefficients. We also show that, for timeinvariant linear systems and some particular cases of sets defined by weighted p-norms, p ∈ {1, 2, ∞} , Nagumo’s
In the remainder of this section we introduce the notations requested by our development and we summarize the key elements of Nagumo’s condition.
condition is equivalent with well-known algebraic properties (Lyapunov inequality, location of Gershgorin’s disks) satisfied by the constant system-matrix. Finally, we evaluate the key ideas of the paper as a potential resource for developing educational materials dedicated to advanced studies in Control Engineering.
A. Notations Throughout the paper we use the following notations: • For a vector x ∈ ℝ n , || x || is an arbitrary vector norm,
I. INTRODUCTION
continuous on ℝ n . The distance between two vectors x, z ∈ ℝ n is given by dist( x, z) = || x − z || .
The nomenclature “Nagumo’s condition” appeared as a consequence of the impact created by the researches of the Japanese mathematician Nagumo developed at the middle of the twentieth century. Today, this terminology is equally used by Mathematics and Engineering works dealing with dynamics / motions described by differential systems on compact sets. The initial formulation of Nagumo’s condition [1] considered a general framework, referring to nonlinear systems and generic sets with arbitrary shapes. The introductory section briefly summarizes this general framework, which includes the connection between Nagumo’s condition and the analysis of invariant sets.
• For a matrix M ∈ ℝ n×n , || M || = sup || Mx || is the || x|| =1
matrix norm induced by the vector norm
µ || || (M ) = lim θ1 [|| I + θ M || −1] θ ↓0
is a matrix measure
based on the matrix norm || i || ([2], pp. 30). For the particular Hölder p-norms corresponding to p ∈ {1, 2, ∞} , the induced matrix measures are given by
{
}
µ || ||1(M ) = max j =1, n m jj + ∑i =1, i ≠ j | mij | , n
µ || ||2 (M ) = max i =1, n {12 λi (M + M T )} ,
The main contribution of the current paper is presented in Section II. It is shown that, for linear time-variant dynamical systems and sets defined by arbitrary norms, Nagumo’s condition has an equivalent form expressing the nonpositiveness of the system-matrix measure. This equivalent form is more attractive for application than the initial formulation since it points out the role of the model coefficients. Section III discusses Nagumo's condition for
{
}
µ || ||∞ (M ) = max i =1, n mii + ∑ j =1, j ≠i | mij | . n
B. Brief overview of Nagumo’s condition Consider the dynamical system in continuous time yɺ (t ) = f (t, y(t )) , y(t0 ) = y0 ∈ D , t0 ∈ I ,
Manuscript received June 20, 2012. This work was supported by UEFISCDI Romania, Grant PN-II-ID-PCE-2011-3-1038. The authors are with the Department of Automatic Control and Applied Informatics of the Technical University “Gheorghe Asachi” of Iasi, Blvd. Mangeron 27, Iasi, RO-700050, Romania (e-mail:
[email protected],
[email protected]). * Corresponding author.
978-1-4673-4505-7/12/$31.00 ©2012 IEEE
|| i || ;
(1)
where I ⊆ ℝ + is an open interval, D ⊆ ℝ n is a domain (nonempty and open set), and f : I × D → ℝ n is a continuous function that, moreover, satisfies the conditions that ensure the existence and uniqueness of the solutions to 849
II. MAIN RESULT
the Cauchy problem (1) – see Exercise 4.1.1 in [3]. Let K ⊆ D be a closed and convex set. Denote by Int{K} the interior of K and by ∂K = K \ Int{K} its
Consider the linear time-variant system yɺ (t ) = C (t ) y(t ) , y(t0 ) = y0 ∈ ℝ n , t, t0 ∈ ℝ + , t ≥ t0 , (4)
boundary. For x ∈ ℝ n , define the distance from x to K by
where C (t ) = cij (t ) denotes a real n × n matrix-valued
dist( x, K) = inf dist( x, z) . z∈K
function, continuous for all t ≥ 0 . In the state space ℝ n consider the norm || || and define the closed convex set
Definition 1. (see Definition 4.1.1. in [3] and Definition 4.1 in [4]) The set K is invariant with respect to system (1) if for each (t0 , ξ ) ∈ I × K , the solution y(t ) :[t0 , T ] → D , T ∈ I , T > t0 , of system (1) corresponding to the initial
{
condition y(t0 ) = ξ , satisfies y(t ) ∈ K for each t ∈ [t 0 , T ] . The solution y(t ) is said to be viable in K on [t0 , T ] .
□
statement f (t, ξ ) ∈ T K (ξ ) by lim θ1 dist(ξ + θ C (t )ξ , K) = 0 , θ ↓0 as per Definition 2. Thus, Nagumo’s condition for system (4) and set (5) is expressed by:
Bouligand-Severi to the set K at the point ξ ∈ K if
θ ↓0
θ
dist(ξ + θη , K) = 0 .
□ (2)
∀t ∈ ℝ + , ∀ξ ∈ ℝ n , || ξ ||= 1: lim θ ↓0
Proposition 1. (see Proposition 2.4.1 in [3]). The set T K (ξ ) of all vectors η ∈ ℝ n which are tangent in the sense of Bouligand-Severi to the set K at the point ξ ∈ K is a closed cone. □ Remark 1. The set T K (ξ ) is called the contingent cone to
1
θ
dist(ξ + θ C (t )ξ , K) = 0 .
continuous function of z ∈ K that reaches its minimum value on K (since K is closed). By using the notation z (θ ) = arg inf || ξ + θ C (t )ξ − z || we can write z∈K
If ξ ∈ ℝ n \ K , then T K (ξ ) = ∅ . Thus, the contingent cone
dist(ξ + θ C (t )ξ , K) = || ξ + θ C (t )ξ − z (θ ) || .
T K (ξ ) is nontrivial only when ξ ∈ ∂K , i.e. on the boundary □
The following theorem characterizes, by equivalence, the invariance of the set K with respect to system (1). Theorem 1 (see Theorem 4.1.1 in [3] and Theorem 4.7, Corollary 4.8 in [4]). The set K is invariant with respect to system (1) if and only if ∀(t, ξ ) ∈ I × ∂K : f (t, ξ ) ∈ T K (ξ ) ,
(7)
We shall prove the equivalence (7) ⇔ (6). Proof for (7) ⇒ (6): Take arbitrary t ≥ 0 and ξ ∈ ∂K , i.e. || ξ || = 1 . For each θ > 0 , || (ξ + θ C (t )ξ ) − z || is a
the set K at the point ξ . If ξ ∈ Int{K} , then T K (ξ ) = ℝ n .
of K .
(6)
Proof: In the general form of Nagumo’s condition (3) we consider the vector function f (t, ξ ) = C (t )ξ and replace
Definition 2. (see Definition 2.4.1 in [3] and Definition 4.6 in [4]). The vector η ∈ ℝ n is tangent in the sense of
1
(5)
Theorem 2. Nagumo’s condition for system (4) and set (5) is equivalent to the inequality:
∀t ∈ ℝ + : µ || ||(C (t )) ≤ 0 .
In order to characterise the invariance of the set K with respect to the dynamical system (1) we need the following prerequisites.
lim
}
K = y ∈ ℝ n || y || ≤ 1 .
Since z (θ ) ∈ K ⇔ || z (θ ) || ≤ 1 , this yields the majorization || ξ + θ C (t )ξ || ≤ || ξ + θ C (t )ξ − z (θ ) || + || z (θ ) || ≤ dist(ξ + θ C (t )ξ , K) + 1 and, consequently, the inequality 1 1 [ || (I + θ C (t ))ξ || −1 ] ≤ dist(ξ + θ C (t )ξ , K) .
θ
θ
1
Since lim dist(ξ + θ C (t )ξ , K) = 0 , for any ξ ∈ ∂K we get θ ↓0 θ
□ (3)
∀ε > 0, ∃δ ξ (ε ) > 0, 0 < θ ≤ δ ξ (ε ):
1
θ
[|| (I + θ C (t))ξ || − 1] ≤ ε .
By taking δ (ε ) = min δ ξ (ε ) , we obtain
Relation (3) represents Nagumo’s condition, also known as the sub-tangency condition. The idea expressed by it is simple and intuitive: A set K is viable under the mapping f if, at any moment t ∈ I and each state x ∈ K , the velocity f (t, x) is tangent to K at x, thus bringing back a
ξ ∈∂K
∀ε > 0, ∃δ (ε ) > 0, 0 < θ ≤ δ (ε ) :
1
θ
solution to the differential equation (1) inside K [5]. 850
[|| (I + θ C (t))ξ || −1] ≤ ε , ∀ξ ∈ ℝ n , || ξ || = 1 .
⌢ On the other hand, for each θ > 0 , there exists ξ ∈ ℝ n , ⌢ ⌢ || ξ || = 1 , such that || I + θ C(t ) || = || (I + θ C(t))ξ || , meaning ∀ε > 0, ∃δ (ε ) > 0, 0 < θ ≤ δ (ε ) :
1
θ
necessarily the same norm that defines the set K in (5). For instance, if in (6) we use the norm different from || || used in (5), then the proof takes into account the inequality ω || x || ≤ x ≤ Ω || x || , x ∈ ℝ n . This allows the connection
[ ||(I + θ C(t ))|| −1] ≤ ε .
between the distance expressed by
The existence of lim θ1 [ || (I + θ C (t ))|| −1] = µ || || (C (t )) is θ ↓0 guaranteed (e.g. [2], pp.30), and we can conclude that µ|| || (C(t)) ≤ 0 , ∀t ≥ 0 .
Corollary 1. A set K of form (5) is invariant with respect to linear system (4) if and only if inequality (6) holds true. □
lim θ1 dist(ξ ∗ + θ C (t ∗ )ξ ∗, K) = 0 is not true. In other words, θ ↓0
we can find ε ∗ > 0 , θ ∗ > 0 , such that for 0 < θ ≤ θ ∗ we get 1 ε ∗ ≤ dist(ξ ∗ + θ C (t ∗ )ξ ∗, K) .
Remark 3. Paper [6] provides a result that may be regarded as equivalent with Corollary 1 limited to linear time-invariant systems, i.e. for C(t) = C ∈ ℝ n×n (constant matrix) in (4) and (6). Actually, paper [6] does not mention any connection between inequalities of form (6) and Nagumo’s condition, but it proves that inequalities of form (6) represent necessary and sufficient conditions for the existence of norm-based Lyapunov functions (defined based on arbitrary norms in ℝ n ). The existence of invariant sets is treated as a consequence, in the sense that the level sets of Lyapunov functions are invariant with respect to the system dynamics. □
θ
For each θ satisfying 0 < θ ≤ θ ∗ we consider the vector 1 v(θ ) = ∗ (ξ ∗ + θ C (t ∗ )ξ ∗ ) || ξ + θ C (t ∗ )ξ ∗ || that belongs to ∂K , since || v(θ ) || = 1 . Consequently, for
dist(ξ ∗ + θ C(t ∗)ξ ∗, K) we obtain the following majorant dist(ξ ∗ + θ C(t ∗ )ξ ∗, K) ≤ || (ξ ∗ + θ C(t ∗ )ξ ∗ ) − v(θ ) || = 1 (ξ ∗ + θ C (t ∗ )ξ ∗ ) || = . || ξ + θ C (t ∗ )ξ ∗ || ∗ ∗ ∗ | || ξ + θ C (t )ξ || −1 | ∗
Remark 4. The sets K (5) considered in Theorem 2 have a large variety of symmetrical shapes. These include the polyhedral shapes corresponding to Hölder 1-norm and ∞norm, and the ellipsoidal shapes corresponding to 2-norm. If || y || is defined as a weighted p-norm || y || = || Gy || p ,
We discuss the sign of the expression || ξ ∗ + θ C(t ∗ )ξ ∗ || −1 . (i) If || ξ ∗ + θ C(t ∗ )ξ ∗ || ≤ 1 , then ξ ∗ + θ C(t ∗)ξ ∗ ∈ K and, therefore, dist(ξ ∗ + θ C(t ∗ )ξ ∗, K) = 0 . We rule out this case, since we assumed that lim θ1 dist(ξ ∗ + θ C (t ∗ )ξ ∗, K) = 0 is θ ↓0 not true. (ii) If || ξ ∗ + θ C(t ∗ )ξ ∗ || > 1 , then dist(ξ ∗ + θ C(t ∗ )ξ ∗, K) ≤
|| ξ ∗ + θ C(t ∗ )ξ ∗ || −1 ≤ || I + θ C(t ∗ ) || −1 , Finally we get
since
with G ∈ ℝ n×n , det G ≠ 0 , for 1 ≤ p ≤ ∞ , the concrete form of K and the length of its symmetry axes depend on the nonsingular weight matrix G ∈ ℝ n×n . In this case, the
|| ξ ∗ ||= 1 .
∀θ , 0 < θ ≤ θ ∗ : 1 1 ε ∗ ≤ dist(ξ ∗ + θ C(t ∗ )ξ ∗, K) ≤ || I + θ C (t ∗ ) || −1 θ θ
induced
matrix
norm
is
|| M || = || GMG −1 || p ,
corresponding matrix measure is µ || || (M ) = µ || || p (GMG −1) , where µ || || p ( i )
the
given by denotes the
matrix measure corresponding to the Hölder p-norm || i || p . □ .
Now, for time-variant systems of form (4), from Theorem 2, we can derive the following result referring to sets K (5) defined by weighted p-norms.
The existence of lim θ1 || (I + θ C (t ∗ )) || −1 = µ || || (C (t )) is θ ↓0 guaranteed (e.g. [2], pp.30), and we can conclude that µ || || (C (t ∗ )) ≥ ε ∗ > 0 . Thus, we contradict inequality (6) and complete the proof.
□
Relying on Theorem 1, we can derive the following corollary of Theorem 2
Proof for (6) ⇒ (7): Assume condition (7) does not hold, ∃ t ∗ ∈ ℝ + , ∃ξ ∗ ∈ ℝ n , || ξ ∗ || = 1 , for which i.e.
= || (ξ ∗ + θ C (t ∗ )ξ ∗ ) −
and all other
constructions that require || || .
Corollary 2. Let 1 ≤ p ≤ ∞ and G ∈ ℝ n×n , det G ≠ 0 . A
□
set
{
}
K = y ∈ ℝ n || Gy || p ≤ 1
III. DISCUSSIONS
(8)
is invariant with respect to the linear time-variant system (4) if and only if the following inequality holds true
Remark 2. The proof of Theorem 2 shows that we can use any relevant norm for expressing the distance of a point from a set. In other words, the norm considered in (6) is not 851
∀t ∈ ℝ + : µ || || p (GC (t )G −1) ≤ 0 .
□ (9)
Remark 5. Paper [7] studies the invariance of timedepending sets defined by weighted Hölder p-norms with respect to linear time-variant systems (4). Corollary 2 represents the equivalence between points (i) and (iii) of Theorem 1 from [7] in the particular case when r = 0 and H (t ) ≡ G , i.e. for constant sets of form (8). Paper [7] does not mention any connection between inequalities based on matrix-measures and Nagumo’s condition, but, generalising paper [6], it proves that inequalities of form (9) represent necessary and sufficient conditions for the existence of norm-based Lyapunov functions. The existence of invariant sets is treated as a consequence, in the sense that the level sets of Lyapunov functions are invariant with respect to the system dynamics. Moreover, Corollary 2 from [7] proves that inequality (9) represents a sufficient conditions for the uniform stability of the equilibrium point {0} of (4). □
⊂ ℂ −0 , j = 1,..., n .
(iii) Let the set K in (8) be defined by the weighted norm || y || = || Gy ||∞ , with G = diag{g1,… , g n} ≻ 0 (positive definite diagonal matrix) and consider the generalized Gershgorin's disks of C, defined for rows by: D ir (GCG −1) = z ∈ ℂ
| z − cii | ≤
gi | cij | , j =1, j ≠ i g j n
∑
for i = 1,..., n . Then Nagumo’s condition is equivalent to n
∪ D ir (GCG −1)
⊂ ℂ −0 ,
i =1
n
gi | cij | , i = 1,..., n g j =1, j ≠ i j
∑
yɺ (t ) = C y(t ) , y(t0 ) = y0 ∈ ℝ n , t, t0 ∈ ℝ + , t ≥ t0 , (10)
−2 1 with C = . 6 −3 (i) Let G = diag{1,
PC + C T P 0
{
1} 6
and define the set
}
Ki = y ∈ ℝ 2 || Gy ||2 ≤ 1 ,
with
T
P = G G , P ≻ 0. This shows that the well-known Lyapunov inequality PC + C T P 0 represents a particular case of Nagumo’s condition (for a time-invariant linear system and a set with ellipsoidal shape). (ii) Let the set K in (8) be defined by the weighted norm || y || = || Gy ||1 , with G = diag{g1,… , g n} ≻ 0 (positive definite diagonal matrix) and consider the generalized Gershgorin's disks of C, defined for columns with by:
D cj (GCG −1) = z ∈ ℂ
| z − c jj | ≤
□
Consider a linear time-invariant system
⇔ the symmetrical matrix GCG −1 + (G −1)T C T GT has nonpositive eigenvalues −1 ⇔ GCG + (G −1)T C T G T 0 (negative semidefinite) ⇔
i =1,i ≠ j
gi | cij | , j = 1,..., n gj
IV. ILLUSTRATIVE EXAMPLES
}
G T GC + C T G T G 0
n
∑
⇔ D cj (GCG −1) ⊂ ℂ −0 , j = 1,..., n .
then for inequality (9) we can write the following equivalent forms: µ || ||2 (GCG −1) = max 12 λi (GCG −1 + (G −1)T C T GT ) ≤ 0
⇔
⇔
D cj (GCG −1)
0 ≥ µ || ||∞ (GCG −1) ⇔ 0 ≥ cii +
|| y || = || Gy ||2 , G ∈ ℝ n×n , det G ≠ 0 ,
i
0 ≥ µ || ||1(GCG −1) ⇔ 0 ≥ c jj +
since
Remark 6. Consider a linear time-invariant system, i.e. C (t) = C ∈ ℝ n×n in (4), and apply Corollary 2 for the constant matrix C and the following particular norms in (8) and (9). (i) If the set K in (8) is defined by the weighted Euclidean norm
{
where ℂ −0 = {z ∈ ℂ Re z ≤ 0 } . The equivalence results from
which is the ellipse represented in Figure 1.
gi | cij | , i =1, i ≠ j g j n
∑
for j = 1,..., n . Then Nagumo’s condition is equivalent to n
∪ D cj (GCG −1) ⊂ ℂ −0 ,
Fig. 1. Graphical representation of the set Ki (11).
j =1
852
(11)
As discussed in Remark 6 (i), Nagumo's condition for the invariance of the set Ki (11) with respect to system (10), i.e.
{
∀ξ ∈ ∂Ki : Cξ ∈ T Ki (ξ ) , is equivalent to PC + C T P 0 ,
where P = G G = diag{1, 16} . (ii) Let G = diag{1, 13} and define the set
{
}
(13)
which is the rectangle represented in Figure 4. As presented in Remark 6 (iii), Nagumo's condition for the invariance of the set Kiii (13) with respect to system (10), i.e. ∀ξ ∈ ∂Kiii : Cξ ∈ T Kiii (ξ ) , is equivalent to
T
Kii = y ∈ ℝ 2 || Gy ||1 ≤ 1 ,
}
Kiii = y ∈ ℝ 2 || Gy ||∞ ≤ 1 .
(12)
D1r (GCG −1) ∪ D2r (GCG −1) ⊂ ℂ −0 ,
where the generalized Gershgorin's disks of C defined for rows are given by D1r (GCG −1) = {z ∈ ℂ | z + 2 | ≤ 2} and
which is the rhomb represented in Figure 2. Based on Remark 6 (ii), Nagumo's condition for the invariance of the set Kii (12) with respect to system (10), i.e. ∀ξ ∈ ∂Kii : Cξ ∈ T Kii (ξ ) , is equivalent to
D2r (GCG −1) = {z ∈ ℂ | z + 3 | ≤ 3} . Note that these disks coincide with the generalized Gershgorin's disks of C defined for columns in Example (ii).
D1c (GCG −1) ∪ D2c (GCG −1) ⊂ ℂ −0 , where the generalized Gershgorin's disks of C defined for columns are given by D1c (GCG −1) = {z ∈ ℂ | z + 2 | ≤ 2}
D2c (GCG −1) = {z ∈ ℂ and represented in Figure 3.
| z + 3 | ≤ 3} ,
and
are
Fig. 4. Graphical representation of the set Kiii (13). V. POTENTIAL SOURCE FOR TRAINING MATERIALS The key ideas of our paper can be used to develop teaching / training materials for Master and / or PhD students specializing in Control Engineering. Theorem 1 and its proof offer an effective and detailed model for studying the connections between the general approach to set invariance in differential equation theory and the algebraic forms of the invariance conditions for linear dynamical systems. Within this context students learn how to handle the concept of “distance” and the vector- or matrix-normbased instruments. The particular cases discussed by Remark 6 play an important role in the profound understanding of the subject because, generally speaking, the Gershgorin's disks and the Lyapunov inequality are familiar topics to Control Engineering students. The two-dimensional examples considered in Section IV ensure relevant “samples” for the generality and, concomitantly, applicability of Nagumo’s condition. The visual support facilitates the illustrative goal of these examples. At the curricula /syllabus level, a material based on the above issues may ensure interesting developments or complements to various course modules addressing system analysis problems.
Fig. 2. Graphical representation of the set Kii (12).
Fig. 3. Graphical representation of the generalized Gershgorin's disks of C defined for columns in Example (ii). (iii) Let G = diag{1, 12} and define the set
853
VI. CONCLUSIONS The paper proves the equivalence between the general formulation of Nagumo’s condition (3) and the algebraic condition (6) expressing the nonpositiveness of the systemmatrix measure. Condition (6) is more attractive for application than the general formulation, since it points out the role of the time-dependent entries of the matrix-valued function C (t ) . The sets K (5) considered by Theorem 2 have a large variety of symmetrical shapes. These include the polyhedral shapes corresponding to p-Holder norm with p = 1 and p = ∞ , and the ellipsoidal shapes corresponding to p = 2 . For these particular cases, the concrete definition of
|| y ||
is based on the weighted p-Holder norm
|| y || = || Gy || p , G ∈ ℝ n×n , det G ≠ 0 (with p = 1 , ∞ and 2, respectively) and Theorem 2 is replaced by Corollary 2.. The paper also shows that, for time-invariant systems (i.e. C (t) = C ∈ ℝ n×n ) and weighted p-norms || y || = || Gy || p , G ∈ ℝ n×n , det G ≠ 0 ,, Nagumo’s condition is equivalent with well-known properties of matrix C, such as: • matrix C solves a Lyapunov inequality (for p = 2 ); • the generalized Gershgorin's disks of matrix C (per columns / rows) are located in the left half-plane of ℂ (for p = 1 / p = ∞ ). Our paper presents a generous amount of information on Nagumo’s condition (basic ideas, results, proof techniques discussions, examples) which may be used in the preparation of teaching materials for Control Engineering students interested in flow-invariance-based approaches. REFERENCES 1.
2. 3. 4. 5. 6.
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