Hausdorff Dimension Estimates for Interconnected Systems With Variable Metrics

In this letter, we develop a framework for estimating the Hausdorff dimension of a compact invariant set for both autonomous and interconnected systems. We first generalize Smith’s method for Hausdorff dimension estimates by using variable metrics in linear matrix inequalities. Then, we study open systems with a characterization similar to the differential dissipativity theory. For linear time-invariant systems, we show that our characterization can be considered as a pure input/output property. This fact would be important because it is independent of internal model representations. Finally, we provide an estimate of the attractor dimension for feedback and interconnected systems. Our estimation is scalable in the sense that the components in an interconnected system can be analyzed independently.


Hausdorff Dimension Estimates for
Interconnected Systems With Variable Metrics Rui Kato and Hideaki Ishii , Fellow, IEEE Abstract-In this letter, we develop a framework for estimating the Hausdorff dimension of a compact invariant set for both autonomous and interconnected systems.We first generalize Smith's method for Hausdorff dimension estimates by using variable metrics in linear matrix inequalities.Then, we study open systems with a characterization similar to the differential dissipativity theory.For linear time-invariant systems, we show that our characterization can be considered as a pure input/output property.This fact would be important because it is independent of internal model representations.Finally, we provide an estimate of the attractor dimension for feedback and interconnected systems.Our estimation is scalable in the sense that the components in an interconnected system can be analyzed independently.

I. INTRODUCTION
I N THE control community, internal and external aspects of dynamical systems have been studied for a long time.However, there is a large theoretical gap between research on closed systems (systems without inputs and outputs) and that on open systems (systems with inputs and outputs).The purpose of this letter is to generalize a framework for estimating the Hausdorff dimension of a compact invariant set, which represents the degree of freedom of asymptotic behaviors, to open systems.According to [1], dimension-like characteristics such as Hausdorff dimension, fractal dimension, and topological entropy play essential roles in dimension theory.These characteristics are useful to understand dynamical systems from various perspectives [2].We mention that our approach is inspired by the recent works [3], [4], [5].
As for closed systems, many researchers have made efforts to understand chaotic phenomena, which are difficult to predict but are important in engineering applications [6].Although many criteria for global stability are available, such results are not applicable to systems which have more than two equilibria.Also, periodic oscillations are in general difficult to analyze in spite of their importance.Furthermore, complex attractors may not be manifolds, and thus, dimension with non-integer values becomes important in the analysis [7].In this letter, for these reasons, we are concerned with finding estimates of Hausdorff dimensions of invariant sets.Our study is further motivated by the relation of Hausdorff dimension with entropy [8], which has recently attracted attentions in control theory [9].
In contrast, open systems are studied by input/output properties such as L 2 -gain and passivity [10], [11].A general framework that links internal stability and input/output properties is the celebrated dissipativity theory [12], [13], [14].This energy-related framework is also useful to analyze stability of large-scale network systems in a scalable way [15].However, relations between internal instability (e.g., chaotic dynamics) and input/output properties are not well understood.Recently, some results in this direction were reported based on the framework of [16], [17].In [3], the existence and stability of limit cycles for interconnected systems were analyzed by transverse contraction.In [4], classical frequency-domain analysis was extended to systems with multiple equilibria and limit cycles.In [5], the authors utilized the inertia for quadratic storage functions to analyze dominating dynamics.In this letter, we consider another direction to explore interconnected systems with Hausdorff dimension. 1iterature Reviews and Contributions: We review some related works on dimension estimates briefly; see also the recent book [19].In the seminal paper [20], a fundamental method to estimate the Hausdorff dimension of a compact invariant set was developed.The so-called Douady-Oesterlé theorem has been further studied with various tools.In [21], the author proposed a computationally efficient method to estimate Hausdorff dimension.In [22], [23], analogues of Lyapunov's second method were explored to improve the estimate of the Douady-Oesterlé theorem.In [24], [25], compound matrices and logarithmic norms were used to study Hausdorff dimension.These results are closely related to kcontraction [26], [27], which is a natural generalization of standard contraction.
In this letter, we focus on Smith's method in [21] to make the conditions easy to verify.We first generalize the results in [21] by employing time-and state-dependent metrics.Another generalization can be found in [28], where k-contraction with respect to L 1 /L 2 /L ∞ norms was studied without calculating compound matrices.In contrast to that work, our results allow for variable metrics, which correspond to the Riemannian metric in differential geometry.We then discuss a methodological relation between Smith's method and the differential Lyapunov framework in [17].This motivates us to extend the framework to open systems.We follow the differential dissipativity theory in [5] to characterize open systems.For linear time-invariant systems, we provide a frequencydomain interpretation of unstable dynamics.Finally, we show that the attractor dimension of an interconnected system can be estimated in a scalable way.We emphasize that our framework is not compatible with that of dominance analysis in [5].This is because our conditions depend on lower bounds of Jacobian's spectrum while those in [5] specify the separation of stable and unstable spectra.
Notations: Throughout this letter, we employ the following notations.Consider a matrix be the eigenvalues of (A + A T )/2 arranged in decreasing order.For a real number d ∈ [0, n], we define the truncated determinant of order d (d-determinant) of A by and the truncated trace of order d (d-trace) of A by We denote by Dϕ the derivative of a smooth map ϕ.
denotes the set of nonnegative real numbers, and S n + denotes the set of positive-definite symmetric matrices of size n.
Let K ⊂ R n be a compact set.Given d ∈ [0, n] and ε > 0, we define where the infimum is taken over all finite coverings of K by balls B i with radii r i .The d-dimensional Hausdorff measure of K is defined by We note that the Hausdorff dimension can be well defined and is less than or equal to n for every compact set in R n .

II. DIMENSION ESTIMATES VIA SMITH'S METHOD
In this section, we extend Smith's method on Hausdorff dimension estimates in [21] and discuss its connection with the differential Lyapunov framework in [17].
A. Generalized Liouville's Formula Consider a linear system where A : R + → R n×n is a continuous function.Recall that a fundamental matrix is an n × n matrix whose columns are linearly independent solutions of (1).A generalization of the well-known Liouville's formula was first obtained in [21] and is stated below.Lemma 1: Let X : R + → R n×n be a fundamental matrix of (1).Then, for each k ∈ {1, . . ., n}, we have The k-determinant of X(t) can be interpreted as the kdimensional volume of the parallelepiped spanned by certain k solutions [27].This result plays a fundamental role in dimension theory [1].However, in general, calculating the k-trace of A(t) is computationally hard.The following proposition generalizes [21, Corollary 1.2], which avoids the computation of eigenvalues.
Proposition 1: Let X : R + → R n×n be a fundamental matrix of (1) normalized as X(0) = I.Suppose that there exist a continuous function λ : R + → R and a continuously differentiable function P : R + → S n + such that for all t ≥ 0, Ṗ(t) + A(t) T P(t) + P(t)A(t) + 2λ(t)P(t) 0. ( Then, for each k ∈ {1, . . ., n}, we have . By the coordinate transformation y = Q(t)x, we transform the system (1) into where Then, the matrix inequality (2) can be rewritten as which implies that η i ( Ã(t)) + λ(t) ≥ 0 for all i ∈ {1, . . ., n}.Thus, we have Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Since Y(t) := Q(t)X(t) is a fundamental matrix of ( 4) such that Y(0) = Q(0), Lemma 1 and (5) imply that Notice that tr Ã(t) = tr A(t) + tr Q(t)Q(t) −1 .From Jacobi's formula, we have tr We integrate both sides over time to obtain Therefore, Finally, it follows from Horn's inequality that and hence, Since Q(t) = P(t) 1/2 , we complete the proof.Remark 1: By limiting the metric P(t) to be a constant matrix, Theorem 1 reduces to [21, Corollary 2.1].Here, P(t) specifies the transformation of (1) into (4).However, the obtained estimate of the exponential growth rate of the kdimensional volume is associated with the coefficient matrix in (1) and not in (4).This estimate can be replaced with which may or may not be less conservative than the original one in (3).It is advantageous to use the estimate in (3) since we can avoid matrix calculations.This point is different from the results in [23], and our bound of the exponential growth rate is easy to calculate.The tightness of the obtained estimate depends on the location of the n − k smallest eigenvalues of Ã(t), as can be seen from ( 5).Thus, the result in Proposition 1 can be conservative especially when k is small.

B. Smith's Method With Variable Metrics
Consider a nonlinear autonomous system ẋ = f (x), (6) where f : R n → R n is a continuously differentiable function.
Let J(x) := Df (x) denote the Jacobian matrix of f (x).A fundamental result for dimension estimates is the so-called Douady-Oesterlé theorem [20].This theorem can be stated as follows: For a given smooth mapping ϕ, if there exists a real number d such that det d Dϕ(x) < 1 holds on a compact invariant set, then its Hausdorff dimension is smaller than d.Note that this estimate can also be used as an upper bound of the fractal dimension [29].
Here, we follow the approach in [21] and provide an extension with variable metrics. 2 The obtained estimate of the Hausdorff dimension can be conservative if d is small for the reason explained in Remark 1.In the theorem below, we use the following notation: . . . . . . . . .
where L f is the Lie derivative of a real-valued function with respect to the vector field f .If P(x) is constant, then we recover [21, Corollary 2.2].Theorem 1: Let A ⊂ R n be a compact invariant set of (6).Suppose that there exist a continuous function λ : R n → R and a continuously differentiable function P : R n → S n + such that for all x ∈ A, Ṗ(x) + J(x) T P(x) + P(x)J(x) + 2λ(x)P(x) 0.
If there exists d ∈ (0, n] such that Proof: The proof is similar to that in [21].Consider a fundamental matrix X of the variational equation δẋ = J(x(t))δx associated with a solution x of (6) starting in A. We apply Proposition 1 to obtain for each k ∈ {1, . . ., n}, Thus, the hypothesis in the theorem implies that the exponential growth rate of det d X(t) is negative, and thus, there is a finite time t for which det d X(t) < 1.Since every linearized flow map on A satisfies the condition of the Douady-Oesterlé theorem, we conclude the proof.
Remark 2: It must be noted that Theorem 1 does not guarantee convergence of the system's trajectories to the invariant set.However, many systems in nature are dissipative in the sense of Levinson [30], and hence, every trajectory of such a system converges to a compact set.Thus, the Hausdorff dimension of the largest invariant set can be used as the maximum degree of freedom of asymptotic behaviors.Several consequences of Theorem 1 are as follows: 1) When d = 1, the system cannot possess any compact invariant set except for a single equilibrium point.2) When d = 2, the system cannot possess any limit cycle.This is because if there is a closed orbit, then there must be a surface which is invariant under the flow.3) When d > 2, chaotic attractors may appear like the famous Lorenz system.
where P : R n → S n + is a continuously differentiable function.Also, we consider the quadratic supply rate w : R m × R p → R of the form w(δu, δy) = δy δu where Q ∈ R p×p , S ∈ R p×m , and R ∈ R m×m are constant matrices with Q and R symmetric.The energy-like property in (7) can now be generalized to the following inequality: where λ : R n → R is a continuous function.
In what follows, we show that the result in Theorem 1 can easily be extended to interconnected systems.This is based on the fact that our condition depends on the diagonal part of the Jacobian.Recall that the trace of the sum of matrices is equal to the sum of their traces.Hence, the trace of the Jacobian of an interconnected system can be replaced by the sum of the traces of the Jacobians of its components.

B. Dimension Estimates for Lur'e Systems
Here, we concentrate on the linear time-invariant system where A, B, and C are constant matrices of appropriate sizes.It is clear that the inequality (10) can be converted to where λ and P do not depend on x.We notice that (12) can be rewritten as Remark 3: From the KYP lemma [33], the above LMI is equivalent to the following frequency-domain inequality: We notice that the trace of A is equal to the sum of all poles of G(s) as long as there is no pole/zero cancellation.Thus, the inequality ( 12) can be regarded as a pure input/output property.We point out that input/output characterization of unstable dynamics has received little attention, and it can be of interest to utilize it in the control-theoretic methods such as data-driven verification.Now, we consider the memoryless feedback of the form u = φ(y), (13) where φ : R p → R m is a continuously differentiable function.
The composite system is called the Lur'e system, which is an important class of nonlinear systems.The following result is motivated from [34] and addresses a more general setting.Theorem 2: Consider the linear time-invariant system (11) such that (12) holds.Suppose that the memoryless feedback (13) satisfies where A is an invariant set of the closed-loop system.If there exists d ∈ (0, n] such that for all x ∈ A, The variational equation of the closed-loop system is described by δẋ = [A + BDφ(Cx(t))C] δx.The inequality (12) together with ( 14) reads as δẋ δx Thus, the closed-loop system satisfies the condition in Theorem 1.The proof is complete.
The following example illustrates the usefulness of the trace in the dimension estimate instead of the truncated trace.
Example 1: Consider a 2-dimensional system with a memoryless feedback u = φ(y) of ( 13) such that Because of the system structure, we have tr BDφ(y)C ≡ 0, and thus, the closed-loop attractor dimension can be estimated independent of φ.This example clarifies robustness of our attractor dimension estimate with respect to uncertainty in φ.

C. Dimension Estimates for Interconnected Systems
Finally, we investigate an interconnected system composed of the following N open systems: where f i : R n i → R n i , B i ∈ R n i ×m i , and C i ∈ R p i ×n i are as before for each i ∈ {1, . . ., N}. Assume that each subsystem i satisfies the inequality (10) with the shift parameter λ i (x i ), the storage function V i (x i , δx i ) = δx T i P i (x i )δx i , and the supply rate We define x := col(x 1 , . . ., x N ), u := col(u 1 , . . ., u N ), and y := col(y 1 , . . ., y N ).The interconnection that we consider is of the form u = My, (16) where is a matrix such that the block-diagonal parts associated with the subsystems are zeros.This assumption does not lose any generality since the blockdiagonal parts can be considered as self-feedbacks. Let Notice that tr BMC = 0 holds because of the structure of M, and hence, the trace of the Jacobian is invariant under the interconnection.This is a key point since the truncated trace of the Jacobian in general varies after interconnections.
In the following theorem, we write (x ), and the same notations are used for Q, S, and R.
Theorem 3: Consider the N subsystems ( 15) such that the hypotheses mentioned above hold.Suppose that the interconnection in (16) satisfies Q+SM +M T S T +M T RM 0. For each i ∈ {1, . . ., N}, let di ∈ (0, n i ] be such that tr J i (x) + (n i − di )λ i (x) < 0, x ∈ R n .
Then, for any compact invariant set A ⊂ R n of ( 17 Using the commutativity between P(x) and (x), we conclude that the dimension estimate d is such that tr J(x)+tr n−d (x) < 0, where J(x) := Df (x).From the definition of di , we can observe that tr J(x) + N i=1 (n i − di )λ i (x i ) < 0, and the proof is complete.
Remark 4: The above result is scalable in the sense that the attractor dimension is estimated from di , which can be computed individually.In addition, tr n−d (x) can be easily computed since (x) is a diagonal matrix.Yet, we must be careful with the dependence on x i of J i (x i ) and λ i (x i ).Thus, we need to solve an infinite system of LMIs.If the system under consideration has a special structure, then the problem can be relaxed (see [35]).In that case, we only need to solve a finite set of LMIs.We here provide an example to demonstrate applicability of the above result.It is known that coupled chaotic systems exhibit various phenomena depending on the coupling parameters [36].Moreover, even if there is no coupling between two chaotic systems, the Hausdorff dimension of the Cartesian product of the two attractors can in general be greater than their sum (see, e.g., [37]).Thus, our analysis of Hausdorff dimension for interconnected systems is meaningful.

[
), it holds true that dimH (A) < d, where d ∈ (0, n] is a solution of tr n−d (x) ≤ N i=1 (n i − di )λ i (x i ), x ∈ A.Proof: We sum up all storage functions to obtainN i=1 Vi (x i , δx i ) + λ i (x i )V i (x i , δx i )] ≥ δy T (Q + SM + M T S T + M T RM)δy.It follows from the assumption in the theorem that δẋ δx