Robust Synchronization in Networks of Compartmental Systems. Milad Alekajbaf

Transcription

1 Robust Synchronization in Networks of Compartmental Systems by Milad Alekajbaf A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto Copyright 204 by Milad Alekajbaf

2 Abstract Robust Synchronization in Networks of Compartmental Systems Milad Alekajbaf Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto 204 This thesis studies robust synchronization in networks of compartmental systems motivated by cellular networks. These networks consist of compartments each of which is composed by interconnected subunits. We investigate conditions on the dynamics of the subunits and on the interconnection topology that guarantee robust synchronization in the presence of external disturbances and possibly nonlinear perturbations on the network. The results are illustrated with several examples, including a network of genetic oscillators. ii

3 Dedication To my dearest parents and sister iii

4 Acknowledgements I would like to express my sincere thanks and gratitude to my supervisors, Professor Luca Scardovi and Professor Bruce Francis for their guidance, patience, and support during the past two years. I gratefully acknowledge Professor Scardovi for his trust, continued assistance, and creative ideas. Furthermore, I thank him for the significant time that he spent to thoroughly review my thesis. I truly appreciate Professor Francis for patiently listening to my ideas and providing his feedback which has improved my research work. Moreover, our discussions inside and outside the office, have been fundamental to my growth as a researcher. I would also like to express my appreciation to all of my friends in the Systems Control Group for their great help and support. My particular thanks go to Farzad and Ashkan for sharing their precious experience and countless stimulating discussions. Last but not least, I am deeply indebted to my parents and sister for their unfailing love and understanding. This work would not have been possible without their continued unconditional support. iv

5 Contents Introduction. Motivating Example Literature Review Thesis Outline Statement of Contributions Background 0 2. Notation Graph Theory Connectivity of Graphs Laplacian Matrix Algebraic Connectivity Stability Theory Diagonal Stability Input-Output Stability and Passivity in L 2 Space Coupled Harmonic Oscillators Introduction Synchronization in The Nominal Network Balanced Graphs General Directed Graphs v

6 3.3 Robustness Analysis Robustness Analysis in The Presence of Disturbances Examples A More General Example Nonlinear Netwroks: An Input-Output Approach Introduction Problem Statement Main Result Example: Network of Goodwin Oscillators Conclusions and Future Work Conclusions Future Work A Goodwin Oscillator 82 Bibliography 83 vi

7 List of Tables 4. Robust synchronization conditions for a network in which only the first species are allowed to diffuse vii

8 List of Figures. Network of four coupled genetic oscillators Block diagram of network of four coupled Goodwin oscillators Some examples for different kind of graphs Block diagram of an isolated harmonic oscillator Block diagram of the nominal network of harmonic oscillators Block diagram of the perturbed network of harmonic oscillators Block diagram of the perturbed network of harmonic oscillators with disturbances Graph representation of G x and G y and their union for Example Asymptotic synchronization of the x components of the states for Example Asymptotic synchronization of the y components of the states for Example Graph representation of G x and G y and their union for Example Asymptotic synchronization of the x components of the states for Example Asymptotic synchronization of the y components of the states for Example Graph representation of G x = G y and G x G y for Example viii

9 3.2 Asymptotic synchronization of the x components of the states for Example Asymptotic synchronization of the y components of the states for Example Asymptotic synchronization of the x components of the states for Example Asymptotic synchronization of the y components of the states for Example Block diagram of an isolated chain of integrators with a negative feedback loop Block diagram of N coupled chain of integrators The input, output, and external input of species k in compartment j Schematic representation of the network An example of the network with cyclic species coupling (I + k jj) output for the input z = 0.5 sin(t) and constants r = 0., m = (I + k jj) output for the input z = 0.5 sin(t) and constants r = 0., m = Robust synchronization of six interconnected Goodwin oscillators with non-vanishing external signals Robust synchronization of six interconnected Goodwin oscillators with vanishing external signals Comparison of ỹ T and ρ ṽ T +ρ 2 for the non-vanishing external signals 78 ix

10 Chapter Introduction At the heart of the universe is a steady, insistent beat: the sound of cycles in sync [55] Steven Strogatz Synchronization phenomena arise in a variety of scenarios in natural sciences, engineering, and social life [39]. Synchronization is ubiquitous in biological systems. For instance, in heart s natural pacemaker, about 0,000 cells work together synchronously with no leader or outside instruction to autonomously produce a rhythmic heartbeat [55]. Abnormal synchronization can also cause critical problems in biological systems, e.g., it is recognized that an anomalous synchronized neural activity can lead to epilepsy and other neurological dysfunctions [29]. Therefore, it is of great interest to understand the conditions under which synchronization can occur in such networks. Synchronization is a dynamical phenomenon and therefore we make use of dynamical systems theory to study it. As we will see in the following, for two or more coupled dynamical systems, their degree of synchrony is quantified as a distance between systems trajectories (or outputs). We will focus on compartmental network models, popular in cellular biochemical

11 Chapter. Introduction 2 networks, where the compartments are diffusively coupled and the network is subjected to external disturbances. of circadian oscillators. A typical example of a compartmental model is a network By means of circadian networks, cells coordinate and share information (via chemical diffusion) in order to obtain a synchronous behaviour. Synchronization in cellular networks is currently an active research topic in distinct research areas such as systems biology, mathematics, physics, engineering, etc. System and control theory is instrumental to study synchronization as it allows the recursive verification of important properties through the use of standard analysis tools. The majority of the existing approaches in systems and control theory assume perfect knowledge of the mathematical model. However, in nature, dealing with uncertainties is crucial as accurate mathematical models are virtually impossible to obtain. Our main goal is to investigate under what conditions synchronization can be proven to be a robust feature of compartmental networks where possibly nonlinear perturbations on the diffusive coupling are present. We gradually approach our main problem in three steps as follows. We first focus our attention on a network of coupled linear oscillators (harmonic oscillators). Our first step (Section 3.2) is to study synchronization in the nominal network, i.e., we assume that the mathematical model is not uncertain. Our second step is to include in the analysis the effect of possibly nonlinear and time-varying perturbations on the coupling and the presence of external disturbances. Sections 3.3 and 3.4 deal with these two problems. Finally, in Chapter 4, we address the general problem. We consider a class of compartmental models where each component of the network (referred to as compartment) consists of interconnected subsystems (referred to as species). Recently, [50] provides sufficient conditions for this type of network to achieve synchronization; however, the coupling is supposed to be linear and certain. We generalize this result by including possibly nonlinear perturbations on the coupling.

12 Chapter. Introduction 3. Motivating Example In a genetic network, DNA information is transcribed letter by letter into RNA language. This process is called transcription. The RNA transcribed from a protein-coding gene is called messenger RNA or mrna. After transcription, mrna travels to a specific location of the cell where translation into protein occurs. Finally, the protein can act as an inhibitor or activator for a specific gene. When the protein represses its own production the resulting negative feedback loop can give rise to oscillations. In Fig.., a graphical representation of a network of four coupled genetic oscillators is depicted where the negative feedback represents repression. In this network, each cell is associated to a compartment and each compartment is composed of several subunits called species. There are two types of couplings in the network, namely, a species coupling and a compartmental coupling. The species coupling is due to the chemical reactions that occur between two distinct species inside the cells, while the compartmental coupling takes into account the chemical diffusion among identical species in different compartments. One of the most popular mathematical models of genetic oscillators is the Goodwin oscillator [7, 8] (see the Appendix for more details). The Goodwin oscillator is the result of a chain of three linear time-invariant first order systems and a static nonlinearity interconnected in negative feedback loop. In contrast with the usual modeling assumptions, in our framework, each compartmental coupling is allowed to be uncertain and can include nonlinear time-varying terms. The block diagram of the network is depicted in Fig..2, where Σ kj for k, j =,..., 4 denote the mathematical model for species k in compartment j, and denotes the uncertainty operator. Although the cells are identical, they may have distinct initial conditions so the operators of the corresponding species are considered to be different. Our goal is to find conditions under which robust synchronization is achieved in perturbed networks such as the one depicted in Fig..2.

13 Chapter. Introduction 4 Figure.: Network of four coupled genetic oscillators I + Σ Σ 2 Σ 3 Σ 4 Σ 2 Σ 22 Σ 32 Σ 42 I + 2 I + 3 I + 4 Σ 4 Σ 24 Σ 34 Σ 44 Σ 3 Σ 23 Σ 33 Σ 43 I + 5 I + 6 Figure.2: Block diagram of network of four coupled Goodwin oscillators

14 Chapter. Introduction 5.2 Literature Review This section reviews some of the existing work reported in the literature. We outline the approaches to synchronization in networks of linear and nonlinear systems. In particular, we review the topics in this area that are more related to our work, including synchronization in second order linear systems, synchronization in nonlinear networks (with an emphasis on the passivity approach), and robust synchronization. Single Integrators When the network is composed by interconnected single integrators, synchronization is mostly referred to as consensus. The idea of relating the communication topology of interconnected systems with graph theory was initiated in [60, 6]. In the last decade, the consensus problem in networks of single integrators became of great interest in systems and control with noteworthy contributions such as [24, 38, 34, 46]. In particular, the authors in [38] studied three network structures, namely, directed balanced networks with fixed topologies, directed balanced networks with switching topology, and undirected networks with a communication time delay. Later, the results were extended to directed graphs having a spanning tree [46]. There are massive developments in analyzing the synchronization problem in networks of first order systems with dynamic communication topologies in different setups such as continuous time, discrete time, and quantized data communications. (see for instance [45, 30, 3, 44, 37, 69] for related works, and [7] for a recent survey in this research area). Second Order Linear Systems The generalization to second order linear systems is not trivial and it is still an active research topic. The main classes of networks considered in the literature are networks of double integrators and networks of harmonic oscillators. The first developments have been reported in [28, 49, 59], where the communication

15 Chapter. Introduction 6 among the agents is assumed to be bidirectional. Later, [44] extended the results to unidirectional communication and showed that both the graph topology and the coupling strength are essential in order to achieve synchronization. The paper [] dealt with the synchronization problem for a network of double integrators where the velocity measurement is not available and the control input has a saturation constraint. The paper [68] investigated some necessary and sufficient conditions for synchronization in a network of double integrators with and without time delay where the coupling among the two components of the states has the same structure but different gains. Recently, [5] provided a necessary and sufficient condition for synchronization in a network of double integrators where both the state components are coupled through two heterogeneous undirected graph topologies. Synchronization of coupled harmonic oscillators under directed fixed and switching topologies was studied in [42], where the oscillators are diffusively coupled only through the second component of the states. The paper [56] dealt with synchronization of coupled harmonic oscillators in a dynamic proximity network popular in the study of flocking behaviour [36]. The papers [6] and [8] made use of average theory and Lyanpunov stability theory on dynamical systems, respectively, to study nonlinearly coupled harmonic oscillators over undirected graph topologies. To the best of our knowledge, synchronization in networks of harmonic oscillators where both components of the states are diffusively coupled with two different communication topologies has not been studied yet. We will present our main results for networks of coupled harmonic oscillators in Chapter 3. High Order Linear Systems In the case of high-order linear systems, [47] dealt with the synchronization problem in the special case of single input linear systems where the graph topologies associated to the components of the states are identical. In this case, it is shown that synchronization

16 Chapter. Introduction 7 is achieved if and only if the directed graph has a spanning tree. Later, by exploiting passivity properties, sufficient conditions are investigated for a more general class of linear systems to achieve synchronization under a possibly time-varying and directed graph topology; however, the graph topologies are assumed to be identical for all the components of the states [5]. In the case of heterogeneous linear systems, [66, 26] adopted an internal model principle approach (see [3]) to take heterogeneities into account in the network, thus generalizing the results in [5]. Lastly, [32] used an integral quadratic constraints approach to address the robust synchronization problem in networks of identical linear systems with nonlinear perturbations. Nonlinear Systems Given a network of nonlinear dynamical systems, synchronization depends on both the dynamics of the subunits and the interconnection topology [40, 2]. Recently, passivitybased techniques proved to be instrumental for the analysis of synchronization in interconnected dynamical systems [20, 2, 9]. In this context, both a state space approach [22, 9] and an input-output approach [50, 2] have been considered and sufficient conditions, based on incremental passivity properties of the subunits, have been investigated. The synchronization conditions in [50] can be seen as a generalization of stability conditions proposed in [3] and [4] which, in turn, build on classical results on the stability of large-scale interconnected systems [35, 65]. The paper [58] studied robust synchronization in networks of identical nonlinear systems with a gain-bounded multiplicative uncertainty in the inputs. The papers [33] and [70] used a state-space approach to derive a bound for the synchronization error in networks of homogeneous systems with linear additive uncertainties and in networks of heterogeneous systems, respectively.

17 Chapter. Introduction 8.3 Thesis Outline The thesis is organized as follows. In the remaining of this chapter we outline the main contributions of the thesis. In Chapter 2, we briefly summarize the notions of graph theory and stability theory that we need throughout the thesis. Chapter 3 focuses on synchronization of coupled harmonic oscillators in both the nominal and the perturbed scenarios. In our framework both components of the harmonic oscillators are diffusively coupled through possibly distinct communication topologies. We claim that our network structure has not been studied before in the literature. We first investigate necessary and sufficient conditions for the nominal network to achieve synchronization. Once the synchronization conditions are established for the nominal network, classical results in stability theory [25] can be used to address robustness of synchronization.we illustrate the main results with several examples. Chapter 4 deals with synchronization in nonlinear networks. Our contribution is an extension of the work in [50]. In particular, [50] adopted an input-output framework to study synchronization in the nominal network, while our main contribution is to investigate under what conditions synchronization is a robust feature of the network with respect to possibly nonlinear perturbations on the diffusive coupling. The result is applied to derive sufficient conditions for synchronization in a network of Goodwin oscillators. Finally, in Chapter 5, we review the main results of the thesis and suggest possible future research..4 Statement of Contributions The main contributions are summarized as follows.. Lemma 3.2 and Lemma 3.4 The lemmae address two useful properties of the kernel of Laplacian matrices as-

18 Chapter. Introduction 9 sociated with directed balanced and general directed graphs. These properties are not only exploited in the main results of Chapter 3, but they are of independent interest. 2. Theorem 3. and Theorem 3.2 Theorem 3. provides a necessary and sufficient condition for synchronization in a network of multi-input harmonic oscillators where both the components of the states are diffusively coupled with possibly distinct directed balanced communication topologies. In Theorem 3.2, the balanced condition for graphs is relaxed; however, only sufficient conditions are provided. 3. Theorem 3.3 This theorem provides a necessary and sufficient condition, based on the spectral properties of our model, for the harmonic oscillators to asymptotically synchronize in the presence of directed communication topologies. Two corollaries provide necessary and sufficient conditions for synchronization. 4. Theorem 3.4 and Theorem 3.5 In Theorem 3.4, a sufficient condition is achieved for robust synchronization of coupled harmonic oscillators, and in Theorem 3.5, an upper bound for the synchronization error of the network in the presence of disturbance signals is obtained. 5. Theorem 4. This Theorem is a generalization of [50, Theorem ] where the coupling is nonlinearly perturbed. Theorem 4. shows that under certain conditions, the network almost synchronizes depending on the level of synchrony of external inputs. As a special case, if the external inputs are in L 2 then, under some technical assumptions, asymptotic synchronization is achieved.

19 Chapter 2 Background In this chapter, we review preliminary mathematical definitions and introduce the background ideas and foundations that are needed for the remaining chapters. Section 2. introduces the notation throughout the thesis. In Section 2.2, we summarize some notions from graph theory. Finally, in Section 2.3, we review some concepts of stability theory including diagonal stability, input-output stability and passivity. 2. Notation The notation used in this thesis is fairly standard. We represent matrices with capital roman letters, while scalars and vectors are represented with lower case letters. Calligraphic fonts (e.g. V, W, etc.) are used to represent sets and in particular graphs and subspaces. Let R and C denote the field of real and complex numbers, respectively. The set of non-negative real numbers is denoted by R +. The notation R n denotes the set of n-dimensional real vector space, and R n m denotes the set of n m matrices with real numbers. Similarly, C n denotes the set of n-dimensional complex vector space. We write f C to show that f is a continuously differentiable function. For x,..., x m R n, col(x,..., x m ) R mn denotes the stacked (column) vector. Let x denote the euclidean 2-norm of x R n. Let I n R n n be the n-dimensional identity 0

20 Chapter 2. Background matrix, and 0 n, n R n be the column vector of all zeros and ones, respectively. We write A = [a ij ] R m n to show that the elements of matrix A are a ij. The transpose of a matrix A is denoted by A T. A positive (negative) definite matrix A is denoted by A 0 (A 0). Similarly, a positive (negative) semi-definite matrix A is denoted by A 0 (A 0). A (block) diagonal matrix B R n n with (block) diagonal elements b,..., b n is denoted by B = diag(b,..., b n ). The (right) kernel and rank of matrix A are respectively denoted by Ker(A) and rank(a). At the end of this section, we briefly introduce the Kronecker product. Given two matrices A := [a ij ] R m n and B R p q, the Kronecker product of A and B is defined as a B... a n B A B = a m B... a mn B The Kronecker product has the following properties [27]: A B C = A (B C) = (A B) C A (B + C) = A B + A C AB CD = (A C)(B D). (2.a) (2.b) (2.c) Furthermore, when A and B are square matrix, the spectrum of square matrix A B can be obtained as follows. Let α,..., α m be the eigenvalues of A R m m and β,..., β n be the eigenvalues of B R n n. The eigenvalues of A B are α i β j for i =,..., m and j =,..., n.

21 Chapter 2. Background Graph Theory Graph theory is an important tool in the analysis of interconnected systems. We can represent the communication topology of a network with a graph. If the information flow is bidirectional, i.e., both systems receive information from each other, then the corresponding graph is undirected. While, if the information flow is unidirectional then the corresponding graph is directed. In this section we review some of the basic notions of graph theory with emphasis on algebraic graph theory. For the reader interested in more details about this section, we recommend [5], [4], [], [67], [53]. A weighted directed graph (or simply digraph) G = {V, E, A} consists of a non-empty finite set V := {,..., n} of elements called nodes, a finite set of edges (or arcs) E V V, and an associated weighted adjacency matrix A := [a ij ] R n n where a ij > 0, (j, i) E a ij = 0, (j, i) / E. Note that the edge (j, i) is graphically represented by an arrow with tail node j and head node i. This means node i receives information from node j, so node i is neighbour (or out-neighbour) of node j. A graph is called undirected if all the pairs in E are unordered, or equivalently, the adjacency matrix is symmetric. Throughout the thesis we assume that the graphs are time invariant, that is, A is a constant matrix. Also, we assume that for i =,..., n the pair (i, i) / E, i.e., the graphs do not have self-edges. The in-degree and out-degree of node i V are defined as d in (i) := n a ij, d out (i) := j= n a ji. j= A directed graph is said to be balanced if for every node the in-degree and the out-degree are equal. So every undirected graph is balanced but not vice versa. Many operations such as union, converse, and different kind of products on directed

22 Chapter 2. Background 3 graphs are defined. Here we only introduce the union operation on weighted directed graphs that will be used in Chapter 3. Let G i = {V, E i, A i } for i =,..., m have the m same node set. The union of G i is defined as G := i G i = {V, i E i, A i }. Different kind of weighted graphs with four nodes are depicted in Fig. 2.. In this figure, G 2 is an undirected graph, while all the other graphs are directed. Note that graph G is a balanced, but G 3, G 4, and G 5 are not balanced. i= (a) G (b) G (c) G (d) G 4 (e) G 5 Figure 2.: Some examples for different kind of graphs 2.2. Connectivity of Graphs A strong path (or directed chain) in a directed graph is a sequence of distinct nodes v 0,..., v k V such that for every i =,..., k, (v i, v i ) E. A weak path in a directed

23 Chapter 2. Background 4 graph is a sequence of distinct nodes v 0,... v k V such that for every i =,..., k, either (v i, v i ) E or (v i, v i ) E. A directed graph is strongly connected if there exists a strong path from every node to every other node. A directed graph has a spanning tree if there exists at least one node, called the root node, that is connected to all the other nodes by a strong path. So a spanning tree has exactly n nodes and n edges, and every node, except the root node, has exactly one incoming edge. In some literature, a graph having a spanning tree is called quasi-strongly connected. A strongly connected graph has a spanning tree but not vice versa. A directed graph is connected (or weakly connected) if there exists a weak path between every two nodes. It can be shown that a connected balanced graph is strongly connected. In Fig. 2., G and G 2 are strongly connected, G 3 is not strongly connected; however, it has a spanning tree with root node. The graph G 4 is connected, whereas G 5 is not connected. Node 4 in G 5 is called isolated node Laplacian Matrix Given a weighted directed graph G = {V, E, A} with set node V = {,..., n} and adjacency matrix A = [a ij ]. Let D := diag(d in (),..., d in (n)) be the in-degree matrix. The Laplacian matrix associated with graph G is defined as L := D A. (2.2) Let L := [l ij ] R n n. It follows from (2.2) that n a ij, i = j l ij = j= a ij, i j. (2.3)

24 Chapter 2. Background 5 To illustrate, consider Fig. 2.. Let L k be the Laplacian matrix associated with G k for k =,..., 5. Then the Laplacian matrices are as follows: L =, L 2 =, L 3 =, L 4 =, L 5 = (2.4) In general, a Laplacian matrix L associated with the graph G has the following properties:. Matrix L is diagonally dominant. So by the Gershgorin disk theorem [23], all the eigenvalues of L are in the closed right half-plane (CRHP). 2. If G is undirected then L is symmetric, and hence it is positive semi-definite. 3. The Laplacian matrix L is a zero sum matrix, i.e., the vector n is in the right kernel of L, i.e., L n = 0 n. Moreover n is also in the left kernel of L if and only if G is balanced. 4. If G is balanced, then L = L + L T is a symmetric Laplacian matrix [38]. Note that in (2.4), L 2 is symmetric due to the fact that G 2 is undirected. Also, since G is balanced, we have T nl = 0 T n. The following well-known lemma is a useful tool for analyzing synchronization problem in interconnected systems. Lemma 2.. [46] Let G be a weighted directed graph and L be the associated Laplacian matrix. Then L has a simple zero eigenvalue and all the other eigenvalues are in the

25 Chapter 2. Background 6 open right half-plane if and only if G has a spanning tree. Lemma 2. implies that rank(l) = n if and only if G has a spanning tree. in Fig. 2., since G, G 2, and G 3 have a spanning tree, we have rank(l ) = rank(l 2 ) = rank(l 3 ) = 3. In contrast, G 4 and G 5 do not have a spanning tree and rank(l 4 ) = rank(l 5 ) = Algebraic Connectivity The connectivity properties of graphs are related to the algebraic properties of the corresponding Laplacian matrices and, in particular, to the notion of algebraic connectivity. Fiedler [0] introduced the algebraic connectivity of an undirected graph as the second smallest eigenvalue of its Laplacian matrix. Lately, this concept was extended to general directed graphs as follows [67]: Definition 2.. Let G be a directed graph with Laplacian matrix L. Let also P := {z R n : z n, z = }. The algebraic connectivity of G is defined as λ := min z P zt Lz. (2.5) The algebraic connectivity of directed graph G with Laplacian L has the following properties:. Let G and H have the same vertex set, and λ and µ be their algebraic connectivity, respectively. Let I := G H with algebraic connectivity γ. Then γ λ + µ. 2. Let λ and λ 2 be the smallest and second smallest eigenvalue of L := L + L T, respectively. Then λ in (2.5) satisfies 2 λ ( L) λ 2 λ 2( L). 3. If G is not connected, then λ 0.

26 Chapter 2. Background 7 4. For a balanced G, λ > 0 if and only if G is connected. 2.3 Stability Theory This section is a review of some standard concepts in stability theory including LaSalle s invariance principle and diagonal stability, input-output stability, and passivity. In Chapter 3 we use LaSalle s invariance principle to study synchronization in networks of harmonic oscillators. Since the network under study in Chapter 3 is linear we make use of the properties of invariant subspace in linear systems. We start with the definition of an invariant subspace under the dynamics of a linear system ẋ = Ax, x R N, x(0) = x 0. (2.6) Definition 2.2. The subspace V R n is said to be invariant under (the dynamics) (2.6) if x 0 V = x(t) V, for every t 0. Theorem 2.. Consider linear system (2.6). Let V R n. Then V is invariant under 2.6 if and only if V is A-invariant, that is, AV V. Now we state the LaSalle s invariance principle for the dynamical system ẋ = f(x), x X R n, (2.7) where f is C and the origin is contained in X. Theorem 2.2. Given the nonlinear system (2.7). Let B X be a domain and Ω B be a compact positively invariant set. Let V : D R be a C function such that for every x Ω, V (x) 0. Let E := {x Ω : V (x) = 0} and M be the largest invariant subset of E. Then, for every x 0 Ω, the solution φ(x, t) M as t.

27 Chapter 2. Background 8 In some cases, such as time-varying systems, the LaSalle invariance principle cannot be applied. An alternative way to characterize the asymptotic behaviour of the solutions is to use the Barbalat lemma: Lemma 2.2. Let f : R + R be a uniformly continuous function for t 0. If lim t t 0 f(t) exists and is finite, then lim t f(t) = 0. The Barbalat lemma implies that if a uniformly continuous signal is in L 2, then the signal converges to zero. We use this fact in Chapter Diagonal Stability Diagonal stability plays an important role in studying stability of network made up passive subsystems. Some of the well-known classical works in this area can be found in [57], [65], and [35]. The definition of diagonal stability is as follows [4]: Definition 2.3. A real square matrix A is said to be diagonally stable if there exists a diagonal matrix D 0 such that A T D + DA 0. According to the above definition, a matrix is diagonally stable if it satisfies the Lyapunov equation with a diagonal matrix. A necessary and sufficient condition for the matrix α 0... β β A = β N α N (2.8) to be diagonally stable is the so called secant condition, as stated in the following lemma [4]:

28 Chapter 2. Background 9 Lemma 2.3. The matrix (2.8) is diagonally stable if and only if α... α [ ( N π )] N < sec. β β N N Lemma 2.3 was extended to other matrix structures in [54] Input-Output Stability and Passivity in L 2 Space This section provides the basic notions of input-output stability and passivity in L 2 space. The material in this section is mostly taken from [52] with some modifications. Definition 2.4. The set L m 2 [0, ) = L m 2 are measurable and satisfy 0 f(t) 2 dt <. consists of all functions f : R + R m which Definition 2.5. Let f : R + R m. Then for every T 0, the truncation of f to the interval [0, T ] is denoted by f T : R + R m and is defined as f(t), 0 t < T f T (t) := 0, t T. We denote by L m 2e the extended space of signals f : R + R m which have the property that f T L m 2 for every T 0. The set L m 2e is called the extended L m 2 space or the extension of L m 2. Given two signals v, w L m 2e and any finite T > 0, we define the inner product of v, w as v, w T := T 0 v(t)t w(t)dt, and w T := w, w T. Definition 2.6. An operator F : L m 2e L m 2e is said to have unbiased finite L 2 gain if there exists δ c 0 such that for every z L m 2e and for every T 0, F z T δ c z T. The L 2 gain of F is defined as δ := inf{δ c }.

29 Chapter 2. Background 20 Now we introduce the notion of passivity in L 2 space. Definition 2.7. An operator F : L m 2e L m 2e is passive if there exists constant β R such that F u, u T β, u L m 2e, T 0. The operator F is input strictly passive if there exist constants β R and δ > 0 such that δ F u, u T u 2 T + β, u L m 2e, T 0. The operator F is output strictly passive if there exist constants β R and ɛ > 0 such that ɛ F u, u T F u 2 T + β, u L m 2e, T 0. Before concluding this chapter, we introduce the notion of relaxed co-coercivity as follows [50]: Definition 2.8. An operator F : L m 2e L m 2e is relaxed co-coercive if there exists constant γ R such that γ F u F u 2 2 T u u 2, F u F u 2 T, u, u 2 L m 2e, T 0. (2.9) If (2.8) holds for γ 0, then F is said to be monotone, and if (2.8) holds for γ > 0, then F is said to be co-coercive. A definition similar to Definition 2.8 can be found in [64]. In the literature, sometimes monotone, co-coercive, and relaxed co-coercive are respectively referred to as incrementally passive, incrementally output strictly passive, and incrementally output feedback passive [20, 22]. In Chapter 4, we define a slightly

30 Chapter 2. Background 2 adapted version of Definition 2.8.

31 Chapter 3 Coupled Harmonic Oscillators 3. Introduction In this chapter, we deal with robust synchronization of diffusively coupled harmonic oscillators. Our framework in this chapter is the simplified version of a nonlinear compartmental network that will be discussed in detail in Chapter 4; however, the results in this chapter are of independent interest in synchronization of multi-input linear systems where many problems have remained open. The purpose of this chapter is to investigate the conditions under which synchronization is achieved in the nominal and perturbed network of harmonic oscillators. Unlike the results in the literature [42, 5], in our framework both state components are coupled through possibly distinct directed communications topologies. This chapter is organized as follows. In Section 3.2, we provide a necessary and sufficient condition for synchronization of the nominal diffusively coupled harmonic oscillators. In Section 3.3, we show that synchronization can be achieved in the network, in the presence of a possibly nonlinear time-varying perturbations on the coupling. In Section 3.4, the effect of disturbance on the network is studied and an upper bound on the norm of the synchronization error is obtained. Finally, the results of this chapter are 22

32 Chapter 3. Coupled Harmonic Oscillators 23 illustrated with several examples in Section Synchronization in The Nominal Network Consider n coupled harmonic oscillators where each isolated one is modelled as ẋ i = y i x i (0) = x i0 ẏ i = x i y i (0) = y i0, x i, y i R, i =,..., n. (3.) The block diagram of system (3.) is depicted in Fig. 3.. Without loss of generality, throughout the thesis, we fix the natural frequency of the harmonic oscillators to one. s x i s y i Figure 3.: Block diagram of an isolated harmonic oscillator Assume that the first components of the states (x), as well as the second components of the states (y), are diffusively coupled as follows: n ẋ i = y i + a ij (x j x i ), ẏ i = x i + j= n b ij (y j y i ), i =,..., n. j= (3.2) We represent the diffusive coupling terms by Laplacian matrices as follows. Let L x := [l x ij] R n n and L y := [l y ij ] Rn n, where n a ij, i = j lij x := j= a ij, i j, n b ij, i = j l y ij := j= b ij, i j.

33 Chapter 3. Coupled Harmonic Oscillators 24 Let G x and G y be the corresponding directed weighted graphs associated with Laplacian matrices L x and L y, respectively. Let x = [x... x n ] T and y = [y... y n ] T. In vector form, we can rewrite (3.2) as ẋ = y L x x x(0) = [x 0... x n0 ] T ẏ = x L y y y(0) = [y 0... y n0 ] T (3.3) or equivalently as ẋ = L x ẏ I }{{} ż } {{ } A I x. (3.4) L y y }{{} z The block diagram of the network is depicted in Fig Our first goal is to find s... s x s... s y L x L y Figure 3.2: Block diagram of the nominal network of harmonic oscillators conditions under which synchronization is achieved asymptotically in the network (3.4). Then, we study robustness in synchronization with respect to nonlinear perturbations on the diffusive coupling, and following that, we study the effect of external inputs on synchronization in the network. In order to do so, first we give an explicit mathematical definition for synchronization in network (3.4) as follows: Definition 3.. For the network of n coupled harmonic oscillators, the harmonic oscil-

34 Chapter 3. Coupled Harmonic Oscillators 25 lators asymptotically synchronize if for every initial condition lim x i(t) x j (t) = 0 t lim y i(t) y j (t) = 0, i, j =,..., n. t From Definition 3., the harmonic oscillators asymptotically synchronize if for every initial condition, the solution of the network converges to the synchronization subspace defined as S := Span n, 0 n. 0 n n To find conditions under which synchronization occurs in the network, we need to study the properties of graphs G x and G y. In this thesis, the weighted directed graphs are taken into account that model the general class of the network structure. First, we study balanced graphs as a particular class of directed graphs and then we relax this condition to general directed graphs. For balanced graphs, we present a necessary and sufficient condition for the network to achieve synchronization, while for the general directed graphs, we only present a sufficient condition; however, we show a necessary and sufficient condition for two special cases of the network structure, namely, when both graphs are identical and when one of them is an empty graph Balanced Graphs In this section we provide a necessary and sufficient condition under which synchronization is achieved. We will use LaSalle s invariance principle which has also been used for the consensus problem in a network of single integrators [69]. We start with two lemmae that are needed to prove the main result of this section. The first lemma [23] is a property

35 Chapter 3. Coupled Harmonic Oscillators 26 of symmetric positive definite matrices and the second lemma shows a useful property of balanced graphs. Lemma 3.. [23] Let Q R n n be symmetric positive semi-definite and let x R n. Then x T Qx = 0 if and only if Qx = 0. Proof : The sufficiency is clear. For the necessity, let x 0 and x T Qx = 0. Consider the following polynomial: p(t) = (tx + Qx) T Q(tx + Qx) = t 2 x T Qx + 2tx T Q 2 x + x T Q 3 x = 2t Qx 2 + x T Q 3 x. Since Q is positive semi-definite, p(t) 0 for every t R. However, if Qx 0 then there exists a negative constant r such that for every t < r we have p(t) < 0. This implies that Qx = 0 and hence Qx = 0. Lemma 3.2. Let L R n n be a Laplacian matrix associated with a directed balanced graph and L := L + L T. Then Ker(L) = Ker( L). Proof : First, we show that Ker(L) Ker( L). Let v Ker(L), i.e., Lv = 0 n. We have 2v T Lv = v T (L + L T )v = 0. Since L is symmetric positive semi-definite (See [38, Theorem 7]), by Lemma 3., we conclude that v Ker( L). Now, we need to show that Ker( L) Ker(L). Let w Ker( L), i.e., Lw = 0 n and w T Lw = 0. (3.5)

36 Chapter 3. Coupled Harmonic Oscillators 27 Let L := [l ij ] for i, j =,..., n where n c ij, i = j l ij := j= c ij, i j. (3.6) We have w T Lw = n i= ( w 2 i n ) c ij j= n c ij w i w j = i,j= n c ij (wi 2 w i w j ), (3.7) i,j= and similarly, n n w T L T w = c ji (wi 2 w i w j ) = c ij (wj 2 w j w i ). (3.8) i,j= i,j= It follows from (3.5), (3.7), and (3.8) that n n w T Lw = w T Lw + w T L T w = c ij (wi 2 + wj 2 2w i w j ) = c ij (w i w j ) 2 = 0. (3.9) i,j= i,j= So, if c ij > 0, then w i = w j. Let w i := w j for every j such that c ij > 0, i.e., the corresponding elements of w for all neighbours of node i have the same value w i. Hence, for every i =,..., n, the i th row of Lw, denoted by L i w, is L i w = n n n l ij w j = l ii w i + l ij w j = w i l ii + w i l ij = w i L i n = 0. j= j=,j i j=,j i Therefore, w Ker(L), i.e., Ker( L) Ker(L). This completes the proof. Now, we are ready to state the main result of this section. Theorem 3.. Let G x and G y be balanced directed weighted graphs. All the harmonic oscillators in network (3.4) asymptotically synchronize if and only if G x G y is connected.

37 Chapter 3. Coupled Harmonic Oscillators 28 Proof : (Sufficiency) We apply LaSalle s invariance principle to prove the sufficiency. Let V (x, y) = x T x + y T y. The derivative of V along the trajectories of the network is as follows: ] V (x, y) = 2 [x T y T A x (3.0) y ] = [x T y T (A + A T ) x (3.) y ] = [x T y T (L x + L T x ) 0 x (3.2) 0 (L y + L T y ) y = x T Lx x y T Ly y 0 (3.3) Let c 0. Consider the compact set Ω := {col(x, y) R 2n : V (x, y) c}. By Nagumo theorem, we conclude that Ω is positively invariant. Let E := {col(x, y) Ω : V (x, y) = 0} and M be the largest invariant subset of E. By LaSalle s invariance principle we conclude that for every initial condition z 0 Ω, the solution Φ(t, z 0 ) M as t. Now, we need to show that if the union of the graphs is connected, then M = S. First, we show that S M, i.e., we want to show that the synchronization subspace as defined in Definition 3. is invariant. It is easy to show that ±j are the eigenvalues of A with the associated eigenvectors col(j n, n ) and col( j n, n ), because they satisfy L x I L x I I j n = n = j j n, L y n j n n I j n = L y n n j n = j j n. n (3.4)

38 Chapter 3. Coupled Harmonic Oscillators 29 So the subspace Span Re j n, Im j n = Span 0 n, n = S n n n 0 n is invariant. Now, we need to show that M S. Let D = Span{ n }. It is clear D Ker(L x ) and D Ker(L y ). Hence, D Ker(L x ) Ker(L y ). Since G x G y is balanced and connected, from Lemma 2., we have rank(l x + L y ) = n and Ker(L x + L y ) = D. Also, we know that for every two matrices, the kernel of their sum contains the intersection of their kernels, i.e., Ker(L x ) Ker(L y ) D. Thus Ker(L x ) Ker(L y ) = D. From (3.3) and Lemma 3. we get E = {col(v, w) R 2n : v Ker( L x ), w Ker( L y )}. By Lemma 3.2, E can be written as E = {col(v, w) R 2n : v Ker(L x ), w Ker(L y )}. Let col(v, w) M E. So we have v Ker(L x ) and w Ker(L y ). By Theorem 2., we know that M is A-invariant, so L x I I v = w M E. L y w v This implies that w Ker(L x ) and v Ker(L y ). Hence v, w Ker(L x ) Ker(L y ) = D and thus col(v, w) S which means M S. So we have M = S, and we can conclude that all the harmonic oscillators asymptotically synchronize.

39 Chapter 3. Coupled Harmonic Oscillators 30 (Necessity) We want to show that if all the harmonic oscillators asymptotically synchronize then G x G y is connected, or equivalently rank(l x + L y ) = n. We argue by contradiction. Suppose v := [v... v n ] T Ker(L x + L y ) and v / D. It follows from Lemma 3. and Lemma 3.2 that v T (L x + L y )v = v T Lx v + v T Ly v = 0. (3.5) Since L x and L y are positive semi-definite, it follows from (3.5) that v T Lx v = v T Ly v = 0, and again by Lemma 3. and Lemma 3.2, one can conclude that v Ker(L x ) and v Ker(L y ). Thus, Ker(L x + L y ) Ker(L x ) and Ker(L x + L y ) Ker(L y ). Similar to (3.4) we can show that col(jv, v) and col( jv, v) are eigenvectors of A associated with eigenvalues ±j, i.e., the algebraic multiplicity and geometric multiplicity of ±j are at least 2. So the subspace V := Span Re jv v, Im jv v = Span 0 n, v v 0 n is invariant, and hence for every nonzero initial condition in V the solution will remain in V and does not converge to zero. From the fact that V S = 0, we conclude that there exists an initial condition such that all the harmonic oscillators do not asymptotically synchronize. This is a contradiction, therefore if all the harmonic oscillators asymptotically synchronize then G x G y is connected. This completes the proof General Directed Graphs In Section 3.2., it is shown that when the graphs are balanced, connectivity of their union is the necessary and sufficient condition for the network to achieve synchronization. Now we consider the general weighted directed graphs. In this case, our simulation results show that, when the union of the graphs has a spanning tree, synchronization is achieved

40 Chapter 3. Coupled Harmonic Oscillators 3 in the network; however, this is only a conjecture. In this section, we investigate sufficient conditions for the harmonic oscillators to asymptotically synchronize. Moreover, for two special network structures, necessary and sufficient conditions for synchronization are provided. We make use of the following two lemmae. These lemmae characterize the left eigenvector of a Laplacian matrix corresponding to the zero eigenvalue. Lemma 3.3 is only for the case where the graph is strongly connected, whereas Lemma 3.4 can be applied to general directed graphs. Lemma 3.3. Let L be a Laplacian matrix associated with a strongly connected digraph. Let p = [p... p n ] T R n be the left eigenvector of L associated with the zero eigenvalue, and let P = diag(p i ), i =..., n. Then P 0 and P L + L T P 0. Proof : See [69, Lemma 6] and [4, Theorem 4.3]. Note that in Lemma 3.3, when the graph is balanced, then p = n, P = I n, and L 0. Lemma 3.4. Let L R n n be a Laplacian matrix associated with a digraph. Suppose there exists p = [p... p n ] T R n such that p i > 0 for i =,... n and p T L = 0 T n. Let P = diag(p i ), i =,..., n. Then, Ker(L T P + P L) Ker(L). Proof : Let v Ker(L T P + P L), i.e., (L T P + P L)v = 0 n and v T (L T P + P L)v = v T P Lv + v T L T P v = 2v T P Lv = 0 (3.6) W need to show that v Ker(L). Let L be defined as (3.6). The quadratic form v T P Lv can be computed as follows: v T P Lv = n n ) n n (v 2i p i c ij p i c ij v i v j = p i c ij (vi 2 v i v j ). (3.7) i= j= i,j= i,j= Since p T L = 0 T n we have p i j= n c ij = n p j c ji (3.8) j=

41 Chapter 3. Coupled Harmonic Oscillators 32 It follows from (3.7) and (3.8) that v T P Lv = = = = n i= n i= n j= v 2 i v 2 i v 2 j n ) (p i c ij j= ( n ) p j c ji j= ( n ) p i c ij n p i c ij v i v j i,j= n p i c ij v i v j i,j= i= i,j= n p i c ij (vj 2 v j v i ). i,j= n p i c ij v i v j (3.9) From (3.6), (3.7), and (3.8) we obtain 2v T P Lv = 2 [ n i= v 2 i n ) (p i c ij j= n = p i c ij (vi 2 v i v j ) + = = i,j= ] n p i c ij v i v j i,j= n p i c ij (vj 2 v j v i ) i,j= n p i c ij (vi 2 + vj 2 2v i v j ) i,j= n p i c ij (v i v j ) 2 = 0 i,j= (3.20) Since p i > 0 for i =,..., n, we can find the possible values of v i similar to Lemma 3.2, that is, if c ij > 0, then v i = v j and conclude that for i =,... n, L i v = 0. Thus, Lv = 0 n which implies that Ker(L T P + P L) Ker(L). This completes the proof. Now, we are ready to state our first theorem for network of diffusively coupled harmonic oscillators with the general directed graph topologies. Theorem 3.2. Let G x and G y be directed graphs, and L x and L y be the associated Laplacian matrices. Assume the following:. G x G y is strongly connected. 2. the intersection of the left kernels of L x and L y is not zero.

42 Chapter 3. Coupled Harmonic Oscillators 33 Then, all the harmonic oscillators in network (3.4) asymptotically synchronize. Proof : Since G x G y is strongly connected, it follows from Lemma 3.3 that there exists p = [p... p n ] T such that p i > 0, i =,..., n and p(l x + L y ) = 0 T n. Note that rank(l x + L y ) = n, so the left kernel of L x + L y has dimension, and it is Span{p}. Also, the intersection of the left kernels of L x and L y is contained in the left kernels L x + L y, and according to our assumption the intersection of the left kernels is not zero, so it is Span{p}. Let P = diag(p i ), i =..., n. Consider ] V (x, y) = [x T y T P 0 x 0 P y The derivative of V along the trajectories of the network is as follows: ] V = [x T y T (A T P + P A) x y ] = [x ( T y T LT x I P 0 + L x I P 0 ) x I L T y 0 P I L y 0 P y ] = [x T y T LT x P P L x 0 x 0 L T y P P Ly y We can compute V by using (3.20) as follows: ( n ) n V = p i a ij (x j x i ) 2 + p i b ij (y j y i ) 2 0. i,j= i,j= Similar to the proof of Theorem 3. consider the compact set Ω := {col(x, y) R 2n : V (x, y) c} where c 0. By Nagumo theorem, we conclude that Ω is positively