STABILITY ANALYSIS OF SWITCHED COUPLED DYNAMICAL SYSTEMS MAHESH SUNKULA (08023)

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1 STABILITY ANALYSIS OF SWITCHED COUPLED DYNAMICAL SYSTEMS A REPORT submitted in partial fulllment of the requirements for the award of the dual degree of Bachelor of Science-Master of Science in MATHEMATICS by MAHESH SUNKULA (08023) DEPARTMENT OF MATHEMATICS INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH BHOPAL BHOPAL April 2013

2 i CERTIFICATE This is to certify that Mahesh Sunkula, BS-MS (Mathematics), has worked on the project entitled ` STABILITY ANALYSIS OF SWITCHED COUPLED DYNAMICAL SYSTEMS' under my supervision and guidance. The content of the project report has not been submitted elsewhere by him/her for the award of any academic or professional degree. April 2013 IISER Bhopal Dr. Nikita Agarwal Committee Member Signature Date

3 ii DECLARATION I hereby declare that this project report is my own work and due acknowledgement has been made wherever the work described is based on the ndings of other investigators. I also declare that I have adhered to all principles of academic honesty and integrity and have not misrepresented or fabricated or falsied any idea/data/fact/source in my submission. April 2013 IISER Bhopal Mahesh Sunkula

4 iii ACKNOWLEDGEMENT At the very onset of this report I wish to extend my sincere gratitude to the following people. Dr. Nikita Agarwal, my project supervisor, for giving me the freedom to explore and work on things I am interested in without any constraints. This project would not have been possible were it not for her guidance and constant supervision. She has had a positive inuence on not only this project, but on me as well. Her eorts to help me develop interaction and presentation skills have been invaluable as has been her role in instilling in me a strong motivation towards research, especially in this eld of Dynamical Systems and Control Theory. Apart from this, I appreciate her down to earth attitude and am indebted to her for recognising my interests before suggesting me pursue a course of study. Dr. Ajit Bhand, and Dr. Saurabh Shrivastava, my committee members, for reading through my report and suggesting necessary corrections and for helping me through their valuable inputs towards improving my presentations. Dr. Anandateertha Mangasuli and all my teachers, whose teachings helped me develop interest in Mathematics, without which the project would never have even started. My friends and batch mates especially Rakesh, Kaustubh and Mukund for making my stay at IISER Bhopal pleasant and memorable. My parents and sister, for all their love, aection, faith and patience. Mahesh Sunkula

5 iv ABSTRACT Most physical systems that we encounter involve interaction between several subsystems, in most cases they evolve according to an interaction between continuous dynamics and discrete events. Dynamical systems that are described by an interaction between continuous and discrete dynamics are called hybrid systems (or, piecewise smooth dynamical systems). An important class of hybrid systems consists of continuous dynamical systems with isolated discrete discontinuities known as switching events. Such systems are known as switched systems. The rule governing the switching events is called a switching signal. The two important classes of switching events are statedependent and time-dependent. The two main problems of interest in the subject are (i) nding conditions that guarantee asymptotic stability of a switched system under arbitrary switching signals, and (ii) identifying the signals which guarantee asymptotic stability in the case where asymptotic stability is not guaranteed for arbitrary switching. The motivation of this project is the work by Daniel J. Stilwell, Erik M. Bollt, and D. Gary Roberson on `Synchronization of time varying networks under fast switching' [7], where they studied coupled cell networks whose network topology (underlying graph) changes with time. They found conditions on the time average of the underlying graph for a fast switching signal to be stable. In this report we will discuss the theory of dynamical systems in detail. We will then study the stability analysis of the hybrid dynamical systems via the approach followed by the control theorists [4]. Finally, we will view a coupled cell network with a time-varying graph structure as a time-dependent switched system and apply the results for switched systems to this problem.

6 Contents 1 Dynamical Systems A Dynamical System Flow and Fixed Points Linear Systems Fundamental Theorem of Linear Systems Matrix Exponential and Solution of Linear Systems Nonlinear Systems Linearization Hartman-Grobman Theorem Stability and Lyapunov Functions Coupled Cell Networks Coupled Cell Network Dynamics Continuous Dynamics Discrete Dynamics Hybrid Dynamics Adjacency Matrix of a Directed Graph Symmetry in Systems Examples of Coupled cell Networks Switched Dynamical Systems Introduction Classication of Hybrid and Switched Systems Stability of Switched Systems Uniform Stability Common Lyapunov Functions and Stability Stability of Switched Linear Systems Stability Under Constrained Switching Stability Under State-Dependent Switching Stability Under Time-Dependent Switching Switched Coupled Cell Dynamical Systems Future Directions 49 A Appendix 51 Bibliography 53

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8 Chapter 1 Dynamical Systems Dynamical systems theory is a study of qualitatively describing the changes over time that occur in physical models. Examples of such systems include the weather, the motion of billiard balls on a billiard table, sugar dissolving in a cup of coee, the stock market. In this chapter we will desribe a smooth dynamical system and its equivalence to the solutions of an ordinary dierential equation. We will then discuss the existence and uniqueness theorems for linear and non-linear dynamical systems. In the last part of the chapter, we will discuss the Hartman-Grobman theorem, the Lyapunov stability theorem and the converse Lyapunov theorem, which are some of the key results in this theory. The discussions in this chapter are based on [3], [5]. 1.1 A Dynamical System Denition 1.1 A smooth dynamical system on E R n is a continuously dierentiable (C 1 ) function Φ : R E E which satises the following two properties: 1. φ 0 : E E is the identity function, φ 0 (x 0 ) = x 0 for all x 0 E, 2. φ t φ s = φ t+s for all t, s R. Here φ t (x) := Φ(t, x). If Φ is a dynamical system on E R n, then the function f(x) = d dt Φ(t, x) (1.1) t=0 denes a vector eld on E and for each x 0 E, Φ(t, x 0 ) is the solution of the initial value problem ẋ = f(x); x(0) = x 0. (1.2) Conversely, if we have a system of ordinary dierential equations, it is not always possible to generate a smooth dynamical system (note that a smooth dynamical system is denes for all time t R). The following two examples illustrate this fact.

9 2 Chapter 1. Dynamical Systems Example 1.2 Cconsider a rst-order dierential equation ẋ = { 1 if x < 0 1 if x 0. The vector eld at a point x 0 R points to the left if x 0 0, and to the right if x 0 < 0. Thus the initial value problem with initial condition x(0) = 0 has no solution. Also, if x 0 > 0, then the solution through x 0 is given by x(t) = x 0 t, but this solution is only valid for t (, x 0 ). Example 1.3 Cconsider a rst-order dierential equation ẋ = 3x 2/3 then any x τ : R R given by x τ (t) = { 0 if t τ (t τ) 3 if t > τ is a solution to the initial value problem with initial condition x(0) = 0. Observe that in Example 1.2, the vector eld is not continuous at 0 and in Example 1.3, the vector eld is not dierentiable at 0. To ensure existence and uniqueness, we need to impose some conditions on the function f, which are discussed in the following theorems. Theorem 1.4 (Existence and Uniqueness of Solutions) If F : R n R n is a C 1 function then the initial value problem ẋ = F (x), x(t 0 ) = x 0, (1.3) with x 0 R n, has an unique solution. More precisely, there exists a constant a > 0 and a unique solution x : (t 0 a, t 0 + a) R n of this dierential equation satisfying x(t 0 ) = x 0. Thus if f is a C 1 function then the solutions satisfy the properties of a smooth manifold for some values of t R. To construct a smooth dynamical systems, we need these propertiesto be satised for all values of t R. The global existence theorem establishes this result. We will rst discuss the denition of topological equivalence and then discuss the theorem. Notation: C 1 (E) is the set of all continuously dierentiable functions on E R n.

10 1.2. Flow and Fixed Points 3 Denition 1.5 Suppose f C 1 (E 1 ) and g C 1 (E 2 ), where E 1 and E 2 are open subsets of R n. Then the two systems of ordinary dierential equations ẋ = f(x) (1.4) ẏ = g(y) (1.5) are said to be topologically equivalent if there is a homeomorphism H : E 1 E 2 which maps the trajectories of (1.4) onto (1.5) and preserves their orientation with time. That is, if x(t) is a solution of (1.4) with initial value x(0) = x 0 and y(t) is a solution of (1.5) with initial value y(0) = H(x 0 ), then y(t) = H(x(t)) for all t R. Theorem 1.6 (Global Existence Theorem) If f C 1 (R n ) and x 0 R n, the initial value problem ẋ = f(x) 1 + f(x) ; x(0) = x 0 (1.6) has a unique solution x(t) dened for all t R, that is, (1.6) denes a dynamical system on R n. Furthermore, (1.6) is topologically equivalent to ẋ = f(x) on R n. This theorem shows that any C 1 vector eld f dened on all of R n leads to a dynamical system on R n. Thus to study the qualitative behaviour of dynamical systems it is enough to nd the solutions of the system of ordinary dierential equations. So, from here on, we will deal with the system of ordinary dierential equations ẋ = f(x) with f C 1 (R n ). Denition 1.7 Phase space of a dynamical system is a space in which all the solution curves (trajectories of all initial conditions) of a system are represented. In particular, if the dimension of the phase space is 2, we call it a phase plane. Denition 1.8 A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase space. Each the trajectories are represented either by a curve, or a point. 1.2 Flow and Fixed Points In this section, we will dene a ow dened by a system of dierential equations and xed points. Fixed points are the points which as the name suggests, remain xed by the vector eld. The study of a dynamical system is mostly focused on the behaviour of the trajectories with initial conditions in a neighbourhood of the xed points.

11 4 Chapter 1. Dynamical Systems Denition 1.9 Let E be an open subset of R n and let f C 1 (E). For x 0 E, let Φ(t, x 0 ) be the solution of the initial value problem ẋ = f(x), x(0) = x 0, dened for all t. Then the set of mappings φ t : E E dened by φ t (x 0 ) = Φ(t, x 0 ) is called the ow of the dierential equation or the ow dened by the dierential equation ẋ = f(x). Denition 1.10 Given a system of ordinary dierential equations ẋ = f(x), a point x R n is called a xed point or an equilibrium point of the system if φ t (x ) = x for all t R, that is, f(x ) = 0. Classication of Fixed points Based on the qualitative behaviour of the trajectories about a xed point, the xed points can be classied as stable or unstable. Denition 1.11 Suppose φ t is the ow of the dierential equation ẋ = f(x) dened for all t R. A xed point x is called stable if for all ɛ > 0 there exists a δ > 0 such that for all x B(x, δ) 1, and t > 0 we have φ t (x) B(x, ɛ). Denition 1.12 A xed point x is asymptotically stable if it is stable and if there exists a δ > 0 such that for all x B(x, δ) we have lim t φ t (x) = x. If for all δ > 0 and x B(x, δ) we have lim t φ t (x) = x, then we say the xed point to be globally asymptotically stable. Denition 1.13 A xed point x is called unstable if it is not stable. Remark 1.14 By denition it is clear that if a xed point is asymptotically stable, then it is stable. But, if a xed point being stable need not necessarily imply it is an asymptotically stable xed point. 1.3 Linear Systems In this section, we will study linear systems of ordinary dierential equations. A linear n-dimensional system is of the form where x R n, A is an n n matrix and ẋ = dx dt = ẋ = Ax, (1.7) 1 B(a, r) is ball of radius r around the point a dx 1 dt. dx n dt.

12 1.3. Linear Systems Fundamental Theorem of Linear Systems Here we will establish that the initial value problem ẋ = Ax; x(0) = x 0 (1.8) has a unique solution for all t R and it is given by x(t) = e At x 0. (1.9) Theorem 1.15 (Fundamental Theorem of Linear Systems) Let A be an n n matrix. Then for a given x 0 R n, the initial value problem (1.8) has a unique solution given by (1.9). Proof If x(t) = e At x 0, then ẋ(t) = d dt eat x 0 = Ae At x 0 = Ax(t) for all t R, Also x(0) = Ix 0 = x 0. Thus x(t) = e At x 0 is a solution. Let x(t) be any solution to the initial value problem and set y(t) = e At x(t). Then ẏ(t) = Ae At x(t) + e At ẋ(t) = Ae At x(t) + e At Ax(t) = 0 for all t R. Thus y is a constant, and as y(0) = x 0, y(t) = x 0. Therefore any solution to the initial value problem is given by x(t) = e At y(t) = e At x Matrix Exponential and Solution of Linear Systems As a consequence of the fundamental theorem of linear systems, to nd the solutions ẋ = Ax, it is sucient to nd the matrix exponential e At. In this section, we will dene the exponential matrix for the following three cases a) the matrix A has all eigenvalues complex, b) the matrix A has only real and may have repeating eigenvalues, c) a general case using Jordan canonical forms. The proofs of all the theorems discussed in this section are given in [3].

13 6 Chapter 1. Dynamical Systems (a) Complex Eigenvalues Theorem 1.16 Let A be a 2n 2n matrix which has all its eigenvalues complex and distinct. Let λ j = a j + ib j and λ j = a j ib j be the eigenvalues of A with the corresponding eigenvectors w j = u j + iv j and w j = u j iv j, j = 1,, n, then {u 1, v 1, u 2, v 2,, u n, v n } is a basis for R 2n, the matrix P = [v 1 u 1 v 2 u 2 v n u n ] is invertible and B 1 0. P 1 AP =.., 0 B n ( ) aj b where B j = j, for all j = 1,, n. b j a j Under the hypothesis of the above theorem, the solution of the initial value problem ẋ = Ax, x(0) = x 0 is given by J 1 0. x(t) = P.. P 1 x 0, 0 J n ( ) where J j = e a jt cos bj t sin b j t, for all j = 1,, n. sin b j t cos b j t (b) Multiple Eigenvalues If all the eigenvalues of A are real and distinct, then A is diagonalizable and it is easy to nd the matrix e At. In this section, we will nd e At when eigenvalues of A are real and repeating. Denition 1.17 Let λ be an eigenvalue of the n n matrix A of multiplicity 1 m n. Then for k = 1,, m, any non-zero solution v of (A λi) k v = 0 is called a generalized eigenvector of A. Denition 1.18 An n n matrix N is said to be nilpotent of order k, if N k 1 0 and N k = 0.

14 1.3. Linear Systems 7 Theorem 1.19 Let A be an n n matrix (real entries) with real eigenvalues λ 1,, λ n repeated according to their multiplicity. Then a )there exists a basis of generalized eigenvectors for R n, b) if v 1,, v n is any basis of generalized eigenvectors for R n, the matrix P = [v 1 v n ] is invertible, A = S + N, λ 1 0. P 1 SP =.., 0 λ n the matrix N = A S is nilpotent of order k n, and SN = NS. Under the hypothesis of the above theorem, the solution of the initial value problem (1.8) is given by e λ 1t 0. x(t) = P. [I. P 1 + Nt + + N ] k 1 t k 1 x 0. (k 1)! 0 e λnt (c) Jordan Forms In general, given a matrix A, its spectrum can contain both real and complex eigenvalues (repeating and/or non-repeating). The Jordan Canonical form in the following theorem will help us to nd e At for a general matrix A. Theorem 1.20 (The Jordan Canonical Form) Let A be a matrix with real eigenvalues λ j, j = 1,, k and complex eigenvalues λ j = a j + ib j and λ j = a j ib j, j = k + 1,..., n. Then there exists a basis {v 1,..., v k, v k+1, u k+1,, v n, u n } for R 2n k, where v j, j = 1,, k and w j = u j + iv j, j = k + 1,, n are generalized eigenvectors of A, such that the matrix P = [v 1,, v k, v k+1, u k+1,, v n, u n ] is invertible and B 1 0. P 1 AP =.. 0 B r where the elementary Jordan blocks B = B j, j = 1,, r are either of the

15 8 Chapter 1. Dynamical Systems form λ λ 1 0 B =... 0 λ for λ as one of the real eigenvalues of A, or of the form D I D I 0 B =... 0 D I 0 0 D ( ) a b where λ = a + ib is an eigenvalue of A, D =, I = ( ) b a ( ) 1 0, 0 = 0 1 With the help of Jordan canonical form and the Fundamental Theorem for Linear Systems, explicit solution of the initial value problem (1.8) can be found, and is given by e B 1t 0. x(t) = P.. P 1 x 0. 0 e Brt Here B i are as given in the above theorem, and e B it can easily be found using the theorems discussed above. 1.4 Nonlinear Systems In the previous section we have studied the linear system ẋ = Ax, and saw that it has an unique solution passing through x 0 in the phase space R n. In this section we will study system of non-linear dierential equations, ẋ = f(x). We have seen in the Section 1.1 the conditions on the function f under which the nonlinear system ẋ = f(x) has a unique solution passing through each point x 0. We will study the local behaviour of the solutions, in particular, we establish the Hartman-Grobman Theorem which gives qualitative behaviour of solutions of the nonlinear system ẋ = f(x) in neighbourhood of xed points. We then establish the Lyapunov theorem and the Converse Lyapunov theorem.

16 1.4. Nonlinear Systems Linearization To analyze the nonlinear system ẋ = f(x), x R n we determine its xed points and analyze the behaviour locally near those points. Denition 1.21 A xed point x 0 is called a hyperbolic xed point if all the eigenvalues of the matrix Df(x 0 ) have non-zero real part. The linear system ẋ = Ax with the matrix A = Df(x 0 ) (the Jacobian of F at x 0 ) is called the linearization of ẋ = f(x) at x 0. Denition 1.22 A xed point x 0 of ẋ = f(x) is called a sink if all of the eigenvalues of the matrix Df(x o ) have negative real part; it is called a source if all of the eigenvalues of Df(x o ) have positive real part; and it is called a saddle if it is a hyperbolic equilibrium point and Df(x o ) has at least one eigenvalue with a positive real part and at least one with a negative real part Hartman-Grobman Theorem The Hartman-Grobman Theorem is a key result, which gives local behaviour of the nonlinear system ẋ = f(x) near a hyperbolic xed point is qualitatively equivalent to that of its linearization ẋ = Df(x 0 ) at the xed point x 0. Theorem 1.23 (Hartman-Grobman Theorem) Let E R n is open which contains the origin and f C 1 (E). Suppose origin is a hyperbolic xed point of the system ẋ = f(x), x E. Then there exists a homeomorphism H of an open set U containing the origin onto an open set V containing the origin such that for each x 0 U there is an open interval I 0 R containing 0 such that for all x 0 U,and t 0 I 0, H φ t (x 0 ) = e At H(x 0 ), where A = Df(x 0 ) is the linearization. Proof Outline of the proof: With out loss of generality we can assume E = R n. Consider the nonlinear system ẋ = f(x) with f C 1 (R n ), f(0) = 0 and A = Df(0). 1. Suppose that the matrix A is written in the form ( ) P 0 A = 0 Q where the eigenvalues of P have negative real part and the eigenvalues of Q have positive real part.

17 10 Chapter 1. Dynamical Systems 2. Let φ t be the ow of the nonlinear system assumed and write the solution ( ) y(t, x0 ) x(t, x 0 ) = φ t (x 0 ) = z(t, x 0 ) where ( ) y0 x 0 = R n, z 0 y 0 belong to the stable subspace of A and z 0 belong to the unstable subspace of A. 3. Dene the functions and Y (y 0, z 0 ) = y(1, y 0, z 0 ) e P y 0 Z (y 0, z 0 ) = z(1, y 0, z 0 ) e P z 0 Then Y = Z (0) = DY (0) = DZ (0) = 0. Since f C 1 (R n ), Y (y(0), z(0)) and Z (y(0), z(0)) are continuously dierentiable. Thus, there exists a constant a > 0 such that and DY (y 0, z 0 ) a DZ (y 0, z 0 ) a on the compact set y z 0 2 s 2 0. The constant a can be chosen as small as we like by choosing s 0 suciently small. Let Y (y 0, z 0 ) and Z(y 0, z 0 ) be smooth functions which are equal to Y (y 0, z 0 ) and Z (y 0, z 0 ) for y z 0 2 (s 0 /2) 2 and zero for y z 0 2 s 2 0. By the Mean Value theorem Y (y 0, z 0 ) a y y 0 2 a( y 0 + z 0 ) and Z(y 0, z 0 ) a y y 0 2 a( y 0 + z 0 ) for all (y 0, z 0 ) R n. Let B = e P and C = e Q. Then if we carry out normalization. we have b = B < 1 and c = C 1 < 1 4. For x = ( ) y R n, z

18 1.4. Nonlinear Systems 11 dene the transformations ( ) By L(y, z) = Cz and T (y, z) = then L(x) = e Ax and locally T (x) = φ 1 (x). ( ) By + Y (y, z) Cz + Z(y, z) Lemma 1.24 There exists a homeomorphism H of an open set U containing the origin onto an open set V containing the origin such that H T = L H. 5. Let H 0 be the homeomorphism dened as in Lemma 1.24 and let L t and T t be the one-parameter families of transformations dened by Dene L t (x 0 ) = e Ax 0 and T t (x 0 ) = φ t (x 0 ) H = 1 0 L s H 0 T s ds. It follows from the above lemma that there exists a neighbourhood of the origin for which L t H T t = = = = t t 0 t 1 0 L t s H 0 T s t ds L s H 0 T s ds L s H 0 T s ds + 1 t L s H 0 T s ds = H 0 L s H 0 T s ds since by the above Lemma 1.24 H 0 = L 1 H 0 T it implies that 0 t L s H 0 T s ds = = 0 t 1 t Thus, H T t = L t H or equivalently H φ t (x 0 ) = e At H(x 0 ) t L s 1 H 0 T s+1 ds L s H 0 T s ds and it can be shown that H is a homeomorphism on R n. With the help of the Hartman-Grobman theorem, we can nd the local behaviour of nonlinear systems locally in a neighbourhood of the hyperbolic xed points.

19 12 Chapter 1. Dynamical Systems Stability and Lyapunov Functions In the previous section we have seen how the behaviour of hyperbolic xed points can be determined. In general the problem of determining the stability of non-hyperbolic xed points is complex. In this section we will study the method by Lyapunov, to study stability of xed points. This method is very useful especially in the case of non-hyperbolic xed points, where the Hartman Grobman Theorem is not applicable. d dv dv Note: V (x(t)) = dt dx ẋ(t) = dx f(x). Theorem 1.25 Given the system ẋ = f(x), where f : R n R n is a locally Lipschitz function. Let origin be a xed point and suppose there exists a positive-denite C 1 function V : R n R whose derivative along solutions of the system satises V = V f(x) 0. (1.10) x Then the origin is a stable xed point. If the derivative of V along solutions of the system satises V = V f(x) < 0, (1.11) x then the origin is asymptotically stable xed point. If in the latter case, V is also radially unbounded, then it is globally asymptotically stable xed point. Proof Suppose V 0 along solutions of the system ẋ = f(x). Given ɛ > 0, choose b < min x =ɛ (V (x)), such a b exists because { x = ɛ} is compact as as V is continuous {V (x) : x = ɛ} is compact. Let S = {x : V (x) b} and B(0, δ) S (can choose such a δ because of continuity). Since V is non increasing along the solutions, therefore, if x(0) B(0, δ) then V (x(t)) b for all t > 0. (1.12) Suppose there exists t 1 > 0 such that x(t 1 ) = ɛ then as b < min x =ɛ (V (x)), we have b < V (x(t 1 )) (1.13) 1.12 and 1.13 are contradictory, therefore for all t 0, x(t) < ɛ, thus we have stability. Now consider V < 0, suppose x(0) δ (δ as above), as V is positive-denite and decreasing along the solutions, it has a limit c 0 as t. We will now show that c = 0. On contrary, suppose that c > 0, then x(t) / {x : V (x) < c}, for all t. In this case the solution evolves in a compact set that does not

20 1.4. Nonlinear Systems 13 contain origin. Suppose the evolution is on the set S = {x : r x ɛ}, r suciently small. Let d := max x S V (x) (1.14) V < 0 and compactness of S implies that d is negative. And as, V d we have V (x(t)) V (x(0)) + d t, thus V eventually becomes smaller than c, which is a contradiction. Now suppose V is radially unbounded, all level sets are bounded and we have x implies f(x). Thus δ as ɛ, thus giving us global asymptotic stability. Denition 1.26 A function V : R n R which satises (1.10) is called a weak Lyapunov function, if it satises (1.11) then it is called Lyapunov function. We will now prove the converse of the above theorem, called the Converse Lyapunov theorem. Theorem 1.27 Suppose there exists β > 0, M > 0 such that each trajectory of ẋ = f(x) satisfy x(t) Me βt x(0), for all t > 0. Then there exists a positive-denite function V : R n R such that along the solutions of ẋ = f(x), V = V x f(x) 0. Proof Fix T > 0 and let z R n. Dene V (z) := T 0 x(t) 2 dt where, x(t) is the solution of ẋ = f(x) with initial condition z, that is, x(0) = z. By hypothesis we have x(t) Me βt z, thus V (z) = T 0 x(t) 2 dt T 0 M 2 2β z 2 M 2 e 2βt z 2 dt

21 14 Chapter 1. Dynamical Systems V (x(τ)) V (x(0)) V = lim τ 0 τ = lim τ 0 τ+t τ 1 = lim τ 0 τ = lim x(t) 2 dt T 0 x(t) 2 dt τ τ 0 1 τ 0 = x(t ) 2 x(0) 2. 0 x(t) 2 1 dt + lim τ 0 τ x(vτ) 2 dv + lim T 1 τ 0 0 τ+t x(t) 2 dt x(vτ + T ) 2 dv As we have x(t) Me βt x(0), for all t > 0, we can choose T suciently large such that x(t ) x(0). Thus we have V < 0.

22 Chapter 2 Coupled Cell Networks In this chapter we will study coupled dynamical systems which are used as models in a wide range of applications including neural networks, biological processes, communications systems, electrical systems. The discussions in this chapter are based on [1]. If we regard the individual unit (cell) in a coupled network as a dynamical system with deterministic dynamics (specied by a vector eld or a map) then a coupled system may be regarded as a set of individual but interacting dynamical systems. Each cell has an output and a number of inputs coming from other cells in the system. A coupled system has a network architecture that can be represented by a directed graph with the vertices corresponding to the cells and each input-output connection as a directed edge. The dierent types of input correspond to dierent edge types in the graph. Consider the graphs shown in Figure 2.1. In Figure 2.1(A), nodes 1 and 3 are dierent A B C Figure 2.1: Coupled Cell Networks A, B, C because node 1 has two edges (one from 2 and one from 3) directed towards it whereas node 3 has only one edge directed towards it. In Figure 2.1(B), all three nodes are same since each node has two edges directed towards it (both edges are of same type). In Figure 2.1(C), all three nodes are same since each node has two edges of same types directed towards it (bold edges are of same type and dotted edges are of same type). To formalize the notion of a coupled network, let us look at the following denitions. Denition 2.1 A cell is a node that admits various types of inputs and that has an output which is uniquely determined by the inputs and the internal state. Denition 2.2 Two cells are of the same class if they give same output, given same inputs and initial state.

23 16 Chapter 2. Coupled Cell Networks Example 2.3 In Figure 2.1(A), each node is a cell. The nodes 1 and 2 are of the same class whereas the nodes 1 and 3 are not of the same class. Remark 2.4 Each cell of a class is represented by the same graphical symbol (circle, triangle, square, etc) dierent from those of other classes of cells. Denition 2.5 Let C = {A, B, C, } be a nite set of (distinct) cell classes. We assume that each X C has a nite (non-zero) number of inputs and suppose the inputs are xed. The collection is called a coupled cell network if 1. C is consistent; that is for every X C, each input type of X comes from a cell class in C, 2. C is indecomposable; that is it is consistent and we cannot write C as a disjoint union of two non-empty consistent sets, 3. There are no unlled inputs. Remark 2.6 A coupled cell network determines an associated directed graph where the nodes of the graph are the cells and there is a directed edge from cell X to cell Y if and only if cell Y receives an input from cell X. Dierent input types will correspond to dierent edge types in the graph. If there are t dierent input types, then there will be t dierent types of edge in the associated graph. 2.1 Coupled Cell Network Dynamics We assume that each individual cell is a dynamical system and see models of continuous, discrete and hybrid dynamics on coupled cell networks Continuous Dynamics Assuming that the cell inputs and outputs depend continuously on time, each cell is modelled by an autonomous ordinary dierential equation. Suppose that we have a coupled cell system with a nite cell set C = {C 1, C 2, C n }, each cell C i C being in the same class and has p-inputs from the cells C i1,, C ip (not necessarily distinct). Then the dynamics of C i will be given by a dierential equation x i = F (x i ; x i1,, x ip ). Here, F : R n (R n ) p R n is a C 1 function and the rst variable x i of vector eld F is the internal variable, while x ip is an input variable from the cell C ip.

24 2.1. Coupled Cell Network Dynamics 17 If there is no dependence on the internal variable, we omit the rst variable x i and write x i = F (x i1,, x ip ). The inputs of similar type are signied by an over line and the vector eld does not change by permuting the such inputs. For example, if x i1, x i2 are of same type and x i3, x i4 and x i5 are of same type then we write Example 2.7 x i = F (x i ; x i1, x i2, x i3, x i4, x i5,, x ip ). 1. If we consider the coupled cell network in the Figure 2.1 A, the continuous time dynamics can be written as follows: x 1 = f(x 1 ; x 2, x 3 ) x 2 = f(x 2 ; x 1, x 3 ) x 3 = g(x 3 ; x 1 ). Here, f : R n (R n ) 2 R n and g : R n R n R n are C 1 functions. 2. If we consider the coupled cell network in the Figure 2.1 B, the continuous time dynamics can be written as follows: x 1 = f(x 1 ; x 2, x 3 ) x 2 = f(x 2 ; x 3, x 1 ) x 3 = f(x 3 ; x 1, x 2 ). Here, f : R n (R n ) 2 R n is a C 1 function. 3. If we consider the coupled cell network in the Figure 2.1 C, the continuous time dynamics can be written as follows: x 1 = f(x 1 ; x 2, x 3 ) x 2 = f(x 2 ; x 3, x 1 ) x 3 = f(x 3 ; x 1, x 2 ). Here, f : R n (R n ) 2 R n is a C 1 function Discrete Dynamics Discrete time coupled cell system is a system of coupled maps updated at regular time intervals. As in the continuous time case, a cell C i can be modelled at time n using a phase space variable x i (n) R N update by x i (n + 1) = F (x i (n); x i1 (n),, x ip (n)). where F : R N (R N ) p R N is a continuous function depending on the internal state x i (n) together with the p inputs to cell C i.

25 18 Chapter 2. Coupled Cell Networks Example If we consider the coupled cell network in the Figure 2.1 A, the discrete time dynamics can be written as follows: x 1 (n + 1) = f(x 1 (n); x 2 (n), x 3 (n)) x 2 (n + 1) = f(x 2 (n); x 1 (n), x 3 (n)) x 3 (n + 1) = g(x 3 (n); x 1 (n)). Here, f : R N (R N ) 2 R N and g : R N R N R n are continuous functions. 2. If we consider the coupled cell network in the Figure 2.1 B, the discrete time dynamics can be written as follows: x 1 (n + 1) = f(x 1 (n); x 2 (n), x 3 (n)) x 2 (n + 1) = f(x 2 (n); x 3 (n), x 1 (n)) x 3 (n + 1) = f(x 3 (n); x 1 (n), x 2 (n)). Here, f : R N (R N ) 2 R N is a continuous function. 3. If we consider the coupled cell network in the Figure 2.1 C, the discrete time dynamics can be written as follows: x 1 (n + 1) = f(x 1 (n); x 2 (n), x 3 (n)) x 2 (n + 1) = f(x 2 (n); x 3 (n), x 1 (n)) x 3 (n + 1) = f(x 3 (n); x 1 (n), x 2 (n)). Here, f : R N (R N ) 2 R N is a continuous function Hybrid Dynamics Most of the physical models of coupled cell networks combine both discrete and continuous dynamics. An example of a hybrid coupled cell system is illustrated below, we will discuss more about the coupled cell networks with hybrid dynamics in the Chapter on Switched Dynamical Systems. Example 2.9 Suppose the cells shown in the Figure 2.2 have one dimensional dynamics (the phase space of each cell is R 1 ). We denote the outputs of C, D by x, y R, respectively. Let the cell C be governed by the ordinary dierential equation ẋ = 1. For discrete dynamics on D, we x x 0 < X R. Let the output of D be given by the map y : R R dened by { x if x < X y(x) = x 0 if x X.

26 2.2. Adjacency Matrix of a Directed Graph 19 x C y D Figure 2.2: Hybrid Coupled Cell System For this system, whenever x(t) < X, x(t) is given as a solution to the ODE ẋ = 1. When x(t) = X, the ODE is reset with initial condition x 0. Necessarily, there is a discontinuity in the solution x(t). 2.2 Adjacency Matrix of a Directed Graph Let G be a directed graph with V (G) = {1,, n} and E(G) = {e 1,, e m }. The adjacency matrix of G denoted by A(G) is an n n matrix whose entries (A(G)) ij are given by (A(G)) ij = number of edges from node i to node j. We refer the eigenvalues of A(G) as the eigenvalues of Graph G. Example 2.10 The adjacency matrix of the coupled cell network B in Figure is given by 1 0 1, and the adjacency matrices of the coupled cell network C in Figure 2.1 for the inputs given by solid arrows and dotted arrows are given by and respectively Symmetry in Systems In this section we will discuss about symmetries in systems. The discussions in this section are based on [2]. Denition 2.11 Let Γ be a group and V be a vector space. An action of Γ in V is a homomorphism ρ : Γ GL(V ). Here, GL(V ) is the group of all automorphisms of the vector space V.

27 20 Chapter 2. Coupled Cell Networks Denition 2.12 Let Γ be a group, γ Γ is a symmetry of the system ẋ = f(x, λ), λ R m and x R n, if for every solution x(t), γx(t) is also a solution. then also; Suppose therefore y(t) = γx(t); ẏ(t) = f(y(t)) = f(γx(t)); ẏ(t) = γ x(t) = γf(x(t)); f(γx(t)) = γf(x(t)) for all the solutions of ẋ = f(x, λ). As the solutions exist for arbitrary initial conditions, the above equation can be re written as f(γx) = γf(x) for all x R n. If Γ act on R n and f : R n R k R n then f is called Γ-invariant if f(γx, α) = γf(x, α) for all γ Γ, x R n. V R n is an invariant subspace of the system ẋ = f(x, λ), λ R m and x R n, if for all x 0 V, φ t (x 0 ) V, for all t R. 2.4 Examples of Coupled cell Networks Considering the coupled cell networks given by the Figures 2.3 and 2.4, we will nd the symmetries, invariant subspaces and linearization of these systems. Let S n the permutation group on n symbols, the action of S n on (R k ) n is as follows: σ((x 1, x 2,, x n )) = (x σ(1), x σ(2),, x σ(n) ) where x i R k. 1 2 n 1 n Figure 2.3: n-cell coupled network

28 2.4. Examples of Coupled cell Networks 21 Example 2.13 The Figure 2.3 is an example of a coupled cell network of n cells, with each cell of the network in the same class. We assume that each cell is a k-dimensional dynamical system, with two inputs of same type (thus the permutation of the inputs of each cell does not change the system). A continuous time dynamical system can be modelled on the network as follows: ẋ 1 = f(x 1 ; x 1, x 2 ) ẋ n = f(x n ; x n 1, x n ) ẋ i = f(x i ; x i 1, x i+1 ), for i {2, 3,, n 1}. Here, x i R k, i = {1, 2,, n} and f : R k (R k ) 2 R k is a C 1 function. If F : (R k ) n (R k ) n is a C 1 function dened by F (x 1, x 2,, x n ) := (f(x 1 ; x 1, x 2 ), f(x 2 ; x 1, x 3 ),, f(x n ; x n 1, x n )), then the above system of ordinary dierential equations can be represented by Ẋ = F (X), where X = (x 1, x 2,, x n ). Symmetries and Invariant spaces Suppose σ S n is a symmetry of the system Ẋ = F (X), then (σx) = F (σx). That is, ẋ σ(1) = f(x σ(1) ; x σ(1), x σ(2) ) ẋ σ(n) = f(x σ(n) ; x σ(n 1), x σ(n) ) ẋ σ(i) = f(x σ(i) ; x σ(i) 1, x σ(i)+1 ), i {2, 3,, n 1}. Thus σ is either the identity element of S n, or is given is given by σ(i) = n+1 i, where i {1, 2,, n}. Hence there are only two invariant subspaces of the system and are given by E 1 = {(x 1, x 2,, x n ) (R k ) n x 1 = x 2 = = x n } and E 2 = {(x 1, x 2,, x n ) (R k ) n x i = x γ(i) ; i {1, 2, n}}. Linearization Since the connections are symmetric we have f(x; y, z) = f(x; z, y), for all x, y, z R k, therefore f = f at all the points. Suppose P y z (Rk ) n is a xed point on the invariant subspace E 1 called the fully synchronized subspace, then P = (p, p,, p), where p R k. If f x p = A and f y p = f z p = B, where both A and B are k k matrices, then the linearization of Ẋ = F (X) about the xed point P is given by Ẋ = J(P )X, where J(P ) is the Jacobian matrix,

29 22 Chapter 2. Coupled Cell Networks given by A + B B B A B J(P ) = B A B B A + B A B B A B 0 B = A B 0 B A B 0 = I A + C B. Here, denotes the Kronecker product (dened in Appendix A), I is n n identity matrix and C =. is the adjacency matrix of the graph dened by the coupled cell network given by Figure 2.3. Suppose λ is an eigenvalue of C and µ an eigenvalue of B, then by the properties of Kronecker product λµ is an eigenvalue of C B. 1 2 n 1 n 2n 2n 1 n+2 n+1 Figure 2.4: 2n-cell coupled network Example 2.14 Let us now consider the coupled cell network of 2n cells given by the Figure 2.4. As in the Example 2.13, we assume that each cell is a k-dimensional dynamical system, with two inputs of same type. A continuous

30 2.4. Examples of Coupled cell Networks 23 time dynamical system can be modelled on the network as follows: ẋ 1 = f(x 1 ; x 2n, x 2 ) x 2n = f(x 2n ; x 2n 1, x 1 ) x i = f(x i ; x i 1, x i+1 ), i {2, 3,, 2n 1}. Here, x i R k, i = {1, 2,, 2n} and f : R k (R k ) 2 R k is a C 1 function. If F : (R k ) 2n (R k ) n is a C 1 function dened by F (x 1, x 2,, x 2n ) := (f(x 1 ; x n, x 2 ), f(x 2 ; x 1, x 3 ),, f(x 2n ; x 2n 1, x 1 )), then the above system can be represented by Ẋ = F (X), where X = (x 1, x 2,, x 2n ). Symmetries and Invariant spaces Suppose σ S n is a symmetry of the system (σx) = F (σx), that is ẋ σ(1) = f(x σ(1) ; x σ(2n), x σ(2) ) ẋ σ(2n) = f(x σ(2n) ; x σ(2n 1), x σ(1) ) Ẋ = F (X), then ẋ σ(i) = f(x σ(i) ; x σ(i) 1, x σ(i)+1 ), i {2, 3,, 2n 1}. Thus, σ takes one of the following two forms σ(i) = σ(i) = { σ(i 1) 1 if σ(i 1) 1 2n if σ(i 1) = 1. { σ(i 1) + 1 if σ(i 1) 2n 1 if σ(i 1) = 2n. The collection of all these symmetries forms a group of 4n elements (dihedral group D 2n ). If σ is a symmetry then E = {(x 1, x 2,, x 2n ) (R k ) 2n x i = x σ(i) i = 1, 2, n} is an invariant subspace of the coupled set network determined by the Figure 2.4. Linearization As seen in the Example 2.13, we have f(x; y, z) = f(x; z, y) for all x, y, z R k, therefore f = f at all the points. Suppose P y z (Rk ) 2n is a xed point on the invariant subspace E 1 = {(x 1, x 2,, x 2n ) (R k ) 2n x 1 = x 2 = = x 2n }, the fully synchronized subspace, then P = (p, p,, p) where p R k. If f x p = A

31 24 Chapter 2. Coupled Cell Networks and f y p = f z p = B, where both A and B are k k matrices, then the linearization of Ẋ = F (X) about the xed point P is given by Ẋ = J(P )X, where J(P ) is the Jacobian matrix, given by A B B B A B J(P ) = B A B B B A A B B 0 A B 0 B = A B 0 B A B B 0 = I A + C B Here denotes the Kronecker product, I is 2n 2n identity matrix and C =. is the adjacency matrix of the graph dened by the coupled cell network given by Figure 2.4. Eigenvalues of C Let S =., then C = S + S t. It is easy to see that S t = S 2n 1 and that the eigenvalues of S are the 2n th roots of unity. Let {ω, ω 2,, ω 2n } be the spectrum of S, then ω k + ω 2n k is an eigenvalue of C, because if v is a corresponding eigenvector for eigenvalue ω k of S then ω 2n k is an eigenvalue of S 2nk k with eigenvector v, therefore ω k + ω 2nk k is an eigenvalue of C = S + S 2n 1 with eigenvector v. We have and ω k = e 2πik 2n ω 2nk k = e 2πi(2n k) 2n = cos( 2πk 2n ) + i sin(2πk 2n ) = cos( 2πk 2n ) i sin(2πk 2n )

32 2.4. Examples of Coupled cell Networks 25 therefore the spectrum of C is {2 cos( 2πk ) k = 1, 2, 2n} which are all real. 2n Theorem 2.15 Suppose λ is an eigenvalue of C with eigenvector v, and µ is an eigenvalue of A + λb with eigenvector v 1 then µ is an eigenvalue of J = I A + C B with eigenvector v v 1. Proof µ is an eigenvalue of A + λb with eigenvector v 1, therefore we have (A + λb)(v 1 ) = µv 1 Av 1 = (µ λb)v 1. Now, (I A + C B)(v v 1 ) = (I A)(v v 1 ) + (C B)(v v 1 ) = (v Av 1 ) + (Cv Bv 1 ) = (v (µv 1 λbv 1 ) + (λv Bv 1 ) = µ(v v 1 ) λ(v Bv 1 ) + λ(v Bv 1 ) = µ(v v 1 ) Therefore the eigenvalues of J(P ) are the eigenvalues of A + λb, where λ is an eigenvalue of the adjacency matrix, of the underlying graph.

33

34 Chapter 3 Switched Dynamical Systems 3.1 Introduction Dynamical systems that are described by an interaction between continuous and discrete dynamics are called hybrid systems (or, piecewise smooth dynamical systems). There is an important class of hybrid systems consists of continuous dynamical systems with isolated discrete discontinuities known as switching events. Such systems are known as switched systems. The rule governing the switching events is called a switching signal. The two important classes of switching events are state-dependent and time-dependent. Let us look at the following examples of hybrid systems. The discussions in this chapter are based on [4]. The motion of an automobile can be described by the follow- Example 3.1 ing model x 1 = x 2 x 2 = f(a, q) where x 1, x 2, a 0 denote the position, velocity, acceleration input, repectively. Also, q {1, 2, 3, 4, 5, 1, 0} denote the gear shift position. When q = 1, the function f is negative and decreasing in a; when q = 0, f is negative and independent of a; and when q > 0, f is decreasing in q. The variables x 1 and x 2 are the continuous state and q is the discrete state. Clearly the discrete transitions aect the continuous trajectory. Example 3.2 Another physical example of a hybrid system is a temperature control system, say a heater and a thermostat. The most important variables that govern this system are the room temperature (continuous) and the operating mode of the heater (discrete - on/o). There is an interaction between these two variables for the proper functioning of the temperature control system. To study hybrid systems, various approaches are followed by scientists in dierent elds. The approach followed in this project is used by control theorists [4], where a hybrid system is by regarded as a continuous system with

35 28 Chapter 3. Switched Dynamical Systems switching events and a greater emphasis is on the properties of the continuous state dynamics. Thus, we are interested in continuous time systems with isolated discrete switching events. Such systems as switched systems Classication of Hybrid and Switched Systems The switched systems can be classied into two classes depending on how the switching events occur state-dependent versus time-dependent; autonomous versus controlled. We will now briey describe all the four types of switching events. In this chapter, we will concentrate on state-dependent and time-dependent switching events State-dependent switching In this type of switching, the switching event is dependent on the state of the system. A switching surface is an (n 1)-dimensional smooth manifold. Suppose the state space is partitioned into operating regions (Ω p, p P, where P an indexing set) by means of switching surfaces (S i, i I, where I is an indexing set). For each p P, let ẋ = f p (x), where f p : R n R n is a smooth map, and let σ : R n R n be a map. Then the state-dependent switched system is specied by a family of switched surfaces: {S i, i I}; a family of operating regions: Ω p, p P, along with a family of continuous-time dynamical systems ẋ = f p (x), p P. These are known as a family of subsystems operating in the respective region Ω p ; a reset map: σ : R n R n. Let x(0) Ω p0, for some p 0 P be a given initial condition. The dynamics of the state-dependent switched system is governed by the continuous-time system ẋ = f p0 (x) until the trajectory hits a switching surface, at a point x 1 say. The reset map σ is applied to x 1 and suppose σ(x 1 ) Ω p1 for some p 1 P. With the initial condition x 1, the dynamics of the state-dependent switched system is governed by the continuous-time system ẋ = f p1 (x) until the trajectory hits another switching surface, and then the process is repeated. The instantaneous jumps by σ as soon as the trajectory hits the switching surface are called impulse eects. A special case is when such impulse eects are absent, that is, the reset map is the identity. In this case, the trajectory is continuous everywhere, but in general loses dierentiability when it passes through a switching surface.

36 3.1. Introduction 29 Ω 1 S 1 Ω 2 S 3 Ω 5 S 2 Ω 4 Ω 3 Figure 3.1: An Example of State Dependent Switching in R 2 Example 3.3 In the Figure 3.1, the curves S 1, S 2, S 3 are the switching curves which divide the phase space R 2 in to regions Ω 1, Ω 2, Ω 3, Ω 4, Ω 5. The solid curves denote the trajectories in the respective region and the dashed curves denote the jumps according to a reset map σ Sliding Modes When there is no impulse eect it is possible that the trajectories get stuck to the switching surfaces and slide along the switching surface. In this section we will explain this behaviour using an example.. x = f 1 (x). x = f 2 (x). x = f 1 (x). x = f 2 (x) S S Figure 3.2: (a) Crossing Switching Surface, (b) Sliding Mode

37 30 Chapter 3. Switched Dynamical Systems Suppose, a state dependent switching is described by a single switching surface S and two subsystems ẋ = f i (x), i = 1, 2, on each side of the switching surface S. If there is no impulse eect and at point where the the trajectory hits the switching surface x S both the vectors f 1 (x) and f 2 (x) point in the same direction as in the Figure 3.2 (a), then the solution curve crosses over the switching surface. But, if the vector elds f 1 and f 2 point towards the switching surface S as in Figure 3.2 (b), then the behaviour of the trajectories cannot be described in the same way as in the case where the trajectories cross over the switching surface. Once the trajectory hits the switching surface S it cannot leave S because of the behaviour of the vector elds, thus the only possible solution is to slide along S. This behaviour is called as a sliding mode Hysteresis Switching To approximate the sliding mode behaviour and to maintain a nite time gap between two consecutive switching events, we follow a method called as hysteresis switching. The following is an illustration of hysteresis switching using the example in the Figure 3.2 (b). We construct two regions Ω 1 and Ω 2 overlapping in the strip between S 1 and S 2 as shown in the Figure 3.3 (a), and follow the subsystem ẋ = f i (x) in the region Ω i. We switch when the trajectory hits the surfaces S 1 or S 2. A typical trajectory obtained by hysteresis switching looks like one in the Figure 3.3(b). Ω1 Ω2 Ω1 Ω2 S S S 1 2 S S S 1 2 Figure 3.3: (a) Switching Regions, (b) A Typical Trajectory

38 3.1. Introduction Time-dependent switching In this type of switching, the switching event is dependent on time. Let {f p : R n R n, p P} be a family of functions which are atleast locally Lipschitz (to ensure existence of solutions). These systems give rise to a family of systems ẋ = f p (x), p P (3.1) When each of the systems in (3.1) are linear, that is, f p (x) = A p x, p P (where A p is a real n n matrix), we have the following ẋ = A p (x), p P (3.2) A switching signal is a piece-wise constant function σ : [0, ) P, which is right continuous everywhere, with the discontinuities known as switching times. A time-dependent switched system is the family of systems (3.1) along with a switching signal. The role of σ is to specify, at each time instant t, the index σ(t) P of the system from the family of subsystems which will be active at time t. Thus a switched system with time-dependent switching is described as follows ẋ(t) = f σ(t) (x(t)). (3.3) In the case when each of the subsystems is linear we have ẋ(t) = A σ(t) x(t). (3.4) σ = 1 σ = 2 σ = 1 σ = 2 t 1 τ 1 t 2 τ 2 t Figure 3.4: An Example of Time Dependent Switching Example 3.4 The Figure 3.4 is an example of a switching signal in the case where P = {1, 2}. Here at t = 0 the system ẋ = f(x) is active and the switching times are t 1, t 1 + τ 1, t 1 + τ 1 + t 2, t 1 + τ 1 + t 2 + τ 2, Autonomous and controlled switching Autonomous switching is a situation where we have no direct control over the switching mechanism that triggers the discrete events. This includes systems with state-dependent switching in which locations of the switching surfaces are not determined, as well as systems with time-dependent switching in which

39 32 Chapter 3. Switched Dynamical Systems the rule that denes the switching signal is unknown. For example, abrupt changes in system dynamics may be caused by unpredictable environmental factors or component failures. In contrast to the above situations, the switching may be imposed by the designer in order to achieve a desired behavior of the system. In this case, we have direct control over the switching mechanism (which can be statedependent or time-dependent) and may adjust it as the system evolves. 3.2 Stability of Switched Systems In this section we discuss the stability analysis of switched systems of the form (3.3). Let us consider an example of state-dependent switching between two systems in the plane. Suppose that the origin (0, 0) is an asympotically stable equillibria for the two individual subsystems, with trajectories as shown in A and B of Figure 3.5. For dierent choices of the switching signal, the switched system can be asymptotically stable or unstable, as shown in C and D of Figure 3.5. In this state-dependent system, the switching surfaces are the coordinate axes and the reset map is the identity map. A B C D Figure 3.5: Switching between stable systems Figure 3.6 illustrates the case when the origin (0, 0) is an unstable equillibria for the two individual subsystems. Again, For dierent choices of the switching signal, the switched system can be asymptotically stable or unstable. In this state-dependent system as well, the switching surfaces are the coordinate axes and the reset map is the identity map. The above example highlights two interesting features of the switched systems: switching may destabilize a switched system even if all individual subsystems are stable;