Existing Methods for Discrete-Time Dynamical Systems

The study of switching discrete-time systems has primarily focused on two different approaches: switching linear systems (“Jump Linear Systems”) and stochastically switching networks of maps (much like the switching networks in flows). We will review both of these approaches, as they both provide important pieces of foundation that we build on later in this work.

1.5.3.1 Jump Linear Systems Jump linear systems are both more specific and more general than our stochastically switching discrete time systems described by (1.7). In

[55], a jump linear systemo _{is described by:}

x(k+ 1) =A(σ(k))x(k), (1.12)

where {σk} is a finite-state Markov chain form process with transition matrix P, and the

individual mode matrices A(1), A(2), . . . , A(N) are real, invertible d×d matrices (where d

is the dimension of the state vector x(k)). This is more general than our setting, because the switching process is a Markov process, it is not assumed to be i.i.d. (though this i.i.d. assumption can be relaxed in our setting in a straightforward way). However, it is also more specific, because the system only switches between linear dynamical regimes, and switching occurs at every time step.

In [54] and [55] Fang and Loparo provide concise recipes for deducing the δ−moment stability and sample path stability of jump linear systems, as well as the relationships be- tween these types of stability. They study the stability of these systems using Lyapunov exponents. In general, a Lypaunov exponent gives a measure of the exponential convergence or divergence of nearby trajectories (that is, whether or not two initial conditions that are close by come together or separate with time). In discrete-time, this Lyapunov exponent is computed as λ= lim n→∞ 1 n n−1 X i=0 ln|f0(x(i))|. (1.13)

For linear systems, like in (1.12), this reduces to

λ= lim n→∞ 1 n n−1 X i=0 ln||A(i)||= lim n→∞ 1 nln|| n−1 Y i=0 A(i)||. (1.14)

The authors concern themselves with two types of stability, almost sure stability, and δ-

moment stability. We say that a stochastically evoloving discrete-time dynamical system is

o_{Indeed, jump linear systems have been extensively studied in the literature, and we can barely scratch}

almost surely stable if

P( lim

k→∞||x(k)||= 0) = 1, (1.15)

where|| · ||is a suitible norm in_RN _{and P denotes a probability measure with respect to the} σ-algebra induced by the switching process. In words, this means that for almost all initial conditions and stochastic sequencesp, the norm of the trajectory converges to 0. This is a convergence problem, hence almost sure stability can be reduced to computing the “top” or “sample path” Lyapunov exponent:

λ= lim

k→∞

1

kln||A(σ(k−1))· · ·A(σ(2))A(σ(1))||. (1.16)

As (1.15) requires showing that almost every stochastic sequence converges, it is the most informative (and therefore least conservative) measure of stability, however it is also the hardest to prove, in general.

Because of the technical challenges in determining almost sure stability of a stochastic system, it is helpful to consider other, more conservative types of stability. One such type of stability, δ-moment stability, holds if

lim

k→∞E

||x(k)||δ

= 0, (1.17)

where E [·] denotes expectation with respect to the σ-algebra induced by the switching pro- cess. When δ = 2 and (1.17) holds, we say that the switching system is asymptotically

mean-square stable. Like almost sure stability, there is aδ-moment stability analogue to the

sample path Lyapunov exponent. The topδ-moment Lyapunov exponent is computed as

g(δ) = lim k→∞ 1 kln E ||x(k)||δ , (1.18)

p_{Almost all in the sense that the set of initial conditions and stochastic sequences for which this does not}

which for jump linear systems becomes g(δ) = lim k→∞ 1 kln E ||A(σ(k−1))· · ·A(σ(2))A(σ(1))||δ .

δ-moment stability is particularly useful in practice, because it reduces the study of a stochas- tic system to the study of a deterministic one (which significantly more tools have been developed for). The authors go on to show the relationships between these two types of stability and under what conditions they are equivalent. Further discussions of these types of stability can be found in Appendix D.

1.5.3.2 Stochastic Synchronization for Coupled Maps Much like [125] and [123] sought to establish a theory for switching networks of ODEs, [114, 115, 1] (to name a few) worked to establish a similar theory for switching networks of discrete-time dynamical systems. The significant seminal work is [114], which aims to present a master stability function for the synchronization of stochastically switching networks of maps (like [109] did for static, diffusively coupled networks of ODEs). It utilizes tools from probability theory, stability theory, and matrix analysis to present concise criteria that ensure the mean-square stability of a switching network of maps. Their setup is as follows:

xi(k+ 1) =F(xi(k))− N

X

j=1

Mij(k)H(xj(k)) for i= 1,2, . . . , N, (1.19)

wherex(k)∈_Rn_{is ann-dimensional vector composed of the states of one node,F:}

Rn →Rn

is the vector-valued function governing the individual dynamics of each node, M(k) is the random (i.i.d.), time-varying connectivity matrix for the network, and H: _Rn _→

Rn is the

vector-valued coupling function between the nodes.

The authors linearize the system about the synchronization solution to construct varia- tional equationsq_{in order to determine the local stability of that solution. Next, the authors}

construct the autocorrelation matrix Ξ(k) =Eξ(k)ξ(k)T, which is a symmetric, positive

q_{Thevariationis denoted by ξ}

semidefinite matrix with traceE[||ξ(k)||2_{]. This trace (the sum of the eigenvalues) converg-}

ing to zero (meaning all of the variation converge to zero, i.e the nodes converge to a common trajectory) means that the system is mean-square stable. The analysis of the autocorrelation matrix reduces the stochastic problem to the stability of a higher dimensional, static system that can be decomposed into the eigendirections transverse to the synchronous manifold, and studied using classical methods in dynamical systems.