Now we study how Lyapunov conditions predict the stochastic stability properties for random solutions associated with the stochastic difference equation x+ = f (x, κ(x), v)
Robust stochastic stability under discontinuous stabilization Chapter 4
(4.4). We could consider random solutions of system (4.4) directly, but there are the following two issues. First, since in Assumption 4.1 we have not assumed that the control law κ :X → U is a measurable function, there is no guarantee that the iteration
xi+1(ω) := f (xi(ω), κ(xi(ω)), vi(ω)), for i∈ Z≥0, (4.10)
x0(ω) := ξ0 ∈ X , yields measurable functions xi : Ω→ X , for i ∈ Z≥0. Secondly, even when the function κ is measurable, the behavior of the random solution that is generated by the iteration (4.10) may not accurately predict the behavior of the system in the presence of small, random or worst-case, perturbations. For these reasons, we choose to define a notion of generalized random solution. Generalized random solutions do not require the control law κ to be measurable and, as we will see, their behavior predicts the behavior of the system under small, random or worst-case, strictly causal perturbations.
This later feature is also present for generalized solutions to non-stochastic differ-ence inclusions as introduced in [7]. In the case of non-stochastic difference equa-tions x+ = f (x, κ(x)), generalized solutions are the solutions of the difference inclusion x+ ∈ f(x, K(x)), with K being the controller regularization as defined in (4.5). It fol-lows from [7] that the existence of a continuous Lyapunov function for x+ = f (x, κ(x)) implies the existence of a continuous Lyapunov function for x+ ∈ f(x, K(x)) and even for an inflation of this later system. However, Example 4.1 suggests that this result does not hold for x+ ∈ f(x, K(x), v) (4.4) in the stochastic case. This fact and the results of the previous section motivate an alternative definition of generalized solutions in the stochastic case, that turns out to generate the same solutions as x+∈ f(x, K(x)) in the non-stochastic case and yet yields a robust Lyapunov result in the stochastic case.
Our strictly causal generalized random solutions are random solutions to the
stochas-Robust stochastic stability under discontinuous stabilization Chapter 4
The types of perturbations to which the behaviors of the solutions will be robust are strictly causal perturbations that appear in the stochastic difference inclusion
where fδ and Kδ are the inflations of f and K respectively, as defined in (4.7), (4.8).
The motivation for considering the above inclusions is that the selection u ∈ Kδ(x)
“does not depend” on the current random input v. This property is what we call strict causality. We notice that, for each (x, u) ∈ X × U, if u ∈ Kδ(x), then we have u+ ∈ Kδ(x+).
Let us first assert certain regularity properties of the set-valued mapping Gδ : X × U × V ⇒ X × U in (4.12), by exploiting Standing Assumption 4.1. The same regularity properties hold for G0 defined in (4.11).
Proposition 4.2 For all continuous functions δ :Rn→ R≥0, the set-valued mappingGδ
defined in (4.12) satisfies the following regularity conditions:
1. for any v ∈ V the mapping (x, u) 7→ Gδ(x, u, v) is outer semicontinuous;
2. the mappingv 7→ graph(Gδ(·, ·, v)) := {(x, u, y) ∈ X ×U ×(X ×U) | y ∈ Gδ(x, u, v)}
Robust stochastic stability under discontinuous stabilization Chapter 4
is measurable;
3. the mapping Gδ is locally bounded.
Since Proposition 4.2 shows that G0 in (4.11) and Gδ in (4.12) have the same regu-larity conditions given in [42, Standing Assumption 1] and Standing Assumption 3.1, we can define the notion of solutions for the stochastic difference inclusion (4.12) having (ex-tended) state variable z := (xu)∈ (X × U). We also define generalized random solutions to (4.4) as the solutions for the regularized stochastic difference inclusion (4.11).
We now show that (4.9) established in Theorem 4.1 is closely related to a Lyapunov condition for the extended stochastic difference inclusion (4.12), with Lyapunov function V :¯ X × U → R≥0 relative to the compact attractor ¯A ⊂ X × U explicitly defined in the following preliminary result.
Lemma 4.3 For any δ ∈ PD(A), the function W : X × U → R≥0 defined as
W (x, u) :=|(x, u)|graph(Kδ) (4.13)
is such that W (x, u) = 0⇐⇒ u ∈ Kδ(x). The set
A := {(x, u) ∈ X × U | x ∈ A, (x, u) ∈ graph(K)} ⊆ X × U¯ (4.14)
is compact. For any δ ∈ PD(A), Γ ∈ K∞ and V : X → R≥0 upper semicontinuous (respectively, continuous), the function ¯V :X × U → R≥0 defined as
V (x, u) := Γ(V (x)) + W (x, u)¯ (4.15)
is upper semicontinuous (respectively, continuous). If there exist α1, α2 ∈ K∞ such that α1(|x|A) ≤ V (x) ≤ α2(|x|A) for all x ∈ X , then there exist ¯α1, ¯α2 ∈ K∞ such that
Robust stochastic stability under discontinuous stabilization Chapter 4
¯
α1(|(x, u)|A¯)≤ ¯V (x, u)≤ ¯α2(|(x, u)|A¯) for all (x, u)∈ (X × U).
We now state the main result of this chapter. Under the conditions of Standing Assumption 4.1, we can establish that the Lyapunov condition in Assumption 4.1 is robust to sufficiently small strictly causal perturbations. In particular, the next result establishes that the Lyaunov conditions in Assumption 4.1 implies the existence of a Lyapunov function for a perturbed version of (4.4).
Theorem 4.2 If Assumption 4.1 holds, then δ ∈ PD(A), Γ ∈ K∞ and % ∈ PD(A) satisfying (4.9), W in (4.13), ¯A in (4.14), and ¯V in (4.15) are such that for all (x, u)∈ X × U we have
Z
V
g∈Gmaxδ(x,u,v)
V (g)µ(dv)¯ ≤ ¯V (x, u)− ¯%(x, u), (4.16)
with %¯∈ PD( ¯A) defined as ¯%(x, u) := W (x, u)/2 + %(x).
Proof: With ¯V := Γ(V ) + W as in (4.15), which is such that ¯V ∈ PD( ¯A) according to Lemma 4.3, the Lyapunov condition (4.16) reads as
Z
V
(gg12)max∈Gδ(x,u,v)
(Γ(V (g1)) + W (g1, g2)) µ(dv) ≤ Γ(V (x)) + W (x, u) − ¯%(x, u). (4.17)
We notice that for any δ ∈ PD(A), we have ¯A = {(x, u) ∈ X × U | x ∈ A, (x, u) ∈ graph(K)} = {(x, u) ∈ X × U | x ∈ A, (x, u) ∈ graph(Kδ)}. Now, for any δ ∈ PD(A), if u /∈ Kδ(x) then by definition (4.12), we get Gδ(x, u, v) =∅, so that max(gg12)∈∅Γ(V (g1))+
W (g1, g2) = 0. Then (4.17) can be trivially satisfied by choosing ¯%(x, u) := W (x, u)/2 +
%(x), so that we get Γ(V (x))− %(x) + W (x, u))/2 ≥ W (x, u))/2 ≥ 0. We notice that
¯
%∈ PD( ¯A). While if u ∈ Kδ(x), then W (x, u) = 0 in view of Lemma 4.3 and, according to (4.12), g2 ∈ Kδ(g1), and hence W (g1, g2) = 0 also in view of Lemma 4.3. Therefore we
Robust stochastic stability under discontinuous stabilization Chapter 4
get max
u∈Kδ(x)
Z
V
g1∈fmaxδ(x,u,v)Γ(V (g1))µ(dv) ≤ Γ(V (x)) − %(x), which is equivalent to (4.9).
It follows from the regularity properties of Gδ established in Proposition 4.2, the inequality (4.16) in Proposition 4.2 and the definition of Lyapunov function that ¯V (4.15) is an Lyapunov function relative to ¯A defined in (4.14) for (4.12). In essence, the above result establishes the robustness of global asymptotic stability in probability property even under the action of a discontinuous control law for the closed loop stochastic system, provided the perturbation is sufficiently small and strictly causal. Similar results for the recurrence property also exist and we refer the reader to [72] for more details.
Chapter 5
Stochastic hybrid systems
5.1 Introduction
Stochastic hybrid systems (SHS) allow continuous-time evolution of the states, discrete-time events and probabilistic behavior. In SHS, randomness can affect the continuous-time dynamics, the discrete-continuous-time dynamics or the transition between the dynamics. Con-sequently, SHS models with varying degrees of complexity are studied in the literature.
Frameworks for modeling SHS are in [23], [25], [24] and [78]. SHS models arise frequently in the context of complex systems like air traffic management systems, networked control systems and systems biology. See [79], [80], [81] and [26] for more details. The recent survey paper [27] presents a unified modeling framework for the various SHS represen-tations in the literature and addresses stability related issues. In particular, important topics that are well studied in the case of non-stochastic hybrid systems like sufficient conditions for stability, weak sufficient conditions for stability, invariance principle, ro-bust stability conditions and converse Lyapunov theorems are analyzed in detail in [27]
for stochastic hybrid systems that produce unique solutions.
In this chapter, the class of systems we study are stochastic hybrid systems modeled by
Stochastic hybrid systems Chapter 5
set-valued mappings for which the randomness is restricted to the discrete-time dynamics.
The system model we study can account for spontaneous transitions, forced transitions and probabilistic resets. We adopt the framework for modeling SHS with non-unique solutions proposed in [25] and [82]. This class of systems covers other frameworks such as piecewise-deterministic Markov processes (PDMP) and Markov jump systems.
The main goal of this chapter is to introduce the reader to a class of stochastic hybrid systems modeled by set-valued mappings and develop results related to the invariance principle. We use the invariance principle to develop weak sufficient conditions for sta-bility and recurrence. As a consequence of the invariance principle we also establish sufficient conditions for stochastic stability properties that rely on Lyapunov-like func-tions satisfying strict decrease properties. The results in this chapter are from [83]. Other aspects related to stability theory like converse theorems and robustness are studied in detail in a subsequent chapter.