Compositional Approximations of Interconnected Stochastic Hybrid
53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA
Compositional Approximations of Interconnected Stochastic Hybrid Systems Majid Zamani
Abstract— This paper provides a compositional approach for approximations of the interconnection of a class of stochastic hybrid systems including both jump linear stochastic systems and linear stochastic hybrid automata. The approximation is based on the recently developed notions of stochastic (bi)simulation functions using which one can quantify the error between original stochastic hybrid systems and their abstractions. Given stochastic (bi)simulation functions between stochastic hybrid subsystems and their corresponding approximations, we provide a stochastic (bi)simulation function between an interconnection of those stochastic hybrid subsystems and that of their corresponding approximations. Consequently, one can leverage the proposed results in this paper to perform the analysis and synthesis over the interconnection of abstract subsystems and then carry the results over the interconnection of concrete ones. We illustrate the effectiveness of the proposed results here by computing a stochastic (bi)simulation function between an interconnection of two identical jump linear stochastic subsystems and that of their corresponding approximations by just using the stochastic (bi)simulation function between one of the jump linear stochastic subsystems and its corresponding approximate abstraction.
I. I NTRODUCTION Stochastic hybrid systems are a general class of dynamical systems consisting of continuous and discrete dynamics subject to probabilistic noise and events. In the past few years, these systems have become ubiquitous in a variety of fields due to providing a rigorous and general framework for modeling of many safety critical applications. Examples of those applications include biochemistry [1], air traffic control [2], systems biology [3], and communication networks [4], that can be modeled and treated as stochastic hybrid systems. Reachability and safety are two fundamental concepts in the study of stochastic hybrid systems that have received significant attentions in the last few years because of their applications in real life systems [5]. There exist some results in the literature such as [6], [7] providing methodologies solving reachability problems, safety ones, or their combinations. Unfortunately, the computational complexity of the existing techniques solving those problems scales exponentially with the dimension of the continuous state space making them intractable for large-scale stochastic hybrid systems. A promising direction to overcome this complexity is to abstract the original concrete system by a simpler one (lower dimension) using an appropriate relation quantifying the error between them. By taking into account the quantified error, one can perform the analysis and synthesis over the simpler system and then carry the results over the concrete one. M. Zamani is with the Department of Electrical Engineering and Information Technology, Technische Universit¨at M¨unchen, D-80290 Munich, Germany. Email: [email protected]. URL: http://www.hcs.ei.tum.de
978-1-4673-6088-3/14/$31.00 ©2014 IEEE
The recent work in [8] provides a notion of stochastic (bi)simulation function providing a metric which quantifies the error between the original stochastic hybrid systems and their approximations. Particularly, the authors in [8] develop a theory of approximate (bi)simulation for a class of stochastic hybrid automata whose continuous dynamics are modeled by continuous-time stochastic differential equations and the switches are modeled as Poisson processes. In this work, we provide a compositional technique for approximation of the interconnection of the same class of stochastic hybrid automata. Given stochastic (bi)simulation functions between each stochastic hybrid subsystem and its corresponding approximation, we provide a stochastic (bi)simulation function between the interconnection of stochastic hybrid subsystems and that of their corresponding approximations. Therefore, one can quantify the error between the interconnection of stochastic hybrid subsystems and that of their corresponding abstractions by just knowing the error between each subsystem and its corresponding abstraction. As a consequence, one can leverage the proposed results here to solve particularly safety/reachability problems over the interconnection of abstract systems and then carry the results over the interconnection of concrete ones. We illustrate the results of this paper by computing a stochastic (bi)simulation function between an interconnection of two identical jump linear stochastic subsystems (3rd order each) and that of their corresponding abstractions (1st order each) by using just the stochastic (bi)simulation function between one of the subsystems and its corresponding approximation. II. S TOCHASTIC H YBRID S YSTEMS A. Notations The identity map on a set A is denoted by 1A . The symbols N, R, R+ , and R+ 0 denote the set of natural, real, positive, and nonnegative real numbers, respectively. Given a vector x ∈ Rn , we denote by xi the i–th element ofpx, and by kxk the Euclidean norm of x, namely, kxk = x21 + x22 + . . . + x2n . We denote by In and 0n the identity matrix and zero vector in Rn×n and Rn , respectively. For any x, y ∈ Rn , the relation x ≤ y is defined by xi ≤ yi for all i ∈ {1, . . . , n}. The relations , and ≥ are defined in the same manner for vectors. Given Pn a matrix P = {pij } ∈ Rn×n , we denote by Tr(P ) = i=1 pii the trace of P and by ρ b (P ) the spectral radius of P , namely, ρb(P ) = max λi , where λi , i ∈ {1, . . . , N }, are eigenvalues of P . i
We denote by diag(a1 , . . . , aN ) a diagonal matrix in RN ×N whose diagonal entries starting from the upper left corner are a1 , . . . , aN . Similarly, we denote by diag(A1 , . . . , AN ) a block diagonal matrix whose diagonal entries starting
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from the upper left corner are matrices A1 , . . . , AN . Given functions fi : Xi Q → Yi , forQany i ∈ {1,Q. . . , N }, their N N N Cartesianproduct i=1 fi : i=1 Xi → i=1 Yi is deQN fined as i=1 fi (x1 , . . . , xN ) = [f1 (x1 ); . . . ; fN (xN )]. n Given a measurable function f : R+ 0 → R , the (essential) supremum of f is denoted by kf k∞ ; we recall that kf k∞ := (ess)sup{kf (t)k, t ≥ 0}. A continuous function + γ : R+ 0 → R0 , is said to belong to class K if it is strictly increasing and γ(0) = 0; γ is said to belong to class K∞ if γ ∈ K and γ(s) → ∞ as s → ∞.
B. Stochastic hybrid systems Let (Ω, F, P) be a probability space endowed with a filtration F = (Fs )s≥0 satisfying the usual conditions of completeness and right continuity [9, p. 48]. Let (Ws )s≥0 be a p-dimensional F-Brownian motion and (Ps )s≥0 be a q-dimensional F-Poisson process. We assume that the Poisson process and the Brownian motion are independent of each other. The Poisson process Ps := Ps1 ; . . . ; Psq model q kinds of events whose occurrences are assumed to be independent of each other. Definition 2.1: The class of stochastic hybrid systems with which we deal in this paper is the tuple Σ = (X, U, U, f, σ, r, Y, h), where n • X ⊆ R is the state set; m • U⊆R is the input set; • U is a subset of the set of all measurable functions of time from intervals of the form [0, ∞[ to U; n • f : X × U → R is the drift term which is globally Lipschitz continuous: there exist constants Lx , Lu ∈ R+ such that: kf (x, u) − f (x0 , u0 )k ≤ Lx kx − x0 k + Lu ku − u0 k for all x, x0 ∈ X and all u, u0 ∈ U; n n×p • σ :R →R is the diffusion term which is globally Lipschitz continuous; n n×q • r : R → R is the reset function which is globally Lipschitz continuous; q˜ • Y ⊆ R is the output set; • h : X → Y is the output map. A stochastic hybrid system Σ satisfies Σ:
d ξ(t) = η(t) =
III. S TOCHASTIC (B I )S IMULATION F UNCTION We recall the notion of stochastic (bi)simulation function, introduced in [8], with a slight modification, which is useful to relate properties of stochastic hybrid systems to those of their abstractions. Definition 3.1: Let Σ = (X, U, U, f, σ, r, Y, h) and Σ = X, U, U, f , σ, r, Y, h be two stochastic hybrid systems such that Y and Y are of equal dimension. A continuous function φ : X × X → R+ 0 ∪ {+∞} is a stochastic simulation function from Σ to Σ if: (i) for every x ∈ X and x ∈ X, φ(x, x) ≥ α h(x) − h (x) for some α ∈ K∞ ; (ii) for any υ ∈ U, there exists υ ∈ U such that the stochastic process φ ξaυ (t), ξ aυ (t) is a supermartingle [11, Appendix C] for any random variable a and a that are measurable1 in F0 .
Stochastic hybrid system Σ is simulated by Σ, or Σ simulates Σ, denoted by Σ S Σ, if there exists a stochastic simulation function from Σ to Σ. The following theorem, inspired by Theorem 3 in [8], clarifies the significance of stochastic simulation functions. Theorem 3.2: Let Σ and Σ be two stochastic hybrid systems and φ be a stochastic simulation function from Σ to Σ. For any solution process ξaυ of Σ, there exists a solution process ξ aυ of Σ such that ( )
φ (a, a) P sup h (ξaυ (t)) − h ξ aυ (t) > ε| (a, a) ≤ . α(ε) + t∈R0 (III.1) Proof: Following the definition of stochastic simulation function in Definition 3.1, for any input signal υ ∈ U, there exists an input signal υ ∈ U such that φ ξaυ (t), ξ aυ (t) is a nonnegative supermartingale. As a result, we have the following chain of inequalities:
P sup h (ξaυ (t)) − h ξ aυ (t) > ε| (a, a) = t∈R+ 0
P sup α h (ξaυ (t)) − h ξ aυ (t) > α (ε) | (a, a) ≤ t∈R+ 0 φ (a, a) P sup φ ξaυ (t), ξ aυ (t) > α (ε) | (a, a) ≤ , t∈R+ α (ε)
f (ξ(t), υ(t)) d t + σ(ξ(t)) d Wt + r(ξ(t)) d Pt , h(ξ(t)), (II.1)
P-almost surely (P-a.s.) for any υ ∈ U, where stochastic process ξ : Ω × [0, ∞[→ X is called a solution process of Σ. We assume that the Poisson processes Psi , for any i ∈ ei . We also write ξaυ (t) to {1, . . . , q}, have the rates of λ denote the value of the solution process at time t ∈ R+ 0 under the input curve υ from initial condition ξaυ (0) = a P-a.s., in which a is a random variable that is measurable in F0 . Let us emphasize that the assumptions on f , σ, and r ensure existence, uniqueness, and strong Markov property of the solution processes [10]. Note that the modeling framework in (II.1) is a special class of the one introduced in [10]. Remark 2.2: We refer the interested readers to the beginning of Section IV in [8] showing how one can cast linear stochastic hybrid automata (LSHA), defined in [8], as jump linear stochastic systems (JLSS) which are a specific class of the ones introduced in Definition 2.1.
0
where the last inequality is implied from φ ξaυ , ξ aυ being a nonnegative supermartingale and [12, Lemma1]. A symmetric version of a stochastic simulation function is called a stochastic bisimulation function as defined next. Definition 3.3: Let Σ = (X, U, U, f, σ, r, Y, h) and Σ = X, U, U, f , σ, r, Y, h be two stochastic hybrid systems such that Y and Y are of equal dimension. A continuous function φ : X×X → R+ 0 ∪{+∞} is a stochastic bisimulation function between Σ and Σ if it is both a stochastic simulation function from Σ to Σ and from Σ to Σ. 1 Note that F can be the trivial sigma-algebra as well, i.e., a and a can 0 be non-probabilistic initial conditions as well.
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Stochastic hybrid system Σ is bisimilar to Σ, denoted by Σ ∼ =S Σ, if there exists a stochastic bisimulation function between Σ and Σ. Similar result as the one of Theorem 3.2 can be established using stochastic bisimulation function as the following. Corollary 3.4: Let Σ and Σ be two stochastic hybrid systems and φ be a stochastic bisimulation function between them. For any solution process ξaυ of Σ, there exists a solution process ξ aυ of Σ and vice versa, such that inequality (III.1) holds. Proof: The proof is similar to the one of Theorem 3.2. Given a complex stochastic hybrid system Σ and its abstraction Σ, one can use the provided results in Theorem 3.2 and Corollary 3.4 to quantify the distance between those systems. Specifically, one can leverage the results in Theorem 3.2 to verify the original system Σ against some complex specifications, e.g. linear temporal logic (LTL), by just verifying its abstraction Σ against the same specifications. For example, Theorem 7 in [8] explaines how an upper bound of the unsafety risk of the complex system Σ can be computed by performing the risk calculation over an abstraction Σ simulating Σ. Moreover, one can use the results in Corollary 3.4 to synthesize control policies enforcing some complex specifications, e.g. LTL, on the original system Σ by just refining the corresponding control policies enforcing the same specifications on its abstraction Σ. The following theorem provides a sufficient condition for the construction of a stochastic (bi)simulation function. Theorem 3.5: Let Σ = (X, U, U, f, σ, r, Y, h) and Σ = X, U, U, f , σ, r, Y, h be two stochastic hybrid systems such that Y and Y are of equal dimension and U = U. Consider a function φ : X × X → R+ 0 ∪ {+∞} that is twice continuously differentiable. If condition (i) in Definition 3.1 is satisfied for some α ∈ K∞ and there exist functions α, ρ ∈ K∞ such that (ii) for any x ∈ X, x ∈ X, and for any u ∈ U, u ∈ U, f (x, u) L φ (x, x) := [∂x φ ∂x φ] f (x, u) h i 1 σ(x) ∂x,x φ + Tr σ T (x) σ T (x) σ (x) ∂x,x φ 2
IV. C OMPOSITIONAL A PPROXIMATIONS FOR I NTERCONNECTED S YSTEMS First we provide a formal definition of interconnection between stochastic hybrid systems. A. Interconnection This subsection introduces the notion of interconnection between N stochastic hybrid subsystems similar to the one introduced in [13]. As an example, Figure 1 illustrates the interconnection of two subsystems.
u1 w12 w21 u2
Fig. 1.
q X
∂x,x φ ∂x,x φ
ui wi1 wiN Fig. 2.
ei (φ (x + r(x)ei , x + r(x)ei ) − φ(x, x)) λ
i=1
where ei ∈ Rq denotes the vector with 1 in the ith coordinate and 0’s elsewhere, then φ is a stochastic bisimulation function between Σ and Σ. u,u In the above theorem, is the infinitesimal generator L of the process ζ = ξ; ξ [11, Section 7.3] and the symbols ∂x and ∂x,x denote the first and the second order partial derivatives with respect to x and x and x, respectively. Proof: Condition (i) in Definition 3.1 is trivially satisfied. Condition (ii) can also be readily verified by using Dynkin’s formula [11], similar to the proof of Theorem 8 in [8]. We refer the interested readers to Lemma 9 in [8] providing equivalent conditions as (i) and (ii) in Theorem 3.5 for
⌃2
y1 z12 z21 y2
Interconnection of two subsystems.
ui = [ui ; wi ] , s.t. wi = [wi1 ; . . . ; wi,i−1 ; wi,i+1 ; . . . ; wiN ], (IV.1) y z yi = hi (xi ) = [hi (xi ); hi (xi )] = [yi ; zi ] , (IV.2) s.t. zi = [zi1 ; . . . ; zi,i−1 ; zi,i+1 ; . . . ; ziN ].
≤ −α (φ (x, x)) + ρ(ku − uk),
⌃1
Consider a complex stochastic hybrid system Σ composed of N stochastic hybrid subsystems Σi interconnected with each other. Here, we assume that any input value ui ∈ Ui , any output value yi ∈ Yi , and correspondingly the output function hi are decomposed to subvectors as depicted in Figure 2 and shown as the following:
u,u
+
two JLSS and a quadratic function φ.
⌃i
yi zi1 ziN
Input/output configuration of subsystem Σi .
The values wi and zi are called internal values, used to construct interconnection between subsystems. On the other hand, the values ui and yi are called external values together with those of other subsystems construct the input/output configuration of the overall interconnected system Σ. Note that the external values remain accessible even after interconnecting subsystems. Now we can formally define the interconnection between stochastic hybrid subsystems. Definition 4.1: Consider N stochastic hybrid subsystems Σi = (Xi , Ui , Ui , fi , σi , ri , Yi , hi ), i ∈ {1, . . . , N }, whose input and output sets and output functions are decomposed as in (IV.1) and (IV.2): Ui = Ui × Wi , Yi = Yi × Zi , and hi = hyi × hzi for some appropriate sets Ui , Wi , Yi , Zi and functions hyi , hzi . Suppose the sizes of subvectors wij and zji are equal for all i ∈ {1, . . . , N }, j ∈ {1, . . . , N } \{i}. The interconnection of Σi ’s, i ∈ {1, . . . , N }, is the stochastic
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hybrid system Σ = (X, U, U, f, σ, r, Y, h), denoted QN QN by Σ = I (Σ , . . . , Σ ), where X = X , U = 1 N i i=1 Ui , U = n o i=1 Q QN QN N + υ ∈ i=1 Ui | υ : R0 → U , f = i=1 fi , σ = i=1 σi , QN QN QN r = i=1 ri , Y = i=1 Yi , h = i=1 hyi , and subject to the constraint: wij = zji , ∀i ∈ {1, . . . , N } , ∀j ∈ {1, . . . , N } \{i}.
(IV.3)
Remark 4.2: Although, without loss of generality, we assumed that only the external output values yi ∈ Yi are accessible after the interconnection, one can assume that some of the internal output values zi ∈ Zi are still accessible after the interconnection and, hence, they can also be considered as external output values as well (cf. the example in Section V). Note that a network of stochastic hybrid systems can be viewed as an interconnection of stochastic hybrid subsystems as explained in the following remark. Remark 4.3: Consider a stochastic hybrid system Σ = (X, U, U, f, σ, r, Y, h), where f (x, u) = [f1 (x, u1 ); . . . ; fN (x, uN )] , σ(x) = [σ1 (x1 ); . . . ; σN (xN )] , r(x) = [r1 (x1 ); . . . ; rN (xN )] , h(x) = [h1 (x1 ); . . . ; hN (xN )] ,
for any x = [x1 ; . . . ; xN ] ∈ X = ΠN i=1 Xi and any u = [u1 ; . . . ; uN ] ∈ U = ΠN U for some appropriate sets i i=1 Xi , Ui . Correspondingly, one can decompose Y = ΠN i=1 Yi for some appropriate sets Yi . One can readily verify that Σ = I (Σ1 , . . . , ΣN ), where
Σi = Xi , Ui ×
N Y
Xj , U × D, fi , σi , ri , Yi × Xi , hi × 1Xi ,
i6=j=1
and D is a subset of the set of all measurable QN functions of time from intervals of the form [0, ∞[ to i6=j=1 Xj . B. Compositional approximations of interconnected systems This subsection contains the main contribution of the paper. We provide a bisimilar approximation of an interconnected system using bisimilar approximations of its subsystems. For showing the main result of the paper, we need the following assumption, inspired by the work in [14]. Assumption 4.4: For any i, j ∈ {1, . . . , N }, there exist K∞ functions γi and constants λi ∈ R+ and δij ∈ R+ 0 such that one has:
between Σi and Σ i for any i ∈ {1, . . . , N }. If Assumption 4.4 and ρb Λ−1 ∆ < 1 hold, then there exist constants µi ∈ PN R+ such that i=1 µi φi is a stochastic bisimulation function between Σ = I (Σ1 , . . . , ΣN ) and Σ = I Σ1 , . . . , ΣN . Proof: As showed in Lemma 3.1 in [14], if ρb Λ−1 ∆ < 1 holds, then there exist constants µi ∈ R+ , i ∈ {1, . . . , N }, such that µT (−Λ + ∆) < 0TN , where µ = [µ1 ; . . . ; µN ]. Let us first show that condition (i) in Definition 3.1 holds. For any x := [x1 ; . . . ; xN ] ∈ X, x := [x1 ; . . . ; xN ] ∈ X, one obtains: N
X y
y
h(x) − h(x) ≤
h (xi ) − hi (xi ) i
i=1
≤
N N X
X
hi (xi ) − hi (xi ) ≤ α−1 (φi (xi , xi )) i
i=1 N X
i=1
µi φi (xi , xi ) µ ˆ
N X µi φi (xi , xi ) ≤ Nα µ ˆ i=1
!
≤
i=1
α−1 i
≤
N X i=1
α
µi φi (xi , xi ) µ ˆ
,
where µ ˆ = min{µ1 , . . . , µN } and α = −1 max α−1 which is a K∞ function. Therefore, 1 , . . . , αN one obtains: N
X α h(x) − h(x) ≤ µi φi (xi , xi ) , i=1
where α(s) = µ ˆα (s/N ), ∀s ∈ R+ 0 , is a K∞ function, and condition (i) in Definition 3.1 is satisfied. Now we show that condition (ii) in Theorem 3.5 is satisfied. Consider any x := [x1 ; . . . ; xN ] ∈ X, x := [x1 ; . . . ; xN ] ∈ X, and any u := [u1 ; . . . ; uN ] ∈ U, u := [u1 ; . . . ; uN ] ∈ U. One obtains the chain of inequalities in (IV.4). In order to show the chain of inequalities in (IV.4), we leverage the following inequality: −1
ρi (s1 + · · · + sN −1 ) ≤
N −1 X
ρi (sj + · · · + sj ),
j=1
which is valid for any ρi ∈ K∞ and any s1 , . . . , sN −1 ∈ R+ 0. By defining α(s) := n o min −µT (−Λ + ∆) Γ (φvec (x, x)) | µT φvec (x, x) = s , (N ) X ρ(s) := max µi ρi (2si ) | k[s1 ; . . . ; sN ]k = s ,
ρi 2(N − 1)α−1 j (s) ≤ δij γj (s), αi (s) ≥ λi γi (s),
i=1
for any s ∈ R+ 0 , where αj , ρi , and αi are the functions appearing in Theorem 3.5, but with indices j, i here. To use in the next theorem, we define Λ := diag(λ1 , . . . , λN ), ∆ := {δij }, and Γ(b s) := [γ1 (s1 ); . . . ; γN (sN )], where sb = [s1 ; . . . ; sN ]. We now have all the ingredients to show the main result of the paper. Theorem 4.5: Consider N stochastic hybrid subsystems Σi and their corresponding approximations Σi , i ∈ {1, . . . , N }, whose input/output sets and output functions are decomposed as in (IV.1) and (IV.2). Assume zji and z ji are of equal dimension, Ui = U i , ∀i ∈ {1, . . . , N }, ∀j ∈ {1, . . . , N }\{i}, and assume there exist stochastic bisimulation functions φi : Xi ×Xi → R+ 0 ∪{+∞} as in Theorem 3.5
where φvec (x, x) = [φ1 (x1 , x1 ); . . . ; φN (xN , xN )], one obtain: Lu,u φ (x, x) ≤ −α (φ (x, x)) + ρ (ku − uk) ,
which completes the proof. It is clear that α, ρ ∈ K∞ . Remark 4.6: Note that if ρ is a linear K∞ function, e.g. ρ(s) = s for any s ∈ R+ 0 , one can remove all the coefficients 2 and N − 1 in the proof of Theorem 4.5 as well as in Assumption 4.4. Remark 4.7: Note that if ∆ is irreducible [15], µ > 0 can be chosen as a left eigenvector of −Λ + ∆ corresponding to the largest eigenvalue, which is real and negative by the Perron-Frobenius theorem [15]. Figure 3 shows schematically the results of Theorem 4.5.
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——————————————————————————————————————————————————Lu,u φ (x, x) =
N X
µi Lui ,ui φi (xi , xi ) ≤
i=1
≤
N X
N X
µi (−αi (φi (xi , xi )) + ρi (k[ui ; wi ] − [ui ; wi ]k))
i=1
µi (−αi (φi (xi , xi )) + ρi (kwi − wi k + kui − ui k)) ≤
i=1
≤
N X
=
≤
≤
≤
≤
kwij − wij k + ρi (2 kui − ui k)
kzji − z ji k + ρi (2 kui − ui k)
i6=j,j=1
N X
µi −αi (φi (xi , xi )) +
ρi (2(N − 1) kzji − z ji k) + ρi (2 kui − ui k)
i6=j,j=1
N X
µi −αi (φi (xi , xi )) +
ρi 2(N − 1) hj (xj ) − hj (xj ) + ρi (2 kui − ui k)
i6=j,j=1
N X
µi −αi (φi (xi , xi )) +
i=1 N X
N X
µi −αi (φi (xi , xi )) + ρi 2
i=1 N X
i6=j,j=1
i=1 N X
N X
µi −αi (φi (xi , xi )) + ρi 2
i=1 N X
µi (−αi (φi (xi , xi )) + ρi (2 kwi − wi k) + ρi (2 kui − ui k))
i=1
i=1 N X
N X
ρi 2(N −
1)α−1 j
(φj (xj , xj )) + ρi (2 kui − ui k)
i6=j,j=1
µi −λi γi (φi (xi , xi )) +
i=1
N X
δij γj (φj (xj , xj )) + ρi (2 kui − ui k)
i6=j,j=1
= µT (−Λ + ∆) Γ (φ1 (x1 , x1 ) ; . . . ; φN (xN , xN )) +
N X
µi ρi (2 kui − ui k) .
(IV.4)
i=1
——————————————————————————————————————————————————⌃1 ui wi1 wiN
yi zi1 ziN
⌃i
⇠ =S
ui wi1 wiN
⌃i
yi z i1 z iN
) Fig. 3.
⌃2
d ξi = ηi =
1 0 −1
For both systems Σi , we assume ui = wi , y1 = y 1 = z 1 (cf. 4.2), and y2 = z 2 . Let us show that φi (ξi , ξ i ) = q Remark T ξi ; ξ i M ξi ; ξ i is a stochastic bisimulation function as in Theorem 3.5 between Σi and Σi , where
1 −1 M = −1 0
" # # " # 1 1 1 T 1 , Bi = 1 , Ci = 0 , 0 −2 0
d ξi = ηi =
−3.5ξ i + υ i d t + 0.5ξ i d Wt + 0.5ξ i d Pt , ξi.
−1 2 2 −1
−1 2 2 −1
0 −1 . −1 1
First let us remark that
and Fi = Ri = 0.5I3 for i = 1, 2. The rate of the Poisson e = 0.5. For both systems Σi , we assume process Pt is λ ui = wi , y1 = y1 = z1 (cf. Remark 4.2), and y2 = z2 . First we show that Σi can be approximated by simpler stochastic hybrid systems Σi (1st order) which are JLSS and whose dynamics are described as the following: Σi :
⌃5
⌃2
Compositionallity results.
(Ai ξi + Bi υi ) d t + Fi ξi d Wt + Ri ξi d Pt , Ci ξ i ,
−2 Ai = 3.5 −1 1
⌃4 ⌃3
⌃5
where "
⌃1
⇠ =S
⌃3
V. E XAMPLE Consider two stochastic hybrid systems Σ1 and Σ2 which are JLSS and whose dynamics are described as the following: Σi :
⌃4
q φi (xi , xi ) = [xi ; xi ]T M [xi ; xi ] q = (xi1 − xi2 − xi3 )2 + (xi2 + xi3 − xi )2 √ √
xi1 xi2 + xi3
≥ 2 |xi1 − xi | = 2 |yi − y i | , = −
xi xi2 + xi3 2 2
where the inequality followes from Lemma 2.3 in [16]. Hence, condition (i) in Definition 3.1 is satisfied with 3399
of the proposed work here to provide finite approximations of interconnected stochastic hybrid systems by using the finite approximations of each stochastic hybrid subsystems proposed in [17], [18].
6 5
|η 1 − η 1 |
4
VII. ACKNOWLEDGEMENT The author would like to thank Peyman Mohajerin Esfahani for fruitful technical discussions.
3 2 1
R EFERENCES
0 0
0.5
1
1.5
time
2
2.5
3
Fig. 4. A few realizations of the error between the output of Σ and of Σ, e.g. |η1 − η 1 |. The dashed line denotes the 93% confidence bound.
αi (s) =
√
2 2 s
for any s ∈ R+ 0 . Moreover, one obtains
Lui ,ui φi (xi , xi ) ≤
2xT M (Ax + Bu) + xT F T M F x + 2φi (xi , xi ) p e xT (RT M R + RT M + M R) x +λ
≤ − 2.86φi (xi , xi ) + |ui − ui |,
where x = [xi ; xi ], u = [ui ; ui ], A = diag(Ai ; [−3.5]), B = diag(Bi ; [1]), F = diag(Fi ; [0.5]), and R = diag(Ri ; [0.5]). Therefore, condition (ii) in Theorem 3.5 is satisfied with αi (s) = 2.86s and ρi (s) = s for any s ∈ R+ 0 . Since ρi is a linear K function (cf. Remark 4.6), by choosing ∞ √ δij = 2, γi (s) = s for any s ∈ R+ 0 , and λi = 2.86 for i, j = 1, 2, one can readily verify that Assumption 4.4 holds. Since ρb(Λ−1 ∆) = 0.99 < 1 and using the results of Theorem 4.5, one concludes that there exists a vector µ = [µ1 ; µ2 ] > 02 such that φ(x, x) = µ1 φ1 (x1 , x1 ) + µ2 φ2 (x2 , x2 ) is a stochastic bisimulation function between Σ = I (Σ1 , Σ2 ) and Σ = I Σ1 , Σ2 , where x = [x1 ; x2 ] and√ x = [x1 ; x2 ]. Using Remark 4.7, one obtains µ1,2 = 22 . In the simulations, the initial states of the interconnected systems Σ and Σ are chosen as x0 = [−0.87; −2.71; −3.28; −0.22; −0.59; −0.71] and x0 = [1.83; 0.32]. In figure 4, we show a few realizations of the error between the output of Σ and of Σ, e.g. |η1 − η 1 |. The dashed line denotes the 93% confidence bound given by the computed stochastic bisimulation function φ as in inequality (III.1). VI. D ISCUSSION In this paper, we provided a compositional approach for approximations of interconnected stochastic hybrid systems. Given stochastic hybrid subsystems and their corresponding approximations and the quantified errors between them, we provided the approximations of an interconnection of those stochastic hybrid subsystems and the overall approximation error. Therefore, one can leverage the proposed techniques in this paper to potentially solve safety/reachability problems for large-scale stochastic hybrid systems which are interconnection of several stochastic hybrid subsystems. Note that the considered approximate abstractions in this paper are still infinite but possibly simpler (lower state space dimension). The author is currently investigating the extensions
[1] A. Singh and J. P. Hespanha, “Stochastic hybrid systems for studying biochemical processes,” Philosophical Transactions of the Royal Society, vol. 368, no. 1930, pp. 4995–5011, November 2010. [2] W. Glover and J. Lygeros, “A stochastic hybrid model for air traffic control simulation,” in Hybrid Systems: Computation and Control, ser. Lecture Notes in Computer Science, R. Alur and G. J. Pappas, Eds. Springer Berlin Heidelberg, March 2004, vol. 2993, pp. 372–386. [3] J. Hu, W. C. Wu, and S. Sastry, “Modeling subtilin production in bacillus subtilis using stochastic hybrid systems,” in Hybrid Systems: Computation and Control, ser. Lecture Notes in Computer Science, R. Alur and G. J. Pappas, Eds. Springer Berlin Heidelberg, March 2004, vol. 2993, pp. 417–431. [4] J. P. Hespanha, “Stochastic hybrid systems: Application to communication networks,” in Hybrid Systems: Computation and Control, ser. Lecture Notes in Computer Science, R. Alur and G. J. Pappas, Eds. Springer Berlin Heidelberg, March 2004, vol. 2993, pp. 387–401. [5] C. G. Cassandras and J. Lygeros, Stochastic hybrid systems, ser. Automation and Control Engineering. Taylor & Francis Group/CRC Press, 2007, vol. 24. [6] P. M. Esfahani, D. Chatterjee, and J. Lygeros, “The stochastic reach-avoid problem and set characterization for diffusions,” arXiv: 1202.4375, October 2012. [7] ——, “On a problem of stochastic reach-avoid set characterization,” In Proceedings of 50th IEEE Conference on Decision and Control and European Control Conference, pp. 7069–7074, December 2011. [8] A. A. Julius and G. J. Pappas, “Approximations of stochastic hybrid systems,” IEEE Transaction on Automatic Control, vol. 54, no. 6, pp. 1193–1203, 2009. [9] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., ser. Graduate Texts in Mathematics. New York: SpringerVerlag, 1991, vol. 113. [10] J. Krystul, H. A. P. Blom, and A. Bagchi, “Stochastic differential equations on hybrid state spaces,” in Stochastic Hybrid Systems, ser. Automation and Control Engineering. Orlando, FL: Taylor & Francis Group/CRC Press, 2006, pp. 15–46. [11] B. K. Oksendal, Stochastic differential equations: An introduction with applications, 5th ed. Springer, November 2002. [12] H. J. Kushner, Stochastic stability and control, ser. ser. Mathematics in Science and Engineering. New York: Academic Press, 1967. [13] Y. Tazaki and J. Imura, “Bisimilar finite abstractions of interconnected systems,” in Proceedings of 11th International Conference on Hybrid Systems: Computation and Control (HSCC), pp. 514–527, 2008. [14] S. Dashkovskiy, H. Ito, and F. Wirth, “On a small gain theorem for ISS networks in dissipative lyapunov form,” European Journal of Control, vol. 17, no. 4, pp. 357–365, 2011. [15] O. Axelsson, Iterative solution methods. Cambridge University Press, 1994. [16] D. Angeli, “A Lyapunov approach to incremental stability properties,” IEEE Transactions on Automatic Control, vol. 47, no. 3, pp. 410–21, March 2002. [17] M. Zamani, P. M. Esfahani, R. Majumdar, A. Abate, and J. Lygeros, “Symbolic control of stochastic systems via approximately bisimilar finite abstractions,” IEEE Transactions on Automatic Control, Special Issue on Control of Cyber-Physical Systems, vol. forthcoming, arXiv: 1302.3868, 2014. [18] M. Zamani, I. Tkachev, and A. Abate, “Bisimilar symbolic models for stochastic control systems without state-space discretization,” in Proceedings of the 17th International Conference on Hybrid Systems: Computation and Control. ACM New York, NY, USA, April 2014, pp. 41–50.
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Compositional Approximations of Interconnected Stochastic Hybrid Systems Majid Zamani
Abstract— This paper provides a compositional approach for approximations of the interconnection of a class of stochastic hybrid systems including both jump linear stochastic systems and linear stochastic hybrid automata. The approximation is based on the recently developed notions of stochastic (bi)simulation functions using which one can quantify the error between original stochastic hybrid systems and their abstractions. Given stochastic (bi)simulation functions between stochastic hybrid subsystems and their corresponding approximations, we provide a stochastic (bi)simulation function between an interconnection of those stochastic hybrid subsystems and that of their corresponding approximations. Consequently, one can leverage the proposed results in this paper to perform the analysis and synthesis over the interconnection of abstract subsystems and then carry the results over the interconnection of concrete ones. We illustrate the effectiveness of the proposed results here by computing a stochastic (bi)simulation function between an interconnection of two identical jump linear stochastic subsystems and that of their corresponding approximations by just using the stochastic (bi)simulation function between one of the jump linear stochastic subsystems and its corresponding approximate abstraction.
I. I NTRODUCTION Stochastic hybrid systems are a general class of dynamical systems consisting of continuous and discrete dynamics subject to probabilistic noise and events. In the past few years, these systems have become ubiquitous in a variety of fields due to providing a rigorous and general framework for modeling of many safety critical applications. Examples of those applications include biochemistry [1], air traffic control [2], systems biology [3], and communication networks [4], that can be modeled and treated as stochastic hybrid systems. Reachability and safety are two fundamental concepts in the study of stochastic hybrid systems that have received significant attentions in the last few years because of their applications in real life systems [5]. There exist some results in the literature such as [6], [7] providing methodologies solving reachability problems, safety ones, or their combinations. Unfortunately, the computational complexity of the existing techniques solving those problems scales exponentially with the dimension of the continuous state space making them intractable for large-scale stochastic hybrid systems. A promising direction to overcome this complexity is to abstract the original concrete system by a simpler one (lower dimension) using an appropriate relation quantifying the error between them. By taking into account the quantified error, one can perform the analysis and synthesis over the simpler system and then carry the results over the concrete one. M. Zamani is with the Department of Electrical Engineering and Information Technology, Technische Universit¨at M¨unchen, D-80290 Munich, Germany. Email: [email protected]. URL: http://www.hcs.ei.tum.de
978-1-4673-6088-3/14/$31.00 ©2014 IEEE
The recent work in [8] provides a notion of stochastic (bi)simulation function providing a metric which quantifies the error between the original stochastic hybrid systems and their approximations. Particularly, the authors in [8] develop a theory of approximate (bi)simulation for a class of stochastic hybrid automata whose continuous dynamics are modeled by continuous-time stochastic differential equations and the switches are modeled as Poisson processes. In this work, we provide a compositional technique for approximation of the interconnection of the same class of stochastic hybrid automata. Given stochastic (bi)simulation functions between each stochastic hybrid subsystem and its corresponding approximation, we provide a stochastic (bi)simulation function between the interconnection of stochastic hybrid subsystems and that of their corresponding approximations. Therefore, one can quantify the error between the interconnection of stochastic hybrid subsystems and that of their corresponding abstractions by just knowing the error between each subsystem and its corresponding abstraction. As a consequence, one can leverage the proposed results here to solve particularly safety/reachability problems over the interconnection of abstract systems and then carry the results over the interconnection of concrete ones. We illustrate the results of this paper by computing a stochastic (bi)simulation function between an interconnection of two identical jump linear stochastic subsystems (3rd order each) and that of their corresponding abstractions (1st order each) by using just the stochastic (bi)simulation function between one of the subsystems and its corresponding approximation. II. S TOCHASTIC H YBRID S YSTEMS A. Notations The identity map on a set A is denoted by 1A . The symbols N, R, R+ , and R+ 0 denote the set of natural, real, positive, and nonnegative real numbers, respectively. Given a vector x ∈ Rn , we denote by xi the i–th element ofpx, and by kxk the Euclidean norm of x, namely, kxk = x21 + x22 + . . . + x2n . We denote by In and 0n the identity matrix and zero vector in Rn×n and Rn , respectively. For any x, y ∈ Rn , the relation x ≤ y is defined by xi ≤ yi for all i ∈ {1, . . . , n}. The relations , and ≥ are defined in the same manner for vectors. Given Pn a matrix P = {pij } ∈ Rn×n , we denote by Tr(P ) = i=1 pii the trace of P and by ρ b (P ) the spectral radius of P , namely, ρb(P ) = max λi , where λi , i ∈ {1, . . . , N }, are eigenvalues of P . i
We denote by diag(a1 , . . . , aN ) a diagonal matrix in RN ×N whose diagonal entries starting from the upper left corner are a1 , . . . , aN . Similarly, we denote by diag(A1 , . . . , AN ) a block diagonal matrix whose diagonal entries starting
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from the upper left corner are matrices A1 , . . . , AN . Given functions fi : Xi Q → Yi , forQany i ∈ {1,Q. . . , N }, their N N N Cartesianproduct i=1 fi : i=1 Xi → i=1 Yi is deQN fined as i=1 fi (x1 , . . . , xN ) = [f1 (x1 ); . . . ; fN (xN )]. n Given a measurable function f : R+ 0 → R , the (essential) supremum of f is denoted by kf k∞ ; we recall that kf k∞ := (ess)sup{kf (t)k, t ≥ 0}. A continuous function + γ : R+ 0 → R0 , is said to belong to class K if it is strictly increasing and γ(0) = 0; γ is said to belong to class K∞ if γ ∈ K and γ(s) → ∞ as s → ∞.
B. Stochastic hybrid systems Let (Ω, F, P) be a probability space endowed with a filtration F = (Fs )s≥0 satisfying the usual conditions of completeness and right continuity [9, p. 48]. Let (Ws )s≥0 be a p-dimensional F-Brownian motion and (Ps )s≥0 be a q-dimensional F-Poisson process. We assume that the Poisson process and the Brownian motion are independent of each other. The Poisson process Ps := Ps1 ; . . . ; Psq model q kinds of events whose occurrences are assumed to be independent of each other. Definition 2.1: The class of stochastic hybrid systems with which we deal in this paper is the tuple Σ = (X, U, U, f, σ, r, Y, h), where n • X ⊆ R is the state set; m • U⊆R is the input set; • U is a subset of the set of all measurable functions of time from intervals of the form [0, ∞[ to U; n • f : X × U → R is the drift term which is globally Lipschitz continuous: there exist constants Lx , Lu ∈ R+ such that: kf (x, u) − f (x0 , u0 )k ≤ Lx kx − x0 k + Lu ku − u0 k for all x, x0 ∈ X and all u, u0 ∈ U; n n×p • σ :R →R is the diffusion term which is globally Lipschitz continuous; n n×q • r : R → R is the reset function which is globally Lipschitz continuous; q˜ • Y ⊆ R is the output set; • h : X → Y is the output map. A stochastic hybrid system Σ satisfies Σ:
d ξ(t) = η(t) =
III. S TOCHASTIC (B I )S IMULATION F UNCTION We recall the notion of stochastic (bi)simulation function, introduced in [8], with a slight modification, which is useful to relate properties of stochastic hybrid systems to those of their abstractions. Definition 3.1: Let Σ = (X, U, U, f, σ, r, Y, h) and Σ = X, U, U, f , σ, r, Y, h be two stochastic hybrid systems such that Y and Y are of equal dimension. A continuous function φ : X × X → R+ 0 ∪ {+∞} is a stochastic simulation function from Σ to Σ if: (i) for every x ∈ X and x ∈ X, φ(x, x) ≥ α h(x) − h (x) for some α ∈ K∞ ; (ii) for any υ ∈ U, there exists υ ∈ U such that the stochastic process φ ξaυ (t), ξ aυ (t) is a supermartingle [11, Appendix C] for any random variable a and a that are measurable1 in F0 .
Stochastic hybrid system Σ is simulated by Σ, or Σ simulates Σ, denoted by Σ S Σ, if there exists a stochastic simulation function from Σ to Σ. The following theorem, inspired by Theorem 3 in [8], clarifies the significance of stochastic simulation functions. Theorem 3.2: Let Σ and Σ be two stochastic hybrid systems and φ be a stochastic simulation function from Σ to Σ. For any solution process ξaυ of Σ, there exists a solution process ξ aυ of Σ such that ( )
φ (a, a) P sup h (ξaυ (t)) − h ξ aυ (t) > ε| (a, a) ≤ . α(ε) + t∈R0 (III.1) Proof: Following the definition of stochastic simulation function in Definition 3.1, for any input signal υ ∈ U, there exists an input signal υ ∈ U such that φ ξaυ (t), ξ aυ (t) is a nonnegative supermartingale. As a result, we have the following chain of inequalities:
P sup h (ξaυ (t)) − h ξ aυ (t) > ε| (a, a) = t∈R+ 0
P sup α h (ξaυ (t)) − h ξ aυ (t) > α (ε) | (a, a) ≤ t∈R+ 0 φ (a, a) P sup φ ξaυ (t), ξ aυ (t) > α (ε) | (a, a) ≤ , t∈R+ α (ε)
f (ξ(t), υ(t)) d t + σ(ξ(t)) d Wt + r(ξ(t)) d Pt , h(ξ(t)), (II.1)
P-almost surely (P-a.s.) for any υ ∈ U, where stochastic process ξ : Ω × [0, ∞[→ X is called a solution process of Σ. We assume that the Poisson processes Psi , for any i ∈ ei . We also write ξaυ (t) to {1, . . . , q}, have the rates of λ denote the value of the solution process at time t ∈ R+ 0 under the input curve υ from initial condition ξaυ (0) = a P-a.s., in which a is a random variable that is measurable in F0 . Let us emphasize that the assumptions on f , σ, and r ensure existence, uniqueness, and strong Markov property of the solution processes [10]. Note that the modeling framework in (II.1) is a special class of the one introduced in [10]. Remark 2.2: We refer the interested readers to the beginning of Section IV in [8] showing how one can cast linear stochastic hybrid automata (LSHA), defined in [8], as jump linear stochastic systems (JLSS) which are a specific class of the ones introduced in Definition 2.1.
0
where the last inequality is implied from φ ξaυ , ξ aυ being a nonnegative supermartingale and [12, Lemma1]. A symmetric version of a stochastic simulation function is called a stochastic bisimulation function as defined next. Definition 3.3: Let Σ = (X, U, U, f, σ, r, Y, h) and Σ = X, U, U, f , σ, r, Y, h be two stochastic hybrid systems such that Y and Y are of equal dimension. A continuous function φ : X×X → R+ 0 ∪{+∞} is a stochastic bisimulation function between Σ and Σ if it is both a stochastic simulation function from Σ to Σ and from Σ to Σ. 1 Note that F can be the trivial sigma-algebra as well, i.e., a and a can 0 be non-probabilistic initial conditions as well.
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Stochastic hybrid system Σ is bisimilar to Σ, denoted by Σ ∼ =S Σ, if there exists a stochastic bisimulation function between Σ and Σ. Similar result as the one of Theorem 3.2 can be established using stochastic bisimulation function as the following. Corollary 3.4: Let Σ and Σ be two stochastic hybrid systems and φ be a stochastic bisimulation function between them. For any solution process ξaυ of Σ, there exists a solution process ξ aυ of Σ and vice versa, such that inequality (III.1) holds. Proof: The proof is similar to the one of Theorem 3.2. Given a complex stochastic hybrid system Σ and its abstraction Σ, one can use the provided results in Theorem 3.2 and Corollary 3.4 to quantify the distance between those systems. Specifically, one can leverage the results in Theorem 3.2 to verify the original system Σ against some complex specifications, e.g. linear temporal logic (LTL), by just verifying its abstraction Σ against the same specifications. For example, Theorem 7 in [8] explaines how an upper bound of the unsafety risk of the complex system Σ can be computed by performing the risk calculation over an abstraction Σ simulating Σ. Moreover, one can use the results in Corollary 3.4 to synthesize control policies enforcing some complex specifications, e.g. LTL, on the original system Σ by just refining the corresponding control policies enforcing the same specifications on its abstraction Σ. The following theorem provides a sufficient condition for the construction of a stochastic (bi)simulation function. Theorem 3.5: Let Σ = (X, U, U, f, σ, r, Y, h) and Σ = X, U, U, f , σ, r, Y, h be two stochastic hybrid systems such that Y and Y are of equal dimension and U = U. Consider a function φ : X × X → R+ 0 ∪ {+∞} that is twice continuously differentiable. If condition (i) in Definition 3.1 is satisfied for some α ∈ K∞ and there exist functions α, ρ ∈ K∞ such that (ii) for any x ∈ X, x ∈ X, and for any u ∈ U, u ∈ U, f (x, u) L φ (x, x) := [∂x φ ∂x φ] f (x, u) h i 1 σ(x) ∂x,x φ + Tr σ T (x) σ T (x) σ (x) ∂x,x φ 2
IV. C OMPOSITIONAL A PPROXIMATIONS FOR I NTERCONNECTED S YSTEMS First we provide a formal definition of interconnection between stochastic hybrid systems. A. Interconnection This subsection introduces the notion of interconnection between N stochastic hybrid subsystems similar to the one introduced in [13]. As an example, Figure 1 illustrates the interconnection of two subsystems.
u1 w12 w21 u2
Fig. 1.
q X
∂x,x φ ∂x,x φ
ui wi1 wiN Fig. 2.
ei (φ (x + r(x)ei , x + r(x)ei ) − φ(x, x)) λ
i=1
where ei ∈ Rq denotes the vector with 1 in the ith coordinate and 0’s elsewhere, then φ is a stochastic bisimulation function between Σ and Σ. u,u In the above theorem, is the infinitesimal generator L of the process ζ = ξ; ξ [11, Section 7.3] and the symbols ∂x and ∂x,x denote the first and the second order partial derivatives with respect to x and x and x, respectively. Proof: Condition (i) in Definition 3.1 is trivially satisfied. Condition (ii) can also be readily verified by using Dynkin’s formula [11], similar to the proof of Theorem 8 in [8]. We refer the interested readers to Lemma 9 in [8] providing equivalent conditions as (i) and (ii) in Theorem 3.5 for
⌃2
y1 z12 z21 y2
Interconnection of two subsystems.
ui = [ui ; wi ] , s.t. wi = [wi1 ; . . . ; wi,i−1 ; wi,i+1 ; . . . ; wiN ], (IV.1) y z yi = hi (xi ) = [hi (xi ); hi (xi )] = [yi ; zi ] , (IV.2) s.t. zi = [zi1 ; . . . ; zi,i−1 ; zi,i+1 ; . . . ; ziN ].
≤ −α (φ (x, x)) + ρ(ku − uk),
⌃1
Consider a complex stochastic hybrid system Σ composed of N stochastic hybrid subsystems Σi interconnected with each other. Here, we assume that any input value ui ∈ Ui , any output value yi ∈ Yi , and correspondingly the output function hi are decomposed to subvectors as depicted in Figure 2 and shown as the following:
u,u
+
two JLSS and a quadratic function φ.
⌃i
yi zi1 ziN
Input/output configuration of subsystem Σi .
The values wi and zi are called internal values, used to construct interconnection between subsystems. On the other hand, the values ui and yi are called external values together with those of other subsystems construct the input/output configuration of the overall interconnected system Σ. Note that the external values remain accessible even after interconnecting subsystems. Now we can formally define the interconnection between stochastic hybrid subsystems. Definition 4.1: Consider N stochastic hybrid subsystems Σi = (Xi , Ui , Ui , fi , σi , ri , Yi , hi ), i ∈ {1, . . . , N }, whose input and output sets and output functions are decomposed as in (IV.1) and (IV.2): Ui = Ui × Wi , Yi = Yi × Zi , and hi = hyi × hzi for some appropriate sets Ui , Wi , Yi , Zi and functions hyi , hzi . Suppose the sizes of subvectors wij and zji are equal for all i ∈ {1, . . . , N }, j ∈ {1, . . . , N } \{i}. The interconnection of Σi ’s, i ∈ {1, . . . , N }, is the stochastic
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hybrid system Σ = (X, U, U, f, σ, r, Y, h), denoted QN QN by Σ = I (Σ , . . . , Σ ), where X = X , U = 1 N i i=1 Ui , U = n o i=1 Q QN QN N + υ ∈ i=1 Ui | υ : R0 → U , f = i=1 fi , σ = i=1 σi , QN QN QN r = i=1 ri , Y = i=1 Yi , h = i=1 hyi , and subject to the constraint: wij = zji , ∀i ∈ {1, . . . , N } , ∀j ∈ {1, . . . , N } \{i}.
(IV.3)
Remark 4.2: Although, without loss of generality, we assumed that only the external output values yi ∈ Yi are accessible after the interconnection, one can assume that some of the internal output values zi ∈ Zi are still accessible after the interconnection and, hence, they can also be considered as external output values as well (cf. the example in Section V). Note that a network of stochastic hybrid systems can be viewed as an interconnection of stochastic hybrid subsystems as explained in the following remark. Remark 4.3: Consider a stochastic hybrid system Σ = (X, U, U, f, σ, r, Y, h), where f (x, u) = [f1 (x, u1 ); . . . ; fN (x, uN )] , σ(x) = [σ1 (x1 ); . . . ; σN (xN )] , r(x) = [r1 (x1 ); . . . ; rN (xN )] , h(x) = [h1 (x1 ); . . . ; hN (xN )] ,
for any x = [x1 ; . . . ; xN ] ∈ X = ΠN i=1 Xi and any u = [u1 ; . . . ; uN ] ∈ U = ΠN U for some appropriate sets i i=1 Xi , Ui . Correspondingly, one can decompose Y = ΠN i=1 Yi for some appropriate sets Yi . One can readily verify that Σ = I (Σ1 , . . . , ΣN ), where
Σi = Xi , Ui ×
N Y
Xj , U × D, fi , σi , ri , Yi × Xi , hi × 1Xi ,
i6=j=1
and D is a subset of the set of all measurable QN functions of time from intervals of the form [0, ∞[ to i6=j=1 Xj . B. Compositional approximations of interconnected systems This subsection contains the main contribution of the paper. We provide a bisimilar approximation of an interconnected system using bisimilar approximations of its subsystems. For showing the main result of the paper, we need the following assumption, inspired by the work in [14]. Assumption 4.4: For any i, j ∈ {1, . . . , N }, there exist K∞ functions γi and constants λi ∈ R+ and δij ∈ R+ 0 such that one has:
between Σi and Σ i for any i ∈ {1, . . . , N }. If Assumption 4.4 and ρb Λ−1 ∆ < 1 hold, then there exist constants µi ∈ PN R+ such that i=1 µi φi is a stochastic bisimulation function between Σ = I (Σ1 , . . . , ΣN ) and Σ = I Σ1 , . . . , ΣN . Proof: As showed in Lemma 3.1 in [14], if ρb Λ−1 ∆ < 1 holds, then there exist constants µi ∈ R+ , i ∈ {1, . . . , N }, such that µT (−Λ + ∆) < 0TN , where µ = [µ1 ; . . . ; µN ]. Let us first show that condition (i) in Definition 3.1 holds. For any x := [x1 ; . . . ; xN ] ∈ X, x := [x1 ; . . . ; xN ] ∈ X, one obtains: N
X y
y
h(x) − h(x) ≤
h (xi ) − hi (xi ) i
i=1
≤
N N X
X
hi (xi ) − hi (xi ) ≤ α−1 (φi (xi , xi )) i
i=1 N X
i=1
µi φi (xi , xi ) µ ˆ
N X µi φi (xi , xi ) ≤ Nα µ ˆ i=1
!
≤
i=1
α−1 i
≤
N X i=1
α
µi φi (xi , xi ) µ ˆ
,
where µ ˆ = min{µ1 , . . . , µN } and α = −1 max α−1 which is a K∞ function. Therefore, 1 , . . . , αN one obtains: N
X α h(x) − h(x) ≤ µi φi (xi , xi ) , i=1
where α(s) = µ ˆα (s/N ), ∀s ∈ R+ 0 , is a K∞ function, and condition (i) in Definition 3.1 is satisfied. Now we show that condition (ii) in Theorem 3.5 is satisfied. Consider any x := [x1 ; . . . ; xN ] ∈ X, x := [x1 ; . . . ; xN ] ∈ X, and any u := [u1 ; . . . ; uN ] ∈ U, u := [u1 ; . . . ; uN ] ∈ U. One obtains the chain of inequalities in (IV.4). In order to show the chain of inequalities in (IV.4), we leverage the following inequality: −1
ρi (s1 + · · · + sN −1 ) ≤
N −1 X
ρi (sj + · · · + sj ),
j=1
which is valid for any ρi ∈ K∞ and any s1 , . . . , sN −1 ∈ R+ 0. By defining α(s) := n o min −µT (−Λ + ∆) Γ (φvec (x, x)) | µT φvec (x, x) = s , (N ) X ρ(s) := max µi ρi (2si ) | k[s1 ; . . . ; sN ]k = s ,
ρi 2(N − 1)α−1 j (s) ≤ δij γj (s), αi (s) ≥ λi γi (s),
i=1
for any s ∈ R+ 0 , where αj , ρi , and αi are the functions appearing in Theorem 3.5, but with indices j, i here. To use in the next theorem, we define Λ := diag(λ1 , . . . , λN ), ∆ := {δij }, and Γ(b s) := [γ1 (s1 ); . . . ; γN (sN )], where sb = [s1 ; . . . ; sN ]. We now have all the ingredients to show the main result of the paper. Theorem 4.5: Consider N stochastic hybrid subsystems Σi and their corresponding approximations Σi , i ∈ {1, . . . , N }, whose input/output sets and output functions are decomposed as in (IV.1) and (IV.2). Assume zji and z ji are of equal dimension, Ui = U i , ∀i ∈ {1, . . . , N }, ∀j ∈ {1, . . . , N }\{i}, and assume there exist stochastic bisimulation functions φi : Xi ×Xi → R+ 0 ∪{+∞} as in Theorem 3.5
where φvec (x, x) = [φ1 (x1 , x1 ); . . . ; φN (xN , xN )], one obtain: Lu,u φ (x, x) ≤ −α (φ (x, x)) + ρ (ku − uk) ,
which completes the proof. It is clear that α, ρ ∈ K∞ . Remark 4.6: Note that if ρ is a linear K∞ function, e.g. ρ(s) = s for any s ∈ R+ 0 , one can remove all the coefficients 2 and N − 1 in the proof of Theorem 4.5 as well as in Assumption 4.4. Remark 4.7: Note that if ∆ is irreducible [15], µ > 0 can be chosen as a left eigenvector of −Λ + ∆ corresponding to the largest eigenvalue, which is real and negative by the Perron-Frobenius theorem [15]. Figure 3 shows schematically the results of Theorem 4.5.
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——————————————————————————————————————————————————Lu,u φ (x, x) =
N X
µi Lui ,ui φi (xi , xi ) ≤
i=1
≤
N X
N X
µi (−αi (φi (xi , xi )) + ρi (k[ui ; wi ] − [ui ; wi ]k))
i=1
µi (−αi (φi (xi , xi )) + ρi (kwi − wi k + kui − ui k)) ≤
i=1
≤
N X
=
≤
≤
≤
≤
kwij − wij k + ρi (2 kui − ui k)
kzji − z ji k + ρi (2 kui − ui k)
i6=j,j=1
N X
µi −αi (φi (xi , xi )) +
ρi (2(N − 1) kzji − z ji k) + ρi (2 kui − ui k)
i6=j,j=1
N X
µi −αi (φi (xi , xi )) +
ρi 2(N − 1) hj (xj ) − hj (xj ) + ρi (2 kui − ui k)
i6=j,j=1
N X
µi −αi (φi (xi , xi )) +
i=1 N X
N X
µi −αi (φi (xi , xi )) + ρi 2
i=1 N X
i6=j,j=1
i=1 N X
N X
µi −αi (φi (xi , xi )) + ρi 2
i=1 N X
µi (−αi (φi (xi , xi )) + ρi (2 kwi − wi k) + ρi (2 kui − ui k))
i=1
i=1 N X
N X
ρi 2(N −
1)α−1 j
(φj (xj , xj )) + ρi (2 kui − ui k)
i6=j,j=1
µi −λi γi (φi (xi , xi )) +
i=1
N X
δij γj (φj (xj , xj )) + ρi (2 kui − ui k)
i6=j,j=1
= µT (−Λ + ∆) Γ (φ1 (x1 , x1 ) ; . . . ; φN (xN , xN )) +
N X
µi ρi (2 kui − ui k) .
(IV.4)
i=1
——————————————————————————————————————————————————⌃1 ui wi1 wiN
yi zi1 ziN
⌃i
⇠ =S
ui wi1 wiN
⌃i
yi z i1 z iN
) Fig. 3.
⌃2
d ξi = ηi =
1 0 −1
For both systems Σi , we assume ui = wi , y1 = y 1 = z 1 (cf. 4.2), and y2 = z 2 . Let us show that φi (ξi , ξ i ) = q Remark T ξi ; ξ i M ξi ; ξ i is a stochastic bisimulation function as in Theorem 3.5 between Σi and Σi , where
1 −1 M = −1 0
" # # " # 1 1 1 T 1 , Bi = 1 , Ci = 0 , 0 −2 0
d ξi = ηi =
−3.5ξ i + υ i d t + 0.5ξ i d Wt + 0.5ξ i d Pt , ξi.
−1 2 2 −1
−1 2 2 −1
0 −1 . −1 1
First let us remark that
and Fi = Ri = 0.5I3 for i = 1, 2. The rate of the Poisson e = 0.5. For both systems Σi , we assume process Pt is λ ui = wi , y1 = y1 = z1 (cf. Remark 4.2), and y2 = z2 . First we show that Σi can be approximated by simpler stochastic hybrid systems Σi (1st order) which are JLSS and whose dynamics are described as the following: Σi :
⌃5
⌃2
Compositionallity results.
(Ai ξi + Bi υi ) d t + Fi ξi d Wt + Ri ξi d Pt , Ci ξ i ,
−2 Ai = 3.5 −1 1
⌃4 ⌃3
⌃5
where "
⌃1
⇠ =S
⌃3
V. E XAMPLE Consider two stochastic hybrid systems Σ1 and Σ2 which are JLSS and whose dynamics are described as the following: Σi :
⌃4
q φi (xi , xi ) = [xi ; xi ]T M [xi ; xi ] q = (xi1 − xi2 − xi3 )2 + (xi2 + xi3 − xi )2 √ √
xi1 xi2 + xi3
≥ 2 |xi1 − xi | = 2 |yi − y i | , = −
xi xi2 + xi3 2 2
where the inequality followes from Lemma 2.3 in [16]. Hence, condition (i) in Definition 3.1 is satisfied with 3399
of the proposed work here to provide finite approximations of interconnected stochastic hybrid systems by using the finite approximations of each stochastic hybrid subsystems proposed in [17], [18].
6 5
|η 1 − η 1 |
4
VII. ACKNOWLEDGEMENT The author would like to thank Peyman Mohajerin Esfahani for fruitful technical discussions.
3 2 1
R EFERENCES
0 0
0.5
1
1.5
time
2
2.5
3
Fig. 4. A few realizations of the error between the output of Σ and of Σ, e.g. |η1 − η 1 |. The dashed line denotes the 93% confidence bound.
αi (s) =
√
2 2 s
for any s ∈ R+ 0 . Moreover, one obtains
Lui ,ui φi (xi , xi ) ≤
2xT M (Ax + Bu) + xT F T M F x + 2φi (xi , xi ) p e xT (RT M R + RT M + M R) x +λ
≤ − 2.86φi (xi , xi ) + |ui − ui |,
where x = [xi ; xi ], u = [ui ; ui ], A = diag(Ai ; [−3.5]), B = diag(Bi ; [1]), F = diag(Fi ; [0.5]), and R = diag(Ri ; [0.5]). Therefore, condition (ii) in Theorem 3.5 is satisfied with αi (s) = 2.86s and ρi (s) = s for any s ∈ R+ 0 . Since ρi is a linear K function (cf. Remark 4.6), by choosing ∞ √ δij = 2, γi (s) = s for any s ∈ R+ 0 , and λi = 2.86 for i, j = 1, 2, one can readily verify that Assumption 4.4 holds. Since ρb(Λ−1 ∆) = 0.99 < 1 and using the results of Theorem 4.5, one concludes that there exists a vector µ = [µ1 ; µ2 ] > 02 such that φ(x, x) = µ1 φ1 (x1 , x1 ) + µ2 φ2 (x2 , x2 ) is a stochastic bisimulation function between Σ = I (Σ1 , Σ2 ) and Σ = I Σ1 , Σ2 , where x = [x1 ; x2 ] and√ x = [x1 ; x2 ]. Using Remark 4.7, one obtains µ1,2 = 22 . In the simulations, the initial states of the interconnected systems Σ and Σ are chosen as x0 = [−0.87; −2.71; −3.28; −0.22; −0.59; −0.71] and x0 = [1.83; 0.32]. In figure 4, we show a few realizations of the error between the output of Σ and of Σ, e.g. |η1 − η 1 |. The dashed line denotes the 93% confidence bound given by the computed stochastic bisimulation function φ as in inequality (III.1). VI. D ISCUSSION In this paper, we provided a compositional approach for approximations of interconnected stochastic hybrid systems. Given stochastic hybrid subsystems and their corresponding approximations and the quantified errors between them, we provided the approximations of an interconnection of those stochastic hybrid subsystems and the overall approximation error. Therefore, one can leverage the proposed techniques in this paper to potentially solve safety/reachability problems for large-scale stochastic hybrid systems which are interconnection of several stochastic hybrid subsystems. Note that the considered approximate abstractions in this paper are still infinite but possibly simpler (lower state space dimension). The author is currently investigating the extensions
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