5 research outputs found
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Compositional Synthesis of not Necessarily Stabilizable Stochastic Systems via Finite Abstractions
In this paper, we propose a compositional framework for the construction of finite abstractions (a.k.a. finite Markov decision processes (MDPs)) for networks of not necessarily stabilizable discrete-time stochastic control systems. The proposed scheme is based on a notion of finite-step stochastic simulation function, using which one can employ an abstractsystem as a substitution of the original one in the controller design process with guaranteed error bounds. In comparison with the existing notions of simulation functions, a finitestep stochastic simulation function needs to decay only after some finite numbers of steps instead of at each time step. In the first part of the paper, we develop a new type of small-gain conditions which are less conservative than the existing ones. The proposed condition compositionally quantifies the distance in probability between the interconnection of stochastic control subsystems and that of their (finite or infinite) abstractions. In particular, using this relaxation via finite-step stochastic simulation functions, it is possible to construct finiteabstractions such that stabilizability of each subsystem is not necessarily required. In the second part of the paper, for the class of linear stochastic control systems, we construct finite MDPs together with their corresponding finite-step stochastic simulation functions. Finally, we demonstrate the effectiveness of the proposed results by compositionally constructing finite MDP of a network of four subsystems such that one of them is not stabilizable.</p
From Small-Gain Theory to Compositional Construction of Barrier Certificates for Large-Scale Stochastic Systems
This paper is concerned with a compositional approach for the construction of
control barrier certificates for large-scale interconnected stochastic systems
while synthesizing hybrid controllers against high-level logic properties. Our
proposed methodology involves decomposition of interconnected systems into
smaller subsystems and leverages the notion of control sub-barrier certificates
of subsystems, enabling one to construct control barrier certificates of
interconnected systems by employing some -type small-gain conditions. The
main goal is to synthesize hybrid controllers enforcing complex logic
properties including the ones represented by the accepting language of
deterministic finite automata (DFA), while providing probabilistic guarantees
on the satisfaction of given specifications in bounded-time horizons. To do so,
we propose a systematic approach to first decompose high-level specifications
into simple reachability tasks by utilizing automata corresponding to the
complement of specifications. We then construct control sub-barrier
certificates and synthesize local controllers for those simpler tasks and
combine them to obtain a hybrid controller that ensures satisfaction of the
complex specification with some lower-bound on the probability of satisfaction.
To compute control sub-barrier certificates and corresponding local
controllers, we provide two systematic approaches based on sum-of-squares (SOS)
optimization program and counter-example guided inductive synthesis (CEGIS)
framework. We finally apply our proposed techniques to two physical case
studies