Compositional Analysis of Hybrid Systems Defined Over Finite Alphabets

Abstract

We consider the stability and the input-output analysis problems of a class of large-scale hybrid systems composed of continuous dynamics coupled with discrete dynamics defined over finite alphabets, e.g., deterministic finite state machines (DFSMs). This class of hybrid systems can be used to model physical systems controlled by software. For such classes of systems, we use a method based on dissipativity theory for compositional analysis that allows us to study stability, passivity and input-output norms. We show that the certificates of the method based on dissipativity theory can be computed by solving a set of semi-definite programs. Nonetheless, the formulation based on semi-definite programs become computationally intractable for relatively large number of discrete and continuous states. We demonstrate that, for systems with large number of states consisting of an interconnection of smaller hybrid systems, accelerated alternating method of multipliers can be used to carry out the computations in a scalable and distributed manner. The proposed methodology is illustrated by an example of a system with 60 continuous states and 18 discrete states.

Distributed Synthesis of Local Controllers for Networked Systems With Arbitrary Interconnection Topologies

“…and x(t), y(t), v(t), u(t), w(t) and σ(t) are the augmented system state, output, coupling input, control input, disturbance and switching signal formed by stacking x i (t), y i (t), v i (t), u i (t), w i (t) and σ i (t) respectively of all N subsystems. As described in [41], the classical form of dissipativity in Definition 1 holds for switched system (22) as well. Along the lines of Section IV, we have the following result on distributed synthesis of local controllers to guarantee dissipativity of the networked switched system T σ N .…”

“…Since Definition 1 holds for switched systems [41], along the lines of proof for Theorem 1, the networked switched system (22) with…”

“…The first class of approaches relies on either exploiting or inducing weak coupling between subsystems in the network to distribute the synthesis problem [3], [12]- [17]. The second class of approaches are based on using numerical techniques like methods of multipliers, subgradient algorithms or distributed invariant set computations to decompose the control synthesis problem into more tractable problems [18]- [22]. The final class of approaches involves dissipativty-based synthesis of subsystem-level controllers to guarantee stability and robustness of the networked system [23]- [26]; however, the dissipativity conditions that the subsystem-level controllers are required to satisfy are computed in a centralized manner.…”