Interconnections of Nonlinear Systems Driven by L₂-ITÔ Stochastic Processes

Fliess operators have been an object of study in connection with nonlinear systems acting on deterministic inputs since the early 1970\u27s. They describe a broad class of nonlinear input-output maps using a type of functional series expansion, but in most applications, a system\u27s inputs have noise components. In such circumstances, new mathematical machinery is needed to properly describe the input-output map via the Chen-Fliess algebraic formalism. In this dissertation, a class of L2-Itô stochastic processes is introduced specifically for this purpose. Then, an extension of the Fliess operator theory is presented and sufficient conditions are given under which these operators are convergent in the mean-square sense. Next, three types of system interconnections are considered in this context: the parallel, product and cascade connections. This is done by first introducing the notion of a formal Fliess operator driven by a formal stochastic process. Then the generating series induced by each interconnection is derived. Finally, sufficient conditions are given under which the generating series of each composite system is convergent. This allows one to determine when an interconnection of Fliess operators driven by a class of L2-Itô stochastic processes is well-defined

Modeling and Stability Analysis of Nonlinear Sampled-Data Systems with Embedded Recovery Algorithms

Computer control systems for safety critical systems are designed to be fault tolerant and reliable, however, soft errors triggered by harsh environments can affect the performance of these control systems. The soft errors of interest which occur randomly, are nondestructive and introduce a failure that lasts a random duration. To minimize the effect of these errors, safety critical systems with error recovery mechanisms are being investigated. The main goals of this dissertation are to develop modeling and analysis tools for sampled-data control systems that are implemented with such error recovery mechanisms. First, the mathematical model and the well-posedness of the stochastic model of the sampled-data system are presented. Then this mathematical model and the recovery logic are modeled as a dynamically colored Petri net (DCPN). For stability analysis, these systems are then converted into piecewise deterministic Markov processes (PDP). Using properties of a PDP and its relationship to discrete-time Markov chains, a stability theory is developed. In particular, mean square equivalence between the sampled-data and its associated discrete-time system is proved. Also conditions are given for stability in distribution to the delta Dirac measure and mean square stability for a linear sampled-data system with recovery logic

On Cascades of Bilinear Systems and Generating Series of Weighted Petri Nets

It has been established in the literature that the cascade interconnection of two bilinear systems does not in general produce another bilinear system. The goals of this paper are two-fold. First, an alternative proof of the sufficient condition for preserving bilinearity under cascades due to Ferfera is presented which is much simpler than the original. Then it is shown that the well known correspondence between rational series and formal power series recognized by weighted finite-state automata can be generalized to produce a correspondence between the generating series of cascades of bilinear systems and a class of weighted Petri nets