The Dynamics of Group Codes: Dual Abelian Group Codes and Systems

Fundamental results concerning the dynamics of abelian group codes (behaviors) and their duals are developed. Duals of sequence spaces over locally compact abelian groups may be defined via Pontryagin duality; dual group codes are orthogonal subgroups of dual sequence spaces. The dual of a complete code or system is finite, and the dual of a Laurent code or system is (anti-)Laurent. If C and C^\perp are dual codes, then the state spaces of C act as the character groups of the state spaces of C^\perp. The controllability properties of C are the observability properties of C^\perp. In particular, C is (strongly) controllable if and only if C^\perp is (strongly) observable, and the controller memory of C is the observer memory of C^\perp. The controller granules of C act as the character groups of the observer granules of C^\perp. Examples of minimal observer-form encoder and syndrome-former constructions are given. Finally, every observer granule of C is an "end-around" controller granule of C.Comment: 30 pages, 11 figures. To appear in IEEE Trans. Inform. Theory, 200

Algebraic Invariance Conditions in the Study of Approximate (Null-)Controllability of Markov Switch Processes

We aim at studying approximate null-controllability properties of a particular class of piecewise linear Markov processes (Markovian switch systems). The criteria are given in terms of algebraic invariance and are easily computable. We propose several necessary conditions and a sufficient one. The hierarchy between these conditions is studied via suitable counterexamples. Equivalence criteria are given in abstract form for general dynamics and algebraic form for systems with constant coefficients or continuous switching. The problem is motivated by the study of lysis phenomena in biological organisms and price prediction on spike-driven commodities.Comment: Mathematics of Control, Signals, and Systems, Springer Verlag (Germany), 2015, online first http://link.springer.com/article/10.1007/s00498-015-0146-

Balanced truncation for linear switched systems

In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems. This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems. Specifically in this paper, we provide a bound on the approximation error in L2 norm for continuous-time and l2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of bal- anced truncation depends only on the input-output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L2 and l2 gains depend only on the input-output map.Comment: We have corrected a number of typos and inconsistencies. In addition, we added new results in Theorem

Controllability Metrics on Networks with Linear Decision Process-type Interactions and Multiplicative Noise

This paper aims at the study of controllability properties and induced controllability metrics on complex networks governed by a class of (discrete time) linear decision processes with mul-tiplicative noise. The dynamics are given by a couple consisting of a Markov trend and a linear decision process for which both the "deterministic" and the noise components rely on trend-dependent matrices. We discuss approximate, approximate null and exact null-controllability. Several examples are given to illustrate the links between these concepts and to compare our results with their continuous-time counterpart (given in [16]). We introduce a class of backward stochastic Riccati difference schemes (BSRDS) and study their solvability for particular frameworks. These BSRDS allow one to introduce Gramian-like controllability metrics. As application of these metrics, we propose a minimal intervention-targeted reduction in the study of gene networks

Model reduction of networked passive systems through clustering

In this paper, a model reduction procedure for a network of interconnected identical passive subsystems is presented. Here, rather than performing model reduction on the subsystems, adjacent subsystems are clustered, leading to a reduced-order networked system that allows for a convenient physical interpretation. The identification of the subsystems to be clustered is performed through controllability and observability analysis of an associated edge system and it is shown that the property of synchronization (i.e., the convergence of trajectories of the subsystems to each other) is preserved during reduction. The results are illustrated by means of an example.Comment: 7 pages, 2 figures; minor changes in the final version, as accepted for publication at the 13th European Control Conference, Strasbourg, Franc