We consider discrete-time plants that interact with their controllers via fixed discrete alphabets. For this class of systems, and in the absence of exogenous inputs, we propose a general, conceptual procedure for constructing a sequence of finite state approximate models starting from finite length sequences of input and output signal pairs. We explicitly derive conditions under which the proposed construct, used in conjunction with a particular generalized structure, satisfies desirable properties of @math approximations thereby leading to nominal deterministic finite state machine models that can be used in certified-by- design controller synthesis. We also show that the cardinality of the minimal disturbance alphabet that can be used in this setting equals that of the sensor output alphabet. Finally, we show that the proposed construct satisfies a relevant semi-completeness property.
Other related research directions make use of symbolic models @cite_32 @cite_22 , approximating automata @cite_34 @cite_21 , and finite quotients of the system @cite_28 @cite_6 .
@cite_22 @cite_28 @cite_21 @cite_32 @cite_6 @cite_34
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Motivated by formal models recently proposed in the context of XML, we study automata and logics on strings over infinite alphabets. These are conservative extensions of classical automata and logics defining the regular languages on finite alphabets. Specifically, we consider register and pebble automata, and extensions of first-order logic and monadic second-order logic. For each type of automaton we consider one-way and two-way variants, as well as deterministic, nondeterministic, and alternating control. We investigate the expressiveness and complexity of the automata and their connection to the logics, as well as standard decision problems. Some of our results answer open questions of Kaminski and Francez on register automata. Linear or one-dimensional reversible second-order cellular automata, exemplified by three cases named as RCA1–3, are introduced. Displays of their evolution in discrete time steps, , from their simplest initial states and on the basis of updating rules in modulo 2 arithmetic, are presented. In these, shaded and unshaded squares denote cells whose cell variables are equal to one and zero respectively. This paper is devoted to finding general formulas for, and explicit numerical evaluations of, the weights N(t) of the states or configurations of RCA1–3, i.e. the total number of shaded cells in tth line of their displays. This is achieved by means of the replacement of RCA1–3 by the equivalent linear first-order matrix automata MCA1–3, for which the cell variables are matrices, instead of just numbers () as for RCA1–3. MCA1–3 are tractable because it has been possible to generalize to them the heavy duty methods already well-developed for ordinary first-order cellular automata like those of Wolfram's Rules 90 and 150. While the automata MCA1–3 are thought to be of genuine interest in their own right, with untapped further mathematical potential, their treatment has been applied here to expediting derivation of a large body of general and explicit results for N(t) for RCA1–3. Amongst explicit results obtained are formulas also for each of RCA1–3 for the total weight of the configurations of the first times, . References.- Technical Foundations.- 1 Introduction.- 2 Graphs and Their Representation.- 3 Graph Planarity and Embeddings.- 4 Graph Drawing Methods.- References.- WilmaScope - A 3D Graph Visualization System.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- Pajek - Analysis and Visualization of Large Networks.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- Tulip - A Huge Graph Visualization Framework.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- Graphviz and Dynagraph - Static and Dynamic Graph Drawing Tools.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- AGD - A Library of Algorithms for Graph Drawing.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- yFiles - Visualization and Automatic Layout of Graphs.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- GDS - A Graph Drawing Server on the Internet.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- BioPath - Exploration and Visualization of Biochemical Pathways.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- DBdraw - Automatic Layout of Relational Database Schemas.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- GoVisual - A Diagramming Software for UML Class Diagrams.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- CrocoCosmos - 3D Visualization of Large Object-oriented Programs.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- ViSta - Visualizing Statecharts.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- visone - Analysis and Visualization of Social Networks.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References.- Polyphemus and Hermes - Exploration and Visualization of Computer Networks.- 1 Introduction.- 2 Applications.- 3 Algorithms.- 4 Implementation.- 5 Examples.- 6 Software.- References. Preface1Difference Equations12Lag Operators253Stationary ARMA Processes434Forecasting725Maximum Likelihood Estimation1176Spectral Analysis1527Asymptotic Distribution Theory1808Linear Regression Models2009Linear Systems of Simultaneous Equations23310Covariance-Stationary Vector Processes25711Vector Autoregressions29112Bayesian Analysis35113The Kalman Filter37214Generalized Method of Moments40915Models of Nonstationary Time Series43516Processes with Deterministic Time Trends45417Univariate Processes with Unit Roots47518Unit Roots in Multivariate Time Series54419Cointegration57120Full-Information Maximum Likelihood Analysis of Cointegrated Systems63021Time Series Models of Heteroskedasticity65722Modeling Time Series with Changes in Regime677A Mathematical Review704B Statistical Tables751C Answers to Selected Exercises769D Greek Letters and Mathematical Symbols Used in the Text786Author Index789Subject Index792