Finite Set Of Inputs Research Articles

Data-Driven Synthesis of Safety Controllers via Multiple Control Barrier Certificates

This work proposes a data-driven framework to synthesize safety controllers for nonlinear systems with finite input sets and unknown mathematical models. The proposed scheme leverages new notions of multiple control barrier certificates (M-CBC) and provides controllers ensuring the safety of systems with confidence 1. While there may not exist a common control barrier certificate with a fixed template, our proposed technique adaptively partitions the state set to potentially find M-CBC of the same template for different regions. In the proposed data-driven framework, we first cast our proposed conditions of M-CBC as a robust optimization program (ROP). Given that the unknown model appears in some of the constraints of the ROP, we propose a sampling approach for collecting data and provide a scenario optimization program (SOP) associated with the proposed ROP. We solve the resulted SOP and construct M-CBC together with safety controllers for the unknown system with 100% correctness guarantee. We apply our results to a nonlinear jet engine compressor with unknown dynamics to illustrate the efficacy of our data-driven approach. In the case study, we show that while there exists no common polynomial-type control barrier certificate of a given degree, there exist polynomial-type M-CBC of the same degree by partitioning the state set to different regions.

Formal Synthesis of Stochastic Systems via Control Barrier Certificates

This paper focuses on synthesizing control policies for discrete-time stochastic control systems together with a lower bound on the probability that the systems satisfy the complex temporal properties. The desired properties of the system are expressed as linear temporal logic (LTL) specifications over finite traces. In particular, our approach decomposes the given specification into simpler reachability tasks based on its automata representation. We then propose the use of so-called \emph{control barrier certificate} to solve those simpler reachability tasks along with computing the corresponding controllers and probability bounds. Finally, we combine those controllers to obtain a hybrid control policy solving the considered problem. Under some assumptions, we also provide two systematic approaches for uncountable and finite input sets to search for control barrier certificates. We demonstrate the effectiveness of the proposed approach on a room temperature control and lane-keeping of a vehicle modeled as a four-dimensional single-track kinematic model. We compare our results with the discretization-based methods in the literature.

Estimating information in time-varying signals.

Across diverse biological systems—ranging from neural networks to intracellular signaling and genetic regulatory networks—the information about changes in the environment is frequently encoded in the full temporal dynamics of the network nodes. A pressing data-analysis challenge has thus been to efficiently estimate the amount of information that these dynamics convey from experimental data. Here we develop and evaluate decoding-based estimation methods to lower bound the mutual information about a finite set of inputs, encoded in single-cell high-dimensional time series data. For biological reaction networks governed by the chemical Master equation, we derive model-based information approximations and analytical upper bounds, against which we benchmark our proposed model-free decoding estimators. In contrast to the frequently-used k-nearest-neighbor estimator, decoding-based estimators robustly extract a large fraction of the available information from high-dimensional trajectories with a realistic number of data samples. We apply these estimators to previously published data on Erk and Ca2+ signaling in mammalian cells and to yeast stress-response, and find that substantial amount of information about environmental state can be encoded by non-trivial response statistics even in stationary signals. We argue that these single-cell, decoding-based information estimates, rather than the commonly-used tests for significant differences between selected population response statistics, provide a proper and unbiased measure for the performance of biological signaling networks.

Synthesizing and tuning stochastic chemical reaction networks with specified behaviours.

Methods from stochastic dynamical systems theory have been instrumental in understanding the behaviours of chemical reaction networks (CRNs) arising in natural systems. However, considerably less attention has been given to the inverse problem of synthesizing CRNs with a specified behaviour, which is important for the forward engineering of biological systems. Here, we present a method for generating discrete-state stochastic CRNs from functional specifications, which combines synthesis of reactions using satisfiability modulo theories and parameter optimization using Markov chain Monte Carlo. First, we identify candidate CRNs that have the possibility to produce correct computations for a given finite set of inputs. We then optimize the parameters of each CRN, using a combination of stochastic search techniques applied to the chemical master equation, to improve the probability of correct behaviour and rule out spurious solutions. In addition, we use techniques from continuous-time Markov chain theory to analyse the expected termination time for each CRN. We illustrate our approach by synthesizing CRNs for probabilistically computing majority, maximum and division, producing both known and previously unknown networks, including a novel CRN for probabilistically computing the maximum of two species. In future, synthesis techniques such as these could be used to automate the design of engineered biological circuits and chemical systems.

The theory of nonlinear systems as an instrument for solving engineering problems

The article outlines theoretical, methodological and practical issues of modern control and optimization theory, as well as the problems of nonlinear systems theory. Theoretical conclusions and results allowed to build mathematical models applicable to the management of objects of different nature with different principles of action, in particular, to the management of complex technical and technological objects that can be considered as nonlinear dynamic systems. The authors find it appropriate to consider nonlinear dynamic integral models as Volterra integro-power series from many functional arguments with multidimensional weight functions and a certain finite set of inputs to the system. The set of multidimensional kernels of integral Volterra operators completely characterizes the nonlinear and dynamic properties, and, consequently, the technical state of the initial system. The application of Volterra series based models allows to take into account the nonlinear and inertial properties of the initial nonlinear dynamic system more fully and accurately, it also makes the model diagnostic of a technical system more universal, raises the reliability of the forecast. The diagnostic procedure in this case is aimed at defining Volterra kernels based on the data of “input-output” experiment and building the diagnostic system of attribute in the space of which the decisive rule of optimal classification is created.

Minimal consistent DFA revisited

We address the classic problem of polynomial computation of a minimal finite-state representation compatible with two finite input sets that contain those strings that have to be accepted and rejected. We extend the previously defined uniformly-complete sample and prove that it is a special case of the sufficient condition we propose.

Uniformity is Weaker than Semi-Uniformity for Some Membrane Systems

We investigate computing models that are presented as families of finite computing devices with a uniformity condition on the entire family. Examples of such models include Boolean circuits, membrane systems, DNA computers, chemical reaction networks and tile assembly systems, and there are many others. However, in such models there are actually two distinct kinds of uniformity condition. The first is the most common and well-understood, where each input length is mapped to a single computing device (e.g. a Boolean circuit) that computes on the finite set of inputs of that length. The second, called semi-uniformity, is where each input is mapped to a computing device for that input (e.g. a circuit with the input encoded as constants). The former notion is well-known and used in Boolean circuit complexity, while the latter notion is frequently found in literature on nature-inspired computation from the past 20 years or so. Are these two notions distinct? For many models it has been found that these notions are in fact the same, in the sense that the choice of uniformity or semi-uniformity leads to characterisations of the same complexity classes. In other related work, we showed that these notions are actually distinct for certain classes of Boolean circuits. Here, we give analogous results for membrane systems by showing that certain classes of uniform membrane systems are strictly weaker than the analogous semi-uniform classes. This solves a known open problem in the theory of membrane systems. We then go on to present results towards characterising the power of these semi-uniform and uniform membrane models in terms of NL and languages reducible to the unary languages in NL, respectively.

Partially-shared zero-suppressed multi-terminal BDDs: concept, algorithms and applications

Multi-Terminal Binary Decision Diagrams (MTBDDs) are a well accepted technique for the state graph (SG) based quantitative analysis of large and complex systems specified by means of high-level model description techniques. However, this type of Decision Diagram (DD) is not always the best choice, since finite functions with small satisfaction sets, and where the fulfilling assignments possess many 0-assigned positions, may yield relatively large MTBDD based representations. Therefore, this article introduces zero-suppressed MTBDDs and proves that they are canonical representations of multi-valued functions on finite input sets. For manipulating DDs of this new type, possibly defined over different sets of function variables, the concept of partially-shared zero-suppressed MTBDDs and respective algorithms are developed. The efficiency of this new approach is demonstrated by comparing it to the well-known standard type of MTBDDs, where both types of DDs have been implemented by us within the C++-based DD-package JINC. The benchmarking takes place in the context of Markovian analysis and probabilistic model checking of systems. In total, the presented work extends existing approaches, since it not only allows one to directly employ (multi-terminal) zero-suppressed DDs in the field of quantitative verification, but also clearly demonstrates their efficiency.

Predictive Control Algorithm Technique for Multilevel Asymmetric Cascaded H-Bridge Inverters

Predictive control algorithms have been proposed for power electronic converters, featuring a high performance in terms of dynamic behavior. This is true due mainly to the accurate modeling and the finite set of inputs or possible switching combinations. However, when it is used in multilevel converters, the dynamic performance is degraded, because the optimization algorithm needs to evaluate and search over a large number of possible switching combinations. In this paper, a predictive control algorithm, which exhibits high dynamic performance for multilevel converters, is proposed. The algorithm reduces the complexity of calculations and the number of possible combinations, allowing the use of predictive control with a large number of switching states. Experimental results on a 27-level asymmetric multicell converter, which validate the proposed algorithm, are presented.

Achieving Maximum Possible Speed on Constrained Block Transmission Systems

We develop a theoretical framework for achieving the maximum possible speed on constrained digital channels with a finite alphabet. A common inaccuracy that is made when computing the capacity of digital channels is to assume that the inputs and outputs of the channel are analog Gaussian random variables, and then based upon that assumption, invoke the Shannon capacity bound for an additive white Gaussian noise (AWGN) channel. In a channel utilizing a finite set of inputs and outputs, clearly the inputs are not Gaussian distributed and Shannon bound is not exact. We study the capacity of a block transmission AWGN channel with quantized inputs and outputs, given the simultaneous constraints that the channel is frequency selective, there exists an average power constraint at the transmitter and the inputs of the channel are quantized. The channel is assumed known at the transmitter. We obtain the capacity of the channel numerically, using a constrained Blahut-Arimoto algorithm which incorporates an average power constraint at the transmitter. Our simulations show that under certain conditions the capacity approaches very closely the Shannon bound. We also show the maximizing input distributions. The theoretical framework developed in this paper is applied to a practical example: the downlink channel of a dial-up PCM modem connection where the inputs to the channel are quantized and the outputs are real. We test how accurate is the bound 53.3 kbps for this channel. Our results show that this bound can be improved upon.

Adaptive properties of living beings: Proposal for a generic mechanism.: (Self-Programming Machines III)

Living systems are capable to have appropriate responses to unpredictable environment. This kind of self-organization seems to operate as a self-programming machine, i.e. an organization able to modify itself. Until now the models of self-organization of living beings proposed are functions solutions of differential systems or transition functions of automata. These functions are fixed and these models are therefore unable to modify their organization. On the other hand, computer science propose a lot of models having the properties of adaptive systems of living beings, but all these models depend on the comparison between a goal and the results and ingenious choices of parameters by programmers, whereas there are no programmer's intention nor choice in the living systems. From two best known examples of adaptive systems of living beings, nervous system and immune system that have in common that the external signals modify the rewriting of their organization and therefore work as self-organizing machines, we devised machines with a finite set of inputs, based upon a recurrence, are able to rewrite their organization (Self-programming machines or m sp ) whenever external conditions vary and have striking properties of adaptation. M sp have similar properties whatever the operation defining the recurrence maybe. These results bring us to make the following statement: adaptive properties of living systems can be explained by their ability to rewrite their organization whenever external conditions vary under the only assumption that the rewriting mechanism be a deterministic constant recurrence in a finite state set. To cite this article: J.-P. Moulin, C. R. Biologies 329 (2006).

A Synthesis Theory for Nonlinear Systems with Plant Uncertainty

A nonlinear plant with parameter uncertainty is imbedded in a feedback system subjected to a finite set of inputs {x. (t)}. For each i there is a specified set of response tolerances. The synthesis procedure guarantees the response tolerances are satisfied over the range of uncertainty, for a large class of plants. The nonlinear plant is converted into an equivalent linear, time-invariant plant with parameter uncertainty, for which exact design is possible. Schauder’s fixed point theorem proves the equivalence is valid. The technique is applicable to any structure for which the equivalent linear, time-invariant problem is solvable.