Observability analysis and model formulation for nonlinear state estimation

A suitable design of state estimators for advanced control requires a detailed and representative mathematical model for capturing the nonlinear process behavior. The system observability, i.e. when the set of measurements provides enough information to estimate all the system states, is not a premise of the derivation of the Kalman filter. However, this propriety can improve the state estimator performance. On the basis of these design tasks, we outline a state estimation tuning strategy for different model formulations and present an algorithm to select the smallest number of measured variables to guarantee the system observability. The Williams–Otto semi-batch reactor was selected as case study, since its model formulation can be represented by two different set of states: (a) a mass basis states set and (b) a mass fraction basis states set. While the process-noise covariance matrix Q in the state estimator can be a diagonal and constant for the first model formulation, the matrix Q is not diagonal and time-varying for the second one due to their highly correlated states. Our results have shown how to convert the tuning matrices between different state definitions so that similar estimation results can be achieved.     

The thermodynamic irreversibility of a mechanical system possessing the property of recurrence

The transition from one thermodynamic state to another by a mechanical system possessing the property of recurrence is investigated. It is established that the concept of a state in phenomenological thermodynamics corresponds to the concept of an attainability set in mechanics. It is shown that the presence of states that are reversible within the framework of classical mechanics, does not contradict the irreversibility of thermodynamic states.     

The set of quantum states and its averaged dynamic transformations

In this paper we consider the set of quantum states and passages to the limit for sequences of quantum dynamic semigroups in the mentioned set. We study the structure of the set of extreme points of the quantum state set and represent an arbitrary state as an integral over the set of one-dimensional orthogonal projectors; the obtained representation is similar to the spectral decomposition of a normal state. We apply the obtained results to the analysis of sequences of quantum dynamic semigroups which occur in the regularization of a degenerate Hamiltonian.     

Finite controllability of linear parabolic systems with stationary final states

The aim of controllability is usually to reach a final state only for a moment. But for practical reasons the trajectory often has to stay in this position for some time. The set of stationary states is analized and controllability for stationary states is investigated with finite dimensional controls.     

Interval reliability for aggregated Markov repairable system with repair time omission

In this paper, Markov models of repairable systems with repair time omission are considered whose finite state space is grouped into two sets, the set of working states, W, and the set of failed states, F. If the system enters failed states from a working state at any instance, and sojourns at the failed states F less than a given nonnegative critical value τ, then the repair interval can be omitted from downtime records. Otherwise, If the system enters failed states from a working state at any instance, and sojourns at the failed states F more than the given nonnegative critical value τ, then the repair interval cannot be omitted from downtime records. In terms of the assumption, a new model is developed. The focus of attention is the new model’s availability, interval reliability and interval unreliability. Several results are derived for these reliability indexes for the new model. Some special cases and numerical examples are given to illustrate the results obtained by using Maple software in the paper.     

Evaluating policies based on their long term average cost

Many decision problems can be characterized by a set of possible states and a cost associated with each possible state transition, hi this paper we discuss how to select a policy from a set of possible policies in the long term. If the cost matrix is not available the transition matrix can be used to compare expected return times to states. In our setting the transition matrix is defined by use of linguistic terms and as a consequence, the expected return times are fuzzy. In the case where the cost matrix is available, fuzzy average costs are computed. The resulting fuzzy quantities are compared by introducing the concept of minimizing sets. Finally, we look at the case where the transition takes place from a state to a state that is known to be an element of some subset of states, but we do not know which one. We use the Dempster–Shafer theory [Shafer 1976] together with techniques of Norton [Norton 1988] and Smetz [Smetz 1976] to approximate the transition probabilities     

Decision making under partial probability information using pair-wise comparisons

In many decision-making situations, decision makers (DMs) have difficulty in specifying their perceived state probability values or even probability value ranges. However, they may find it easier to tell how much more likely is the occurrence of a given state when compared with other states. An approach is proposed to identify the efficient strategies of a decision-making situation where the DMs involved declare their perceived relative likelihood of the occurrence of the states by pair-wise comparisons. The pair-wise comparisons of all the states are used to construct a judgment matrix, which is transformed into a probability matrix. The columns of the transformed matrix delineate a convex cone of the state probabilities. Next, an efficiency linear program (ELP) is formulated for each available strategy, whose optimal solution determines whether or not that strategy is efficient within the probability region defined by the cone. Only an efficient strategy can be optimum for a given set of state probability values. Inefficient strategies are never used irrespective of state probability values. The application of the approach is demonstrated using examples where DMs offer differing views on the occurrence of the states.     

Data-driven modeling of the temporal evolution of breakers’ states in the French electrical transmission grid

In electrical transmission grids, it is common to observe the states of circuit breakers. While they are known at irregular times, system modeling and grid state estimation are of the highest importance to ensure secure operations. This paper proposes a richer method to estimate the grid state over its reference configurations based on the temporal evolution of its breakers’ states. The first contribution consists in developing a general multi-observation continuous-time finite-state Hidden Markov Model with filter-based parameter estimation to infer the hidden state (e.g., the grid reference configuration) handling multiple observed processes with irregular “jump” times (e.g., the breakers’ states). As a second contribution, we build a numerical scheme with no discretization error adapted to all state jumps generated by the observed processes. Finally, we apply our model to simulated and real data to illustrate the approach’s performance. The available data consists of historical records of breakers’ states during the electrical transmission grid operated normally. For this real-data-driven application, we also present a clustering approach to identify the set of grid reference configurations.     

State feedback control of condition/event systems

We consider the problem of synthesizing state feedback control policies for condition/event (C/E) systems where the control signals can be both conditions, which enable and inhibit state transitions, and events, which force state transitions. To represent the causal structure of condition and event feedback, a new feedback composition is defined. An algorithm is presented for synthesizing state feedback policies that keep the state of the C/E system out of a given set of forbidden states. The problem formulation and solution is illustrated for a process control example.     

Stochastic applications of media theory: Random walks on weak orders or partial orders

This paper presents the axioms of a real time random walk on the set of states of a medium and some of their consequences, such as the asymptotic probabilities of the states. The states of the random walk coincide with those of the medium, and the transitions of the random walk are governed by a probability distribution on the set of token-events, together with a Poisson process regulating the arrivals of such events. We examine two special cases. The first is the medium on strict weak orders on a set of three elements, the second the medium of strict partial orders on the same set. Thus, in each of these cases, a state of the medium is a binary relation. We also consider tune in-and-out extensions of these two special cases. We review applications of these models to opinion poll data pertaining to the 1992 United States presidential election. Each strict weak order or strict partial order is interpreted as being the implicit or explicit opinion of some individual regarding the three major candidates in that election, namely, Bush, Clinton and Perot. In particular, the strict partial order applications illustrate our notion of a response function that provides the link between theory and data in situations where, in contrast to previous papers, the permissible responses do not span the entire set of permissible states of the medium.     

On the notion of weak stability and related issues of hybrid diffusion systems

This work is concerned with the weak stability of hybrid diffusion systems, which consist of a number of diffusions modulated by a jump process. First, we show that when the jump component is nearly completely decomposable into a number of ergodic jump processes, the overall system is still weakly stable (or positive recurrent). Next we show that even if the process contains transient states, the positive recurrence can still be preserved. Then, we examine the asymptotic distribution when only one ergodic group of states is involved in the jump process and the state space of the continuous state belongs to a compact set. Our attention is devoted to asymptotic distribution in this case. The distribution is obtained by utilizing a spectrum gap property of the underlying process.     

Relative controllability of linear stationary completely integrable Pfaff systems

We consider a process described by stationary completely integrable Pfaff systems linear in the input and the control as well as in the output and the state. We study the state controllability properties of the Pfaff system for the case in which an arbitrary initial state (a constant vector) and an arbitrary terminal state (either a constant vector, or an analytic vector function, or a constant vector and an analytic vector function) are given. The terminal states belong to a subspace of the state space of the system. The entire set of Pfaff systems splits into three classes, in each of which we obtain conditions for the controllability of Pfaff systems in the class of analytic controls. The conditions are stated in terms of the rank of some matrices formed from the given matrices of the Pfaff system and the matrix that specifies the subspace of terminal states.     

Parallel sequential Monte Carlo samplers and estimation of the number of states in a Hidden Markov Model

The majority of modelling and inference regarding Hidden Markov Models (HMMs) assumes that the number of underlying states is known a priori. However, this is often not the case and thus determining the appropriate number of underlying states for a HMM is of considerable interest. This paper proposes the use of a parallel sequential Monte Carlo samplers framework to approximate the posterior distribution of the number of states. This requires no additional computational effort if approximating parameter posteriors conditioned on the number of states is also necessary. The proposed strategy is evaluated on a comprehensive set of simulated data and shown to outperform the state of the art in this area: although the approach is simple, it provides good performance by fully exploiting the particular structure of the problem. An application to business cycle analysis is also presented.     

A survey of algorithmic methods for partially observed Markov decision processes

A partially observed Markov decision process (POMDP) is a generalization of a Markov decision process that allows for incomplete information regarding the state of the system. The significant applied potential for such processes remains largely unrealized, due to an historical lack of tractable solution methodologies. This paper reviews some of the current algorithmic alternatives for solving discrete-time, finite POMDPs over both finite and infinite horizons. The major impediment to exact solution is that, even with a finite set of internal system states, the set of possible information states is uncountably infinite. Finite algorithms are theoretically available for exact solution of the finite horizon problem, but these are computationally intractable for even modest-sized problems. Several approximation methodologies are reviewed that have the potential to generate computationally feasible, high precision solutions.     

The computation of stability boundaries in state space for a class of biochemical engineering systems

The stability of a class of biochemical processes defined by a set of m biochemical reactions involving n components is analysed. The processes operate in a continuous mode and possess at least two stable equilibrium states: the normal operating point and a biological wash out state. Using a canonical state representation of the process dynamics the geometric structure of the operating point's stability boundary is characterized. Numerical algorithms are developed to evaluate this boundary and to visualize its extent in state space. The proposed technique is illustrated with a representative engineering example.     

Quasi‐conscious multivariate systems

Conscious experience is awash with underlying relationships. Moreover, for various brain regions such as the visual cortex, the system is biased toward some states. Representing this bias using a probability distribution shows that the system can define expected quantities. The mathematical theory in this article links these facts using expected float entropy (efe), which is a measure of the expected amount of information needed, to specify the state of the system, beyond what is already known about the system from relationships that appear as parameters. Under the requirement that the relationship parameters minimize efe, the brain defines relationships. It is proposed that when a brain state is interpreted in the context of these relationships the brain state acquires meaning in the form of the relational content of the associated experience. For a given set, the theory represents relationships using weighted relations which assign continuous weights, from 0 to 1, to the elements of the Cartesian product of that set. The relationship parameters include weighted relations on the nodes of the system and on their set of states. Examples obtained using Monte‐Carlo methods (where relationship parameters are chosen uniformly at random) suggest that efe distributions with long left tails are most important. © 2015 Wiley Periodicals, Inc. Complexity 21: 125–147, 2016     

模糊自动机的强连通性及群自动机

为了更好地研究模糊自动机的结构和性质,采用代数的方法,在传统的模糊有限状态自动机的基础上,通过定义状态集合为代数群的自动机,讨论了这一类自动机的连通性和正则性,这丰富了模糊自动机理论．     

动态联盟形成与谈判集

在具有可转让效用的合作博弈中,T oen A rno ld和U lrich Schw a lbe提出了动态联盟形成及其吸收状态的模型:在联盟形成的动态过程中,当核心非空时,大联盟和核心是吸收状态.在此基础上,我们进一步推广了该模型:证明了Zhou L in谈判集是动态联盟形成过程的吸收状态之一,弥补了当核心是空集时,除了大联盟之外再无吸收状态的不足.该结果使得原模型更趋完善.     

Quantum model of emerging grammars

A special class of quantum recurrent nets (QRNs) simulating Markov chains with absorbing states is introduced. The absorbing states are exploited for pattern recognition: each class of patterns is attracted to a unique absorbing state. Due to quantum interference of patterns, each combination of patterns acquires its own meaning: it is attracted to a certain combination of absorbing states which is different from those of individual attractions. This fundamentally new effect can be interpreted as formation of a grammar, i.e., a set of rules assigning certain meaning to different combinations of patterns. It appears that there exists a class of unitary operators in which each member gives rise to a different artificial language with associated grammar.