scholarly journals Time triggered stochastic hybrid system with nonlinear continuous dynamics

2021 ◽  
Author(s):  
Zahra Vahdat ◽  
Abhyudai Singh
Keyword(s):  
State Space ◽  
Cell Size ◽  
Time Evolution ◽  
Continuous Time ◽  
Linear Time ◽  
The State ◽  
Closed Form Expression ◽  
Model Parameters ◽  
Continuous Growth ◽  
Piecewise Deterministic Markov Processes

Time triggered stochastic hybrid systems (TTSHS) constitute a class of piecewise-deterministic Markov processes (PDMP), where continuous-time evolution of the state space is interspersed with discrete stochastic events. Whenever a stochastic event occurs, the state space is reset based on a random map. Prior work on this topic has focused on the continuous-time evolution being modeled as a linear time- invariant system, and in this contribution, we generalize these results to consider nonlinear continuous dynamics. Our approach relies on approximating the nonlinear dynamics between two successive events as a linear time-varying system and using this approximation to derive analytical solutions for the state space’s statistical moments. The TTSHS framework is used to model continuous growth in an individual cell’s size and its subsequent division into daughters. It is well known that exponential growth in cell size, together with a size- independent division rate, leads to an unbounded variance in cell size. Motivated by recent experimental findings, we consider nonlinear growth in cell size based on a Michaelis- Menten function and show that this leads to size homeostasis in the sense that the variance in cell size remains bounded. Moreover, we provide a closed-form expression for the variance in cell size as a function of model parameters and validate it by performing exact Monte Carlo simulations. In summary, our work provides an analytical approach for characterizing moments of a nonlinear stochastic dynamical system that can have broad applicability in studying random phenomena in both engineering and biology.

NEURAL NETWORK-BASED DIGITAL REDESIGN APPROACH FOR CONTROL OF UNKNOWN CONTINUOUS-TIME CHAOTIC SYSTEMS

2005 ◽  
Vol 15 (08) ◽  
pp. 2433-2455
Author(s):  
JOSE I. CANELON ◽  
LEANG S. SHIEH ◽  
SHU M. GUO ◽  
HEIDAR A. MALKI
Keyword(s):  
Neural Network ◽  
State Space ◽  
Continuous Time ◽  
Digital Control ◽  
Chaotic Systems ◽  
Linear Time ◽  
The State ◽  
Control Law ◽  
Analog Control ◽  
Digital Redesign

This paper presents a neural network-based digital redesign approach for digital control of continuous-time chaotic systems with unknown structures and parameters. Important features of the method are that: (i) it generalizes the existing optimal linearization approach for the class of state-space models which are nonlinear in the state but linear in the input, to models which are nonlinear in both the state and the input; (ii) it develops a neural network-based universal optimal linear state-space model for unknown chaotic systems; (iii) it develops an anti-digital redesign approach for indirectly estimating an analog control law from a fast-rate digital control law without utilizing the analog models. The estimated analog control law is then converted to a slow-rate digital control law via the prediction-based digital redesign method; (iv) it develops a linear time-varying piecewise-constant low-gain tracker which can be implemented using microprocessors. Illustrative examples are presented to demonstrate the effectiveness of the proposed methodology.

scholarly journals Maximum approximate likelihood estimation of general continuous-time state-space models

2022 ◽  
pp. 1471082X2110657
Author(s):  
Sina Mews ◽  
Roland Langrock ◽  
Marius Ötting ◽  
Houda Yaqine ◽  
Jost Reinecke
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Likelihood Estimation ◽  
The State ◽  
State Space Models ◽  
State Transitions ◽  
Model Parameters ◽  
State Process ◽  
Approximate Likelihood ◽  
Non Gaussian

Continuous-time state-space models (SSMs) are flexible tools for analysing irregularly sampled sequential observations that are driven by an underlying state process. Corresponding applications typically involve restrictive assumptions concerning linearity and Gaussianity to facilitate inference on the model parameters via the Kalman filter. In this contribution, we provide a general continuous-time SSM framework, allowing both the observation and the state process to be non-linear and non-Gaussian. Statistical inference is carried out by maximum approximate likelihood estimation, where multiple numerical integration within the likelihood evaluation is performed via a fine discretization of the state process. The corresponding reframing of the SSM as a continuous-time hidden Markov model, with structured state transitions, enables us to apply the associated efficient algorithms for parameter estimation and state decoding. We illustrate the modelling approach in a case study using data from a longitudinal study on delinquent behaviour of adolescents in Germany, revealing temporal persistence in the deviation of an individual's delinquency level from the population mean.

Maximum-a-posteriori estimation of LTI state-space models via efficient Monte-Carlo sampling

2021 ◽  
pp. 1-6
Author(s):  
Manas Mejari ◽  
Dario Piga
Keyword(s):  
Monte Carlo ◽  
State Space ◽  
Posterior Distribution ◽  
Linear Time ◽  
Monte Carlo Sampling ◽  
The State ◽  
Maximum A Posteriori ◽  
Model Parameters ◽  
A Posteriori ◽  
State Sequence

Abstract This paper addresses Maximum-A-Posteriori (MAP) estimation of Linear Time-Invariant State-Space (LTI-SS) models. The joint posterior distribution of the model matrices and the unknown state sequence is approximated by using Rao-Blackwellized Monte-Carlo sampling algorithms. Specifically, the conditional distribution of the state sequence given the model parameters is derived analytically, while only the marginal posterior distribution of the model matrices is approximated using a Metropolis-Hastings Markov-Chain Monte-Carlo sampler. From the joint distribution, MAP estimates of the unknown model matrices as well as the state sequence are computed. The performance of the proposed algorithm is demonstrated on a numerical example and on a real laboratory benchmark dataset of a hair dryer process.

scholarly journals Block Method for SolvingState-Space Equations of Linear Continuous-Time Control Systems

2007 ◽  
Vol 4 (2) ◽  
pp. 318-329
Author(s):  
Baghdad Science Journal
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Mimo Systems ◽  
Multiple Input Multiple Output ◽  
Method Comparison ◽  
The State ◽  
Block Method ◽  
Single Input Single Output ◽  
Single Output ◽  
Time Control

This paper presents a newly developed method with new algorithms to find the numerical solution of nth-order state-space equations (SSE) of linear continuous-time control system by using block method. The algorithms have been written in Matlab language. The state-space equation is the modern representation to the analysis of continuous-time system. It was treated numerically to the single-input-single-output (SISO) systems as well as multiple-input-multiple-output (MIMO) systems by using fourth-order-six-steps block method. We show that it is possible to find the output values of the state-space method using block method. Comparison between the numerical and exact results has been given for some numerical examples for solving different types of state-space equations using block method for conciliated the accuracy of the results of this method.

Regularity conditions for semi-Markov and Markov chains in continuous time

1983 ◽  
Vol 20 (03) ◽  
pp. 505-512
Author(s):  
Russell Gerrard
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
The State ◽  
Holding Time ◽  
Regularity Conditions ◽  
Regularity Properties ◽  
Classical Condition

The classical condition for regularity of a Markov chain is extended to include semi-Markov chains. In addition, for any given semi-Markov chain, we find Markov chains which exhibit identical regularity properties. This is done either (i) by transforming the state space or, alternatively, (ii) by imposing conditions on the holding-time distributions. Brief consideration is given to the problem of extending the results to processes other than semi-Markov chains.

Aggregated Markov processes with negative exponential time interval omission

1990 ◽  
Vol 22 (04) ◽  
pp. 802-830 ◽  
Author(s):  
Frank Ball
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Single Channel ◽  
The State ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Negative Exponential Distribution ◽  
Finite State ◽  
Negative Exponential ◽  
Finite State Space

We consider a time reversible, continuous time Markov chain on a finite state space. The state space is partitioned into two sets, termed open and closed, and it is only possible to observe whether the process is in an open or a closed state. Further, short sojourns in either the open or closed states fail to be detected. We consider the situation when the length of minimal detectable sojourns follows a negative exponential distribution with mean μ–1. We show that the probability density function of observed open sojourns takes the form , where n is the size of the state space. We present a thorough asymptotic analysis of f O(t) as μ tends to infinity. We discuss the relevance of our results to the modelling of single channel records. We illustrate the theory with a numerical example.

Regularity conditions for semi-Markov and Markov chains in continuous time

1983 ◽  
Vol 20 (3) ◽  
pp. 505-512 ◽  
Author(s):  
Russell Gerrard
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
The State ◽  
Holding Time ◽  
Regularity Conditions ◽  
Regularity Properties ◽  
Classical Condition

The classical condition for regularity of a Markov chain is extended to include semi-Markov chains. In addition, for any given semi-Markov chain, we find Markov chains which exhibit identical regularity properties. This is done either (i) by transforming the state space or, alternatively, (ii) by imposing conditions on the holding-time distributions. Brief consideration is given to the problem of extending the results to processes other than semi-Markov chains.

scholarly journals Exact and approximate hidden Markov chain filters based on discrete observations

2015 ◽  
Vol 32 (3-4) ◽  
pp. 159-176
Author(s):  
Nicole Bäuerle ◽  
Igor Gilitschenski ◽  
Uwe Hanebeck
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Discrete Time ◽  
Continuous Time ◽  
Hidden Markov ◽  
Numerical Study ◽  
Recursion Formula ◽  
The State ◽  
Continuous Time Markov Chain ◽  
Discrete Observations

Abstract We consider a Hidden Markov Model (HMM) where the integrated continuous-time Markov chain can be observed at discrete time points perturbed by a Brownian motion. The aim is to derive a filter for the underlying continuous-time Markov chain. The recursion formula for the discrete-time filter is easy to derive, however involves densities which are very hard to obtain. In this paper we derive exact formulas for the necessary densities in the case the state space of the HMM consists of two elements only. This is done by relating the underlying integrated continuous-time Markov chain to the so-called asymmetric telegraph process and by using recent results on this process. In case the state space consists of more than two elements we present three different ways to approximate the densities for the filter. The first approach is based on the continuous filter problem. The second approach is to derive a PDE for the densities and solve it numerically. The third approach is a crude discrete time approximation of the Markov chain. All three approaches are compared in a numerical study.

Hydraulic-seasonal-time-based state space model for displacement monitoring of high concrete dams

2021 ◽  
pp. 014233122110183
Author(s):  
Shaowei Wang ◽  
Cong Xu ◽  
Chongshi Gu ◽  
Huaizhi Su ◽  
Bangbin Wu
Keyword(s):  
State Space ◽  
Elastic Deformation ◽  
State Space Model ◽  
State Equation ◽  
Time Effect ◽  
The State ◽  
Arch Dam ◽  
Model Parameters ◽  
Concrete Dams ◽  
Space Model

Displacement is the most intuitive reflection of the comprehensive behavior of concrete dams, especially the time effect displacement, which is a key index for the evaluation of the structural behavior and health status of a dam in long-term service. The main purpose of this paper is to establish a state space model for separating causal components from the measured dam displacement. This approach is conducted by initially proposing two equations, which are the state and observation equations, and model parameters are then optimized by the Kalman filter algorithm. The state equation is derived according to the creep deformation of dam concrete and foundation rock and is used to preliminarily predict the dam time effect displacement. Considering the generally recognized three components of dam displacement, the hydraulic-seasonal-time (HST) model is used to establish the observation equation, which is used to update the time effect displacement. The efficiency and rationality of the established state space model is verified by an engineering example. The results show that the hydraulic component separated by the state space model only contains the instantaneous elastic hydraulic deformation, while the hysteretic elastic hydraulic deformation is divided into the time effect component. The inverted elastic modulus of dam body concrete is an instantaneous value for the state space model but a comprehensive reflection of the instantaneous and hysteretic elastic deformation ability for the HST model, where the hysteretic elastic deformation is a part of the hydraulic component. For the Xiaowan arch dam, the inverted values are 42.9 and 36.7 GPa for the state space model and HST model, respectively. The proposed state space model is useful to improve the interpretation ability of the separated displacement components of concrete dams.

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