Estimation algorithm for stochastic linear hybrid systems with quadratic guard conditions

Hybrid systems consist of both continuous state evolutions and discrete state (or mode) transitions. In many hybrid systems, mode transitions are governed by guard conditions that are dependent on the continuous state. These kind of mode transitions can be described by continuous-state-dependent mode transition probabilities. However, most multiple model Kalman filter based algorithms, such as the well-known interacting multiple model (IMM) algorithm, assume constant mode transition probabilities. We propose a hybrid estimation algorithm based on the IMM approach for the stochastic hybrid systems with continuous-state-dependent transitions. We first develop two models to describe the guard conditions for various stochastic hybrid systems. From the models of the guard conditions, we derive the corresponding continuous-state-dependent mode transition probabilities, which are expressed in terms of Gaussian probability density functions (pdfs) or Gaussian cumulative density functions (cdfs). We then propose a hybrid estimation algorithm for the stochastic hybrid systems based on the IMM algorithm. Like the IMM algorithm, our algorithm utilizes Gaussian distributions to represent the continuous state pdfs which are updated analytically at each time step. We also use analytical integration methods to reduce the computational cost of updating the mode probabilities. As a result, the proposed algorithm is computationally efficient and is applicable to high-dimensional problems. The performance of the proposed algorithm is illustrated with two examples in air traffic control tracking applications.

In this study a landing control of an acrobat robot is considered and Q-learning method is applied for falling avoidance control. Since the dynamics of the system is changed according to contact conditions to the ground, the system is a typical variable constraint and hybrid system. The state space for the Q-learning consists of discrete mode variable and continuous states. It is shown by numerical simulations that taking a step motion is automatically generated and falling down is avoided properly.

Jump Markov linear systems are linear systems whose parameters evolve with time according to a finite-state Markov chain. Given a set of observations, our aim is to estimate the states of the finite-state Markov chain and the continuous (in space) states of the linear system. The computational cost in computing conditional mean or maximum a posteriori (MAP) state estimates of the Markov chain or the state of the jump Markov linear system grows exponentially in the number of observations. We present three globally convergent algorithms based on stochastic sampling methods for state estimation of jump Markov linear systems. The cost per iteration is linear in the data length. The first proposed algorithm is a data augmentation (DA) scheme that yields conditional mean state estimates. The second proposed scheme is a stochastic annealing (SA) version of DA that computes the joint MAP sequence estimate of the finite and continuous states. Finally, a Metropolis-Hastings DA scheme based on SA is designed to yield the MAP estimate of the finite-state Markov chain. Convergence results of the three above-mentioned stochastic algorithms are obtained. Computer simulations are carried out to evaluate the performances of the proposed algorithms. The problem of estimating a sparse signal developing from a neutron sensor based on a set of noisy data from a neutron sensor and the problem of narrow-band interference suppression in spread spectrum code-division multiple-access (CDMA) systems are considered.

The interplay of random phenomena with discrete-continuous dynamics deserves increased attention in many systems of growing importance. Their verification needs to consider both stochastic behaviour and hybrid dynamics. In the verification of classical hybrid systems, one is often interested in deciding whether unsafe system states can be reached. In the stochastic setting, we ask instead whether the probability of reaching particular states is bounded by a given threshold. In this thesis, we consider stochastic hybrid systems and develop a general abstraction framework for deciding such problems. This gives rise to the first mechanisable technique that can, in practice, formally verify safety properties of systems which feature all the relevant aspects of nondeterminism, general continuous-time dynamics, and probabilistic behaviour. Being based on tools for classical hybrid systems, future improvements in the effectiveness of such tools directly carry over to improvements in the effectiveness of our technique. We extend the method in several directions. Firstly, we discuss how we can handle continuous probability distributions. We then consider systems which we are in partial control of. Next, we consider systems in which probabilities are parametric, to analyse entire system families at once. Afterwards, we consider systems equipped with rewards, modelling costs or bonuses. Finally, we consider all orthogonal combinations of the extensions to the core model.

Continuous-time birth-death Markov processes serve as useful models in population biology. In this paper, we present a procedure for constructing approximate stochastic models for these processes. This is done by representing the population of a species as the continuous state of a stochastic hybrid system (SHS). This SHS is characterized by reset maps that account for births and deaths, transition intensities that correspond to the birth-death rates, and trivial continuous dynamics. It has been shown that for this type of SHS the statistical moments of the continuous state evolve according to an infinite-dimensional linear ordinary differential equation (ODE). However, for analysis purposes it is convenient to approximate this infinite-dimensional linear ODE by a finite-dimensional nonlinear one. This procedure generally approximates some higher-order moments by a nonlinear function of lower-order moments and it is called moment closure. We obtain moment closures by matching time derivatives of the infinite-dimensional ODE with its finite-dimensional approximation at some time t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . This guarantees a good approximation, at-least locally in time. We provide explicit formulas to construct these approximations and compare this technique with alternative moment-closure methods available in the literature. This comparison takes into account both the transient and the steady-state regimens

A bottleneck to the flow of air traffic is in the traffic around airports. Accurate tracking of aircraft movements around airports is important to improve traffic capacity and safety. We propose a hybrid estimation algorithm that uses the knowledge of departure or arrival aircraft flight plans and nominal flight profiles to improve tracking of aircraft movements around airports. The algorithm also provides an accurate estimation of aircraft flight modes that is often used in conflict detection, intent inference, and trajectory prediction algorithms. In the algorithm, we first develop a methodology to model flight mode transitions based on typical aircraft landing profiles, takeoff profiles, and standard flight routes around airports. Based on the model, we derive a conditional mode transition matrix that is dependent (or conditioned) on the continuous state of the aircraft. The aircraft-tracking problem is then solved using a hybrid estimation algorithm. We also present an analytical result that reduces the dimension, and hence the computational cost, of a multivariate integral that arises due to the continuous-state-dependent mode transition matrix. We present two simulation examples to illustrate the performance of our aircraft-tracking algorithm.

Hybrid dynamical systems are those with interaction between continuous and discrete dynamics. For the analysis and control of such systems concepts and theories from either the continuous or the discrete domain are typically readapted. In this thesis the ideas from perturbation theory are readapted for approximating a hybrid system using a continuous one. To this purpose, hybrid systems that possess a two-time scale property, i.e. discrete states evolving in a fast time-scale and continuous states in a slow time-scale, are considered. Then, as in singular perturbation or averaging methods, the system is approximated by a slow continuous time system. Since the hybrid nature of the process is removed by averaging, such a procedure is referred to as dehybridization in this thesis. It is seen that fast transitions required for dehybridization correspond to fast switching in all but one of the discrete states (modes). Here, the notion of dominant mode is defined and the maximum time interval spent in the non-dominant modes is considered as the 'small' parameter which determines the quality of approximation. It is shown that in a finite time interval, the solutions of the hybrid model and the continuous averaged one stay 'close' such that the error between them goes to zero as the 'small' parameter goes to zero. To utilize the ideas of dehybridization for control purposes, a cascade control design scheme is proposed, where the inner-loop artificially creates the two-time scale behavior, while the outer-loop exponentially stabilizes the approximate continuous system. It is shown that if the origin is a common equilibrium point for all modes, then for sufficiently small values of the 'small' parameter, exponential stability of the hybrid model can be guaranteed. However, it is shown that if the origin is not an equilibrium point for some modes, then the trajectories of the hybrid model are ultimately bounded, the bound being a function of the 'small' parameter. The analysis approach used here defines the hybrid system as a perturbation of the averaged one and works along the lines of robust stability. The key technical diffierence is that though the norm of the perturbation is not small, the norm of its time integral is small. This thesis was motivated by the stick-slip drive, a friction-based micropositioning setup, which operates in two distinct modes 'stick' and 'slip'. It consists of two masses which stick together when the interfacial force is less than the Coulomb frictional force, and slips otherwise. The proposed methodology is illustrated through simulation and experimental results on the stick-slip drive.

The subject of modeling and stability analysis of dc-dc resonant converters is still a challenge. The conventional large signal nonlinear model of the resonant converter is derived using the sinusoidal approximation and averaging followed by linearization about an operating point. Models obtained with such method involve considerable approximation, and produce results that are limited for higher performance designs. Therefore, it is essential to investigate the stability of the resonant converters using a more sophisticated model. Because of semiconductors switching, dc-dc resonant converters are intrinsically hybrid systems consist of discrete input and continuous states. The complexity of the stability analysis and controller design is increased when the system variables are a combination of discrete and continuous variables. In this paper a novel hybrid modeling framework is proposed for the DC-DC series resonant converters that is effective for operating both above and below resonance and when the converter operates either in continuous or discontinuous conduction modes. By defining the discrete input based on continuous states, a new hybrid system is obtained which has both continuous input and continuous states.

This work is concerned with the safety controller synthesis of stochastic hybrid systems, in which continuous evolutions are described by stochastic differential equations with both Brownian motions and Poisson processes, and instantaneous jumps are governed by stochastic difference equations with additive noises. Our proposed framework leverages the notion of control barrier certificates (CBC), as a discretization-free approach, to synthesize safety controllers for stochastic hybrid systems while providing safety guarantees in finite time horizons. In our proposed scheme, we first provide an augmented framework to characterize each stochastic hybrid system containing continuous evolutions and instantaneous jumps with a unified system covering both scenarios. We then introduce an augmented control barrier certificate (ACBC) for augmented systems and propose sufficient conditions to construct an ACBC based on CBC of original hybrid systems. By utilizing the constructed ACBC, we quantify upper bounds on the probability that the stochastic hybrid system reaches certain unsafe regions in a finite time horizon. The proposed approach is verified over a nonlinear case study.

Approximate Model Checking of Stochastic Hybrid Systems

This paper presents a rapid prototyping technique that focuses on transition management in hybrid systems. In particular, it integrates hybrid modeling and simulation (for specifying and validating transition strategies) with a generic transition management pattern built on a Real-Time CORBA-based platform for reconfigurable control systems. Code generation is performed from the hybrid models to transition management code. This allows hybrid systems and their reconfiguration strategies to be prototyped and validated at an early stage in development and for the strategies to be directly transferred into the control system software without a separate, manual reimplementation step. The application of this work is found in most complex control systems, including the control of unmanned aerial vehicle (UAVs) performing extreme maneuvers, which is our driving application area.

Explicit and tractable characterizations of the dynamical behavior of virus particles are pivotal for a thorough understanding of the infection mechanisms of viruses. This thesis deals with the problem of extracting symbolic representations of the dynamical behavior of fluorescent particles from fluorescence microscopy image sequences. The focus is on the behavior of virus particles such as fusion with the cell membrane. A numerical representation is obtained by tracking the particles in the image sequences. We have investigated probabilistic tracking approaches, including approaches based on the Kalman filter as well as based on particle filters. For reasons of efficiency and robustness, we developed a tracking approach based on the probabilistic data association (PDA) algorithm in combination with an ellipsoidal sampling scheme that exploits effectively the image data via parametric appearance models. To track objects in close proximity, we compute the support that each image position provides to each tracked object relative to the support provided to the object's neighbors. After tracking, the problem of mapping the trajectory information computed by the tracking approaches to symbolic representations of the behavior arises. To compute symbolic representations of behaviors related to the fusion of single virus particles with the cell membrane based on their intensity over time, we developed a layered probabilistic approach based on stochastic hybrid systems as well as hidden Markov models (HMMs). We use a maxbelief strategy to efficiently combine both representations. The layered approach describes the intensity, intensity models, and behaviors of single virus particles. We introduce models for the evolution of the intensity and the behavior. To compute estimates for the intensity, intensity models, and behaviors we use a hybrid particle filter and the Viterbi algorithm. The developed approaches have been applied to synthetic images as well as to real microscopy image sequences displaying human immunodeficiency virus (HIV-1) particles. We have performed an extensive quantitative evaluation of the performance and a comparison with several existing approaches. It turned out that our approaches outperform previous ones, thus yielding more accurate and more reliable information about the behavior of virus particles. Moreover, we have successfully applied our tracking approaches to 3D image sequences displaying herpes simplex virus (HSV) replication compartments. We also applied the tracking approaches to image data displaying microtubule tips and analyzed their motion. In addition, our tracking approaches were successfully applied to the 2D and 3D image data of a Particle Tracking Challenge.

The increasing volume of traffic in the air transportation is leading to excessive workload on air traffic controllers. Developing automated air traffic management (ATM) tools is a critical technology in reducing the workload of air traffic controllers and hence increasing the airspace capacity. The existing approaches to automated ATM either use overly-simplified air traffic and aircraft dynamics models to reduce computational complexity or end up being computationally intractable for large-scale ATM scenarios. This paper presents a new hybrid system description of modeling the decision process of the air traffic controllers in en-route and approach operations. The model is based on the domain expertise provided by the State Airport Authority and Air Navigation Service Provider (ANSP) of Turkey. The emulation of air traffic controller decision process in the hybrid model provides realistic conflict resolution maneuvers and separation assurance in 3D, while being computationally tractable. The algorithm has polynomial iteration complexity in the number of waypoints of the aircraft, which makes it scalable to large-scale ATM scenarios with more than 100 aircraft. The algorithm is validated on the real air traffic data over the Istanbul region extracted from the ALLFT+ dataset provided by EUROCONTROL, which includes over 7000 flights in a 96-hour period. The developed algorithm is also integrated into a Boeing 737-800 flight deck simulator with a custom radar display to demonstrate the applicability to existing avionics systems.

Hybrid system consists of discrete and continuous state. Fault diagnosis and prediction of hybrid system is a new and critical issue in hybrid system study. Utilizing the advantage of particle filter, which can estimate the discrete and continuous states simultaneously, a new method for fault diagnosing and predicting of hybrid system is proposed. After giving the design steps of particle filter based fault diagnosis and prediction, we discuss the application of the method to the mobile robot fault diagnosis and prediction system. The simulation results show the effectiveness of the proposed approach.

FADS based aerodynamic parameters estimation for mars entry considering fault detection and tolerance