Hybrid singular systems of differential equations

This work develops hybrid models for large-scale singular differential system and analyzes their asymptotic properties. To take into consideration the discrete shifts in regime across which the behavior of the corresponding dynamic systems is markedly different, our goals are to develop hybrid systems in which continuous dynamics are intertwined with discrete events under random-jump disturbances and to reduce complexity of large-scale singular systems via singularly perturbed Markov chains. To reduce the complexity of large-scale hybrid singular systems, two-time scale is used in the formulation. Under general assumptions, limit behavior of the underlying system is examined. Using weak convergence methods, it is shown that the systems can be approximated by limit systems in which the coefficients are averaged out with respect to the quasi-stationary distributions. Since the limit systems have fewer states, the complexity is much reduced.     

A Limit Theorem for Quantum Markov Chains Associated to Beam Splittings

We study special quantum Markov chains on a Fock space related to iterated beam splittings as introduced in [23]. Besides a characterization of the position distributions of the chain, we show some kind of weak convergence of such discrete time quantum Markov chains to a kind of continuous time quantum Markov process. Furthermore, we provide existence and uniqueness for the solution of a quantum stochastic differential equation related to this quantum Markov process both on an exponential domain and, on a larger domain, in a pointwise sense.     

Review of several stochastic speech unit models

Hidden Markov models are commonly used for speech unit modelling. This type of model is composed of a non-observable or “hidden” process, representing the temporal structure of the speech unit, and an observation process linking the hidden process with the acoustic parameters extracted from the speech signal.Different types of hidden processes (Markov chain, semi-Markov chain, “expanded-state” Markov chain) as well as different types of observation processes (discrete, continuous, semi-continuous—multiple processes) are reviewed, showing their relationships. The maximum likelihood estimation of two-stage stochastic process parameters is presented in an a posteriori probability formalism. An intepretation of the expectation-maximization algorithm is proposed and the practical learning algorithms for hidden Markov models and hidden semi-Markov models are compared in terms of computation structure, probabilistic justification and complexity.This presentation is illustrated by experiments on a multi-speaker 130 isolated word recognition system. The implementation techniques are detailed and the different combinations of state occupancy modelling techniques and observation modelling techniques are studied from a practical point of view.     

Sojourn time distributions and time scale separation results for multiclass state-dependent processor sharing systems with a finite service pool and exponential service requirements

The paper addresses multiclass processor sharing systems with general state-dependent service rates, exponential service requirements and a finite service pool. By considering the amount of service received by a permanent customer and associating this service with the evolution of a Markov Reward process, the sojourn time distribution is formulated in terms of a matrix exponential expression. When the service rates are balanced, this expression can be diagonalized. Tail asymptotics are also discussed. The matrix exponential expression is subsequently exploited towards the study of time scale separation regimes. Unlike the standard practice of assuming a distinct time scale per class, the paper groups more realistically all customer classes in two time scales. Provably tight approximations, of a known, small degree of error, are developed for the sojourn time distribution of a given class (with either fast or slow dynamics), in terms of reduced models containing only the customer classes operating in the same time scale. The approximation for the fast classes gives rise to further characterization of the tail behavior. Additionally, the paper studies another, more specialized variant of the time scale separation regime, in which the service rates take a special form that leads to even simpler approximations. Finally, it is shown that the essence of the main results applies also to the more general setting of service requirement distributions with Markovian phase-type form.     

Hierarchical aggregation of linear systems with multiple time scales

In this paper we carry out a detailed analysis of the multiple time scale behavior of singularly perturbed linear systems of the formdot{x}^{epsilon}(t) = A(epsilon)x^{epsilon}(t)whereA(epsilon)is analytic in the small parameter ε. Our basic result is a uniform asymptotic approximation toexp A(epsilon)tthat we obtain under a certain multiple semistability condition. This asymptotic approximation gives a complete multiple time scale decomposition of the above system and specifies a set of reduced order models valid at each time scale. Our contribution is threefold. 1) We do not require that the state variables be chosen so as to display the time scale structure of the system. 2) Our formulation can handle systems with multiple ( > 2) time scales and we obtain uniform asymptotic expansions for their behavior on [0, infty]. 3) We give an aggregation method to produce increasingly simplified models valid at progressively slower time scales.     

Spectral factorization using the delta operator

In recent years many papers have been published abouth the γ-operator, mostly caused by the better numerical properties and the rapprochement between continuous and discrete time. A major problem within the LQG-design of a δ-based input-output relation has been how to spectral-factorize in an efficient way. The discrete-time method of Kuc era will not be applied since numerical word-length characteristics will be poor for fast sampling rates. In this paper a new approach is considered. A new γ-operator (Tustin operator) is introduced, in order to make an iterative and numerical stable solution to the spectral factorization problem. The key idea is to use the γ-operator resembled by its behavior to the differential operator.     

Rationalization of singularly perturbed continuous systems with time delays

The state equation of a continuous time system with time delay is nonlinear. Therefore, its transfer function is irrational, which is difficult to solve. However, the transfer function of a discrete time state equation with time delay is rational. Hence, the problem can be solved easily. A rationalization of the singularly perturbed continuous system of single and multiple time delays is presented by setting a delay coefficient. The technique is a powerful tool when solving the singularly perturbed continuous equation with single and multiple time delays instead of when using numerical methods such as the Runge-Kutta algorithm.     

Risk-sensitive probability for Markov chains

The probability distribution of a Markov chain is viewed as the information state of an additive optimization problem. This optimization problem is then generalized to a product form whose information state gives rise to a generalized notion of probability distribution for Markov chains. The evolution and the asymptotic behavior of this generalized or “risk-sensitive” probability distribution is studied in this paper and a conjecture is proposed regarding the asymptotic periodicity of risk-sensitive probability and proved in the two-dimensional case. The relation between a set of simultaneous non-linear and the set of periodic attractors is analyzed.     

Control of continuous‐time Markov chains with safety constraints

In this work the controlled continuous‐time finite‐state Markov chain with safety constraints is studied. The constraints are expressed as a finite number of inequalities, whose intersection forms a polyhedron. A probability distribution vector is called safe if it is in the polyhedron. Under the assumptions that the controlled Markov chain is completely observable and the controller induces a unique stationary distribution in the interior of the polyhedron, the author identifies the supreme invariant safety set (SISS) where a set is called an invariant safety set if any probability distribution in the set is initially safe and remains safe as time evolves. In particular, the necessary and sufficient condition for the SISS to be the polyhedron itself is given via linear programming formulations. A closed‐form expression for the condition is also derived as the constraints impose only upper and/or lower bounds on the components of the distribution vectors. If the condition is not satisfied, a finite time bound is identified and used to characterize the SISS. Numerical examples are provided to illustrate the results. Copyright © 2011 John Wiley & Sons, Ltd.     

Decentralized networked control of discrete‐time systems with local networks

This paper considers discrete‐time large‐scale networked control systems with multiple local communication networks connecting sensors, controllers, and actuators. The local networks operate asynchronously and independently of each other in the presence of variable sampling intervals, transmission delays, and scheduling protocols (from sensors to controllers). The time‐delay approach that was recently suggested to decentralized stabilization of large‐scale networked systems in the continuous time is extended to decentralized control in the discrete time. An appropriate Lyapunov‐Krasovskii method is presented that leads to efficient LMI conditions for the exponential stability and l₂‐gain analysis of the closed loop large‐scale system. Differences from the continuous‐time results are discussed. A numerical example of decentralized control of 2 coupled cart‐pendulum systems illustrates the efficiency of the results.     

Fluid computation of passage-time distributions in large Markov models

Recent developments in the analysis of large Markov models facilitate the fast approximation of transient characteristics of the underlying stochastic process. Fluid analysis makes it possible to consider previously intractable models whose underlying discrete state space grows exponentially as model components are added. In this work, we show how fluid-approximation techniques may be used to extract passage-time measures from performance models. We focus on two types of passage measure: passage times involving individual components, as well as passage times which capture the time taken for a population of components to evolve.Specifically, we show that for models of sufficient scale, global passage-time distributions can be well approximated by a deterministic fluid-derived passage-time measure. Where models are not of sufficient scale, we are able to generate upper and lower approximations for the entire cumulative distribution function of these passage-time random variables, using moment-based techniques. Additionally, we show that, for passage-time measures involving individual components, the cumulative distribution function can be directly approximated by fluid techniques.Finally, using the GPA tool, we take advantage of the rapid fluid computation of passage times to show how a multi-class client-server system can be optimised to satisfy multiple service level agreements.     

The equivalence of time-scale decomposition techniques used in the analysis and design of linear systems

Linear system models can be decomposed into reduced-order ‘ fast’ and ‘ slow ’ subsystems using ‘ spectrum separating ’ solutions to dual Riccati type equations. Eigenspace power iterations provide globally convergent iterative schemes for obtaining these solutions. From this basis, an equivalence of previous decomposition techniques is then achieved including the classic two-time-scale asymptotic series decomposition of singularly perturbed systems. The result is a unified theory for the decomposition of time scales in linear systems applicable to both continuous and discrete time models.     

Multi‐surface sliding control for fast finite‐time leader–follower consensus with high order SISO uncertain nonlinear agents

In this paper, multi surface sliding cooperative control scheme is presented and new multiple sliding surfaces are proposed. It is proven that, for the setup that each agent is described by a chain of integrators, where the last integrator is perturbed by a bounded disturbance, leader–follower consensus can be achieved on these sliding surfaces if the communication graph has a directed spanning tree. Also, sliding variables can be driven to the sliding surfaces in fast finite time by the nonsmooth control law. The fast finite‐time Lyapunov stability theorem, the terminal sliding control technique, and the adding a power integrator design approach are used in our proposed control. Simulation results demonstrate the effectiveness of the proposed scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Dynamic behavior of the discrete‐time double integrator with saturated locally stabilizing linear state feedback laws

The main contribution of this paper is to completely characterize the dynamic behavior of the discrete‐time double integrator with a saturated locally stabilizing linear state feedback law. In continuous‐time setting, any linear state feedback control law that locally stabilizes the double integrator also globally stabilizes the system in the presence of actuator saturation. In discrete‐time setting, the equivalent of the double integrator does not have the same property. In this paper, we completely characterize the global behavior of saturated locally stabilizing linear state feedback laws for the discrete‐time double integrator. Copyright © 2012 John Wiley & Sons, Ltd.     

Asymptotically optimal controls of hybrid linear quadratic regulators in discrete time

This work develops asymptotically optimal controls for a class of discrete-time hybrid systems involving singularly perturbed Markov chains having weak and strong interactions. The state space of the underlying Markov chain is decomposed into a number of recurrent classes and a group of transient states. Using a hierarchical control approach, by aggregating the states in each recurrent class into a single state, a continuous-time quadratic limit control problem in which the resulting limit Markov chain has much smaller state space is derived. Using the optimal control for the limit problem, a control for the original problem is constructed, which is shown to be nearly optimal. Finally, a numerical example is given to demonstrate the effectiveness of the approximation scheme.     

Fluctuations of a stochastic approximation procedure with diffusion perturbation

A stochastic approximation procedure with a singularly perturbed regression function is considered. The form of the limit process is obtained that corresponds to the fluctuation of the stochastic approximation procedure in the neighborhood of an equilibrium point. A generator for the limit process is also constructed. The solution of the singular perturbation problem is given for the asymptotic representation of the generator of a Markov renewal process. The results obtained allow one to extend the possibilities of investigation of the asymptotic behavior of the procedure itself.     

用时序和逻辑规则形成动态故障树的紧缩Markov链

针对枚举底事件的所有时序状态来形成Markov链的弊端,借鉴静态故障树的最小割集(minimum cut set, MCS)思想,考虑形成紧缩的Markov链（即最简顺序割集）。其重要目的是剔除冗余的Markov链,在保证不影响系统失效分析的同时减小状态空间,最大限度地避免陷入组合爆炸的危险。其显著特点在于用时序规则和逻辑规则来描述动态逻辑门。以此为基础,用BDD（binary decision diagram）技术首先形成基于逻辑规则的动态系统的MCS,其次对每个MCS运用时序规则,组合成必要的Mark     

Characteristic of exit moments and models of enlargement of states for finite markov chains in terms of global memory functionals

A notion of local- and global-memory functionate for discrete-type distributions is introduced by analogy with the notion of the memory for continuous-type distributions introduced in Muth’s papers. In the class of PH-distributions (i.e., of distributions of the time of Markov chain exit from a subset of states) the necessary and sufficient conditions are obtained for the case where the exit time has an exponential (continuous-time) or a geometric (discrete time) distribution. A new notion of a global memory functional for decomposition of the state space of a finite Markov chain is introduced. Its properties as a measure of quality of decomposition and enlargement of a state space are studied. The asymptotic optimality is proved. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 110–120, May–June, 2000.     

A logic for reasoning about time and reliability

We present a logic for stating properties such as, after a request for service there is at least a 98% probability that the service will be carried out within 2 seconds. The logic extends the temporal logic CTL by Emerson, Clarke and Sistla with time and probabilities. Formulas are interpreted over discrete time Markov chains. We give algorithms for checking that a given Markov chain satisfies a formula in the logic. The algorithms require a polynomial number of arithmetic operations, in size of both the formula and the Markov chain. A simple example is included to illustrate the algorithms.This paper is a revised and extended version of a paper that has appeared under the title A Framework for Reasoning about Time and Reliability in the Proceeding of the 10 ^th IEEE Real-time Systems Symposium, Santa Monica CA, December 1989. The work presented here was performed while the authors were employed by the Swedish Institute of Computer Science (SICS), and partially supported by the Swedish Board for Technical Development (ESPRIT/BRA project 3096, SPEC) and the Swedish Telecommunication Administration (project: PROCOM).