Discrete-time Case Research Articles

Potential difference games and applications

This paper concerns noncooperative difference games with infinite horizon discounted payoffs. More precisely, it is a sequel to a previous paper [A. Fonseca-Morales and O. Hern'andez-Lerma, Potential differential games, Dynamic Games Appl., 8(2) (2018), pp. 254–279.] where the notion of continuous-time potential games was introduced. That is, a noncooperative differential game to which we can associate a continuous-time optimal control problem (OCP) whose solutions are Nash equilibria for the original game. Thus, finding or analysing the properties of Nash equilibria for the game reduces to that of the optimal solution of an OCP. Here, we study difference games, that is, the discrete-time case. First, we give several mild conditions for which a difference game is a potential difference game (PDG). Then, we illustrate our results with several examples and applications.

Demographic inference from multiple whole genomes using a particle filter for continuous Markov jump processes.

Demographic events shape a population’s genetic diversity, a process described by the coalescent-with-recombination model that relates demography and genetics by an unobserved sequence of genealogies along the genome. As the space of genealogies over genomes is large and complex, inference under this model is challenging. Formulating the coalescent-with-recombination model as a continuous-time and -space Markov jump process, we develop a particle filter for such processes, and use waypoints that under appropriate conditions allow the problem to be reduced to the discrete-time case. To improve inference, we generalise the Auxiliary Particle Filter for discrete-time models, and use Variational Bayes to model the uncertainty in parameter estimates for rare events, avoiding biases seen with Expectation Maximization. Using real and simulated genomes, we show that past population sizes can be accurately inferred over a larger range of epochs than was previously possible, opening the possibility of jointly analyzing multiple genomes under complex demographic models. Code is available at https://github.com/luntergroup/smcsmc.

One Version of the Theorem on Stability by the First Approximation in the Discrete-Time Case

One Version of the Theorem on Stability by the First Approximation in the Discrete-Time Case

Markov evolution algebras

ABSTRACT We study Markov evolution algebras, that is, evolution algebras having Markov structure matrices. We first consider the discrete-time case, and delve into their algebraic structure for later application to continuous-time Markov evolution algebras that arise defined by standard stochastic semigroups. The study of embeddable Markov evolution algebras, that is, of Markov evolution algebras with structure matrices existing within standard stochastic semigroups is then equivalent to the embedding problem for Markov matrices.

Sharp Error Term in Local Limit Theorems and Mixing for Lorentz Gases with Infinite Horizon

We obtain sharp error rates in the local limit theorem for the Sinai billiard map (one and two dimensional) with infinite horizon. This result allows us to further obtain higher order terms and thus, sharp mixing rates in the speed of mixing of dynamically Hölder observables for the planar and tubular infinite horizon Lorentz gases in the map (discrete time) case. We also obtain an asymptotic estimate for the tail probability of the first return time to the initial cell. In the process, we study families of transfer operators for infinite horizon Sinai billiards perturbed with the free flight function and obtain higher order expansions for the associated families of eigenvalues and eigenprojectors.

Stability and gain performance analysis for singular switched positive systems under MDMDT switching signal

For delayed singular switched positive systems(SSPSs) under mode-dependent minimum dwell time(MDMDT) switching signal, the stability and standard gain performance in the sense of L1-gain and L∞-gain are analyzed in this paper, and both continuous-time and discrete-time cases are taken into account. Firstly, a sufficient stability condition for delayed continuous-time SSPSs under MDMDT switching signal is provided, and meanwhile, to analyze the disturbance attenuation performance, the standard gain performance conditions in the sense of L1-gain and L∞-gain are proposed, respectively. Next, considering delayed discrete-time SSPSs under MDMDT switching signal, sufficient stability condition and standard gain performance conditions in the sense of L1-gain and L∞-gain are put forward, respectively. Moreover, for delayed SSPSs with minimum dwell time switching signal, stability and standard gain performance are also analyzed. At last, the validity of theoretical results are proven via two examples.

LMI stability test for multidimensional linear state–space models

Stability is a basic property of dynamical systems. In this paper we analyze the stability of multidimensional systems and present new sufficient conditions for the asymptotic stability in terms of linear matrix inequalities. We treat both the discrete-time and continuous-time cases and also propose variants that require linear matrix inequalities of more moderate size.

Dual formulation of causal gain-scheduled output feedback controller design using parameter-dependent Lyapunov functions

This paper is concerned with the design problem of causal gain-scheduled output feedback controllers for linear parameter-varying systems using parameter-dependent Lyapunov functions. This research topic has already been addressed by several researchers, and several effective methods have been correspondingly proposed in terms of parametrically dependent matrix inequalities. This paper addresses the dual formulation of one of the methods, and successfully derives the counterpart result in continuous-time case but points out the intractability of deriving the counterpart result in discrete-time case. It is also shown, in discrete-time case, that the intractability disappears and a design method similar to an existing method is successfully derived if the causality with respect to scheduling parameters is abandoned, namely, if one-step ahead scheduling parameters are available. A toy example is introduced to confirm that, in continuous-time case, the derived dual formulation gives the same performance as the previously proposed method for the dual state-space representation of the example.

A Subgradient Algorithm for Data-Rate Optimization in the Remote State Estimation Problem

In the remote state estimation problem, an observer tries to reconstruct the state of a dynamical system at a remote location, where no direct sensor measurements are available. The observer only has access to information sent through a digital communication channel with a finite capacity. The recently introduced notion of restoration entropy provides a way to determine the smallest channel capacity above which an observer can be designed that observes the system without a degradation of the initial observation quality. In this paper, we propose a subgradient algorithm to estimate the restoration entropy via the computation of an appropriate Riemannian metric on the state space, which allows us to determine the approximate value of the entropy from the time-one map (in the discrete-time case) or the generating vector field (for ODE systems), respectively.

McKean--Vlasov SDEs in Nonlinear Filtering

Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in Crisan & Xiong (2010) and Clark & Crisan (2005). We consider three filters that have been proposed in the literature and use this framework to derive It\^{o} representations of their limiting forms as the approximation parameter $\delta \rightarrow 0$. All filters require the solution of a Poisson equation defined on $\mathbb{R}^{d}$, for which existence and uniqueness of solutions can be a non-trivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.

Стохастическое восстановление квадратично интегрируемых функций

The purpose of the study was to solve the problem of stochastic recovery of square-integrable (with respect to the Lebesgue measure) functions defined on the real line from observations with additive white Gaussian noise, for the case of discrete time. The problem is a nonparametric, i.e., infinite-dimensional, estimation problem. The study substantiates the procedure of optimal recovery, in the mean-square sense, with respect to the product of the Lebesgue measure and the Gaussian measure, and describes an algorithm for recovering such square-integrable functions. Findings of research show that the constructed procedure for nonparametric recovery of a square-integrable function gives an unbiased and consistent recovery of an unknown function. This result has not been previously described. In addition, for smooth reconstructed functions, an almost optimal reconstruction procedure is introduced and substantiated, which gives an unimprovable (in order of magnitude) estimate of the dependence of the number of orthogonal functions on the number of observations. The error of the constructed almost optimal recovery procedure in relation to the optimal recovery procedure is no more than 50 %

Passive linear continuous-time systems: Characterization through structure

We here show that the family of finite-dimensional, continuous-time, passive, linear, time-invariant systems can be characterized through the structure of maximal matrix-convex cones, closed under inversion. Moreover, this observation unifies three setups: (i) differential inclusions, (ii) matrix-valued rational functions, (iii) realization arrays associated with rational functions. It turns out that in the discrete-time case, the corresponding structure is of a maximal matrix-convex set, closed under multiplication among its elements.

INEFFICIENT BUBBLES AND EFFICIENT DRAWDOWNS IN FINANCIAL MARKETS

At odds with the common “rational expectations” framework for bubbles, economists like Hyman Minsky, Charles Kindleberger and Robert Shiller have documented that irrational behavior, ambiguous information or certain limits to arbitrage are essential drivers for bubble phenomena and financial crises. Following this understanding that asset price bubbles are generated by market failures, we present a framework for explosive semimartingales that is based on the antagonistic combination of (i) an excessive, unstable pre-crash process and (ii) a drawdown starting at some random time. This unifying framework allows one to accommodate and compare many discrete and continuous time bubble models in the literature that feature such market inefficiencies. Moreover, it significantly extends the range of feasible asset price processes during times of financial speculation and frenzy and provides a strong theoretical background for future model design in financial and risk management problem settings. This conception of bubbles also allows us to elucidate the status of rational expectation bubbles, which, by design, suffer from the paradox that a rational market should not allow for misvaluation. While the discrete time case has been extensively discussed in the literature and is most criticized for its failure to comply with rational expectations equilibria, we argue that this carries over to the finite time “strict local martingale”-approach to bubbles.

Model Reduction of Power System by Modified Balanced Truncation Method

In this article, we are exploring and implementing the new model order reduction (MOR) method for Large-Scale Linear Dynamic System (LSLDS) to achieve reduced order. These technologies are designed to better understand and explain LSLDS based on the Modified Balanced Truncation Method (BTM). This refers to continuous/discrete LTI structures that are minimal / non-minimum. This reduced method allows MOR to preserve complete parameters with reasonable accuracy. The approach is based on the maintenance of dominant system modes and a relatively small state truncation. As the reduced-order model (ROM) is derived from the retention of dominant modes, the reduction remains stable. The main demerit of the balanced truncation method is that the ROM, stable states, does not match the original structures. By modified BTM to narrow the deviations in the ROM transfer function matrix, a gain factor is added to adjust the steady-state values of the reduction system without altering the dynamic behaviour of the system. The proposed method has been successfully applied to a real-time single area power system with ease of extension to a discrete-time case and the results obtained show the efficacy of the method. Application model and the results obtained indicate the effectiveness of the methodology. The time response of the system has been demonstrated by the proposed method, which proves to be excellent match, effectiveness and superiority compared to the response of other approaches in the literature review of the original system.

Exponential stabilization of switched linear systems subject to actuator saturation with stabilizable and unstabilizable subsystems

This paper is concerned with the exponential stabilization of switched linear systems subject to actuator saturation with both stabilizable subsystems and unstabilizable subsystems for continuous-time case and discrete-time case, respectively. Sufficient conditions for the exponential stabilization under dwell time switching under the cases of continuous-time and discrete-time are established by using a novel class of multiple time-varying Lyapunov function. The existence conditions for stabilizing controllers are presented in terms of linear matrix inequalities (LMIs) for the continuous-time case and the discrete-time case, respectively. Two optimization problems are proposed for obtaining the maximal attraction region. The problem of exponential stabilization for switched system subject to actuator saturation with asynchronous switching controller is also studied. Several numerical examples are presented to prove the validity of the obtained results.

Reach almost sure consensus via Lp-norm group information

This paper continues our previous consensus scheme with only group information when neither absolute states nor inter-agent relative states are available, by considering two special nonlinear cases, L∞-norm and L2-norm. In the scheme, all the agents are partitioned into two subgroups at each time by certain probability, and then use the relative group information that concerns all agents’ states inside for state updating process. It is then shown that almost sure consensus can be achieved under the proposed scheme in both discrete time and continuous time for the two nonlinear cases. When for L∞-norm group information, almost sure consensus is achieved if and only if the weighting parameter is greater than one in discrete-time case, and the weighting parameter is positive in continuous-time case. When for L2-norm group information, the sufficient conditions to guarantee system consensus almost surely are the weighting parameter greater than n/2 and positive in discrete-time case and continuous-time case, respectively. Finally, numerical simulations are proposed to validate the theoretical results.

Event-triggered control of positive Markov jump systems without/with input saturation

This paper proposes an event-triggered control for positive Markov jump systems without/with input saturation. First, an event-triggering condition is established in the form of 1-norm for positive Markov jump systems. Under the event-triggering condition, the term related to error signal between the sample state and the actual state is transformed into a form of interval uncertainty. Then, the positivity and stability of the systems are discussed by using the lower and upper bounds of interval uncertain systems, respectively. An event-triggered control law for the underlying systems is proposed by decomposing controller gain matrix into non-positive and non-negative parts. For positive Markov jump systems with input saturation, a design approach to the controller gain matrices and the corresponding controller auxiliary gain is presented. All conditions are described in terms of linear programming. Furthermore, the obtained approaches are developed for the discrete-time case. Finally, the effectiveness of the proposed design is verified via three examples.

Using Semi-Markov Chains to Solve Semi-Markov Processes

This article provides a novel method to solve continuous-time semi-Markov processes by algorithms from discrete-time case, based on the fact that the Markov renewal function in discrete-time case is a finite series. Bounds of approximate errors due to discretization for the transition function matrix of the continuous-time semi-Markov process are investigated. This method is applied to a reliability problem which refers to the availability analysis of the system subject to sequential cyber-attacks. Two cases where sojourn times follow exponential and Weibull distributions are considered and computed in order to verify and illustrate the proposed method.

Event-triggered Control of Positive Systems With State Saturation Using Linear Programming

This paper proposes the event-triggered control of positive systems with state saturation both in discrete-time and continuous-time cases. A 1-norm based event-triggered mechanism is established for positive systems. Under the event-triggered mechanism, the error term between actual and sample states is transformed into interval uncertain form. Together with the properties of saturation, the systems with state saturation are transformed into interval uncertain systems and the corresponding lower and upper bounds of the system matrices are obtained, respectively. Using a linear co-positive Lyapunov function, the event-triggered controller and the auxiliary gain matrix of the domain of attraction are designed in terms of linear programming, respectively. Then, the systems with state and input saturation are also described via interval uncertain systems. An event-triggered controller is designed and thus the closed-loop systems are positive and stable under the designed controller. Furthermore, the presented event-triggered control approach is extended to the continuous-time case. Compared to existing control approaches, the event-triggered control can reduce energy consumption and increase the practicability. Finally, several numerical examples are given to illustrate the effectiveness of the proposed design.

Hankel Singular Values and LQG Characteristic Values of Discrete-Time Linear Systems in Cascade With Inner Systems

Recent results have shown that, for continuous-time systems obtained by cascading an asymptotically stable system with an inner system, the first $n$ Hankel singular values are greater than or equal to those of the original system. Similarly, cascading a system in minimal form with an inner system, the same property holds for the linear quadratic Gaussian (LQG) characteristic values. In this article, we consider the discrete-time case and demonstrate that the property also holds for these systems. A very important consequence stemming from these results is that the Hankel singular values and the LQG characteristic values of input-delayed discrete systems are greater than or equal to those of their zero-delay counterpart.