Discrete-time Dynamical Systems Research Articles

Optimal measurement budget allocation for Kalman prediction over a finite time horizon by genetic algorithms

In this paper, we address the problem of optimal measurement budget allocation to estimate the state of a linear discrete-time dynamical system over a finite horizon. More precisely, our aim is to select the measurement times in order to minimize the variance of the estimation error over a finite horizon. In addition, we investigate the closely related problem of finding a trade-off between number of measurements and signal to noise ratio.First, the optimal measurement budget allocation problem is reduced to a deterministic combinatorial program. Then, we propose a genetic algorithm implementing a count preserving crossover to solve it. On the theoretical side, we provide a one-dimensional analysis that indicates that the benefit of using irregular measurements grows when the system is unstable or when the process noise becomes important. Then, using the duality between estimation and control, we show that the problem of selecting optimal control times for a linear quadratic regulator can be reduced to our initial problem.Finally, numerical implementations demonstrate that using measurement times optimized by our genetic algorithm gives better estimate than regularly spaced measurements. Our method is applied to a discrete version of a continuous-time system and the impact of the discretization time step is studied. It reveals good convergence properties, showing that our method is well suited to both continuous-time and discrete-time setups.

Formal controller synthesis from specifications given by discrete-time hybrid automata

This paper deals with formal controller synthesis for discrete-time dynamical systems. We consider a specification provided under the form of a discrete-time hybrid automaton with external inputs, which can represent, for instance, instructions or informations received from a human user or from another system. The hybrid automaton describes the intended behavior of the system and we first consider the problem of synthesizing a controller such that the maximal trajectories of the closed-loop system are also maximal trajectories of the hybrid automaton. We show that the existence of an alternating simulation relation from the specification to the open-loop system is a necessary and sufficient condition for the existence of such controllers. To be able to solve this problem using symbolic (i.e. finite-state) abstractions, we provide a method to compute a symbolic specification that under-approximates the behavior of the hybrid automata. Then, we extend our approach to consider additional safety or reachability requirements so that some unsafe (e.g. blocking) states are avoided or some target states are reached, respectively. The originality of the problem is that these additional requirements are not formulated over the states of the system but over the states of the specification. Finally, we demonstrate the effectiveness of our approach with two illustrative examples from autonomous vehicle control.

An automated system repair framework with signal temporal logic

We present an automated system repair framework for cyber-physical systems. The proposed framework consists of three main steps: (1) system simulation and fault detection to generate a labeled dataset, (2) identification of the repairable temporal properties leading to the faulty behavior and (3) repairing the system to avoid the occurrence of the cause identified in the second step. We express the cause as a past time signal temporal logic (ptSTL) formula and present an efficient monotonicity-based method to synthesize a ptSTL formula from a labeled dataset. Then, in the third step, we modify the faulty system by removing all behaviors that satisfy the ptSTL formula representing the cause of the fault. We apply the framework to two rich modeling formalisms: discrete-time dynamical systems and timed automata. For both of them, we define repairable formulae, the corresponding repair procedures, and illustrate them over case studies.

Dissipativity, inverse optimal control, and stability margins for nonlinear discrete-time stochastic feedback regulators

ABSTRACT In this paper, we derive stability margins for optimal and inverse optimal stochastic feedback regulators. Specifically, gain, sector, and disk margin guarantees are obtained for discrete-time nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal controllers that minimise a nonlinear-nonquadratic performance criterion. Furthermore, using the newly developed notion of stochastic dissipativity we derive a return difference inequality to provide connections between stochastic dissipativity and optimality of nonlinear controllers for discrete-time stochastic dynamical systems. In particular, using extended Kalman-Yakubovich-Popov conditions characterising stochastic dissipativity we show that our optimal feedback control law satisfies a return difference inequality predicated on a difference operator of a controlled Markov dispersion process and is stochastically dissipative with respect to a specific quadratic supply rate.

Set-Membership Filter for Discrete-Time Nonlinear Systems Using State-Dependent Coefficient Parameterization

In this article, a recursive set-membership filtering algorithm for discrete-time nonlinear dynamical systems subject to unknown but bounded process and measurement noises is proposed. The nonlinear dynamics are represented in a pseudolinear form using the state-dependent coefficient (SDC) parameterization. Matrix Taylor expansions are utilized to expand the state-dependent matrices about the state estimates. Upper bounds on the norms of remainders in the matrix Taylor expansions are calculated online using a nonadaptive random search algorithm at each time step. Utilizing these upper bounds and the ellipsoidal set description of the uncertainties, a two-step filter is derived that utilizes the “correction–prediction” structure of the standard Kalman filter variants. At each time step, correction and prediction ellipsoids are constructed that contain the true state of the system by solving the corresponding semidefinite programs. Finally, a simulation example is included to illustrate the effectiveness of the proposed approach.

Controllability of a Linear System With Nonnegative Sparse Controls

This paper studies controllability of a discrete-time linear dynamical system using nonnegative and sparse inputs. These constraints on the control input arise naturally in many real-life systems where the external influence on the system is unidirectional, and activating each input node adds to the cost of control. We derive the necessary and sufficient conditions for controllability of the system, without imposing any constraints on the system matrices. Unlike the well-known Kalman rank based controllability criteria, the conditions presented in this paper can be verified in polynomial time, and the verification complexity is independent of the sparsity level. The proof of the result is based on the analytical tools concerning the properties of a convex cone. Our results also provide a closed-form expression for the minimum number of control nodes to be activated at every time instant to ensure controllability of the system using positive controls.

Hybrid Neural Adaptive Control for Practical Tracking of Markovian Switching Networks.

While neural adaptive control is widely used for dealing with continuous- or discrete-time dynamical systems, less is known about its mechanism and performance in hybrid dynamical systems. This article develops analytical tools to investigate the neural adaptive tracking control of the hybrid Markovian switching networks with heterogeneous nonlinear dynamics and randomly switched topologies. A gradient-descent adaptation law built on neural networks (NNs) is presented for efficient distributed adaptive control. It is shown that the proposed control scheme can guarantee a stable closed-loop error system for any positive control gain and tuning gain. The tracking error is demonstrated to be practically uniformly exponentially stable with a threshold in the mean-square sense. This study further reveals how the topological structure affects the NN function, by measuring the influence of the switched topologies on the learning performance.

Algorithm for solving optimal open-loop terminal control problem for spacecraft rendezvous with account of constraints on the state

The paper proposes an algorithm for solving the optimal open-loop terminal control problem of two spacecraft rendezvous with constraints on their states. A system of nonlinear differential equations that describes the dynamics of the active (maneuvering) spacecraft relative to the passive spacecraft (station) in the central gravitational field of the Earth in the orbital coordinate system of coordinates related to the passive spacecraft center-of-mass is considered as an initial model. The obtained nonlinear model of the active spacecraft dynamics is linearized relative to the specified reference state trajectory of the passive spacecraft, and then it is discretized and reduced to linear recurrence relations. Mathematical formalization of the spacecraft rendezvous problem under consideration is carried out at a specified final moment of time for the obtained discrete-time controlled dynamical system. The quality of solving the problem is estimated by a convex functional taking into account the geometric constraints on the active spacecraft states and the associated control actions in the form of convex polyhedral-compacts in the appropriate finite dimensional vector space. We propose a solution of the problem of optimal terminal control over the approach of the active spacecraft relative to the passive spacecraft in the form of a constructive algorithm on the basis of the general recursive algebraic method for constructing the availability domains of linear discrete controlled dynamic systems, taking into account specified conditions and constraints, as well as using the methods of direct and inverse constructions. In the final part of the paper, the computer modeling results are presented and conclusions about the effectiveness of the proposed algorithm are made.

Describing and Controlling Multivariate Nonlinear Dynamics: A Boolean Network Approach

We introduce a discrete-time dynamical system method, the Boolean network method, that may be useful for modeling, studying, and controlling nonlinear dynamics in multivariate systems, particularly when binary time-series are available. We introduce the method in three steps: inference of the temporal relations as Boolean functions, extraction of attractors and assignment of desirability based on domain knowledge, and design of network control to direct a psychological system toward a desired attractor. To demonstrate how the Boolean network can describe and prescribe control for emotion regulation dynamics, we applied this method to data from a study of how children use bidding to an adult and/or distraction to regulate their anger during a frustrating task (N = 120, T = 480 seconds). Network control strategies were designed to move the child into attractors where anger is OFF. The sample shows heterogeneous emotion regulation dynamics across children in 22 distinct Boolean networks, and heterogeneous control strategies regarding which behavior to perturb and how to perturb it. The Boolean network method provides a novel method to describe nonlinear dynamics in multivariate psychological systems and is a method with potential to eventually inform the design of interventions that can guide those systems toward desired goals.

On Sufficient Conditions for Chaotic Behavior of Multidimensional Discrete Time Dynamical System

In this work, we look at the extension of classical discrete dynamical system to multidimensional discrete-time dynamical system by characterizing chaos notions on {mathbb {Z}}^d-action. The {mathbb {Z}}^d-action on a space X has been defined in a very general manner, and therefore we introduce a {mathbb {Z}}^d-action on X which is induced by a continuous map, f:{mathbb {Z}}times X rightarrow X and denotes it as T_f:{mathbb {Z}}^d times X rightarrow X. Basically, we wish to relate the behavior of origin discrete dynamical systems (X, f) and its induced multidimensional discrete-time (X,T_f). The chaotic behaviors that we emphasized are the transitivity and dense periodicity property. Analogues to these chaos notions, we consider k-type transitivity and k-type dense periodicity property in the multidimensional discrete-time dynamical system. In the process, we obtain some conditions on (X,T_f) under which the chaotic behavior of (X,T_f) is inherited from the original dynamical system (X, f). The conditions varies whenever f is open, totally transitive or mixing. Some examples are given to illustrate these conditions.

Virtual unmodeled dynamic compensation and data-driven nonlinear generalized predictive control

For a class of discrete-time nonlinear dynamic systems, a nonlinear generalized predictive control method is proposed, which combines the conventional generalized predictive control with data-driven control, the feedback control with feedforward compensation, and the signal compensation with the regulator principle. The input-output dynamic equations of the closed-loop system are derived, and the performance of the proposed control algorithm is analyzed. The advantages and effectiveness of the proposed control algorithm are demonstrated through comparative experiments.

Identifying human postural dynamics and control from unperturbed balance

BackgroundUpright standing requires control of an inherently unstable multi-joint human body within a small base of support, despite biological motor and / or sensory noise which challenge balance. Without applying perturbations, system identification methods have been regarded as inadequate, because the relevant internal biological noise processes are not accessible to direct measurement. As a result, unperturbed balance studies have been limited to investigation of behavioral patterns rather than possible underlying control strategies.MethodsIn this paper, we present a mathemathically rigorous system identification method that is applicable to study the dynamics and control of unperturbed balance. The method is derived from autocorrelation matrices with non-zero time lags and identifies the system matrix of a discrete-time dynamic system in the presence of unknown noise processes, without requiring any information about the strength of the noise.ResultsUnlike reasonable ‘least-squares’ approaches, the performance of the new method is consistent across a range of different combinations of internal and measurement noise strengths, even when measurement noise is substantial. We present a numerical example of a model that simulates human upright balancing and show that its dynamics can be identified accurately. With a biomechanically reasonable choice of state and input variables, a state feedback controller can also be identified.ConclusionsThis study provides a new method to correctly identify the dynamics of human standing without the need for known external perturbations. The method was numerically validated using simulation that included realistic features of human balance. This method avoids potential issues of adaptation or possible reflex responses evoked by external perturbations, and does not require expensive in-lab, high-precision measurement equipment. It may eventually enable diagnosis and treatment of individuals with impaired balance, and the development of safe and effective assistive and / or rehabilitative technologies.

Applications of the Poincaré–Hopf Theorem: Epidemic Models and Lotka–Volterra Systems

This article focuses on properties of equilibria and their associated regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincaré–Hopf theorem is used to derive a general result providing a sufficient condition for the system to have a unique equilibrium. The condition involves the Jacobian of the system at possible equilibria and ensures that the system is in fact locally exponentially stable. We apply this result to the susceptible–infected–susceptible (SIS) networked epidemic model, and a generalized Lotka–Volterra system. We use the result further to extend the SIS model via the introduction of decentralized feedback controllers, which significantly change the system dynamics, rendering existing Lyapunov-based approaches invalid. Using the Poincaré–Hopf approach, we identify a necessary and sufficient condition, under which the controlled SIS system has a unique nonzero equilibrium (a diseased steady state), and monotone systems theory is used to show that this nonzero equilibrium is attractive for all nonzero initial conditions. A counterpart condition for the existence of a unique equilibrium for a nonlinear discrete-time dynamical system is also presented.

Statistical and Nonstatistical Methods for Estimating the Impulse Characteristic of a Dynamical System

We consider the problem of estimating the impulse characteristic of a linear time-invariant discrete-time dynamical system, using data on its input and output signals. We propose versions of estimation methods for asymptotically stable dynamical systems in the context of various approaches to the presentation of parametric uncertainty, including approaches based on the interval analysis and the fuzzy set theory. A comparative analysis and numerical illustration of the methods are presented.

Asymptotics for the second-largest Lyapunov exponent for some Perron–Frobenius operator cocycles

Given a discrete-time random dynamical system represented by a cocycle of non-singular measurable maps, we may obtain information on dynamical quantities by studying the cocycle of Perron–Frobenius operators associated to the maps. Of particular interest is the second-largest Lyapunov exponent for the cocycle of operators, λ 2, which can tell us about mixing rates and decay of correlations in the system. We prove a generalized Perron–Frobenius theorem for cocycles of bounded linear operators on Banach spaces that preserve and occasionally contract a cone; this theorem shows that the top Oseledets space for the cocycle is one-dimensional, and there is a lower bound for the gap between the largest Lyapunov exponents λ 1 and λ 2 (that is, an upper bound for λ 2 which is strictly less than λ 1) explicitly in terms of quantities related to cone contraction. We then apply this theorem to the case of cocycles of Perron–Frobenius operators arising from a parametrized family of maps to obtain an upper bound on λ 2; to the best of our knowledge, this work is the first time λ 2 has been upper-bounded for a family of maps. In doing so, we utilize a new balanced Lasota–Yorke inequality. We also examine random perturbations of a fixed map within the family with two invariant densities and show that as the perturbation is scaled back down to the unperturbed map, λ 2 is at least asymptotically linear in the scale parameter. Our estimates are sharp, in the sense that there is a sequence of scaled perturbations of the fixed map that are all Markov, such that λ 2 is asymptotic to −2 times the scale parameter.

Poncelet property and quasi-periodicity of the integrable Boltzmann system

We study the motion of a particle in a plane subject to an attractive central force with inverse-square law on one side of a wall at which it is reflected elastically. This model is a special case of a class of systems considered by Boltzmann which was recently shown by Gallavotti and Jauslin to admit a second integral of motion additionally to the energy. By recording the subsequent positions and momenta of the particle as it hits the wall, we obtain a three-dimensional discrete-time dynamical system. We show that this system has the Poncelet property: If for given generic values of the integrals one orbit is periodic, then all orbits for these values are periodic and have the same period. The reason for this is the same as in the case of the Poncelet theorem: The generic level set of the integrals of motion is an elliptic curve, and the Poincaré map is the composition of two involutions with fixed points and is thus the translation by a fixed element. Another consequence of our result is the proof of a conjecture of Gallavotti and Jauslin on the quasi-periodicity of the integrable Boltzmann system, implying the applicability of KAM perturbation theory to the Boltzmann system with weak centrifugal force.

Controlled Discrete-Time Semi-Markov Random Evolutions and Their Applications

In this paper, we introduced controlled discrete-time semi-Markov random evolutions. These processes are random evolutions of discrete-time semi-Markov processes where we consider a control. applied to the values of random evolution. The main results concern time-rescaled weak convergence limit theorems in a Banach space of the above stochastic systems as averaging and diffusion approximation. The applications are given to the controlled additive functionals, controlled geometric Markov renewal processes, and controlled dynamical systems. We provide dynamical principles for discrete-time dynamical systems such as controlled additive functionals and controlled geometric Markov renewal processes. We also produce dynamic programming equations (Hamilton–Jacobi–Bellman equations) for the limiting processes in diffusion approximation such as controlled additive functionals, controlled geometric Markov renewal processes and controlled dynamical systems. As an example, we consider the solution of portfolio optimization problem by Merton for the limiting controlled geometric Markov renewal processes in diffusion approximation scheme. The rates of convergence in the limit theorems are also presented.

A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks

The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.

Stability, Chaos Diagnose and Adaptive Control of Two Dimensional Discrete - Time Dynamical System

In this paper 2D discrete time dynamical system is presented. The fixed points were found. The stability of fixed points is measured by characteristic roots, jury criteria, Lyapunov function. All show that the system is unstable, and analyzing the dynamic behavior of the system finds bifurcation diagrams at the bifurcation parameter. Newton’s Raphson numerical method was used the roots of the system with the minimum error. Then, chaoticity is measured by the phase space; maximum Lyapunov exponent is obtain as (Lmax=2.394569); Lyapunov dimension is obtain as (DL=3.366413); binary test (0 - 1) is obtain as (k = 0.982). All show that the system is chaotic. Finally, the adaptive control was performed. Moreover, theoretical and graphical results of the system after control show the system is stable and Lyapunov exponent is obtained as: L1=-0.390000, L2=-0.500000, so the system is regular.

Inspecting debt servicing mechanism in Nigeria using ARMAX model of the Koyck-kind

The burden of external debt affects the wellbeing of an economy (or a country) by making the economy vulnerable to external shocks and crowding out investment. When dealing with debt management in indebted poor countries like Nigeria, the rational approach is to allocate a portion of export earnings for debt service payments. Along this line, there is a need to identify the link between debt servicing and export earnings. Hence, the current and long-run effects of export earnings on debt service payments is modelled as a single-input-single-output discrete-time dynamical system within the framework of Autoregressive Moving Average Explanatory Input model of the Koyck-kind (KARMAX). The KARMAX model is identified for Nigeria, using data from the World Bank database from 1970 to 2018 based on the maximum likelihood (ML) method, and the obtained results are compared to the prediction error and the instrumental variable methods. From a theoretical perspective, the KARMAX specification identified by the ML method is more ideal and inspiring. By doing so, this article contributes to the literature on the econometrics of public debt management.