Discrete-time Nonlinear Dynamical Systems Research Articles

AN OPTIMIZATION OF MAXIMAL INVARIANCE IN A CLASS OF MULTIPLE VALUED ITERATIVE DYNAMICS MODELS OF NONLINEAR DISTURBED CONTROL SYSTEMS

We discuss an optimization problem on the maximal invariance of a class of multiple-valued iterative dynamics (MVID) models of discrete-time nonlinear control dynamical systems with singular disturbance. We study the inner and outer bounds of the maximal invariance, between which all noninterfering MVID models of nonlinear discrete-time control dynamical systems with singular disturbance reside. We also study the invariant fractal structure and an optimization of Lyapunov multipliers associated to it.

A proportional integral extremum-seeking control approach for discrete-time nonlinear systems

ABSTRACTThis paper proposes a proportional-integral extremum-seeking control technique for a class of discrete-time nonlinear dynamical systems with unknown dynamics. The technique is a generalisation of existing time-varying extremum-seeking control (ESC) techniques that provides fast transient performance of the closed-loop system to the optimum equilibrium of a measured objective function. The main contribution of the proposed technique is the addition of a proportional action that can be used to minimise the impact of a time-scale separation on the transient performance of the ESC system. The integral action fulfils the role of standard ESC techniques to identify optimal equilibrium conditions. The effectiveness of the proposed approach is demonstrated using a simulation example.

Heterogeneous Expectations, Boom-Bust Housing Cycles, and Supply Conditions: A Nonlinear Dynamics Approach

We combine a standard stock-flow housing market model, incorporating explicit relationships between house prices, the housing stock, and the rent level, with a parsimonious expectation formation scheme of housing market investors, reflecting an evolving mix of extrapolative and regressive expectation rules. The model results in a two-dimensional discrete-time nonlinear dynamical system. Based on realistic parameters, the model is able to generate endogenous boom-bust housing market dynamics with lasting periods of overvaluation and overbuilding. We thus exploit our model to investigate how real forces, in particular supply conditions, interact with expectations-driven housing market fluctuations.

Heterogeneous expectations, boom-bust housing cycles, and supply conditions: A nonlinear economic dynamics approach

We combine a standard stock-flow housing market model, incorporating explicit relationships between house prices, the housing stock, and the rent level, with a parsimonious expectation formation scheme of housing market investors, reflecting an evolving mix of extrapolative and regressive expectation rules. The model results in a two-dimensional discrete-time nonlinear dynamical system. Based on realistic parameters, the model is able to generate endogenous boom-bust housing market dynamics with lasting periods of overvaluation and overbuilding. We thus exploit our model to investigate how real forces, in particular supply conditions, interact with expectations-driven housing market fluctuations.

Neural estimation using a stable discrete-time MLP observer for a class of discrete-time uncertain MIMO nonlinear systems

This paper has proposed a novel class of the stable discrete-time multilayer perceptron (MLP) neural observer for a class of discrete-time multiple-input multiple-output uncertain nonlinear systems. It is trained online and is used to estimate the states of the system especially when there is no prior knowledge about the dynamics of the system and can be applied as a basis for designing an intelligent control. The weight updating for the MLP is the developed backpropagation algorithm. Online training, convergence of the observer error to neighborhood of origin, robustness against uncertainties and disturbance, low computational expense and fast convergence rate are the main qualities of the suggested method. Lyapunov’s direct method is used to guarantee the stability of the proposed system. To demonstrate the performance of the discrete-time MLP observer, two discrete-time nonlinear dynamic systems are simulated in MATLAB/Simulink. Simulation results confirm the proficiency of the system even at different operating conditions and the presence of disturbance and measurement noise.

FCPN Approach for Uncertain Nonlinear Dynamical System with Unknown Disturbance

In this work, we have used a fuzzy counter-propagation network (FCPN) model to control different discrete-time, uncertain nonlinear dynamic systems with unknown disturbances. Fuzzy competitive learning (FCL) is used to process the weight connection and make adjustments between the instar and the outstar of the network. FCL paradigm adopts the principle of learning, used for calculation of the Best Matched Node (BMN) in the instar–outstar network. FCL provides a control of discrete-time uncertain nonlinear dynamic systems having dead zone and backlash. The errors like mean absolute error (MAE), mean square error (MSE), and best fit rate, etc. of FCPN are compared with networks like dynamic network (DN) and back propagation network (BPN). The FCL foretells that the proposed FCPN method gives better results than DN and BPN. The success and enactments of the proposed FCPN are validated through simulations on different discrete-time uncertain nonlinear dynamic systems and Mackey–Glass univariate time series data with unknown disturbances over BPN and DN.

Stabilization of nonlinear discrete-time dynamic control systems with a parameter and state-dependent coefficients

A numerical-analytical algorithm for designing nonlinear stabilizing regulators for the class of nonlinear discrete-time control systems is proposed that significantly reduces computational costs. The resulting regulator is suboptimal with respect to the constructed quadratic functional with state-dependent coefficients. The conditions for the stability of the closed-loop system are established, and a stability result is stated. Numerical results are presented showing that the nonlinear regulator designed is superior to the linear one with respect to both nonlinear and standard time-invariant cost functionals. An example demonstrates that the closed-loop system is uniformly asymptotically stable.

Stochastic Averaging in Discrete Time and its Applications to Extremum Seeking

We investigate stochastic averaging theory for locally Lipschitz discrete-time nonlinear systems with stochastic perturbation and its applications to convergence analysis of discrete-time stochastic extremum seeking algorithms. Firstly, by defining two average systems (one is continuous time, the other is discrete time), we develop discrete-time stochastic averaging theorem for locally Lipschitz nonlinear systems with stochastic perturbation. Our results only need some simple and applicable conditions, which are easy to verify, and remove a significant restriction present in existing results: global Lipschitzness of the nonlinear vector field. Secondly, we provide a discrete-time stochastic extremum seeking algorithm for a static map, in which measurement noise is considered and an ergodic discrete-time stochastic process is used as the excitation signal. Finally, for discrete-time nonlinear dynamical systems, in which the output equilibrium map has an extremum, we present a discrete-time stochastic extremum seeking scheme and, with a singular perturbation reduction, we prove the stability of the reduced system. Compared with classical stochastic approximation methods, while the convergence that we prove is in a weaker sense, the conditions of the algorithm are easy to verify and no requirements (e.g., boundedness) are imposed on the algorithm itself.

An Iterative Learning Control Design Method for Nonlinear Discrete-Time Systems with Unknown Iteration-Varying Parameters and Control Direction

An iterative learning control (ILC) scheme is designed for a class of nonlinear discrete-time dynamical systems with unknown iteration-varying parameters and control direction. The iteration-varying parameters are described by a high-order internal model (HOIM) such that the unknown parameters in the current iteration are a linear combination of the counterparts in the previous certain iterations. Under the framework of ILC, the learning convergence condition is derived through rigorous analysis. It is shown that the adaptive ILC law can achieve perfect tracking of system state in presence of iteration-varying parameters and unknown control direction. The effectiveness of the proposed control scheme is verified by simulations.

Neural networks for tracking of unknown SISO discrete-time nonlinear dynamic systems

This article presents a Lyapunov function based neural network tracking (LNT) strategy for single-input, single-output (SISO) discrete-time nonlinear dynamic systems. The proposed LNT architecture is composed of two feedforward neural networks operating as controller and estimator. A Lyapunov function based back propagation learning algorithm is used for online adjustment of the controller and estimator parameters. The controller and estimator error convergence and closed-loop system stability analysis is performed by Lyapunov stability theory. Moreover, two simulation examples and one real-time experiment are investigated as case studies. The achieved results successfully validate the controller performance.

The Effect of Foreign Direc t Investment in Economic Growth from the Perspective of Nonlinear Dynamics

In today’s globalized economy one of the most crucial factors for the economic growth of a country, especially of a developing country, is the foreign direct investment, not only because of the transfer of capital but also of technology. In this work, the effect of foreign direct investments in a county’s economic growth by using tools of nonlinear dynamics is studied. As a model of the economic growth of a country, a well-known nonlinear discrete-time dynamical system, the Logistic map, is used. The system under study consists of two countries with a strong economic relationship. The source country of foreign direct investments is an industrialized, economically powerful and technologically advanced country that makes significant investments in the host country, which is a developing country and strong dependent from the source country. Simulation results of system’s behavior and especially the bifurcation diagrams reveal the strong connection between the countries of the proposed system and the effect of foreign direct investments in the economic growth of the host country.

Modeling and Identification of Discrete-Time Nonlinear Dynamic Cascade Systems with Input Hysteresis

A simple approach to modeling and identification of discrete-time nonlinear dynamic systems having an input hysteresis in cascade with a linear dynamic system is presented. A special form of Coleman-Hodgdon model for the hysteresis is considered, which is linear in parameters. For the cascade system parameter estimation, an iterative method with internal variable estimation is proposed. Simulation studies of cascade systems identification using special periodic inputs are included.

Game formulation of multiteam target assignment and suppression mission

Formulating control commands for decision makers in an extended complex mission that involves adversarial behavior has recently received significant attention. Missions of this type are typically modeled by a nonlinear discrete time dynamic system. The state is controlled by two decision makers, each with a different objective function and hierarchy of decision-making structure. To illustrate such a mission, we derive a state space dynamic model of an extended complex military operation that involves two opposing forces engaged in a battle. The model assumes a number of fixed targets that one force is attacking and the other is defending. Due to the number of control commands, the options for each force, and the steps during which the two forces could be engaged, the optimal solution for such a complicated dynamic game over all stages is computationally extremely difficult, if not impossible, to prove. As an alternative, we propose an expeditious suboptimal solution for this type of adversarial engagement. We discuss a solution approach where the decisions are decomposed hierarchically, and the task allocation is separated from cooperation decisions. This novel decoupled solution, although suboptimal in the global sense, is useful when decisions should be taken swiftly in the presence of adversaries. Some example scenarios illustrating this military model and our solution approach are presented.

On the Robustness of Nominal Nonlinear Minimum-Time Control and Extension to Non-Robustly Controllable Target Sets

This work deals with the analysis and the design of minimum-time control laws for a class of nonlinear discrete-time dynamical systems characterized by K-continuous transition maps and bounded control inputs. In the paper, it is shown that the reachability properties of the target set, even if not robust positively controllable in one state transition, can be exploited to assess the existence of a robust positively controllable set including the target in its interior. This result allows the formulation of a robustified minimum-time control policy, based on iterated online optimizations and guaranteeing the ultimate boundedness of the state-trajectories in the presence of bounded uncertainties, even if the target set is not robust positively controllable.

Mean-square analysis of stochastic cycles in nonlinear discrete-time systems with parametric noise

We study a discrete-time nonlinear dynamical system forced by parametric noise. A method of mean-square analysis of the dispersion of random solutions near deterministic cycle is elaborated. A problem of the existence of the stable periodic solution of the closed system for second moments is studied in detail. This problem is reduced to the estimation of the spectral radius of some positive operator. A constructive method of the spectral majorants is suggested. The accuracy of our mathematical technique is demonstrated in the analysis of stochastically forced periodic regimes for the Henon model.

Design of an adaptive fuzzy sliding mode control for uncertain discrete-time nonlinear systems based on noisy measurements

This paper presents the design of an adaptive fuzzy sliding mode control (AFSMC) for uncertain discrete-time nonlinear dynamic systems. The dynamic systems are described by a discrete-time state equation with nonlinear uncertainties, and the uncertainties include the modelling errors and the external disturbances to be unknown but nonlinear with the bounded properties. The states are measured by the restriction of measurement sensors and the contamination with independent measurement noises. The nonlinear uncertainties are approximated by using the fuzzy IF-THEN rules based on the universal approximation theorem, and the approximation error is compensated by adding an adaptive complementary term to the proposed AFSMC. The fuzzy inference approach based on the extended single input rule modules is proposed to reduce the number of the fuzzy IF-THEN rules. The estimates for the un-measurable states and the adjustable parameters are obtained by using the weighted least squares estimator and its simplified one. It is proved that under some conditions the estimation errors will remain in the vicinity of zero as time increases, and the states are ultimately bounded subject to the proposed AFSMC. The effectiveness of the proposed method is indicated through the simulation experiment of a simple numerical system.

Approximating Solutions for Nonlinear Dynamic Tracking Games

This paper presents the OPTGAME algorithm developed to iteratively approximate equilibrium solutions of ‘tracking games’, i.e. discrete-time nonzero-sum dynamic games with a finite number of players who face quadratic objective functions. Such a tracking game describes the behavior of decision makers who act upon a nonlinear discrete-time dynamical system, and who aim at minimizing the deviations from individually desirable paths of multiple states over a joint finite planning horizon. Among the noncooperative solution concepts, the OPTGAME algorithm approximates feedback Nash and Stackelberg equilibrium solutions, and the open-loop Nash solution, and the cooperative Pareto-optimal solution.

Auto-regressive moving-average discrete-time dynamical systems and autocorrelation functions on real-valued Riemannian matrix manifolds

The present research paper proposes an extension of the classical scalar Auto-Regressive Moving-Average (ARMA) model to real-valued Riemannian matrix manifolds. The resulting ARMA model on matrix manifolds is expressed as a non-linear discrete-time dynamical system in state-space form whose state evolves on the tangent bundle associated with the underlying manifold. A number of examples are discussed within the present contribution that aim at illustrating the numerical behavior of the proposed ARMA model. In order to measure the degree of temporal dependency between the state-values of the ARMA model, an extension of the classical autocorrelation function for scalar sequences is suggested on the basis of the geometrical features of the underlying real-valued matrix manifold.

Discrete-time Stochastic Extremum Seeking

We present discrete-time stochastic extremum seeking algorithms and prove their convergence using stochastic averaging theory that we recently developed. First, we provide a discrete stochastic extremum seeking algorithm for a static map, in which measurement noise is considered and an ergodic discrete-time stochastic process is used as the excitation signal. Second, for discrete-time nonlinear dynamical systems, in which the output equilibrium map has an extremum, we present a discrete-time stochastic extremum seeking scheme and, with a singular perturbation reduction, we prove the stability of the reduced system. Compared with classical stochastic approximation methods, while the convergence that we prove is in a weaker sense, the conditions of the algorithm are easy to verify and no requirements (e.g., boundedness) are imposed on the algorithm itself.

Calculation Method of Local Bifurcation Point in Piecewise Nonlinear Discrete-Time Dynamical Systems

This paper proposes a calculation method of local bifurcation points for discrete-time dynamical systems with piecewise nonlinear characteristic (PNDDS). First, an n-dimensional PNDDS, which has two piecewise nonlinear maps, is shown and its variational equation is derived. Next, the calculation method of the local bifurcation points which utilizes the conditional equation of the periodic solution and the characteristic equation is proposed. Then, we have to calculate derivatives of the map with an initial value and with a bifurcation parameter to obtain the bifurcation points continuously in the parameter space. The above calculation process is a key of the proposed method, and we explain it in detail. Finally, we apply the proposed method to a two-dimensional PNDDS and calculate the local bifurcation points for confirming its validity.