General Dynamical Systems Research Articles

Unifying framework for information processing in stochastically driven dynamical systems

A dynamical system can be regarded as an information processing apparatus that encodes input streams from the external environment to its state and processes them through state transitions. The information processing capacity (IPC) is an excellent tool that comprehensively evaluates these processed inputs, providing details of unknown information processing in black box systems; however, this measure can be applied to only time-invariant systems. This paper extends the applicable range to time-variant systems and further reveals that the IPC is equivalent to coefficients of polynomial chaos (PC) expansion in more general dynamical systems. To achieve this objective, we tackle three issues. First, we establish a connection between the IPC for time-invariant systems and PC expansion, which is a type of polynomial expansion using orthogonal functions of input history as bases. We prove that the IPC corresponds to the squared norm of the coefficient vector of the basis in the PC expansion. Second, we show that an input following an arbitrary distribution can be used for the IPC, removing previous restrictions to specific input distributions. Third, we extend the conventional orthogonal bases to functions of both time and input history and propose the IPC for time-variant systems. To show the significance of our approach, we demonstrate that our measure can reveal information representations in not only machine learning networks but also a real, cultured neural network. Our generalized measure paves the way for unveiling the information processing capabilities of a wide variety of physical dynamics which has been left behind in nature.

Scheduling and Power Control for Wireless Multicast Systems via Deep Reinforcement Learning.

Multicasting in wireless systems is a natural way to exploit the redundancy in user requests in a content centric network. Power control and optimal scheduling can significantly improve the wireless multicast network’s performance under fading. However, the model-based approaches for power control and scheduling studied earlier are not scalable to large state spaces or changing system dynamics. In this paper, we use deep reinforcement learning, where we use function approximation of the Q-function via a deep neural network to obtain a power control policy that matches the optimal policy for a small network. We show that power control policy can be learned for reasonably large systems via this approach. Further, we use multi-timescale stochastic optimization to maintain the average power constraint. We demonstrate that a slight modification of the learning algorithm allows tracking of time varying system statistics. Finally, we extend the multi-time scale approach to simultaneously learn the optimal queuing strategy along with power control. We demonstrate the scalability, tracking and cross-layer optimization capabilities of our algorithms via simulations. The proposed multi-time scale approach can be used in general large state-space dynamical systems with multiple objectives and constraints, and may be of independent interest.

The turnpike property in nonlinear optimal control—A geometric approach

This paper presents, using dynamical system theory, a framework for investigating the turnpike property in nonlinear optimal control. First, it is shown that a turnpike-like property appears in general dynamical systems with hyperbolic equilibrium and then, apply it to optimal control problems to obtain sufficient conditions for the turnpike behavior to occur. The approach taken is geometric and gives insights for the behaviors of controlled trajectories as well as links between the turnpike property and stability and/or stabilizability in nonlinear control theory. It also allows us to find simpler proofs for existing results on the turnpike properties. Attempts to remove smallness restrictions for initial and target states are also discussed based on the geometry of (un)stable manifolds and a recent result on exponential stabilizability of nonlinear control systems obtained by one of the authors.

Performance Residual Based Fault Detection for Feedback Control Systems

This brief is devoted to the issues of performance supervised fault detection (FD) for two types of feedback control systems by adopting the novel performance residual as the process evaluator. Specifically, for a class of uncertain state feedback control systems, the boundaries of the performance residual caused by the model uncertainties are analyzed theoretically, then the randomized algorithm is exploited to construct the threshold, aiming at avoiding the worst-case handling and improving the FD performance. The second part considers the general dynamic output feedback control systems with external reference signals. For the FD purpose, the performance residual is derived based on the backward computation, and the state variables of the closed-loop system are constructed by the accessible input, output, and reference signals to release the strict requirements for full state measurements. Finally, the effectiveness of the developed FD approaches is demonstrated on the model of a DC motor system.

The weak Smale horseshoe and mean hyperbolicity

The paper concerns the mean hyperbolicity of the skew-product flows induced by the perturbation equations driven by varieties of non-periodic forcings. The weakly averaged horseshoe can be constructed as a mean hyperbolic invariant set in the Poincare section for high dimensional phase space. Due to the non-periodic property, the Poincare return map restricted to the weakly averaged horseshoe region can semi-conjugate to the full Bernoulli shift on infinite symbols, which implies the infinitely many independent choices on the length of return time. As a direct application of mean hyperbolicity, we extend the shadowing lemma due to Liao to the general nonautonomous dynamical systems.

High-order fully actuated system approaches: Part X. Basics of discrete-time systems

A basic introduction to high-order fully actuated (HOFA) approaches for discrete-time systems is given. Firstly, it is shown that, different from the continuous-time systems, general dynamical discrete-time systems can be represented by two types of discrete-time HOFA models, namely, the step forward HOFA models and the step backward HOFA models. Secondly, controllers for the step backward HOFA models are designed, which result in constant linear closed-loop systems with arbitrarily assignable eigenstructures. The related problem of feasibility is also discussed. The well-known discrete-time feedback linearisable systems and strict-feedback systems are shown to be equivalent to both the step forward and step backward HOFA models. Finally, a generalised step backward HOFA model containing control vectors of different time instants is proposed and investigated and demonstrated with a type of proposed pseudo feed-forward systems. The contribution in this paper has laid a fundamental basis for discrete-time HOFA approaches. Further analysis and design problems can be naturally established parallel to the continuous-time system case.

High-order fully-actuated system approaches: Part IX. Generalised PID control and model reference tracking

Tracking control of general dynamical systems in high-order fully actuated (HOFA) system representation is solved. For the case of tracking a constant or slow time-varying signal in the presence of constant or slow time-varying disturbances, a generalised PID control scheme is proposed, which realises asymptotical tracking to a prescribed signal and also guarantees that the state derivatives of certain orders converge to the origin. For the case that the signal to be tracked is generated by a reference model, a model reference tracking (MRT) controller is presented, which relies on the solution to a type of generalised Sylvester matrix equations and guarantees the desired asymptotical tracking requirement. Due to the full-actuation property of the HOFA models, closed-loop systems under both control schemes are constant and linear. Furthermore, based on a general parametric solution to the type of Sylvester matrix equations and a general parametric eigenstructure assignment result, simple and complete parameterisation of the two types of control designs are provided, and feasibility conditions in terms of the system initial values for sub-fully actuated systems are also derived. An illustrative example is presented to demonstrate the application of the proposed approaches and their effects.

Stochastic facilitation in heteroclinic communication channels.

Biological neural systems encode and transmit information as patterns of activity tracing complex trajectories in high-dimensional state spaces, inspiring alternative paradigms of information processing. Heteroclinic networks, naturally emerging in artificial neural systems, are networks of saddles in state space that provide a transparent approach to generate complex trajectories via controlled switches among interconnected saddles. External signals induce specific switching sequences, thus dynamically encoding inputs as trajectories. Recent works have focused either on computational aspects of heteroclinic networks, i.e., Heteroclinic Computing, or their stochastic properties under noise. Yet, how well such systems may transmit information remains an open question. Here, we investigate the information transmission properties of heteroclinic networks, studying them as communication channels. Choosing a tractable but representative system exhibiting a heteroclinic network, we investigate the mutual information rate (MIR) between input signals and the resulting sequences of states as the level of noise varies. Intriguingly, MIR does not decrease monotonically with increasing noise. Intermediate noise levels indeed maximize the information transmission capacity by promoting an increased yet controlled exploration of the underlying network of states. Complementing standard stochastic resonance, these results highlight the constructive effect of stochastic facilitation (i.e., noise-enhanced information transfer) on heteroclinic communication channels and possibly on more general dynamical systems exhibiting complex trajectories in state space.

Chaotic Dynamics of Partial Difference Equations with Polynomial Maps

In this paper, chaotic dynamics of a class of partial difference equations are investigated. With the help of the coupled-expansion theory of general discrete dynamical systems, two chaotification schemes for partial difference equations with polynomial maps are established. These controlled equations are proved to be chaotic either in the sense of Li–Yorke or in the sense of both Li–Yorke and Devaney. One example is provided to illustrate the theoretical results with computer simulations for demonstration.

Value added services at intermodal inland terminals and the importance of choosing a moderate innovation path

After a stage of growth, in which terminals competed primarily by expanding their capacity, we observe a transition into a stage of quality and differentiation by offering value added services (VAS). This paper presents a generic System Dynamics model in order to analyse the impact of offering VAS on a terminal's operational performance. The developed Stock & Flow model allows understanding mid- and long-term consequences of strategic decisions as it deals with the characteristics of a growing system. We investigate the implementation of two VAS, train OCR gates and a fast lane service for trucks. The results show that terminals benefit from an increase in demand, due to reputational gains caused by improved efficiency. The efficiency increase outweighs the downside of higher investment and transhipment costs. However, it is vital to choose a moderate innovation path and to invest step by step to cope with an increase in demand.

Prediction of temperature barriers in weakly collisional plasmas by a Lagrangian coherent structures computational tool

Analysis of Lagrangian Coherent Structures (LCSs) has been shown to be a valid mathematical approach to explain the formation of transport barriers in magnetized plasmas. Such LCSs, borrowed from fluid dynamics theory, can be considered as the hidden skeleton of the system and can be used for studying a wide spectrum of transport mechanisms, even in plasmas. An LCS can be considered as a generalization for a finite time interval and for a general dynamical system of what manifolds are for Hamiltonian systems. In this paper, we demonstrate that such structures can be particularly useful for underlying the hidden paths governing the motion of magnetic field lines in chaotic magnetic fields.To perform such an analysis, we developed a numerical tool able to detect LCSs for a general dynamical system. The tool is able to deal with general coordinate systems and it is shown to match with other techniques already used to analyse chaotic magnetic fields, e.g. connection length. After the description of the computational tool, we focus on the heat transport equation and the comparison between temperature profile and topology of the LCSs. We provide evidence that numerical simulations are able to reproduce the temperature profiles similar to those observed in reversed field pinch experiments and that our tool successfully predicts the location of temperature gradients. The results suggest that, inside the chaotic region, the field-lines motion is far from stochastic and that the presence of hidden patterns allows the development of high temperature gradients.

Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton’s Methods

We investigate i.i.d. random complex dynamical systems generated by probability measures on finite unions of the loci of holomorphic families of rational maps on the Riemann sphere. We show that under certain conditions on the families, for a generic system, (especially, for a generic random polynomial dynamical system,) for all but countably many initial values $z$ in the Riemann sphere, for almost every sequence of maps $\gamma =(\gamma_{1},\gamma_{2},\ldots )$, the Lyapunov exponent of $\gamma $ at $z$ is negative. Also, we show that for a generic system, for every initial value $z$ in the Riemann sphere, the orbit of the Dirac measure at $z$ under the iteration of the dual map of the transition operator tends to a periodic cycle of measures in the space of probability measures on the Riemann sphere. Note that these are new phenomena in random complex dynamics which cannot hold in deterministic complex dynamical systems. We apply the above theory and results of random complex dynamical systems to finding roots of any polynomials by random relaxed Newton's methods and we show that for any polynomial $g$, for any initial value $z$ in the complex plane which is not a root of $g'$, the random orbit starting with $z$ tends to a root of $g$ almost surely, which is the virtue of the effect of randomness.

On the application of the Ważewski method to the problem of global stabilization

We consider a possible application of the Ważewski topological method to feedback control systems and to more general dynamical systems. We show how this method can be used to prove the impossibility of global stabilization in such problems. Moreover, we give sufficient conditions for the existence of a solution such that its trajectory never leaves a subset of the extended phase space of the system and does not tend asymptotically to a given equilibrium. We illustrate our result with various real-life systems including the Furuta pendulum and the wheeled inverted pendulum.

A Revisit to the Controllability and Observability of Networked Dynamic Systems

In this article, the controllability and observability of a general networked dynamic system (NDS) are reinvestigated. While some computationally efficient conditions have been established in previous works, difficulties arise when a full normal rank condition is not satisfied for some subsystems. To address this problem, the subsystem connection topology of the NDS is utilized and a sufficient and necessary condition is derived for its controllability and observability in this article. It is observed that the effects of the transmission zeros and the uncontrollable/unobservable modes of subsystems should be eliminated by subsystem connections to guarantee the controllability/observability of an NDS. Moreover, a sufficient condition and a necessary condition are provided for constructing controllable/observable NDSs in the view of subsystem connection topology and subsystem internal input/output signal, respectively. Finally, the sensor placement of a mechanical system is discussed, which shows the effectiveness of our results.

Sustainability of religious communities.

This article focuses on the relationship between church population and sustainability. We carried out the study on a sample of Presbyterian churches in South Korea, and implemented dynamic optimization of the church population based on the Susceptible-Infected-Recovered (SIR) epidemic model. In particular, System Dynamics (SD) and Agent-Based Model (ABM) simulations are performed for a prototype model with key parameters that contribute to church growth. Potential parameters reflecting sustainability for churches trigger dramatic growth in church populations. We categorized five dimensions of sustainability with various multi-dimensional indicators in order to measure the level of sustainability, and we obtained the values of the indicators by analyzing a number of news articles searched with a text mining technique. As time-dependent values of sustainability are imposed on the generic SD model for church population dynamics as sustainable potential parameters, the optimized result reproduces specific features for the church population. We discuss the roles of key parameters for sustainable church growth, and the contributions of the churches to sustainability.

High-order fully actuated system approaches: part VII. Controllability, stabilisability and parametric designs

In this paper, high-order fully actuated (HOFA) models with multiple orders for general dynamical control systems are firstly proposed, for which controllers can be easily designed such that the closed-loop systems are constant linear ones with completely assignable eigenstructures. Based on this special feature of the type of HOFA models, controllability of general dynamical control systems is proposed. It is revealed that a general dynamical control system can be represented by a controllable subsystem described by a HOFA model, and an extra uncontrollable one or supplementary one. While a general dynamical control system is called stabilisable if its uncontrollable (autonomous) subsystem does not exist, or is stable in a certain sense. Finally, two parametric design approaches for control of the type of general HOFA models are proposed, which provide all the design degrees of freedom. Several examples, including ones of sub-fully actuated systems, are treated, which demonstrate the effect of the proposed theories.

ON THE STABILITY OF DYNAMIC SYSTEMS WITH CERTAIN SWITCHINGS, WHICH CONSISTS OF LINEAR SUBSYSTEMS WITHOUT DELAY

This work is devoted to the further development of the study of the stability of dynamic systems with switchings. There are many different classes of dynamical systems described by switched equations. The authors of the work divide systems with switches into two classes. Namely, on systems with definite and indefinite switchings. In this paper, the system with certain switching, namely a system composed of differential and difference sub-systems with the condition of decreasing Lyapunov function. One of the most versatile methods of studying the stability of the zero equilibrium state is the second Lyapunov method, or the method of Lyapunov functions. When using it, a positive definite function is selected that satisfies certain properties on the solutions of the system. If a system of differential equations is considered, then the condition of non-positiveness (negative definiteness) of the total derivative due to the system is imposed. If a difference system of equations is considered, then the first difference is considered by virtue of the system. For more general dynamical systems (in particular, for systems with switchings), the condition is imposed that the Lyapunov function does not increase (decrease) along the solutions of the system. Since the paper considers a system consisting of differential and difference subsystems, the condition of non-increase (decrease of the Lyapunov function) is used.For a specific type of subsystems (linear), the conditions for not increasing (decreasing) are specified. The basic idea of using the second Lyapunov method for systems of this type is to construct a sequence of Lyapunov functions, in which the level surfaces of the next Lyapunov function at the switching points are either «stitched» or «contain the level surface of the previous function».

On fuzzifications of non-autonomous dynamical systems

In this paper, we derive a sharp condition on the equivalence of topological transitivity among an interval autonomous dynamical system, its induced set-valued system and induced normal fuzzified system. We also prove that their sensitivity (resp., total transitivity) are equivalent. For a general non-autonomous dynamical system, we show the equivalence of topological mixing (resp., mild mixing, cofinite sensitivity, multi-sensitivity and syndetic sensitivity) among the non-autonomous system and its two induced systems. In contrast, we construct a non-autonomous system that is weakly mixing but neither of its two induced systems is weakly mixing. We extend the topological equi-conjugacy between two non-autonomous systems to their two induced systems. Finally, we verify some basic properties of topological entropy among a non-autonomous system and its two induced systems, and establish some sufficient conditions for the topological equi-conjugacy between the fuzzification of a non-autonomous system and a subshift of finite type.

An Efficient Weighted Residual Time Integration Family

A new family of time integration methods is formulated. The recommended technique is useful and robust for the loads with large variations and the systems with nonlinear damping behavior. It is also applicable for the structures with lots of degrees of freedom, and can handle general nonlinear dynamic systems. By comparing the presented scheme with the fourth-order Runge–Kutta and the Newmark algorithms, it is concluded that the new strategy is more stable. The authors’ formulations have good results on amplitude decay and dispersion error analyses. Moreover, the family orders of accuracy are [Formula: see text] and [Formula: see text] for even and odd values of [Formula: see text], respectively. Findings demonstrate the superiority of the new family compared to explicit and implicit methods and dissipative and non-dissipative algorithms.

Stable Adaptive Controller Based on Generalized Regression Neural Networks and Sliding Mode Control for a Class of Nonlinear Time-Varying Systems

Finding synergy between a variety of control and estimation approaches can lead to effective solutions for controlling nonlinear dynamic systems in an efficient and systematic manner. In this paper, a novel controller design consisting of generalized regression neural networks (GRNNs) and sliding mode control (SMC) is proposed to control nonlinear multi-input and multi-output (MIMO) dynamic systems. The proposed design transforms GRNN from an offline regression model to an online adaptive controller. The suggested controller does not require any pretraining and it learns quickly from scratch. It uses a low computational complexity algorithm to provide accurate and stable performance. The proposed controller (GRNNSMC) performance is verified with a generic MIMO nonlinear dynamic system and a hexacopter model with a variable center of gravity. The results are compared with the standard PID controller. In addition, the stability of the GRNNSMC controller is verified using the Lyapunov stability method.