ISSUE INFORMATION

Modeling social‐ecological systems is difficult due to the complexity of ecosystems and of individual and collective human behavior. Key components of the social‐ecological system are often over‐simplified or omitted. Generalized modeling is a dynamical systems approach that can overcome some of these challenges. It can rigorously analyze qualitative system dynamics such as regime shifts despite incomplete knowledge of the model's constituent processes. Here, we review generalized modeling and use a recent study on the Baltic Sea cod fishery's boom and collapse to demonstrate its application to modeling the dynamics of empirical social‐ecological systems. These empirical applications demand new methods of analysis suited to larger, more complicated generalized models. Generalized modeling is a promising tool for rapidly developing mathematically rigorous, process‐based understanding of a social‐ecological system's dynamics despite limited knowledge of the system.     

Other than chance and necessity; intention and purpose by design

Very little success has been enjoyed by decision makers in managing the life threatening problems found in the world today using techniques borrowed from the natural sciences and humanities. New promise, however, can be found in the expanded and transdisciplinary notion of the Systems Approach as developed by C. West Churchman. Using Churchman's work as a base for transdisciplinary inquiry, the new image of the dynamics of complex systems presented in the work of Ilya prigogine and others is brought to bear on the most complex system of all; social systems and the roles played by two types of social system participants; designers and planners. Designers provide the trigger in social systems leading to the dynamic system behavior described by prigogine as bifurcation which leads to rebirth and renewal in complex social systems (progress). However, unlike Prigogine's random perturbation, the designed trigger is full of intent and purpose. The complementary role of the planner is to determine the ideal level of systems performance compared to the actual level of performance and to determine how to more closely approximate the ideal (improvement).     

Updating of a lumped model for an experimental web tension control system using a multivariable optimization method

The modelling and the control of web handling systems have been studied for a long time; correct modelling is necessary in order to design a better control system or to identify the plant parameters experimentally. On the web dynamics itself, lumped parameters expressions may be used to designate a web section between two adjacent drive rolls, and there is the necessity of incorporating the property of viscoelasticity to the web.     

Stabilization and tracking controller for a class of nonlinear discrete-time systems

In this paper, stabilization and tracking control problem for parametric strict feedback class of discrete time systems is addressed. Recursive design of control function based on contraction theory framework is proposed instead of traditional Lyapunov based method. Explicit structure of controller is derived for the addressed class of nonlinear discrete-time systems. Conditions for exponential stability of system states are derived in terms of controller parameters. At each stage of recursive procedure a specific structure of Jacobian matrix is ensured so as to satisfy conditions of stability. The closed loop dynamics in this case remains nonlinear in nature. The proposed algorithm establishes global stability results in quite a simple manner as it does not require formulation of error dynamics. Problem of stabilization and output tracking control in case of single link manipulator system with actuator dynamics is analyzed using the proposed strategy. The proposed results are further extended to stabilization of discrete time chaotic systems. Numerical simulations presented in the end show the effectiveness of the proposed approach.     

Modeling complex systems macroscopically: Case/agent-based modeling,synergetics, and the continuity equation

Recently, the continuity equation (also known as the advection equation) has been used to study stability properties of dynamical systems, where a linear transfer operator approach was used to examine the stability of a nonlinear equation both in continuous and discrete time (Vaidya and Mehta, IEEE Trans Autom Control 2008, 53, 307–323; Rajaram et al., J Math Anal Appl 2010, 368, 144–156). Our study, which conducts a series of simulations on residential patterns, demonstrates that this usage of the continuity equation can advance Haken's synergetic approach to modeling certain types of complex, self-organizing social systems macroscopically. The key to this advancement comes from employing a case-based approach that (1) treats complex systems as a set of cases and (2) treats cases as dynamical vsystems which, at the microscopic level, can be conceptualized as k dimensional row vectors; and, at the macroscopic level, as vectors with magnitude and direction, which can be modeled as population densities. Our case-based employment of the continuity equation has four benefits for agent-based and case-based modeling and, more broadly, the social scientific study of complex systems where transport or spatial mobility issues are of interest: it (1) links microscopic (agent-based) and macroscopic (structural) modeling; (2) transforms the dynamics of highly nonlinear vector fields into the linear motion of densities; (3) allows predictions to be made about future states of a complex system; and (4) mathematically formalizes the structural dynamics of these types of complex social systems.     

Classical social processes: Attractor and computational models*

Attractor models provide a generalized way to represent processes found throughout science. A fuller articulation of the attractor framework requires that it be addressed qualitatively and conceptually as a nonlinear mathematical order residing between cyclical and random processes. Many significant nonlinear social processes have been identified and analyzed in classical social theory. These include the circulation of the elites (Pareto), cultural dynamics (Sorokin), social differentiation (Durkheim) and rationalization in modern institutions (Weber). The present discussion develops a qualitative consideration of such classical social processes as attractor systems, and discusses possible applications of such models in computational social science.     

Punctuated equilibrium in a non-linear system of action

Coleman’s equilibrium model of social development, the Linear System of Action, is extended to cover the dynamics of societal transitions. The model implemented has the characteristics of a dissipative system. A variation and selection algorithm favoring the retention of relatively dependent actors forces the system away from equilibrium, while exchange of control, according to Coleman the driving force behind social action, accounts for dissipation, pulling the social system back to equilibrium. This Non-linear System of Action self-organizes into a critical state, as confirmed by the robust power law distribution of exchange of control for a wide range of model sizes. Related punctuated equilibrium dynamics and structural change are of special interest, as these are closely connected to hypotheses on social dynamics developed in the literature on societal transitions.     

The generation of a social welfare function under ordinal preferences

Against the background of recent impossibility theorems, the paper establishes a number of rules for the formation of social preferences to ensure the generation of a Bergson-Samuelson social welfare function of the desired ‘individualistic’ form. It demonstrates that such welfare functions can indeed exist under conditions of simply ordinality of individual preferences and a lack of interpersonal comparability, for the general n-person case with only weak restrictions upon individual preferences. A valuable tool of social welfare theory for the single-profile case is therefore restored under a wide set of conditions. By examining Kemp and Ng's justification for their A3 condition in this context, we are able also to highlight a number of key differences between the conditions satisfied by many political constitutions and the requirements for achieving the positive existence of a Bergson-Samuelson social welfare function. In particular the paper focusses here on the weighting systems on individual preferences that are consistent with generating an individualistic welfare function and examines the information content of individual ordinal preferences which must be utilized if the desired welfare function is to result.     

Stability analysis and uniform ultimate boundedness control synthesis for a class of nonlinear switched difference systems

In this paper, we deal with stability analysis of a class of nonlinear switched discrete-time systems. Systems of the class appear in numerical simulation of continuous-time switched systems. Some linear matrix inequality type stability conditions, based on the common Lyapunov function approach, are obtained. It is shown that under these conditions the system remains stable for any switching law. The obtained results are applied to the analysis of dynamics of a discrete-time switched population model. Finally, a continuous state feedback control is proposed that guarantees the uniform ultimate boundedness of switched systems with uncertain nonlinearity and parameters.     

Using asymptotic invariance methods for output stabilization of a class of uncertain systems with stable zero dynamics

Conclusion We have introduced the notion of exponential invariance and have shown that output stabilization of uncertain systems of first relative order with a stable zero dynamics reduces to the design of an exponentially invariant (in the generalized sense) system. We have derived sufficient conditions of exponential invariance and applied them to calculate the parameters of large-gain stabilizing algorithms and nonlinear dynamic feedbacks. Transient bounds have been derived. It has been shown that the properties of a closed-loop control system depend only on the exponential invariance parameters and can be specified in advance, before the control is chosen. Examples have been examined that demonstrate the constructivity of the proposed approach. The approach is shown to be applicable to a wide class of uncertain controlled systems, including both phase-minimal and phase-nonminimal plants. Translated from Metody Analiza Nelineinykh Sistem, Moscow University, pp. 87–101, 1997.     

Visualizing the “heartbeat” of a city with tweets

Describing the dynamics of a city is a crucial step to both understanding the human activity in urban environments and to planning and designing cities accordingly. Here, we describe the collective dynamics of New York City (NYC) and surrounding areas as seen through the lens of Twitter usage. In particular, we observe and quantify the patterns that emerge naturally from the hourly activities in different areas of NYC, and discuss how they can be used to understand the urban areas. Using a dataset that includes more than 6 million geolocated Twitter messages we construct a movie of the geographic density of tweets. We observe the diurnal “heartbeat” of the NYC area. The largest scale dynamics are the waking and sleeping cycle and commuting from residential communities to office areas in Manhattan. Hourly dynamics reflect the interplay of commuting, work and leisure, including whether people are preoccupied with other activities or actively using Twitter. Differences between weekday and weekend dynamics point to changes in when people wake and sleep, and engage in social activities. We show that by measuring the average distances to a central location one can quantify the weekly differences and the shift in behavior during weekends. We also identify locations and times of high Twitter activity that occur because of specific activities. These include early morning high levels of traffic as people arrive and wait at air transportation hubs, and on Sunday at the Meadowlands Sports Complex and Statue of Liberty. We analyze the role of particular individuals where they have large impacts on overall Twitter activity. Our analysis points to the opportunity to develop insight into both geographic social dynamics and attention through social media analysis. © 2015 Wiley Periodicals, Inc. Complexity 21: 280–287, 2016     

Optimal processes in smooth-convex minimization problems

The paper elaborates a general method for studying smooth-convex conditional minimization problems that allows one to obtain necessary conditions for solutions of these problems in the case where the image of the mapping corresponding to the constraints of the problem considered can be of infinite codimension. On the basis of the elaborated method, the author proves necessary optimality conditions in the form of an analog of the Pontryagin maximum principle in various classes of quasilinear optimal control problems with mixed constraints; moreover, the author succeeds in preserving a unified approach to obtaining necessary optimality conditions for control systems without delays, as well as for systems with incommensurable delays in state coordinates and control parameters. The obtained necessary optimality conditions are of a constructive character, which allows one to construct optimal processes in practical problems (from biology, economics, social sciences, electric technology, metallurgy, etc.), in which it is necessary to take into account an interrelation between the control parameters and the state coordinates of the control object considered. The result referring to systems with aftereffect allows one to successfully study many-branch product processes, in particular, processes with constraints of the “bottle-neck” type, which were considered by R. Bellman, and also those modern problems of flight dynamics, space navigation, building, etc. in which, along with mixed constraints, it is necessary to take into account the delay effect. The author suggests a general scheme for studying optimal process with free right endpoint based on the application of the obtained necessary optimality conditions, which allows one to find optimal processes in those control systems in which no singular cases arise. The author gives an effective procedure for studying the singular case (the procedure for calculating a singular control in quasilinear systems with mixed constraints. Using the obtained necessary optimality conditions, the author constructs optimal processes in concrete control systems. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 42, Optimal Control, 2006.     

Extending satisficing control strategy to slowly varying nonlinear systems

Based on the satisficing control strategy, a novel approach to design a stabilizing control law for nonlinear time varying systems with slowly varying parameters (slowly varying systems) is presented. The satisficing control strategy has been originally introduced for time-invariant systems; however, this technique does not have any stability proof for time varying systems. In this paper, first, a parametric version of the satisficing control strategy is developed. Then, by considering the time as a frozen parameter, the parametric satisficing control strategy is utilized. Finally, a theorem is presented which suggested a stabilizing satisficing control law for the slowly varying control systems. Moreover, in this theorem, the maximum admissible rate of change of the system dynamics is evaluated. The efficiency of the proposed approach is demonstrated by a computer simulation.     

Chaos and order in stateless societies: Intercommunity exchange as a factor impacting the population dynamical patterns

The once abstract notions of dynamical chaos now appear naturally in various systems [Kaplan D, Glass L. Understanding nonlinear dynamics. New York: Springer; 1995]. As a result, future trajectories of the systems may be difficult to predict. In this paper, we demonstrate the appearance of chaotic dynamics in model human communities, which consist of producers of agricultural product and producers of agricultural equipment. In the case of a solitary community, the horizon of predictability of the human population dynamics is shown to be dependent on both intrinsic instability of the dynamics and the chaotic attractor sizes. Since a separate community is usually a part of a larger commonality, we study the dynamics of social systems consisting of two interacting communities. We show that intercommunity barter can lead to stabilization of the dynamics in one of the communities, which implies persistence of stable equilibrium under changes of the maximum value of the human population growth rate. However, in the neighboring community, the equilibrium turns into a stable limit cycle as the maximum value of the human population growth rate increases. Following an increase in the maximum value of the human population growth rate leads to period-doubling bifurcations resulting in chaotic dynamics. The horizon of predictability of the chaotic oscillations is found to be limited by 5 years. We demonstrate that the intercommunity interaction can lead to the appearance of long-period harmonics in the chaotic time series. The period of the harmonics is of order 100 and 1000 years. Hence the long-period changes in the population size may be considered as an intrinsic feature of the human population dynamics.     

Chaotic response and bifurcation analysis of a flexible rotor supported by porous and non-porous bearings with nonlinear suspension

This study presents numerical work investigating the dynamic responses of a flexible rotor supported by porous journal bearings. Both porous and non-porous bearing types are taken into consideration in this study. The rotating speed ratios and imbalance parameters are also presented and proved to be important control parameters. Many non-periodic responses to chaotic and quasi-periodic motions are found, too. From the bifurcation diagrams in this paper, it is also evidenced that the vibration behaviors would be improved by porous bearings. The modeling result obtained here can be employed to predict the dynamics of bearing–rotor systems, and undesirable behavior of the rotor and bearing orbits can be avoided. Also, this could help engineers and researchers in designing and studying bearing–rotor systems or some turbo-machinery in the future.     

Event-triggered delayed impulsive control for input-to-state stability of nonlinear impulsive systems

This paper studies the input-to-state stability (ISS) and integral input-to-state stability (iISS) of nonlinear impulsive systems in the framework of event-triggered impulsive control (ETIC), where the stabilizing effect of time delays in impulses is fully considered. Some sufficient conditions which can avoid Zeno behavior and guarantee the ISS/iISS property of impulsive systems are proposed, where external inputs are considered in both the continuous dynamics and impulsive dynamics. A novel event-triggered delayed impulsive control (ETDIC) strategy which establishes a relationship among event-triggered parameters, impulse strength and time delays in impulses is presented. It is shown that time delays in impulses can contribute to the stabilization of impulsive systems in ISS/iISS sense. Finally, the effectiveness of the proposed theoretical results is illustrated by two numerical examples.     

Evasion with a single measurement of the state vector

We consider a plant whose dynamics is described by a system of linear ordinary differential equations with an unknown noise. The considered differential game has fixed terminal time. The players’ control parameters are subjected to geometric constraints in the form of closed balls. The aim of the control is to evade zero. The problem is solved with the use of control laws with a single measurement of the state vector. We present necessary conditions and a construction algorithm for the optimal measurement instant. The results are illustrated by some control systems of a special form. In particular, we consider an example in which one measurement can guarantee the result not worse than that for a refined grid of measurement points. We discuss whether this situation is typical. We also consider an example with infinitely many points at which the unique measurement can be performed.     

Robust control of multiple robots handling a constrained load

Multiple-robot systems are usually confronted with uncertainties,such as uncertainties in the manipulators and load parameters,and unmodelled dynamics. In this paper, the problem of controllingmultiple manipulators handling a constrained load is addressed.A reduced-order dynamic model of the system is first derived,and several properties of this model are established. Usingthe reduced-order model, a robust control law is proposed. Thiscontroller guarantees the uniform ultimate boundedness of theposition error, the internal-force error, and the constraint-forceerror. The proposed control law requires only the bounds onthe uncertainties of the parameters. Simulation results of twoplanar robots moving a load along a horizontal plane are givento illustrate the theoretical developments.     

Variable structure control systems: synthesis of scalar and vector systems by state and output feedback

Modern views are applied to examine synthesis of variable structure systems (VSS) for SISO and MIMO systems of general position in the presence of information on the full phase vector or only on the output vector. In the latter case we consider separately the cases of organizing a sliding mode in plant output coordinates and in arbitrary coordinates. The latter case requires the construction of asymptotic observers for linear systems with uncertainty in the parameters or in inputs, and the corresponding observation theory is sketched. Special attention is given to the case of so-called hyperoutput systems, i.e., multiply connected systems in which the number of outputs exceeds the number of inputs. For hyperoutput systems under fairly general conditions we can arbitrarily choose the observer dynamics, which is important for improving the performance of the control system as a whole. We show that the observer dynamics does not change the VSS stability.     

Data-driven optimal tracking control of switched linear systems

This paper addresses the optimal tracking control for switched linear systems with unknown dynamics. We convert the problem into an optimal control problem of the augmented switched systems. In view of the augmented systems, we propose a data-driven switched linear quadratic regular algorithm for obtaining the optimal switching signal under unknown system dynamics. It is proved that the optimal switching signal will not cause Zeno behavior and can make the system stable. Besides, with the proposed algorithm, we just need to identify an autonomous system instead of the original systems, which has fewer parameters to be determined. A numerical example is given to illustrate the validity of the main results.