As previously, in FY 2011 the research conducted by the System Analysis Laboratory falls in the category of Complexity & System Science, which is a novel interdisciplinary branch of science studding emergent phenomena met in a wide variety of systems different in nature, spanning from traditional objects of the inanimate world up to social, economic, and ecological systems, where human or living beings play a crucial role. Our main interest, first, is concerned with the basic principles governing systems with motivation and the appropriate mathematical formalism required for their description. Systems with motivation comprise various statistical ensembles of elements with social behavior, human being or social animals, whose dynamics is impacted by memory effects, decisionmaking process, perception, recognition, prediction, and learning, in other words, by various stimuli motivating the system elements to behave in a particular way. Traffic and pedestrian flows, interacting market agents, as well as bird flocks and fish schools, animal foraging can be regarded as characteristic examples of such systems. Second direction is mathematical analysis of complex behavior exhibited by multi-element systems and the feasibility of their control.
In FY2011 the main attention of the laboratory members was focused on
constructing mathematical formalism for modeling systems with motivation that takes into account the basic features of human cognition;
nonlinear stochastic processes as a key element in modeling behavior of human and living beings;
effects of human memory and development of the relevant basic notions necessary for their description;
structure of the phase space required for describing systems with motivation;
various linear system compensator designs with compact parameters to obtain optimal compensator effectively;
control in discrete-time linear shift-invariant multi-input multi-output systems with approximate Riccati equation.
Among the results gained in FY2011 we note the following.
The basic methodological aspects of studying social systems were considered in detail with the main focused being on the fact that hybrid human-computer simulation referred to as “virtual experiments” should be one its cornerstones. The special attention was paid to systems with motivation that can be treated as a characteristic representative of statistical social systems. The dynamics of such objects is governed by cooperative phenomena, which enables us to introduce the notion of characteristic elements. Some examples of possible virtual experiments enabling us to investigate the basic properties of human perception, memory effects, learning and adaptation to environment changing in time were also proposed.
The Lévy type statistics as a mathematical formalism is of considerable interest for modeling systems with motivation because it characterizes a large number of phenomena met in social and economics systems, in motion of animals, etc. Nevertheless, up to now the theory of Lévy type stochastic processes contains many challenging problems. In the reported year we made next steps toward constructing a theory of the Lévy random walks based on continuous Markovian nonlinear stochastic processes. In particular, they include the following.
An novel classification of random trajectories was constructed. It has enabled us, first, to describe Lévy type stochastic processes as a sequence of peaks in the velocity time pattern of wandering particles whose statistical properties determine the basic characteristics of Lévy flights. Second, in this way we have established a relationship between the classical discrete representation of Lévy flights and Lévy random walks, the so-called Continuous Time Random Walks, and our approach to describing such anomalous stochastic processes.
The previously developed Markovian model for Lévy flights was modified such that now it can also describe truncated Lévy flights. It is valuable because only truncated Lévy flights can be implemented in the reality.
The concept of dynamical traps implementing the effects of human fuzzy rationality in controlling the systems with a certain instability was elaborated in detail. In particular,
it was demonstrated that the model of oscillator with dynamical traps is the direct generalization of the concept of the stationary point for the systems with partial equilibrium;
a sequence of complex phase transitions of the first order was found numerically in the chain of oscillators with dynamical traps;
a new evidence for the dynamical trap model was found based on experiments on balancing a virtual pendulum;
a generalized concept of dynamical traps in the space of human actions was proposed to describe the bounded capacity of human cognition in a choosing the appropriate strategy of behavior in governing the system dynamics.
The two-stage compensator designs of linear system has been investigated in the framework of the factorization approach. Our first result of this academic year was to obtain “full feedback” two-stage compensator design. This means that we have made a feedback system by using all inputs and outputs of the original feedback system, in which we use the feedback twice.
Obtained “full feedback” two-stage compensator design has a high generality. This design method can be reduced many “partial feedback” two-stage compensator designs. This method has restriction in the parametrization. However, this is more compact than previous and is to give a large number of two-stage compensator designs. As a result, we are effectively able to obtain the optimal compensator.
We have presented a method of solving an optimal control problem approximately in the case of discrete-time linear shift-invariant multi-input multi-output systems. The method we have applied is the approximate linear quadratic (LQ) method, which is to solve approximately linear quadratic equations. Our method leaves multiple parameters as their symbols in the evaluation function, that is, postpones the determination of the evaluation function until obtaining the optimal input. Thereby, the relation between the evaluation function and the input-output relation becomes explicitly clear.
I. A. Lubashevsky. Truncated Lévy flights and generalized Cauchy processes. Eur. Phys. J. B, 82:189-195, 2011.
A continuous Markovian model for truncated Lévy flights is proposed. It generalizes the approach developed previously by Lubashevsky et al. [Phys. Rev. E 79, 011110 (2009); Phys. Rev. E 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010)] and allows for nonlinear friction in wandering particle motion as well as saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and, as shown in the paper, individually give rise to a cutoff in the generated random walks meeting the Lévy type statistics on intermediate scales. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta method. The obtained numerical data were employed to analyze the statistics of the particle displacement during a given time interval, namely, to calculate the geometric mean of this random variable and to construct its distribution function. It is demonstrated that the time dependence of the geometric mean comprises three fragments following one another as the time scale increases that can be categorized as the ballistic regime, the Lévy type regime (superballistic, quasiballistic, or superdiffusive one), and the standard motion of Brownian particles. For the intermediate Lévy type part the distribution of the particle displacement is found to be of the generalized Cauchy form with cutoff. Besides, the properties of the random walks at hand are shown to be determined mainly by a certain ratio of the friction coefficient and the noise intensity rather than their characteristics individually.
I. Lubashevsky. Dynamical traps caused by fuzzy rationality as a new emergence mechanism. Advances in Complex Systems, 15:1250045 (25 pages), 2012.
A new emergence mechanism related to the human fuzzy rationality is considered. It assumes that individuals (operators) governing the dynamics of a certain system try to follow an optimal strategy in controlling its motion but fail to do this perfectly because similar strategies are indistinguishable for them. The main attention is focused on the systems where the optimal dynamics implies the stability of a certain equilibrium point in the corresponding phase space. In such systems the fuzzy rationality gives rise to some neighborhood of the equilibrium point, the region of dynamical traps, wherein each point is regarded as an equilibrium one by the operators. So, when the system enters this region and while it is located in it, maybe for a long time, the operator control is suspended. To elucidate a question as to whether the dynamical traps on their own can cause emergent phenomena, the stochastic factors are eliminated from consideration. In this case the system can leave the dynamical trap region only because of the mismatch between actions of different operators. By way of example, a chain of oscillators with dynamical traps is analyzed numerically. As demonstrated, the dynamical traps do induce instability and complex behavior of such systems.
K. Mori. Parameterisation of stabilising controllers with precompensators. IET Control Theory and Applications, 6(2):297-304, 2012.
Within the framework of the factorisation approach, the author presents parameterisation methods for stabilising controllers. The parameterisations of this study are characterised by an ideal finitely generated by some strictly causal stable transfer functions. Several typical examples of the parameterisations are also presented, namely (i) all stabilising controllers such that their relative degrees are more than or equal to some fixed number in the continuous-time linear time-invariant (LTI) system model; (ii) all stabilising controllers such that they must have at least some fixed number of delay operators in the discrete-time LTI system model; (iii) all stabilising controllers such that they must have some delay operators in the multidimensional system model. The parameterisation method of this study is also applied to (iv) a system that is stabilisable but that does not admit a doubly coprime factorisation.
I. Lubashevsky. Virtual experiments as a third cornerstone of social physics. In V. Klyuev and A. Vazhenin, editors, Proceedings of the 2012 Joint International Conference on Human-Centered Computer Environments, Aizu-Wakamatsu, Japan : March 08 - 13, 2012, number doi: 10.1145/2160749.2160794, pages 220-224. ACM New York, NY, USA, 2012.
In the present paper we discuss the basic methodological aspects of studying social systems and attract attention to the fact that hybrid human-computer simulation referred to as “virtual experiment” should be one its cornerstones. The special attention is paid to systems with motivation that can be treated as a characteristic representative of statistical social systems. The dynamics of such objects is governed by cooperative phenomena, which enables us to introduce the notion of characteristic element. Some examples of possible virtual experiments enabling us to investigate the basic properties of human perception, memory effects, learning and adaptation to environment changing in time are also discussed.
D. Parfenov, I. Lubashevsky, and N. Goussein-zade. Non-potential phase transitions induced by dynamical traps. In V. Klyuev and A. Vazhenin, editors, Proceedings of the 2012 Joint International Conference on HumanCentered Computer Environments, Aizu-Wakamatsu, Japan — March 08 13, 2012, number doi: 10.1145/2160749.2160775, pages 118-123. ACM New York, NY, USA, 2012.
Phase transitions in a chain of oscillators with dynamical traps is studied numerically. The notion of dynamical traps mimics the basic features caused by the bounded capacity of human cognition in decision-making. In mathematical terms the dynamical traps form a “low” dimensional region in the phase space of a given system where its dynamics is stagnated. It is demonstrated that in the system under consideration noise gives rise to complex emergent phenomena as its intensity grows.
I. Lubashevsky and S. Kanemoto. Complex dynamics of systems with motivation caused by scale-free memory. In H. Sayama, A. Minai, D. Braha, and Y. Bar-Yam, editors, Unifying Themes in Complex Systems Volume VIII: Proceedings of the Eighth International Conference on Complex Systems. New England Complex Systems Institute Series on Complexity (NECSI Knowledge Press, 2011), pages 245-248. New England Complex Systems Institute, Cambridge, MA, USA, ISBN 978-0-9656328-4-3, 2011.
First, we pose a general question as to what mathematical formalism is required for describing systems where human factor is essential and justify the necessity of an individual formalism appealing to the difference between social and physical objects in basic properties. Then, the main attention is focused on the effects of human memory that cause the dynamics of social systems to depend on their whole history. In other words, due to human memory the characteristics of system dynamics at the current moment of time have to be determined by the set of all the states attained by a system in the past rather then its state at a certain “initia” moment of time. It should be noted that in the Newtonian mechanics the notion of initial conditions play a crucial role. As a result, in the general case the appropriate governing equations should be of an integral form specified by the properties of human memory.
Different models for human memory are considered from this point of view. In particular, it is the exponential decay of human memory retention and a power-law, i.e., scale free decay. We demonstrate that only for the exponential decay the system dynamics admits a description matching directly the formalism of the Newtonian mechanics, i.e., that can be represented as a collection of differential equations subject to the corresponding initial conditions. The available psychological data enable us to claim that human memory is scale free and, thus, in the general case the governing equations for social objects are not reduced to the standard differential equations and the notion of initial conditions can become inapplicable.
At the current state of art in order to construct the appropriate mathematical formalism it is worthwhile to study in detail particular objects that exemplify the basic features of social systems. Keeping the latter in mind we, finally, present the results of simulations for two characteristic objects of social nature. Each of them can be regarded as a multi-element ensemble where human motives for actions among with physical regularities govern the system dynamics. In particular, the latter feature is the reason of calling such objects systems with motivation.
The first object under consideration is the model for multi-agent reinforcement learning with scale free memory. For example, it mimics the observed complex evolution of ecological trimorphic populations. The agents are assumed to act independently of one another in optimizing their choice of possible actions via trial-and-error search. To gain awareness about the action value the agents accumulate in their memory the rewards obtained from taking an action at each moment of time. The contribution of the rewards in the past to the agent current perception of action value is described by an integral operator with a power-law kernel. Finally a fractional differential equation governing the system dynamics is obtained and a generalized notion of initial conditions is introduced. As a specific example of systems with non-transitive interactions, a two agent and three agent systems of the rock-paper-scissors type are analyzed in detail. Scale-free memory is demonstrated to cause complex dynamics of the systems at hand. In particular, it is shown that there can be simultaneously two modes of the system instability undergoing subcritical and supercritical bifurcation, with the latter one exhibiting anomalous oscillations with the amplitude and period growing in time. Besides, the instability onset via this supercritical mode may be regarded as “altruism self-organization.” For the three agent system the instability dynamics is found to be rather irregular and can be composed of alternate fragments of oscillations different in their properties.
The second object is an ensemble of cars following one another on a road. The widely used approach to modeling such systems assumes drivers to respond with a certain fixed delay to two stimuli related with controlling the car velocity and maintaining a safe headway distance. Appealing to the properties of human memory the model under consideration assumes these stimuli to be accumulated in the driver memory again with a power-law decay, so the driver response is determined by all the states of traffic flow in the past rather then some events at a fixed time leg. The instability of traffic flow and the temporal-spatial patterns caused by the instability of the steady state motion are analyzed depending on the driver memory properties.
In conclusion, we state that the equations with fractional derivatives of the Caputo type are one of the basic elements of the mathematical formalism in describing systems with motivation. In addition, the complete description of such an objects requires the introduction of a generalized phase space where a trajectory of the system motion as a whole determines its current dynamics.
A. Zgonnikov, I. Lubashevsky, and M. Mozgovoy. Computer simulation of stick balancing: action point analysis. In V. Klyuev and A. Vazhenin, editors, Proceedings of the 2012 Joint International Conference on HumanCentered Computer Environments, Aizu-Wakamatsu, Japan — March 08 13, 2012, number doi: 10.1145/2160749.2160783, pages 162-164. ACM New York, NY, USA, 2012.
We analyze data collected during the series of experiments aimed at elucidation of basic properties of human perception, namely, the limited capacity of ordering events, actions, etc. according to their preference. Previously it was shown that in a wide class of human-controlled systems small deviations from the equilibrium position do not cause any actions of the system's operator, so any point in a certain neighborhood of equilibrium position is treated as an equilibrium one. This phenomenon can be described by the notion of dynamical traps that was introduced to denote a region in the system phase space where the object under consideration cannot clearly determine the most preferable of the positions that are similar in some sense. According to this concept, the motion of the system in the dynamical trap region is mainly not affected by the operator. The moments of time when the system leaves the dynamical trap region, or in other words, when the operator decides to start or stop the control over the system, are called action points . These moments are seem to be determined intuitively by the operator, and the purpose of our work is to understand the nature of such intuitive decision making process by investigating the action points data obtained from the experiments.
I. Lubashevsky. Continuous Markovian description of Lévy flights. In T. E. Simos, G. Psihoyios, Ch. Tsitouras, and Z. Anastassi, editors, Numerical Analysis and Applied Mathematics ICNAAM 2011, Halkidiki, (Greece) 1925 September 2011, volume AIP Conf. Proc. 1389, pages 1882-1885; doi: 10.1063/1.3636978. American Institute of Physics, 978-0-7354-0956-9, 2011.
We discuss a new approach to describing Lévy flights in terms of continuous Markovian processes admitting the introduction of continuous trajectories of particle motion
K. Mori. Capability of Two-Stage Compensator Designs — SingleInput Single-Output Case —. In Proceedings of SICE Annual Conference (SICE 2011), pages 288-293, Tokyo, 2011.
The two-stage compensator designs for single-input single-ouput plants are investigated. The first result (Theorem2) is an extension of the previous result. From this result, we pay attention to the form of its feedback. By generalizing the form of feedback, the second result (Theorem3) is obtained as an extension of Theorem2. Based on Theorem3, various types of the two-stage compensator designs are given.
K. Mori. Approximate Riccati Equation and Its Application to Optimal Control in Discrete-Time Systems. In Proceedings of IAENG International Conference on Control and Automation (ICCA'11), pages 808-812, Hong Kong, 2011.
We present a method of solving an linear quadratic regulator problem approximately in the case of discrete-time systems. This method leaves parameters as symbols in the evaluation function. We introduce the concept of the approximate linear quadratic regulator problem. We also propose a computation method to solve the problem. A numerical example of the approximate linear quadratic regulator problem is also presented.
K. Mori. Two-Stage Compensator Designs with Partial Feedbacks. In Proceedings of International Conference on Control, Automation, Robotics and Vision (ICCARV 2012), pages 283-289, Kualalumpur, Malaysia, 2012.
The two-stage compensator designs of linear system are investigated in the framework of the factorization approach. First, we give "full feedback" two-stage compensator design. Based on this result, various types of the two-stage compensator designs with partial feedbacks are derived.
K. Mori. Capability of Two-Stage Compensator Designs. In Proceedings of The 9th IEEE International Conference on Control & Automation (IEEE ICCA11), pages 1231-1236, Santiago, Chile, 2011.
The two-stage compensator designs are investigated. The first result (Theorem 2) is an extension of the previous result. From this result, we pay attention to the form of its feedback. By generalizing the form of feedback, the second result (Theorem 3) is obtained as an extension of Theorem 2. Based on Theorem 3, various types of the two-stage compensator designs are given.
K. Mori. Multi-Stage Compensator Design | A Factorization Approach |. In Proceedings of The 8th Asian Control Conference (ASCC 2011), pages 1012-1017, Kaohsiung, Taiwan, 2011.
In the framework of the factorization approach, we investigate the multi-stage compensator design. So far, the results of the two-stage compensator design use the norm algebras as well as the factorization approach, which reduces the attractiveness of the factorization approach. We further consider alternative two-stage compensator designs. The results of this paper are based on the factorization approach only.
I. Lubashevsky. Continuous and discrete realization of Lévy flights. One-dimensional process. e-prints http://arxiv.org (Cornell University server), arXiv:1203.5860:1-27, 2012.
submitted to Physica A: The present paper is focused on constructing a relationship between continuous Markovian models for one-dimensional Lévy flights as random motion of a wandering particle with stochastic self-acceleration and their discrete representation that may be treated as a generalized version of continuous time random walks (CTRW). For this purpose a notion of random motion inside a certain neighborhood of the particle velocity axis and outside it is developed. In this way a continuous particle trajectory is reduced to a collection of discrete steps of particle spatial displacement determined mainly by particle motion within individual peaks forming the time pattern of the velocity fluctuations. The obtained discrete random walks, indeed, may be treated as some generalization of CTRW because the individual duration of their steps and the corresponding particle displacement are random variables correlated in part with each other. The main difference between the standard approach and the constructed one is due to no assumption similar to uniform motion of a particle between the terminal points of one step is adopted. In addition, using the developed trajectory classification a certain parameter-free core stochastic process is constructed, so all the characteristics of Lévy flights similar to the exponent of the Lévy scaling law are no more than the parameters of the corresponding transformation. In this way the validity of the continuous Markovian model for all the regimes of Lévy flights is explained. Based on the obtained results an efficient “one-peak” approximation is constructed that enables one to find the basic characteristics of Lévy flights using the extreme values of the velocity fluctuations and the shape of the most probable trajectories of particle motion.