In general, 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, decision-making 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; ffl structure of the phase space required for describing systems with motivation;
multi-stage compensator design for linear system theory to clarify its capability;
optimal control in discrete-time linear systems with approximate Riccati equation.
Among the results gained in FY2010 we note the following.
It has been demonstrated and discussed in detail that practically all basic elements of the mathematical formalism of systems with motivation are now developed or under construction in modern applied mathematics and synergistics. So in order to elucidate understanding their properties and feasibilities, at the current state of art it is worthwhile to focus attention on particular phenomena and to study in detail specific systems that can exemplify their features.
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. One of them is the Lévy random walks in heterogeneous media or systems with boundaries because no continuous trajectories of particle motion can be ascribed to these random walks in the general case. In present year we made a next step toward constructing a theory of the Lévy random walks based on continuous Markovian nonlinear stochastic processes. This theory is expected to open a gate to investigation of spatially heterogeneous Lévy type processes. In particular, it has been demonstrated that the model developed previously for the Lévy random walks with the superdiffusive regime, in fact, allows for all the possible regimes, namely, the superdiffusive, ballistic, and superballistic ones.
A new approach to modeling the Lévy flights observed in animal foraging has been proposed. It is based on the previously developed theory of continuous Markovian nonlinear stochastic processes of the generalized Cauchy type and introduces the notion of an extended phase space, where a new phase variable, the acceleration of particle motion, is included. Exactly via controlling this variable a self-driving particle can govern its motion. This model mimics complex motion of animals, birds, or fish that have to keep up a certain speed to move.
The human memory endows social systems with significant dependence on their history. Unlike objects of the inanimate world, in the general case the history as a whole affects the system dynamics. In this case it becomes impossible to introduce the notion of initial conditions. To be specific we have studied the memory effects in the frameworks of a multi-agent model for reinforcement learning. It assumes the agents to gain awareness about the action value via accumulating in their memory the previous rewards obtained, which is described by an integral operator with a power-law kernel. Finally a fractional differential equation governing the system dynamics is obtained. 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 with 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, which is rather similar to the observed complex phenomena in phenotype evolution of trimorphic populations. 5. In some design problems, one uses a so-called two-state procedure for selecting an appropriate stabilizing compensator. Given a plant, the first stage consists of selecting a stabilizing compensator for the plant. The second stage consists of selecting a stabilizing controller for the new closed-loop system that also achieves some other design objectives such as decoupling, sensitivity minimization, etc. The rationale behind this procedure is that the design problems are often easier to solve when the plant is stable. So far, the results of the two-stage compensator design use the norm algebras as well as the factorization approach. Because the analysis by the norm algebra is based on a concrete specified model, this reduces the attractiveness of the factorization approach. In our study, we use the factorization approach only, which means that the results can be applied to numerous linear systems.
We have clarified the relationship between two-stage compensator design and the Youla-parametrization. In its process, we also obtained alternative three "two-stage compensator designs." We have clarified their capability about the parametrization. We have also proposed new design method based on multiple "two-stage compensator designs." This can give any stabilizing controller, which enables us to obtain, in general, an optimal stabilizing controller in a given class of stabilizing controllers.
Optimal control problem is one of the most essential problems in the control theory. However, the solution of an optimal control problem for a linear system is generally based on the maximization or minimization of an evaluation function. However, it is not straightforwardly clear what sort of influence the evaluation function has to the input-output relation. Hence, it is necessary to do a control design with a trial evaluation function in order to obtain the appropriate evaluation function we need.
We have presented a method of solving an optimal control problem approximately in the case of discrete-time linear systems. Our method leaves parameters as 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.
We have considered the linear quadratic method and proposed the notion of approximate linear quadratic (LQ) method. To include parameters, we have introduced the notion of the approximate Riccati equation. With numerical examples, the effectiveness of the proposed method has been evaluated.
Ihor Lubashevsky, Andreas Heuer, Rudolf Friedrich, and Ramil Usmanov. Continuous Markovian model for Lévy random walks with superdiffusive and superballistic regimes. The European Physical Journal B: Condensed Matter and Complex Systems, 78:207-216, November 2010.
We consider a previously derived model describing Lévy random walks [I. Lubashevsky, R. Friedrich, A. Heuer, Phys. Rev. E 79, 011110 (2009); I. Lubashevsky, R. Friedrich, A. Heuer, Phys. Rev. E 80, 031148 (2009)]. It is demonstrated numerically that the given model describes Lévy random walks with superdiffusive, ballistic, as well as superballistic dynamics. Previously only the superdiffusive regime has been analyzed. In this model the walker velocity is governed by a nonlinear Langevin equation. Analyzing the crossover from small to large time scales we find the time scales on which the velocity correlations decay and the walker motion essentially exhibits Lévy statistics. Our analysis is based on the analysis of the geometric means of walker displacements and allows us to tackle probability density functions with power-law tails and, correspondingly, divergent moments.
Ihor Lubashevsky and Shigeru Kanemoto. Scale-free memory model for multiagent reinforcement learning. Mean field approximation and rockpaper-scissors dynamics. The European Physical Journal B: Condensed Matter and Complex Systems, 76:69-85, July 2010.
A continuous time model for multiagent systems governed by reinforcement learning with scale-free memory is developed. 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 a specific 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. The agents are considered to interact with one another implicitly via the reward of one agent depending on the choice of the other agents. The pairwise interaction model is adopted to describe this effect. 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, including the stability analysis and numerical simulation. Scalefree 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 with 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.
Tanja Mues, Andreas Heuer, Martin Burger, and Ihor Lubashevsky. Model of oscillatory zoning in two dimensions: Simulation and mode analysis. Physical Review E: Statistical, Nonliear, and Soft-Matter Physics, 81:051605(1 10), May 2010.
Oscillatory zoning OZ occurs in all major classes of minerals and also in a wide range of geological environments. It is caused by self-organization and describes fluctuations of the spatial chemical composition profile of the crystal. We present here a two-dimensional model of OZ based on our previous one-dimensional 1D analysis and investigate whether the results of the 1D stability analysis remain valid. With the additional second dimension we were able to study the origin of the spatially homogeneous layer formation by linear stability analysis. Numerical solutions of the model are presented and the results of a Fourier analysis delivers a detailed understanding of the crystal growth behavior as well as the limits of the model. Effects beyond linear stability analysis are important to finally understand the final structure formation.
Kazuyoshi Mori. Parametrization of Stabilizing Controllers with Precompensators. IET Control Theory and Applications, 2010.
Ihor Lubashevsky. Towards description of social systems as a novel class of physical problems. In Vitaly V. Klyuev and Michael Cohen, editors, The 13th International Conference on 'Human and Computers', HC 2010, pages 106-113, University of Aizu, Aizu-Wakamatsu, Fukushima-ken 9658580, Japan, December 2010. University of Aizu Press.
We discuss fundamental problems of mathematical description of social systems based on physical concepts, with so-called statistical social systems being the main subject of consideration. Basic properties of human beings and human societies that distinguish social and natural systems from each other are listed to make it clear that individual mathematical formalism and physical notions should be developed to describe such objects rather then can be directly inherited from classical mechanics and statistical physics. As a particular example, systems with motivation are considered. Their characteristic features are analyzed individually and the appropriate mathematical constructions are proposed. Finally we conclude that the basic elements necessary for describing statistical social systems or, more rigorously, systems with motivation are available or partly developed in modern physics and applied mathematics.
Kazuyoshi 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, 2010.
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.
Kazuyoshi Mori. Parametrization of All Stabilizing Controllers with a Precompensator. In Proceedings of The 30th IASTED International Conference on Modelling, Identification, and Control (MIC 2010), pages 394-399, 2010.
In the framework of the factorization approach, we give a parameterization of all stabilizing controllers with some stabilizing controllers with some ?xed precompensator. As an example, we present xed precompensator. As an example, we present a parameterization of all causal stabilizing controllers which has the integrator for the classical continuous-time system model.
Kazuyoshi Mori. Parametrization of Stabilizing Controllers with Fixed Precompensators. In Proceedings of 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010), pages 1629-1633, 2010.
In the framework of the factorization approach, we give a parameterization of a class of stabilizing controllers. This class is characterized by some fixed strictly causal precompensators. As applications, we present the parameterization of all causal stabilizing controllers including the some fixed number or more integrators, and the parameterization of all strictly causal stabilizing controllers which has the some fixed number or more delay operators.
Kazuyoshi Mori. A New Parametrization of A Class of Causal Stabilizing Controllers. In Proceedings of SICE Annual Conference (SICE 2010), pages 3457-3460, 2010.
Within the framework of the factorization approach, we present new parametrizations of a class of causal stabilizing controllers. This will result stabilizing controllers that have the multiple-order lag elements for continuous-time LTI systems and stabilizing controllers that have multiple-delay elements for discrete-time LTI systems.
Kazuyoshi MORI, 2010.
Reviewer of IET Control Theory & Applications
Kazuyoshi MORI, 2010.
Mathematical Reviews, ACM
Kazuyoshi MORI, 2010.
Reviewer of IEEE Transactions on Automatic Control
Kazuyoshi MORI, 2010.
Reviewer of J. of SICE