## Issue 1 (195), article 4

Kibern. vyčisl. teh., 2018, Issue 1 (195), pp.

Milyavsky Y.L., Senior Lecturer,
Department of the Mathematical Methods of System Analysis
e-mail: yuriy.milyavsky@gmail.com

National Technical University of Ukraine “I. Sikorsky Kyiv Polytechnic Institute”
37 Peremohy av., Kyiv, 03056, Ukraine

IDENTIFICATION IN COGNITIVE MAPS IN IMPULSE PROCESS MODE WITH INCOMPLETE MEASUREMENT OF NODES COORDINATES

Introduction. Cognitive map is a popular way of modeling complex multivariate systems. Usually weights coefficients of edges connecting the cognitive map nodes are suggested by experts. But such a method is always inaccurate. In case when nodes coordinates are measured, there is the possibility for mathematical identification of these coefficients. However, the issue is that often not all nodes coordinates of a cognitive map are measured, but only a few of them. In this case the problem of identification is much more complicated.
The purpose of the article is to research and develop a method for identifying weights of cognitive map nodes in case when number of nodes is known, but not all of them are measured.
Results. Identification method based on 4SID method is suggested. It allows finding some realization of the system equivalent to the original cognitive map in its outputs, with the control observation matrices remaining unchanged.Invariants of the original and identified systems are analyzed. Practical example of identifying a cognitive map of an IT company is considered. It is shown what the accuracy of the suggested method depends on and under which conditions it is applicable.
Conclusions. As demonstrated in the research, the proposed method of identifying cognitive maps achieves almost full coincidence of measured coordinates between the original and the identified systems, although the incidence matrices themselves may not be equal. Invariants of the system, specifically eigenvalues, are identified with sufficient precision if the problem is well-conditioned, i.e. with sufficient number of measurable coordinates, sufficient number of measurement periods and low level of measurement noise. If these conditions are not fulfilled, the identification results become incorrect.

Keywords: cognitive map, identification, 4SID method, unmeasurable coordinates.

REFERENCES

1 Roberts F. Discrete Mathematical Models with Applications to Social, Biological, and Environmental Problems. Englewood Cliffs: Prentice-Hall, 1976. 559p.

2 V. Gubarev, V. Romanenko, Y. Milyavsky. Identification in cognitive maps in the impulse process mode with full information.Problems of control and informatics. 2018. N 4. P. 30-43 (in Russian).

3 Verhagen M., Dewilde P., Subspace model identification. Part I: the output-error state space model identification class of algorithms. Int. J. Control. 1992,. N56. P. 1187-1210. https://doi.org/10.1080/00207179208934363

4 V. Romanenko, Y. Milyavsky, M. Polyakov, Y. Letser, G. Shevchenko. Research of scenarios of IT company development based on decision-making in cognitive maps impulse process control mode. Proceedings of 1st international scientific and practical forum “Science and business”. (29-30 of June, 2015, Dnipropetrovsk), Dnipropetrovsk, 2015. P. 233-237.2015. P. 233-237.

Recieved 27.11.2018

## Issue 184, article 4

KVT, 2016, Issue 184, pp.44-56

UDC 681.5

CONTROL METHOD IN COGNITIVE MAPS BASED ON WEIGHTS INCREMENTS

Educational and Scientific Complex “Institute for Applied Systems Analysis” of National Technical University of Ukraine “Kyiv Polytechnic Institute”, Kiev, Ukraine

Introduction. Cognitive maps are widely used for modeling large multidimensional systems. These are weighted oriented graphs that represent concepts and relations between them. When external or internal disturbances affect the system impulse process is initiated. It is described by first-order equation in increments of vertices coordinates. A number of articles solved a problem of control in cognitive map’s impulse process by means of control theory methods. But all of them used external control inputs, i.e. resources of the vertices, for this purpose.

The purpose of the article is to develop new method of control where cognitive map’s edges weights are used as controls for impulse process stabilisation.

Results. New method of control of cognitive maps was developed. It is based on varying of the map’s edges weights. It was supposed that some of the vertices may affect other ones in different way, i.e. stronger or weaker. After presenting impulse process model in full coordinates weights increments were added to the difference equation. They were considered as control inputs which were generated according to the control law developed based on quadratic criterion. Stability of the closed-loop system was demonstrated. To verify the results, method was simulated using cognitive map of student’s socio-educational process. Finally we obtained that for stable cognitive map vertices’ coordinates are quickly stabilised at new levels via edges’ weights varying.

Conclusion. Applying the proposed method of control based on weights varying to impulse process of cognitive map allows setting vertices coordinates on desired levels.

Keywords: cognitive map, control law, weights increments, stabilisation at new levels.

References:

1 Axelrod R. The Structure of Decision: Cognitive Maps of Political Elites. Princeton University Press, 1976, 404 p.

2 Roberts F. Discrete Mathematical Models with Applications to Social, Biological, and Environmental Problems. Englewood Cliffs, Prentice-Hall, 1976, 559 p.

3 Gorelova G.V., Zakharova E.N., Radchenko S.A. Research of semi-structured problems in socio-economic systems. Cognitive approach. Rostov-na-Donu: Publisher RSU, 2006, 332 p. (in Russian).

4 Romanenko V.D., Milyavskiy Y.L. Stabilizing of impulse processes in cognitive maps based on state-space models. System Research & Information Technologies, 2014, No1, pp. 26–42 (in Russian).

5 Romanenko V.D., Milyavskiy Y.L., Reutov A.A. Adaptive Control Method for Unstable Impulse Processes in Cognitive Maps Based on Reference Models. Journal of Automation and Information Sciences, 2015, No 2 pp. 35–45 (in Russian). https://doi.org/10.1615/JAutomatInfScien.v47.i3.20

6 Romanenko V.D., Milyavskiy Y.L. Coordinates ratio control for cognitive model of a complex system under unstable impulse process. System Research & Information Technologies, 2015, No1, pp. 121–129 (in Russian).

7 Romanenko V.D., Milyavskiy Y.L. Impulse processes stabilisation in cognitive maps of complex systems based on modal state controllers. Kibernetika i vycislitel’naa tehnika, 2015, No179, pp. 43–55 (in Russian).

8 Romanenko V.D., Milyavskiy Y.L. Adaptive coordinating control of interacting cognitive maps vertices’ ratios in impulse mode. System Research & Information Technologies, 2015, No3. pp. 109–120 (in Russian).

## ISSUE 179, article 4

Kibern. vyčisl. teh., 2015, Issue 179, pp 43-55.

Romanenko Victor D., Dr (Engineering), Prof., Deputy Director for Scientific and Pedagogical Work, Institute for Applied Systems Analysis of National Technical University of Ukraine “Kiev Polytechnic Institute”, Peremogy Ave. 37, Kiev, 03056, Ukraine, email: ipsa_mmsa@ukr.net

Milyavskiy Yurii L., PhD (Engineering),Assistant of the Department of Mathematical Methods of System Analysis of the Institute for Applied Systems Analysis of National Technical University of Ukraine “Kiev Polytechnic Institute”, Peremogy Ave. 37, Kiev, 03056, Ukraine, email: yuriy.milyavsky@gmail.com

IMPULSE PROCESSES STABILISATION IN COGNITIVE MAPS OF COMPLEX SYSTEMS BASED ON MODAL STATE REGULATORS

Introduction. Cognitive modelling is one of the most widespread approaches for ill-structured socio-economic systems research nowadays. It is usually used when subject of enquiry is a complex high-dimensional system; in fact, most of financial, economical, social, political systems belong to this category. Cognitive map (CM) is a directed graph, where vertices represent concepts, directed edges represent the causal effect relationships between concepts, and the weights of edges represent the degree of the causal effect. When CM switches to transition process as a result of external or internal impulse, so called «impulse process» is described by difference first-order equation for increments. It was previously shown by the authors that impulse process can also be equivalently expressed by state-space model. One of the most important questions is how to stabilise unstable CM. For this purpose control inputs should be added to the system. Then the problem of regulators design arises.

Purpose of the paper is to investigate possibility of applying modal control methods for state regulators design (with single and multiple controls) to stabilise unstable impulse process in CM.

Results. Different methods of modal control were investigated and applied for CM impulse process stabilisation. CM dynamics was presented by state space model. Problem of external control inputs for CM was discussed. It was demonstrated that as opposed to «input — output» models, state space models allow to use smaller number of controls for stabilisation (if the system is controllable). Cases with single and multiple inputs were discussed. Impulse process in the CM for commercial bank was simulated, and different approaches to modal regulators design were applied for this cognitive model. Simulation results demonstrated efficiency of the proposed approach to CM stabilisation. It was also shown that modal control with multiple inputs is preferable where possible because it allows to get quicker response and smoother transition process with lower amplitude of control inputs.

Conclusion. Applying of modal control methods allows effective stabilising of CM. Using multiple control inputs helps to increase performance and make transition process smoother and easier to implement.

Keywords: cognitive map, modal control, impulse process stabilisation, closed loop pole placement.

References

1 Axelrod R. The Structure of Decision: Cognitive Maps of Political Elites. Princeton University Press. 1976. 404 p.

2 Roberts F. Discrete Mathematical Models with Applications to Social, Biological, and Environmental Problems. Englewood Cliffs, Prentice-Hall. 1976. 559 p.

3 Avdeeva Z.K., Kovriga S.V., Makarenko D.I., Maksimov V.I. Cognitive approach in control. Control problems, 2002, no. 3, pp. 2–8 (in Russian).

4 Maksimov V.I. Structural–target analysis of socio-economic situations development. Control problems, 2005, no. 3, pp. 30–38 (in Russian).

5 Gorelova G.V., Zakharova E.N., Radchenko S.A. Research of semi-structured problems in socio-economic systems. Cognitive approach. Rostov-na-Donu: Publisher RSU, 2006, 332 p. (in Russian).

6 Romanenko V.D., Milyavskiy Y.L. Stabilizing of impulse processes in cognitive maps based on state-space models. System research & information technologies, 2014, no. 1, pp. 26–42 (in Russian).

7 Isermann R. Digital control systems. Berlin: Springer-Verlag. 1981. 566 p. https://doi.org/10.1007/978-3-662-02319-8

8 Yegupov N.D., ed. Methods of classic and modern automatic control theory. Handbook. Vol. 2: Regulators design and optimization theory of automated control systems. Moscow: MSTU, 2000. 736 p. (in Russian).

9 Ackermann J. Sampled-Data Control Systems. Berlin: Springer-Verlag. 1985. 596 p. https://doi.org/10.1007/978-3-642-82554-5

10 Valasek M., Olgac N. Efficient Eigenvalue Assignments for General Linear MIMO Systems. Automatica, 1995, vol. 31, pp. 1605–1617. https://doi.org/10.1016/0005-1098(95)00091-A

11 Romanenko V.D., Milyavskiy Y.L., Reutov A.A. Adaptive Control Method for Unstable Impulse Processes in Cognitive Maps Based on Reference Models. Journal of Automation and Information Sciences, 2015, no. 2, pp. 35–45 (in Russian). https://doi.org/10.1615/JAutomatInfScien.v47.i3.20