Suscripción a Biblioteca: Guest
SJR: 0.232 SNIP: 0.464 CiteScore™: 0.27

ISSN Imprimir: 1064-2315
ISSN En Línea: 2163-9337

# Journal of Automation and Information Sciences

DOI: 10.1615/JAutomatInfScien.v51.i8.40
pages 43-57

## Walsh Functions in Linear-Quadratic Optimization Problems of Linear Nonstationary Systems

Alexander A. Stenin
National Technical University of Ukraine "Igor Sikorsky Kiev Polytechnic Institute", Kiev
Yuriy A. Timoshin
National Technical University of Ukraine "Igor Sikorsky Kiev Polytechnic Institute", Kiev
Irina G. Drozdovich
Institute of Telecommunication and Global Information Space of National Academy of Sciences of Ukraine, Kiev

### SINOPSIS

Currently the solution of problems of analytical design of an optimal controller (ADOC) for stationary dynamic objects has been well studied, and a number of works has been devoted to them. At the same time the synthesis of optimal control laws of nonstationary dynamic objects in a general case is quite complex task which often cannot be solved in an analytical form. This is primarily due to difficulty of solving a nonstationary nonlinear vector-matrix Riccati equation. This article deals with linear-quadratic problems of synthesis of a closed optimal control law for one class of linear nonstationary systems. Determination of the optimal control law within the framework of ADOC problem is based on the Pontryagin maximum principle. The fundamental matrix of the system of simplified canonical equations is used to establish the connection between the auxiliary vector and the state vector. It is worth noting that in a general case it is not possible to obtain an analytical expression of the fundamental matrix for linear nonstationary systems. This article proposed to find the fundamental matrix of the system of simplified canonical equations by means of approximate integration of the linear matrix differential equation of state, which it satisfies, using the mathematical apparatus of Walsh functions. In this case the elements of the matrix of the optimal control law are also determined in the form of Walsh series the constant coefficients of which are found from the system of algebraic equations. Since elements of the matrix of the optimal control law are piecewise constant functions this essentially simplified their practical implementation in comparison with nonstationary matrices of optimal control obtained based on the solution of Riccati equation. The accuracy of the obtained approximate optimal solution is achieved by choosing the appropriate number of terms of Walsh series expansion.

### REFERENCIAS

1. Morozova T.Yu., IvanovaI.A., NikonovV.V., GrishinA.A., Upgrading of control system of nonstationary complex technical objects, Tekhnicheskie nauki, 2012, No. 3, 22-30, DOI: 10.12731/ WSD-2015-6-10. .

2. GabasovR., KirillovaF.M., Qualitative theory of optimal processes [in Russian], Nauka, Moscow, 1971. .

3. EgupovN.D., Nonstationary systems of automatic control: analysis, synthesis and optimization [in Russian], MGU im. Baumana, Moscow, 2007. .

4. Mastaliev R.O., On problem of optimal control of linear system with variable structure, Vladikavkazskiy matematicheskiy zhurnal, 2016, 18, No. 1, 63-70. DOI: 10.23671/VNC.2016.1.5953. .

5. Bystrov S.V., Grigoriev V.V., PershinI.M., Mansurova O.K., Synthesis of linear-quadratic control laws for continuous dynamic objects, Mezhdunarodnyi nauchno-issledovatelskiy zhurnal, 2017, No. 2 (56), Ch. 3, 97-100, DOI: 10.23670/ IRJ.2017.56.052. .

6. Letov A.M., Analytical design of controllers, Avtomatika i telemekhanika, 1960, No. 4, 436-441; No. 5, 561-568; No. 6, 661-665; 1961, No.4, 425-435. .

7. KalmanR., Contribution to the theory of optimal control, Bul. Soc. Mech. Mat, 1960, 12, No. 2, 102-119. .

8. PontryaginL.S., Boltyanskiy V.G., Gamkrelidze R.V., Mishchenko E.V., Mathematical theory of optimal processes [in Russian], Fizmatgiz, Moscow, 1961. .

9. Michael Athans, Peter L. Falb, Optimal control: An introduction to the theory and its applications, Courier Corporation, 2006. .

10. Zalmanzon L.A., Fourie, Walsh, Haar transforms and their application in control, communication and other areas [in Russian], Nauka, Moscow, 1989. .

11. GolubovB.I., Efimov A.V., Skvortsov V.A., Walsh series and transforms: theory and applications [in Russian], Nauka, Moscow, 1987. .

12. ChenC.F., Hsiao C.H., Walsh series analysys in optimal control, Int. J. Control, 1975, 21, No. 6, 881-897. .