Decomposition of Descriptor Control Systems
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DECOMPOSITION OF DESCRIPTOR CONTROL SYSTEMS L. A. Vlasenko,1† A. G. Rutkas,1‡ V. V. Semenets,1†† and A. A. Chikrii2
UDC 517.9
Abstract. We establish the conditions to decompose a complex descriptor control system into simpler subsystems. The system state and input are described by equations not solved with respect to the derivative of the state. We consider two types of decompositions: sequential and parallel ones. The decomposition conditions are formulated in terms of the existence of invariant pairs of subspaces for operator pencils consisting of system coefficients. The results are illustrated by the example of a descriptor system that describes transient states in a radio-engineering filter. We perform the cascade–parallel decomposition of forth-order filter into the simplest first-order filters, each containing one inertial element. Keywords: descriptor control system, characteristic operator pencil, invariant pair of subspaces, sequential decomposition, parallel decomposition, cascade–parallel decomposition of radio engineering filter. The present study deals with systems whose state and output can be described by equations not solved with respect to the derivative of the state. In control theory, such systems are called descriptor ones. In Sec. 1, we will outline the main provisions and assumptions. In Secs. 2 and 3, we will establish the conditions whereby the descriptor system admits both sequential and parallel decompositions into simpler subsystems of smaller dimensions. In Sec. 4, we will show how these results can be applied to the analysis of radio engineering systems. To this end, we implemented cascade–parallel decomposition of a complex quadripolar filter into elementary quadripoles with one inertial element. 1. THE MAIN CONCEPTS AND ASSUMPTIONS Methods of the analysis of various classes of control systems F = F( u, x , u ) with input (control) u( t ) , state x ( t ), and output u( t ) depend on the mathematical model that describes relations between the vector functions u( t ), x ( t ), and u( t ) . We consider a linear descriptor control system that can be described by the equations
d ( Ax ) + Bx ( t ) = Fu( t ), 0 £ t £ T , dt d u( t ) = ( Mx ) + Nx + Ku, 0 £ t £ T , dt
(1) (2)
for the state and output, respectively, as well as by the initial state
x (0) = x 0 .
(3)
Here, A, B ÎC m ´m , M , N ÎC p ´m , F ÎC m ´n , and K ÎC p ´n , where C m ´n denotes the set of m ´ n matrices with complex elements. These matrices should be understood as matrices of linear operators A, B : X ® Y , F :U ® Y , 1 Kharkiv National University of Radio Electronics, Kharkiv, Ukraine, †[email protected]; ‡[email protected]; [email protected]. 2V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 6, November–December, 2020, pp. 75–85. Original article submitted May 19, 2020. ††
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1060-0396/20/5606-0924 ©2020 Springer Science+Business Media, LLC
M , N : X ® V , K :U ® V acting in finite-di
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