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Tham khảo tài liệu 'time delay systems part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | On Stable Periodic Solutions of One Time Delay System Containing Some Nonideal Relay Nonlinearities 109 One can easy verify that k1 r2 V1 0 k2 r1 V2 0 k3 T v 0 kị r1 Vị 0. Denote M1 i - k-1v1r 2 eA1 T1 M3 I - k-iv3T 2 eA1T3 M2 i - k-1v2rQ eA2T2 M4 l - k-1 v4rQ eA2Tị. and l M 4 n M i i K 0.3033 1. So as dsk 1 Mdsk the periodic solution under consideration is orbitally asymptotically stable. Similar results can be obtained in case of nonlinearity 3 . 6. Perturbed system Consider a system x Ax c q t u t - t 10 where q t is scalar Tq-periodic continuous function of time. Let f is given by 3 . Consider a special case of the previous system see Nelepin 2002 Kamachkin Shamberov 1995 . Let n 2 ý g1 ý g2 ý u t - t q i 11 here ý t Ễ R is sought-for time variable g1 2 are real constants a X1ý 2ý a 1 2 are real constants. Let us rewrite system 11 in vector form. Denote z ý ý in that case z Pz q q t u t - t u t - t f a t - t a a z 12 where P L44 -g2 -g1 0 1 q a Suppose that characteristic determinant D s det P - si has real simple roots A1 2 and vectors q Pq are linearly independent. In that case system 12 may be reduced to the form 10 where A 0 aU by means of nonsingular linear transformation N A N A d 2 z Tx T D A1 A A D Aj d-D s Nj s t qiDij s 13 D A1 D Ả2 s A i 1 Dij s is algebraic supplement for element lying in the intersection of i-th row and j-th column of determinant D s . 110 Time-Delay Systems Note that a Y x Y T a. Furthermore since Yi - D Xi -1 j Nj Xi i 1 2. j 1 then Y1 X1 - X2 1 a1 2X1 Y2 X2 - X1 1 a1 2 X2 . Transformation 13 leads to the following system X1 X1X1 f a t- t y t IX x2X2 f a t- t t . If for example 1 -X1 2 then Y1 0 Y2 2 a Y2 X2. Function f in that case is independent of variable X1 and a X2a Y2 f Y2 X2 t - t y t . 14 Solution of the latest equation when f u where u m1 m2 or 0 has the following form a t to ao u eX2 t-to ao Y2eX2t 1 e-X2s u p s ds. J to Let us trace out necessary conditions for existing of periodic solution of the system 10 3 having .
