tailieunhanh - AUTOMATION & CONTROL - Theory and Practice Part 5

Tham khảo tài liệu 'automation & control - theory and practice part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Nonlinear Analysis and Design of Phase-Locked Loops 91 i t - 2 t C1 VT e 0 T 4 Wj t R V t e 0 T 5 are satisfied. Here we assume that the quantity Ỗ is sufficiently small with respect to the fixed numbers T C C1 the number R is sufficiently great with respect to the number Ô. The latter means that on the small intervals t t J the functions y t and Wj t are almost constants and the functions fj t rapidly oscillate as harmonic functions. Consider two block diagrams shown in Fig. 2 and Fig. 3. Fig. 2. Multiplier and filter. o. t . 1 G t PD Filter M Fig. 3. Phase detector and filter. Here 9j t Wj t t fij are phases of the oscillations fj t PD is a nonlinear block with the characteristic ty 9 being called a phase detector or discriminator . The phases 9j t are the inputs of PD block and the output is the function ty 91 t Ỡ2Ơ . The shape of the phase detector characteristic is based on the shape of input signals. The signals f1 t f2 t and ty 91 t 92 t are inputs of the same filters with the same impulse transient function y t . The filter outputs are the functions g t and G t respectively. A classical PLL synthesis is based on the following result Theorem 1. Viterbi 1966 If conditions 3 - 5 are satisfied and we have 1 v 9 2 A1 A2 cos 9 then for the same initial data of filter the following relation IG t g t l C2Ô Vt e 0 T is satisfied. Here C2 is a certain number being independent of Ỗ. 92 AUTOMATION CONTROL - Theory and Practice Proof of Theorem 1 Leonov 2006 For t Ễ 0 T we obviously have g i - G t t r Ị y t s A1A sin 1 s s 1 sin 2 s s 2 0 p 1 s s 2 s s 1 2 ds . . t A1A2 27Y t s cos I 1 s 2 s s 1 2 ds. Consider the intervals k5 k 1 5 where k 0 . m and the number m is such that t Ễ m5 m 1 5 . From conditions 3 5 it follows that for any s Ễ k5 k 1 5 the relations y t s y t k5 0 5 6 1 s 2 s 1 kô 2 k5 0 5 7 are valid on each interval k5 k 1 5 . Then by 7 for any s Ễ k5 k 1 5 the estimate cos I 1 s 2 s s 1 2 cos I 1 k5 2 k5 s 1 2 0 5 8 is valid. Relations 6 and 8 imply that t