# Systems, Structure and Control 2012 Part 3

## Tham khảo tài liệu 'systems, structure and control 2012 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Asymptotic Stability Analysis of Linear Time-Delay Systems Delay Dependent Approach 33 Conclusion Stojanovic Debeljkovic 2006 Eq. 4 expressed through matrix R can be written in a different form as follows R- A0 -e-RTA1 0 8 and there follows det r- A0 -e-RTA1 0 9 Substituting a matrix variable R by scalar variable s in 7 the characteristic equation of the system 1 is obtained as f s det sI- Ao - e sTA1 0 10 Let us denote E s f s 0 11 a set of all characteristic roots of the system 1 . The necessity for the correctness of desired results forced us to propose new formulations of Theorem . Theorem Stojanovic Debeljkovic 2006 Suppose that there exist s the solution s T 0 e Qt of 4 . Then the system 1 is asymptotically stable if and only if any of the two following statements holds 1. For any matrix Q Q 0 there exists matrix P0 P0 0 such that 2 holds for all solutions T 0 eftp of 4 . 2. The condition 7 holds for all solutions R A1 T 0 e Qr of 8 . Conclusion Stojanovic Debeljkovic 2006 Statement Theorem require that condition 2 is fulfilled for all solutions T 0 e Qt of 4 . In other words it is requested that condition 7 holds for all solution R of 8 especially for R Rmax where the matrix Rm e QR is maximal solvent of 8 that contains eigenvalue with a maximal real part Xm eE Re Xm maxRes . Therefore from 7 follows condition Re Xi Rm 0. These seE matrix condition is analogous to the following known scalar condition of asymptotic stability System 1 is asymptotically stable if and only if the condition Res 0 holds for all solutions s of 10 especially for s Xm . On the basis of Conclusion it is possible to reformulate Theorem in the following way. Theorem Stojanovic Debeljkovic 2006 Suppose that there exists maximal solvent R m of 8 . Then the system 1 is asymptotically stable if and only if any of the two following equivalent statements holds 1. For any matrix Q Q 0 there exists matrix Po Po 0 such that 6 holds for the solution

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