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Systems, Structure and Control 2012 Part 4

Tham khảo tài liệu 'systems, structure and control 2012 part 4', 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 53 Âm e E if it exists as a solution of the system of equations 73 we arrive at maximal solvent R Ịm. Necessary condition. If system 67 is asymptotically stable then VÂị eE Xi 1. Since X R -m cE it follows that p R -m 1 therefore the positive definite solution of Lyapunov matrix equation 67 exists. Corollary 3.2.1 Suppose that for the given 1 N there exists matrix R Ị being solution of SMPE 73 . If system 67 is asymptotically stable then matrix R ị is discrete stable p R Ị 1 . Proof. If system 67 is asymptotically stable then V z e s z 1. Since X RỊ c s it follows that VXeX R Ị x 1 i.e. matrix RI is discrete stable. Conclusion 3.2.1 It follows from the aforementioned that it makes no difference which of the matrices R Im 1 I N we are using for examining the asymptotic stability of system 67 . The only condition is that there exists at least one matrix for at least one Í. Otherwise it is impossible to apply Theorem 3.2.2. Conclusion 3.2.2 The dimension of system 67 amounts to Ne s j hmj 1 . Conversely if there exists a maximal solvent the dimension of R m is multiple times smaller and amounts to n . That is why our method is superior over a traditional procedure of examining the stability by eigenvalues of matrix A. The disadvantage of this method reflects in the probability that the obtained solution need not be a maximal solvent and it can not be known ahead if maximal solvent exists at all. Hence the proposed methods are at present of greater theoretical than of practical significance. 3.2.4 Numerical example Example 3.2.1 Consider a large-scale linear discrete time-delay systems consisting of three subsystems described by Lee Radovic 1987 S1 xi k 1 Aixi k Biui k Ai2x2 k - h12 S2 x2 k 1 A2x2 k B2u2 k A21x1 k h21 A23x3 k h23 S 3 x3 k 1 A3x3 k B3u3 k A31x1 k h31 A1 0.8 0.6 A2 0.7 0.1 0 6 0.5 0.1 B1 0.1 A12 0.1 0 0.1 b2 0 0.1 0.1 0.2 0.4 0.9 2 0.1 12 0.1 0 0.1 - 0.6 1 0.8 L J L J _ 0