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Analysis in the Frequency Domain
Published in Wai-Kai Chen, Circuit Analysis and Feedback Amplifier Theory, 2018
If the highest power of the polynomial is odd, then at least one real root will exist. The other roots may be either real or complex, but if they are complex, then they always appear in complex conjugate pairs. The roots of the numerator are called zeros, and those of the denominator are called poles. We denote the zeros by zi=ai+jbi
Complex numbers
Published in John Bird, Bird's Engineering Mathematics, 2021
The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. Hence the complex conjugate of a+jb is a−jb. The product of a complex number and its complex conjugate is always a real number.
Fundamentals of Digital Signal Processing
Published in Wai-Kai Chen, Analog and VLSI Circuits, 2018
It is important to note that the unit pulse response is a linear combination of exponential functions with behavior that is determined by the roots of the characteristic equation. These roots can be real or they can occur in complex conjugate pairs. For example, if a root y is real, then h(n) will contain the term Kγn that may decrease exponentially to zero if ‖γ‖<1, remain at unity if γ = 1, or increase exponentially if ‖γ‖>1. Or, if a complex conjugate pair of roots occurs, then h(n) will contain the sum of terms Kγn+K*(γ*)n=2‖K‖‖γ‖ncos(n∠γ+∠K) that will oscillate and increase or decrease exponentially. If we require that limn→∞h(n) = 0, then all roots of the characteristic equation must satisfy ‖γi‖<1, i = 1,2,…,N.
Stability analysis of networked control systems under DoS attacks in frequency domain via game theory strategy
Published in International Journal of Systems Science, 2021
Lingli Cheng, Huaicheng Yan, Xisheng Zhan, Sha Fan, Kaibo Shi
Notations: The notations in this paper are fairly standard. For any matrix U, its Euclidean norm, Frobenius norm, conjugation and transposition are , , and , respectively, and , where is transfer function. Given any complex number , its complex conjugate transpose is . : = means defined as, γ is SNR, inf defines the infimum, represents the set of rational transfer function matrices, means the probability of · occurring, Γ is the maximum input energy of communication networked channel, I denotes an unit matrix, and means the mathematical expectation. The open right-half plane and the open left-half plane are defined as: Hardy space and its corresponding orthogonal complementary space are defined as follows:
H 2 input load disturbance rejection controller design for synchronised output regulation of time-delayed multi-agent systems with frequency domain method
Published in International Journal of Control, 2019
is the set of n × n real matrices, and is the set of n × 1 real vectors. The asterisk (*) denotes the complex conjugate transpose of a matrix. In and 0n are the identity matrix and zero matrix of dimension n. Imn = diag{Im, 0n − m}. Denote by 1 and 0 column vectors with all entries equal to one and zero. denotes the degree of the polynomial F(s). λmax (A)(λmin (A)) represents the maximum (minimum) eigenvalue of the matrix A. A⊗B denotes the Kronecker product of matrices A and B. For a set of N matrices {A1,… , AN}, we define the direct sum as . ‖ · ‖ stands for either the Euclidean vector norm or its induced matrix 2-norm. The 2-norm of transfer function F(s) is defined by
Robustness analysis of the feedback interconnection of discrete-time negative imaginary systems via integral quadratic constraints
Published in International Journal of Control, 2021
Qian Zhang, Liu Liu, Yufeng Lu
Let and be the complex plane and the set of the positive integers, respectively. Let be the n-vectors over . is the set of complex matrices. Given a matrix , , and denote the complex conjugate, the transpose and the complex conjugate transpose of M, respectively. M is Hermitian if . A Hermitian matrix M is positive definite (M>0) if for all non-zero , it is positive semi-definite () if for all non-zero . is the largest eigenvalue of a square complex matrix M with real eigenvalues. denotes the largest singular value of M. The real part of M is . The kernel of M is denoted by .