By Jinzhi Wang, Zhisheng Duan, Ying Yang, Lin Huang

ISBN-10: 9812814698

ISBN-13: 9789812814692

Nonlinear platforms with desk bound units are vital simply because they hide loads of sensible platforms in engineering. prior research has been in keeping with the frequency-domain for this type of structures. in spite of the fact that, few effects on robustness research and controller layout for those platforms are simply available.

This e-book offers the research in addition to equipment in keeping with the worldwide homes of structures with desk bound units in a unified time-domain and frequency-domain framework. the focal point is on multi-input and multi-output structures, in comparison to earlier courses which thought of in basic terms single-input and single-output structures. The regulate equipment awarded during this e-book could be important for examine on nonlinear platforms with desk bound sets.

Contents: Linear platforms and Linear Matrix Inequalities; LMI method of H keep an eye on; research and regulate of optimistic genuine platforms; Absolute balance and Dichotomy of Lur e structures; Pendulum-Like suggestions platforms; Controller layout for a category of Pendulum-Like platforms; Controller Designs for platforms with enter Nonlinearities; research and regulate for doubtful suggestions Nonlinear platforms; keep watch over of Periodic Oscillations in Nonlinear platforms; Interconnected platforms; Chua s Circuits.

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**Extra info for Analysis And Control Of Nonlinear Systems With Stationary Sets: Time-Domain and Frequency-Domain Methods**

**Sample text**

E. H → X is a function, which will be denoted by Ric. 3) are complementary. This solution will be called the stabilizing solution. Thus, X = Ric(H) and Ric : dom(Ric) ⊂ R2n×2n → Rn×n . 3. Suppose H ∈ dom(Ric) and X = Ric(H). Then (i) X is real symmetric; (ii) X satisfies the algebraic Riccati equation AT X + XA + XRX + Q = 0; (iii) A + RX is stable. 1) under certain restrictions on the matrix R. 4. Suppose H has no imaginary eigenvalues and R is either positive semi-definite or negative semi-definite.

Let Q be a given matrix. 7) has a unique solution if and only if λi + λj = 0, ∀λi , λj ∈ Λ(A). 11) holds, then P = P T when Q = QT . 1. Suppose that all the eigenvalues of the matrix A have negative real parts. 7) has a unique solution ∞ P = T eA t QeAt dt. 9 is necessary and sufficient for a unique solution to exist. 11) is not satisfied. The following example explains this situation. 1. Consider Lyapunov equation AT P + P A + I = 0, where −1 0 A= . 9 is not 0 1 1 α satisfied. But it is easy to test that any matrix in the form P = 2 1 α −2 is the solution to above Lyapunov equation for any α ∈ R.

1≤k≤N This value is usually read directly from a Bode singular value plot. The L∞ norm can also be computed from the state space realization if G is rational. 1. Let γ > 0 and G= AB CD ∈ RL∞ . Then G ∞ < γ if only if σ ¯ (D) < γ and the Hamiltonian matrix H has no eigenvalues on the imaginary axis, where H := A + BR−1 DT C −C T (I + DR−1 DT )C BR−1 B T −(A + BR−1 DT C)T and R = γ 2 I − DT D. A bisection algorithm is presented by Boyd and Balakrishnan [Boyd et al. (1989)] and Robel [Robel (1989)], to compute the RL∞ norm with graranteed accuracy, using the relation between the singular values of the transfer function matrix and the eigenvalues of a related Hamiltonian matrix as follows (a) select an upper bound γu and a lower bound γl such that γu ≤ G ∞ ≤ γl ; (b) if (γu − γl )/γl ≤specified level, stop; G ∞ ≈ (γu + γl )/2, otherwise go to the next step; (c) set γ = (γu + γl )/2; (d) test if G ∞ < γ by calculating the eigenvalues of H for the given γ: if H has an eigenvalue on jR, set γl = γ; otherwise set γu = γ; go back to step (b).

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