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

Nonlinear platforms with desk bound units are vital simply because they hide loads of useful structures in engineering. past research has been in line with the frequency-domain for this category of platforms. notwithstanding, few effects on robustness research and controller layout for those platforms are simply available.

This e-book provides the research in addition to equipment in response to 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 merely single-input and single-output platforms. The keep watch over tools provided during this e-book could be necessary for learn on nonlinear structures with desk bound sets.

Contents: Linear platforms and Linear Matrix Inequalities; LMI method of H regulate; 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 structures; regulate of Periodic Oscillations in Nonlinear platforms; Interconnected platforms; Chua s Circuits.

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

**Example text**

5 Bounded real lemma For a system G with the following realization G= AB CD the bounded real lemma gives the equivalent matrix characterization of the stability of A and the H∞ norm constraint G ∞ < γ. 7 (Bounded real lemma). Let γ > 0, G ∈ RH∞ , and H := A + BR−1 DT C −C T (I + DR−1 DT )C BR−1 B T −(A + BR−1 DT C)T where R = γ 2 I − DT D. 5. Bounded real lemma 35 (iii) σ ¯ (D) < γ and H has no eigenvalues on the imaginary axis; (iv) σ ¯ (D) < γ and H ∈ dom(Ric); (v) σ ¯ (D) < γ and H ∈ dom(Ric) and Ric(H) ≥ 0(Ric(H) > 0 if (A, C) is observable); (vi) σ ¯ (D) < γ and there exists an X ≥ 0 such that X(A + BR−1 DT C) + (A + BR−1 DT C)T X + XBR−1 B T X +C T (I + DR−1 DT )C = 0 and A+BR−1 DT C +BR−1 B T X has no eigenvalues on the imaginary axis; (vii) σ ¯ (D) < γ and there exists an X > 0 such that X(A + BR−1 DT C) + (A + BR−1 DT C)T X + XBR−1 B T X +C T (I + DR−1 DT )C < 0.

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).

1) in terms of invariant subspaces of H. 1. Let V ⊂ C2n be an n-dimensional invariant subspace of H, and let X1 , X2 ∈ Cn×n be two complex matrices such that X1 V = Im . 1) and Λ(A + RX) = Λ(H|V ). Furthermore, the solution X is independent of a specific choice of bases of V. The converse of the theorem also holds. 2. 1), then there exist matrices X1 , X2 ∈ Cn×n , with X1 invertible, such that X1 X = X2 X1−1 and the columns of form a basis of an n-dimensional X2 invariant subspace of H. December 14, 2008 17:15 World Scientific Book - 9in x 6in 32 ws-book9x6˙wang LMI Approach to H∞ Control Assume H has no eigenvalues on the imaginary axis.