By Jinzhi Wang, Zhisheng Duan, Ying Yang, Lin Huang
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
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).
Analysis And Control Of Nonlinear Systems With Stationary Sets: Time-Domain and Frequency-Domain Methods by Jinzhi Wang, Zhisheng Duan, Ying Yang, Lin Huang