Download e-book for kindle: Absolute Stability of Nonlinear Control Systems, 2nd Edition by Xiaoxin Liao, Pei Yu

By Xiaoxin Liao, Pei Yu

ISBN-10: 1402084811

ISBN-13: 9781402084812

ISBN-10: 140208482X

ISBN-13: 9781402084829

Following the hot advancements within the box of absolute balance, Prof. Xiaoxin Liao, together with Prof. Pei Yu, has created a moment version of his seminal paintings at the topic. Liao starts off with an advent to the Lurie challenge and Lurie regulate process, sooner than relocating directly to the easy algebraic adequate stipulations for absolutely the balance of self sustaining and non-autonomous ODE structures, in addition to a number of distinctive periods of Lurie-type structures. the focal point of the publication then shifts towards the hot effects and learn that experience seemed within the decade because the first variation was once released. This e-book is aimed for use by way of undergraduates within the components of utilized arithmetic, nonlinear keep an eye on structures, and chaos keep watch over and synchronisation, yet can also be helpful as a reference for researchers and engineers. The ebook is self-contained, even though a simple wisdom of calculus, linear approach and matrix conception, and usual differential equations is a prerequisite.

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Additional info for Absolute Stability of Nonlinear Control Systems, 2nd Edition (Mathematical Modelling: Theory and Applications)

Example text

9), for any x0 ∈ Rn , we get ϕ ( x(t,t0 ; x0 ) ) ≤ V (x(t,t0 ; x0 )) ≤ V (x0 ), thus x(t,t0 ; x0 ) ≤ ϕ −1 (V (x0 )) := M. Hence, according to the Weierstrass’s accumulation principle, we see that the set Ω (x0 ) is nonempty and bounded. Now we proceed to prove that Ω (x0 ) = {0}. If this is not true, then there is a sequence {tn } satisfying tn → +∞ as n → +∞ such that lim x(tn ,t0 ; x0 ) = x∗ = 0. n→+∞ In virtue of the positive definiteness of V (x) and dV (x(t)) ≤ 0, we know that dt V (x(t,t0 ; x0 )) is monotone nonincreasing, continuous, and nonnegative.

I=1 Clearly, m V (x) ≥ ∑ ci |xi | := ϕ (y) → +∞ y → +∞, as i=1 and ϕ (y) is positive definite. 18) ≤ ≤ n ∑ − c j | f j j (x j )| + ∑ − c j | f j j (x j )| + j=1 m j=1 <0 n ∑ ci | fi j (x j )| ∑ ci | fi j (x j )| i=1,i= j n i=1,i= j if y = 0. t. the partial variable y. 34. 18) satisfies the following conditions: 1. 31; f (x ) 2. , i = j, i, j = 1, . . , n; 3. ⎤ ⎡ 1 −b21 · · · −bn1 ⎢ −b21 1 · · · −bn2 ⎥ A˜ 11 A˜ 12 ⎥ ⎢ , A˜ := ⎢ . .. ⎥ := A˜ A˜ . ⎣ . 21 22 . ⎦ −b1n −b2n · · · 1 where A˜ 11 , A˜ 12 , A˜ 21 , and A˜ 22 are m × m, m × p, p × m, and p × p matrices, ˜ ˜ −1 ˜ respectively, and A˜ 11 , A˜ 22 , I − A˜ −1 11 A12 A22 A21 are all M matrices.

N) such that ⎧ n ⎪ ⎪ −c |a | + ci |ai j | < 0, j = 1, . . , m, ⎪ j j j ∑ ⎨ ⎪ ⎪ ⎪ ⎩ −c j |a j j | + i=1,i= j n ∑ ci |ai j | ≤ 0, j = m + 1, . . t. the partial variable y. Proof. We construct the Lyapunov function n V (x) = ∑ ci |xi |. t. y. 34. 38. 20) satisfies the following conditions: 1. fi (xi ) xi > 0 for xi = 0, aii < 0, i = 1, 2, . . , n, and fi (xi ) xi ≥ 0, i = m + 1, . . , n; 2. ⎡ a21 an1 ⎤ 1 − ··· − ⎢ a11 a11 ⎥ ⎢ a an2 ⎥ 21 ⎥ ⎢ 1 ··· − ⎥ ⎢ − ˜ ˜ a a ⎥ := A11 A12 , ⎢ ˜ 22 22 A := ⎢ ⎥ ˜ 21 A˜ 22 .

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Absolute Stability of Nonlinear Control Systems, 2nd Edition (Mathematical Modelling: Theory and Applications) by Xiaoxin Liao, Pei Yu


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