By Xiaoxin Liao, Pei Yu
Following the hot advancements within the box of absolute balance, Prof. Xiaoxin Liao, along with Prof. Pei Yu, has created a moment variation of his seminal paintings at the topic. Liao starts with an advent to the Lurie challenge and Lurie keep an eye on process, prior to relocating directly to the straightforward algebraic adequate stipulations for absolutely the balance of independent and non-autonomous ODE platforms, in addition to numerous particular periods of Lurie-type structures. the point of interest of the ebook then shifts towards the recent effects and learn that experience seemed within the decade because the first version was once released. This ebook is aimed for use by way of undergraduates within the parts of utilized arithmetic, nonlinear keep watch over platforms, and chaos keep an eye on and synchronisation, yet can also be beneficial as a reference for researchers and engineers. The ebook is self-contained, although a simple wisdom of calculus, linear method and matrix idea, and traditional differential equations is a prerequisite.
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Extra info for Absolute Stability of Nonlinear Control Systems, 2nd Edition (Mathematical Modelling: Theory and Applications)
For any symmetric matrix B, we have cT b + d TB−1 d ≥ 0, where d= 1 T (A c + 2 Pb), 2 Proof. 2) σ 0 P A + ATP = − B. f (σ ) dσ . Then we have x f (σ ) = T B −d − d T − cT b x f (σ ) . 19) Since B is positive deﬁnite, det B > 0. Then, det B−1 0 = det B−1 = (det B)−1 > 0, 0T 1 and det B−1 0 0T 1 B −d − d T − cT b Thus, det = det I − B−1 d = − cT b − d TB−1 d. − d T − cT b B −d = (− cT b − d TB−1 d) det B. 20) is an important formula and has many applications. t. (x, f (σ )), which is, however, impossible as shown above.
N; 2. There exist constants ci > 0 (i = 1, 2, . . , m), c j ≥ 0 ( j = m + 1, . . , 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 .
The partial variable y. Proof. 26 to complete the proof. 33. 18) satisﬁes 1. 31; 2. There exist n functions ci > 0 (i = 1, 2, . . , m) and c j ≥ 0 ( j = m + 1, . . , n) such that ⎧ n ⎪ ⎪ −c j | f j j (x j )| + ∑ ci | f j j (x j )| < 0 for x j = 0, j = 1, . . , m, ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −c j | f j j (x j )| + i=1,i= j n ∑ ci | f j j (x j )| ≤ 0, j = m+1, . . t. the partial variable y. Proof. We construct the Lyapunov function n V (x) = ∑ ci |xi |. i=1 Clearly, m V (x) ≥ ∑ ci |xi | := ϕ (y) → +∞ y → +∞, as i=1 and ϕ (y) is positive deﬁnite.
Absolute Stability of Nonlinear Control Systems, 2nd Edition (Mathematical Modelling: Theory and Applications) by Xiaoxin Liao, Pei Yu