Read e-book online Active disturbance rejection control for nonlinear systems : PDF
By Bao-Zhu Guo, Zhi-Liang Zhao
A concise, in-depth creation to energetic disturbance rejection keep watch over conception for nonlinear structures, with numerical simulations and obviously labored out equations
- Provides the basic, theoretical origin for functions of energetic disturbance rejection control
- Features numerical simulations and obviously labored out equations
- Highlights some great benefits of energetic disturbance rejection keep watch over, together with small overshooting, speedy convergence, and effort savings
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Additional info for Active disturbance rejection control for nonlinear systems : an introduction
210) Therefore, Vδ (t, x) is well defined and is of class C ∞ on (−1 + δ, +∞) × Rn . In addition, Vδ (t, x) → VL (t, x) uniformly on S as δ → 0. 207) if δ is small enough. To complete the proof of the lemma, it remains to show that there exists δ0 > 0 such that for any δ ∈ (0, δ0 ), (t0 , x0 ) ∈ S and v ∈ F2 (t0 , x0 ), ∂Vδ (t0 , x0 ) 3 + ∇x Vδ (t0 , x0 ), v ≤ − λVL (t0 , x0 ). 211) Let δ1 > 0 be a small constant so that S + Bδ1 (0) ⊂ U and let L > 0 be a positive constant so that for any pairs (t1 , x1 ), (t2 , x2 ) ∈ S + Bδ1 (0).
X2n ). Proof. Let B = (bij ) satisfy → − − → (A ⊗ I + I ⊗ A ) B = C . 61) It follows from the Gramer law that bij = det(Δij ) , i, j = 1, 2, . . , n, det(Δ) where Δij is the matrix where the (i − 1)nj th column (the number of bij ’s coefficient column) → − in Δ is replaced by C and other columns are the same as in Δ. Then n V (x) = x Bx = bij xi xj . 62) i,j=1 On the other hand, a direct computation shows that n n det(Δij ) 0 X 1 = xx = b xx . 5. 53) is globally asymptotical stable. Proof.
35) is asymptotically stable and the attracting basin is Ω ⊂ Rn , where Ω is a connected domain and 0 ∈ Ω◦ . If f ∈ C(Ω, Rn ) is locally Lipschitz continuous, then there exist Lyapunov functions V ∈ C 1 (Ω, [0, ∞)) and W ∈ C(Ω, [0, ∞)) such that dV (x) dt ≤ −W (x), ∀ x ∈ Ω, lim V (x) = +∞. 6. The well-known Lasalle invariance principle is a powerful tool for verifying stability for autonomous systems. 45), f (0) = 0 and Ω = Br (0) ⊂ Rn is a connected domain. The function V ∈ C 1 (Ω, [0, ∞)) is positive definite and satisfies Lf V (x) ≤ 0, ∀x ∈ Ω.
Active disturbance rejection control for nonlinear systems : an introduction by Bao-Zhu Guo, Zhi-Liang Zhao