Active disturbance rejection control for nonlinear systems : by Bao-Zhu Guo, Zhi-Liang Zhao

By Bao-Zhu Guo, Zhi-Liang Zhao

A concise, in-depth creation to lively disturbance rejection keep an eye on concept for nonlinear platforms, with numerical simulations and obviously labored out equations

  • Provides the basic, theoretical beginning 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, quick convergence, and effort savings

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1⎟ ⎠ 0 0 ··· 0 .. 82) 30 Active Disturbance Rejection Control for Nonlinear Systems: An Introduction n is Hurwitz, then there exists θ∗ ∈ n+1 , 1 such that for any θ ∈ (θ∗ , 1), the system following is finite-time stable: ⎧ x˙ 1 (t) = x2 (t) − k1 [x1 (t)]θ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪x˙ 2 (t) = x3 (t) − k2 [x1 (t)]2θ−1 , ⎪ ⎪ ⎨ .. 83) . ⎪ ⎪ ⎪ ⎪ ⎪ x˙ n (t) = xn+1 (t) − kn [x1 (t)]nθ−(n−1) , ⎪ ⎪ ⎪ ⎩ x˙ n+1 (t) = −kn+1 [x1 (t)](n+1)θ−n . Proof. 83) is (θ − 1) degree homogeneous with weights {(i − n 1)θ − (i − 2)}n+1 i=1 .

This completes the proof of the lemma. Now we use δ(t, x) to denote any continuous function defined on [−1, ∞) × Rn → Rn such that, for any (t, x) ∈ [−1, ∞) × Rn , δ(t, x) ≥ 0 and δ(t, x) = 0 if and only if x = 0. 118) where E = ([−1, ∞) \ N0 ) × Rn . 11. 119) x(t) ˙ ∈ F1 (t, x(t)), t ≥ −1. 95), there exists a class KL function β : [0, ∞) × [0, ∞) → [0, ∞) such that x(t0 + h) ≤ β(h, x0 ). We say that β(t, s) is the class KL function if, for any given t, β(t, s) is the class K∞ function with respect to s and, for any given s, β(t, s) is decreasing with respect to t and limt→∞ β(t, s) = 0.

Case 1: Vq (t0 , x0 ) = 0. In this case, the proof is accomplished by the following two claims. Claim 1: If η0 is small enough, then Vq (t, x) = 0 for any (t, x) ∈ Bη0 (t0 , x0 ). Let ϕ0 ∈ St0 ,x0 satisfy Vq (t0 , x0 ) − e2λτ Gq ( ϕ0 (t0 + τ ) ∞) < Vq (t0 , x0 ) , τ ∈ (0, T ]. 155) Then ϕ0 (t0 + τ ) > 1/q. 11 shows that m−1 1 q < ϕ0 (t0 + h) ∞ < m(R + 1), ∀ h ∈ [0, τ ]. 156) We assume without loss of generality that η0 < τ . 156) holds true for h ∈ [−η0 , τ and [t0 − η0 , t0 + τ ] ⊂ 42 Active Disturbance Rejection Control for Nonlinear Systems: An Introduction [−(R + 1)/(R + 2), R + T + 1].

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