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

By Bao-Zhu Guo, Zhi-Liang Zhao

A concise, in-depth creation to energetic disturbance rejection keep an eye on idea for nonlinear platforms, 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 the benefits of energetic disturbance rejection keep an eye on, together with small overshooting, speedy convergence, and effort savings

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Example text

If Vq (t, x) > 0, then for any σ ∈ (0, Vq (t, x)), there exists a solution ϕ ∈ St;x and τ ∈ [0, T ] such that Vq (t, x) ≤ e2λτ Gq ( ϕ(t + τ ) ∞) + σ. 182) Once again we divide the remaining proof into two cases. Case (a): t0 < t + τ . In this case, since ϕ(t) is defined on [t − η0 , +∞), it is also well-defined at t0 . Since ϕ(t + τ ) ∞ > 1/q, we have m−1 (1/q) < ϕ(s) ∞ < m(R + 1) for any s ∈ [t − η0 , t + τ ]. 154), ϕ(t0 ) − x0 ≤ (M + 1)η0 < η¯0 . 11 that there exists a solution ψ ∈ St0 ;x0 such that ∞ ≤ ϕ(t0 ) − ϕ(t) ψ(t + τ ) − ϕ(t + τ ) ∞ ∞ + x − x0 ≤ ψ(t0 ) − ϕ(t0 ) ≤ ( x0 − x ∞ ∞ K|t+τ −t0 | ∞e + ϕ(t) − ϕ(t0 ) K(T +1) ∞ )e ≤ (M + 1)eK(T +1) (t − t0 , x − x0 ) This yields from Vq (t0 , x0 ) = 0 that Gq ( ψ(t + τ ) Vq (t, x) ≤ e2λτ (Gq ( ϕ(t + τ ) ∞) ∞) ∞.

Let x(t; x0 ) be the solution of the initial value problem following x(t) ˙ = f (x(t)), x(0) = x0 . 69) 28 Active Disturbance Rejection Control for Nonlinear Systems: An Introduction Then dV (x(t; x0 )) = Lf V (x(t; x0 )) ≤ −CV α (x(t; x0 )). 70) Solve the following initial value problem: z(t) ˙ = −C|z(t)|α sign(z(t)), z(0) = V (x0 ), to obtain ⎧ ⎪ ⎪ ⎨ t+ z(t) = ⎪ ⎪ ⎩0, 1 (V (x0 ))1−α C(1 − α) 1 1−α 1 (V (x0 ))1−α , C(1 − α) 1 (V (x0 ))1−α . 72) This together with the comparison principle of the ordinary differential equations gives V (x(t; x0 )) = 0, ∀ t ≥ 1 (V (x0 ))1−α .

90) This shows that V (x) is k-degree homogenous with weights {r1 , . . , rn }. Furthermore, we can find that there exist l, L > 0 such that V˜ (μr1 x1 , . . , μrn xn ) ≤ 1 ∀ x ∈ Rn , 12 ≤ x ≤ 2, μ ≤ l, V˜ (μr1 x1 , . . , μrn xn ) ≥ 2 ∀ x ∈ Rn , 12 ≤ x ≤ 2, μ ≥ L. 91) Therefore, for any x ∈ Rn , 1/2 ≤ x ≤ 2, L V (x) = l 1 1 (α ◦ V˜ )(μr1 x1 , . . , μrn xn )dμ + . 92) 32 Active Disturbance Rejection Control for Nonlinear Systems: An Introduction It is easy to see that V (x) is of C ∞ on {x ∈ R : 1/2 < x < 2} and ∂V (x) = ∂xi L l μri ∂ V˜ (y1 , .

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