By Bao-Zhu Guo, Zhi-Liang Zhao
A concise, in-depth creation to lively disturbance rejection regulate conception for nonlinear platforms, with numerical simulations and obviously labored out equations
- Provides the basic, theoretical starting place for purposes of energetic disturbance rejection control
- Features numerical simulations and obviously labored out equations
- Highlights the benefits of energetic disturbance rejection keep watch over, together with small overshooting, quick convergence, and effort savings
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Extra info for Active disturbance rejection control for nonlinear systems : an introduction
1, we need the following elementary lemma. 10 Let V (x) be a function defined on a set K = ni=1 [ai , bi ] ⊂ Rn . Assume that there exists a constant L > 0 such that for any x0 ∈ K there exists η0 > 0 satisfying |V (x) − V (x0 )| ≤ x − x0 ∞, ∀ x ∈ Bη0 (x0 ) ∩ K. 138) Then V (x) is Lipschitz continuous on K with the Lipschitz constant nL. Proof. Let x1 = (x11 , x12 , . . , x1n ) ∈ K, x2 = (x21 , x22 , . . , x2n ) ∈ K. 139) Then n |V (x1 ) − V (x2 )| ≤ |V (x11 , . . , x1j , x2j+1 , . . , x2n ) − V (x11 , .
54) The Kronecker product of A and B is an (ml) × (ns) matrix, which is defined as follows: ⎞ ⎛ a11 B a12 B · · · a1n B ⎟ ⎜ ⎜a B a B ··· a B ⎟ ⎜ 21 22 2n ⎟ ⎟ . 55) A⊗B =⎜ ⎜ .. .. ⎟ .. ⎟ ⎜ . . ⎠ ⎝ am1 B am2 B · · · amn B (ml)×(ns) The straightening operator is a 1 × (nm) matrix given by − → A = (a11 , . . , a1n , a21 , . . a2n , . . , an1 , . . , ann ) . 56) We can verify that the Kronecker product and straightening operator have the following properties.
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. 5 that there exists a positive definite symmetrical matrix PA such that A PA + PA A = −In×n , where In×n is the n × n identity matrix. Let V (ν) = ν PA ν for all ν ∈ Rn . A direct computation shows that dV (x(t; x0 )) dt = (x(t; x0 )) (A PA + PA A)x(t; x0 ) = − x(t; x0 ) 2 . 6 then follows by setting W (ν) = ν 2 for all ν ∈ Rn .