Transition in Nonlinear Systems: Tracking & Control Strategies, Slides of Nonlinear Control Systems

Download Transition in Nonlinear Systems: Tracking & Control Strategies and more Slides Nonlinear Control Systems in PDF only on Docsity! Nonlinear Systems and Control Lecture # 36 Tracking Equilibrium-to-Equilibrium Transition – p. 1/12 Docsity.com η̇ = f0(η, ξ) ξ̇i = ξi+1, 1 ≤ i ≤ ρ − 1 ξ̇ρ = L ρ fh(x) ︸ ︷︷ ︸ fb(η,ξ) + LgL ρ−1 f h(x) ︸ ︷︷ ︸ gb(η,ξ) u y = ξ1 Equilibrium point: 0 = f0(η̄, ξ̄) 0 = ξ̄i+1, 1 ≤ i ≤ ρ − 1 0 = fb(η̄, ξ̄) + gb(η̄, ξ̄)ū ȳ = ξ̄1 – p. 2/12 Docsity.com η(0) = 0, e1(0) = −ȳ, ei(0) = 0 for i ≥ 2 The shape of the transient response depends on the solution of ė = (Ac − BcK)e in feedback linearization or the solution of       ė1 ė2 ... ėρ−1       =       1 . . . 1 −k1 −kρ−1             e1 e2 ... eρ−1       +      1      s in sliding mode control What is the impact of the reaching phase? – p. 5/12 Docsity.com Second Approach: Take r(t) as the zero-state response of a Hurwitz transfer function driven by y∗ Typical Choice: aρ sρ + a1sρ−1 + · · · + aρ−1s + aρ Choose the parameters a1 to aρ to shape the response of r r(0) = 0 ⇒ e1(0) = 0 ⇒ e(0) = 0 Feedback Linearization: Sliding Mode Control: – p. 6/12 Docsity.com The derivatives of r are generated by the pre-filter ż =       1 . . . 1 −aρ −a1       z +      aρ      y∗ r = [ 1 ] z r = z1, ṙ = z2, . . . . . . r (ρ−1) = zρ r(ρ) = − ρ ∑ i=1 aρ−i+1zi + aρy ∗ Does r(t) satisfy the assumptions imposed last lecture? – p. 7/12 Docsity.com 0 0.5 1 1.5 2 0 0.5 1 1.5 2 O ut pu t τ = 0.05 0 0.5 1 1.5 2 0 0.5 1 1.5 2 O ut pu t τ = 0.25 output reference 0 0.5 1 1.5 2 −2 −1 0 1 2 τ = 0.05 C on tr ol Time output reference 0 0.5 1 1.5 2 −2 −1 0 1 2 Time C on tr ol τ = 0.25 – p. 10/12 Docsity.com Third Approach: Plan a trajectory (r(t), ṙ(t), . . . , r(ρ)(t)) to move from (0, 0, . . . , 0) to (ȳ, 0, . . . , 0) in finite time T Example: ρ = 2 TT/20 a −a r(2) at a(T−t) r(1) at2/2 −aT 2/4+aTt−at2/2 aT2/4 r – p. 11/12 Docsity.com r(t) =    at2 2 for 0 ≤ t ≤ T 2 −aT 2 4 + aT t − at2 2 for T 2 ≤ t ≤ T aT 2 4 for t ≥ T a = 4ȳ T 2 ⇒ r(t) = ȳ for t ≥ T – p. 12/12 Docsity.com