Download Nonlinear Systems and Control: An Introduction to Second-Order Systems and Phase Portraits and more Slides Nonlinear Control Systems in PDF only on Docsity! Nonlinear Systems and Control Lecture # 3 Second-Order Systems – p. 1/?? Docsity.com ẋ1 = f1(x1, x2) = f1(x) ẋ2 = f2(x1, x2) = f2(x) Let x(t) = (x1(t), x2(t)) be a solution that starts at initial state x0 = (x10, x20). The locus in the x1–x2 plane of the solution x(t) for all t ≥ 0 is a curve that passes through the point x0. This curve is called a trajectory or orbit The x1–x2 plane is called the state plane or phase plane The family of all trajectories is called the phase portrait The vector field f(x) = (f1(x), f2(x)) is tangent to the trajectory at point x because dx2 dx1 = f2(x) f1(x) – p. 2/?? Docsity.com Numerical Construction of the Phase Portrait: Select a bounding box in the state plane Select an initial point x0 and calculate the trajectory through it by solving ẋ = f(x), x(0) = x0 in forward time (with positive t) and in reverse time (with negative t) ẋ = −f(x), x(0) = x0 Repeat the process interactively Use Simulink or pplane – p. 5/?? Docsity.com Qualitative Behavior of Linear Systems ẋ = Ax, A is a 2 × 2 real matrix x(t) = M exp(Jrt)M −1x0 Jr = [ λ1 0 0 λ2 ] or [ λ 0 0 λ ] or [ λ 1 0 λ ] or [ α −β β α ] x(t) = Mz(t) ż = Jrz(t) – p. 6/?? Docsity.com Case 1. Both eigenvalues are real: λ1 6= λ2 6= 0 M = [v1, v2] v1 & v2 are the real eigenvectors associated with λ1 & λ2 ż1 = λ1z1, ż2 = λ2z2 z1(t) = z10e λ1t, z2(t) = z20e λ2t z2 = cz λ2/λ1 1 , c = z20/(z10) λ2/λ1 The shape of the phase portrait depends on the signs of λ1 and λ2 – p. 7/?? Docsity.com x2 x 1 v1 v2 (b) x1 x 2 v1 v2 (a) Stable Node Unstable Node – p. 10/?? Docsity.com λ2 < 0 < λ1 eλ1t → ∞, while eλ2t → 0 as t → ∞ Call λ2 the stable eigenvalue (v2 the stable eigenvector) and λ1 the unstable eigenvalue (v1 the unstable eigenvector) z2 = cz λ2/λ1 1 , λ2/λ1 < 0 Saddle – p. 11/?? Docsity.com z1 z2 (a) x 1 x 2 v1v2 (b) Phase Portrait of a Saddle Point – p. 12/?? Docsity.com Effect of Perturbations A → A + δA (δA arbitrarily small) The eigenvalues of a matrix depend continuously on its parameters A node (with distinct eigenvalues), a saddle or a focus is structurally stable because the qualitative behavior remains the same under arbitrarily small perturbations in A A stable node with multiple eigenvalues could become a stable node or a stable focus under arbitrarily small perturbations in A – p. 15/?? Docsity.com A center is not structurally stable [ µ 1 −1 µ ] Eigenvalues = µ ± j µ < 0 ⇒ Stable Focus µ > 0 ⇒ Unstable Focus – p. 16/?? Docsity.com
