Nonlinear Control Systems Homework: Dynamical Systems, Predator-Prey Model, Assignments of Electrical and Electronics Engineering

Download Nonlinear Control Systems Homework: Dynamical Systems, Predator-Prey Model and more Assignments Electrical and Electronics Engineering in PDF only on Docsity! ECSE 6420: Nonlinear Control Systems Homework set 1. Due date: 5 Feb 2009 Points: Problem 1 = 10pts, Problem 2 = 5+10+10+5 pts, Problem 3 = 20pts, Problem 4 = 5+10 pts, Problem 5 = 10+15 pts 1. Consider an autonomous dynamical system ẋ = f(x), x(0) = x0, x ∈ R n. (1) Assume that f(·) is such that (1) has a unique solution for all positive time1. Such a system can be thought of as a map from x0 to the solution of (1). That is, if we denote the map as H, then H(x0)(t) is the differentiable solution of (1). The system is linear if and only if H is a linear map, i.e. ∀x1, x2 ∈ R n,∀a1, a2 ∈ R,H(a1x1 + a2x2) = a1H(x1) + a2H(x2). (2) Prove that f(x) can always be written as Ax, for some constant matrix A. 2. The predator-prey model that we discussed in class assumes unlimited resources that enable the prey to grow unboundedly in the absence of the predators. A more realistic model that model competition within the same species is posed below: ẋ = ax − bxy − λx2, (3) ẏ = cxy − dy − µy2. (4) (a) Prove that the first quadrant {(x, y)|x ≥ 0, y ≥ 0} is invariant. (b) Specify all the equilibria in the first quadrant and simulate for a = b = c = d = 1, λ = 0.5, µ = 1. Use Hartman-Grobman Theorem (when applicable) to classify the types of the equilibria (c) Specify all the equilibria in the first quadrant and simulate for a = b = c = d = 1, λ = 2, µ = 1. Use Hartman-Grobman Theorem (when applicable) to classify the types of the equilibria (d) Discuss the difference in the behavior of the system in (b) and (c). 3. Hartman-Grobman Theorem asserts that local linearization of a nonlinear system ẋ = f(x) around an equilibrium x0 can qualitatively describe the local nonlinear dynamics, provided that the eigenvalues of the Jacobian matrix at x0 exclude the imaginary axis. This is because when there 1Later in the course we are going to study conditions on f(·) such that this is true. 1