Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 a...
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and k >a. Consider the following Lyapunov Function: Where p >0. Answer the following questions: . Is V(x) a good candidate Lyapunov function? Explain 2. Is the origin at least stable? Explain (Hint: set p c) 3. Show that the system is Globally Asymptotically Stable. Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and k >a. Consider the following Lyapunov...
Problem 2 (25 pts): Consider the following non-linear autonomous system Consider a quadratic Lyapunov function in the form And study the stability of the system as function of the parameter k. More specifically 1. Show that the origin is Globally Asymptotically Stable for k 0. 2. Assume kヂ0. Is the origin still stable? Provide an interpretation. Problem 2 (25 pts): Consider the following non-linear autonomous system Consider a quadratic Lyapunov function in the form And study the stability of the...
Problem 7: Consider the following non-linear, non-autonomous system Here, g(t) is a continuous, differentiable and bounded function with g(t) 2 k>0 for all t2 0. Consider the quadratic Lyapunov function and answer the following questions: 1. Is the origin of the system uniformly asymptotically stable (UAS)?
Consider the following nonlinear dynamic system, with a possible potential candidate function. Use the given Lyapunov function, us such function (Lyapunov Direct) approach to; (5 Marks): Show that the system is globally stable around the origin (5 Marks): The origin is globally asymptotically stable. (5 Marks): Only SKETCH a possible Phase Plan, as based on (a), (b). a. b. c. Consider the following nonlinear dynamic system, with a possible potential candidate function. Use the given Lyapunov function, us such function...
3 [15 pts Consider the Lorenz system given by xy-B2, z = where σ, ρ, β > 0 are constants. For ρ (0.1), using the Lyapunov function V(x, y, z) = ρ「2 + ơy2 + ơz?, show that the origin is globally asymptotically stable. (Hint. You may need to use the Invariance Principle as well.) στ 3 [15 pts Consider the Lorenz system given by xy-B2, z = where σ, ρ, β > 0 are constants. For ρ (0.1), using...
Problems: (1) Answer True or False to each of the following. You must substantiate your answers. (A) A differentiable function is always globally Lipschitz. (B) The trajectory of the system , r(0) is bounded for all t 0 (C) A linear tine-varying system á(t) A(t)a(t) is asymptotically stable around the origin if and only if it is uniformly exponentially stable around the origin. (D) Given the equation x f(x), and suppose that xe 0 is an exponentially stable equilibrium point...
Problem 3 Consider the following system: 2 213+w. where w denotes control input. Here we design a control system based on passivity. (a) Suppose that w =-r1 + x2 + 2.123 + u for a new control input u. Show that the state equation can be written as the following cascade form: i fa(2) +F(z)y, 22u yT2, where z = [ri, r3]T e R2. Find the expression for fa (z) and F(z). (b) Show that when y0, the origin 0...
Consider the nonlinear second-order differential equation where k > 0 is a constant. Answer to the following questions (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: {stable / saddle / unstable(not saddle)) (c) Show that there is no periodic solution in a simply connected region (Hint: Use the corollary to Theorem 11.5.1) Consider the nonlinear second-order differential equation where...
Consider an autonomous ODE y = f(y) where f(y) = y2 - 1 A. (1 pt) Draw the graph of function f(y) in the y,y plane and specify the points y where f(y) is singular (that is, f(y) takes an infinite value). B. (1 pt) Finf the equilibrium solution and determine its type: stable, un- stable or semi-stable. Indicate on which side it attracts/repels nearby solutions. C. (2 pt) By separating y and t, specify the general solution y=yt,C) t+C...
Consider the nonlinear second-order differential equation x4 3(x')2 + k2x2 - 1 = 0, _ where k > 0 is a constant. Answer to the following questions. (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: stable / saddle unstable(not saddle)} (c) Show that there is no periodic solution in a simply connected region {(r, y) R2< 0} R =...