Homework Answers
1) yes V(X) is the good candidate.
2) no , origin is not stable.The function is not get a stable point after solving that define the actual settling zone.Thats why we come fact that the fuction have no stable point.
Add Answer to:
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and...
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 systerm Where a 0,b0.c O,d > 0 and k> a. Consider the following Lyapunov Function Where p >0. Answer the following questions: 1. 1s V (x) a good candidate Lyapunov function? Explairn 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 systerm Where a 0,b0.c O,d > 0 and k>...
Problem 2 (25 pts): Consider the following non-linear autonomous system Consider a quadratic Lyap...
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...
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 th...
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 a...
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...
Closed loop Controller - Dynamical System Consider the following continuous non-linear dynamical system: x1 = (11-2x1)ex1...
Closed loop Controller - Dynamical System
Consider the following continuous non-linear dynamical system: x1 = (11-2x1)ex1 2(2x1-4x2)e*z The system is driven by the following closed-loop controller: 1. For all values of K, find the equilibrium points of the closed loop system, i.e. find the equilibrium point as K varies between-co and +co 2. Consider the origin of the system. Determine the character of the origin for all values of the parameter K. Determine specifically for what values of K the...
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...
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...
Exercise 5.5. Consider the linear system 2 as in (5.44) with A-4 0 C [1 0...
Exercise 5.5. Consider the linear system 2 as in (5.44) with A-4 0 C [1 0 -1 4 1 a. Show that the system is not (internally) asymptotically stable b. Show that the system is both controllable and observable. c. Find matrices F e R1x2 and GE R2x1 such that o(A+ BF) C C_ and o (A GC) C C_ d. Find matrices (K, L, M, N) such that the feedback controller w(t) Kw(t) Ly(t) u(t) Mw(t)Ny(t) is internally stabilizing...
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...
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 questio...
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...
Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 and k> a. Consider the following Lyapunov Function Where p >0. Answer the following questions: 1. 1s V (x) a good candidate Lyapunov function? Explairn 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 systerm Where a 0,b0.c O,d > 0 and k>...
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...
Closed loop Controller - Dynamical System
Consider the following continuous non-linear dynamical system: x1 = (11-2x1)ex1 2(2x1-4x2)e*z The system is driven by the following closed-loop controller: 1. For all values of K, find the equilibrium points of the closed loop system, i.e. find the equilibrium point as K varies between-co and +co 2. Consider the origin of the system. Determine the character of the origin for all values of the parameter K. Determine specifically for what values of K the...
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...
Exercise 5.5. Consider the linear system 2 as in (5.44) with A-4 0 C [1 0 -1 4 1 a. Show that the system is not (internally) asymptotically stable b. Show that the system is both controllable and observable. c. Find matrices F e R1x2 and GE R2x1 such that o(A+ BF) C C_ and o (A GC) C C_ d. Find matrices (K, L, M, N) such that the feedback controller w(t) Kw(t) Ly(t) u(t) Mw(t)Ny(t) is internally stabilizing...
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...
Most questions answered within 3 hours.
-
What mechanisms Drive speciation??
(I.e. what was Dawins theory on the orgin of species, and how...
asked 1 hour ago -
The manager at a car assembly plant believes that the mean
assembly time for a car...
asked 2 hours ago -
Which of the following is true of electron capture?
A) It decreases the nuclide's mass number...
asked 3 hours ago -
Assuming an efficiency of 43.10%, calculate the actual yield of
magnesium nitrate formed from 114.9 g...
asked 4 hours ago -
The highly pathogenic bacterium Clostridium
perfringens causes gangrene, a disease that results in the
destruction of...
asked 6 hours ago -
In the context of situation analysis, which of the following is
a category for analysis in...
asked 6 hours ago -
In a study of the gas phase decomposition of sulfuryl chloride
at 600 K SO2Cl2(g)SO2(g) +...
asked 6 hours ago -
75 g of 2-propanol (C3H8O) and 25 g of pentane are mixed in a
200 mL...
asked 6 hours ago -
The 2800-turn coil in a dc motor has an area per turn of 1.1 ×
10-2...
asked 6 hours ago -
Draw a combinational logic circuit diagram with a symbol inside
the box for two I/P of...
asked 6 hours ago -
The cliché we use quite a lot in finance is: there is a need to
maximize...
asked 6 hours ago -
In class we discussed the addition of HCl to alpha pinene. Would
you expect one or...
asked 6 hours ago