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27.(a) State and prove Liapunov's theorem for a continuous-time dynamical system. (b) By finding a suitable Liapunov function show that the origin is a stable fixed point for the dynamical system...
7. Consider the system 1 2 y (a) Show that the origin is a fixed point, and determine its stability (b) Show that the origin is the only fixed point. Hint: Argue using a theorem or result based on pr...
7. Consider the system 1 2 y (a) Show that the origin is a fixed point, and determine its stability (b) Show that the origin is the only fixed point. Hint: Argue using a theorem or result based on properties of the matrix.
7. Consider the system 1 2 y (a) Show that the origin is a fixed point, and determine its stability (b) Show that the origin is the only fixed point. Hint: Argue using a theorem or result...
36. Show that the fixed point at the origin of the system 4 2,2,.2 is unstable by using the function for a suitable choice of the constants α and β. 36. Show that the fixed point at the origin o...
36. Show that the fixed point at the origin of the system 4 2,2,.2 is unstable by using the function for a suitable choice of the constants α and β.
36. Show that the fixed point at the origin of the system 4 2,2,.2 is unstable by using the function for a suitable choice of the constants α and β.
Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b)...
Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b) Describe dynamics of the system. (c) Sketch the phase portrait. y(k + 1)「L 3y(A)-42(k)
Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b) Describe dynamics of the system. (c) Sketch the phase portrait. y(k + 1)「L 3y(A)-42(k)
(1 point) Consider the discrete-time dynamical system x+1 = 2x,(1 - x). If x = 1,...
(1 point) Consider the discrete-time dynamical system x+1 = 2x,(1 - x). If x = 1, Xr+1 = 0.5 Is x = an equilibrium for this system (yes/no)? yes What is the updating function f(x)? Compute the derivative: f'(x) = Evaluate the derivative at the equilibrium: Is the equilibruim stable, unstable or neither? stable
Prove that the following two-point boundary-value problem has a UNIQUE solution. Thank you Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and...
Prove that the following two-point boundary-value problem has a
UNIQUE solution.
Thank you
Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00<s< 00. Assume that on this domain THEOREM4 11. Prove that the following two-point boundary-value problem has a unique solution: "(t3 5)x +sin t
Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00
F1. need help solving this problem. 1. (25 pts) Here's a neat theorem. Suppose that f la, b] [a, b] is continuous; then f will always map some s-value to itself (a so-called fixed point): i.e....
F1.
need help solving this problem.
1. (25 pts) Here's a neat theorem. Suppose that f la, b] [a, b] is continuous; then f will always map some s-value to itself (a so-called fixed point): i.e. 3 c E (a, b) for which f(c)-c (a) Give a "visual proof" of this theorem. Hint: take your inspiration from our "visual proofs" of Theorem 15 and IVT And notice here that the domain and range of f are the same interval; this...
Question 3: A continuous-time system is modelled by the following differential equation y" ()+27' ()+ y(t)...
Question 3: A continuous-time system is modelled by the following differential equation y" ()+27' ()+ y(t) = x(1-1) (a) Find the transfer function and frequency response of this system, (10 marks) (b) Find the impulse response of this system. (10 marks) (e) Is the system stable? Explain (5 marks)
5 Stripes Consider the following dynamical system. State space: 2 Dynamical map: Each 0 that's followed...
5 Stripes Consider the following dynamical system. State space: 2 Dynamical map: Each 0 that's followed by a 0 turns into a 1, and each 1 that's followed by a 1 turns into a o Let's call this map E. As a demonstration, here's what E does to one point in 2N 001110011011110000101110 E(u) 100010001000011110100010 a. Find two fixed points of E, and convince the grader they're the only two. (Corrected The previous version claimed, incorrectly, that there was only...
Given the system x'=y+ux and y'= -x + uy - x2y, with u representing the greek letter mu: a) Show that the origin is a fixed point b) Linearize the system near the fixed point and determine the...
Given the system x'=y+ux and y'= -x + uy - x2y, with u representing the greek letter mu: a) Show that the origin is a fixed point b) Linearize the system near the fixed point and determine the eigenvalues c) Show that a bifurcation exists at u=0 and determine the behavior of the fixed point for u<0, u=0, and u>0
10. Prove the following theorem Theorem 1 Let H and H denote the input-output transfer functions for the continuous tim...
10. Prove the following theorem Theorem 1 Let H and H denote the input-output transfer functions for the continuous time systems associated with state matrices (A, B, C) and (A, B,C), respectively. Thus the systems have state representations (t) = Ar(t)+Bu(t) t)C(t) 1(t y(t) and Ci(t) = Assume system (A B. C) and (A. B,C) are equivalent representations, and hence there erists an invertible matriz P such that i(t) = Pa (t) defines a coordinate transformation between the two systems...
7. Consider the system 1 2 y (a) Show that the origin is a fixed point, and determine its stability (b) Show that the origin is the only fixed point. Hint: Argue using a theorem or result based on properties of the matrix.
7. Consider the system 1 2 y (a) Show that the origin is a fixed point, and determine its stability (b) Show that the origin is the only fixed point. Hint: Argue using a theorem or result...
36. Show that the fixed point at the origin of the system 4 2,2,.2 is unstable by using the function for a suitable choice of the constants α and β.
36. Show that the fixed point at the origin of the system 4 2,2,.2 is unstable by using the function for a suitable choice of the constants α and β.
Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b) Describe dynamics of the system. (c) Sketch the phase portrait. y(k + 1)「L 3y(A)-42(k)
Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b) Describe dynamics of the system. (c) Sketch the phase portrait. y(k + 1)「L 3y(A)-42(k)
(1 point) Consider the discrete-time dynamical system x+1 = 2x,(1 - x). If x = 1, Xr+1 = 0.5 Is x = an equilibrium for this system (yes/no)? yes What is the updating function f(x)? Compute the derivative: f'(x) = Evaluate the derivative at the equilibrium: Is the equilibruim stable, unstable or neither? stable
Prove that the following two-point boundary-value problem has a
UNIQUE solution.
Thank you
Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00<s< 00. Assume that on this domain THEOREM4 11. Prove that the following two-point boundary-value problem has a unique solution: "(t3 5)x +sin t
Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00
F1.
need help solving this problem.
1. (25 pts) Here's a neat theorem. Suppose that f la, b] [a, b] is continuous; then f will always map some s-value to itself (a so-called fixed point): i.e. 3 c E (a, b) for which f(c)-c (a) Give a "visual proof" of this theorem. Hint: take your inspiration from our "visual proofs" of Theorem 15 and IVT And notice here that the domain and range of f are the same interval; this...
Question 3: A continuous-time system is modelled by the following differential equation y" ()+27' ()+ y(t) = x(1-1) (a) Find the transfer function and frequency response of this system, (10 marks) (b) Find the impulse response of this system. (10 marks) (e) Is the system stable? Explain (5 marks)
5 Stripes Consider the following dynamical system. State space: 2 Dynamical map: Each 0 that's followed by a 0 turns into a 1, and each 1 that's followed by a 1 turns into a o Let's call this map E. As a demonstration, here's what E does to one point in 2N 001110011011110000101110 E(u) 100010001000011110100010 a. Find two fixed points of E, and convince the grader they're the only two. (Corrected The previous version claimed, incorrectly, that there was only...
10. Prove the following theorem Theorem 1 Let H and H denote the input-output transfer functions for the continuous time systems associated with state matrices (A, B, C) and (A, B,C), respectively. Thus the systems have state representations (t) = Ar(t)+Bu(t) t)C(t) 1(t y(t) and Ci(t) = Assume system (A B. C) and (A. B,C) are equivalent representations, and hence there erists an invertible matriz P such that i(t) = Pa (t) defines a coordinate transformation between the two systems...
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