Question

Problem 4 : Linearization of a Nonlinear System The simple pendulum has the following equation of motion: J?? + (mgl/2)sin? = T, where J is the pendulum's moment of inertia about the point of rotation and T is the applied torque. (a) Let the state variables be x = [x1; x2] = [?; ??] and the input to the system be u = T. Put the equation of motion into the nonlinear state space form, x? = f(x, u). (b) Suppose that u = 0. Find the 2 equilibrium points xeq,1 and xeq,2 of the system. (c) Linearize the nonlinear state space model about x? = [0; 0] and about the input u? = 0 to obtain the linear state space model x?? = Ax? + Bu?, where x? = x - x? and u? = u - u?. Do this by deriving the Jacobian matrices A and B. (d) Suppose that there is no applied torque (T = 0) and the system starts at rest with the pendulum angle at ? = ?/2. Use the MATLAB function ode45 to solve both the four 1st-order nonlinear equations of motion and the 1st-order linearized equations of motion from part (c). Use the parameters J = 1kgm^2, m = 1kg, and l = 1m. Plot ? over time from the nonlinear and linearized models in one figure, and plot ?? over time from both models in another figure. Hand in your code with your homework.

          Problem 4 : Linearization of a Nonlinear System
The simple pendulum has the following equation of motion:
J?? + (mgl/2)sin? = T,
where J is the pendulum's moment of inertia about the point of rotation and T is the applied torque.
(a) Let the state variables be x = [x1; x2] = [?; ??] and the input to the system be u = T. Put the equation of motion into the nonlinear state space form, x? = f(x, u).
(b) Suppose that u = 0. Find the 2 equilibrium points xeq,1 and xeq,2 of the system.
(c) Linearize the nonlinear state space model about x? = [0; 0] and about the input u? = 0 to obtain the linear state space model x?? = Ax? + Bu?, where x? = x - x? and u? = u - u?. Do this by deriving the Jacobian matrices A and B.
(d) Suppose that there is no applied torque (T = 0) and the system starts at rest with the pendulum angle at ? = ?/2. Use the MATLAB function ode45 to solve both the four 1st-order nonlinear equations of motion and the 1st-order linearized equations of motion from part (c). Use the parameters J = 1kgm^2, m = 1kg, and l = 1m. Plot ? over time from the nonlinear and linearized models in one figure, and plot ?? over time from both models in another figure. Hand in your code with your homework.

Problem 4 : Linearization of a Nonlinear System
The simple pendulum has the following equation of motion:
J?? + (mgl/2)sin? = T,
where J is the pendulum's moment of inertia about the point of rotation and T is the applied torque.
(a) Let the state variables be x = [x1; x2] = [?; ??] and the input to the system be u = T. Put the equation of motion into the nonlinear state space form, x? = f(x, u).
(b) Suppose that u = 0. Find the 2 equilibrium points xeq,1 and xeq,2 of the system.
(c) Linearize the nonlinear state space model about x? = [0; 0] and about the input u? = 0 to obtain the linear state space model x?? = Ax? + Bu?, where x? = x - x? and u? = u - u?. Do this by deriving the Jacobian matrices A and B.
(d) Suppose that there is no applied torque (T = 0) and the system starts at rest with the pendulum angle at ? = ?/2. Use the MATLAB function ode45 to solve both the four 1st-order nonlinear equations of motion and the 1st-order linearized equations of motion from part (c). Use the parameters J = 1kgm^2, m = 1kg, and l = 1m. Plot ? over time from the nonlinear and linearized models in one figure, and plot ?? over time from both models in another figure. Hand in your code with your homework.

Added by Eric H.

Question

Please give Ace some feedback