Approrimate the delay function) It is sometimes necessary to approximate the delay function y(t)ut- T) by...
Problem 3. (40 points) For the process described by the transfer function 10(1-2s)e2s Y(s) U(s) (10s+1)(4s+ 1)(s +1) (a) Find an approximate transfer function of first-order-plus-time-delay form that describes this process (b) Determine and plot the response y(t) of the approximate model, obtained in part (a), for a unit ramp using Skogestad's "Half Rule"; change in u(t) (U(s)
Problem 3. (40 points) For the process described by the transfer function 10(1-2s)e2s Y(s) U(s) (10s+1)(4s+ 1)(s +1) (a) Find an approximate...
(a) Approximate the function f(x) = cos(z) using the first three non-zero terms of its Taylor series centered at a = 0. The potential energy of a spring system can be written as U (t) = KA2 cosa(wt), where t is time, w is the frequency of the spring, A is the amplitude and k is the spring constant. Use the Taylor approximation you obtained to show that near the beginning of the spring's trajectory, the potential energy can be...
Given a two-variable function f(x, y), if P(x0,yo) is a critical point, then the behavior of f around P can be approximated by its second order terms according to Taylor series, that is, f(x,y) = f(P) + F(x – xo)?H (x, y) , where H(x, y) = fyy(P)(=%)2 + 2 fxy(P) (?=%) + fxx(P). (a). If H(x, y) > 0 for all x,y, is P a local max, local min or saddle point? (b). Let s = (4=90). Then, H(x,...
Given an input signal (t)nd the output signal is y(t)-(e2 e)u(t), compute the transfer function H(s). Determine whether the system is stable and causal or not.
Problem 2: (40 pts) Part A: (20pts) A third-order system has an of Y(s)-L[y(t) corresponding to a unit step input u(t) is known to be input of u(t) and an output of y(t). The forced response portion 1 Ys) (3 +3s2+ 4s +5) = a) Determine the input-output differential equation for the system b) From your result in a), determine the transformed free response Yee (s) corresponding to initial conditions of: y(0)= y(0) = 0 and ý(0)-6 Part B (20pts)...
aliasing? A continuous-time system is given by the input/output differential equation 4. H(s) v(t) dy(t) dt dx(t) + 2 (+ x(t 2) dt (a) Determine its transfer function H(s)? (b) Determine its impulse response. (c) Determine its step response. (d) Is the stable? (a) Give two reasons why digital filters are favored over analog filters 5. (b) What is the main difference between IIR and FIR digital filters? (c) Give an example of a second order IIR filter and FIR...
Please solve Q 7 & 8
7. 14+6 marks] Consider the initial value problem y_y2, 2,y(1) = 1 y'= 1-t (a) Assuming y(t) is bounded on [1, 2], Show that f(t,v)--satisfies Lipschitz condition with respect to y. (b) Use second order Taylor method with h 0.2 to approximate y(1.2), then use the Runge- Kutta method: to compute an approximation of y(1.4). 8. [4 marks) Assuming that a1, o2 are non negative constants, determine the parameters o and β1 of the...
Exam 2018s1] Consider the function f R2 R, defined by f(x,y) =12y + 3y-2 (a) Find the first-order Taylor approximation at the point Xo-(1,-2) and use it to find an approximate value for f(1.1,-2.1 (b) Calculate the Hessian 1 (x-4)' (Hr(%)) (x-%) at X-(1-2) c) Find the second-order Taylor approximation at xo- (1,-2) and use it to find an approximate value for f(1.1,-2.1 Use the calculator to compute the exact value of the function f(11,-2.1)
Exam 2018s1] Consider the function...
Question 4 Consider the initial value problem Y = 1+(t- y) with y(2) = 1 and 2 st s 3. dt Apply Taylor series method of order two to find y, and y, using step length h = 0.25.
2y"(t) + 3 y' (t) + y(t)=x"(t) +x'(t) - x(t), y(0) = -2, y'(0) = 0, u(t) is the step function. 1. Write an expression for Y(s); at first leave U(s) symbolic. Identify which part is the zero-state and which part is the zero-input frequency-domain solution. Identify which part is the transfer function and which part is the initial condition polynomial. You will need to use the following transform pairs or properties, noting that they apply to the input as...