The response of a first-order system is written as x(t) = 12e-3t+a, where a = 24....
Problem 5a (10 points): In class, you have derived the response of a first-order system to a unit-step input. Given a first-order system of the form G(s) = K / ( 1 + T s), where T is the time-constant, and K is the constant, find i) The time-response to a unit-ramp input r(t) = t. ii) The steady-state error for error measured as e(t) = r(t) - c(t). (Hint: the steady-state error is measured as t tends to infinity).
5. The solution to the system X-Ax can be written as etAX(O), where the matrix function etA İS defined by the series etAA)tA) Given the matrix tA- etA (part of Lebl 3.8.2)? 3t t t 3t , what is 3! 2! 5. The solution to the system X-Ax can be written as etAX(O), where the matrix function etA İS defined by the series etAA)tA) Given the matrix tA- etA (part of Lebl 3.8.2)? 3t t t 3t , what is...
Question One (a) The Impulse Response of a second order system is given by h(t) where: h (t) 4000 e 3000 e20 where the time, t, is given in milliseconds (ms) and h(t) is considered to be the resulting voltage in volts. (0) Derive the Transfer Function, the Laplace Transform H(s) of h(t). (i) Using part (0, write out the Frequency Response, H(jo), of the second order (ii) Express the Frequency Response obtained in part (i) as a single response...
2). Solve the following first order system response to the ramp function (2t) as described in the equation below. (The initial condition is zero). Derive an expression for the steady state error. At what time does it reach 98% of its value? Plot a graph containing the input and system response. 4x + 2x = 21 2). Solve the following first order system response to the ramp function (2t) as described in the equation below. (The initial condition is zero)....
3. The input and output corresponding to the steady (vibratory) response of a first order system have been recorded below. (a) Use the phase difference to find the time constant of the system. (b) Use the magnitude ratio between both curves to find the time constant of the system and compare your answer to that of part (a) if the equation is known to be ai + bx = Csin(wt) where the input is Csin(wt), the output is x, and...
Consider the linear system of first order differential equations x' = Ax, where x= x(t), t > 0, and A has the eigenvalues and eigenvectors below. 4 2 11 = -2, V1 = 2 0 3 12 = -3, V2= 13 = -3, V3 = 1 7 2 i) Identify three solutions to the system, xi(t), xz(t), and x3(t). ii) Use a determinant to identify values of t, if any, where X1, X2, and x3 form a fundamental set of...
1. The change of position of the center of mass of a rigid body in a mechanical system is being monitored. At time t 0, when the initial conditions of the system were x = 0.1 m and x -0m/s, a step input of size 10 N began to apply to the system. The response of the system was represented by this differential equation: 2r + 110x + 500 x = 10 a) Write the order of the system, its...
Problem 2.31: Please complete all of the following Problem 2.31: An underdamped mass-spring-dashpot system is subject to a periodic force F(t) of a period T and a saw-tooth form, as shown in Fig. P2.31. Assume ζ 0.1. AF(t) T" 2T 3T Figure P2.31 Periodic loading of saw-tooth shape (a) Obtain the Fourier series expansion for the force. (b) Find the Fourier series expansion of the system's steady-state response. (c) For T/T, = 0.5, where T, is the natural period of...
5. Fourier Transform and System Response (12 pts) A signal æ(t) = (e-t-e-3t)u(t) is input to an LTI system T with impulse response h(t) and the output has frequency content Y(jw) = 3;w – 4w2 - jw3 (a) (10 pts) Find the Fourier transform H(jw) = F{h(t)}, i.e., the frequency response of the system. (b) (2 pts) What operation does the system T perform on the input signal x(t)?
Question One (a) The Impulse Response of a second order system is given by h(t) where: h(t) 4000e 3000 c0, where the time, t, is given in milliseconds (ms) and h(t) is considered to be the resulting voltage in volts. 0) Derive the Transfer Function, the Laplace Transform H(s) of h(t). () Using part (0). write out the Frequency Response, HGo), of the second order (ii) Express the Frequency Response obtained in part (i) as a single response system. and...