Problem 4 (Analytical and Computational-20 points) Given a second-order ordinary differential equation: d2f(t) df(t) with the following initial conditions: (O) 1 and ait 0 (Analytical-10 points)...
Matlab code for the following problems. Consider the differential equation y(t) + 69(r) + 5y( Q3. t)u(t), where y(0) (0)0 and iu(t) is a unit step. Deter- mine the solution y(t) analytically and verify by co-plotting the analytic solution and the step response obtained with the step function. Consider the mechanical system depicted in Figure 4. The input is given by f(t), and the output is y(t). Determine the transfer function from f(t) to y(t) and, using an m-file, plot...
Problem 3: Insights into Differential Equations a. Consider the differential equation 습 +4 = f(t), where f(t) = e-u, 12 0. Please write the forms of the natural and forced solution for this differential equation. You DO NOT need to solve. (7 points) b. Again consider the differential equation f(t), where f(t) is an input and y(t) is the output (response) of interest. Please write the differential equation in state-space form. (10 points) c. The classical method for solving differential...
Consider the linear system given by the following differential equation y(4) + 3y(3) + 2y + 3y + 2y = ů – u where u = r(t) is the input and y is the output. Do not use MATLAB! a) Find the transfer function of the system (assume zero initial conditions)? b) Is this system stable? Show your work to justify your claim. Note: y(4) is the fourth derivative of y. Hint: Use the Routh-Hurwitz stability criterion! c) Write the...
Problem 1 Given the circuit shown below in Fig. 1.1: Write the ordinary differential equation (ODE) for the capacitor voltage. Find the zero-state unit step responses of v(t) and i(t) if vs-u(t) V using each of the following three methods of solving the ODE: a. b. i. ii. Solve the ODE by integrating for the solution; Solve the ODE by assuming homogeneous and particular solutions; Solve the ODE by using the general form solution for a 1st order ODE. iii....
Question: given a differential equation: a. initial conditions for the plan and input are zero, derive plan's transfer function in Laplace transform b. using inverse Laplace transform, find the solution for the differential equation for the plan (find function y(t)). c. derive state-space model of the plan d. Assume open-loop system with no controller added to the plant, analyse the steady-state value of the system using final value theorem and step input e. Calculate value of the overshoot, rise time...
Please write and include a matlab code for the following: Given the following differential equation 10?̈+ 20?̇ + 250? = ?(?) a. Plot response to f(t) = 200sin(4t) b. Plot response to a step f(t) = 200 that starts at t = 0 s and stops at t = 2s; note that the response goes past the 2 seconds of input. Part a.) I need matlab code to generate the plot for 10x''+20x'+250x=200sin(4t) Part b.) I need matlab code to...
4. Problem 4. Consider the following system of first order coupled ordinary differential equations, where r (t) and a) Rewrite the initial value problem (IVP) in a matrix form aAi, where ? r (0) +v()() b) Find the three distinct (real) eįgrivalus {A] c) Verify that, satisfies the IVP where the constant ακ fficients c1 c2 and C3 can be detennined from the three given initial conditions. P BIVPn initial 5. Problem 5 (challenge problem): Sinultaneous diagonalization of commuting matrices...
The state space model of an interconnected three tank water storage system is given by the following equation: -3 1 0 1rh dt os lo 0 3] 10 1-3 The heights of water in the tanks are, respectively, h,h2,h3. Each tank has an independent input flow; the volume flow rates of input water into the three tanks are, respectively, qǐ1,W2,4a. Each tank also has a water discharge outlet and the volume flow rates of water coming out of the tanks...
Problem Thre: 125 points) Consider the following initial value problem: dy-2y+ t The y(0) -1 ea dt ical solution of the differential equation is: y(O)(2-2t+3e-2+1)y fr exoc the differential equation numerically over the interval 0 s i s 2.0 and a step size h At 0.5.A Apply the following Runge-Kutta methods for each of the step. (show your calculations) i. [0.0 0.5: Euler method ii. [0.5 1.0]: Heun method. ii. [1.0 1.5): Midpoint method. iv. [1.5 2.0): 4h RK method...
The state space model of an interconnected three tank water storage system is given by the following equation -3 1 o 1[hi] dt The heights of water in the tanks are, respectively, hi, h2, hz. Each tank has an independent input flow; the volume flow rates of input water into the three tanks are, respectively, qii,qi2, Oi3. Each tank also has a water discharge outlet and the volume flow rates of water coming out of the tanks are, respectively, qo1,...