Problem 4 (10 Points) Consider a system described by ☺ + 2y – 3y = 1...
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 (25 points): Consider a system described by the differential equation: +0)-at)y(t) = 3ú(1); where y) is the system output, u) is the system input, and a(t)is a function of time t. o) (10 points): Is the system linear? Why? P(15 points): Ifa(t) 2, find the state space equations?
Given a system described by ODE y + 3y' + 2y = 30% + 2x, what is the transfer function for the system (Laplace Domain)? What is the impulse response of the system (Time Domain)? (10 points)
Problem 5. Consider the system = I-? – ry, = 3y - xy - 2y? Please answer the following questions. (a) Determine all critical points of the system of equations. (b) Find the corresponding linear system near each critical point. (c) Discuss the stability of each critical point for the nonlinear system.
Question 4 (a) A feedback control system with a proportional controller is shown in Figure Q4 (a). (i) Sketch the root locus of the system, (ii) Design the proportional controller (choose the value of K) such that the damping ratio does not exceed 0.5 and the time constant is less than 1 second. [All necessary steps of root locus construction and controller design must be shown). C(s) R(S) + s(s+4)(s + 10) Figure Q4 (a). A feedback control system [11...
(10 points) For the differential equation y(6) - 2y (5) – 3y(4) + 2y(3) + 10y" – 8y = 0. Find the fundamental solution set to the DE if the characteristic equation in factored form is given by (r – 2) (r2 + 2r + 2) (r - 1) (r + 1) = 0
Problem 4. (20 points): Consider a causal LTI system that is described by the difference equation Find the impulse response sequence h[n] by computing the system function H(S2)
For a system whose dynamics are expressed by the differential equation: 2° + 2y + 3y = 3u By selecting the sampling time 0.1 second and using the zero- order holder (ZOH), obtain the discrete time transfer function of the system: G (z) = Y (2) / U (Z).
Problem 7. [13 points; 4, 4, 5.] Consider the function f(r, y) 2y ln(r- ). (i) Find the unit direction of steepest increase for f at the point P (2, 1) (ii) Find the directional derivative of f at the point P(2,1) in the direction u = S (iii) Linearly approximate the value f((2,1)00) Problem 7. [13 points; 4, 4, 5.] Consider the function f(r, y) 2y ln(r- ). (i) Find the unit direction of steepest increase for f at...
Problem N° 1 114 points] Consider two spinless particles with orbital angular momenta quantum numbers l-1 and 122. If the state of the two- particle system is described by the wave function 4 (a). Find the constant A 12 points]. (b). Find the probability that, as a result of a measurement, the system is found in a state of the form |1 1>121>112 points). Problem N° 1 114 points] Consider two spinless particles with orbital angular momenta quantum numbers l-1...