Question

A system has the following model y+ 2y – 8y = 4u a. (15 points) Find the free response if y(0=1 and 9(0=0. b. (15 points) Find the forced response that results from a unit ramp input (u(t)=t). c. (5 points) Is the system stable, unstable, or marginally stable? Justify your answer.

Answer 1

To solve the given differential equation, we use the initial conditions to find the free response. For the forced response, we substitute the unit ramp input into the equation. The stability of the system can be determined by analyzing the roots of the characteristic equation.

Step-by-step explanation:

To find the free response, we can solve the differential equation y'' + 2y' - 8y = 4u. Given that y(0) = 1 and y'(0) = 0, we can solve this equation using the initial conditions. To find the forced response resulting from a unit ramp input, we can substitute u(t) = t into the differential equation and solve for y. To determine the stability of the system, we need to analyze the roots of the characteristic equation.

As mentioned earlier, the characteristic roots are r1 = -4 and r2 = 2. The system is stable if the real parts of all characteristic roots are negative. In this case, both roots have negative real parts, so the system is stable. The forced response is the response of the system due to the input (in this case, u(t) = t). We can find it using the method of undetermined coefficients.

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