Question

Compute by hand and plot the Zero State response for a Unit Impulse and the Zero Input response for the initial state value X(0)=[1; 1;1] for the system: dX/dt=AX+Bu where A=[2 0 0; 0 0 1; 0 -2 -3]; B= [0; 0;1];

Answer 1

Compute the Zero Input response using the homogeneous equation with initial conditions and determine the Zero State response to the unit impulse input, especially considering the third state affected by the impulse. Plot these responses over the indicated time intervals.

Step-by-step explanation:

To compute by hand the Zero State response for a Unit Impulse and the Zero Input response for the given system dX/dt=AX+Bu where A=[2 0 0; 0 0 1; 0 -2 -3] and B= [0; 0;1], and initial condition X(0)=[1; 1; 1], we carry out the following steps:

1. Determine the zero input response by solving the homogeneous equation (dX/dt=AX) with the given initial conditions X(0).
2. For the zero state response, we need to calculate the response to the unit impulse input. However, given the matrix B does not affect the first two states, the impulse response in this case only affects the third state. Therefore we find the response by considering the effect of the impulse input on the third state alone.
3. Finally, we plot the Zero State response and Zero Input response over the given time intervals. Since you've provided a time-based instruction for ū to be equal to AX, it implies that we are looking at the response when the system is driven by its own dynamics, and so, we need to plot how X evolves over time without any extra input once the initial impulse has been applied.