I have an LED strip with 5 red LEDs whose brightnesses I want to set. These LEDs are addressed as a queue: at each time step, I can push a new brightness command between 0 and 255 to the left-most LED Each of the following LEDs will then take on the brightness previously displayed by the LED mmediately to its left.

(a) What shoukd we use for our state vector? What does it mean that this is a state vector? What is our input?

(b) Is our systelinear? Assume that the input of the system will never exceed 255 or go below 0. If it is linear, write out the state equations in matrix form. Please choose a reasonable order for the state variables in the state vector.

(c) Is this system controllable? Explain intuitively what this system's controllability means in terms of LED brightnesses

(d) Is this system stable?

e) Starting from the pattern of brightnesses (from left to right) [0, 127, 0, 255, 0). can we maintain this pattern for all future time steps? Can we display any fixed pattern of brightnesses for all time?