A House for Horses

Consider the following system (call it {\cal H}):

Part 1

For what values of G is this system stable?

Minimum stable G: 

Maximum stable G: 

Part 2

Now, imagine we have a value of G outside the stable range above. We can attempt to stabilize the system {\cal H} by hooking it up as follows:

If G = 1.5, for what values of k is the composite system above stable?

Minimum stable k for G=1.5: 

Maximum stable k for G=1.5: 

Part 3

If G = 1.5, for what value of k does the system's unit sample response converge most quickly? Enter your answer as a Python number.

k for fastest convergence with G=1.5: 

Part 4

Note that it might be possible to choose G such that no value of k will cause the system to stabilize.

For what values of G is it possible to stabilize the system by adjusting k?

Minimum G for which system can be stabilized: 

Maximum G for which system can be stabilized: