Alternates from 10 to 11 until the change signal comes up. Then alternates from 8 to 9 until the change sinal comes up again.
Despite having an external input, this is still a Moore machine, as the input only affects the state transition
C - change signal
A - current state bit 1
B - current state bit 0
Y - next state
S - output
Y1 = A*!B + A*!C + !A*B
Y0 = !B
S0 = B
S1 = !A
S2 = 0
S3 = 1
Logic table:
C AB Y S
0 00 01 1010
0 01 00 1011
0 10 11 1000
0 11 10 1001
1 00 01 1010
1 01 10 1011
1 10 11 1000
1 11 00 1001