Loading Document1-a Define the controllability and observability.
A system is said to be controllable at time t0 if it is possible by means of a control signal to transfer the system from any initial state x(t0) to any other state in a finite interval of time.
A system is said to be output controllable if it is possible to construct a control signal u(t) that will transfer any given initial output y(t0) to any final output y(t1) in a finite time interval.
A system is said to be observable at time t0 if it is possible to determine the initial state x(t0) from the observation of the output over a finite time interval.
1-c For the system shown
x˙=⎣⎡acbd⎦⎤x+⎣⎡11⎦⎤u and y=[10]x
Determine the conditions on a,b,c and d. so that the system is completely controllable and observable.
controllability condition ∣∣∣BAB∣∣∣=0
A=⎣⎡acbd⎦⎤,B=⎣⎡11⎦⎤
∣∣∣∣∣∣11a+bc+d∣∣∣∣∣∣=(c+d)−(a+b)
the complete controllability condition is (a+b)=(c+d)
observability condition ∣∣∣C∗A∗C∗∣∣∣=0
A∗=⎣⎡abcd⎦⎤,C∗=⎣⎡10⎦⎤
∣∣∣∣∣∣10ab∣∣∣∣∣∣=b
the observability condition is b=0
condition for the system to be completely controllable and observable:
(a+b)=(c+d) and b=0
4-c For the system shown
x˙=⎣⎡1011⎦⎤x+⎣⎡k1k2⎦⎤u and y=[c1c2]x
Determine the conditions on k1,k2,c1 and c2. so that the system is completely controllable and observable.
controllability condition ∣∣∣BAB∣∣∣=0
A=⎣⎡1011⎦⎤,B=⎣⎡k1k2⎦⎤
∣∣∣∣∣∣k1k2k1+k2k2∣∣∣∣∣∣=k1k2−(k1+k2)k2
the complete controllability condition is k1k2−(k1+k2)k2=0⟹k2=0
observability condition ∣∣∣C∗A∗C∗∣∣∣=0
A∗=⎣⎡1101⎦⎤,C∗=⎣⎡c1c2⎦⎤
∣∣∣∣∣∣c1c2c1c1+c2∣∣∣∣∣∣=(c1+c2)c1−c1c2=0
the observability condition is c1=0
condition for the system to be completely controllable and observable:
k2=0 and c1=0