Question 1: The ability to balance actively is a key ingredient in the mobility of a device that hops and runs on the one springy leg, as shown in gure. The control of attitude of device uses a gyroscope and a feedback such that u (t) = Kx (t) ; where
K =
k
0
and
0
2k
x (t) = Ax (t) + Bu (t)
where
A =
0
1
;B=I
1
0
Determine K so that response of the system is critically damped? (Use = 1)
Question 2: A system is described by the equations
x (t) =
0
5
x (t) +
2
u (t)
0
1
0
y (t) = 0 2 x (t)
determine controllabilty and observability.
Question 3: A system is described by the equations
x (t) =
1
2
x (t) +
1
u (t)
0
1
0
y (t) = 1 0 x (t)
determine controllabilty and observability.
Question 4: Hydraulic power actuators were used to drive the dinosaurs of the movie Jurassic park, the motions of the large monsters required high power actuators requiring 1200 watts. One speci c limb motion has dynamics represented by
x (t) =
1
1
x (t) +
0
u (t)
4
0
1
y (t) = 0 1 x (t) + 0 u (t)
We want to place the closed loop poles at S = 1 j3. Determine the required state variable feedback using any method (ackermans formula or coe cient comparison). Assume that complete state vector is available for feedback.
Question 5: Consider
x (t) = AX (t) + BU (t)
y (t) = CX (t) + DU (t)
Where;
5
10 ;B=
1 ;
A =
1
4
0
C=1 4;D=0
a) Comment on observability.
b) Design full state observer placing poles at S1;2 =
1.
c) Plot
the response
of the estimations error e (t)
= x (t) x^ (t) with initial estimation error
e (0) =
1 1 0 using Matlab and attach your result.
Question 6: A system is described by the equations
x (t) =
5
20
x (t) +
0
u (t)
1
0
100
y (t) = 1 0 x (t)
Determine observer gains to place the observer poles at S1;2 = 5.
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