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Q.1. Consider the mechanical system shown in the figure below. Here u (t) is an external force applied to the mass M, y (t) is the displacement of the mass with respect to the position when the spring is relaxed. The spring force and friction force are given respectively by

Homework assignment 1 (due in one week)

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Q.2. Consider the electric circuit network in the figure below. Let the input be vi(t) and output be vo(t).

1. Derive the state and output equation of the network.

2. Find the transfer function of the network.

Assuming that R1 =R2 =R3 =1, C1 =C2 =1 F and L1 =1H,

3. Find the unit step response of the network.

4. Find the unit impulse response of the network.

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Q.3.

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Q.4.

Q.5.

Q.6. Show that the pendulum system is a BIBO unstable system even though it was proved to be internally marginally stable. Identify a bounded input signal such that when it is applied to the pendulum, the resulting output response will go unbounded.

For simplicity, you can assume that 2 1 and .M L g L

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Homework assignment 2 (due in one week)

Q.1. It was shown in the section of dynamic modeling that the two-cart system can be described by the following state space model:

(a)

(b)

1 1

1 1

2 2

2 2

0 1 0 0 01

,0 0 0 1 0

0

x xx xK F K F

fx xx xK F K F

1

1

2

2

0 0 1 0

xx

yxx

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Q.2.

Q.3.

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Q.4.

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Q.5. Verify the result in Q.4 for the following systems:

Q.6. Given ARn, BRm, show than if the pair (A, B ) is controllable (detectable) if and only if ( AT, BT) is observable (stabilizable).

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Homework assignment 3 (due in two weeks)

Q.1. It was showed earlier that the invariant zeros of linear systems are invariant under state feedback. More specifically, for a system characterized by

with a state feedback u = F x + v, it gives a closed-loop system

We have showed that if a scalar is an invariant zero of the original system, it is also an invariant zero of the new one as well.

x A x Buy C x Du

( )( )

x A BF x B vy C DF x D v

(a) Show that the state feedback law does not change the controllability property of the given system either.

(b) Show by a simple example that the state feedback law, however, may change the observability property of the given system.

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Q.2.

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Q.3.

Q.4.

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Q.5. Given an unsensed system characterized by a matrix pair in the CSD form

Let the output equation be given as . Verify that the resulting system has

(a) No invariant zero if C = [ 1 0 0 ];

(b) One invariant zero if C = [ 0 1 0 ]; and

(c) Two invariant zero if C = [ 0 0 1 ].

Q.6. Given the matrix pair (A, B) as that in Q.5, determine an appropriate state feedback gain matrix F such that A +B F has its eigenvalues at –1, –1 j, respectively. Show that such an F is unique.

Show by an example that solutions to the pole placement problem for a multiple input system is non-unique. Hint: put the pair in the CSD form.

0 1 0 0, with 0 0 1 , 0

2 1 1 1x A x Bu A B

y C x