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Problem Set # 1

 

ECE 515: Control System Theory and Design

 

Instructor: Sadegh Bolouki, [email protected]

 

Teaching Assistant: Ivan T. Abraham, [email protected]

 

Due date: Thursday, Sep 1, 2016, 14:00

 

Readings

 

Sections 1.1, 1.2, 1.3, and 1.5 of

 

Tamer Ba¸sar, Sean Meyn, and William R. Perkins. Lecture Notes on Control system Theory

 

and Design. 2013.

 

Section 4.3 and Theorem 4.2 of

 

Chi-Tsong Chen. Linear system theory and design. 3rd edition. Oxford University Press.

 

1999.

 

Problems

 

1. For the network shown below,

 


 

(a) Find the transfer function.

 

(b) Find a state-space representation of the network.

 

(c) Specify whether the system is linear or nonlinear, and time-invariant or time-varying.

 

2. Exercise 1.7.7 of

 

Tamer Ba¸sar, Sean Meyn, and William R. Perkins. Lecture Notes on Control system Theory

 

and Design. 2013.

 

Consider a satellite in planar orbit about the earth. The situation is modeled as a point

 

mass m in an inverse square law force field, as sketched below. The satellite is capable of

 

1

 


 

thrusting (using gas jets, for example) with a radial thrust u1 and a tangential (? direction)

 

thrust u2 . Recalling that acceleration in polar coordinates has a radial component (¨

 

r ? r??2 ),

 

? Newton?s Law gives

 

and a tangential component (r?¨ + 2r? ?),

 

k

 

m(¨

 

r ? r??2 ) = ? 2 + u1

 

r

 

? = u2

 

m(r?¨ + 2r? ?)

 

where k is a gravitational constant.

 

(a) Convert these equations to (nonlinear) state-space form using x1 = r, x2 = r,

 

? x3 = ?,

 

?

 

and x4 = ?.

 

(b) Consider a nominal circular trajectory r(t) = r0 and ?(t) = ?0 t, where r0 and ?0

 

are constants. Assume the inputs are zero, i.e., u1 = u2 = 0, for any t ? 0. Obtain

 

expressions for the nominal state variables corresponding to the circular trajectory.

 

How do r0 and ?0 relate to k and m?

 

(c) Linearize the state equation obtained in (a) around the trajectory obtained in (b). Express the equation in matrix form in terms of r0 , ?0 , and m (eliminate k).

 

3. Which one(s) of the transfer functions below is realizable? Explain why.

 

(a)

 


 

2s2 ?3s+1

 

s3 +s+2

 


 

(b)

 


 

s3 +2s2 ?3s+1

 

s3 +s+2

 


 

(c)

 


 

s4 +2s2 ?3s+1

 

s3 +s+2

 


 

2

 


 

 







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