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i need this answerplease teach me this is ece515

i  need this answer?please teach me? this is ece515

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




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.






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





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? ?),






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




? = 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.





2s2 ?3s+1


s3 +s+2






s3 +2s2 ?3s+1


s3 +s+2






s4 +2s2 ?3s+1


s3 +s+2







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