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(Solved) ECE1653: Hybrid Systems and Control Applications Homework

Please help me to solve this assignment. There are two files in the attachment. HWK3 is the requirement for the homework, and the Lecture Notes is the information knowledge. Thanks for your help.

ECE1653: Hybrid Systems and Control Applications

Homework 3

March 16, 2016

Due Date: April 4, 2016

1. Let {w1 , . . . , wr } be a linearly independent set of vectors in Rn . Suppose we are given integers

p and q such that 1 ? q ? p ? r. De?ne the subspaces of Rn :

V1

=

{

v1 , . . . , v p }

V2

=

{

vq , . . . , v r } .

Show that V1 ? V2 ={ vq , . . . , vp }.

2. Prove the following lemma.

Lemma 1. Suppose (R1)-(R4) hold. Also suppose that B ? C(vr1 ) ? sp{b1 , . . . , br1 ?1 }, where

sp{b1 , . . . , br1 ?1 } is the unique minimal subspace containing B ? C(vr1 ) and generated by the

linearly independent vectors {b1 , . . . , br1 ?1 , br1 +1 , . . . , bm+1 }. There exists br1 ? B ? C(vr1 )

such that

br1 = c1 b1 + ? ? ? + cr1 ?1 br1 ?1 .

3. Consider the a?ne control system

x = Ax + Bu + a .

?

(1)

Suppose OS = co{v1 , . . . , v?+1 } and let IOS := {1, . . . , ? + 1}. Let B = sp{b1 , . . . , bm | bi ?

B ? C(vi )} be a maximal linearly independent set with respect to OS , in the sense discussed

in the lectures. De?ne the a?ne feedback transformation

u = K1 x + g1 + G1 w

where w is a new exogenous input. We obtain the new system

?

?

x = (A + BK1 )x + BG1 w + (Bg1 + a) =: Ax + Bu + a .

?

?

1

(2)

We assume K1 , G1 , and g1 are selected so that

?

Avi + a

?

=

0,

?

Avi + a

?

?

C(vi ) ,

B

=

[ b1 b2 ? ? ? bm ] .

i ? IOS

i ? {? + 1, . . . , n}

Prove or disprove the following statements:

?

(a) OS := {x ? Rn | Ax + a ? B}.

?

S

?

(b) If there exists u = Kx+g such that S ?? F0 for system (1), then there exists w = Kx+?

g

S

such that S ?? F0 for system (2).

4. Let S be determined by v0 = (0, 0), v1 = (0, 1) and v2 = (1, 0), and consider the a?ne

dynamics

x=

?

?3

1

0

?2

x+

?2

1

u+

1

1

.

(a) Find B, O, and OS .

(b) Compute B ? C(v0 ).

(c) Show that the RCP is not solvable by continuous state feedback.

(d) Compute the reach control indices.

(e) Using the algorithm shown in lecture, solve the RCP by discontinuous PWA feedback.

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in. Additionally, while you are allowed to discuss the problems with others, the expectation is that

the work performed will be your own, and you are solely responsible for your own work. For more

information about academic integrity, please visit http://academicintegrity.utoronto.ca/.

2

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