Cvičení k přednášce 2 - DCE FEL ČVUT v Praze

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Cvičení k přednášce 2 - DCE FEL ČVUT v Praze

Cvičení k přednášce 2

Exercises for Lecture 2

1. Determine the matrix A 100 for

using two different methods.

2. Calculate At

e for

using two different methods.

3. Given is the system of equations,

⎡ x

1 ⎤

⎢ =

x


⎣ 2


⎡1

A = ⎢

⎣0

⎡1

A =



0

⎢⎣

0

⎡−


⎣ 0

4

2

0

1 ⎤

−1



1 1

10⎤

0



.

2 ⎥⎦

0⎤

⎡ x ⎤ ⎡ 1 ⎤

+ u

1

⎥ ⎢

x

⎥ ⎢ ⎥

⎦ ⎣ 2 ⎦ ⎣−

1⎦

with u( t)

= 1(

t)

, the unit step. Calculate the components of the solution for different initial

conditions

⎡ ⎤

= ⎢ ⎥

⎣b⎦

a

x ( 0)

and investigate the changes in the asymptotic behavior of solutions.

4. Consider the system x

= Ax + Bu,

y = Cx + Du with B = 0, D = 0 and

⎡−1

1 0⎤

A =


0 1 0



− , C = [ 1 1 1].

⎢⎣

0 0 2⎥



= −t

Select x(0) in such a manner that y(

t)

te

, t ≥ 0.

5. Determine the discrete-time system

x ( k + 1)

= A x(

k)

+ Bu

( k)

, y(

k)

= Cx

( k)

+ Du

( k)

that is obtained by skew sampling of the continuous-time system

x

= Ax + Bu,

y = Cx + Du

given by


with period T and shift α.

⎡ 0 1⎤

⎡0⎤

A = ⎢ = =

1 0

⎥ , B ⎢

1

⎥ , C D

⎣−

⎦ ⎣ ⎦

[ 1 0]

, = 0

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