exercises 05

Příklady k přednášce 5 Exercises for Lecture 5 1. Consider the state equation . 1 1 0 0 1 0 ) ( ), ( 1 0 0 0 0 0 1 0 1 ) 1 ( ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = + k y k x k x Determine all states that are (a) unobservable and (b) unconstructible. 2. Suppose that for the system ) ( 1 0 0 1 1 0 ) ( ), ( 1 0 0 0 1 0 0 1 1 ) 1 ( k x k y k x k x ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = + it is known that . 1 1 ) 3 ( ) 2 ( ) 1 ( ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = = = y y y Based on this information, what can be said about the initial condition x(0)? 3. Consider the system , where , Cx y Ax x = = [ ] . 0 1 0 , 1 1 0 1 2 1 1 1 0 = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − = C A Which eigenvalues of the system are unobservable? 4. Consider the state equation , , Cx y Bu Ax x = + = where , 0 1 0 0 0 0 0 1 , 1 0 0 0 0 1 0 0 , 0 0 2 0 1 0 0 0 2 0 0 3 0 0 1 0 2 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = C B w w w A which is obtained by linearizing the nonlinear equations of motion of an orbiting satellite about a steady-state solution. In this equation, w is a nonzero constant. In the state T , x 1 is the differential radius, while x 3 is the differential angle. In the input vector T , u 1 is the radial thrust and u 2 is the tangential thrust. The output vector T comprises the measurable state variables x 1 and x 3 . ] [ 4 3 2 1 x x x x x = u = [ 2 1 y y y = ] [ 2 1 u u ] (a) Is this system controllable from u? Is the system observable from y? (b) Can the system be controlled if the radial thruster fails? What if the tangential thruster fails? (c) Is the system observable from y 1 only? From y 2 only?

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Příklady k přednášce 5 Exercises for Lecture 5 1. Consider the state equation

.110010

)(),(100000101

)1( ⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=+ kykxkx

Determine all states that are (a) unobservable and (b) unconstructible. 2. Suppose that for the system

)(100110

)(),(100010011

)1( kxkykxkx ⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=+

it is known that

.11)3()2()1( ⎥⎦⎤

⎢⎣⎡=== yyy

Based on this information, what can be said about the initial condition x(0)?

3. Consider the system , where , CxyAxx ==

[ ].010,

110

121

110

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

= CA

Which eigenvalues of the system are unobservable?

4. Consider the state equation ,, CxyBuAxx =+= where

,01000001

10000100

00201000

20030010

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

= CB

wwA

which is obtained by linearizing the nonlinear equations of motion of an orbiting satellite about a steady-state solution. In this equation, w is a nonzero constant. In the state

T, x1 is the differential radius, while x3 is the differential angle. In the input vector T, u1 is the radial thrust and u2 is the tangential thrust. The output vector T comprises the measurable state variables x1 and x3.

][ 4321 xxxxx =u =

[ 21 yyy =][ 21 uu

]

(a) Is this system controllable from u? Is the system observable from y? (b) Can the system be controlled if the radial thruster fails? What if the tangential thruster

fails? (c) Is the system observable from y1 only? From y2 only?