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Find state equations of a discrete-time LTI system with system function
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STATE SPACE ANALYSIS
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[CHAP. 7
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From the definition of the system function [Eq. (4.4111
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we have (1
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+ a , z - ' + ~ , Z - ~ ) Y ( Z = ( 6 , + b,z-I + b )
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, ~ - ~ ) ~ ( z )
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Rearranging the above equation, we get
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Y ( z ) = - a , z - ' Y ( z ) - ~ , z - ~ Y ( + b o X ( z ) + b , z - I ~ ( z+ b , ~ - ~ ~ ( z ) z) )
from which the simulation diagram in Fig. 7-8 can be drawn. Choosing the outputs of unit-delay elements as state variables as shown in Fig. 7-8, we get
Y ~ = 9I , [ n l + b , x [ n l
q , [ n+ 1 1
-a,y[nI +q2bI +b,x[nl
In matrix form
Note that in the simulation diagram in Fig. 7-8 the number of unit-delay elements is 2 (the order of the system) and is the minimum number required. Thus, Fig. 7-8 is known as the canonical simulation of the first form and Eq. (7.85) is known as the canonical state representation of the first form.
Fig. 7-8 canonical simulation of the first form.
CHAP. 71
STATE SPACE ANALYSIS
7.10. Redo Prob. 7.9 by expressing H ( z ) as
H ( z =H,(z)H,(z )
Then we have
W ( z )+ a , z - ' W ( z ) Y ( z )= b , W ( z )
~ z - ~ W X z z) ) =( (
+ b , z - ' ~ ( z+ b , z - ' ~ ( z ) )
Rearranging Eq. (7.881, we get
W ( z ) = - a , z - ' W ( z ) - a , ~ - ~ ~ ( z( ) z ) +~
From Eqs. (7.89) and (7.90) the simulation diagram in Fig. 7-9 can be drawn. Choosing the outputs of unit-delay elements as state variables as shown in Fig. 7-9, we have
u l [ n+ l ] = I u , [ n + 11
I ~ [ ~ ]
- a , ~ ! , [ n- a , u , [ n ] + x [ n ] ]
Y [ ~ = b,u,[nI + b,u,[nI + b,u,[n I
+ 11
( b 2 - b o a 2 ) u , [ n l + ( 6 1- b , a , ) c 2 t n I +b&l
Fig. 7-9 Canonical simulation of the second form.
STATE SPACE ANALYSIS
[CHAP. 7
In matrix form
The simulation in Fig. 7-9 is known as the canonical simulation of the second form, and Eq. (7.91) is known as the canonical state representation of the second form.
7.11. Consider a discrete-time LTI system with system function
H(z)=
2 z 2 - 3z
Find a state representation of the system. Rewriting H( z ) as
Comparing Eq. (7.93)with Eq. (7.84) in Prob. 7.9, we see that
bo = 0 b, = f Substituting these values into Eq. (7.85) in Prob. 7.9, we get
a1 = - 2
=1 2
b2 = 0
7.12. Consider a discrete-time LTI system with system function
H ( z )=
2 z 2 - 32 + 1
(7.95)
2 ( z - l ) ( z - $)
Find a state representation of the system such that its system matrix A is diagonal. First we expand H ( z ) in partial fractions as
where Let Then or
(1 - p k z - ' ) Y k ( z ) = a k X ( z )
Y k ( z )= p k z - I Y k ( z )+ a k X ( z )
from which the simulation diagram in Fig. 7-10 can be drawn. Thus, H( z ) = HI(z ) + H2(z ) can
CHAP. 71
STATE SPACE ANALYSIS
Fig. 7-10
Fig. 7-11
be simulated by the diagram in Fig. 7-11 obtained by parallel connection of two systems. Choosing the outputs of unit-delay elements as state variables as shown in Fig. 7-11, we have
In matrix form
Note that the system matrix A is a diagonal matrix whose diagonal elements consist of the poles of M z ) .
7.13. Sketch a block diagram of a discrete-time system with the state representation
STATE SPACE ANALYSIS
[CHAP. 7
We rewrite Eq. (7.98) as
4 d n + 11 = q z b l q2[n + 11 = & l [ n ] + : q , [ n ] + x [ n ]
Y ~ = 3q,[nl - 2q2bI I
from which we can draw the block diagram in Fig. 7-12.
Fig. 7-12
STATE EQUATIONS OF CONTINUOUS-TIME LTI SYSTEMS DESCRIBED BY DIFFERENTIAL EQUATIONS 7.14. Find state equations of a continuous-time LTI system described by
y(t)
+3 j ( t )+2y(t)=x(t)
sdt)=y(t) 92(1) = ~ ( t )
(7.100)
Choose the state variables as
Then from Eqs. (7.100) and (7,101)we have
4 1 ( ~ )q 2 ( l ) =
42(t) = - 2 q l ( f ) -392(1) + x ( t ) Y ( t )= 4 d t )
In matrix form
CHAP. 71
STATE SPACE ANALYSIS
7.15. Find state equations of a continuous-time LTI system described by
y(t)
+ 3y(t) + 2 y ( t )= 4x(t) + x ( t )
(7.103)
Because of the existence of the term 4 1 ( t ) on the right-hand side of Eq. (7.103), the selection of y ( t ) and y ( t ) as state variables will not yield the desired state equations of the system. Thus, in order to find suitable state variables we construct a simulation diagram of Eq. (7.103) using integrators, amplifiers, and adders. Taking the Laplace transforms of both sides of Eq. (7.103),we obtain
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