Digital Signal Processing © Dr. Fred J. Taylor, Professor
STATE_Determined_SYS  1
Lesson Title: State Determined Systems
Lesson Number [17] / Lecture [19]
What’s it all about?
ã How do state variables support the analysis of linear systems?
ã What is a state transition matrix?
State Variable Model
The singleinput singleoutput Nth order LTI system shown in Figure 1 can be modeled in terms
of a state 4tuple [A, b, c, d] which defines an N
th
order state variable model:
Digital Signal Processing
Dr. Fred J. Taylor, Professor
STATE_Determined_SYS  1
Lesson Title: State Determined SystemsLesson Number [17] / Lecture [19]
What’s it all about?
ã
How do state variables support the analysis of linear systems?
ã
What is a state transition matrix?
State Variable Model
The singleinput singleoutput
N
th order LTI system shown in Figure 1 can be modeled in termsof a state 4tuple [
A
,
b
,
c
, d] which defines an
N
th
order state variable model:State equation:][][]1[
k u k k
bAxx
+=+
1Initial condition:
0
]0[
xx
=
2Output equation: ][][][
k du k k
T
+=
xcy
. 3where
x
[
k
] is the state
n
vector,
u
[
k
] is the input,
y
[
k
] is the output,
A
is an
n
x
n
state matrix,
b
isan 1x
n
input vector,
c
is a 1xn output vector, and d is a scalar. At some sample index
k
=
k
0
, thenext state can be computed to be:][][]1[
000
k u k k
bAxx
+=+
,
4At sample instant
k
=
k
0
+1, the next state is given by:].1[][][]1+[]1[]2[
0002000
+++=++=+
k u k u k k u k k
bAbxAbAxx
5Continuing this pattern, at sample
k
=
k
0
+
n
–1 the next state vector is given by:].1+[]2[]1+[][][][
00020100
−+−+++++=+
−−
nk u nk u k u k u k nk
nnn
bAbbAbAxAx
L
6Equation 6 consists of two parts that define the homogeneous and inhomogeneous solutions.The homogeneous solution is driven only by the initial conditions(
x
[
k
0
]), whereas theinhomogeneous solution is forced by the input
u
[
k
]. Equation 6 can be rearranged andexpressed as:
∑
−=−−−
+=
110
00
][][][
k k i i k k k
i u k k
bAxAx
. 7
Digital Signal Processing
Dr. Fred J. Taylor, Professor
STATE_Determined_SYS  2Upon substituting Equation 7 into the output equation (Equation 3), the generalized output of thestate determined system is:
++=+=
∑
−=−−−
110
00
][][][][][][
k k i i k T k k T T
k du k u k k du k k
bAcxAcxcy
. 8For notational convenience, it shall be assumed that
k
0
=0. The term
ΦΦΦΦ
[
k
]=
A
k
, is called the statetransition matrix and is seen to influence both the homogenous and inhomogeneous solutions.After simplification, Equation 8 can be recast as:
[ ] [ ]
k u d i k k k
k i T T
[1]0[][
10
+−−Φ+Φ=
∑
−=
bcxcy
. 9The state transition matrix can be computed using CayleyHamilton method,
z
transform withthe later normally implemented using Leverrier’s algorithm. It is generally accepted, however,that computing y[k] using timedomain convolution methods, as described in Equation 9, iscomputationally intensive. Instead, transformdomain techniques are often championed.Observe that the
z
transform of Equations 1 and 3 are:
( ) ( ) ( )( ) ( ) ( )
z dU z z
z U z zx z z
T
==+=−
Xc YbAXX
0
10The homogenous response can be determined by setting
U
(
z
)=0 and solving for
Y
(
z
) inEquation 10. This results in:
( )( )
001001
)()(
;)()(
xcxAIc Y
xxAIX
z z z z
z z z z
T T
Φ=−=
Φ=−=
−−
. 11This can be seen to be consistent in form with the results given in Equation 9. Theinhomogeneous solution can be computed by setting
x
0
=
0
and solving for
Y
(
z
) using:).()()()(
)()()(
11
z dU z U z z Y
z U z z
T
+−=−=
−−
bAIc
bAIX
12The system’s transfer function is associated with the inhomogeneous solution of an
atrest
system, and is defined by
H
(
z
)=
Y
(
z
)/
U
(
z
). It therefore follows that:
d z z H
T
+−=
−
bAIc
1
)()(. 13
Ab c
T
Tdu[k]
Σ
x[k+1]x[k]y[k]
Σ
Figure 1: State Determined System
Digital Signal Processing
Dr. Fred J. Taylor, Professor
STATE_Determined_SYS  3Notice the similarity between the homogeneous solution (Equation 11) and the inhomogeneoussolution (Equation 13). The homogeneous response is defined in terms of the state transitionmatrix
( )
1
)(
−
−=Φ
AI
z z z
while the inhomogeneous solution is defined in terms of convolutionwith a term having the form
1
)(
−
−
AI
z
in the transform domain. Notice that
( )
z z z z z z
Φ=−=−
−−−−
1111
)(()(
AIAI
or matrix transition matrix delayed by one sample in thetime domain.
Example: Transfer Function
An atrest 2
nd
order linear timeinvariant (LTI) system is assumed to be given by the state 4tuple (
A
,
b
,
c
, d):1;11;10;1110
====
d
cbA
.Computing (
z
I
–
A
)
–1
:
( )
−−−=−
−
z z z z z
11111
21
AI
.Using Equation 10, the transfer function can then be determined and it is:
[ ]
.11111101111111=)()(
22221
−−=+−−+=+−−−+−=
−
z z z z z z z z z z d z z H
T
bAIc
Example: State Determined Systems
The response of a linear system, characterized by the state 4tuple [
A
,
b
,
c
, d], is modeled as:
[ ] [ ]
i bu k
k i i k
∑
−=−−
=
101
Ax
[ ] [ ]
][
101
k du i bu k
k i i k T
+=
∑
−=−−
Acy
In the zdomain. Tthe solution can be described as:
( )
)()(
1
z U zI z
bAX
−
−=
( ) ( )
( )
( ) ( ) ( )
z U z H z U d zI z dU z U zI z
T T
=+−=+−=
−−
bAcbAc Y
11
)()()(
Suppose a specific 2
nd
order system is given by:
[ ]
1;011.0;10;111.0
10
====
d c b A
The architecture associated with this manifestation [
A
,
b
,
c
,d] is shown in Figure 2.
Digital Signal Processing
Dr. Fred J. Taylor, Professor
STATE_Determined_SYS  4Figure 2: State induced architectureThe location of the system’s poles are defined by
∆
(
z
)=det(
z
I

A
)=det(
−−−
111.0
1
z z
)=
z
2

z
0.11=(
z
+0.1)(
z
1.1)Therefore the filter's poles are located at z=0.1 and z=1.1. The system is therefore seen to beunstable based on the unit circle criterion. Continuing, the state transition matrix denoted
Φ
(z) inthe zdomain, is given by:
( )( ) ( )( ) ( )( )( )( )
−−+=−=ΦΦΦΦ=Φ
−
z z z z z AzI z z z z z z
11.0111.11.0
22
1211211
Using Heaviside's method to inverting each term (4 required), the state transition in the timedomain is obtained for k
≥
0, and it is:
[ ]
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
[ ]
I k A
k k k k
k k k k
k
==Φ+−+−−
+−−+−
=Φ=
10010;1.112111.01211.1121.11.0121.11.112101.012101.11211.01211 The instability of the system can be immediately seen by the presence of the term (1.1)
k
.Suppose the system is initially
atrest
and the input to the atrest system is
u
[
k
]=
δ
[
k
]. Then
X
(
z
)= (
zI

A
)
1
bU
(
z
)= (
zI

A
)
1
b
and it follows that:
( ) ( )( )( )
( )
( )( )( )( )
[ ][ ]
00001111.11.0
1011.0111.11.0
1
221222112111
>=−Φ−Φ↔ΦΦ=−+==−−+=−=
−−−−
k if k if k k z z z z z z z z z z z z z b AzI z X
Z
For example, direct state computation at
k
=1 and 2, yields:u[k]b=1
Σ
x
2
[k+1]x
2
[k]x
1
[k+1]x
1
[k]
Σ
a
21
=0.11y[k]c
1
=0.11d=1
Σ
a
22
=1a
12
=1
TT