The document provides an overview of state-space representation of linear time-invariant (LTI) systems. It defines key concepts such as state variables, state vector, state equations, and output equations. Examples are given to show how to derive the state-space models from differential equations describing dynamical systems. Specifically, it shows how to 1) select state variables, 2) write first-order differential equations as state equations, and 3) obtain output equations to fully represent LTI systems in state-space form.
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Modern Control - Lec07 - State Space Modeling of LTI Systems
2. Agenda
State Variables of a Dynamical System
State Variable Equation
Why State space approach
Derive Transfer Function from State Space Equation
Time Response and State Transition Matrix
2
3. Introduction
The classical control theory and methods (such as root locus) that we
have been using in class to date are based on a simple input-output
description of the plant, usually expressed as a transfer function. These
methods do not use any knowledge of the interior structure of the
plant, and limit us to single-input single-output (SISO) systems, and as
we have seen allows only limited control of the closed-loop behavior
when feedback control is used.
Modern control theory solves many of the limitations by using a much
“richer” description of the plant dynamics. The so-called state-space
description provide the dynamics as a set of coupled first-order
differential equations in a set of internal variables known as state
variables, together with a set of algebraic equations that combine the
state variables into physical output variables.
3
4. Definition of System State
State: The state of a dynamic system is the smallest set of variables
(𝒙 𝟏, 𝒙 𝟐, … … , 𝒙 𝒏) (called State Variables or State Vector) such that knowledge of
these variables at 𝑡 = 𝑡0, together with knowledge of the input for 𝑡 ≥ 𝑡0 ,
completely determines the behavior of the system for any time t to t0 .
The number of state variables to completely define the dynamics of the system is
equal to the number of integrators involved in the system (System Order).
Assume that a multiple-input, multiple-output system involves n integrators (State
Variables).
Assume also that there are r inputs u1(t), u2(t),……. ur(t) and p outputs y1(t),
y2(t), …….. yp(t).
4
Inner state variables
nxxx ,, 21
)(1 tu
)(2 tu
)(tur
)(1 ty
)(2 ty
)(typ
8. Input/Output Models vs State-Space Models
State Space Models:
consider the internal behavior of a system
can easily incorporate complicated output variables
have significant computation advantage for computer simulation
can represent multi-input multi-output (MIMO) systems and nonlinear
systems
Input/Output Models:
are conceptually simple
are easily converted to frequency domain transfer functions that are more
intuitive to practicing engineers
are difficult to solve in the time domain (solution: Laplace transformation)
8
9. Some definitions
System variable: any variable that responds to an input or initial
conditions in a system
State variables: the smallest set of linearly independent system
variables such that the values of the members of the set at time t0
along with known forcing functions completely determine the value of
all system variables for all t ≥ t0
State vector: a vector whose elements are the state variables
State space: the n-dimensional space whose axes are the state variables
State equations: a set of first-order differential equations with b
variables, where the n variables to be solved are the state variables
Output equation: the algebraic equation that expresses the output
variables of a system as linear combination of the state variables and
the inputs.
10. General State Representation
1. Select a particular subset of all possible system variables, and call
state variables.
2. For nth-order, write n simultaneous, first-order differential equations
in terms of the state variables (state equations).
3. If we know the initial condition of all of the state variables at 𝑡0 as
well as the system input for 𝑡 ≥ 𝑡0, we can solve the equations
11. State-Space Representation of nth-Order Systems of Linear
Differential Equations
Consider the following nth-order system:
𝒚
(𝒏)
+ 𝒂 𝟏 𝒚
(𝒏−𝟏)
+ … + 𝒂 𝒏−𝟏 𝒚 + 𝒂 𝒏 𝒚 = 𝒖
where y is the system output and u is the input of the System.
The system is nth-order, then it has n-integrators (State Variables)
Let us define n-State variables
11
12. State-Space Representation of nth-Order Systems of Linear
Differential Equations (Cont.)
Then the last Equation can be written as
12
13. State-Space Representation of nth-Order Systems of Linear
Differential Equations (Cont.)
Then, the stat-space state equation is
where
13
14. State-Space Representation of nth-Order Systems of Linear
Differential Equations (Cont.)
Since, the output equation is
Then, the stat-space output Equation is
where
14
15. Example #1
From the diagram, the system Equation is
𝑀 𝑦 + 𝐵 𝑦 + 𝐾𝑦 = 𝑓(𝑡)
This system is of second order. This means that the system involves two
integrators (State Variables).
Let us define the state variables
𝑥1 = 𝑦
𝑥2 = 𝑦
Then, we obtain
𝑥1 = 𝑦 = 𝑥2
𝑥2 = 𝑦 =
1
𝑀
−𝐵 𝑦 + 𝐾𝑦 −
1
𝑀
𝑓 𝑡 =
−𝐵
𝑀
𝑥2 −
𝐾
𝑀
𝑥1 −
1
𝑀
𝑓 𝑡
15
y
K
M
B
f(t)
16. Example #1 (Cont.)
Then, the State Space equation is
𝑥1
𝑥2
=
0 1
−𝐾
𝑀
−𝐵
𝑀
𝑥1
𝑥2
+
0
1
𝑀
𝑓(𝑡)
The output Equation is
𝑦 = 1 0
𝑥1
𝑥2
The System Block diagram is
16
17. Example #2
17
LR
c)(tei )(tec
+
- )(ti
+
-
t
i tedtti
cdt
tdi
LtRi
0
)()(
1)(
)(
dttitx
titx
let
)()(
)()(
2
1
)()( tity
2
1
2
1
2
1
01)(
)(
0
1
01
1
x
x
ty
teLx
x
LCL
R
x
x
i
)()(ˆ
)()(ˆ
2
1
tetx
titx
let
c
)()( tity
2
1
2
1
2
1
ˆ
ˆ
01)(
)(
0
1
ˆ
ˆ
01ˆ
ˆ
x
x
ty
teL
x
x
L
L
R
L
R
x
x
i
Remark : the choice of states is not unique.
19. Example #4
Find the state space model for a system that described by the following
differential equation
Solution:
The system is 3rd order, then it has three states as follows
The output equation is
rcccc 2424269
cx 1
cx 2
cx 3
21 xx
32 xx
rxxxx 2492624 3213
1xcy
differentiation
22. State-Space Representation in Canonical Forms
We here consider a system defined by
where u is the control input and y is the output. We can write this
equation as
we shall present state-space representation of the system defined by
(1) and (2) in controllable canonical form, observable canonical form,
and diagonal canonical form.
22
23. Controllable Canonical Form
We consider the following state-space representation, being called a
controllable canonical form, as
Note that the controllable canonical form is important in discussing the
pole-placement approach to the control system design.
23
24. Observable Canonical Form
We consider the following state-space representation, being called an
observable canonical form, as
24
25. Diagonal Canonical Form
Diagonal Canonical Form greatly simplifies the task of computing the
analytical solution to the response to initial conditions.
We here consider the transfer function system given by (2). We have the
case where the dominator polynomial involves only distinct roots. For
the distinct root case, we can write (2) in the form of
25
26. Diagonal Canonical Form (Cont.)
The diagonal canonical form of the state-space representation of this
system is given by
26
27. Example #5
Obtain the state-space representation of the transfer function system
(16) in the controllable canonical form.
Solution: From the transfer function (16), we obtain the following
parameters: b0 = 1, b1 = 3, b2 = 3, a1 = 2, and a2 = 1. The resulting
state-space model in controllable canonical form is obtained as
27
28. Example #6
Find the state-space representation of the following transfer function
system (13) in the diagonal canonical form.
Solution: Partial fraction expansion of (13) is
Hence, we get A = −1 and B = 3. We now have two distinct poles. For
this, we can write the transfer function (13) in the following form:
28
30. The state space model
by Laplace transform
Then, the transfer function is
sBUsAXssX
sDUsCXsY
sBUAsIsX
1
BuAxx
DuCxy
sUDBAsICsY
1
DBAsIC
sU
sY
sT
1
State Space model to Transfer Function
31. Example (2)
Find the transfer function from the following transfer function
Solution:
uxx
0
0
10
321
100
010
xy 001
321
10
01
s
s
s
AsI
)det(
)(1
AsI
AsIadj
AsI
123
)12(
)3(1
13)23(
23
2
2
sss
sss
sss
sss
33. System Poles from State Space model
poles and check the stability of the following state space Example find the
System model
Solution:
Since
To find the poles
Then the poles are {-1, -2 }, the system is stable
uxx
0
5
31
20
xy 01
02)3(
31
2
ss
s
s
AsI
31
2
s
s
AsI
35. State-Space Modeling with MATLAB
MATLAB uses the controllable canonical form by default when converting from
a state space model to a transfer function. Referring to the first example
problem, we use MATLAB to create a transfer function model and then convert
it to find the state space model matrices:
35
36. State-Space Modeling with MATLAB
Note that this does not match the result we obtained in the first example. See
below for further explanation. No we create an LTI state space model of the
system using the matrices found above:
36
37. State-Space Modeling with MATLAB
we can generate the observable and controllable models as follows:
37
39. Introduction
The behavior of x(t) and y(t):
1) Homogeneous solution of x(t).
2) Non-homogeneous solution of x(t).
39
)()()(
)()()(
tDutCxty
tButAxtx
dt
d
67. Introduction
The main objective of using state-space equations to model systems is
the design of suitable compensation schemes to control these systems.
Typically, the control signal u(t) is a function of several measurable
state variables. Thus, a state variable controller, that operates on the
measurable information is developed.
State variable controller design is typically comprised of three steps:
Assume that all the state variables are measurable and use them to design a
full-state feedback control law. In practice, only certain states or
combination of them can be measured and provided as system outputs.
An observer is constructed to estimate the states that are not directly
sensed and available as outputs. Reduced-order observers take advantage of
the fact that certain states are already available as outputs and they don’t
need to be estimated.
Appropriately connecting the observer to the full-state feedback control law
yields a state-variable controller, or compensator. 67
68. Introduction
a given transfer function G(s) can be realized using infinitely many
state-space models
certain properties make some realizations preferable to others
one such property is controllability
68
69. Motivation1: Controllability
69
2
1
2
1
2
1
01
)(
0
1
10
12
x
x
y
tu
x
x
x
x
1
s 1
s 1
1 2
u y1x2x
s
x )0(2
s
x )0(1
1 1x2x
1
controllable
uncontrollable
70. Controllability and Observability
Plant:
Definition of Controllability
70
DuCxy
RxBuAxx n
,
A system is said to be (state) controllable at time , if
there exists a finite such for any and any ,
there exist an input that will transfer the state
to the state at time , otherwise the system is said to
be uncontrollable at time .
0t
01 tt )( 0tx 1x
][ 1,0 ttu )( 0tx
1x 1t
0t
71. Controllability Matrix
Consider a single-input system (u ∈ R):
The Controllability Matrix is defined as
We say that the above system is controllable if its controllability matrix
𝐶(𝐴, 𝐵) is invertible.
As we will see later, if the system is controllable, then we may assign
arbitrary closed-loop poles by state feedback of the form 𝑢 = −𝐾𝑥.
Whether or not the system is controllable depends on its state-space
realization.
71
BABAABBBAC
nCrankBA
n 12
),(
,)(leControllab,
72. Example: Computing 𝐶(𝐴, 𝐵)
Let’s get back to our old friend:
Here,
Is this system controllable?
72
74. Proof of controllability matrix
74
)1(
)2(
1
)1()2(1
21
)1()2(1
21
1
2
12
112
1
)(
nk
nk
k
n
k
n
nk
nknkk
n
k
n
k
n
nk
nknkk
n
k
n
k
n
nk
kkkkkkk
kkk
kkk
u
u
u
BABBAxAx
BuABuBuABuAxAx
BuABuBuABuAxAx
BuABuxABuBuAxAx
BuAxx
BuAxx
Initial condition
75. Motivation2: Observability
75
2
1
2
1
2
1
01
)(
1
3
10
02
x
x
y
tu
x
x
x
x
1
s 1
s 1
1 2
u y1x2x
s
x )0(2
s
x )0(1
1 1x2x
3
observable
unobservable
76. Controllability and Observability
Plant:
Definition of Observability
76
DuCxy
RxBuAxx n
,
A system is said to be (completely state) observable at
time , if there exists a finite such that for any
at time , the knowledge of the input and the
output over the time interval suffices to
determine the state , otherwise the system is said to be
unobservable at .
0t 01 tt )( 0tx
][ 1,0 ttu
],[ 10 tt
0x
0t
0t
][ 1,0 tty
77. Observability Matrix
77
Example: An Unobservable System
xy
uxx
40
1
0
20
10
※ State is unobservable.1x
1)(sU -1
s
-1
s 1x2x
2
4
)(sY
nVrankCA )(Observable, 0)det( V
1
2
MatrixityObservabil
n
CA
CA
CA
C
V
Ry if
78. Proof of observability matrix
78
)1()2()3(11
1
)1()2(1
321
1
111
111
1
)(),2(),1(
)(
)2()(
)1(
nknknkkkkkk
k
n
nknkk
n
k
n
k
n
nk
kkkkkkk
kkk
kkk
kkk
DuCBuCABuDuCBuyDuy
x
CA
CA
C
n
nDuCBuBuCABuCAxCAy
DuCBuCAxDuBuAxCy
DuCxy
DuCxy
BuAxx
Inputs & outputs
79. Example
Plant:
Hence the system is both controllable and observable.
79
10,
1
0
,
01
10
CBA
DuCxy
RxBuAxx n
,
01
10
MatrixtyObervabili
01
10
MatrixilityControllab
CA
C
N
ABBV
2)()( NrankVrank
80. Controllability and Observability
80
Theorem I
)()()( tuBtxAtx cccc
Controllable canonical form Controllable
Theorem II
)()(
)()()(
txCty
tuBtxAtx
oo
oooo
Observable canonical form Observable
A system in Controller Canonical Form (CCF) is always controllable!!
A system in Observable Canonical Form (OCF) is always controllable!!
81. Example
81
c
cc
xy
uxx
12
1
0
32
10
Controllable canonical form
12
12
31
10
CA
C
V
ABBU
nVrank
nUrank
1][
2][
o
oo
xy
uxx
10
1
2
31
20
Observable canonical form
31
10
11
22
CA
C
V
ABBU
nVrank
nUrank
2][
1][
)2)(1(
2
)(
ss
s
sT
82. Linear system (Analysis)
82
Theorem III
)()()(
)()()(
tDutCxty
tButJxtx
Jordan form
321
3
2
1
3
2
1
CCCC
B
B
B
B
J
J
J
J
Jordan block
Least row
has no zero
row
First column has no zero column
87. Kalman Canonical Decomposition
Diagonalization: &
All the Eigenvalues of A are distinct, i.e.
There exists a coordinate transform such that
System in z-coordinate becomes
Homogeneous solution of the above state equation is
87
BuAxx DuCxy
n 321
Txz
.
0
0
where
1
1
n
mm AATTA
zCy
uBzAz
m
mm
)0()0()( 11
1
n
t
n
t
zevzevtz n
mnmm
mn
m
m
ccCTC
b
b
BTB
1
1
1
observableandlecontrollabismode0,and0If i mimi cb
88. How to construct coordinate transformation matrix for diagonalization
All the Eigenvalues of A are distinct, i.e.
The coordinate matrix for diagonalization
Consider diagonalized system
88
][ 21 n,v,, vvT
t.independenare,rs,Eigenvecto 21 n,v,, vv
n 321
ubzλz
ubzz
ubzz
mnnnn
m
m
2222
1111
nmnmm zczczcy 2211
89. Transfer function is
H(s) has pole-zero cancellation.
89
n
mnmnmm
n
i i
mimi
s
bc
s
bc
s
bc
sH
1
11
1
)(
le,unobservaborableuncontrollismode0,or0If i mimi cb
1
1mc1mb
2
2mc2mb
n
mncmnb
∑
)(tu
)(ty
91. Kalman Canonical Decomposition: State Space Equation
91
xCCy
u
B
B
x
x
x
x
A
A
A
A
x
x
x
x
OCCO
OC
CO
OC
OC
OC
CO
OC
OC
OC
CO
OC
OC
OC
CO
00
0
0
000
000
000
000
(5.X)
93. Pole-zero Cancellation in Transfer Function
From Sec. 5.2, state equation
may be transformed to
93
n
mnmnmm
n
i i
mimi
s
bC
s
bC
s
bC
sH
1
11
1
)(
Hence, the T.F. represents the controllable and observable
parts of the state variable equation.
BuAxx
DuCxy
T.F..invanishesandableuncontrollismode,0If imib
T.F..invanishesandleunobservabismode0,If imic
94. Example
Plant:
Transfer Function
94
BAsICsH
sU
sY 1
)(
)(
)(
4
1
2
22
10
42
1
1
2
41
02
10
42
1
s
s
s
ss
s
s
ss
4,2 21
xy
uxx
10
1
2
21
04
T.F..invanishes"-2"Mode
95. Example 5.6
Plant:
Transfer Function
95
uxx
1
2
11
60
xy 10
3
1
)(
)(
)( 1
s
BAsICsT
sU
sY
T.F..invanishes"2"Mode
-3,2 21
96. Minimum Realization
Realization:
Realize a transfer function via a state space equation.
Example
Realization of the T.F.
Method 1:
Method 2:
There is infinity number of realizations for a given T.F. .
96
3
1
)(
s
sT
1 1)(sU )(sY
3
1 1)(sU )(sY-1
s
3
2
-1
s
-1
s
1
3
1
)(
)(
)(
s
sT
sU
sY
2
2
3
1
)(
)(
)(
s
s
s
sT
sU
sY
97. Minimum Realization
Minimum realization:
Realize a transfer function via a state space equation with elimination of its
uncontrollable and unobservable parts.
Example 5.8
Realization of the T.F.
97
3
5
)(
)(
)(
s
sT
sU
sY
1 5)(sU )(sY
3
-1
s
3
5
)(
s
sT