State Variables Represent the Degrees of Freedom of Dynamic
Systems. The variables x
1
; ; x
n
are called the state variables of the dynamic
system de®ned by Equation 2.2. They are collected into a single n-vector
xtx
1
t x
2
t x
3
t ÁÁÁ x
n
t
T
2:3
called the state vector of the dynamic system. The n-dimensional domain of the state
vector is called the state space of the dynamic system. Subject to certain continuity
conditions on the functions f
i
and u
i
; the values x
i
t
0
at some initial time t
0
will
uniquely determine the values of the solutions x
i
t on some closed time interval
t Pt
0
; t
f
with initial time t
0
and ®nal time t
f
[57]. In that sense, the initial value of
each state variable represents an independent degree of freedom of the dynamic
system. The n values x
1
t
0
; x
2
t
0
; x
3
t
0
; ; x
n
t
0
can be varied independently, and
they uniquely determine the state of the dynamic system over the time interval
t Pt
0
; t
f
.
EXAMPLE 2.3: State Space Model of the Harmonic Resonator For the
second-order differential equation introduced in Example 2.2, let the state variables
x
1
d and x
2
_
d. The ®rst state variable represents the displacement of the mass
from static equilibrium, and the second state variable represents the instantaneous
velocity of the mass. The system of ®rst-order differential equations for this dynamic
system can be expressed in matrix form as
d
dt
x
1
t
x
2
t
F
c
x
1
t
x
2
t
;
F
c
01
À
k
s
m
À
k
d
m
"#
;
where F
c
is called the coef®cient matrix of the system of ®rst-order linear differential
equations. This is an example of what is called the companion form for higher order
linear differential equations expressed as a system of ®rst-order differential equa-
tions.
2.2.3 Continuous Time and Discrete Time
The dynamic system de®ned by Equation 2.2 is an example of a continuous system,
so called because it is de®ned with respect to an independent variable t that varies
continuously over some real interval t Pt
0
; t
f
. For many practical problems,
however, one is only interested in knowing the state of a system at a discrete set
of times t Pft
1
; t
2
; t
3
; g. These discrete times may, for example, correspond to the
times at which the outputs of a system are sampled (such as the times at which Piazzi
recorded the direction to Ceres). For problems of this type, it is convenient to order
the times t
k
according to their integer subscripts:
t
0
< t
1
< t
2
< ÁÁÁt
kÀ1
< t
k
< t
k1
< ÁÁÁ:
2.2 DYNAMIC SYSTEMS 29
That is, the time sequence is ordered according to the subscripts, and the subscripts
take on all successive values in some range of integers. For problems of this type, it
suf®ces to de®ne the state of the dynamic system as a recursive relation,
xt
k1
f xt
k
; t
k
; t
k1
; 2:4
by means of which the state is represented as a function of its previous state. This is
a de®nition of a discrete dynamic system. For systems with uniform time intervals Dt
t
k
kDt:
Shorthand Notation for Discrete-Time Systems. It uses up a lot of ink if
one writes xt
k
when all one cares about is the sequence of values of the state
variable x. It is more ef®cient to shorten this to x
k
, so long as it is understood that it
stands for xt
k
, and not the kth component of x. If one must talk about a particular
component at a particular time, one can always resort to writing x
i
t
k
to remove any
ambiguity. Otherwise, let us drop t as a symbol whenever it is clear from the context
that we are talking about discrete-time systems.
2.2.4 Time-Varying Systems and Time-Invariant Systems
The term ``physical plant'' or ``plant'' is sometimes used in place of ``dynamic
system,'' especially for applications in manufacturing. In many such applications, the
dynamic system under consideration is literally a physical plantÐa ®xed facility
used in the manufacture of materials. Although the input ut may be a function of
time, the functional dependence of the state dynamics on u and x does not depend
upon time. Such systems are called time invariant or autonomous. Their solutions
are generally easier to obtain than those of time-varying systems.
2.3 CONTINUOUS LINEAR SYSTEMS AND THEIR SOLUTIONS
2.3.1 Input±Output Models of Linear Dynamic Systems
The block diagram in Figure 2.1 represents a linear continuous system with three
types of variables:
Inputs, which are under our control, and therefore known to us, or at least
measurable by us. (In the next chapter, however, they will be assumed to be
known only statistically. That is, individual samples of u are random but with
known statistical properties.)
State variables, which were described in the previous section. In most
applications, these are ``hidden variables,'' in the sense that they cannot
generally be measured directly but must be somehow inferred from what can
be measured.
Outputs, which are those things that can be known through measurements.
These concepts are discussed in greater detail in the following subsections.
30 LINEAR DYNAMIC SYSTEMS
2.3.2 Dynamic Coef®cient Matrices and Input Coupling Matrices
The dynamics of linear systems are represented by a set of n ®rst-order linear
differential equations expressible in vector form as
_
xt
d
dt
xt
FtxtCtut; 2:5
where the elements and components of the matrices and vectors can be functions of
time:
Ft
f
11
t f
12
t f
13
t ÁÁÁ f
1n
t
f
21
t f
22
t f
23
t ÁÁÁ f
2n
t
f
31
t f
32
t f
33
t ÁÁÁ f
3n
t
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
f
n1
t f
n2
t f
n3
t ÁÁÁ f
nn
t
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
;
Ct
c
11
t c
12
t c
13
t ÁÁÁ c
1r
t
c
21
t c
22
t c
23
t ÁÁÁ c
2r
t
c
31
t c
32
t c
33
t ÁÁÁ c
3r
t
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
c
n1
t c
n2
t c
n3
t ÁÁÁ c
nr
t
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
;
utu
1
t u
2
t u
3
t ÁÁÁ u
r
t
T
:
The matrix Ft is called the dynamic coef®cient matrix, or simply the dynamic
matrix. Its elements are called the dynamic coef®cients. The matrix Ct is called the
input coupling matrix, and its elements are called input coupling coef®cients. The
r-vector u is called the input vector.
Fig. 2.1 Block diagram of a linear dynamic system.
2.3 CONTINUOUS LINEAR SYSTEMS AND THEIR SOLUTIONS 31
EXAMPLE 2.4: Dynamic Equation for a Heating/Cooling System Consider
the temperature T in a heated enclosed room or building as the state variable of a
dynamic system. A simpli®ed plant model for this dynamic system is the linear
equation
_
TtÀk
c
TtÀT
o
t k
h
ut;
where the constant ``cooling coef®cient'' k
c
depends on the quality of thermal
insulation from the outside, T
o
is the temperature outside, k
h
is the heating=cooling
rate coef®cient of the heater or cooler, and u is an input function that is either u 0
(off) or u 1 (on) and can be de®ned as a function of any measurable quantities.
The outside temperature T
o
, on the other hand, is an example of an input function
which may be directly measurable at any time but is not predictable in the future. It is
effectively a random process.
2.3.3 Companion Form for Higher Order Derivatives
In general, the nth-order linear differential equation
d
n
yt
dt
n
f
1
t
d
nÀ1
yt
dt
nÀ1
ÁÁÁf
nÀ1
t
dyt
dt
f
n
tytut2:6
can be rewritten as a system of n ®rst-order differential equations. Although the state
variable representation as a ®rst-order system is not unique [56], there is a unique
way of representing it called the companion form.
Companion Form of the State Vector. For the nth-order linear dynamic
system shown above, the companion form of the state vector is
xt yt;
d
dt
yt;
d
2
dt
2
yt; ;
d
nÀ1
dt
nÀ1
yt
T
: 2:7
Companion Form of the Differential Equation. The nth-order linear differ-
ential equation can be rewritten in terms of the above state vector xt as the vector
differential equation
d
dt
x
1
t
x
2
t
.
.
.
x
nÀ1
t
x
n
t
2
6
6
6
6
6
4
3
7
7
7
7
7
5
01 0ÁÁÁ 0
00 1ÁÁÁ 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
00 0ÁÁÁ 1
Àf
n
tÀf
nÀ1
tÀf
nÀ2
t ÁÁÁ Àf
1
t
2
6
6
6
6
4
3
7
7
7
7
5
x
1
t
x
2
t
x
3
t
.
.
.
x
n
t
2
6
6
6
6
6
4
3
7
7
7
7
7
5
0
0
.
.
.
0
1
2
6
6
6
6
4
3
7
7
7
7
5
ut:
2:8
32 LINEAR DYNAMIC SYSTEMS
When Equation 2.8 is compared with Equation 2.5, the matrices Ft and Ct are
easily identi®ed.
The Companion Form is Ill-conditioned. Although it simpli®es the relation-
ship between higher order linear differential equations and ®rst-order systems of
differential equations, the companion matrix is not recommended for implementa-
tion. Studies by Kenney and Liepnik [185] have shown that it is poorly conditioned
for solving differential equations.
2.3.4 Outputs and Measurement Sensitivity Matrices
Measurable Outputs and Measurement Sensitivities. Only the inputs and
outputs of the system can be measured, and it is usual practice to consider the
variables z
i
as the measured values. For linear problems, they are related to the state
variables and the inputs by a system of linear equations that can be represented in
vector form as
ztHtxtDtut; 2:9
where
ztz
1
t z
2
t z
3
t ÁÁÁ z
`
t
T
;
Ht
h
11
t h
12
t h
13
t ÁÁÁ h
1n
t
h
21
t h
22
t h
23
t ÁÁÁ h
2n
t
h
31
t h
32
t h
33
t ÁÁÁ h
3n
t
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
h
`1
t h
`2
t h
`3
t ÁÁÁ h
`n
t
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
;
Dt
d
11
t d
12
t d
13
tÁÁÁd
1r
t
d
21
t d
22
t d
23
tÁÁÁd
2r
t
d
31
t d
32
t d
33
tÁÁÁd
3r
t
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
d
`1
t d
`2
t d
`3
tÁÁÁd
`r
t
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
:
The `-vector zt is called the measurement vector, or the output vector of the
system. The coef®cient h
ij
t represents the sensitivity (measurement sensor scale
factor) of the ith measured output to the jth internal state. The matrix Ht of these
values is called the measurement sensitivity matrix, and Dt is called the input±
output coupling matrix. The measurement sensitivities h
ij
t and input=output
coupling coef®cients d
ij
t; 1 i `; 1 j r, are known functions of time. The
state equation 2.5 and the output equation 2.9 together form the dynamic equations
of the system shown in Figure 2.1.
2.3 CONTINUOUS LINEAR SYSTEMS AND THEIR SOLUTIONS 33
2.3.5 Difference Equations and State Transition Matrices (STMs)
Difference equations are the discrete-time versions of differential equations. They
are usually written in terms of forward differences xt
k1
Àxt
k
of the state variable
(the dependent variable), expressed as a function c of all independent variables or of
the forward value xt
k1
as a function f of all independent variables (including the
previous value as an independent variable):
xt
k1
Àxt
k
ct
k
; xt
k
; ut
k
;
or
xt
k1
ft
k
; xt
k
; ut
k
; 2:10
ft
k
; xt
k
; ut
k
xt
k
ct
k
; xt
k
; ut
k
:
The second of these (Equation 2.10) has the same general form of the recursive
relation shown in Equation 2.4, which is the one that is usually implemented for
discrete-time systems.
For linear dynamic systems, the functional dependence of xt
k1
on xt
k
and
ut
k
can be represented by matrices:
xt
k1
Àxt
k
Ct
k
xt
k
Ct
k
ut
k
;
x
k1
F
k
x
k
C
k
u
k
;
F
k
I Ct
k
;
2:11
where the matrices C and F replace the functions c and f, respectively. The matrix
F is called the state transition matrix (STM). The matrix c is called the discrete-time
input coupling matrix, or simply the input coupling matrixÐif the discrete-time
context is already established.
2.3.6 Solving Differential Equations for STMs
A state transition matrix is a solution of what is called the ``homogeneous''
3
matrix
equation associated with a given linear dynamic system. Let us de®ne ®rst what
homogeneous equations are, and then show how their solutions are related to the
solutions of a given linear dynamic system.
Homogeneous Systems. The equation
_
xtFtxt is called the homoge-
neous part of the linear differential equation
_
xtFtxtCtut. The solution
of the homogeneous part can be obtained more easily than that of the full equation,
and its solution is used to de®ne the solution to the general (nonhomogeneous) linear
equation.
3
This terminology comes from the notion that every term in the expression so labeled contains the
dependent variable. That is, the expression is homogeneous with respect to the dependent variable.
34 LINEAR DYNAMIC SYSTEMS
Fundamental Solutions of Homogeneous Equations. An n  n matrix-
valued function Ft is called a fundamental solution of the homogeneous equation
_
xtFtxt on the interval t P0; T if
_
FtFtFt and F0I
n
, the n Ân
identity matrix. Note that, for any possible initial vector x0, the vector
xtFtx0 satis®es the equation
_
xt
d
dt
Ftx0 2:12
d
dt
Ft
x02:13
FtFtx02:14
FtFtx0 2:15
Ftxt: 2:16
That is, xtFtx0 is the solution of the homogeneous equation
_
x Fx with
initial value x0.
EXAMPLE 2.5 The unit upper triangular Toeplitz matrix
Ft
1 t
1
2
t
2
1
1 Á 2 Á 3
t
3
ÁÁÁ
1
n À 1!
t
nÀ1
01 t
1
2
t
2
ÁÁÁ
1
n À 2!
t
nÀ2
00 1 t ÁÁÁ
1
n À 3!
t
nÀ3
00 0 1 ÁÁÁ
1
n À 4!
t
nÀ4
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
00 0 0 ÁÁÁ 1
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
is the fundamental solution of
_
x Fx for the strictly upper triangular Toeplitz
dynamic coef®cient matrix
F
010ÁÁÁ 0
001ÁÁÁ 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
000ÁÁÁ 1
000ÁÁÁ 0
2
6
6
6
6
4
3
7
7
7
7
5
;
which can be veri®ed by showing that F0I and
_
F FF. This dynamic
coef®cient matrix, in turn, is the companion matrix for the nth-order linear
homogeneous differential equation d=dt
n
yt0.
2.3 CONTINUOUS LINEAR SYSTEMS AND THEIR SOLUTIONS 35
Existence and Nonsingularity of Fundamental Solutions. If the elements
of the matrix Ft are continuous functions on some interval 0 t T, then the
fundamental solution matrix Ft is guaranteed to exist and to be nonsingular on an
interval 0 t 4t for some t > 0. These conditions also guarantee that Ft will be
nonsingular on some interval of nonzero length, as a consequence of the continuous
dependence of the solution Ft of the matrix equation on its (nonsingular) initial
conditions [F0I] [57].
State Transition Matrices. Note that the fundamental solution matrix Ft
transforms any initial state x0 of the dynamic system to the corresponding state
xt at time t.IfFt is nonsingular, then the products F
À1
txtx0 and
FtF
À1
txtxt. That is, the matrix product
Ft; tFtF
À1
t2:17
transforms a solution from time t to the corresponding solution at time t,as
diagrammed in Figure 2.2. Such a matrix is called the state transition matrix
4
for the
associated linear homogeneous differential equation. The state transition matrix
Ft; t represents the transition to the state at time t from the state at time t.
Properties of STMs and Fundamental Solution Matrices. The same
symbol (F) has been used for fundamental solution matrices and for state transition
matrices, the distinction being made by the number of arguments. By convention,
then,
Ft; 0Ft:
Other useful properties of F include the following:
1. Ft; tF0I,
2. F
À1
t; tFt; t,
3. Ft; sFs; tFt; t,
4. @=@tFt; tFtFt; t,
4
Formally, an operator Ft; t
0
; xt
0
such that xtFt; t
0
; xt
0
is called an evolution operator for a
dynamic system with state x. A state transition matrix is a linear evolution operator.
Φ
–1
(t )
Φ(τ, t )
Φ(τ)
x(t )
x(0)
x(τ)
0
t
τ
Fig. 2.2 The STM as a composition of fundamental solution matrices.
36 LINEAR DYNAMIC SYSTEMS
and
5. @=@tFt; tÀFt; tFt.
EXAMPLE 2.6: Fundamental Solution Matrix for the Underdamped Harmo-
nic Resonator The general solution of the differential equation. In Examples 2.2
and 2.3, the displacement d of the damped harmonic resonator was modeled by the
state equation
x
d
_
d
"#
;
_
x Fx;
F
01
À
k
s
m
À
k
d
m
2
4
3
5
:
The characteristic values of the dynamic coef®cient matrix F are the roots of its
characteristic polynomial
detlI À Fl
2
k
d
m
l
k
s
m
;
which is a quadratic polynomial with roots
l
1
1
2
À
k
d
m
k
2
d
m
2
À
4k
s
m
r
!
;
l
2
1
2
À
k
d
m
À
k
2
d
m
2
À
4k
s
m
r
!
:
The general solution for the displacement d can then be written in the form
dtae
l
1
t
be
l
2
t
;
where a and b are (possibly complex) free variables.
The underdamped solution. The resonator is considered underdamped if the
discriminant
k
2
d
m
2
À
4k
s
m
< 0;
2.3 CONTINUOUS LINEAR SYSTEMS AND THEIR SOLUTIONS 37
in which case the roots are a conjugate pair of nonreal complex numbers and the
general solution can be rewritten in ``real form'' as
dtae
Àt= t
cosotbe
Àt=t
sinot;
t
2m
k
d
;
o
k
s
m
À
k
2
d
4m
2
r
;
where a and b are now real variables, t is the decay time constant, and o is the
resonator resonant frequency. This solution can be expressed in state-space form in
terms of the real variables a and b:
dt
_
dt
e
Àt=t
cosot sinot
À
cosot
t
À o sinot o cosotÀ
sinot
t
2
4
3
5
a
b
"#
:
Initial value constraints. The initial values
d0a;
_
d0À
a
t
ob
can be solved for a and b as
a
b
10
1
ot
1
o
2
4
3
5
d0
_
d0
:
This can then be combined with the solution for xt in terms of a and b to yield the
fundamental solution
xtFtx0;
Ft
e
Àt= t
ot
2
tot cosotsinot t
2
sinot
À1 o
2
tsinotÀot cosotsinot
"#
in terms of the damping time constant and the resonant frequency.
38 LINEAR DYNAMIC SYSTEMS
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