G( s) = msm + bm− 1 sm− 1 + · · · + b 0 =
sn + an− 1 sn− 1 + · · · + a 0
U ( s)
where n > m. Then the system may be represented in the controller form
−an− 1 −an− 2 · · · −a 1 −a 0
1
1
0
· · ·
0
0
0
˙ x =
0
1
· · ·
0
0
.
x + .. u
(2.3.6)
.
.
.
.
..
..
. .
..
0
0
0
· · ·
1
0
0
y = [0 , 0 , . . . , bm, . . . , b 1 , b 0] x
or in the observer form
−a
0
n− 1
1 0 · · · 0
.
−an− 2 0 1 · · · 0
.
.
˙ x =
.
.
.
.
..
..
. . .. x + bm u
(2.3.7)
−a
.
1
0 0 · · · 1
..
−a 0
0 0 · · · 0
b 0
y = [1 , 0 , . . . , 0] x
2.3. INPUT/OUTPUT MODELS
37
One can go on and generate many different state-space representations
describing the I/O properties of the same system. The canonical forms in
(2.3.6) and (2.3.7), however, have some important properties that we will use
in later chapters. For example, if we denote by ( Ac, Bc, Cc) and ( Ao, Bo, Co)
the corresponding matrices in the controller form (2.3.6) and observer form
(2.3.7), respectively, we establish the relations
[adj( sI − Ac)] Bc = [ sn− 1 , . . . , s, 1] = αn− 1( s)
(2.3.8)
Co adj( sI − Ao) = [ sn− 1 , . . . , s, 1] = αn− 1( s)
(2.3.9)
whose right-hand sides are independent of the coefficients of G( s). Another
important property is that in the triples ( Ac, Bc, Cc) and ( Ao, Bo, Co), the
n+ m+1 coefficients of G( s) appear explicitly, i.e., ( Ac, Bc, Cc) (respectively
( Ao, Bo, Co)) is completely characterized by n + m + 1 parameters, which are
equal to the corresponding coefficients of G( s).
If G( s) has no zero-pole cancellations then both (2.3.6) and (2.3.7) are
minimal state-space representations of the same system. If G( s) has zero-
pole cancellations, then (2.3.6) is unobservable, and (2.3.7) is uncontrollable.
If the zero-pole cancellations of G( s) occur in Re[ s] < 0, i.e., stable poles are
cancelled by stable zeros, then (2.3.6) is detectable, and (2.3.7) is stabilizable.
Similarly, a system described by a state-space representation is unobservable
or uncontrollable, if and only if the transfer function of the system has zero-
pole cancellations. If the unobservable or uncontrollable parts of the system
are asymptotically stable, then the zero-pole cancellations occur in Re[ s] < 0.
An alternative approach for representing the differential equation (2.3.1)
is by using the differential operator
d( ·)
p( ·) = dt
which has the following properties:
( i) p( x) = ˙ x;
( ii) p( xy) = ˙ xy + x ˙ y
where x and y are any differentiable functions of time and ˙ x = dx( t) .
dt
The inverse of the operator p denoted by p− 1 or simply by 1 is defined
p
as
1
t
( x) =
x( τ ) dτ + x(0) ∀t ≥ 0
p
0
38
CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS
where x( t) is an integrable function of time. The operators p, 1 are related
p
to the Laplace operator s by the following equations
L {p( x) }|
= sX( s)
x(0)=0
1
1
L{ ( x) } |
X( s)
p
x(0)=0= s
where L is the Laplace transform and x( t) is any differentiable function of
time. Using the definition of the differential operator, (2.3.1) may be written
in the compact form
R( p)( y) = Z( p)( u)
(2.3.10)
where
R( p) = pn + an− 1 pn− 1 + · · · + a 0
Z( p) = bmpm + bm− 1 pm− 1 + · · · + b 0
are referred to as the polynomial differential operators [226].
Equation (2.3.10) has the same form as
R( s) Y ( s) = Z( s) U ( s)
(2.3.11)
obtained by taking the Laplace transform on both sides of (2.3.1) and as-
suming zero initial conditions. Therefore, for zero initial conditions one can
go from representation (2.3.10) to (2.3.11) and vice versa by simply replacing
s with p or p with s appropriately. For example, the system
s + b
Y ( s) =
0 U( s)
s 2 + a 0
may be written as
( p 2 + a 0)( y) = ( p + b 0)( u)
with y(0) = ˙ y(0) = 0 , u(0) = 0 or by abusing notation (because we never
defined the operator ( p 2 + a 0) − 1) as
p + b
y( t) =
0 u( t)
p 2 + a 0
Because of the similarities of the forms of (2.3.11) and (2.3.10), we will use
s to denote both the differential operator and Laplace variable and express
the system (2.3.1) with zero initial conditions as
Z( s)
y =
u
(2.3.12)
R( s)
2.3. INPUT/OUTPUT MODELS
39
where y and u denote Y ( s) and U ( s), respectively, when s is taken to be the Laplace operator, and y and u denote y( t) and u( t), respectively, when s is taken to be the differential operator.
We will often refer to G( s) = Z( s) in (2.3.12) as the filter with input u( t)
R( s)
and output y( t).
Example 2.3.1 Consider the system of equations describing the motion of the cart
with the two pendulums given in Example 2.2.1, where y = θ 1 is the only measured
output. Eliminating the variables θ 1 , θ 2, and ˙ θ 2 by substitution, we obtain the
fourth order differential equation
y(4) − 1 . 1( α 1 + α 2) y(2) + 1 . 2 α 1 α 2 y = β 1 u(2) − α 1 β 2 u where αi, βi, i = 1 , 2 are as defined in Example 2.2.1, which relates the input u with
the measured output y.
Taking the Laplace transform on each side of the equation and assuming zero
initial conditions, we obtain
[ s 4 − 1 . 1( α 1 + α 2) s 2 + 1 . 2 α 1 α 2] Y ( s) = ( β 1 s 2 − α 1 β 2) U( s) Therefore, the transfer function of the system from u to y is given by
Y ( s)
β
=
1 s 2 − α 1 β 2
= G( s)
U ( s)
s 4 − 1 . 1( α 1 + α 2) s 2 + 1 . 2 α 1 α 2
For l 1 = l 2, we have α 1 = α 2 , β 1 = β 2, and
β
β
G( s) =
1( s 2 − α 1)
=
1( s 2 − α 1)
s 4 − 2 . 2 α 1 s 2 + 1 . 2 α 21
( s 2 − α 1)( s 2 − 1 . 2 α 1)
has two zero-pole cancellations. Because α 1 > 0, one of the zero-pole cancellations
occurs in Re[ s] > 0 which indicates that any fourth-order state representation of
the system with the above transfer function is not stabilizable.
2.3.2
Coprime Polynomials
The I/O properties of most of the systems studied in this book are repre-
sented by proper transfer functions expressed as the ratio of two polynomials
in s with real coefficients, i.e.,
Z( s)
G( s) =
(2.3.13)
R( s)
40
CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS
where Z( s) = bmsm + bm− 1 sm− 1 + · · · + b 0 , R( s) = sn + an− 1 sn− 1 + · · · + a 0
and n ≥ m.
The properties of the system associated with G( s) depend very much
on the properties of Z( s) and R( s). In this section, we review some of
the general properties of polynomials that are used for analysis and control
design in subsequent chapters.
Definition 2.3.1 Consider the polynomial X( s) = αnsn + αn− 1 sn− 1 + · · · +
α 0 . We say that X( s) is monic if αn = 1 and X( s) is Hurwitz if all the roots of X( s) = 0 are located in Re[ s] < 0 . We say that the degree of X( s) is n if the coefficient αn of sn satisfies αn = 0 .
Definition 2.3.2 A system with a transfer function given by (2.3.13) is
referred to as minimum phase if Z( s) is Hurwitz; it is referred to as stable
if R( s) is Hurwitz.
As we mentioned in Section 2.3.1, a system representation is minimal
if the corresponding transfer function has no zero-pole cancellations, i.e., if
the numerator and denominator polynomials of the transfer function have
no common factors other than a constant. The following definition is widely
used in control theory to characterize polynomials with no common factors.
Definition 2.3.3 Two polynomials a( s) and b( s) are said to be coprime (or
relatively prime ) if they have no common factors other than a constant.
An important characterization of coprimeness of two polynomials is given
by the following Lemma.
Lemma 2.3.1 (Bezout Identity) Two polynomials a( s) and b( s) are co-
prime if and only if there exist polynomials c( s) and d( s) such that
c( s) a( s) + d( s) b( s) = 1
For a proof of Lemma 2.3.1, see [73, 237].
The Bezout identity may have infinite number of solutions c( s) and d( s)
for a given pair of coprime polynomials a( s) and b( s) as illustrated by the
following example.
2.3. INPUT/OUTPUT MODELS
41
Example 2.3.2 Consider the coprime polynomials a( s) = s + 1 , b( s) = s + 2. Then the Bezout identity is satisfied for
c( s) = sn + 2 sn− 1 − 1 , d( s) = −sn − sn− 1 + 1
and any n ≥ 1.
Coprimeness is an important property that is often exploited in control
theory for the design of control schemes for LTI systems. An important the-
orem that is very often used for control design and analysis is the following.
Theorem 2.3.1 If a( s) and b( s) are coprime and of degree na and nb, re-
spectively, where na > nb, then for any given arbitrary polynomial a∗( s) of
degree na∗ ≥ na, the polynomial equation
a( s) l( s) + b( s) p( s) = a∗( s)
(2.3.14)
has a unique solution l( s) and p( s) whose degrees nl and np, respectively,
satisfy the constraints np < na, nl ≤ max( na∗ − na, nb − 1) .
Proof From Lemma 2.3.1, there exist polynomials c( s) and d( s) such that
a( s) c( s) + b( s) d( s) = 1
(2.3.15)
Multiplying Equation (2.3.15) on both sides by the polynomial a∗( s), we obtain
a∗( s) a( s) c( s) + a∗( s) b( s) d( s) = a∗( s) (2.3.16)
Let us divide a∗( s) d( s) by a( s), i.e.,
a∗( s) d( s)
p( s)
= r( s) +
a( s)
a( s)
where r( s) is the quotient of degree na∗ + nd − na; na∗, na, and nd are the degrees of a∗( s) , a( s), and d( s), respectively, and p( s) is the remainder of degree np < na.
We now use
a∗( s) d( s) = r( s) a( s) + p( s)
to express the right-hand side of (2.3.16) as
a∗( s) a( s) c( s) + r( s) a( s) b( s) + p( s) b( s) = [ a∗( s) c( s) + r( s) b( s)] a( s) + p( s) b( s) 42
CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS
and rewrite (2.3.16) as
l( s) a( s) + p( s) b( s) = a∗( s)
(2.3.17)
where l( s) = a∗( s) c( s) + r( s) b( s). The above equation implies that the degree of l( s) a( s) = degree of ( a∗( s) − p( s) b( s)) ≤ max {na∗, np + nb}. Hence, the degree of l( s), denoted by nl, satisfies nl ≤ max {na∗ − na, np + nb − na}. We, therefore, established that polynomials l( s) and p( s) of degree nl ≤ max {na∗ − na, np +
nb − na} and np < na respectively exist that satisfy (2.3.17). Because np < na