sn
z =
y( n) =
y
Λ( s)
Λ( s)
α
α
φ =
n− 1( s) u, − n− 1( s) y
Λ( s)
Λ( s)
and
Λ( s) = sn + λn− 1 sn− 1 + · · · + λ 0
50
CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS
is an arbitrary Hurwitz polynomial in s. It is clear that the scalar signal z and
vector signal φ can be generated, without the use of differentiators, by simply
filtering the input u and output y with stable proper filters si , i = 0 , 1 , . . . n.
Λ( s)
If we now express Λ( s) as
Λ( s) = sn + λ αn− 1( s)
where λ = [ λn− 1 , λn− 2 , . . . , λ 0] , we can write
sn
Λ( s) − λ α
α
z =
y =
n− 1( s) y = y − λ
n− 1( s) y
Λ( s)
Λ( s)
Λ( s)
Therefore,
α
y = z + λ
n− 1( s) y
Λ( s)
Because z = θ∗ φ = θ∗ 1 φ 1 + θ∗ 2 φ 2, where
θ∗ 1 = [ bn− 1 , bn− 2 , . . . , b 0] , θ∗ 2 = [ an− 1 , an− 2 , . . . , a 0]
α
α
φ
n− 1( s)
n− 1( s)
1 =
u, φ
y
Λ( s)
2 = −
Λ( s)
it follows that
y = θ∗ 1 φ 1 + θ∗ 2 φ 2 − λ φ 2
Hence,
y = θ∗λ φ
(2.4.9)
where θ∗λ = [ θ∗ 1 , θ∗ 2 − λ ] . Equations (2.4.8) and (2.4.9) are represented
by the block diagram shown in Figure 2.2.
A state-space representation for generating the signals in (2.4.8) and
(2.4.9) may be obtained by using the identity
[adj( sI − Λ c)] l = αn− 1( s)
where Λ c, l are given by
−λn− 1 −λn− 2 · · · −λ 0
1
1
0
· · ·
0
0
Λ
c =
.
.
.
,
l = .
..
. .
..
..
0
· · ·
1
0
0
2.4. PLANT PARAMETRIC MODELS
51
φ
✲
u αn− 1( s)
✲
1
y
θ∗
✲
+ ❧
Σ
✲
Λ( s)
1
− +
✻
❆❑
φ 2 −α
❆
θ∗
n− 1( s)
✛
✛
2
Λ( s)
λ
✛
❄
✲
+ +
❧ z
Σ
✲
Figure 2.2 Plant Parameterization 1.
which implies that
α
det( sI − Λ
n− 1( s)
c) = Λ( s) ,
( sI − Λ c) − 1 l = Λ( s)
Therefore, it follows from (2.4.8) and Figure 2.2 that
˙ φ 1 = Λ cφ 1 + lu, φ 1 ∈ Rn
˙ φ 2 = Λ cφ 2 − ly, φ 2 ∈ Rn
y = θ∗λ φ
(2.4.10)
z = y + λ φ 2 = θ∗ φ
Because Λ( s) = det( sI − Λ c) and Λ( s) is Hurwitz, it follows that Λ c is a
stable matrix.
The parametric model (2.4.10) is a nonminimal state-space representa-
tion of the plant (2.4.3). It is nonminimal because 2 n integrators are used
to represent an n th-order system. Indeed, the transfer function Y ( s) /U ( s)
computed using (2.4.10) or Figure 2.2, i.e.,
Y ( s)
Z( s) Λ( s)
Z( s)
=
=
U ( s)
R( s) Λ( s)
R( s)
involves n stable zero-pole cancellations.
The plant (2.4.10) has the same I/O response as (2.4.3) and (2.4.1)
provided that all state initial conditions are equal to zero, i.e., x 0 = 0,
φ 1(0) = φ 2(0) = 0. In an actual plant, the state x in (2.4.1) may represent
physical variables and the initial state x 0 may be different from zero. The
52
CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS
effect of the initial state x 0 may be accounted for in the model (2.4.10) by
applying the same procedure to equation (2.4.2) instead of equation (2.4.3).
We can verify (see Problem 2.9) that if we consider the effect of initial con-
dition x 0, we will obtain the following representation
˙ φ 1 = Λ cφ 1 + lu, φ 1(0) = 0
˙ φ 2 = Λ cφ 2 − ly, φ 2(0) = 0
y = θ∗λ φ + η 0
(2.4.11)
z = y + λ φ 2 = θ∗ φ + η 0
where η 0 is the output of the system
˙ ω = Λ cω,
ω(0) = ω 0
η 0 = C 0 ω
(2.4.12)
where ω ∈ Rn, ω 0 = B 0 x 0 and C 0 ∈ Rn, B 0 ∈ Rn×n are constant matrices
that satisfy C 0 { adj( sI − Λ c) }B 0 = C { adj( sI − A) }.
Because Λ c is a stable matrix, it follows from (2.4.12) that ω, η 0 converge
to zero exponentially fast. Therefore, the effect of the nonzero initial condi-
tion x 0 is the appearance of the exponentially decaying to zero term η 0 in
the output y and z.
Parameterization 2
Let us now consider the parametric model (2.4.9)
y = θ∗λ φ
and the identity Wm( s) W − 1
m ( s) = 1, where Wm( s) = Zm( s) /Rm( s) is a
transfer function with relative degree one, and Zm( s) and Rm( s) are Hurwitz
polynomials. Because θ∗λ is a constant vector, we can express (2.4.9) as
y = Wm( s) θ∗λ W − 1
m ( s) φ
If we let
1
α
α
ψ =
φ =
n− 1( s)
u, −
n− 1( s)
y
Wm( s)
Wm( s)Λ( s)
Wm( s)Λ( s)
2.4. PLANT PARAMETRIC MODELS
53
ψ
✲
u
αn− 1( s)
✲
1
y
θ∗
✲
+ ❧
Σ
✲ Wm( s)
✲
Λ( s) W
1
m( s)
− +
✻
❆❑
ψ 2
−αn− 1( s)
❆
✛
✛
θ∗ 2
Λ( s) Wm( s)
λ
✛
Figure 2.3 Plant Parameterization 2.
we have
y = Wm( s) θ∗λ ψ
(2.4.13)
Because all the elements of αn− 1( s) are proper transfer functions with
Λ( s) Wm( s)
stable poles, the state ψ = [ ψ 1 ,ψ 2 ] , where
α
α
ψ
n− 1( s)
n− 1( s)
1 =
u, ψ
y
W
2 = −
m( s)Λ( s)
Wm( s)Λ( s)
can be generated without differentiating y or u. The dimension of ψ depends
on the order n of Λ( s) and the order of Zm( s). Because Zm( s) can be
arbitrary, the dimension of ψ can be also arbitrary.
Figure 2.3 shows the block diagram of the parameterization of the plant
given by (2.4.13). We refer to (2.4.13) as Parameterization 2. In [201],
Parameterization 2 is referred to as the model reference representation and
is used to design parameter estimators for estimating θ∗λ when Wm( s) is a
strictly positive real transfer function (see definition in Chapter 3).
A special case of (2.4.13) is the one shown in Figure 2.4 where
1
Wm( s) = s + λ 0
and ( s + λ 0) is a factor of Λ( s), i.e.,
Λ( s) = ( s + λ 0)Λ q( s) = sn + λn− 1 sn− 1 + · · · + λ 0
where
Λ q( s) = sn− 1 + qn− 2 sn− 2 + · · · + q 1 s + 1
54
CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS
ψ
✲
u
αn− 1( s)
✲
1
y
θ∗
✲
+ ❧
Σ
✲
1
✲
Λ
1
s+ λ 0
q ( s)
− +
✻
❆❑
ψ 2
−αn− 1( s)
❆
✛
✛
θ∗ 2
Λ q( s)
λ
✛
Figure 2.4
Plant Parameterization 2 with Λ( s) = ( s + λ 0)Λ q( s) and
Wm( s) = 1 .
s+ λ 0
The plant Parameterization 2 of Figure 2.4 was first suggested in [131],
where it was used to develop stable adaptive observers. An alternative para-
metric model of the plant of Figure 2.4 can be obtained by first separating
the biproper elements of αn− 1( s) as follows:
Λ q( s)
For any vector c = [ cn− 1 , cn− 2 , . . . , c 1 , c 0] ∈ Rn, we have
c αn− 1( s)
c
¯ c α
= n− 1 sn− 1 +
n− 2( s)
(2.4.14)
Λ q( s)
Λ q( s)
Λ q( s)
where ¯ c = [ cn− 2 , . . . , c 1 , c 0] , αn− 2 = [ sn− 2 , . . . , s, 1] . Because Λ q( s) =
sn− 1 + ¯
q αn− 2( s), where ¯
q = [ qn− 2 , . . . , q 1 , 1] , we have sn− 1 = Λ q( s) −
¯
q αn− 2, which after substitution we obtain
c αn− 1( s)
(¯ c − c
= c
n− 1 ¯
q) αn− 2( s)
(2.4.15)
Λ
n− 1 +
q( s)
Λ q( s)
We use (2.4.15) to obtain the following expressions:
α
α
θ∗
n− 1( s)
n− 2( s)
1
u = b
u
Λ
n− 1 u + ¯
θ∗ 1
q( s)
Λ q( s)
α
α
−( θ∗
n− 1( s)
n− 2( s)
2
− λ )
y = ( λ
y (2.4.16)
Λ
n− 1 − an− 1) y − ¯
θ∗ 2
q( s)
Λ q( s)
where ¯
θ∗ 1 = ¯ b − bn− 1¯ q, ¯ θ∗ 2 = ¯ a − ¯ λ − ( an− 1 − λn− 1)¯ q and ¯ a = [ an− 2,
. . . , a 1 , a 0] , ¯ b = [ bn− 1 , . . . , b 1 , b 0] , ¯ λ = [ λn− 2 , . . . , λ 1 , λ 0] . Using (2.4.16), Figure 2.4 can be reconfigured as shown in Figure 2.5.
2.4. PLANT PARAMETRIC MODELS
55
¯
ψ
1
¯
x
✲
u
αn− 2( s)
✲
1
¯
+
1 = y
θ∗
✲ ❧
Σ
✲
✲
Λ
1
+ +
s + λ
q ( s)
0
✁✕+✻
❆❑
✁
❆
¯
ψ
✲
2
−α
b
✁
❆
¯
n− 2( s)
✛
✛
n− 1
θ∗ 2
Λ q( s)
✛
λn− 1 −an− 1
Figure 2.5 Equivalent plant Parameterization 2.
A nonminimal state space representation of the plant follows from Fig-
ure 2.5, i.e.,
˙¯ x 1 = −λ 0¯ x 1 + ¯ θ∗ ¯
ψ,
¯
x 1 ∈ R 1
˙¯
ψ
¯
1
= ¯
Λ cψ 1 + ¯ lu,
¯
ψ 1 ∈ Rn− 1
˙¯
ψ
¯
2
= ¯
Λ cψ 2 − ¯ ly,
¯
ψ 2 ∈ Rn− 1
(2.4.17)
y = ¯
x 1
where ¯
θ∗ = [ bn− 1 , ¯
θ∗ 1 , λn− 1 − an− 1 , ¯ θ∗ 2 ] , ¯
ψ = [ u, ¯
ψ 1 , y, ¯