2
2 m 2
2
574
CHAPTER 8. ROBUST ADAPTIVE LAWS
Because η ∈ L
+ σ). Because
m
∞ it follows that V, ˜
θ ∈ L∞ and m ∈ S( η 2
m 2
˙˜ θ = γ m u − γσθ and u ∈ L
m
m
∞, we can establish as in Section 8.4.1 that
˙˜ θ ∈ S( η 2 + σ). The boundedness of ˜ θ, u , η also implies by (8.4.49) that
m 2
m m
, m, ˙ θ ∈ L∞. The properties of (8.4.48) with w = σ are therefore given by
• (i) , m, θ, ˙ θ ∈ L∞
• (ii) , m, ˙ θ ∈ S( η 2 + σ)
m 2
In a similar manner, other choices for w( t) and modifications, such as
dead zone and projections, may be used together with the normalizing signal
(8.4.45) to design adaptive laws that are robust with respect to the dynamic
uncertainty η.
The use of dynamic normalization was first introduced in [48] to handle
the effects of bounded disturbances which could also be handled by static
normalization. The use of dynamic normalization in dealing with dynamic
uncertainties in adaptive control was first pointed out in [193] where it was
used to develop some of the first global results in robust adaptive control for
discrete-time plants. The continuous-time version of the dynamic normal-
ization was first used in [91, 113] to design robust MRAC schemes for plants
with additive and multiplicative plant uncertainties. Following the work of
[91, 113, 193], a wide class of robust adaptive laws incorporating dynamic
normalizations are developed for both continuous- and discrete-time plants.
Remark 8.4.2 The leakage, projection, and dead-zone modifications are
not necessary for signal boundedness when u in the parametric model
(8.4.1) is bounded and PE and η ∈ L∞, as indicated in Section 8.3.1.
The PE property guarantees exponential stability in the absence of
modeling errors which in turn guarantees bounded states in the pres-
ence of bounded modeling error inputs provided the modeling error
term doesnot destroy the PE property of the input. In this case the
steady state bounds for the parameter and estimation error are of the
order of the modeling error, which implies that phenomena, such as
bursting, where the estimation error assumes values larger than the
order of the modeling error at steady state, are not present. The use
of PE signals to improve robustness and performance is discussed in
8.5. ROBUST ADAPTIVE LAWS
575
subsequent sections. When u, η are not necessarily bounded, the above
remarks hold provided u ∈ L
m
∞ is PE.
8.5
Robust Adaptive Laws
In Chapter 4 we developed a wide class of adaptive laws for estimating
on-line a constant parameter vector θ∗ in certain plant parametric models.
The vector θ∗ could contain the unknown coefficients of the plant transfer
function or the coefficients of various other plant parameterizations. The
adaptive laws of Chapter 4 are combined with control laws in Chapters
6, 7 to form adaptive control schemes that are shown to meet the control
objectives for the plant model under consideration.
A crucial assumption made in the design and analysis of the adaptive
schemes of Chapters 4 to 7 is that the plant model is an ideal one, i.e., it
is free of disturbances and modeling errors and the plant parameters are
constant for all time.
The simple examples of Section 8.3 demonstrate that the stability prop-
erties of the adaptive schemes developed in Chapters 4 to 7 can no longer
be guaranteed in the presence of bounded disturbances, unmodeled plant
dynamics and parameter variations. The main cause of possible instabili-
ties in the presence of modeling errors is the adaptive law that makes the
overall adaptive scheme time varying and nonlinear. As demonstrated in
Section 8.4 using a simple example, the destabilizing effects of bounded dis-
turbances and of a class of dynamic uncertainties may be counteracted by
using simple modifications that involve leakage, dead-zone, projection, and
dynamic normalization. In this section we extend the results of Section 8.4
to a general class of parametric models with modeling errors that may arise
in the on-line parameter estimation problem of a wide class of plants.
We start with the following section where we develop several parametric
models with modeling errors that are used in subsequent sections to develop
adaptive laws that are robust with respect to uncertainties. We refer to such
adaptive laws as robust adaptive laws.
576
CHAPTER 8. ROBUST ADAPTIVE LAWS
8.5.1
Parametric Models with Modeling Error
Linear Parametric Models
Let us start with the plant
y = G 0( s)(1 + ∆ m( s)) u + d
(8.5.1)
where
Z( s)
G 0( s) =
(8.5.2)
R( s)
represents the dominant or modeled part of the plant transfer function with
Z( s) = bn− 1 sn− 1 + bn− 2 sn− 2 + . . . + b 1 s + b 0 and R( s) = sn + an− 1 sn− 1 +
. . . + a 1 s + a 0; ∆ m( s) is a multiplicative perturbation and d is a bounded disturbance.
We would like to express (8.5.1) in the form where the coefficients of
Z( s) , R( s) lumped in the vector
θ∗ = [ bn− 1 , bn− 2 , . . . , b 0 , an− 1 , an− 2 , . . . , a 0]
are separated from signals as done in the ideal case, where ∆ m = 0 , d = 0,
presented in Section 2.4.1. From (8.5.1) we have
Ry = Zu + Z∆ mu + Rd
(8.5.3)
As in Section 2.4.1, to avoid the presence of derivatives of signals in the
parametric model, we filter each side of (8.5.3) with
1 , where Λ( s) is a
Λ( s)
monic Hurwitz polynomial of degree n, to obtain
R
Z
y =
u + η
Λ
Λ
m
where
Z∆
R
η
m
m =
u +
d
Λ
Λ
is the modeling error term because of ∆ m, d. If instead of (8.5.1), we consider
a plant with an additive perturbation, i.e.,
y = G 0( s) u + ∆ a( s) u + d
we obtain
R
Z
y =
u + η
Λ
Λ
a
8.5. ROBUST ADAPTIVE LAWS
577
where
R
R
ηa =
∆
d
Λ au + Λ
Similarly, if we consider a plant with stable factor perturbations, i.e.,
N
y =
0( s) + ∆1( s) u + d
D 0( s) + ∆2( s)
where
Z( s)
R( s)
N 0( s) =
, D
Λ( s)
0( s) = Λ( s)
and N 0 , D 0 are proper stable transfer functions that are coprime, we obtain
R
Z
y =
u + η
Λ
Λ
s
where
R
ηs = ∆1 u − ∆2 y + (
+ ∆
Λ
2) d
Therefore, without loss of generality we can consider the plant parameteri-
zation
R
Z
y =
u + η
(8.5.4)
Λ
Λ
where
η = ∆ y( s) y + ∆ u( s) u + d 1
(8.5.5)
is the modeling error with ∆ y, ∆ u being stable transfer functions and d 1
being a bounded disturbance and proceed as in the ideal case to obtain a
parametric model that involves θ∗. If we define
sn
α
α
z =
y, φ =
n− 1( s) u, − n− 1( s) y
Λ( s)
Λ( s)
Λ( s)
where αi( s) = [ si, si− 1 , . . . , s, 1] , as in Section 2.4.1, we can rewrite (8.5.4)
in the form
z = θ∗ φ + η
(8.5.6)
or in the form
y = θ∗λ φ + η
(8.5.7)
where θ∗λ = [ θ∗ 1 , θ∗ 2 −λ ] , θ∗ 1 = [ bn− 1 , . . . , b 0] , and θ∗ 2 = [ an− 1 , . . . , a 0] ; λ = [ λn− 1 , . . . , λ 0] is the coefficient vector of Λ( s) −sn = λn− 1 sn− 1+ · · ·+ λ 0.
578
CHAPTER 8. ROBUST ADAPTIVE LAWS
The effect of possible nonzero initial conditions in the overall plant state-
space representation may be also included in (8.5.6), (8.5.7) by following
the same procedure as in the ideal case. It can be shown that the initial
conditions appear as an exponentially decaying to zero term η 0, i.e.,
z = θ∗ φ + η + η 0
y = θ∗λ φ + η + η 0
(8.5.8)
where η 0 is the output of the system
˙ ω = Λ cω,
ω(0) = ω 0
η 0 = C 0 ω
Λ c is a stable matrix whose eigenvalues are equal to the poles of ∆ m( s) or
∆ a( s), or ∆1( s) and ∆2( s) and Λ( s), and the degree of ω is equal to the
order of the overall plant.
The parametric models given by (8.5.6) to (8.5.8) correspond to Param-
eterization 1 in Section 2.4.1. Parameterization 2 developed in the same
section for the plant with η = 0 may be easily extended to the case of η = 0
to obtain
y = Wm( s) θ∗λ ψ + η
z = Wm( s) θ∗ ψ + η
(8.5.9)
where Wm( s) is a stable transfer function with relative degree 1 and ψ =
W − 1
m ( s) φ.
Parametric models (8.5.6), (8.5.7) and (8.5.9) may be used to estimate
on-line the parameter vector θ∗ associated with the dominant part of the
plant characterized by the transfer function G 0( s). The only signals available
for measurements in these parametric models are the plant output y and the
signals φ, ψ, and z that can be generated by filtering the plant input u
and output y as in Section 2.4.1. The modeling error term η due to the
unmodeled dynamics and bounded disturbance is unknown, and is to be
treated as an unknown disturbance term that is not necessarily bounded.
If, however, ∆ y, ∆ u are proper then d 1 , y, u ∈ L∞ will imply that η ∈ L∞.
The properness of ∆ y, ∆ u may be established by assuming that G 0( s) and
the overall plant transfer function are proper. In fact ∆ y, ∆ u can be made
strictly proper by filtering each side of (8.5.4) with a first order stable filter
without affecting the form of the parametric models (8.5.6) to (8.5.9).
8.5. ROBUST ADAPTIVE LAWS
579
Bilinear Parametric Models
Let us now extend the bilinear parametric model of Section 2.4.2 developed
for the ideal plant to the plant that includes a multiplicative perturbation
and a bounded disturbance, i.e., consider the plant
y = G 0( s)(1 + ∆ m( s)) u + d
(8.5.10)
where
Z
G
p( s)
0( s) = kp Rp( s)
kp is a constant, Rp( s) is monic and of degree n, Zp( s) is monic Hurwitz of
degree m < n and kp, Zp, Rp satisfy the Diophantine equation
RpQ + kpZpP = ZpA
(8.5.11)
where
Q( s) = sn− 1 + q αn− 2( s) , P ( s) = p αn− 1( s)
and A( s) is a monic Hurwitz polynomial of degree 2 n− 1. As in Section 2.4.2,
our objective is to obtain a parameterization of the plant in terms of the
coefficient vectors q, p of Q, P , respectively, by using (8.5.11) to substitute
for Zp( s) , Rp( s) in (8.5.10). This parameterization problem appears in direct
MRAC where q, p are the controller parameters that we like to estimate on-
line. Following the procedure of Section 2.4.2, we express (8.5.10) in the
form
Rpy = kpZpu + kpZp∆ mu + Rpd
which implies that
QRpy = kpZpQu + kpZpQ∆ mu + QRpd
(8.5.12)
From (8.5.11) we have QRp = ZpA − kpZpP , which we use in (8.5.12) to
obtain
Zp( A − kpP ) y = kpZpQu + kpZpQ∆ mu + QRpd
Because Zp( s) , A( s) are Hurwitz we can filter both sides of the above equa-
tion with 1 /( ZpA) and rearrange the terms to obtain
P
Q
Q∆
QR
y = k
m
p
p
y + k
u + k
u +
d
A
p A
p
A
AZp
580
CHAPTER 8. ROBUST ADAPTIVE LAWS
Substituting for P ( s) = p αn− 1( s) , Q( s) = sn− 1 + q αn− 2( s), we obtain Λ( s)
α
α
sn− 1
y =
k
n− 1( s) y + q
n− 2( s) u +
u + η
(8.5.13)
A( s) p p
Λ( s)
Λ( s)
Λ( s)
where Λ( s) is a Hurwitz polynomial of degree nλ that satisfies 2 n− 1 ≥ nλ ≥
n − 1 and
k
QR
η = pQ∆ m u +
p d
(8.5.14)
A
AZp
is the modeling error resulting from ∆ m, d. We can verify that Q∆ m , QRp
A
AZp
are strictly proper and biproper respectively with stable poles provided the
overall plant transfer function is strictly proper. From (8.5.13), we obtain
the bilinear parametric model
y = W ( s) ρ∗( θ∗ φ + z 0) + η
(8.5.15)
where ρ∗ = kp,
Λ( s)
W ( s) = A( s)
is a proper transfer function with stable poles and zeros and
α
α
sn− 1
θ∗ = [ q , p ] , φ =
n− 2( s) u, n− 1( s) y
, z
u
Λ( s)
Λ( s)
0 = Λ( s)
If instead of (8.5.10), we use the plant representation with an additive plant
perturbation, we obtain (8.5.15) with
A − k
η =
pP (∆
A
au + d)
and in the case of a plant with stable factor perturbations ∆1 , ∆2 we obtain
(8.5.15) with
Λ Q
Λ Q
R
η =
(∆
p + ∆
AZ
1 u − ∆2 y) +
2
d
p
AZp
Λ
where Λ is now restricted to have degree n. If we assume that the overall
plant transfer function is strictly proper for the various plant representations
with perturbations, then a general form for the modeling error term is
η = ∆ uu + ∆