0
if m < −g
m
0
Adaptive law
g =
− g 0
if m > g
0
m
−
if | m| ≤ g 0
z − θ φ
=
m 2
Normalizing signal
As in Table 8.1
Assumptions
As in Table 8.1
Design variables
g 0 > |η|; Γ = Γ > 0
m
Properties
(i) , m, θ, ˙ θ ∈ L∞; (ii) , m, ˙ θ ∈ S( η 2 /m 2 + g 0);
(iii) lim t→∞ θ( t) = ¯
θ
8.10 Consider the closed-loop adaptive control scheme of Section 8.3.2, i.e.,
˙ x = ax + z − u
µ ˙ z = −z + 2 u
y = x
u = −kx,
˙ k = γx 2
Show that there exists a region of attraction D whose size is of O( 1 ) for
µα
some α > 0 such that for x(0) , z(0) , k(0) ∈ D we have x( t) , z( t) → 0 and k( t) −→ k( ∞) as t → ∞. (Hint: use the transformation η = z − 2 u and choose V = x 2 + ( k−k∗)2 + µ ( x+ η)2 where k∗ > a.)
2
2 γ
2
630
CHAPTER 8. ROBUST ADAPTIVE LAWS
Table 8.5 Robust adaptive law with leakage for the bilinear
model
Parametric
z = ρ∗( θ∗ φ + z 1) + η, η = ∆ u( s) u + ∆ y( s) y + d model
˙ θ = Γ φ sgn( ρ∗) − w 1Γ θ
Adaptive law
˙ ρ = γ ξ − w 2 γρ
z − ρξ
=
, ξ = θ φ + z
m 2
1
Normalizing
As in Table 8.1.
signal
Leakage wi
i = 1 , 2
As in Table 8.1
Assumptions
(i) ∆ u, ∆ y are as in Table 8.1; (ii) φ = H( s)[ u, y] ,
z 1 = h 1( s) u + h 2( s) y, where H( s) , h 1( s) , h 2( s) are strictly proper and analytic in Re[ s] ≥ −δ 0 / 2
Design
Γ = Γ > 0 , γ > 0; the constants in wi are as defined
variables
in Table 8.1
Proporties
(i) , m, ρ, θ, ˙ ρ, ˙ θ ∈ L∞; (ii) , m, ˙ ρ, ˙ θ ∈ S( η 2 /m 2 +
f 0), where f 0 is as defined in Table 8.1
8.11 Perform simulations to compare the properties of the various choices of leakage
given in Section 8.4.1 using an example of your choice.
8.12 Consider the system
y = θ∗u + η
η = ∆( s) u
where y, u are available for measurement, θ∗ is the unknown constant to
be estimated and η is a modeling error signal with ∆( s) being proper and
analytic in Re[ s] ≥ − 0 . 5. The input u is piecewise continuous. Design an
adaptive law with a switching- σ to estimate θ∗.
8.7. PROBLEMS
631
Table 8.6 Robust adaptive law with projection for the bilinear
model
Parametric
Same as in Table 8.5
model
Γ
i φi
if |θi| < Mi
˙ θ
or if |θi|= Mi and (Γ i φi) θi ≤ 0
i =
Adaptive
I − Γ iθiθi
Γ
θ Γ
i φi
otherwise
i
iθi
law
i = 1 , 2 with
θ 1 = θ, θ 2 = ρ, φ 1 = φ sgn( ρ∗) , φ 2 = ξ, Γ1 = Γ , Γ2 = γ
= z−ρξ , ξ = θ φ + z
m 2
1
Assumptions
As in Table 8.5
Normalizing
As in Table 8.1
signal
Design
|θ(0) | ≤ M 1 , |ρ(0) | ≤ M 2; Γ = Γ > 0 , γ > 0
variables
Properties
(i) , m, ρ, θ, ˙ ρ, ˙ θ ∈ L∞; (ii) , m, ˙ ρ, ˙ θ ∈ S( η 2 /m 2)
8.13. The linearized dynamics of a throttle angle θ to vehicle speed V subsystem
are given by the 3rd order system
bp
V =
1 p 2
θ + d
( s + a)( s + p 1)( s + p 2)
where p 1 , p 2 > 20 , 1 ≥ a > 0 and d is a load disturbance.
(a) Obtain a parametric model for the parameters of the dominant part of
the system.
(b) Design a robust adaptive law for estimating on-line these parameters.
(c) Simulate your estimation scheme when a = 0 . 1 , b = 1 , p 1 = 50 , p 2 = 100
and d = 0 . 02 sin 5 t.
632
CHAPTER 8. ROBUST ADAPTIVE LAWS
Table 8.7 Robust adaptive law with dead zone for the bilinear
model
Parametric
Same as in Table 8.5
model
˙ θ = Γ φ( + g)sgn( ρ∗)
˙ ρ = γξ( + g)
g
0
if m < −g
m
0
Adaptive law
g =
− g 0
if m > g
0
m
−
if | m| ≤ g 0
= z−ρξ , ξ = θ φ + z
m 2
1
Assumptions
As in Table 8.5
Normalizing
Same as in Table 8.1
signal
Design
g 0 > |η|; Γ = Γ > 0 , γ > 0
m
variables
Properties
(i)
,
m, ρ, θ, ˙ ρ, ˙ θ ∈ L∞; (ii) ,
m, ˙ ρ,
˙ θ ∈ S( η 2 /m 2 + g 0); (iii) lim t→∞ θ( t) = ¯ θ
8.14 Consider the parameter error differential equation
˙˜ θ = −γu 2˜ θ+ γdu
that arises in the estimation problem of Section 8.3.1 in the presence of a
bounded disturbance d.
(a) Show that for d = 0 and u =
1
, the equilibrium ˜
θ
1
e = 0 is u.s and
(1+ t) 2
a.s but not u.a.s. Verify that for
5
d( t) = (1 + t) − 14
− 2(1 + t) − 14
4
8.7. PROBLEMS
633
u = (1 + t) − 12 and γ = 1 we have y → 0 as t → ∞ and ˜
θ( t) → ∞ as
t → ∞.
(b) Repeat the same stability analysis for u = u 0 where u 0 = 0 is a constant,
and show that for d = 0, the equilibrium ˜
θe = 0 is u.a.s. Verify that ˜
θ( t)
is bounded for any bounded d and obtain an upper bound for |˜
θ( t) |.
8.15 Repeat Problem 8.12 for an adaptive law with (i) dead zone; (ii) projection.
8.16 Consider the dynamic uncertainty
η = ∆ u( s) u + ∆ y( s) y
where ∆ u, ∆ y are proper transfer functions analytic in Re[ s] ≥ − δ 0 for some
2
known δ 0 > 0.
(a) Design a normalizing signal m that guarantees η/m ∈ L∞ when
(i) ∆ u, ∆ y are biproper.
(ii) ∆ u, ∆ y are strictly proper.
In each case specify the upper bound for |η| .
m
(b) Calculate the bound for |η|/m when
e−τs − 1
s 2
(i) ∆ u( s) =
, ∆
s + 2
y ( s) = µ ( s + 1)2
µs
µs
(ii) ∆ u( s) =
, ∆
µs + 2
y ( s) = ( µs + 1)2
where 0 < µ
1 and 0 < τ
1.
8.17 Consider the system
e−τsb
y =
u
( s + a)( µs + 1)
where 0 < τ
1 , 0 < µ
1 and a, b are unknown constants. Obtain a
parametric model for θ∗ = [ b, a] by assuming τ ≈ 0 , µ ≈ 0. Show the effect
of the neglected dynamics on the parametric model.
Chapter 9
Robust Adaptive Control
Schemes
9.1
Introduction
As we have shown in Chapter 8, the adaptive control schemes of Chapters 4
to 7 may go unstable in the presence of small disturbances or unmodeled
dynamics. Because such modeling error effects will exist in any implementa-
tion, the nonrobust behavior of the schemes of Chapters 4 to 7 limits their
applicability.
The purpose of this chapter is to redesign the adaptive schemes of the
previous chapters and establish their robustness properties with respect to a
wide class of bounded disturbances and unmodeled dynamics that are likely
to be present in most practical applications.
We start with the parameter identifiers and adaptive observers of Chap-
ter 5 and show that their robustness properties can be guaranteed by de-
signing the plant input to be dominantly rich. A dominantly rich input is
sufficiently rich for the simplified plant model, but it maintains its richness
outside the high frequency range of the unmodeled dynamics. Furthermore,
its amplitude is higher than the level of any bounded disturbance that may
be present in the plant. As we will show in Section 9.2, a dominantly rich
input guarantees exponential convergence of the estimated parameter errors
to residual sets whose size is of the order of the modeling error.
While the robustness of the parameter identifiers and adaptive observers
634
9.2. ROBUST IDENTIFIERS AND ADAPTIVE OBSERVERS
635
of Chapter 5 can be established by simply redesigning the plant input with-
out having to modify the adaptive laws, this is not the case with the adaptive
control schemes of Chapters 6 and 7 where the plant input is no longer a
design variable. For the adaptive control schemes presented in Chapters 6
and 7, robustness is established by simply replacing the adaptive laws with
robust adaptive laws developed in Chapter 8.
In Section 9.3, we use the robust adaptive laws of Chapter 8 to mod-
ify the MRAC schemes of Chapter 6 and establish their robustness with
respect to bounded disturbances and unmodeled dynamics. In the case of
the MRAC schemes with unnormalized adaptive laws, semiglobal bounded-
ness results are established. The use of a dynamic normalizing signal in the
case of MRAC with normalized adaptive laws enables us to establish global
boundedness results and mean-square tracking error bounds. These bounds
are further improved by modifying the MRAC schemes using an additional
feedback term in the control input. By choosing a certain scalar design pa-
rameter τ in the control law, the modified MRAC schemes are shown to
guarantee arbitrarily small L∞ bounds for the steady state tracking error
despite the presence of input disturbances. In the presence of unmodeled
dynamics, the choice of τ is limited by the trade-off between nominal per-
formance and robust stability.
The robustification of the APPC schemes of Chapter 7 is presented in
Section 9.5. It is achieved by replacing the adaptive laws used in Chapter 7
with the robust ones developed in Chapter 8.
9.2
Robust Identifiers and Adaptive Observers
The parameter identifiers and adaptive observers of Chapter 5 are designed
for the SISO plant model
y = G 0( s) u
(9.2.1)
where G 0( s) is strictly proper with stable poles and of known order n. In
this section we apply the schemes of Chapter 5 to a more realistic plant
model described by
y = G 0( s)(1 + ∆ m( s))( u + du)
(9.2.2)
where G( s) = G 0( s)(1 + ∆ m( s)) is strictly proper of unknown degree; ∆ m( s) is an unknown multiplicative perturbation with stable poles and du is a
636
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
bounded disturbance. Our objective is to estimate the coefficients of G 0( s)
and the states of a minimal or nonminimal state representation that cor-
responds to G 0( s), despite the presence of ∆ m( s), du. This problem is,
therefore, similar to the one we would face in an actual application, i.e.,
(9.2.1) represents the plant model on which our adaptive observer design is
based and (9.2.2) the plant to which the observer will be applied.
Most of the effects of ∆ m( s) , du on the robustness and performance of
the schemes presented in Chapter 5 that are designed based on (9.2.1) may
be illustrated and understood using the following simple examples.
Example 9.2.1 Effect of bounded disturbance. Let us consider the simple
plant model
y = θ∗u + d
where d is an external bounded disturbance, i.e., |d( t) | ≤ d 0 , ∀t ≥ 0 , u ∈ L∞ and θ∗
is the unknown constant to be identified. The adaptive law based on the parametric
model with d = 0 is
˙ θ = γ 1 u,
1 = y − θu
(9.2.3)
which for d = 0 guarantees that 1 , θ ∈ L∞ and 1 , ˙ θ ∈ L 2. If, in addition, u is PE,
then θ( t) → θ∗ as t → ∞ exponentially fast. When d = 0, the error equation that
describes the stability properties of (9.2.3) is
˙˜ θ = −γu 2˜ θ+ γud
(9.2.4)
where ˜
θ = θ − θ∗. As shown in Section 8.3, if u is PE, then the homogeneous part
of (9.2.4) is exponentially stable, and, therefore, the bounded input γud implies
bounded ˜
θ. When u is not PE, the homogeneous part of (9.2.4) is only u.s. and
therefore a bounded input does not guarantee bounded ˜
θ. In fact, as shown in
Section 8.3, we can easily find an input u that is not PE, and a bounded disturbance
d that will cause ˜
θ to drift to infinity. One way to counteract parameter drift and
establish boundedness for ˜
θ is to modify (9.2.3) using the techniques of Chapter 8.
In this section, we are concerned with the parameter identification of stable plants
which, for accurate parameter estimates, requires u to be PE independent of whether
we have disturbances or not. Because the persistent excitation of u guarantees
exponential convergence, we can establish robustness without modifying (9.2.3).
Let us, therefore, proceed with the analysis of (9.2.4) by assuming that u is PE
with some level α 0 > 0, i.e., u satisfies
t+ T
u 2 dτ ≥ α 0 T,
∀t ≥ 0 , for some T > 0
t
9.2. ROBUST ADAPTIVE OBSERVERS
637
Then from (9.2.4), we obtain
k
|˜
θ( t) | ≤ k
t
1
t
1 e−γα 0 | ˜
θ(0) | +
(1 − e−γα 0 ) sup |u( τ ) d( τ ) |
α 0
τ ≤t
for some constants k 1 , α 0 > 0, where α 0 depends on α 0. Therefore, we have
k
k
lim sup |˜
θ( τ ) | ≤ 1 lim sup |u( τ ) d( τ ) | = 1 sup |u( τ ) d(