The extension of this example to the general case follows from the mate-
rial presented in Section 9.3.3 for the direct case and that in Chapter 6 for
indirect MRAC and is left as an exercise for the reader.
9.4. PERFORMANCE IMPROVEMENT OF MRAC
693
9.4
Performance Improvement of MRAC
In Chapter 6 we have established that under certain assumptions on the
plant and reference model, we can design MRAC schemes that guarantee
signal boundedness and asymptotic convergence of the tracking error to
zero. These results, however, provide little information about the rate of
convergence and the behavior of the tracking error during the initial stages
of adaptation. Of course if the reference signal is sufficiently rich we have
exponential convergence and therefore more information about the asymp-
totic and transient behavior of the scheme can be inferred. Because in most
situations we are not able to use sufficiently rich reference inputs without
violating the tracking objective, the transient and asymptotic properties of
the MRAC schemes in the absence of rich signals are very crucial.
The robustness modifications, introduced in Chapter 8 and used in the
previous sections for robustness improvement of MRAC, provide no guaran-
tees of transient and asymptotic performance improvement. For example,
in the absence of dominantly rich input signals, the robust MRAC schemes
with normalized adaptive laws guarantee signal boundedness for any finite
initial conditions, and a tracking error that is of the order of the modeling
error in m.s.s. Because smallness in m.s.s. does not imply smallness point-
wise in time, the possibility of having tracking errors that are much larger
than the order of the modeling error over short time intervals at steady state
cannot be excluded. A phenomenon known as “bursting,” where the track-
ing error, after reaching a steady-state behavior, bursts into oscillations of
large amplitude over short intervals of time, have often been observed in
simulations. Bursting cannot be excluded by the m.s.s. bounds obtained in
the previous sections unless the reference signal is dominantly rich and/or
an adaptive law with a dead zone is employed. Bursting is one of the most
annoying phenomena in adaptive control and can take place even in sim-
ulations of some of the ideal MRAC schemes of Chapter 6. The cause of
bursting in this case could be the computational error which acts as a small
bounded disturbance. There is a significant number of research results on
bursting and other undesirable phenomena, mainly for discrete-time plants
[2, 68, 81, 136, 203, 239]. We use the following example to explain one of
the main mechanisms of bursting.
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CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
Example 9.4.1 (Bursting) Let us consider the following MRAC scheme:
Plant
˙ x = ax + bu + d
Reference Model
˙ xm = −xm + r,
xm(0) = 0
Adaptive Controller
u = θ ω,
θ = [ θ 0 , c 0] , ω = [ x, r]
˙ θ = P r[ −e 1 ω sgn( b)] , e 1 = x − xm
(9.4.1)
The projection operator in (9.4.1) constrains θ to lie inside a bounded set, where
|θ| ≤ M 0 and M 0 > 0 is large enough to satisfy |θ∗| ≤ M 0 where θ∗ = − a+1 , 1
b
b
is the desired controller parameter vector. The input d in the plant equation is an
arbitrary unknown bounded disturbance. Let us assume that |d( t) | ≤ d 0 , ∀t ≥ 0 for
some d 0 > 0.
If we use the analysis of the previous sections, we can establish that all signals
are bounded and the tracking error e 1 ∈ S( d 20) , i.e.,
t
e 21 dτ ≤ d 20( t − t 0) + k 0 , ∀t ≥ t 0 ≥ 0
(9.4.2)
t 0
where k 0 depends on initial conditions. Furthermore, if r is dominantly rich, then
e 1, ˜
θ = θ − θ∗ converge exponentially to the residual set
S 0 = e 1 , ˜
θ |e 1 | + |˜
θ| ≤ cd 0
Let us consider the tracking error equation
˙ e 1 = −e 1 + b˜
θ ω + d
(9.4.3)
and choose the following disturbance
d = h( t)sat {−b(¯
θ − θ∗) ω}
where h( t) is a square wave of period 100 π sec and amplitude 1,
x
if |x| < d 0
sat {x} =
d
0
if x ≥ d 0
−d 0 if x ≤ −d 0
and ¯
θ is an arbitrary constant vector such that |¯
θ| < M 0. It is clear that |d( t) | < d 0
∀t ≥ 0. Let us consider the case where r is sufficiently rich but not dominantly
rich, i.e., r( t) has at least one frequency but |r|
d 0. Consider a time interval
[ t 1 , t 1 + T 1] over which |e 1( t) | ≤ d 0. Such an interval not only exists but is also large due to the uniform continuity of e 1( t) and the inequality (9.4.2). Because |e 1 | ≤ d 0
9.4. PERFORMANCE IMPROVEMENT OF MRAC
695
and 0 < |r|
d 0, we could have |b(¯
θ − θ∗) ω| < d 0 for some values of |¯
θ| < M 0
over a large interval [ t 2 , t 2 + T 2] ⊂ [ t 1 , t 1 + T ]. Therefore, for t ∈ [ t 2 , t 2 + T 2] we have d = −h( t) b(¯
θ − θ∗) ω and equation (9.4.3) becomes
˙ e 1 = −e 1 + b( θ − ¯
θ) ¯
ω
∀t ∈ [ t 2 , t 2 + T 2]
(9.4.4)
where ¯
ω = h( t) ω. Because r is sufficiently rich, we can establish that ¯
ω is PE and
therefore θ converges exponentially towards ¯
θ. If T 2 is large, then θ will get very
close to ¯
θ as t → t 2 + T 2. If we now choose ¯
θ to be a destabilizing gain or a gain
that causes a large mismatch between the closed-loop plant and the reference model,
then as θ → ¯
θ the tracking error will start increasing, exceeding the bound of d 0.
In this case d will reach the saturation bound d 0 and equation (9.4.4) will no longer
hold. Since (9.4.2) does not allow large intervals of time over which |e 1 | > d 0, we
will soon have |e 1 | ≤ d 0 and the same phenomenon will be repeated again.
The simulation results of the above scheme for a = 1, b = 1 , r = 0 . 1 sin 0 . 01 t, d 0 = 0 . 5, M 0 = 10 are given in Figures 9.4 and 9.5. In Figure 9.4, a stabilizing
¯
θ = [ − 3 , 4] is used and therefore no bursting occurred. The result with ¯
θ = [0 , 4] ,
where ¯
θ corresponds to a destabilizing controller parameter, is shown in Figure 9.5.
The tracking error e 1 and parameter θ 1( t), the first element of θ, are plotted as
functions of time. Note that in both cases, the controller parameter θ 1( t) converges
to ¯
θ 1, i.e., to − 3 (a stabilizing gain) in Figure 9.4, and to 0 (a destabilizing gain)
in Figure 9.5 over the period where d = −h( t) b(¯
θ − θ∗) ω. The value of ¯
θ 2 = 4 is
larger than θ∗ 2 = 1 and is responsible for some of the nonzero values of e 1 at steady
state shown in Figures 9.4 and 9.5.
Bursting is not the only phenomenon of bad behavior of robust MRAC.
Other phenomena such as chaos, bifurcation and large transient oscillations
could also be present without violating the boundedness results and m.s.s.
bounds developed in the previous sections [68, 81, 136, 239].
One way to eliminate most of the undesirable phenomena in MRAC is to
use reference input signals that are dominantly rich. These signals guarantee
a high level of excitation relative to the level of the modeling error, that in
turn guarantees exponential convergence of the tracking and parameter error
to residual sets whose size is of the order of the modeling error. The use
of dominantly rich reference input signals is not always possible especially
in the case of regulation or tracking of signals that are not rich. Therefore,
by forcing the reference signal to be dominantly rich, we eliminate bursting
and other undesirable phenomena at the expense of destroying the tracking
properties of the scheme in the case where the desired reference signal is not
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CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
0.3
0.2
0.1
tracking error
0
-0.10
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
time (sec)
2
0
-2
-4
estimated parameter
-60
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
time (sec)
Figure 9.4 Simulation results for Example 9.4.1: No bursting because of
the stabilizing ¯
θ = [ − 3 , 4] .
0.4
0.2
0
tracking error
-0.20
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
time (sec)
1
0
-1
-2
-3
estimated parameter
-40
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
time (sec)
Figure 9.5 Simulation results for Example 9.4.1: Bursting because of the
destabilizing ¯
θ = [0 , 4] .
9.4. PERFORMANCE IMPROVEMENT OF MRAC
697
rich. Another suggested method for eliminating bursting is to use adaptive
laws with dead zones. Such adaptive laws guarantee convergence of the
estimated parameters to constant values despite the presence of modeling
error provided of course the size of the dead zone is higher than the level
of the modeling error. The use of dead zones, however, does not guarantee
good transient performance or zero tracking error in the absence of modeling
errors.
In an effort to improve the transient and steady-state performance of
MRAC, a high gain scheme was proposed in [148], whose gains are switched
from one value to another based on a certain rule, that guarantees arbi-
trarily good transient and steady state tracking performance. The scheme
does not employ any of the on-line parameter estimators developed in this
book. The improvement in performance is achieved by modifying the MRC
objective to one of “approximate tracking.” As a result, non-zero tracking
errors remained at steady state. Eventhough the robustness properties of the
scheme of [148] are not analyzed, the high-gain nature of the scheme is ex-
pected to introduce significant trade-offs between stability and performance
in the presence of unmodeled dynamics.
In the following sections, we propose several modified MRAC schemes
that guarantee reduction of the size of bursts and an improved steady-state
tracking error performance.
9.4.1
Modified MRAC with Unnormalized Adaptive Laws
The MRAC schemes of Chapter 6 and of the previous sections are designed
using the certainty equivalence approach to combine a control law, that
works in the case of known parameters, with an adaptive law that provides
on-line parameter estimates to the controller. The design of the control law
does not take into account the fact that the parameters are unknown, but
blindly considers the parameter estimates provided by the adaptive laws to
be the true parameters. In this section we take a slightly different approach.
We modify the control law design to one that takes into account the fact
that the plant parameters are not exactly known and reduces the effect of
the parametric uncertainty on stability and performance as much as possi-
ble. This control law, which is robust with respect to parametric uncertainty,
can then be combined with an adaptive law to enhance stability and per-
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CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
formance. We illustrate this design methodology using the same plant and
control objective as in Example 9.4.1, i.e.,
Plant
˙ x = ax + bu + d
Reference Model
˙ xm = −xm + r,
xm(0) = 0
Let us choose a control law that employs no adaptation and meets the
control objective of stability and tracking as close as possible even though
the plant parameters a and b are unknown.
We consider the control law
u = ¯
θ 0 x + ¯ c 0 r + uα
(9.4.5)
where ¯
θ 0, ¯ c 0 are constants that depend on some nominal known values of
a, b if available, and uα is an auxiliary input to be chosen. With the input
(9.4.5), the closed-loop plant becomes
a + 1
˙ x = −x + b ¯
θ 0 +
x + b¯ c
b
0 r + buα + d
(9.4.6)
and the tracking error equation is given by
˙ e 1 = −e 1 + b˜¯
θ 0 e 1 + b˜¯
θ 0 xm + b˜¯ c 0 r + buα + d
(9.4.7)
where ˜¯
θ 0 = ¯
θ 0 + a+1 , ˜¯ c
are the constant parameter errors. Let us
b
0 = c 0 − 1 b
now choose
s + 1
uα = −
sgn( b) e
τ s
1
(9.4.8)
where τ > 0 is a small design constant. The closed-loop error equation
becomes
τ s
e 1 =
[ b˜¯
θ 0 xm + b˜¯ c 0 r + d]
(9.4.9)
τ s 2 + ( τ + |b| − τ b˜¯
θ 0) s + |b|
If we now choose τ to satisfy
1
0 < τ <
(9.4.10)
|˜¯
θ 0 |
the closed-loop tracking error transfer function is stable which implies that
e 1 , 1 e
s 1 ∈ L∞ and therefore all signals are bounded.
9.4. PERFORMANCE IMPROVEMENT OF MRAC
699
Another expression for the tracking error obtained using x = e 1 + xm
and (9.4.7) is
τ s
e 1 =
[ b˜¯
θ
( s + 1)( τ s + |b|)
0 x + b˜
¯ c 0 r + d]
(9.4.11)
or for |b| = τ we have
τ
|b|
1
e 1 =
−
( w + d)
(9.4.12)
|b| − τ τ s + |b|
s + 1
where w = b(˜¯
θ 0 x + ˜¯ c 0 r) is due to the parametric uncertainty. Because
x ∈ L∞, for any given τ ∈ (0 , 1 /|˜¯
θ 0 |), we can establish that there exists a
constant w 0 ≥ 0 independent of τ such that sup t |w( t) | ≤ w 0. It, therefore, follows from (9.4.12) that
τ
|b|
2 τ
|e
t
1( t) | ≤
e− |b|
τ
− e−t |e
( w
|b| − τ
τ
1(0) | + |b| − τ 0 + d 0)
(9.4.13)
where d 0 is the upper bound for |d( t) | ≤ d 0 , ∀t ≥ 0. It is, therefore, clear
that if we use the modified control law
s + 1
u = ¯
θ 0 x + ¯ c 0 r −
sgn( b) e
τ s
1
with
1
0 < τ < min |b|,
(9.4.14)
|˜¯
θ 0 |
then the tracking error will converge exponentially fast to the residual set
2 τ
Se = e 1 |e 1 | ≤
( w
|b| − τ
0 + d 0)
(9.4.15)
whose size reduces to zero as τ → 0.
The significance of the above control law is that no matter how we choose
the finite gains ¯
θ 0 , ¯ c 0, there always exist a range of nonzero design parameter
values τ for stability. Of course the further ¯
θ 0 is away from the desired
θ∗ 0 where θ∗ 0 = −a+1, the smaller the set of values of τ for stability as
b
indicated by (9.4.14