or
∆
e
m( s)( s + a)
1 = W ( s) ˜
θy +
b
( s + a
mr
(9.3.18)
m)( s + am − θ∗∆ m( s))
The ideal error equation obtained by setting ∆ m( s) ≡ 0 is
1
e
˜
1 =
θy
s + am
which, based on the results of Chapter 4, suggests the adaptive law
˙ θ = ˙˜ θ = −γe 1 y
(9.3.19)
due to the SPR property of
1
. As we showed in Chapter 4, the adaptive
s+ am
control law (9.3.16), (9.3.19) meets the control objective for the plant (9.3.15)
provided ∆ m( s) ≡ 0.
The presence of ∆ m( s) introduces an external unknown input that de-
pends on ∆ m, r and acts as a bounded disturbance in the error equation.
Furthermore ∆ m( s) changes the SPR transfer function that relates e 1 to ˜
θy
from
1
to W ( s) as shown by (9.3.18). As we showed in Chapter 8, adap-
s+ am
tive laws such as (9.3.19) may lead to instability when applied to (9.3.15) due
to the presence of the disturbance term that appears in the error equation.
One way to counteract the effect of the disturbance introduced by ∆ m( s) = 0
and r = 0 is to modify (9.3.19) using leakage, dead zone, projection etc. as
shown in Chapter 8. For this example, let us modify (9.3.19) using the fixed
σ-modification, i.e.,
˙ θ = −γe 1 y − γσθ
(9.3.20)
where σ > 0 is small.
Even with this modification, however, we will have difficulty establishing
global signal boundedness unless W ( s) is SPR. Because W ( s) depends on
∆ m( s) which is unknown and belongs to the class defined by (9.3.17), the
SPR property of W ( s) cannot be guaranteed unless ∆ m( s) ≡ 0. Below we
treat the following cases that have been considered in the literature of robust
adaptive control.
9.3. ROBUST MRAC
659
Case I: W ( s) Is SPR–Global Boundedness
Assume that ∆ m( s) is such that W ( s) is guaranteed to be SPR. The error
equation (9.3.18) may be expressed as
e 1 = W ( s)(˜
θy + d)
(9.3.21)
where d = 1 − W − 1( s) b
s+ a
mr is guaranteed to be bounded due to r ∈ L∞
m
and the stability of W − 1( s) which is implied by the SPR property of W ( s).
The state-space representation of (9.3.21) given by
˙ e = Ace + Bc(˜
θy + d)
(9.3.22)
e 1 = Cc e
where Cc ( sI − Ac) − 1 Bc = W ( s) motivates the Lyapunov-like function
e P
˜
θ 2
V =
ce +
2
2 γ
with Pc = Pc > 0 satisfies the equations in the LKY Lemma. The time
derivative of V along the solution of (9.3.20) and (9.3.22) is given by
˙
V = −e qq e − νce Lce + e 1 d − σ ˜
θθ
where νc > 0, Lc = Lc > 0 and q are defined by the LKY Lemma. We have
˙
V ≤ −νcλc|e| 2 + c|e||d| − σ|˜
θ| 2 + σ|˜
θ||θ∗|
where c = C
, λc = λmin( Lc). Completing the squares and adding and
subtracting αV , we obtain
˙
c 2 |d| 2
σ|θ∗| 2
V ≤ −αV +
+
(9.3.23)
2 νcλc
2
where α = min{ νcλc , σγ} and λ
λ
p = λmax( Pc), which implies that V and,
p
therefore, e, θ, e 1 ∈ L∞ and that e 1 converges to a residual set whose size
is of the order of the disturbance term |d| and the design parameter σ. If,
instead of the fixed- σ, we use the switching σ or the projection, we can verify
that as ∆ m( s) → 0, i.e., d → 0, the tracking error e 1 reduces to zero too.
Considerable efforts have been made in the literature of robust adaptive
control to establish that the unmodified adaptive law (9.3.19) can be used to
660
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
establish stability in the case where y is PE [85, 86]. It has been shown [170]
that if W ( s) is SPR and y is PE with level α 0 ≥ γ 0 ν 0 + γ 1 where ν 0 is an upper bound for |d( t) | and γ 0 , γ 1 are some positive constants, then all signals
in the closed-loop system (9.3.18), (9.3.19) are bounded. Because α 0 , ν 0 are
proportional to the amplitude of r, the only way to generate the high level
α 0 of PE relative to ν 0 is through the proper selection of the spectrum
of r. If ∆ m( s) is due to fast unmodeled dynamics, then the richness of r
should be achieved in the low frequency range, i.e., r should be dominantly
rich. Intuitively the spectrum of r should be chosen so that |W ( jω) |
|W ( jω) −
1
| for all ω in the frequency range of interest.
jω+ am
An example of ∆ m( s) that guarantees W ( s) to be SPR is ∆ m( s) = µs
1+ µs
where µ > 0 is small enough to guarantee that ∆ m( s) satisfies (9.3.17).
Case II: W ( s) Is Not SPR–Semiglobal Stability
When W ( s) is not SPR, the error equation (9.3.18) may not be the appro-
priate one for analysis. Instead, we express the closed-loop plant (9.3.15),
(9.3.16) as
1
1
y =
( θ∗y + b
∆
s + a
mr + ˜
θy) + s + a m( s) u
and obtain the error equation
1
e 1 =
(˜
θy + ∆
s + a
m( s) u)
m
or
˙ e 1 = −ame 1 + ˜
θy + η
(9.3.24)
η = ∆ m( s) u = ∆ m( s)( θy + bmr)
Let us analyze (9.3.24) with the modified adaptive law (9.3.20)
˙ θ = −γe 1 y − γσθ
The input η to the error equation (9.3.24) cannot be guaranteed to be
bounded unless u is bounded. But because u is one of the signals whose
boundedness is to be established, the equation η = ∆ m( s) u has to be ana-
lyzed together with the error equation and adaptive law. For this reason, we
need to express η = ∆ m( s) u in the state space form. This requires to assume
some structural information about ∆ m( s). In general ∆ m( s) is assumed to
be proper and small in some sense. ∆ m( s) may be small at all frequencies,
9.3. ROBUST MRAC
661
i.e., ∆ m( s) = µ∆1( s) where µ > 0 is a small constant whose upper bound
is to be established and ∆1( s) is a proper stable transfer function. ∆ m( s)
could also be small at low frequencies and large at high frequencies. This is
the class of ∆ m( s) often encountered in applications where ∆ m( s) contains
all the fast unmodeled dynamics which are usually outside the frequency
range of interest. A typical example is
2 µs
∆ m( s) = −
(9.3.25)
1 + µs
which makes (9.3.15) a second-order nonminimum-phase plant and W ( s) in
(9.3.18) nonminimum-phase and, therefore, non-SPR. To analyze the stabil-
ity of the closed-loop plant with ∆ m( s) specified in (9.3.25), we express the
error equation (9.3.24) and the adaptive law (9.3.20) in the state-space form
˙ e 1 = −ame 1 + ˜
θy + η
µ ˙ η = −η − 2 µ ˙ u
(9.3.26)
˙˜ θ = −γe 1 y − γσθ
Note that in this representation µ = 0 ⇒ η = 0, i.e., the state η is a
good measure of the effect of the unmodeled dynamics whose size is charac-
terized by the value of µ. The stability properties of (9.3.26) are analyzed
by considering the following positive definite function:
e 2
˜
θ 2
µ
V ( e
1
1 , η, ˜
θ) =
+
+ ( η + e
2
2 γ
2
1)2
(9.3.27)
For each µ > 0 and some constants c 0 > 0 , α > 0, the inequality
V ( e 1 , η, ˜
θ) ≤ c 0 µ− 2 α
defines a closed sphere L( µ, α, c 0) in R 3 space. The time derivative of V
along the trajectories of (9.3.26) is
˙
V
= −ame 21 − σ˜ θθ − η 2 + µ[ ˙ e 1 − 2 ˙ u]( η + e 1)
σ ˜
θ 2
σθ∗ 2
≤ −a
˜
me 2
1 −
− η 2 +
+ µc[ e 4
θ 2 + |e
2
2
1 + |e 1 | 3 + e 2
1 + e 2
1 | ˜
θ| + e 21
1 | ˜
θ 2
+ |e 1 ||˜
θ| + |e 1 ||˜
θ||η| + |e 1 | 3 |η| + |e 1 | + |η| + η 2 + |e 1 ||η|
+ |η||˜
θ| + |η|˜
θ 2 + e 2
˜
1 |η| + |e 1 θ 2 ||η| + | ˜
θ|η 2 + |e 1 || ˙ r| + |η|| ˙ r|]
(9.3.28)
662
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
for some constant c ≥ 0, where the last inequality is obtained by substituting
for ˙ e 1, ˙ u = ˙ θy + θ ˙ y + bm ˙ r and taking bounds. Using the inequality 2 αβ ≤
α 2 + β 2, the multiplicative terms in (9.3.28) can be manipulated so that after
some tedious calculations (9.3.28) is rewritten as
˙
a
σ ˜
θ 2
η 2
V
≤ − me 21 −
−
− η 2 1 − µc( |˜
θ| + 1)
2
4
2
2
a
− e 2
m
1
− µc( e 2
2
1 + |e 1 | + | ˜
θ| + ˜
θ 2 + |η| + |e 1 ||η| + 1)
σ|θ∗| 2
− ˜
θ 2 σ − µc( |e
(9.3.29)
4
1 | + 1 + |e 1 ||η|) + µc| ˙
r| 2 + µc +
2
Inside L( µ, α, c 0) , |e 1 |, |˜
θ| can grow up to O( µ−α) and |η| can grow up to
O( µ− 1 / 2 −α). Hence, there exist positive constants k 1 , k 2 , k 3 such that inside L( µ, α, c 0), we have
|e 1 | < k 1 µ−α,
|˜
θ| < k 2 µ−α,
|η| < k 3 µ− 1 / 2 −α
For all e 1 , η, ˜
θ inside L( µ, α, c 0) , (9.3.29) can be simplified to
˙
a
η 2
σ
V
≤ − m e 2
−
˜
θ 2 − η 2 1 − β
4 1 − 2
4
2
2 µ 1 −α
a
−e 2
m
1
− µ 1 / 2 − 2 αβ
− β
2
1
− ˜
θ 2 σ 4
3 µ 1 −α
(9.3.30)
σθ∗ 2
+ µc| ˙ r| 2 + µc + 2
for some positive constants β 1 , β 2 , β 3. If we now fix σ > 0 then for 0 < α <
1 / 4, there exists a µ∗ > 0 such that for each µ ∈ (0 , µ∗]
am
1
σ
≥ µ 1 / 2 − 2 αβ
≥ β
> β
2
1 ,
2
2 µ 1 −α,
4
3 µ 1 −α
Hence, for each µ ∈ (0 , µ∗] and e 1 , η, ˜
θ inside L( µ, α, c 0), we have
˙
a
η 2
σ
σθ∗ 2
V < − m e 2
−
˜
θ 2 +
+ µc| ˙ r| 2 + µc
(9.3.31)
2 1 − 2
4
2
On the other hand, we can see from the definition of V that
e 2
˜
θ 2
µ
a
η 2
σ
V ( e
1
m
˜
1 , η, ˜
θ) =
+
+ ( η + e
e 2
+ θ 2
2
2 γ
2
1)2 ≤ c 4
2 1 + 2
4
9.3. ROBUST MRAC
663
where c 4 = max{ 1+2 µ, 2 , 2 µ}. Thus, for any 0 < β ≤ 1 /c
a
4, we have
m
γσ
˙
σθ∗ 2
V < −βV +
+ µc| ˙ r| 2 + µc
2
Because r, ˙ r are uniformly bounded, we define the set
1 σθ∗ 2
D 0( µ) = e 1 , ˜
θ, η V ( e 1 , η, ˜
θ) <
+ µc| ˙