10
15
20
25
30
35
40
45
50 sec
(b)
Figure 6.10 Response of the MRAC scheme for Example 6.4.2 with r( t) =
unit step function.
Remark 6.4.4 The control law (6.4.30) is a modification of the certainty
equivalence control law up = θ ω and is motivated from stability con-
siderations. The additional term ˙ θ φ = −φ Γ φe 1 sgn( ρ∗) is a nonlin-
ear one that disappears asymptotically with time, i.e., up = θ ω + ˙ θ φ
converges to the certainty equivalence control law as t → ∞. The num-
ber and complexity of the additional terms in the certainty equivalence
control law increase with the relative degree n∗ as we demonstrate in
the next section.
362
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
5
0
tracking error
-50
5
10
15
20
25
30
35
40
45
50 sec
(a)
6
4
2
0
parameter error -2
-4
0
5
10
15
20
25
30
35
40
45
50 sec
(b)
Figure 6.11 Response of the MRAC scheme for Example 6.4.2 with
r( t) = 3 sin 4 . 9 t + 0 . 5 sin 0 . 7 t.
Remark 6.4.5 The proof of Theorem 6.4.2 may be accomplished by using
a minimal state space realization for the error equation
e 1 = Wm( s)( s + p 0) ρ∗ ˜
θ φ
The details of such an approach are left as an exercise for the reader.
6.4.3
Relative Degree n∗ = 3
As in the case of n∗ = 2, the transfer function Wm( s) of the reference
model cannot be chosen to be SPR because according to assumption (M2),
6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 363
Wm( s) should have the same relative degree as the plant transfer function.
Therefore, the choice of up = θ ω in the error equation
e 1 = Wm( s) ρ∗( up − θ∗ ω)
(6.4.33)
will not lead to the desired error equation where the tracking error is related
to the parameter error through an SPR transfer function. As in the case
of n∗ = 2, let us rewrite (6.4.33) in a form that involves an SPR transfer
function by using the techniques of Chapter 4, i.e., we express (6.4.33) as
e 1 = Wm( s)( s + p 0)( s + p 1) ρ∗ uf − θ∗ φ
(6.4.34)
where
1
1
uf =
u
ω
( s + p
p,
φ =
0)( s + p 1)
( s + p 0)( s + p 1)
and Wm( s) , p 0 , p 1 are chosen so that ¯
Wm( s) = Wm( s)( s + p 0)( s + p 1) is
SPR, which is now possible because the relative degree of ¯
Wm( s) is 1. For
simplicity and without loss of generality let us choose
1
Wm( s) = ( s + p 0)( s + p 1)( s + q 0)
for some q 0 > 0 so that
1
e 1 =
ρ∗( u
s + q
f − θ∗ φ)
(6.4.35)
0
The estimate of ˆ
e 1 of e 1 based on the estimates ρ, θ is given by
1
ˆ
e 1 =
ρ( u
s + q
f − θ φ)
(6.4.36)
0
If we proceed as in the case of n∗ = 2 we would attempt to choose
uf = θ φ
(6.4.37)
to make ˆ
e 1 = 1 [0] and obtain the error equation
s+ q 0
1
e 1 =
ρ∗ ˜
θ φ
(6.4.38)
s + q 0
364
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
The adaptive law
˙ θ = −Γ e 1 φ sgn( ρ∗)
(6.4.39)
will then follow by using the standard procedure. Equation (6.4.37), how-
ever, implies the use of the control input
up = ( s + p 0)( s + p 1) uf = ( s + p 0)( s + p 1) θ φ
(6.4.40)
which involves ¨
θ, that is not available for measurement. Consequently the
control law (6.4.40) cannot be implemented and the choice of uf = θ φ is
not feasible.
The difficulty of not being able to extend the results for n∗ = 1 , 2 to n∗ ≥
3 became the major obstacle in advancing research in adaptive control during
the 1970s. By the end of the 1970s and early 1980s, however, this difficulty
was circumvented and several successful MRAC schemes were proposed using
different approaches. Efforts to extend the procedure of n∗ = 1 , 2 to n∗ ≥ 3
continued during the early 1990s and led to new designs for MRAC . One such
design proposed by Morse [164] employs the same control law as in (6.4.40)
but the adaptive law for θ is modified in such a way that ¨
θ becomes an
available signal. This modification, achieved at the expense of a higher-order
adaptive law, led to a MRAC scheme that guarantees signal boundedness
and convergence of the tracking error to zero.
Another successful MRAC design that has it roots in the paper of Feuer
and Morse [54] is proposed in [162] for a third order plant with known high
frequency gain. In this design, the adaptive law is kept unchanged but the
control law is chosen as
up = θ ω + ua
where ua is designed based on stability considerations. Below we present
and analyze a very similar design as in [162].
We start by rewriting (6.4.35), (6.4.36) as
1
1
e 1 =
ρ∗(˜
θ φ + r
ρr
s + q
0) ,
ˆ
e 1 =
0
(6.4.41)
0
s + q 0
where r 0 = uf − θ φ and ˜
θ = θ − θ∗.
Because r 0 cannot be forced to be equal to zero by setting uf = θ φ, we
will focus on choosing up so that r 0 goes to zero as t → ∞. In this case,
6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 365
the estimation error 1 = e 1 − ˆ e 1 is not equal to e 1 because ˆ e 1 = 0 owing to
ρr 0 = 0. However, it satisfies the error equation
1
1 = e 1 − ˆ
e 1 =
( ρ∗ ˜
θ φ − ˜
ρr
s + q
0)
(6.4.42)
0
that leads to the adaptive law
˙ θ = −Γ 1 φ sgn( ρ∗) , ˙ ρ = γ 1 r 0
(6.4.43)
where Γ = Γ and γ > 0 by considering the Lyapunov-like function
2
˜
θ Γ − 1 ˜
θ
˜
ρ 2
V = 1 +
|ρ∗| +
2
2
2 γ
We now need to choose ua in up = θ ω + ua to establish stability for the
system (6.4.41) to (6.4.43). Let us now express r 0 as
1
r 0 = uf − θ φ =
u
s + p
1 − ˙
θ φ − θ φ 1
0
where
1
1
u 1 =
u
ω
s + p
p,
φ 1 = ( s + p 0) φ =
1
s + p 1
i.e.,
˙ r 0 = −p 0 r 0 + u 1 − ˙ θ φ − θ φ 1
(6.4.44)
Substituting for ˙ θ, we obtain
˙ r 0 = −p 0 r 0 + u 1 + φ Γ φ 1 sgn( ρ∗) − θ φ 1
(6.4.45)
If we now choose u 1 = −φ Γ φ 1sgn( ρ∗) + θ φ 1 then ˙ r 0 = −p 0 r 0 and r 0
converges to zero exponentially fast. This choice of u 1, however, leads to a
control input up that is not implementable since up = ( s + p 1) u 1 will involve
the first derivative of u 1 and, therefore, the derivative of e 1 that is not
available for measurement. Therefore, the term φ Γ φ 1 sgn( ρ∗) in (6.4.45)
cannot be eliminated by u 1. Its effect, however, may be counteracted by
introducing what is called a “nonlinear damping” term in u 1 [99]. That is,
we choose
2
u 1 = θ φ 1 − α 0 φ Γ φ r 0
(6.4.46)
366
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
where α 0 > 0 is a design constant, and obtain
2
˙ r 0 = − p 0 + α 0 φ Γ φ
r 0 + φ Γ φ 1sgn( ρ∗)
The purpose of the nonlinear term ( φ Γ φ)2 is to “damp out” the possible
destabilizing effect of the nonlinear term φ Γ φ 1 as we show in the analysis
to follow. Using (6.4.46), the control input up = ( s + p 1) u 1 is given by
up = θ ω + ˙ θ φ 1 − ( s + p 1) α 0( φ Γ φ)2 r 0
(6.4.47)
If we now perform the differentiation in (6.4.47) and substitute for the
derivative of r 0 we obtain
2
up = θ ω + ˙ θ φ 1 − 4 α 0 φ Γ φ φ Γ ˙ φ r 0 − α 0 ( p 1 − p 0) φ Γ φ r 0
4
3
+ α 20 φ Γ φ r 0 − α 0 φ Γ φ
1sgn( ρ∗)
(6.4.48)
where ˙ φ is generated from
˙
s
φ =
ω
( s + p 0)( s + p 1)
which demonstrates that up can be implemented without the use of differ-
entiators.
We summarize the main equations of the MRAC scheme in Table 6.3.
The stability properties of the proposed MRAC scheme listed in Table
6.3 are summarized as follows.
Theorem 6.4.3 The MRAC scheme of Table 6.3 guarantees that
(i) All signals in the closed-loop plant are bounded and r 0( t) , e 1( t) → 0 as
t → ∞.
(ii) If kp is known, r is sufficiently rich of order 2 n and Zp, Rp are coprime,
then the parameter error |˜
θ| = |θ − θ∗| and tracking error e 1 converge
to zero exponentially fast.
(iii) If r is sufficiently rich of order 2 n and Zp, Rp are coprime, then |˜
θ| and
e 1 converge to zero asymptotically (not necessarily exponentially fast).
(iv) The estimate ρ converges to a constant ¯
ρ asymptotically independent of
the richness of r.
6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 367
Table 6.3 MRAC scheme: n∗ = 3
Plant
y
Zp( s)
p = kp
u
R
p,
n∗ = 3
p( s)
Reference
y
model
m = Wm( s) r
˙ ω 1 = F ω 1 + gup, ω 1(0) = 0
˙ ω 2 = F ω 2 + gyp,
ω 2(0) = 0
˙ r
Control law
0 = −( p 0 + α 0( φ Γ φ)2) r 0 + φ Γ φ 1sgn( ρ∗)
up = θ ω + ua
ua = ˙ θ φ 1 −α 0( p 1 −p 0)( φ Γ φ)2 r 0 − 4 α 0 φ Γ φ( φ Γ ˙ φ) r 0
+ α 20( φ Γ φ)4 r 0 − α 0( φ Γ φ)3 1sgn( ρ∗)
˙ θ = −Γ 1 φ sgn( ρ∗) , ˙ ρ = γ 1 r 0
1 = e 1 − ˆ
e 1 , ˆ e 1 = 1 ρr
s+ q
0
Adaptive law
0
φ =
1
ω, ω = ω
( s+ p 0)( s+ p 1)
1 , ω 2 , yp, r
φ 1 = 1 ω, e
s+ p
1 = yp − ym
1
Γ = Γ > 0 , γ > 0 , α 0 > 0 are arbitrary design con-
Design
stants; Wm( s)( s + p 0)( s + p 1) is strictly proper and
variables
SPR; F, g are as in the case of n∗ = 1; Zp( s) , Rp( s)
and Wm( s) satisfy assumptions P1 to P4, M1 and M2,
respectively; sgn( ρ∗) = sgn( kp/km)
Proof (i) The equations that describe the stability properties of the closed-loop
plant are
˙1 = −q 0 1 + ρ∗ ˜
θ φ − ˜
ρr 0
˙ r 0 = −( p 0 + α 0( φ Γ φ)2) r 0 + φ Γ φ 1 sgn( ρ∗) (6.4.49)
˙˜ θ = −Γ 1 φ sgn( ρ∗) , ˙˜ ρ = γ 1 r 0
368
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
We propose the Lyapunov-like function
2
Γ − 1
˜
ρ 2
r 2
V = 1 + |ρ∗|˜
θ
˜
θ +
+ γ 0
2
2
2 γ
0 2
where γ 0 > 0 is a constant to be selected. The time derivative of V along the
trajectories of (6.4.49) is given by
˙
V
= −q 2
0 1 − γ 0 p 0 r 2
0 − γ 0 α 0 r 2
0 ( φ Γ φ)2 + γ 0 1 r 0 φ Γ φ sgn( ρ∗)
≤ −q 2
0 1 − γ 0 p 0 r 2
0 − γ 0 α 0 r 2
0 ( φ Γ φ)2 + γ 0 | 1 | |r 0 |φ Γ φ
By completing the squares we obtain
2
2
˙
q
|r
r 2
V ≤ −q 1
0
0 |φ Γ φ
0 ( φ Γ φ)2
0
−
|
+ γ 2
− γ
2
2
1 | − γ 0
q
0
0 p 0 r 2
0 − γ 0 α 0 r 2
0 ( φ Γ φ)2
0
2 q 0
2
γ