laws for estimating θ∗ by using the results of Chapter 4. We should note
that (6.4.5) and (6.4.6) hold for any relative degree and will also be used in
later sections.
The estimate ˆ
e 1( t) of e 1( t) based on θ( t), the estimate of θ∗ at time t, is given by
ˆ
e 1 = Wm( s) ρ up − θ ω
(6.4.7)
where ρ is the estimate of ρ∗. Because the control input is given by
up = θ ( t) ω
it follows that ˆ
e 1 = Wm( s)[0]; therefore, the estimation error 1 defined in
Chapter 4 as 1 = e 1 − ˆ e 1 may be taken to be equal to e 1, i.e., 1 = e 1.
Consequently, (6.4.7) is not needed and the estimate ρ of ρ∗ does not have
to be generated. Substituting for the control law in (6.4.5), we obtain the
error equation
˙ e = Ace + ¯
Bcρ∗ ˜
θ ω, e(0) = e 0
e 1 = Cc e
(6.4.8)
where
¯
Bc = Bcc∗ 0
or
e 1 = Wm( s) ρ∗ ˜
θ ω
which relates the parameter error ˜
θ = θ( t) − θ∗ with the tracking error e 1.
Because Wm( s) = Cc ( sI − Ac) − 1 Bcc∗ 0 is SPR and Ac is stable, equation (6.4.8) is in the appropriate form for applying the SPR-Lyapunov design
approach.
348
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
We therefore proceed by proposing the Lyapunov-like function
e P
˜
θ Γ − 1 ˜
θ
V ˜
θ, e =
ce +
|ρ∗|
(6.4.9)
2
2
where Γ = Γ > 0 and Pc = Pc > 0 satisfies the algebraic equations
PcAc + Ac Pc = −qq − νcLc
P ¯
cBc
= Cc
where q is a vector, Lc = Lc > 0 and νc > 0 is a small constant, that are
implied by the MKY lemma. The time derivative ˙
V of V along the solution
of (6.4.8) is given by
˙
e qq e
ν
V = −
− c e L
¯
B
2
2
ce + e Pc
cρ∗ ˜
θ ω + ˜
θ Γ − 1 ˙˜
θ|ρ∗|
Because e P ¯
cBc = e 1 and ρ∗ = |ρ∗| sgn( ρ∗), we can make ˙
V ≤ 0 by choosing
˙˜ θ = ˙ θ = −Γ e 1 ω sgn( ρ∗)
(6.4.10)
which leads to
˙
e qq e
ν
V = −
− c e L
2
2
ce
(6.4.11)
Equations (6.4.9) and (6.4.11) imply that V and, therefore, e, ˜
θ ∈ L∞.
Because e = Yc −Ym and Ym ∈ L∞, we have Yc ∈ L∞, which implies that
yp, ω 1 , ω 2 ∈ L∞. Because up = θ ω and θ, ω ∈ L∞ we also have up ∈ L∞.
Therefore all the signals in the closed-loop plant are bounded. It remains to
show that the tracking error e 1 = yp − ym goes to zero as t → ∞.
From (6.4.9) and (6.4.11) we establish that e and therefore e 1 ∈ L 2.
Furthermore, using θ, ω, e ∈ L∞ in (6.4.8) we have that ˙ e, ˙ e 1 ∈ L∞. Hence,
e 1 , ˙ e 1 ∈ L∞ and e 1 ∈ L 2, which, by Lemma 3.2.5, imply that e 1( t) → 0 as t → ∞.
We summarize the main equations of the MRAC scheme in Table 6.1.
The stability properties of the MRAC scheme of Table 6.1 are given by
the following theorem.
Theorem 6.4.1 The MRAC scheme summarized in Table 6.1 guarantees
that:
6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 349
Table 6.1 MRAC scheme: n∗ = 1
Plant
y
Zp( s)
p = kp
u
R
p,
n∗ = 1
p( s)
Reference
y
Zm( s)
model
m = Wm( s) r,
Wm( s) = km Rm( s)
˙ ω 1 = F ω 1 + gup, ω 1(0) = 0
˙ ω
Control law
2 = F ω 2 + gyp,
ω 2(0) = 0
up = θ ω
ω = [ ω 1 , ω 2 , yp, r] , ω 1 ∈ Rn− 1 , ω 2 ∈ Rn− 1
˙ θ = −Γ e
Adaptive law
1 ω sgn( ρ∗)
e 1 = yp − ym, sgn( ρ∗) = sgn( kp/km)
Zp, Rp and Wm( s) satisfy assumptions P1 to P4,
Assumptions
and M1 and M2, respectively; Wm( s) is SPR; ( sI −
F ) − 1 g = α( s) , α( s) = [ sn− 2 , sn− 3 , . . . s, 1] , where Λ( s)
Λ = Λ0 Zm is Hurwitz, and Λ0( s) is of degree n− 1 −qm,
qm is the degree of Zm( s); Γ = Γ > 0 is arbitrary
(i) All signals in the closed-loop plant are bounded and the tracking error
e 1 converges to zero asymptotically with time for any reference input
r ∈ L∞.
(ii) If r is sufficiently rich of order 2 n, ˙ r ∈ L∞ and Zp( s) , Rp( s) are relatively coprime, then the parameter error |˜
θ| = |θ − θ∗| and the tracking
error e 1 converge to zero exponentially fast.
Proof
(i) This part has already been completed above.
(ii) Equations (6.4.8) and (6.4.10) have the same form as (4.3.30) and (4.3.35)
with n 2 s = 0 in Chapter 4 whose convergence properties are established by Corollary
350
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
4.3.1. Therefore, by using the same steps as in the proof of Corollary 4.3.1 we can
establish that if ω, ˙ ω ∈ L∞ and ω is PE then ˜
θ( t) → 0 exponentially fast. If ˙ r ∈ L∞
then it follows from the results of part (i) that ˙ ω ∈ L∞. For the proof to be
complete, it remains to show that ω is PE.
We express ω as
( sI − F ) − 1 gG− 1
p ( s) yp
( sI − F ) − 1 gy
ω =
p
y
(6.4.12)
p
r
Because yp = ym + e 1 = Wm( s) r + e 1 we have
ω = ωm + ¯
ω
(6.4.13)
where
ωm = H( s) r,
¯
ω = H 0( s) e 1
and
( sI − F ) − 1 gG− 1
p ( s) Wm( s)
( sI − F ) − 1 gG− 1
p ( s)
( sI − F ) − 1 gW
( sI − F ) − 1 g
H( s) =
m( s)
W
, H 0( s) =
m( s)
1
1
0
The vector ¯
ω is the output of a proper transfer matrix whose poles are stable and
whose input e 1 ∈ L 2 ∩ L∞ and goes to zero as t → ∞. Hence, from Corollary 3.3.1
we have ¯
ω ∈ L 2 ∩ L∞ and |¯
ω( t) | → 0 as t → ∞. It then follows from Lemma 4.8.3
that ω is PE if ωm is PE.
It remains to show that ωm is PE when r is sufficiently rich of order 2 n.
Because r is sufficiently rich of order 2 n, according to Theorem 5.2.1, we can
show that ωm is PE by proving that H( jω 1) , H( jω 2) , . . . , H( jω 2 n) are linearly independent on C 2 n for any ω 1 , ω 2 , . . . , ω 2 n ∈ R with ωi = ωj for i = j.
From the definition of H( s), we can write
α( s) Rp( s) kmZm( s)
1
α( s) k
1
H( s) =
pZp( s) kmZm( s)
H
k
=
1( s)
pZp( s)Λ( s) Rm( s)
Λ( s) kpZp( s) kmZm( s)
kpZp( s)Λ( s) Rm( s)
Λ( s) kpZp( s) Rm( s)
(6.4.14)
Because all the elements of H 1( s) are polynomials of s with order less than or equal
to that of Λ( s) Zp( s) Rm( s), we can write
sl
sl− 1
H 1( s) = ¯
H
.
(6.4.15)
..
1
6.4. DIRECT MRAC WITH UNNORMALIZED ADAPTIVE LAWS 351
where l = 2 n − 1 + qm is the order of the polynomial Λ( s) Zp( s) Rm( s), qm is the degree of Zm( s) and ¯
H ∈ R 2 n×( l+1) is a constant matrix.
We now prove by contradiction that ¯
H in (6.4.15) is of full rank, i.e., rank( ¯
H) =
2 n. Suppose rank( ¯
H) < 2 n, i.e., there exists a constant vector C ∈ R 2 n with C = 0
such that
C ¯
H = 0
or equivalently
C H 1( s) = 0
(6.4.16)
for all s ∈ C. Let C = [ C 1 , C 2 , c 3 , c 4] , where C 1 , C 2 ∈ Rn− 1 , c 3 , c 4 ∈ R 1, then (6.4.16) can be written as
C 1 α( s) Rp( s) kmZm( s) + C 2 α( s) kpZp( s) kmZm( s)
+ c 3Λ( s) kpZp( s) kmZm( s) + c 4Λ( s) Rm( s) kpZp( s) = 0
(6.4.17)
Because the leading coefficient of the polynomial on the left hand side is c 4, for
(6.4.17) to hold, it is necessary that c 4 = 0. Therefore,
[ C 1 α( s) Rp( s) + C 2 α( s) kpZp( s) + c 3Λ( s) kpZp( s)] kmZm( s) = 0
or equivalently
C 1 α( s) Rp( s) + C 2 α( s) kpZp( s) + c 3Λ( s) kpZp( s) = 0
(6.4.18)
Equation (6.4.18) implies that
Z
C
k
p( s)
1 α( s)
p
= −
(6.4.19)
Rp( s)
c 3Λ( s) + C 2 α( s)
Noting that c 3Λ( s) + C 2 α( s) is of order at most equal to n − 1, (6.4.19) contradicts our assumption that Zp( s) , Rp( s) are coprime. Therefore ¯
H must be of full rank.
Now consider the 2 n × 2 n matrix L( ω 1 , . . . , ω 2 n) = [ H( jω 1) , , . . . , H( jω 2 n)].
Using (6.4.14) and (6.4.15), we can express L( ω 1 , ω 2 , . . . , ω 2 n) as
( jω 1) l
( jω 2) l
. . .
( jω 2 n) l
( jω 1) l− 1
( jω 2) l− 1 . . . ( jω 2 n) l− 1
L( ω 1 , . . . , ω 2 n) =
¯
H
.
.
.
..
..
..
1
1
. . .
1
1
0
. . .
0
D( jω 1)
0
1
. . .
0
D( jω
×
2)
.
(6.4.20)
0
. .
0
0
0
· · ·
1
D( jω 2 n)
352
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
where D( s) = kpZp( s)Λ( s) Rm( s). Note that the matrix in the middle of the right-hand side of (6.4.20) is a submatrix of the Vandermonte matrix, which is always
nonsingular for ωi = ωk, i = k; i, k = 1 , . . . , 2 n. We, therefore, conclude from (6.4.20) that L( ω 1 , . . . , ω 2 n) is of full rank which implies that H( jω 1) , . . . , H( jω 2 n) are linearly independent on C 2 n and the proof is complete.
✷
Example 6.4.1 Let us consider the second order plant
k
y
p( s + b 0)
p =
u
( s 2 + a
p
1 s + a 0)
where kp > 0 , b 0 > 0 and kp, b 0 , a 1 , a 0 are unknown constants. The desired performance of the plant is specified by the reference model
1
ym =
r
s + 1
Using Table 6.1, the