Any adaptive law from Tables 6.5, 6.6
Assumptions
Plant and reference model satisfy assumptions P1
to P4 and M1, M2 respectively.
F, g chosen so that ( sI − F ) − 1 g = α( s) , where
Design variables
Λ( s)
α( s) = [ sn− 2 , sn− 3 , . . . , s, 1] for n ≥ 2 and α( s) =
0 for n = 1; Λ( s) = sn− 1 + λn− 2 sn− 2 + · · · + λ 0 is
Hurwitz.
Figure 6.14 is obtained by rewriting up as
up = θ∗ ω + ˜
θ ω
where ˜
θ = θ − θ∗, and by using the results of Section 6.4.1, in particular,
equation (6.4.6) to absorb the term θ∗ ω.
For ˜
θ = θ − θ∗ = 0, the closed-loop MRAC scheme shown in Figure 6.14
reverts to the one in the known parameter case shown in Figure 6.5. For
˜
θ = 0, the stability of the closed-loop MRAC scheme depends very much on
the properties of the input 1 ˜
θ ω, which, in turn, depend on the properties
c∗ 0
of the adaptive law that generates the trajectory ˜
θ( t) = θ( t) − θ∗.
6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS
387
Table 6.5 Adaptive laws based on e 1 = Wm( s) ρ∗( up − θ∗ ω)
A. Based on the SPR-Lyapunov approach
Parametric
e 1 = Wm( s) L( s)[ ρ∗( uf − θ∗ φ)]
model
˙ θ = −Γ φ sgn( kp/km)
˙ ρ = γ ξ,
= e 1 − ˆ e 1 − Wm( s) L( s)( n 2 s)
Adaptive law
ˆ
e 1 = Wm( s) L( s)[ ρ( uf − θ φ)]
ξ = uf − θ φ, φ = L− 1( s) ω
uf = L− 1( s) up, n 2 s = φ φ + u 2 f
Assumptions
Sign ( kp) is known
Design
Γ = Γ > 0 , γ > 0; Wm( s) L( s) is proper and SPR;
variables
L− 1( s) is proper and has stable poles
B. Gradient algorithm with known sgn( kp)
Parametric
e 1 = ρ∗( uf − θ∗ φ)
model
˙ θ = −Γ φ sgn( kp/km)
˙ ρ = γ ξ
= e 1 −ˆ e 1
m 2
Adaptive law
ˆ
e 1 = ρ( uf − θ φ)
φ = Wm( s) ω, uf = Wm( s) up
ξ = uf − θ φ
m 2 = 1 + φ φ + u 2 f
Design
Γ = Γ > 0 , γ > 0
variables
388
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
Table 6.5 (Continued)
C. Gradient algorithm with unknown sgn( kp)
Parametric
e 1 = ρ∗( uf − θ∗ φ)
model
˙ θ = −N( x 0)Γ φ
˙ ρ = N ( x 0) γ ξ
N ( x 0) = x 20 cos x 0
Adaptive law
x 0 = w 0 + ρ 2 ,
˙
w
2 γ
0 = 2 m 2 , w 0(0) = 0
= e 1 −ˆ e 1 , ˆ
e
m 2
1 = N ( x 0) ρ( uf − θ φ)
φ = Wm( s) ω, uf = Wm( s) up
ξ = uf − θ φ, m 2 = 1 + n 2 s, n 2 s = φ φ + u 2 f
Design
Γ = Γ > 0 , γ > 0
variables
The following theorem gives the stability properties of the MRAC scheme
shown in Figures 6.13 and 14 when the adaptive laws in Tables 6.5 and 6.6
are used to update θ( t) on-line.
Theorem 6.5.1 The closed-loop MRAC scheme shown in Figure 6.13 and
described in Table 6.4 with any adaptive law from Tables 6.5 and 6.6 has the
following properties:
(i) All signals are uniformly bounded.
(ii) The tracking error e 1 = yp − ym converges to zero as t → ∞.
(iii) If the reference input signal r is sufficiently rich of order 2 n, ˙ r ∈ L∞
and Rp, Zp are coprime, the tracking error e 1 and parameter error
˜
θ = θ − θ∗ converge to zero for the adaptive law with known sgn( kp) .
The convergence is asymptotic in the case of the adaptive law of Table
6.5(A, B) and exponential in the case of the adaptive law of Table 6.6.
6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS
389
Table 6.6 Adaptive laws based on Wm( s) up = θ∗ φp
g( θ) = c
Constraint
0 − c 0 sgn c 0 ≤ 0
θ = [ θ 1 , θ 2 , θ 3 , c 0]
Projection
Γ x
if |c 0( t) | > c 0 or
operator
P
if |c
r[Γ x]=
0( t) | = c
0 and (Γ x) ∇g ≤ 0
Γ x−Γ ∇g∇g Γ x otherwise
∇g Γ ∇g
A. Gradient Algorithm
Adaptive law
˙ θ = P r[Γ φp]
Design variable
Γ = Γ > 0
B. Integral gradient aAlgorithm
˙ θ = P r[ −Γ( Rθ + Q)]
Adaptive law
˙
R = −βR + φpφp ,
R(0) = 0
m 2
˙
Q = −βQ − φp z, Q(0) = 0
m 2
z = Wm( s) up
Design
Γ = Γ > 0 , β > 0
variable
C. Least-squares with covariance resetting
˙ θ = P r[ P φp]
Adaptive law
−P φ
pφp P
if |c 0( t) | > c
˙
m 2
0 or
P=
if |c
0( t) | = c 0 and ( P φp)
g ≤ 0
0
otherwise
P ( t+
r ) = P 0 = ρ 0 I
Design
P (0) = P (0) > 0, tr is the time for which
variable
λmin( P ( t)) ≤ ρ 1 , ρ 1 > ρ 0 > 0
Common signals and variables
= Wm( s) up−ˆ z, ˆ
z = θ φ
m 2
p, m 2 = 1+ φp φp
φp =[ Wm( s) ω 1 , Wm( s) ω 2 , Wm( s) yp, yp]
|c 0(0) | ≥ c 0 , 0 < c 0 ≤ |c∗ 0 |, sgn( c 0(0)) = sgn( kp/km) 390
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
Table 6.7 Hybrid MRAC
Parametric
Wm( s) up = θ∗ φp
model
˙ ω 1 = F ω 1 + gup
Control law
˙ ω 2 = F ω 2 + gyp
up( t) = θk ω( t) , t ∈ ( tk, tk+1]
θk+1= ¯
P r θk+Γ tk+1
t
( τ ) φp( τ) dτ ; k=0 , 1 , 2 , . . .
k
Hybrid
= Wm( s) up−ˆ z, ˆ
z( t) = θ
adaptive law
m 2
k φp,
t ∈ ( tk, tk+1]
m 2 = 1 + βφp φp
where ¯
P r[ ·] is the discrete-time projection
¯
x
if |x
P r{x} =
0 | ≥ c 0
x + γ 0 ( c
γ 00
0 − x 0)sgn( c 0)
otherwise
where x 0 is the last element of the vector x; γ 0 is
the last column of the matrix Γ, and γ 00 is the last
element of γ 0
Design
β > 1; tk = kTs, Ts > 0; Γ = Γ > 0
variables
2 − Tsλmax(Γ) > γ for some γ > 0
Outline of Proof The proof follows the same procedure as that for the ex-
amples presented in Sections 6.5.1, 6.5.2. It is completed in five steps.
Step 1. Express the plant input and output in terms of the adaptation error
˜
θ ω. Using Figure 6.14 we can verify that the transfer function between the input
r + 1 ˜
θ ω and the plant output y
c∗
p is given by
0
1
y
˜
p = Gc( s)
r +
θ ω
c∗ 0
where
c∗
G
0 kpZp
c( s) = (1 − θ∗ 1 ( sI − F) − 1 g) Rp − kpZp[ θ∗ 2 ( sI − F) − 1 g + θ∗ 3]
Because of the matching equations (6.3.12) and (6.3.13), and the fact that ( sI −
F ) − 1 g = α( s) , we have, after cancellation of all the stable common zeros and poles
Λ( s)
6.5. DIRECT MRAC WITH NORMALIZED ADAPTIVE LAWS
391
✲ W
ym
m( s)
− ❄
❧
e
Σ ✲
1
✒
r
u
+
✲ c
+
p
yp
✻
0
✲ ❧
Σ
✲ Gp( s)
+ +
✒✂✍ ✻
+
❄
❄
✂
✂
( sI − F ) − 1 g
( sI − F ) − 1 g
✒
θ
✛ ω 1
1
✒
✄
✄
ω 2
θ
✛
2
✒
✄
θ
✛
3
Adaptive Law
from
✛
yp, ym, up
Tables 6.5 to 6.7
Figure 6.13 Block diagram of direct MRAC with normalized adaptive
law.
in Gc( s), that Gc( s) = Wm( s). Therefore, the plant output may be written as
1
y
˜
p = Wm( s)
r +
θ ω
(6.5.42)
c∗ 0
Because yp = Gp( s) up and G− 1
p ( s) has stable poles, we have
1
u
˜
p = G− 1
p ( s) Wm( s)
r +
θ ω
(6.5.43)
c∗ 0
where G− 1
p ( s) Wm( s) is biproper Because of Assumption M2.
We now define the fictitious normalizing signal mf as
m 2
2
2
f = 1 + up
+ yp
(6.5.44)
where
·
denotes the L 2 δ-norm for some δ > 0. Using the properties of the L 2 δ
norm it follows that
mf ≤ c + c ˜
θ ω
(6.5.45)
392
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
1 ˜
θ ω
c∗ 0
1
˜
θ
Adaptive
✛
× ✛
✛
ω
c∗ 0
Law
✻
+
❄
ω
r ✲ ❧
+
u
y
Σ ✲ c∗
✲ ❧
p
✲
p
✲
0
Σ
Gp( s)
+
+ +
✒✂✍ ✻
❄
❄
+
✂
✂
( sI − F ) − 1 g
( sI − F ) − 1 g
θ∗ ✛ ω 1
1
ω 2
θ∗ ✛
2
θ∗ ✛
3
Figure 6.14 Equivalent representation of the MRAC scheme of Table 6.4.
where c is used to denote any finite constant and δ > 0 is such that Wm( s −
δ ) , G− 1
) have stable poles. Furthermore, for θ ∈ L
2
p ( s − δ
2
∞ (guaranteed by the
adaptive law), the signal mf bounds most of the signals and their derivatives from
above.
Step 2. Use the Swapping Lemmas and properties of the L 2 δ norm to upper
bound
˜
θ ω with terms that are guaranteed by the adaptive law to have finite
L 2 gains. This is the most complicated step and it involves the use of Swapping
Lemmas A.1 and A.2 to obtain the inequality
˜
c
θ ω ≤
m
α
f + cαn∗
0
˜
gmf
(6.5.46)
0
where ˜
g 2 = 2 n 2 s + | ˙ θ| 2 + 2 and ˜ g is guaranteed by the adaptive law to belong to L 2 and α 0 > 0 is an arbitrary constant to be chosen.
Step 3. Use the B-G Lemma to establish boundedness. From (6.5.45) and
(6.5.46), it follows that
c
m 2
2
f ≤ c +
m 2
(6.5.47)
α 2 f + cα 2 n∗
0
˜
gmf
0
or
t
m 2 f ≤ c + c
α 2 n∗