✄
ω 2
θ
✛
2
✒
✄
θ
✛
3
Controller Parameter
Plant
Calculation Using
θp
✛
✛
u
Parameter
p
Mapping of
✛ y
Estimation
p
(6.3.12),(6.3.16),(6.3.17)
Figure 6.15 Block diagram of an indirect MRAC scheme.
with
a
b
k∗ = m + a ,
l∗ = m
(6.6.4)
b
b
could be used to meet the control objective. In the unknown parameter case,
we propose
u = −k( t) x + l( t) r
(6.6.5)
where k( t) , l( t) are the on-line estimates of k∗, l∗ at time t, respectively. In direct adaptive control, k( t) , l( t) are generated directly by an adaptive law.
In indirect adaptive control, we follow a different approach. We evaluate
k( t) , l( t) by using the relationship (6.6.4) and the estimates ˆ a, ˆ b of the unknown parameters a, b as follows:
a
b
k( t) = m + ˆ a( t) , l( t) = m
(6.6.6)
ˆ b( t)
ˆ b( t)
6.6. INDIRECT MRAC
399
where ˆ a, ˆ b are generated by an adaptive law that we design.
Adaptive Law The adaptive law for generating ˆ a, ˆ b is obtained by following
the same procedure as in the identification examples of Chapter 4, i.e., we
rewrite (6.6.1) as
1
x =
[( a + a
s + a
m) x + bu]
m
and generate ˆ
x, the estimate of x, from
1
ˆ
x =
[(ˆ a + a
s + a
m) x + ˆ
bu] = xm
(6.6.7)
m
where the last equality is obtained by using (6.6.5), (6.6.6). As in Section
6.2.2, the estimation error 1 = x − xm = e 1 is the same as the tracking error
and satisfies the differential equation
˙ e 1 = −ame 1 − ˜ ax − ˜ bu
(6.6.8)
where
˜ a = ˆ a − a,
˜ b = ˆ b − b
are the parameter errors. Equation (6.6.8) motivates the choice of
1
˜ a 2
˜ b 2
V =
e 2
+
(6.6.9)
2
1 + γ 1
γ 2
for some γ 1 , γ 2 > 0, as a potential Lyapunov-like function candidate for
(6.6.8). The time derivative of V along any trajectory of (6.6.8) is given by
˜
˙
˜ a ˙˜ a
b˙˜ b
V = −ame 21 − ˜ axe 1 − ˜ bue 1 +
+
(6.6.10)
γ 1
γ 2
Hence, for
˙
˙
˜ a = ˙ˆ a = γ
ˆ
1 e 1 x,
˙˜ b = b = γ 2 e 1 u
(6.6.11)
we have
˙
V = −ame 21 ≤ 0
which implies that e 1 , ˆ a, ˆ b ∈ L∞ and that e 1 ∈ L 2 by following the usual
arguments. Furthermore, xm, e 1 ∈ L∞ imply that x ∈ L∞. The bound-
edness of u, however, cannot be established unless we show that k( t) , l( t)
400
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
are bounded. The boundedness of 1 and therefore of k( t) , l( t) cannot be
ˆ b
guaranteed by the adaptive law (6.6.11) because (6.6.11) may generate esti-
mates ˆ b( t) arbitrarily close or even equal to zero. The requirement that ˆ b( t)
is bounded away from zero is a controllability condition for the estimated
plant that the control law (6.6.5) is designed for. One method for avoid-
ing ˆ b( t) going through zero is to modify the adaptive law for ˆ b( t) so that
adaptation takes place in a closed subset of R 1 which doesnot include the
zero element. Such a modification is achieved by using the following a priori
knowledge:
The sgn( b) and a lower bound b 0 > 0 for |b| is known
(A2)
Applying the projection method with the constraint ˆ b sgn( b) ≥ b 0 to the
adaptive law (6.6.11), we obtain
if |ˆ b| > b 0 or
˙
˙
γ
ˆ a = γ
ˆ
2 e 1 u
1 e 1 x,
b =
if |ˆ b| = b
(6.6.12)
0 and e 1 u sgn( b) ≥ 0
0
otherwise
where ˆ b(0) is chosen so that ˆ b(0)sgn( b) ≥ b 0.
Analysis
It follows from (6.6.12) that if ˆ b(0)sgn( b) ≥ b 0, then whenever
ˆ
˙
b( t)sgn( b) = |ˆ b( t) | becomes equal to b
ˆ
0 we have bˆ
b ≥ 0 which implies that
|ˆ b( t) | ≥ b 0 , ∀t ≥ 0. Furthermore the time derivative of (6.6.9) along the
trajectory of (6.6.8), (6.6.12) satisfies
˙
−a
V =
me 2
1
if |ˆ b| > b 0 or |ˆ b| = b 0 and e 1 u sgn( b) ≥ 0
−ame 21 − ˜ be 1 u if |ˆ b| = b 0 and e 1 u sgn( b) < 0
Now for |ˆ b| = b 0, we have (ˆ b − b)sgn( b) < 0. Therefore, for |ˆ b| = b 0 and e 1 u sgn( b) < 0, we have
˜ be 1 u = (ˆ b − b) e 1 u = (ˆ b − b)sgn( b)( e 1 u sgn( b)) > 0
which implies that
˙
V ≤ −ame 21 ≤ 0 , ∀t ≥ 0
Therefore, the function V given by (6.6.9) is a Lyapunov function for the
system (6.6.8), (6.6.12) since u, x in (6.6.8) can be expressed in terms of e 1
and xm where xm( t) is treated as an arbitrary bounded function of time.
6.6. INDIRECT MRAC
401
Reference Model
bm
✲
xm
s + am
− ❄
❧ e
Plant
Σ
✲
1
✒
r
+
b
+
✲ l( t) ✲ ❧
x
✻
Σ
✲
−
✒
s − a
✻
k( t) ✛
✁
Adaptive
✛
✲ Law for ˆ a, ˆ b ✛
(6.6.12)
ˆ a ❄
❄
ˆ b
k( t)
Controller Parameter
l( t)
Calculation
(6.6.6)
Figure 6.16 Block diagram for implementing the indirect MRAC scheme
given by (6.6.5), (6.6.6), and (6.6.12).
Hence the equilibrium e 1 e = 0 , ˆ ae = a, ˆ be = b is u.s. and e 1 , ˆ b, ˆ a ∈ L∞.
Using the usual arguments, we have e 1 ∈ L 2 and ˙ e 1 ∈ L∞ which imply that
˙
e
ˆ
1( t) = x( t) − xm( t) → 0 as t → ∞ and therefore that ˙
ˆ a( t) , b( t) → 0 as
t → ∞.
As in the direct case it can be shown that if the reference input signal
r( t) is sufficiently rich of order 2 then ˜ b, ˜ a and, therefore, ˜
k, ˜ l converge to
zero exponentially fast.
Implementation The proposed indirect MRAC scheme for (6.6.1) described
by (6.6.5), (6.6.6), and (6.6.12) is implemented as shown in Figure 6.16.
6.6.2
Indirect MRAC with Unnormalized Adaptive Laws
As in the case of direct MRAC considered in Section 6.5, we are interested
in extending the indirect MRAC scheme for the scalar plant of Section 6.6.1
to a higher order plant. The basic features of the scheme of Section 6.6.1 is
that the adaptive law is driven by the tracking error and a single Lyapunov
function is used to design the adaptive law and establish signal boundedness.
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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
In this section, we extend the results of Section 6.6.1 to plants with
relative degree n∗ = 1. The same methodology is applicable to the case of
n∗ ≥ 2 at the expense of additional algebraic manipulations. We assign these
more complex cases as problems for the ambitious reader in the problem
section.
Let us start by considering the same plant and control objective as in
the direct MRAC scheme of Section 6.4.1 where the relative degree of the
plant is assumed to be n∗ = 1. We propose the same control law
˙ ω 1 = F ω 1 + gup,
ω 1(0) = 0
˙ ω 2 = F ω 2 + gyp,
ω 2(0) = 0
(6.6.13)
up = θ ω
as in the direct MRAC case where θ( t) is calculated using the estimate of kp
and the estimates of the coefficients of the plant polynomials Zp( s) , Rp( s),
represented by the vector θp( t), at each time t. Our goal is to develop an
adaptive law that generates the estimate θp( t) and specify the mapping from
θp( t) to θ( t) that allows us to calculate θ( t) at each time t. We start with the mapping that relates the unknown vectors θ∗ = θ∗ 1 , θ∗ 2 , θ∗ 3 , c∗ 0
and
θ∗p specified by the matching equations (6.3.12), (6.3.16), and (6.3.17) (with
Q( s) = 1 due to n∗ = 1), i.e.,
k
c∗
m
0
=
kp
θ∗ 1 α( s) = Λ( s) − Zp( s) ,
(6.6.14)
R
θ∗
p( s) − Λ0( s) Rm( s)
2 α( s) + θ∗
3Λ( s)
=
kp
To simplify (6.6.14) further, we express Zp( s), Rp( s), Λ( s), Λ0( s) Rm( s) as
Zp( s) = sn− 1 + p 1 αn− 2( s)
Rp( s) = sn + an− 1 sn− 1 + p 2 αn− 2( s)
Λ( s) = sn− 1 + λ αn− 2( s)
Λ0( s) Rm( s) = sn + rn− 1 sn− 1 + ν αn− 2( s)
where p 1 , p 2 ∈Rn− 1 , an− 1 are the plant parameters, i.e., θ∗p =[ kp, p 1 , an− 1 , p 2 ] ; λ, ν ∈ Rn− 1 and rn− 1 are the coefficients of the known polynomials Λ( s),
6.6. INDIRECT MRAC
403
Λ0( s) Rm( s) and αn− 2( s) = sn− 2 , sn− 3 , . . . , s, 1 , which we then substitute in (6.6.14) to obtain the equations
k
c∗
m
0
=
kp
θ∗ 1 = λ − p 1
p
θ∗
2 − an− 1 λ + rn− 1 λ − ν
2
=
(6.6.15)
kp
a
θ∗
n− 1 − rn− 1
3
=
kp
If we let ˆ
kp( t) , ˆ
p 1( t) , ˆ
p 2( t) , ˆ an− 1( t) be the estimate of kp, p 1 , p 2 , an− 1 re-
spectively at each time t, then θ( t) = [ θ 1 , θ 2 , θ 3 , c 0] may be calculated as
k
c
m
0( t)
= ˆ kp( t)
θ 1( t) = λ − ˆ
p 1( t)
ˆ
p
θ
2( t) − ˆ
an− 1( t) λ + rn− 1 λ − ν
2( t)
=
(6.6.16)
ˆ
kp( t)
ˆ a
θ
n− 1( t) − rn− 1
3( t)
=
ˆ
kp( t)
provided |ˆ
kp( t) | = 0 , ∀t ≥ 0.
The adaptive laws for generating ˆ
p 1 , ˆ
p 2 , ˆ an− 1 , ˆ kp on-line can be devel-
oped by using the techniques of Chapter 4. In this section we concentrate
on adaptive laws that are driven by the tracking error e 1 rather than the
normalized estimation error, and are developed using the SPR-Lyapunov
design approach. We start with the parametric model given by equation
(6.4.6), i.e.,
e 1 = Wm( s) ρ∗( up − θ∗ ω)
(6.6.17)
where ρ∗ = 1 = kp . As in the direct case, we choose W
c∗
k
m( s), the transfer
0
m
function of the reference model, to be SPR with relative degree n∗ = 1.
The adaptive law for θp = [ˆ kp, ˆ
p 1 , ˆ an− 1 , ˆ p 2 ] is developed by first relating
e 1 with the parameter error ˜
θp = θp − θ∗p through the SPR transfer function
Wm( s) and then proceeding with the Lyapunov design approach as follows:
404
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
We rewrite (6.6.17) as
1
e 1 = Wm( s)
k
k
pup − kpθ∗ ω − ˆ
kpup + ˆ kpθ ω
(6.6.18)
m
where −ˆ
kpup + ˆ kpθ ω = 0 because of (6.6.13). If we now substitute for
kpθ∗, ˆ kpθ from (6.6.15) and (6.6.16), respectively, in (6.6.18) we obtain
1
e
˜
1= Wm( s)
k
k
p λ ω 1 −up +