cannot be implemented. A possible procedure to follow in the unknown
parameter case is to use the same control law as given in (6.2.2) but with k∗
replaced by its estimate k( t), i.e., we use
u = −k( t) x
(6.2.3)
and search for an adaptive law to update k( t) continuously with time.
Adaptive Law The adaptive law for generating k( t) is developed by view-
ing the problem as an on-line identification problem for k∗. This is ac-
complished by first obtaining an appropriate parameterization for the plant
(6.2.1) in terms of the unknown k∗ and then using a similar approach as in
Chapter 4 to estimate k∗ on-line. We illustrate this procedure below.
We add and subtract the desired control input −k∗x in the plant equation
to obtain
˙ x = ax − k∗x + k∗x + u.
Because a − k∗ = −am we have
˙ x = −amx + k∗x + u
or
1
x =
( u + k∗x)
(6.2.4)
s + am
Equation (6.2.4) is a parameterization of the plant equation (6.2.1) in terms
of the unknown controller parameter k∗. Because x, u are measured and
am > 0 is known, a wide class of adaptive laws may be generated by simply
using Tables 4.1 to 4.3 of Chapter 4. It turns out that the adaptive laws
developed for (6.2.4) using the SPR-Lyapunov design approach without nor-
malization simplify the stability analysis of the resulting closed-loop adaptive
control scheme considerably. Therefore, as a starting point, we concentrate
on the simple case and deal with the more general case that involves a wide
class of adaptive laws in later sections.
Because
1
is SPR we can proceed with the SPR-Lyapunov design
s+ am
approach of Chapter 4 and generate the estimate ˆ
x of x as
1
1
ˆ
x =
[ kx + u] =
(0)
(6.2.5)
s + am
s + am
6.2. SIMPLE DIRECT MRAC SCHEMES
317
where the last equality is obtained by substituting the control law u = −kx.
If we now choose ˆ
x(0) = 0, we have ˆ
x( t) ≡ 0 , ∀t ≥ 0, which implies that the
estimation error 1 defined as 1 = x − ˆ x is equal to the regulation error,
i.e., 1 = x, so that (6.2.5) does not have to be implemented to generate
ˆ
x. Substituting for the control u = −k( t) x in (6.2.4), we obtain the error
equation that relates the parameter error ˜
k = k − k∗ with the estimation
error 1 = x, i.e.,
˙1 = −am 1 − ˜ kx,
1 = x
(6.2.6)
or
1
1 =
−˜
kx
s + am
As demonstrated in Chapter 4, the error equation (6.2.6) is in a convenient
form for choosing an appropriate Lyapunov function to design the adaptive
law for k( t). We assume that the adaptive law is of the form
˙˜ k = ˙ k = f 1 ( 1 , x, u)
(6.2.7)
where f 1 is some function to be selected, and propose
2
˜
k 2
V
1
1 , ˜
k =
+
(6.2.8)
2
2 γ
for some γ > 0 as a potential Lyapunov function for the system (6.2.6),
(6.2.7). The time derivative of V along the trajectory of (6.2.6), (6.2.7) is
given by
˜
˙
kf
V = −a
2
1
m 1 − ˜
k 1 x +
(6.2.9)
γ
Choosing f 1 = γ 1 x, i.e.,
˙ k = γ 1 x = γx 2 , k(0) = k 0
(6.2.10)
we have
˙
V = −a
2
m 1 ≤ 0
(6.2.11)
Analysis
Because V is a positive definite function and ˙
V ≤ 0, we have
V ∈ L∞, which implies that 1 , ˜ k ∈ L∞. Because 1 = x, we also have
that x ∈ L∞ and therefore all signals in the closed-loop plant are bounded.
318
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
Furthermore, 1 = x ∈ L 2 and ˙1 = ˙ x ∈ L∞ (which follows from (6.2.6) )
imply, according to Lemma 3.2.5, that 1( t) = x( t) → 0 as t → ∞. From
x( t) → 0 and the boundedness of k, we establish that ˙ k( t) → 0 , u( t) →
0 as t → ∞.
We have shown that the combination of the control law (6.2.3) with the
adaptive law (6.2.10) meets the control objective in the sense that it forces
the plant state to converge to zero while guaranteeing signal boundedness.
It is worth mentioning that as in the simple parameter identification
examples considered in Chapter 4, we cannot establish that k( t) converges
to k∗, i.e., that the pole of the closed-loop plant converges to the desired one
given by −am. The lack of parameter convergence is less crucial in adaptive
control than in parameter identification because in most cases, the control
objective can be achieved without requiring the parameters to converge to
their true values.
The simplicity of this scalar example allows us to solve for 1 = x explic-
itly, and study the properties of k( t) , x( t) as they evolve with time. We can
verify that
2 ce−ct
1( t)
= c + k 0 − a + ( c − k 0 + a) e− 2 ct 1(0) , 1 = x
c ( c + k
k( t) = a +
0 − a) e 2 ct − ( c − k 0 + a)
(6.2.12)
( c + k 0 − a) e 2 ct + ( c − k 0 + a)
where c 2 = γx 20 + ( k 0 − a)2, satisfy the differential equations (6.2.6) and
(6.2.10) of the closed-loop plant. Equation (6.2.12) can be used to investigate
the effects of initial conditions and adaptive gain γ on the transient and
asymptotic behavior of x( t) , k( t). We have lim t→∞ k( t) = a + c if c > 0 and lim t→∞ k( t) = a − c if c < 0, i.e.,
lim k( t) = k∞ = a + γx 2
t→∞
0 + ( k 0 − a)2
(6.2.13)
Therefore, for x 0 = 0 , k( t) converges to a stabilizing gain whose value de-
pends on γ and the initial condition x 0 , k 0. It is clear from (6.2.13) that
the value of k∞ is independent of whether k 0 is a destabilizing gain, i.e.,
0 < k 0 < a, or a stabilizing one, i.e., k 0 > a, as long as ( k 0 − a)2 is the same.
The use of different k 0, however, will affect the transient behavior as it is
obvious from (6.2.12). In the limit as t → ∞, the closed-loop pole converges
6.2. SIMPLE DIRECT MRAC SCHEMES
319
r = 0 ✲ ❧ ✲
u
1
x
✲
+ Σ
s − a
✻
−
✒
❄
✛
k( t)
( •)2
k
1
✛
γ
✛
s
✻
k(0)
Figure 6.2
Block diagram for implementing the adaptive controller
(6.2.14).
to − ( k∞ − a), which may be different from −am. Because the control ob-
jective is to achieve signal boundedness and regulation of the state x( t) to
zero, the convergence of k( t) to k∗ is not crucial.
Implementation
The adaptive control scheme developed and analyzed
above is given by the following equations:
u = −k( t) x,
˙ k = γ 1 x = γx 2 , k(0) = k 0
(6.2.14)
where x is the measured state of the plant. A block diagram for implementing
(6.2.14) is shown in Figure 6.2.
The design parameters in (6.2.14) are the initial parameter k 0 and the
adaptive gain γ > 0. For signal boundedness and asymptotic regulation of
x to zero, our analysis allows k 0 , γ to be arbitrary. It is clear, however, from
(6.2.12) that their values affect the transient performance of the closed-loop
plant as well as the steady-state value of the closed-loop pole. For a given
k 0 , x 0 = 0, large γ leads to a larger value of c in (6.2.12) and, therefore,
to a faster convergence of x( t) to zero. Large γ, however, may make the
differential equation for k “stiff” (i.e., ˙ k large) that will require a very small
step size or sampling period to implement it on a digital computer. Small
sampling periods make the adaptive scheme more sensitive to measurement
noise and modeling errors.
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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
Remark 6.2.1 In the proceeding example, we have not used any reference
model to describe the desired properties of the closed-loop system. A
reasonable choice for the reference model would be
˙ xm = −amxm,
xm(0) = xm 0
(6.2.15)
which, by following exactly the same procedure, would lead to the
adaptive control scheme
u = −k( t) x,
˙ k = γe 1 x
where e 1 = x − xm. If xm 0 = x 0, the use of (6.2.15) will affect the
transient behavior of the tracking error but will have no effect on the
asymptotic properties of the closed-loop scheme because xm converges
to zero exponentially fast.
6.2.2
Scalar Example: Adaptive Tracking
Consider the following first order plant:
˙ x = ax + bu
(6.2.16)
where a, b are unknown parameters but the sign of b is known. The control
objective is to choose an appropriate control law u such that all signals in
the closed-loop plant are bounded and x tracks the state xm of the reference
model given by
˙ xm = −amxm + bmr
i.e.,
b
x
m
m =
r
(6.2.17)
s + am
for any bounded piecewise continuous signal r( t), where am > 0, bm are
known and xm( t) , r( t) are measured at each time t. It is assumed that
am, bm and r are chosen so that xm represents the desired state response of
the plant.
Control Law For x to track xm for any reference input signal r( t), the
control law should be chosen so that the closed-loop plant transfer function
6.2. SIMPLE DIRECT MRAC SCHEMES
321
from the input r to output x is equal to that of the reference model. We
propose the control law
u = −k∗x + l∗r
(6.2.18)
where k∗, l∗ are calculated so that
x( s)
bl∗
b
x
=
=
m
= m( s)
(6.2.19)
r( s)
s − a + bk∗
s + am
r( s)
Equation (6.2.19) is satisfied if we choose
b
a
l∗ = m ,
k∗ = m + a
(6.2.20)
b
b
provided of course that b = 0, i.e., the plant (6.2.16) is controllable. The
control law (6.2.18), (6.2.20) guarantees that the transfer function of the
closed-loop plant, i.e., x( s) is equal to that of the reference model. Such
r( s)
a transfer function matching guarantees that x( t) = xm( t) , ∀t ≥ 0 when
x(0) = xm(0) or |x( t) − xm( t) | → 0 exponentially fast when x(0) = xm(0), for any bounded reference signal r( t).
When the plant parameters a, b are unknown, (6.2.18) cannot be imple-
mented. Therefore, instead of (6.2.18), we propose the control law
u = −k( t) x + l( t) r
(6.2.21)
where k( t) , l( t) is the estimate of k∗, l∗, respectively, at time t, and search for an adaptive law to generate k( t) , l( t) on-line.
Adaptive Law
As in Example 6.2.1, we can view the problem as an on-
line identification problem of the unknown constants k∗, l∗. We start with
the plant equation (6.2.16) which we express in terms of k∗, l∗ by adding and
subtracting the desired input term −bk∗x + bl∗r to obtain
˙ x = −amx + bmr + b ( k∗x − l∗r + u)
i.e.,
b
b
x =
m
r +
( k∗x − l∗r + u)
(6.2.22)
s + am
s + am
Because xm = bm r is a known bounded signal, we express (6.2.22) in terms
s+ am
of the tracking error defined as e = x − xm, i.e.,
b
e =
( k∗x − l∗r + u)
(6.2.23)
s + am
322
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
Because b is unknown, equation (6.2.23) is in the form of the bilinear
parametric model considered in Chapter 4, and may be used to choose an
adaptive law directly from Table 4.4 of Chapter 4.
Following the procedure of Chapter 4, the estimate ˆ
e of e is generated as
1
1
ˆ
e =
ˆ b ( kx − lr + u) =
(0)
(6.2.24)
s + am
s + am
where the last identity is obtained by substituting for the control law
u = −k( t) x + l( t) r
Equation (6.2.24) implies that the estimation error, defined as 1 = e − ˆ e,
can be simply taken to be the tracking error, i.e., 1 = e, and, therefore,
there is no need to generate ˆ
e. Furthermore, since ˆ
e is not generated, the
estimate ˆ b of b is not required.
Substituting u = −k( t) x + l( t) r in (6.2.23) and defining the parameter errors ˜
k = k − k∗, ˜ l= l − l∗, we have
b
1 = e =
−˜
kx + ˜ lr
s + am
or
˙1 = −am 1 + b −˜ kx + ˜ lr ,
1 = e = x − xm
(6.2.25)
As shown in Chapter 4, the development of the differential equation (6.2.25)
relating the estimation error with the parameter error is a significant step
in deriving the adaptive laws for updating k( t) , l( t). We assume that the
structure of the adaptive law is given by
˙ k = f 1 ( 1 , x, r, u) , ˙ l = f 2 ( 1 , x, r, u)
(6.2.26)
where the functions f 1 , f 2 are to be designed.
As shown in Example 6.2.1, however, the use of the SPR-Lyapunov ap-
proach without normalization allows us to design an adaptive law for k, l
and analyze the stability properties of the closed-loop system using a single
Lyapunov function. For this reason, we proceed with the SPR-Lyapunov ap-
proach without normalization and postpone the use of other approaches that
are based on the use of the normalized estimation error for later sections.
6.2. SIMPLE DIRECT MRAC SCHEMES
323
Consider the function
2
˜
k 2
˜ l 2
V