whose understanding helps find ways to counteract them by redesigning the
adaptive schemes of the previous chapters.
8.3.1
Parameter Drift
Let us consider the same plant as in the example of Section 4.2.1, where the
plant output is now corrupted by some unknown bounded disturbance d( t),
i.e.,
y = θ∗u + d
The adaptive law for estimating θ∗ derived in Section 4.2.1 for d( t) = 0
∀t ≥ 0 is given by
˙ θ = γ 1 u,
1 = y − θu
(8.3.1)
where γ > 0 and θ( t) is the on-line estimate of θ∗. We have shown that for
d( t) = 0 and u, ˙ u ∈ L∞, (8.3.1) guarantees that θ, 1 ∈ L∞ and 1( t) → 0 as t → ∞. Let us now analyze (8.3.1) when d( t) = 0. Defining ˜
θ = θ − θ∗, we
have
1 = − ˜
θu + d
and
˙˜ θ = −γu 2˜ θ+ γdu
(8.3.2)
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CHAPTER 8. ROBUST ADAPTIVE LAWS
We analyze (8.3.2) by considering the function
˜
θ 2
V (˜
θ) =
(8.3.3)
2 γ
Along the trajectory of (8.3.2), we have
˜
˙
θ 2 u 2
1
d 2
V = −˜
θ 2 u 2 + d˜
θu = −
− (˜
θu − d)2 +
(8.3.4)
2
2
2
For the class of inputs considered, i.e., u ∈ L∞ we cannot conclude that
˜
θ is bounded from considerations of (8.3.3), (8.3.4), i.e., we cannot find a
constant V 0 > 0 such that for V > V 0, ˙ V ≤ 0. In fact, for θ∗ = 2 , γ = 1, u = (1 + t) − 12 ∈ L∞ and
5
d( t) = (1 + t) − 14
− 2(1 + t) − 14
→ 0 as t → ∞
4
we have
5
y( t) = (1 + t) − 14 → 0 as t → ∞
4
1
1( t) =
(1 + t) − 14 → 0 as t → ∞
4
and
1
θ( t) = (1 + t) 4 → ∞ as t → ∞
i.e., the estimated parameter θ( t) drifts to infinity with time. We refer to
this instability phenomenon as parameter drift.
It can be easily verified that for u( t) = (1 + t) − 12 the equilibrium ˜
θe = 0
of the homogeneous part of (8.3.2) is u.s. and a.s., but not u.a.s. Therefore,
it is not surprising that we are able to find a bounded input γdu that leads
to an unbounded solution ˜
θ( t) = θ( t) − 2. If, however, we restrict u( t) to be
PE with level α 0 > 0, say, by choosing u 2 = α 0, then it can be shown (see
Problem 8.4) that the equilibrium ˜
θe = 0 of the homogeneous part of (8.3.2)
is u.a.s. (also e.s.) and that
d
lim sup |˜
θ( τ ) | ≤ 0 ,
d 0 = sup |d( t) u( t) |
t→∞ τ≥t
α 0
t
i.e., ˜
θ( t) converges exponentially to the residual set
d
D
0
θ =
˜
θ |˜
θ| ≤ α 0
8.3. INSTABILITY PHENOMENA IN ADAPTIVE SYSTEMS
547
indicating that the parameter error at steady state is of the order of the
disturbance level.
Unfortunately, we cannot always choose u to be PE especially in the case
of adaptive control where u is no longer an external signal but is generated
from feedback.
Another case of parameter drift can be demonstrated by applying the
adaptive control scheme
u = −kx,
˙ k = γx 2
(8.3.5)
developed in Section 6.2.1 for the ideal plant ˙ x = ax + u to the plant
˙ x = ax + u + d
(8.3.6)
where d( t) is an unknown bounded input disturbance. It can be verified that
for k(0) = 5 , x(0) = 1 , a = 1 , γ = 1 and
d( t) = (1 + t) − 15 5 − (1 + t) − 15 − 0 . 4(1 + t) − 65 → 0 as t → ∞
the solution of (8.3.5), (8.3.6) is given by
x( t) = (1 + t) − 25 → 0 as t → ∞
and
1
k( t) = 5(1 + t) 5 → ∞ as t → ∞
We should note that since k(0) = 5 > 1 and k( t) ≥ k(0) ∀t ≥ 0 parameter
drift can be stopped at any time t by switching off adaptation, i.e., setting
γ = 0 in (8.3.5), and still have x, u ∈ L∞ and x( t) → 0 as t → ∞. For
example, if γ = 0 for t ≥ t 1 > 0, then k( t) = ¯ k ∀t ≥ t 1, where ¯ k ≥ 5 is a stabilizing gain, which guarantees that u, x ∈ L∞ and x( t) → 0 as t → ∞.
One explanation of the parameter drift phenomenon may be obtained by
solving for the “quasi” steady state response of (8.3.6) with u = −kx, i.e.,
d
xs ≈ k − a
Clearly, for a given a and d, the only way for xs to go to zero is for k → ∞.
That is, in an effort to eliminate the effect of the input disturbance d and send
x to zero, the adaptive control scheme creates a high gain feedback. This
high gain feedback may lead to unbounded plant states when in addition to
bounded disturbances, there are dynamic plant uncertainties, as explained
next.
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CHAPTER 8. ROBUST ADAPTIVE LAWS
8.3.2
High-Gain Instability
Consider the following plant:
y
1 − µs
1
2 µs
=
=
1 −
(8.3.7)
u
( s − a)(1 + µs)
s − a
1 + µs
where µ is a small positive number which may be due to a small time constant
in the plant.
A reasonable approximation of the second-order plant may be obtained
by approximating µ with zero leading to the first-order plant model
¯
y
1
=
(8.3.8)
u
s − a
where ¯
y denotes the output y when µ = 0. The plant equation (8.3.7) is in
the form of y = G 0(1 + ∆ m) u where G 0( s) = 1 and ∆
. It
s−a
m( s) = − 2 µs
1+ µs
may be also expressed in the singular perturbation state-space form discussed
in Section 8.2, i.e.,
˙ x = ax + z − u
µ ˙ z = −z + 2 u
(8.3.9)
y = x
Setting µ = 0 in (8.3.9) we obtain the state-space representation of (8.3.8),
i.e.,
˙¯ x = a¯ x + u
(8.3.10)
¯
y = ¯
x
where ¯
x denotes the state x when µ = 0. Let us now design an adaptive
controller for the simplified plant (8.3.10) and use it to control the actual
second order plant where µ > 0. As we have shown in Section 6.2.1, the
adaptive law
u = −k¯
x,
˙ k = γ 1¯ x,
1 = ¯
x
can stabilize (8.3.10) and regulate ¯
y = ¯
x to zero. Replacing ¯
y with the actual
output of the plant y = x, we have
u = −kx,
˙ k = γ 1 x,
1 = x
(8.3.11)
which, when applied to (8.3.9), gives us the closed-loop plant
˙ x = ( a + k) x + z
(8.3.12)
µ ˙ z = −z − 2 kx
8.3. INSTABILITY PHENOMENA IN ADAPTIVE SYSTEMS
549
whose equilibrium xe = 0 , ze = 0 with k = k∗ is a.s. if and only if the
eigenvalues of (8.3.12) with k = k∗ are in the left-half s-plane, which implies
that
1 − a > k∗ > a
µ
Because ˙ k ≥ 0, it is clear that
1
1
k(0) >
− a ⇒ k( t) >
− a, ∀t ≥ 0
µ
µ
that is, from such a k(0) the equilibrium of (8.3.12) cannot be reached.
Moreover, with k > 1 − a the linear feedback loop is unstable even when the
µ
adaptive loop is disconnected, i.e., γ = 0.
We refer to this form of instability as high-gain instability. The adaptive
control law (8.3.11) can generate a high gain feedback which excites the
unmodeled dynamics and leads to instability and unbounded solutions. This
type of instability is well known in robust control with no adaptation [106]
and can be avoided by keeping the controller gains small (leading to a small
loop gain) so that the closed-loop bandwidth is away from the frequency
range where the unmodeled dynamics are dominant.
8.3.3
Instability Resulting from Fast Adaptation
Let us consider the second-order plant
˙ x = −x + bz − u
µ ˙ z = −z + 2 u
(8.3.13)
y = x
where b > 1 / 2 is an unknown constant and µ > 0 is a small number. For
u = 0, the equilibrium xe = 0 , ze = 0 is e.s. for all µ ≥ 0. The objective
here, however, is not regulation but tracking. That is, the output y = x is
required to track the output xm of the reference model
˙ xm = −xm + r
(8.3.14)
where r is a bounded piecewise continuous reference input signal. For µ = 0,
the reduced-order plant is
˙¯ x = −¯ x + (2 b − 1) u
(8.3.15)
¯
y = ¯
x
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CHAPTER 8. ROBUST ADAPTIVE LAWS
which has a pole at s = − 1, identical to that of the reference model, and an
unknown gain 2 b − 1 > 0. The adaptive control law
u = lr,
˙ l = −γ 1 r,
1 = ¯
x − xm
(8.3.16)
guarantees that ¯
x( t) , l( t) are bounded and |¯
x( t) − xm( t) | → 0 as t → ∞ for
any bounded reference input r. If we now apply (8.3.16) with ¯
x replaced by
x to the actual plant (8.3.13) with µ > 0, the closed-loop plant is
˙ x = −x + bz − lr
µ ˙ z = −z + 2 lr
(8.3.17)
˙ l = −γr( x − xm)
When γ = 0 that is when l is fixed and finite, the signals x( t) and z( t) are
bounded. Thus, no instability can occur in (8.3.17) for γ = 0.
Let us assume that r = constant. Then, (8.3.17) is an LTI system of the
form
˙
Y = AY + Bγrxm
(8.3.18)
where Y = [ x, z, l] , B = [0 , 0 , 1] ,
− 1
b
−r
A =
0
− 1 /µ 2 r/µ
−γr
0
0
and γrxm is treated as a bounded input. The characteristic equation of
(8.3.18) is then given by
1
1
γ
det( sI − A) = s 3 + (1 + ) s 2 + ( − γr 2) s + r 2(2 b − 1)
(8.3.19)
µ
µ
µ
Using the Routh-Hurwitz criterion, we have that for
1 (1 + µ)
γr 2 >
(8.3.20)
µ (2 b + µ)
two of the roots of (8.3.19) are in the open right-half s-plane. Hence given any
µ > 0, if γ, r satisfy (8.3.20), the solutions of (8.3.18) are unbounded in the
sense that |Y ( t) | → ∞ as t → ∞ for almost all initial conditions. Large γr 2
increases the speed of adaptation, i.e., ˙ l, which in turn excites the unmodeled
dynamics and leads to instability. The effect of ˙ l on the unmodeled dynamics
8.3. INSTABILITY PHENOMENA IN ADAPTIVE SYSTEMS
551
can be seen more clearly by defining a new state variable η = z − 2 lr, called
the parasitic state [85, 106], to rewrite (8.3.17) as
˙ x = −x + (2 b − 1) lr + bη
µ ˙ η = −η + 2 µγr 2( x − xm) − 2 µl ˙ r
(8.3.21)
˙ l = −γr( x − xm)
where for r = constant, ˙ r ≡ 0. Clearly, for a given µ, large γr 2 may lead to
a fast adaptation and large parasitic state η which acts as a disturbance in
the dominant part of the plant leading to false adaptation and unbounded
solutions. For stability and bounded solutions, γr 2 should be kept small, i.e.,
the speed of adaptation should be slow relative to the speed of the parasitics
characterized by 1 .
µ
8.3.4
High-Frequency Instability
Let us now assume that in the adaptive control scheme described by (8.3.21),
both γ and |r| are kept small, i.e., adaptation is slow. We consider the case
where r = r 0 sin ωt and r 0 is small but the frequency ω can be large. At
lower to middle frequencies and for small γr 2 , µ, we can approximate µ ˙ η ≈ 0
and solve (8.3.21) for η, i.e., η ≈ 2 µγr 2( x − xm) − 2 µl ˙ r, which we substitute into the first equation in (8.3.21) to obtain
˙ x = −(1 − 2 µbγr 2) x + ( gr − 2 bµ ˙ r) l − 2 µbγr 2 xm
˙
(8.3.22)
l = −γr( x − xm)
where g = 2 b − 1. Because γr is small we can approximate l( t) with a
constant and solve for the sinusoidal steady-state response of x from the
first equation of (8.3.22) where the small µγr 2 terms are neglected, i.e.,
r
x
0
ss =
[( g − 2 bµω 2) sin ωt − ω( g + 2 bµ)cos ωt] l
1 + ω 2
Now substituting for x = xss in the second equation of (8.3.22) we obtain
˙
γr 2
l
0 sin ωt
s = −
( g − 2 bµω 2) sin ωt − ω( g + 2 bµ)cos ωt l
1 + ω 2
s + γxmr 0 sin ωt
i.e.,
˙ ls = α( t) ls + γxmr 0 sin ωt
(8.3.23)
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CHAPTER 8. ROBUST ADAPTIVE LAWS
γr 2