p
p( s)
Λ p( s)
Λ p( s)
θp = θb , θa
ˆ
Zp( s, t)= θb ( t) αn− 1( s) , ˆ
Rp( s, t)= sn+ θa( t) αn− 1( s)
˙ˆ e = ˆ
Aˆ
e + ˆ
B ¯
up − ˆ
Ko( t)( C ˆ e − e 1) , ˆ e ∈ Rn+ q
In+ q− 1
ˆ
A=
−θ 1( t) − − −− , ˆ
B= θ 2( t) , C =[1 , 0 , . . . , 0]
0
State observer
θ 1 ∈Rn+ q is the coefficient vector of ˆ
RpQm−sn+ q
θ 2 ∈ Rn+ q is the coefficient vector of ˆ
ZpQ 1
ˆ
Ko = α∗ − θ 1, and α∗ is the coefficient vector of
A∗o( s) − sn+ q
Calculation of
controller
Solve for ˆ
Kc pointwise in time the equation
parameters
det( sI − ˆ
A + ˆ
B ˆ
Kc) = A∗c( s)
Control law
¯
up = − ˆ
Kc( t)ˆ e, up = Q 1 ¯ u
Q
p
m
Qm( s) monic of degree q with nonrepeated roots on
the jω-axis; Q 1( s) monic Hurwitz of degree q; A∗ 0( s)
Design
monic Hurwitz of degree n+ q; A∗c( s) monic Hurwitz of
variables
degree n + q and with Q 1( s) as a factor; Λ p( s) monic
Hurwitz of degree n
7.4. INDIRECT APPC SCHEMES
485
also included in A∗c( s). Therefore the condition that guarantees the existence
and uniform boundedness of ˆ
Kc is that ˆ
Zp( s, t), ˆ
Rp( s, t) Qm( s) are strongly
coprime at each time t. As we mentioned earlier, such a condition cannot
be guaranteed by any one of the adaptive laws developed in Chapter 4 with-
out additional modifications, thus giving rise to the so-called stabilizability
or admissibility problem to be discussed in Section 7.6. In this section, we
assume that the polynomials ˆ
Zp( s, t), ˆ
Rp( s, t) Qm( s) are strongly coprime at
each time t and proceed with the analysis of the APPC scheme of Table 7.5.
We relax this assumption in Section 7.6 where we modify the APPC schemes
to handle the possible loss of stabilizability of the estimated plant.
Theorem 7.4.2 Assume that the polynomials ˆ
Zp, ˆ
RpQm are strongly co-
prime at each time t. Then all the signals in the closed-loop APPC scheme
of Table 7.5 are uniformly bounded and the tracking error e 1 converges to
zero asymptotically with time. The same result holds if we replace the gra-
dient algorithm with any other adaptive law from Tables 4.2 to 4.4 that is
based on the plant parametric model (7.4.2) or (7.4.3).
Outline of Proof
Step 1. Develop the state error equations for the closed-loop APPC scheme,
i.e.,
˙ˆ e = Ac( t)ˆ e + ˆ
KoC eo
˙ eo = Aoeo + ˜
θ 1 e 1 − ˜
θ 2 ¯ up
(7.4.46)
yp = C eo + C ˆ e + ym
up = W 1( s) ˆ
Kc( t)ˆ e + W 2( s) yp
¯
up = − ˆ
Kcˆ e
where eo = e − ˆ e is the observation error, Ao is a constant stable matrix, W 1( s) and W 2( s) are strictly proper transfer functions with stable poles, and Ac( t) = ˆ
A− ˆ
B ˆ
Kc.
Step 2. Establish e.s. for the homogeneous part of (7.4.46). The gain ˆ
Kc is
chosen so that the eigenvalues of Ac( t) at each time t are equal to the roots of the
Hurwitz polynomial A∗c( s). Because ˆ
A, ˆ
B ∈ L∞ (guaranteed by the adaptive law)
and ˆ
Zp, ˆ
RpQm are strongly coprime (by assumption), we conclude that ( ˆ
A, ˆ
B) is
stabilizable in a strong sense and ˆ
Kc, Ac ∈ L∞. Using ˙ θa, ˙ θb ∈ L 2, guaranteed by
the adaptive law, we have ˙ˆ
Kc, ˙
Ac ∈ L 2. Therefore, applying Theorem 3.4.11, we
have that Ac( t) is a u.a.s. matrix. Because Ao is a constant stable matrix, the e.s.
of the homogeneous part of (7.4.46) follows.
486
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
Step 3. Use the properties of the L 2 δ norm and the B-G Lemma to establish
signal boundedness. We use the properties of the L 2 δ norm and equation (7.4.46)
to establish the inequality
m 2
2
f ≤ c gmf
+ c
where g 2 = 2 m 2 + | ˙ θ
2
2
a| 2 + | ˙
θb| 2 and m 2 = 1 + u
+ y
is the fictitious
f
p
p
normalizing signal. Because g∈ L 2, it follows that mf ∈ L∞ by applying the B-G
Lemma. Using mf ∈ L∞, we establish the boundedness of all signals in the closed-
loop plant.
Step 4. Establish the convergence of the tracking error e 1 to zero. This is done
by following the same procedure as in Example 7.4.2.
✷
The details of the proof of Theorem 7.4.2 are given in Section 7.7.
7.4.4
Adaptive Linear Quadratic Control (ALQC)
The linear quadratic (LQ) controller developed in Section 7.3.4 can be made
adaptive and used to meet the control objective when the plant parameters
are unknown. This is achieved by combining the LQ control law (7.3.28) to
(7.3.32) with an adaptive law based on the plant parametric model (7.4.2)
or (7.4.3).
We demonstrate the design and analysis of ALQ controllers using the
following examples:
Example 7.4.3 We consider the same plant and control objective as in Example
7.3.3, given by
˙ x = −ax + bup
yp = x
(7.4.47)
where the plant input up is to be chosen to stabilize the plant and regulate yp to
zero. In contrast to Example 7.3.3, the parameters a and b are unknown constants.
The control law up = − 1 bpy
λ
p in Example 7.3.3 is modified by replacing the
unknown plant parameters a, b with their on-line estimates ˆ a and ˆ b generated by
the same adaptive law used in Example 7.4.2, as follows:
Adaptive Law
˙ θp = Γ φ, Γ = Γ > 0
z − θ
s
=
p φ , m 2 = 1 + φ φ, z =
y
m 2
s + λ
p
0
7.4. INDIRECT APPC SCHEMES
487
ˆ b
1
u
θ
p
p
=
,
φ =
ˆ a
s + λ 0
−yp
where λ 0 > 0 is a design constant.
Control Law
1
u
ˆ
p = −
b( t) p( t) y
λ
p
(7.4.48)
Riccati Equation Solve the equation
p 2( t)ˆ b 2( t)
− 2ˆ a( t) p( t) −
+ 1 = 0
λ
at each time t for p( t) > 0, i.e.,
−λˆ a +
λ 2ˆ a 2 + ˆ b 2 λ
p( t) =
> 0
(7.4.49)
ˆ b 2
As in the previous examples, for the solution p( t) in (7.4.49) to be finite, the estimate
ˆ b should not cross zero. In fact, for p( t) to be uniformly bounded, ˆ b( t) should
satisfy |ˆ b( t) | ≥ b 0 > 0 , ∀t ≥ 0 for some constant b 0 that satisfies |b| ≥ b 0. Using the knowledge of b 0 and sgn( b), the adaptive law for ˆ b can be modified as before
to guarantee |ˆ b( t) | ≥ b 0 , ∀t ≥ 0 and at the same time retain the properties that
θp ∈ L∞ and , m, ˙ θp ∈ L 2 ∩ L∞. The condition |ˆ b( t) | ≥ b 0 implies that the estimated plant, characterized by the parameters ˆ a, ˆ b, is strongly controllable at
each time t, a condition required for the solution p( t) > 0 of the Riccati equation
to exist and be uniformly bounded.
Analysis For this first-order regulation problem, the analysis is relatively simple
and can be accomplished in the following four steps:
Step 1. Develop the closed-loop error equation. The closed-loop plant can be
written as
ˆ b 2 p
˙ x = −(ˆ a +
) x + ˜ ax − ˜ bu
λ
p
(7.4.50)
by adding and subtracting ˆ ax − ˆ bup and using up = −ˆ bpx/λ. The inputs ˜ ax, ˜ bup are due to the parameter errors ˜ a = ˆ a − a, ˜ b = ˆ b − b.
Step 2. Establish the e.s. of the homogeneous part of (7.4.50). The eigenvalue
of the homogeneous part of (7.4.50) is
ˆ b 2 p
−(ˆ a +
)
λ
488
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
which is guaranteed to be negative by the choice of p( t) given by (7.4.49), i.e.,
ˆ b 2 p
ˆ b 2
b
−(ˆ a +
) = − ˆ a 2 +
≤ − 0
√ < 0
λ
λ
λ
provided, of course, the adaptive law is modified by using projection to guarantee
|ˆ b( t) | ≥ b 0 , ∀t ≥ 0. Hence, the homogeneous part of (7.4.50) is e.s.
Step 3. Use the properties of the L 2 δ norm and B-G Lemma to establish bound-
edness. The properties of the input ˜ ax − ˜ bup in (7.4.50) depend on the properties
of the adaptive law that generates ˜ a and ˜ b. The first task in this step is to establish
the smallness of the input ˜ ax − ˜ bup by relating it with the signals ˙˜ a, ˙˜ b, and εm that are guaranteed by the adaptive law to be in L 2.
We start with the estimation error equation
1
1
εm 2 = z − θp φ = −˜ θp φ = ˜ a
x − ˜ b
u
s + λ
p
(7.4.51)
0
s + λ 0
Operating with ( s + λ 0) on both sides of (7.4.51) and using the property of differ-
entiation, i.e., sxy = x ˙ y + ˙ xy where s = d is treated as the differential operator, dt
we obtain
1
1
( s + λ 0) εm 2 = ˜ ax − ˜ bup + ˙˜ a
x − ˙˜ b
u
s + λ
p
(7.4.52)
0
s + λ 0
Therefore,
1
1
˜ ax − ˜ bup = ( s + λ 0) εm 2 − ˙˜ a
x + ˙˜ b
u
s + λ
p
0
s + λ 0
which we substitute in (7.4.50) to obtain
ˆ b 2 p
1
1
˙ x = −(ˆ a +
) x + ( s + λ
x + ˙˜ b
u
λ
0) εm 2 − ˙
˜ a s + λ
p
(7.4.53)
0
s + λ 0
If we define ¯
e = x − εm 2, (7.4.53) becomes
ˆ b 2 p
ˆ b 2 p
1
1
˙¯ e = −(ˆ a +
)¯
e + ( λ
) εm 2 − ˙˜ a
x + ˙˜ b
u
λ
0 − ˆ
a − λ
s + λ
p (7.4.54)
0
s + λ 0
x = ¯
e + m 2
Equation (7.4.54) has a homogeneous part that is e.s. and an input that is small in
some sense because of εm, ˙˜ a, ˙˜ b ∈ L 2 .
Let us now use the properties of the L 2 δ norm, which for simplicity is denoted
by
·
to analyze (7.4.54). The fictitious normalizing signal mf satisfies
m 2
2
2
f = 1 + yp
+ up
≤ 1 + c x 2
(7.4.55)
7.4. INDIRECT APPC SCHEMES
489
for some δ > 0 because of the control law chosen and the fact that ˆ b, p ∈ L∞.
Because x = ¯
e + εm 2, we have x ≤ ¯
e + εm 2 , which we use in (7.4.55) to
obtain
m 2 f ≤ 1 + c ¯ e 2 + c m 2 2
(7.4.56)
From (7.4.54), we have
¯
e 2 ≤ c m 2 2 + c ˙˜ a¯
x 2 + c ˙˜ b¯
u 2
p
(7.4.57)
where ¯
x =
1
x, ¯
u
u
s+ λ
p =
1
p. Using the properties of the L 2 δ norm, it can be
0
s+ λ 0
shown that mf bounds from above m, ¯ x, ¯ up and therefore it follows from (7.4.56),
(7.4.57) that
m 2
2
2
2
f ≤ 1 + c
mmf
+ c ˙˜ amf
+ c ˙˜ bmf
(7.4.58)
which implies that
t
m 2 f ≤ 1 + c
e−δ( t−τ) g 2( τ ) m 2 f( τ) dτ
0<