y
e
y
s + λ p = ( s + ˆ a) s + λ 1 + ˙ˆ a s + λ p
which we substitute in (7.4.19) to obtain
1
1
˙
1
s( m 2) = ( s + ˆ a) s + ˆ b(ˆ
p
ˆ
1 s + ˆ
p 0)
e
y
b
u
s + λ 1 + ˙ˆ a s + λ p − s + λ p
Using (7.4.10) we have ( s + ˆ a) s + ˆ b(ˆ
p 1 s + ˆ
p 0) = ( s + 1)2 and therefore
( s + 1)2
1
˙
1
s( m 2) =
e
y
ˆ b
u
s + λ
1 + ˙
ˆ a s + λ p − s + λ p
or
s( s + λ)
s + λ
1
s + λ ˙ 1
e
˙
ˆ
1 =
m 2 −
ˆ a
y
b
u
( s + 1)2
( s + 1)2 s + λ p + ( s + 1)2 s + λ p
(7.4.20)
˙
Because u
ˆ
p, yp, m,
∈ L∞ and ˙ˆ a, b, m ∈ L∞
L 2, it follows from Corollary 3.3.1
that e 1 ∈ L∞
L 2. Hence, if we show that ˙ e 1 ∈ L∞, then by Lemma 3.2.5 we can
conclude that e 1 → 0 as t → ∞. Since ˙ e 1 = ˙ yp = ayp + bup ∈ L∞, it follows that e 1 → 0 as t → ∞.
˙
We can continue our analysis and establish that , ˙ˆ a, ˆ b, ˙ˆ
p 0 , ˙ˆ p 1 → 0 as t → ∞.
There is no guarantee, however, that ˆ
p 0 , ˆ
p 1, ˆ a, ˆ b will converge to the actual values
p 0 , p 1 , a, b respectively unless the reference signal ym is sufficiently rich of order 2,
which is not the case for the example under consideration.
474
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
As we indicated earlier the calculation of the controller parameters ˆ
p 0( t) , ˆ
p 1( t)
at each time is possible provided the estimated plant polynomials ( s + ˆ a( t)) , ˆ b( t) are strongly coprime, i.e., provided |ˆ b( t) | ≥ b 0 > 0 ∀t ≥ 0. This condition implies that at each time t, the estimated plant is strongly controllable. This is not surprising
because the control law is calculated at each time t to meet the control objective
for the estimated plant. As we will show in Section 7.6, the adaptive law without
projection cannot guarantee that |ˆ b( t) | ≥ b 0 > 0 ∀t ≥ 0. Projection requires the knowledge of b 0 and sgn( b) and constrains ˆ b( t) to be in the region |ˆ b( t) | ≥
b 0 where controllability is always satisfied. In the higher-order case, the problem
of controllability or stabilizability of the estimated plant is more difficult as we
demonstrate below for a general nth-order plant.
General Case
Let us now consider the nth-order plant
Z
y
p( s)
p =
u
R
p
p( s)
where Zp( s) , Rp( s) satisfy assumptions P1, P2, and P3 with the same control
objective as in Section 7.3.1, except that in this case the coefficients of Zp, Rp
are unknown. The APPC scheme that meets the control objective for the
unknown plant is formed by combining the control law (7.3.12), summarized
in Table 7.1, with an adaptive law based on the parametric model (7.4.2)
or (7.4.3). The adaptive law generates on-line estimates θa, θb of the coeffi-
cient vectors, θ∗a of Rp( s) = sn + θ∗a αn− 1( s) and θ∗b of Zp( s) = θ∗b αn− 1( s) respectively, to form the estimated plant polynomials
ˆ
Rp( s, t) = sn + θa αn− 1( s) , ˆ
Zp( s, t) = θb αn− 1( s)
The estimated plant polynomials are used to compute the estimated con-
troller polynomials ˆ
L( s, t) , ˆ
P ( s, t) by solving the Diophantine equation
ˆ
LQm · ˆ
Rp + ˆ
P · ˆ
Zp = A∗
(7.4.21)
for ˆ
L, ˆ
P pointwise in time or the algebraic equation
ˆ
S ˆ
lβl = α∗
l
(7.4.22)
7.4. INDIRECT APPC SCHEMES
475
for ˆ
βl, where ˆ
Sl is the Sylvester matrix of ˆ
RpQm, ˆ
Zp; ˆ
βl contains the coeffi-
cients of ˆ
L, ˆ
P ; and α∗l contains the coefficients of A∗( s) as shown in Table 7.1.
The control law in the unknown parameter case is then formed as
1
1
up = (Λ − ˆ
LQm) u
( y
Λ p − ˆ
P Λ p − ym)
(7.4.23)
Because different adaptive laws may be picked up from Tables 4.2 to 4.5, a
wide class of APPC schemes may be developed. As an example, we present
in Table 7.4 the main equations of an APPC scheme that is based on the
gradient algorithm of Table 4.2.
The implementation of the APPC scheme of Table 7.4 requires that
the solution of the polynomial equation (7.4.21) for ˆ
L, ˆ
P or of the alge-
braic equation (7.4.22) for ˆ
βl exists at each time. The existence of this
solution is guaranteed provided that ˆ
Rp( s, t) Qm( s) , ˆ
Zp( s, t) are coprime at
each time t, i.e., the Sylvester matrix ˆ
Sl( t) is nonsingular at each time t.
In fact for the coefficient vectors l, p of the polynomials ˆ
L, ˆ
P to be uni-
formly bounded for bounded plant parameter estimates θp, the polynomials
ˆ
Rp( s, t) Qm( s) , ˆ
Zp( s, t) have to be strongly coprime which implies that their
Sylvester matrix should satisfy
| det( Sl( t)) | ≥ ν 0 > 0
for some constant ν 0 at each time t. Such a strong condition cannot be
guaranteed by the adaptive law without any additional modifications, giv-
ing rise to the so called “stabilizability” or “admissibility” problem to be
discussed in Section 7.6. As in the scalar case, the stabilizability problem
arises from the fact that the control law is chosen to stabilize the estimated
plant (characterized by ˆ
Zp( s, t) , ˆ
Rp( s, t)) at each time. For such a control law
to exist, the estimated plant has to satisfy the usual observability, controlla-
bility conditions which in this case translate into the equivalent condition of
ˆ
Rp( s, t) Qm( s) , ˆ
Rp( s, t) being coprime. The stabilizability problem is one of
the main drawbacks of indirect APPC schemes in general and it is discussed
in Section 7.6. In the meantime let us assume that the estimated plant is
stabilizable, i.e., ˆ
RpQm, ˆ
Zp are strongly coprime ∀t ≥ 0 and proceed with
the analysis of the APPC scheme presented in Table 7.4.
Theorem 7.4.1 Assume that the estimated plant polynomials ˆ
RpQm, ˆ
Zp are
strongly coprime at each time t. Then all the signals in the closed-loop
476
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
Table 7.4 APPC scheme: polynomial approach.
yp = Zp( s) u
R
p
p( s)
Plant
Zp( s) = θ∗b αn− 1( s)
Rp( s) = sn + θ∗a αn− 1( s)
αn− 1( s) = [ sn− 1 , sn− 2 , . . . , s, 1] , θ∗p = [ θ∗b , θ∗a ]
Reference
Q
signal
m( s) ym = 0
Gradient algorithm from Table 4.2
˙ θp = Γ φ, Γ = Γ > 0
= ( z − θp φ) /m 2 , m 2 = 1 + φ φ
Adaptive law
α
( s)
α
( s)
φ = [ n− 1
u
n− 1
y
Λ
p, −
p]
p( s)
Λ p( s)
z = sn y
Λ
p,
θp = [ θ
p( s)
b , θa ]
ˆ
Zp( s, t) = θb αn− 1( s) , ˆ
Rp( s, t) = sn + θa αn− 1( s)
Solve for ˆ
L( s, t) = sn− 1 + l αn− 2( s),
ˆ
P ( s, t) = p αn+ q− 1( s) the polynomial equation:
ˆ
L( s, t) ·Qm( s) · ˆ
Rp( s, t) + ˆ
P ( s, t) · ˆ
Zp( s, t) = A∗( s)
or solve for ˆ
βl the algebraic equation
ˆ
S ˆ
lβl = α∗
l
Calculation
where ˆ
Sl is the Sylverster matrix of ˆ
RpQm, ˆ
Zp
ˆ
βl =[ lq , p ] ∈R 2( n+ q) , lq =[0 , . . . , 0 , 1 , l ] ∈Rn+ q q
A∗( s) = s 2 n+ q− 1 + α∗ α 2 n+ q− 2( s)
α∗l = [0 , . . . , 0 , 1 , α∗ ] ∈ R 2( n+ q)
q
Control law
up = (Λ − ˆ
LQm) 1 u
( y
Λ p − ˆ
P 1Λ p − ym)
A∗( s) monic Hurwitz; Λ( s) monic Hurwitz of degree
Design
n + q − 1; for simplicity, Λ( s) = Λ p( s)Λ q( s), where
variables
Λ p( s) , Λ q( s) are monic Hurwitz of degree n, q − 1,
respectively
7.4. INDIRECT APPC SCHEMES
477
APPC scheme of Table 7.4 are u.b. and the tracking error converges to zero
asymptotically with time. The same result holds if we replace the gradient
algorithm in Table 7.4 with any other adaptive law from Tables 4.2 and 4.3.
Outline of Proof: The proof is completed in the following four steps as in Example
7.4.1:
Step 1. Manipulate the estimation error and control law equations to express
the plant input up and output yp in terms of the estimation error. This step leads
to the following equations:
˙ x = A( t) x + b 1( t) m 2 + b 2 ¯ ym
up = C 1 x + d 1 m 2 + d 2¯ ym
(7.4.24)
yp = C 2 x + d 3 m 2 + d 4¯ ym
where ¯
ym ∈ L∞; A( t) , b 1( t) are u.b. because of the boundedness of the estimated
plant and controller parameters (which is guaranteed by the adaptive law and the
stabilizability assumption); b 2 is a constant vector; C 1 and C 2 are vectors whose
elements are u.b.; and d 1 to d 4 are u.b. scalars.
Step 2. Establish the e.s. of the homogeneous part of (7.4.24). The matrix A( t)
has stable eigenvalues at each frozen time t that are equal to the roots of A∗( s) = 0.
In addition ˙ θp, ˙ l, ˙ p ∈ L 2 (guaranteed by the adaptive law and the stabilizability
assumption), imply that
˙
A( t) ∈ L 2. Therefore, using Theorem 3.4.11, we conclude
that the homogeneous part of (7.4.24) is u.a.s.
Step 3. Use the properties of the L 2 δ norm and B-G Lemma to establish
boundedness. Let m 2 = 1 + u 2 + y 2 where · denotes the L
f
p
p
2 δ norm. Using
the results established in Steps 1 and 2 and the normalizing properties of mf , we
show that
m 2
2
f ≤ c
mmf
+ c
(7.4.25)
which implies that
t
m 2 f ≤ c
e−δ( t−τ) 2 m 2 m 2 fdτ + c
0
Because m ∈ L 2, the boundedness of mf follows by applying the B-G lemma.
Using the boundedness of mf , we can establish the boundedness of all signals in
the closed-loop plant.
Step 4. Establish that the tracking error e 1 converges to zero. The convergence
of e 1 to zero follows by using the control and estimation error equations to express
e 1 as the output of proper stable LTI systems whose inputs are in L 2 ∩ L∞.
The details of the proof of Theorem 7.4.1 are given in Section 7.7.
✷
478
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
7.4.3
APPC Schemes: State-Variable Approach
As in Section 7.4.2, let us start with a scalar example to illustrate the design
and analysis of an APPC scheme formed by combining the control law of
Section 7.3.3 developed for the case of known plant parameters with an
adaptive law.
Example 7.4.2 We consider the same plant as in Example 7.3.2, i.e.,
b
yp =
u
s + a p
(7.4.26)
where a and b are unknown constants with b = 0 and up is to be chosen so that the
poles of the closed-loop plant are placed at the roots of A∗( s) = ( s + 1)2 = 0 and
yp tracks the reference signal ym = 1. As we have shown in Example 7.3.2, if a, b
are known, the following control law can be used to meet the control objective:
˙
−a 1
1
ˆ
e =
ˆ
e +
b¯
u
0
0
1
p − Ko([1 0]ˆ
e − e 1)
s + 1
¯
up = −Kcˆ e, up =
¯
u
s
p
(7.4.27)
where Ko, Kc are calculated by solving the equations
det( sI − A + BKc) = ( s + 1)2
det( sI − A + KoC ) = ( s + 5)2
where
−a 1
1
A =
, B =
b, C = [1 , 0]
0
0