(7.6.21)
0
if t ∈ [ tk, tk + jkτ]
where
• k = 1 , 2 , . . .
• t 1 is the first time instant for which Cd( θp( t 1)) = ν(1) > 0 , where C
ν(1)
ν( k) =
d( θ 0) =
, k = 1 , 2 , . . .
(7.6.22)
2 k
k
• tk ( k ≥ 2) is the first time instant after t = tk− 1 + jk− 1 τ for which
k− 1
ν(1)
Cd( θp( tk)) = ν( k +
ji) =
(7.6.23)
( k +
k− 1
i=1
i=1 ji)
and jk = 1 , 2 , . . . is the smallest integer for which
k
Cd( θp( tk + jkτ)) > ν( k +
ji)
(7.6.24)
i=1
where τ > 0 is a design constant.
• uc( t) is the certainty equivalence control given in Section 7.4.
• ur( t) is any bounded stationary signal which is sufficiently rich of order
2 n. For example, one can choose
n
ur( t) =
Ai sin ωit
i=1
512
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
where Ai = 0 , i = 1 , . . . , n, and ωi = ωj for i = j.
From (7.6.18), we see that in the time intervals ( tk + jkτ, tk+1), the sta-
bilizability degree Cd( θp( t)) is above the threshold ν( k +
k
i=1 ji) and the
adaptive control system includes a normalized least-squares estimator and
a pole placement controller. In the time intervals [ tk+1 , tk+1 + jk+1 τ], the
control input is equal to an external exciting input ur( t) and the parameter
vector estimate θp( t) is generated by an unnormalized least-squares estima-
tor.
The switching (at time t = tk) from the pole placement control uc to the
external rich signal ur occurs when the stabilizability degree Cd( θp( t)) of the
estimated model reaches the threshold ν( k +
k
i=1 ji). We keep applying u =
ur during successive time intervals of fixed length τ, until time t = tk + jkτ
for which the condition Cd( θp( tk + jkτ)) > ν( k +
k
i=1 ji) is satisfied and u
switches back to the CEC law. The idea behind this approach is that when
the estimated model is stabilizable, the control objective is pole placement
and closed-loop stabilization; but when the estimation starts to deteriorate,
the control priority becomes the “improvement” of the quality of estimation,
so that the estimated parameters can cross the hypersurfaces that contain
the points where Cd( θp) is close to zero.
The following theorem establishes the stability properties of the proposed
adaptive pole placement scheme.
Theorem 7.6.1 All the signals in the closed-loop (7.3.1) and (7.6.18) to
(7.6.24) are bounded and the tracking error converges to zero as t → ∞.
Furthermore, there exist finite constants ν∗, T ∗ > 0 such that for t ≥ T ∗, we
have Cd( θp( t)) ≥ ν∗ and u( t) = uc.
The proof of Theorem 7.6.1 is rather long and can be found in [63].
The design of the switching logic in the above modified controllers is
based on a simple and intuitive idea that when the quality of parameter es-
timation is “poor,” the objective changes from pole placement to parameter
identification. Parameter identification is aided by an external open-loop
sufficiently rich signal that is kept on until the quality of the parameter es-
timates is acceptable for control design purposes. One of the advantages of
the switched-excitation algorithm is that it is intuitive and easy to imple-
ment. It may suffer, however, from the same drawbacks as other switching
algorithms, that is, the transient performance may be poor during switching.
7.7. STABILITY PROOFS
513
The adaptive control scheme (7.6.18) to (7.6.24) may be simplified when a
lower bound ν∗ > 0 for Cd( θ∗p) is known. In this case, ν( k) = ν∗ ∀k. In the proposed scheme, the sequence ν( k) converges to ν∗ and therefore the lower
bound for Cd( θ∗p) is also identified. The idea behind the identification of ν∗
is due to [189] where very similar to the switched-excitation approach meth-
ods are used to solve the stabilizability problem of APPC for discrete-time
plants.
7.7
Stability Proofs
In this section we present all the long proofs of the theorems of the previous sub-
sections. In most cases, these proofs follow directly from those already presented
for the simple examples and are repeated for the sake of completeness.
7.7.1
Proof of Theorem 7.4.1
Step 1. Let us start by establishing the expressions (7.4.24). We rewrite the control
law (7.4.23) and the normalized estimation error as
ˆ
1
1
LQm u
( y
Λ p = − ˆ
P Λ p − ym)
(7.7.1)
1
1
m 2 = z − θp φ = ˆ
Rp
y
u
Λ
p − ˆ
Zp
p
(7.7.2)
p
Λ p
where Λ( s) , Λ p( s) are monic, Hurwitz polynomials of degree n+ q− 1, n, respectively, ˆ
Rp = sn+ θa αn− 1( s) , ˆ
Zp = θ α
b
n− 1( s). From Table 7.4, we have Λ( s) = Λ p( s)Λ q ( s),
where Λ q( s) is a monic Hurwitz polynomial of degree q− 1. This choice of Λ simplifies
the proof. We should point out that the same analysis can also be carried out with
Λ , Λ p being Hurwitz but otherwise arbitrary, at the expense of some additional
algebra.
Let us define
1
1
uf =
u
y
Λ p, yf = Λ p
and write (7.7.1), (7.7.2) as
ˆ
P yf + ˆ
LQmuf = ym 1 , ˆ
RpΛ qyf − ˆ
ZpΛ quf = m 2
(7.7.3)
where ym 1 = ˆ
P 1 y
Λ m ∈ L∞. By expressing the polynomials ˆ
Rp( s)Λ q( s), ˆ
Zp( s)Λ q( s),
ˆ
P ( s), ˆ
L( s) Qm( s) as
ˆ
Rp( s)Λ q( s) = sn+ q− 1 + ¯
θ 1 αn+ q− 2( s) , ˆ
Zp( s)Λ q( s) = ¯
θ 2 αn+ q− 2( s)
514
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
ˆ
P ( s) = p 0 sn+ q− 1 + ¯
p αn+ q− 2( s) , ˆ
L( s) Qm( s) = sn+ q− 1 + ¯ l αn+ q− 2( s)
we can rewrite (7.7.3) in the form of
y( n+ q− 1)
= −¯
θ
f
1 αn+ q− 2( s) yf + ¯
θ 2 αn+ q− 2( s) uf + m 2
u( n+ q− 1)
= ( p ¯
θ
¯
θ
f
0 1 − ¯
p) αn+ q− 2( s) yf − ( p 0 2 + ¯ l) αn+ q− 2( s) uf (7.7.4)
−p 0 m 2 + ym 1
where the second equation is obtained from (7.7.3) by substituting for y( n+ q− 1).
f
Defining the state x = y( n+ q− 2) , . . . , ˙ y
, . . . , ˙ u
f
f , yf , u( n+ q− 2)
f
f , uf
, we obtain
˙ x = A( t) x + b 1( t) m 2 + b 2 ym 1
(7.7.5)
where
−¯
θ 1
|
¯
θ 2
− − − − − − −
− − − − − − −
0
1
0
.
.
.
I
.
| O
.
.
n+ q− 2
.
( n+ q− 2) ×( n+ q− 1)
.
.
0
0
0
A( t)=
− − − − − − −
− − − − − − − , b 1( t)= −p 0 , b 2 = 1
¯
¯
p 0 θ 1 − ¯ p
|
−p 0 θ 2 − ¯ l
0
0
− − − − − − −
− − − − − − −
.
.
.
.
.
.
0
.
0
0
O( n+ q− 2) ×( n+ q− 1) |
In+ q− 2
..
0
O( n+ q− 2) ×( n+ q− 1) is an ( n + q − 2) by ( n + q − 1) matrix with all elements equal to zero. Now, because up = Λ uf = u( n+ q− 1) + λ α
f
n+ q− 2( s) uf , yp = Λ yf =
y( n+ q− 1) + λ α
f
n+ q− 2( s) yf where λ is the coefficient vector of Λ( s) − sn+ q− 1, we have
up = [0 , . . . , 0 , 1 , 0 , . . . , 0] ˙ x + [0 , . . . , 0 , λ ] x n+ q− 1
n+ q− 1
n+ q− 1
yp = [1 , 0 , . . . , 0 , 0 , . . . , 0] ˙ x + [ λ , 0 , . . . , 0] x (7.7.6)
n+ q− 1
n+ q− 1
n+ q− 1
Step 2. Establish the e.s. of the homogeneous part of (7.7.5). Because ˆ
P , ˆ
L
satisfy the Diophantine equation (7.4.21), we can show that for each frozen time1 t,
det( sI − A( t)) = ˆ
Rp Λ q · ˆ
LQm + ˆ
P · ˆ
ZpΛ q = A∗Λ q
(7.7.7)
1 X · Y denotes the algebraic product of two polynomials that may have time-varying
coefficients.
7.7. STABILITY PROOFS
515
i.e., A( t) is a stable matrix for each frozen time t. One way to verify (7.7.7) is to
consider (7.7.3) with the coefficients of ˆ
Rp, ˆ
Zp, ˆ
P , ˆ
L frozen at each time t. It follows
from (7.7.3) that
1
yf =
( ˆ
LQ
ˆ
m m 2 + ˆ
ZpΛ qym 1)
RpΛ q · ˆ
LQm + ˆ
P · ˆ
ZpΛ q
whose state space realization is given by (7.7.5). Because
ˆ
RpΛ q · ˆ
LQm + ˆ
P · ˆ
ZpΛ q = A∗Λ q
(7.7.7) follows.
We now need to show that
˙
A( t) ∈ L 2 , A( t) ∈ L∞ from the properties
θp ∈ L∞ and ˙ θp ∈ L 2 which are guaranteed by the adaptive law of Table 7.4. Using
the assumption that the polynomials ˆ
RpQm and ˆ
Zp are strongly coprime at each
time t, we conclude that the Sylvester matrix ˆ
Sl (defined in Table 7.4) is uniformly
nonsingular, i.e., | det( ˆ
Sl) | > ν 0 for some ν 0 > 0, and thus θp ∈ L∞ implies that
S
ˆ
l, S− 1 ∈ L
β
which
l
∞. Therefore, the solution ˆ
βl of the algebraic equation ˆ
Sl l = α∗l
can be expressed as
ˆ
βl = ˆ
S− 1 α∗
l
l
is u.b. On the other hand, because θp ∈ L∞ and ˙ θp ∈ L 2, it follows from the
definition of the Sylvester matrix that ˙ˆ
Sl( t) ∈ L 2. Noting that
˙ˆ β
˙ˆ ˆ
l = − ˆ
S− 1 S S− 1 α∗
l
l l
l
˙
we have ˆ
βl ∈ L 2 which is implied by ˆ
Sl, ˆ
S− 1 ∈ L
l
∞ and
˙ˆ
Sl( t) ∈ L 2.
Because the vectors ¯
θ 1 , ¯
θ 2 , ¯
p, ¯ l are linear combinations of θp, ˆ
βl and all elements
in A( t) are uniformly bounded, we have
˙
˙
A( t) ≤ c( | ˆ
βl( t) | + | ˙ θp( t) |)
which implies that
˙
A( t) ∈ L 2. Using Theorem 3.4.11, it follows that the homo-
geneous part of (7.7.5) is e.s.
Step 3. Use the properties of the L 2 δ norm and B-G Lemma to establish
boundedness. As before, for clarity of presentation, we denote the L 2 δ norm as