( τ ) φ( τ ) dτ, k = 0 , 1 , . . .
k
α
( s)
α
( s)
φ =
n− 1
u
n− 1
y
y
Λ
p, −
p
,
z = sn
p
p( s)
Λ p( s)
Λ p( s)
z−θ
φ
=
pk
, m 2 = 1 + φ φ, ∀t ∈ [ t
Adaptive law
m 2
k, tk+1)
θpk = [ θbk, θak]
ˆ
Rp( s, tk) = sn + θ
α
a( k− 1) n− 1( s)
ˆ
Zp( s, tk) = θ
α
b( k− 1) n− 1( s)
T
Design variables
s = tk+1 − tk > Tm; 2 − Tsλ max(Γ) > γ, for some
γ > 0; Λ p( s) is monic and Hurwitz with degree n
with respect to such computational real time delays can be considerably
improved by using a hybrid adaptive law for parameter estimation. The
sampling rate of the hybrid adaptive law may be chosen appropriately to
allow for the computations of the control law to be completed within the
sampling interval.
Let Tm be the maximum time for performing the computations required
to calculate the control law. Then the sampling period Ts of the hybrid
adaptive law may be chosen as Ts = tk+1 − tk > Tm where {tk : k =
1 , 2 , . . .} is a time sequence. Table 7.7 presents a hybrid adaptive law based
on parametric model (7.4.2). It can be used to replace the continuous-time
496
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
Table 7.8 Hybrid APPC scheme: polynomial approach
Z
Plant
y
p( s)
p =
u
R
p
p( s)
Reference signal
Qm( s) ym = 0
Adaptive law
Hybrid adaptive law of Table 7.7
Algebraic
Solve for ˆ
L( s, tk)= sn− 1+ l ( tk) αn− 2( s)
êquation
P ( s, tk) = p ( tk) αn+ q− 1( s) from equation
ˆ
L( s, tk) Qm( s) ˆ
Rp( s, tk) + ˆ
P ( s, tk) ˆ
Zp( s, tk) = A∗( s)
Control law
Λ( s) − ˆ
L( s, t
ˆ
P ( s, t
u
k) Qm( s)
k)
p =
u
( y
Λ( s)
p −
Λ( s)
p − ym)
A∗ monic Hurwitz of degree 2 n + q − 1; Qm( s)
Design variables
monic of degree q with nonrepeated roots on the
jω axis; Λ( s) = Λ p( s)Λ q( s); Λ p( s) , Λ q( s) monic and Hurwitz with degree n, q − 1, respectively
adaptive laws of the APPC schemes discussed in Section 7.4 as shown in
Tables 7.8 to 7.10.
The controller parameters in the hybrid adaptive control schemes of Ta-
bles 7.8 to 7.10 are updated at discrete times by solving certain algebraic
equations. As in the continuous-time case, the solution of these equations ex-
ist provided the estimated polynomials ˆ
Rp( s, tk) Qm( s) , ˆ
Zp( s, tk) are strongly
coprime at each time tk.
The following theorem summarizes the stability properties of the hybrid
APPC schemes presented in Tables 7.8 to 7.10.
Theorem 7.5.1 Assume that the polynomials ˆ
Rp( s, tk) Qm( s) , ˆ
Zp( s, tk) are
strongly coprime at each time t = tk. Then the hybrid APPC schemes given
7.5. HYBRID APPC SCHEMES
497
Table 7.9 Hybrid APPC scheme: state variable approach
Plant
Z
y
p( s)
p =
u
R
p
p( s)
Reference signal
Qm( s) ym = 0
Adaptive law
Hybrid adaptive law of Table 7.7.
˙ˆ e = ˆ
Ak− 1ˆ e + ˆ
Bk− 1¯ up − ˆ
Ko( k− 1)[ C ˆ e − e 1] ˆ
Ak− 1 =
In+ q− 1
−θ 1( k− 1) − − −−
ˆ
Bk− 1 = θ 2( k− 1) , C
=
State observer
0
[1 , 0 , . . . , 0] ˆ
Ko( k− 1) = a∗− θ 1( k− 1) θ 1( k− 1), θ 2( k− 1)
are the coefficient vectors of ˆ
Rp( s, tk) Qm( s) −sn+ q,
ˆ
Zp( s, tk) Q 1( s), respectively, a∗ is the coefficient
vector of A∗o( s) − sn+ q
Algebraic
Solve for ˆ
Kc( k− 1) the equation det[ sI − ˆ
Ak− 1 +
equation
ˆ
B
ˆ
k− 1 Kc( k− 1)] = A∗c( s)
Control law
Q
¯
u
1( s)
p = − ˆ
Kc( k− 1)ˆ e, up =
¯
u
Q
p
m( s)
Design variables
Choose Qm, Q 1 , A∗o, A∗c, Q 1 as in Table 7.5
in Tables 7.8 to 7.10 guarantee signal boundedness and convergence of the
tracking error to zero asymptotically with time.
The proof of Theorem 7.5.1 is similar to that of the theorems in Sec-
tion 7.4, with minor modifications that take into account the discontinuities
in the parameters and is given in Section 7.7.
Table 7.10 Hybrid adaptive LQ control scheme
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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
Z
Plant
y
p( s)
p =
u
R
p
p( s)
Reference signal
Qm( s) ym = 0
Adaptive law
Hybrid adaptive law of Table 7.7.
˙ˆ e = ˆ
A
State observer
k− 1 ˆ
e + ˆ
Bk− 1¯ up − ˆ
Ko( k− 1)[ C ˆ e − e 1]
ˆ
Ko( k− 1) , ˆ
Ak− 1 , ˆ
Bk− 1 , C as in Table 7.9
Solve for P
Riccati equation
k− 1 = Pk− 1 > 0 the equation
ˆ
A
ˆ
ˆ
ˆ
k− 1 Pk− 1+ Pk− 1 Ak− 1 − 1 P
B
B
λ k− 1 k− 1 k− 1 Pk− 1+ CC = 0
1
Q
Control law
¯
u
ˆ
1( s)
p = −
B
¯
u
λ k− 1 Pk− 1ˆ
e, up = Q
p
m( s)
Design variables
Choose λ, Q 1( s) , Qm( s) as in Table 7.6
The major advantage of the hybrid adaptive control schemes described in
Tables 7.7 to 7.10 over their continuous counterparts is the smaller computa-
tional effort required during implementation. Another possible advantage is
better robustness properties in the presence of measurement noise, since the
hybrid scheme does not respond instantaneously to changes in the system,
which may be caused by measurement noise.
7.6
Stabilizability Issues and Modified APPC
The main drawbacks of the APPC schemes of Sections 7.4 and 7.5 is that
the adaptive law cannot guarantee that the estimated plant parameters or
polynomials satisfy the appropriate controllability or stabilizability condi-
tion at each time t, which is required to calculate the controller parameter
vector θc. Loss of stabilizability or controllability may lead to computational
problems and instability.
7.6. STABILIZABILITY ISSUES AND MODIFIED APPC
499
In this section we concentrate on this problem of the APPC schemes and
propose ways to avoid it. We call the estimated plant parameter vector θp at
time t stabilizable if the corresponding algebraic equation is solvable for the
controller parameters. Because we are dealing with time-varying estimates,
uniformity with respect to time is guaranteed by requiring the level of sta-
bilizability to be greater than some constant ε∗ > 0 . For example, the level
of stabilizability can be defined as the absolute value of the determinant of
the Sylvester matrix of the estimated plant polynomials.
We start with a simple example that demonstrates the loss of stabiliz-
ability that leads to instability.
7.6.1
Loss of Stabilizability: A Simple Example
Let us consider the first order plant
˙ y = y + bu
(7.6.1)
where b = 0 is an unknown constant. The control objective is to choose u
such that y, u ∈ L∞, and y( t) → 0 as t → ∞.
If b were known then the control law
2
u = − y
(7.6.2)
b
would meet the control objective exactly. When b is unknown, a natural
approach to follow is to use the certainty equivalence control (CEC) law
2
uc = − y
(7.6.3)
ˆ b
where ˆ b( t) is the estimate of b at time t, generated on-line by an appropriate
adaptive law.
Let us consider the following two adaptive laws:
(i) Gradient
˙ˆ b = γφε , ˆ b(0) = ˆ b 0 = 0
(7.6.4)
where γ > 0 is the constant adaptive gain.
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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
(ii) Pure Least-Squares
˙ˆ b = Pφε , ˆ b(0) = ˆ b 0 = 0
˙
P
= −P 2
φ 2
, P (0) = p
1 + β
0 > 0
(7.6.5)
0 φ 2
where P, φ ∈ R 1,
z − ˆ bφ
1
1
ε =
, y
y , φ =
u
1 + β
f =
0 φ 2
s + 1
s + 1
z =
˙ yf − yf
(7.6.6)
It can be shown that for β 0 > 0, the control law (7.6.3) with ˆ b generated
by (7.6.4) or (7.6.5) meets the control objective provided that ˆ b( t) = 0 ∀t ≥ 0 .
Let us now examine whether (7.6.4) or (7.6.5) can satisfy the condition
ˆ b( t) = 0 , ∀t ≥ 0 .
From (7.6.1) and (7.6.6), we obtain
˜ bφ
ε = −
(7.6.7)
1 + β 0 φ 2
where ˜ b = ˆ b − b is the parameter error. Using (7.6.7) in (7.6.4), we have
˙ˆ
φ 2
b = −γ
(ˆ b − b) , ˆ b(0) = ˆ b
1 + β
0
(7.6.8)
0 φ 2
Similarly, (7.6.5) can be rewritten as
˙ˆ
φ 2
b = −P
(ˆ b − b) , ˆ b(0) = ˆ b
1 + β
0
0 φ 2 p
P ( t) =
0
, p 0 > 0
(7.6.9)
1 + p
t
φ 2
0
dτ
0 1+ β 0 φ 2
˙
It is clear from (7.6.8) and (7.6.9) that for ˆ b(0) = b, ˆ b( t) = 0 and ˆ b( t) =
b, ∀t ≥ 0; therefore, the control objective can be met exactly with such an
initial condition for ˆ b.
˙
If φ( t) = 0 over a nonzero finite time interval, we will have ˆ b = 0 , u =
y = 0, which is an equilibrium state (not necessarily stable though) and the
control objective is again met.
7.6. STABILIZABILITY ISSUES AND MODIFIED APPC
501
10
9
8
7
6
5
Output y(t)
4
3
2
1
0
-1
-0.5
0
0.5
1
1.5
2
2.5
Estimate b(t)
^
Figure 7.5 Output y( t) versus estimate ˆ b( t) for different initial conditions
y(0) and ˆ b(0) using the CEC uc = − 2 y/ˆ b.
For analysis purposes, let us assume that b > 0 (unknown to the designer).
For φ = 0, both (7.6.8), (7.6.9) imply that
˙
sgn(ˆ b) = − sgn(ˆ b( t) − b)
and, therefore, for b > 0 we have
˙ˆ
˙
b( t) > 0 if ˆ b(0) < b and ˆ b( t) < 0 if ˆ b(0) > b Hence, for ˆ b(0) < 0 < b, ˆ b( t) is monotonically increasing and crosses zero
leading to an unbounded control uc.
Figure 7.5 shows the plots of y( t) vs ˆ b( t) for different initial conditions
ˆ b(0), y(0), demonstrating that for ˆ b(0) < 0 < b, ˆ b( t) crosses zero leading to unbounded closed-loop signals. The value of b = 1 is used for this simulation.
The above example demonstrates that the CEC law (7.6.3) with (7.6.4)
or (7.6.5) as adaptive laws for generating ˆ b is not guaranteed to meet the
control objective. If the sign of b and a lower bound for |b| are known,
then the adaptive laws (7.6.4), (7.6.5) can be modified using projection to
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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
constrain ˆ b( t) from changing sign. This projection approach works for this
simple example but its extension to the higher order case is awkward due to
the lack of any procedure for constructing the appropriate convex parameter
sets for projecting the estimated parameters.
7.6.2
Modified APPC Schemes
The stabilizability problem has attracted considerable interest in the adap-
tive control community and several solutions have been proposed. We list
the most important ones below with a brief explanation regarding their ad-
vantages and drawbacks.
(a) Stabilizability is assumed. In this case, no modifications are introduced
and stabilizability is assumed to hold for all t ≥ 0 . Even though there is no
theoretical justification for such an assumption to hold, it has been often
argued that in most simulation studies, no stabilizability problems usually
arise. The example presented above illustrates that no stabilizability prob-