control laws that can meet the pole placement control objective when the
plant parameters are known exactly. The form of these control laws as well
as the mapping between the controller and plant parameters will be used
in Section 7.4 to form indirect APPC schemes for plants with unknown
parameters.
7.3.1
Problem Statement
Consider the SISO LTI plant
Z
y
p( s)
p = Gp( s) up,
Gp( s) =
(7.3.1)
Rp( s)
where Gp( s) is proper and Rp( s) is a monic polynomial. The control objec-
tive is to choose the plant input up so that the closed-loop poles are assigned
to those of a given monic Hurwitz polynomial A∗( s). The polynomial A∗( s),
referred to as the desired closed-loop characteristic polynomial, is chosen
based on the closed-loop performance requirements. To meet the control
objective, we make the following assumptions about the plant:
P1. Rp( s) is a monic polynomial whose degree n is known.
P2. Zp( s) , Rp( s) are coprime and degree( Zp) < n.
Assumptions (P1) and (P2) allow Zp, Rp to be non-Hurwitz in contrast to the
MRC case where Zp is required to be Hurwitz. If, however, Zp is Hurwitz,
the MRC problem is a special case of the general pole placement problem
defined above with A∗( s) restricted to have Zp as a factor. We will explain
the connection between the MRC and the PPC problems in Section 7.3.2.
In general, by assigning the closed-loop poles to those of A∗( s), we can
guarantee closed-loop stability and convergence of the plant output yp to zero
provided there is no external input. We can also extend the PPC objective
to include tracking, where yp is required to follow a certain class of reference
signals ym, by using the internal model principle discussed in Chapter 3 as
follows: The reference signal ym ∈ L∞ is assumed to satisfy
Qm( s) ym = 0
(7.3.2)
7.3. PPC: KNOWN PLANT PARAMETERS
449
where Qm( s), the internal model of ym, is a known monic polynomial of
degree q with nonrepeated roots on the jω-axis and satisfies
P3. Qm( s) , Zp( s) are coprime.
For example, if yp is required to track the reference signal ym = 2 + sin(2 t),
then Qm( s) = s( s 2 + 4) and, therefore, according to P3, Zp( s) should not
have s or s 2 + 4 as a factor.
The effect of Qm( s) on the tracking error e 1 = yp − ym is explained
in Chapter 3 for a general feedback system and it is analyzed again in the
sections to follow.
In addition to assumptions P1 to P3, let us also assume that the co-
efficients of Zp( s) , Rp( s), i.e., the plant parameters are known exactly and
propose several control laws that meet the control objective. The knowledge
of the plant parameters is relaxed in Section 7.4.
7.3.2
Polynomial Approach
We consider the control law
Qm( s) L( s) up = −P ( s) yp + M( s) ym
(7.3.3)
where P ( s) , L( s) , M ( s) are polynomials (with L( s) monic) of degree q + n −
1 , n − 1 , q + n − 1, respectively, to be found and Qm( s) satisfies (7.3.2) and
assumption P3.
Applying (7.3.3) to the plant (7.3.1), we obtain the closed-loop plant
equation
Z
y
pM
p =
y
LQ
m
(7.3.4)
mRp + P Zp
whose characteristic equation
LQmRp + P Zp = 0
(7.3.5)
has order 2 n + q − 1. The objective now is to choose P, L such that
LQmRp + P Zp = A∗
(7.3.6)
is satisfied for a given monic Hurwitz polynomial A∗( s) of degree 2 n + q − 1.
Because assumptions P2 and P3 guarantee that QmRp, Zp are coprime, it
450
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
follows from Theorem 2.3.1 that L, P satisfying (7.3.6) exist and are unique.
The solution for the coefficients of L( s) , P ( s) of equation (7.3.6) may be
obtained by solving the algebraic equation
Slβl = α∗l
(7.3.7)
where Sl is the Sylvester matrix of QmRp, Zp of dimension 2( n+ q) × 2( n+ q) βl = [ lq , p ] , a∗l = [0 , . . . , 0 , 1 , α∗ ]
q
lq = [0 , . . . , 0 , 1 , l ] ∈ Rn+ q
q
l = [ ln− 2 , ln− 3 , . . . , l 1 , l 0] ∈ Rn− 1
p = [ pn+ q− 1 , pn+ q− 2 , . . . , p 1 , p 0] ∈ Rn+ q
α∗ = [ a∗ 2 n+ q− 2 , a∗ 2 n+ q− 3 , . . . , a∗ 1 , a∗ 0] ∈ R 2 n+ q− 1
li, pi, a∗i are the coefficients of
L( s) = sn− 1 + ln− 2 sn− 2 + · · · + l 1 s + l 0 = sn− 1 + l αn− 2( s) P ( s) = pn+ q− 1 sn+ q− 1 + pn+ q− 2 sn+ q− 2 + · · · + p 1 s + p 0 = p αn+ q− 1( s) A∗( s) = s 2 n+ q− 1+ a∗ 2 n+ q− 2 s 2 n+ q− 2+ · · ·+ a∗ 1 s+ a∗ 0 = s 2 n+ q− 1+ α∗ α 2 n+ q− 2( s) The coprimeness of QmRp, Zp guarantees that Sl is nonsingular; therefore,
the coefficients of L( s) , P ( s) may be computed from the equation
βl = S− 1 α∗
l
l
Using (7.3.6), the closed-loop plant is described by
Z
y
pM
p =
y
A∗
m
(7.3.8)
Similarly, from the plant equation in (7.3.1) and the control law in (7.3.3)
and (7.3.6), we obtain
R
u
pM
p =
y
A∗
m
(7.3.9)
7.3. PPC: KNOWN PLANT PARAMETERS
451
ym
−e 1
P ( s)
u
✲ ❧
p
yp
+ Σ
✲
✲ G
✲
Q
p( s)
m( s) L( s)
✻
−
Figure 7.2 Block diagram of pole placement control.
Because ym ∈ L∞ and ZpM
A∗ , RpM
A∗
are proper with stable poles, it follows
that yp, up ∈ L∞ for any polynomial M( s) of degree n + q − 1. Therefore,
the pole placement objective is achieved by the control law (7.3.3) without
having to put any additional restrictions on M ( s) , Qm( s). When ym = 0,
(7.3.8), (7.3.9) imply that yp, up converge to zero exponentially fast.
When ym = 0, the tracking error e 1 = yp − ym is given by
Z
Z
LR
e
pM − A∗
p
p
1 =
y
( M − P ) y
Q
A∗
m = A∗
m − A∗ mym
(7.3.10)
For zero tracking error, (7.3.10) suggests the choice of M ( s) = P ( s) to null
the first term in (7.3.10). The second term in (7.3.10) is nulled by using
Qmym = 0. Therefore, for M( s) = P ( s), we have
Z
LR
e
p
p
1 =
[0] −
[0]
A∗
A∗
Because Zp
A∗ , LRp
A∗
are proper with stable poles, it follows that e 1 converges
exponentially to zero. Therefore, the pole placement and tracking objective
are achieved by using the control law
QmLup = −P ( yp − ym)
(7.3.11)
which is implemented as shown in Figure 7.2 using n + q − 1 integrators
to realize C( s) =
P ( s)
. Because L( s) is not necessarily Hurwitz, the
Qm( s) L( s)
realization of (7.3.11) with n+ q − 1 integrators may have a transfer function,
namely C( s), with poles outside C−. An alternative realization of (7.3.11) is
obtained by rewriting (7.3.11) as
Λ − LQ
P
u
m
p =
u
( y
Λ
p − Λ p − ym)
(7.3.12)
452
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
−e
u
1
P ( s)
y
p
yp
m
✲
+ ❧
Σ ✲
✲
+ ❧
Σ
✲ G
✲
Λ( s)
p( s)
✻
−
✻
+
Λ( s) − Qm( s) L( s) ✛
Λ( s)
Figure 7.3 An alternative realization of the pole placement control.
where Λ is any monic Hurwitz polynomial of degree n+ q− 1. The control law
(7.3.12) is implemented as shown in Figure 7.3 using 2( n + q − 1) integrators
to realize the proper stable transfer functions Λ −LQm , P . We summarize the
Λ
Λ
main equations of the control law in Table 7.1.
Remark 7.3.1 The MRC law of Section 6.3.2 shown in Figure 6.1 is a
special case of the general PPC law (7.3.3), (7.3.6). We can obtain the
MRC law of Section 6.3.2 by choosing
R
Q
mΛ0
m = 1 ,
A∗ = ZpΛ0 Rm,
M ( s) =
kp
L( s) = Λ( s) − θ∗ 1 αn− 2( s) , P ( s) = −( θ∗ 2 αn− 2( s) + θ∗ 3Λ( s)) Z
Λ = Λ
m
0 Zm,
ym = km
r
Rm
where Zp, Λ0 , Rm are Hurwitz and Λ0 , Rm, kp, θ∗ 1 , θ∗ 2 , θ∗ 3 , r are as defined in Section 6.3.2.
Example 7.3.1 Consider the plant
b
yp =
u
s + a p
where a and b are known constants. The control objective is to choose up such that
the poles of the closed-loop plant are placed at the roots of A∗( s) = ( s + 1)2 and
yp tracks the constant reference signal ym = 1. Clearly the internal model of ym is
Qm( s) = s, i.e., q = 1. Because n = 1, the polynomials L, P, Λ are of the form
L( s) = 1 ,
P ( s) = p 1 s + p 0 ,
Λ = s + λ 0
7.3. PPC: KNOWN PLANT PARAMETERS
453
Table 7.1 PPC law: polynomial approach
Plant
yp = Zp( s) u
R
p
p( s)
Reference input
Qm( s) ym = 0
Solve
for
L( s)
=
sn− 1 + l αn− 2( s) and
P ( s) = p αn+ q− 1( s) the polynomial equation
L( s) Qm( s) Rp( s) + P ( s) Zp( s) = A∗( s) Calculation
or solve for βl the algebraic equation Slβl = α∗l,
where Sl is the Sylvester matrix of RpQm, Zp
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
u
u
e
Control law
p = Λ −LQm
Λ
p − P
Λ 1
e 1 = yp − ym
A∗( s) is monic Hurwitz; Qm( s) is a monic poly-
nomial of degree q with nonrepeated roots on jω
Design variables
axis; Λ( s) is a monic Hurwitz polynomial of degree
n + q − 1
where λ 0 > 0 is arbitrary and p 0 , p 1 are calculated by solving
s( s + a) + ( p 1 s + p 0) b = ( s + 1)2
(7.3.13)
Equating the coefficients of the powers of s in (7.3.13), we obtain
2 − a
1
p 1 =
,
p
b
0 = b
454
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
Equation (7.3.13) may be also written in the form of the algebraic equation (7.3.7),
i.e., the Sylvester matrix of s( s + a) , b is given by
1 0 0 0
a 1 0 0
S
l = 0 a b 0
0 0 0 b
and
0
0
1
1
β
l = p , α∗l =
1
2
p 0
1
Therefore, the PPC law is given by
( s + λ
2 − a
1
1
u
0 − s)
p
=
u
s +
e
s + λ
p −
1
0
b
b s + λ 0
λ
(2 − a) s + 1
=
0
u
e