MRC Schemes: Known Plant Parameters
In addition to assumptions P1 to P4 and M1, M2, let us also assume
that the plant parameters, i.e., the coefficients of Gp( s) are known exactly.
Because the plant is LTI and known, the design of the MRC scheme is
achieved using linear system theory.
The MRC objective is met if up is chosen so that the closed-loop transfer
function from r to yp has stable poles and is equal to Wm( s), the transfer
function of the reference model. Such a transfer function matching guaran-
tees that for any reference input signal r( t), the plant output yp converges
to ym exponentially fast.
A trivial choice for up is the cascade open-loop control law
k Z
R
u
m
m( s)
p( s)
p = C( s) r,
C( s) =
(6.3.6)
kp Rm( s) Zp( s)
which leads to the closed-loop transfer function
yp
k Z R k
= m m
p
pZp = W
r
k
m( s)
(6.3.7)
p Rm Zp Rp
This control law, however, is feasible only when Rp( s) is Hurwitz. Other-
wise, (6.3.7) may involve zero-pole cancellations outside C−, which will lead
to unbounded internal states associated with non-zero initial conditions [95].
In addition, (6.3.6) suffers from the usual drawbacks of open loop control
such as deterioration of performance due to small parameter changes and
inexact zero-pole cancellations.
Instead of (6.3.6), let us consider the feedback control law
α( s)
α( s)
up = θ∗ 1
u
y
Λ( s) p + θ∗ 2 Λ( s) p + θ∗ 3 yp + c∗ 0 r
(6.3.8)
shown in Figure 6.4 where
α( s) = αn− 2( s) = sn− 2 , sn− 3 , . . . , s, 1
for n ≥ 2
α( s) = 0
for n = 1
c∗ 0 , θ∗ 3 ∈ R 1; θ∗ 1 , θ∗ 2 ∈ Rn− 1 are constant parameters to be designed and Λ( s) is an arbitrary monic Hurwitz polynomial of degree n− 1 that contains Zm( s)
as a factor, i.e.,
Λ( s) = Λ0( s) Zm( s)
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CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
r✲ c∗
✲
+ ♥
up
yp
0
Σ
✲ Gp( s)
✲
+ +
✂✂✍ ❆
❆❑
✻
+
α( s)
✂
❆
θ∗
✛
1
✂
Λ( s)
✂
✂
α( s)
θ∗
✛
✂
2
Λ( s)
✂
✂
✛
θ∗ 3
Figure 6.4 Structure of the MRC scheme (6.3.8).
which implies that Λ0( s) is monic, Hurwitz and of degree n 0 = n − 1 − qm.
The controller parameter vector
θ∗ = θ∗ 1 , θ∗ 2 , θ∗ 3 , c∗ 0
∈ R 2 n
is to be chosen so that the transfer function from r to yp is equal to Wm( s).
The I/O properties of the closed-loop plant shown in Figure 6.4 are
described by the transfer function equation
yp = Gc( s) r
(6.3.9)
where
c∗
G
0 kpZpΛ2
c( s) =
(6.3.10)
Λ Λ − θ∗ 1 α( s) Rp − kpZp θ∗ 2 α( s) + θ∗ 3Λ
We can now meet the control objective if we select the controller param-
eters θ∗ 1 , θ∗ 2 , θ∗ 3 , c∗ 0 so that the closed-loop poles are stable and the closed-loop
transfer function Gc( s) = Wm( s), i.e.,
c∗ 0 kpZpΛ2
Z
= k
m
(6.3.11)
Λ Λ − θ∗
m R
1 α Rp − kpZp θ∗
2 α + θ∗
3Λ
m
is satisfied for all s ∈ C. Because the degree of the denominator of Gc( s) is
np + 2 n − 2 and that of Rm( s) is pm ≤ n, for the matching equation (6.3.11)
to hold, an additional np + 2 n − 2 − pm zero-pole cancellations must occur in
Gc( s). Now because Zp( s) is Hurwitz by assumption and Λ( s) = Λ0( s) Zm( s) 6.3. MRC FOR SISO PLANTS
335
is designed to be Hurwitz, it follows that all the zeros of Gc( s) are stable
and therefore any zero-pole cancellation can only occur in C−. Choosing
k
c∗
m
0 =
(6.3.12)
kp
and using Λ( s) = Λ0( s) Zm( s) the matching equation (6.3.11) becomes
Λ − θ∗ 1 α Rp − kpZp θ∗ 2 α + θ∗ 3Λ = ZpΛ0 Rm
(6.3.13)
or
θ∗ 1 α( s) Rp( s) + kp θ∗ 2 α( s) + θ∗ 3Λ( s) Zp( s) = Λ( s) Rp( s) − Zp( s)Λ0( s) Rm( s) (6.3.14)
Equating the coefficients of the powers of s on both sides of (6.3.14), we can
express (6.3.14) in terms of the algebraic equation
S ¯
θ∗ = p
(6.3.15)
where ¯
θ∗ = θ∗ 1 , θ∗ 2 , θ∗ 3 , S is an ( n + np − 1) × (2 n − 1) matrix that depends on the coefficients of Rp, kpZp and Λ, and p is an n + np − 1 vector
with the coefficients of Λ Rp−ZpΛ0 Rm. The existence of ¯
θ∗ to satisfy (6.3.15)
and, therefore, (6.3.14) will very much depend on the properties of the matrix
S. For example, if n > np, more than one ¯
θ∗ will satisfy (6.3.15), whereas if
n = np and S is nonsingular, (6.3.15) will have only one solution.
Remark 6.3.2 For the design of the control input (6.3.8), we assume that
n ≥ np. Because the plant is known exactly, there is no need to assume
an upper bound for the degree of the plant, i.e., because np is known
n can be set equal to np. We use n ≥ np on purpose in order to use
the result in the unknown plant parameter case treated in Sections 6.4
and 6.5, where only the upper bound n for np is known.
Remark 6.3.3 Instead of using (6.3.15), one can solve (6.3.13) for θ∗ 1 , θ∗ 2 , θ∗ 3
as follows: Dividing both sides of (6.3.13) by Rp( s), we obtain
Z
∆ ∗
Λ − θ∗
p
1 α − kp
( θ∗
R
2 α + θ∗
3Λ) = Zp
Q + kp
p
Rp
336
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
where Q( s) (of degree n − 1 − mp) is the quotient and kp∆ ∗ (of degree
at most np − 1) is the remainder of Λ0 Rm/Rp, respectively. Then the
solution for θ∗i, i = 1 , 2 , 3 can be found by inspection, i.e.,
θ∗ 1 α( s) = Λ( s) − Zp( s) Q( s)
(6.3.16)
Q( s) R
θ∗
p( s) − Λ0( s) Rm( s)
2 α( s) + θ∗
3Λ( s) =
(6.3.17)
kp
where the equality in the second equation is obtained by substituting
for ∆ ∗( s) using the identity Λ0 Rm = Q + kp∆ ∗ . The parameters θ∗
Rp
Rp
i , i =
1 , 2 , 3 can now be obtained directly by equating the coefficients of the
powers of s on both sides of (6.3.16), (6.3.17).
Equations (6.3.16) and (6.3.17) indicate that in general the controller
parameters θ∗i, i = 1 , 2 , 3 are nonlinear functions of the coefficients of
the plant polynomials Zp( s) , Rp( s) due to the dependence of Q( s) on
the coefficients of Rp( s). When n = np and n∗ = 1, however, Q( s) = 1
and the θ∗i’s are linear functions of the coefficients of Zp( s) , Rp( s).
Lemma 6.3.1 Let the degrees of Rp, Zp, Λ , Λ0 and Rm be as specified in
(6.3.8). Then (i) The solution ¯
θ∗ of (6.3.14) or (6.3.15) always exists.
(ii) In addition if Rp, Zp are coprime and n = np, then the solution ¯
θ∗ is
unique.
Proof Let Rp = ¯
Rp( s) h( s) and Zp( s) = ¯
Zp( s) h( s) and ¯
Rp( s) , ¯
Zp( s) be coprime,
where h( s) is a monic polynomial of degree r 0 (with 0 ≤ r 0 ≤ mp). Because Zp( s) is Hurwitz, it follows that h( s) is also Hurwitz. If Rp, Zp are coprime, h( s) = 1, i.e., r 0 = 0. If Rp, Zp are not coprime, r 0 ≥ 1 and h( s) is their common factor. We can now write (6.3.14) as
θ∗ 1 α ¯
Rp + kp( θ∗ 2 α + θ∗ 3Λ) ¯
Zp = Λ ¯
Rp − ¯
ZpΛ0 Rm
(6.3.18)
by canceling h( s) from both sides of (6.3.14). Because h( s) is Hurwitz, the cancella-
tion occurs in C−. Equation (6.3.18) leads to np + n−r 0 − 2 algebraic equations with
2 n− 1 unknowns. It can be shown that the degree of Λ ¯
Rp− ¯
ZpΛ0 Rm is np+ n−r 0 − 2
because of the cancellation of the term snp+ n−r 0 − 1. Because ¯
Rp, ¯
Zp are coprime,
it follows from Theorem 2.3.1 that there exists unique polynomials a 0( s) , b 0( s) of
degree n − 2 , np − r 0 − 1 respectively such that
a 0( s) ¯
Rp( s) + b 0( s) ¯
Zp( s) = Λ( s) ¯
Rp( s) − ¯
Zp( s)Λ0( s) Rm( s)
(6.3.19)
6.3. MRC FOR SISO PLANTS
337
is satisfied for n ≥ 2. It now follows by inspection that
θ∗ 1 α( s) = f( s) ¯
Zp( s) + a 0( s)
(6.3.20)
and
kp( θ∗ 2 α( s) + θ∗ 3Λ( s)) = −f( s) ¯
Rp( s) + b 0( s)
(6.3.21)
satisfy (6.3.18), where f ( s) is any given polynomial of degree nf = n − np + r 0 − 1.
Hence, the solution θ∗ 1 , θ∗ 2 , θ∗ 3 of (6.3.18) can be obtained as follows: We first solve
(6.3.19) for a 0( s) , b 0( s). We then choose an arbitrary polynomial f( s) of degree nf = n − np + r 0 − 1 and calculate θ∗ 1 , θ∗ 2 , θ∗ 3 from (6.3.20), (6.3.21) by equating coefficients of the powers of s. Because f ( s) is arbitrary, the solution θ∗ 1 , θ∗ 2 , θ∗ 3 is not unique. If, however, n = np and r 0 = 0, i.e., Rp, Zp are coprime, then f( s) = 0
and θ∗ 1 α( s) = a 0( s), kp( θ∗ 2 α( s) + θ∗ 3Λ( s)) = b 0( s) which implies that the solution θ∗ 1 , θ∗ 2 , θ∗ 3 is unique due to the uniqueness of a 0( s) , b 0( s). If n = np = 1, then α( s) = 0, Λ( s) = 1, θ∗ 1 = θ∗ 2 = 0 and θ∗ 3 given by (6.3.18) is unique.
✷
Remark 6.3.4 It is clear from (6.3.12), (6.3.13) that the control law (6.3.8)
places the poles of the closed-loop plant at the roots of the polynomial
Zp( s)Λ0( s) Rm( s) and changes the high frequency gain from kp to km
by using the feedforward gain c∗ 0. Therefore, the MRC scheme can be
viewed as a special case of a general pole placement scheme where the
desired closed-loop characteristic equation is given by
Zp( s)Λ0( s) Rm( s) = 0
The transfer function matching (6.3.11) is achieved by canceling the
zeros of the plant, i.e., Zp( s), and replacing them by those of the
reference model, i.e., by designing Λ = Λ0 Zm. Such a cancellation
is made possible by assuming that Zp( s) is Hurwitz and by designing
Λ0 , Zm to have stable zeros.
We have shown that the control law (6.3.8) guarantees that the closed-
loop transfer function Gc( s) of the plant from r to yp has all its poles in C−
and in addition, Gc( s) = Wm( s). In our analysis we assumed zero initial
conditions for the plant, reference model and filters. The transfer function
matching, i.e., Gc( s) = Wm( s), together with zero initial conditions guar-
antee that yp( t) = ym( t) , ∀t ≥ 0 and for any reference input r( t) that is 338
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
bounded and piecewise continuous. The assumption of zero initial condi-
tions is common in most I/O control design approaches for LTI systems and
is valid provided that any zero-pole cancellation in the closed-loop plant
transfer function occurs in C−. Otherwise nonzero initial conditions may
lead to unbounded internal states that correspond to zero-pole cancellations
in C+.
In our design we make sure that all cancellations in Gc( s) occur in C−
by assuming stable zeros for the plant transfer function and by using stable
filters in the control law. Nonzero initial conditions, however, will affect the
transient response of yp( t). As a result we can no longer guarantee that
yp( t) = ym( t) ∀t ≥ 0 but instead that yp( t) → ym( t) exponentially fast with a rate that depends on the closed-loop dynamics. We analyze the effect of
initial conditions by using state space representations for the plant, reference
model, and controller as follows: We begin with the following state-space
realization of the control law (6.3.8):
˙ ω 1 = F ω 1 + gup,
ω 1(0) = 0
˙ ω 2 = F ω 2 + gyp,
ω 2(0) = 0
(6.3.22)
up = θ∗ ω
where ω 1 , ω 2 ∈ Rn− 1,
θ∗ = θ∗ 1 , θ∗ 2 , θ∗ 3 , c∗ 0
, ω = [ ω 1 , ω 2 , yp, r]
−λ
n