y
Λ
p, φ 2 = − αn− 1( s) p
p( s)
Λ p( s)
φ 1 = [ φ 0 , ¯
φ 1 ] , φ 0 ∈ Rn−m, ¯
φ 1 ∈ Rm
φ 1 m ∈ R 1 is the last element of φ 0
ˆ
p 1 = [ˆ bm− 1 , . . . , ˆ b 0] , ˆ
p 2 = [ˆ an− 1 , . . . , ˆ a 0]
ˆ¯
Zp( s, t)= ˆ kpsm+ ˆ
p 1 αm− 1( s) , ˆ
Rp( s, t)= sn+ ˆ
p 2 αn− 1( s)
c 0( t) = km
ˆ
kp( t)
θ
ˆ¯
1 ( t) αn− 2( s) = Λ( s) −
1 Z
ˆ
k
p( s, t) · ˆ
Q( s, t)
p( t)
Calculation
θ 2 ( t) αn− 2( s) + θ 3( t)Λ( s)
of θ
= 1 ( ˆ
Q( s, t) · ˆ
R
ˆ
k
p( s, t) − Λ0( s) Rm( s))
p
ˆ
Q( s, t) = quotient of Λ0( s) Rm( s) / ˆ
Rp( s, t)
k 0: lower bound for |kp| ≥ k 0 > 0; Λ p( s): monic Hurwitz
of degree n; For simplicity, Λ
Design
p( s) = ( s + λ 0)Λ( s), λ 0 > 0;
Λ( s) = Λ
variables
0( s) Zm( s); Γ1 = Γ1 > 0, Γ2 = Γ2 > 0; Γ1 ∈ Rm×m,
Γ2 ∈ Rn×n; λp ∈ Rn is the coefficient vector of Λ p( s) −sn
6.6. INDIRECT MRAC
411
where a is the only unknown parameter. The output yp is required to track the
output of ym of the reference model
1
ym =
r
( s + 2)3
The control law is given by
s
1
s
1
up = θ 11
u
u
y
y
( s + λ
p + θ 12
p + θ 21
p + θ 22
p + θ 3 yp + c 0 r
1)2
( s + λ 1)2
( s + λ 1)2
( s + λ 1)2
where θ = [ θ 11 , θ 12 , θ 21 , θ 22 , θ 3 , c 0] ∈ R 6. In direct MRAC, θ is generated by a sixth-order adaptive law. In indirect MRAC, θ is calculated from the adaptive law
as follows:
Using Table 6.9, we have
θp = [0 , 0 , 1 , ˆ a, 0 , 0]
˙ˆ a = γaφa
z − ˆ
z
=
, ˆ
z = θ
1 + φ φ
p φ, z = yp + λp φ 2
[ s 2 , s, 1]
[ s 2 , s, 1]
φ = [ φ 1 , φ 2 ] , φ 1 =
u
y
( s + λ
p,
φ 2 = −
p
1)3
( s + λ 1)3
where Λ p( s) is chosen as Λ p( s) = ( s + λ 1)3, λp = [3 λ 1 , 3 λ 21 , λ 31] .
s 2
φa = [0 , 0 , 0 , 1 , 0 , 0 , ] φ = −
y
( s + λ
p
1)3
and γa > 0 is a constant. The controller parameter vector is calculated as
s
c 0 = 1 , θ 1
= ( s + λ
1
1)2 − ˆ
Q( s, t)
s
θ 2
+ θ
1
3( s + λ 1)2 = ˆ
Q( s, t) · [ s 3 + ˆ as 2] − ( s + λ 1)2( s + 2)3
where ˆ
Q( s, t) is the quotient of ( s+ λ 1)2( s+2)3 .
s 3+ˆ
as 2
The example demonstrates that for the plant (6.6.26), the indirect scheme
requires a first order adaptive law whereas the direct scheme requires a sixth-
order one.
412
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
6.7
Relaxation of Assumptions in MRAC
The stability properties of the MRAC schemes of the previous sections are
based on assumptions P1 to P4, given in Section 6.3.1. While these as-
sumptions are shown to be sufficient for the MRAC schemes to meet the
control objective, it has often been argued whether they are also necessary.
We have already shown in the previous sections that assumption P4 can be
completely relaxed at the expense of a more complex adaptive law, therefore
P4 is no longer necessary for meeting the MRC objective. In this section we
summarize some of the attempts to relax assumptions P1 to P3 during the
1980s and early 1990s.
6.7.1
Assumption P1: Minimum Phase
This assumption is a consequence of the control objective in the known
parameter case that requires the closed-loop plant transfer function to be
equal to that of the reference model. Since this objective can only be achieved
by cancelling the zeros of the plant and replacing them by those of the
reference model, Zp( s) has to be Hurwitz, otherwise zero-pole cancellations
in C+ will take place and lead to some unbounded state variables within the
closed-loop plant. The assumption of minimum phase in MRAC has often
been considered as one of the limitations of adaptive control in general, rather
than a consequence of the MRC objective, and caused some confusion to
researchers outside the adaptive control community. One of the reasons for
such confusion is that the closed-loop MRAC scheme is a nonlinear dynamic
system and zero-pole cancellations no longer make much sense. In this case,
the minimum phase assumption manifests itself as a condition for proving
that the plant input is bounded by using the boundedness of other signals
in the closed-loop.
For the MRC objective and the structures of the MRAC schemes pre-
sented in the previous sections, the minimum phase assumption seems to be
not only sufficient, but also necessary for stability. If, however, we modify
the MRC objective not to include cancellations of unstable plant zeros, then
it seems reasonable to expect to be able to relax assumption P1. For exam-
ple, if we can restrict ourselves to changing only the poles of the plant and
tracking a restricted class of signals whose internal model is known, then we
may be able to allow plants with unstable zeros. The details of such designs
6.7. RELAXATION OF ASSUMPTIONS IN MRAC
413
that fall in the category of general pole placement are given in Chapter 7.
It has often been argued that if we assume that the unstable zeros of the
plant are known, we can include them to be part of the zeros of the reference
model and design the MRC or MRAC scheme in a way that allows only the
cancellation of the stable zeros of the plant. Although such a design seems
to be straightforward, the analysis of the resulting MRAC scheme requires
the incorporation of an adaptive law with projection. The projection in turn
requires the knowledge of a convex set in the parameter space where the es-
timation is to be constrained. The development of such a convex set in the
higher order case is quite awkward, if possible. The details of the design and
analysis of MRAC for plants with known unstable zeros for discrete-time
plants are given in [88, 194].
The minimum phase assumption is one of the main drawbacks of MRAC
for the simple reason that the corresponding discrete-time plant of a sampled
minimum phase continuous-time plant is often nonminimum phase [14].
6.7.2
Assumption P2: Upper Bound for the Plant Order
The knowledge of an upper bound n for the plant order is used to determine
the order of the MRC law. This assumption can be completely relaxed if the
MRC objective is modified. For example, it has been shown in [102, 159, 160]
that the control objective of regulating the output of a plant of unknown
order to zero can be achieved by using simple adaptive controllers that are
based on high-gain feedback, provided the plant is minimum phase and the
plant relative degree n∗ is known. The principle behind some of these high-
gain stabilizing controllers can be explained for a minimum phase plant with
n∗ = 1 and arbitrary order as follows: Consider the following minimum phase
plant with relative degree n∗ = 1:
k
y
pZp( s)
p =
u
R
p
p( s)
From root locus arguments, it is clear that the input
up = −θyp sgn( kp)
with sufficiently large gain θ will force the closed loop characteristic equation
Rp( s) + |kp|θZp( s) = 0
414
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
to have roots in Re [ s] < 0. Having this result in mind, it can be shown that
the adaptive control law
up = − θyp sgn ( kp) ,
˙ θ = y 2 p
can stabilize any minimum phase plant with n∗ = 1 and of arbitrary or-
der. As n∗ increases, the structure and analysis of the high gain adaptive
controllers becomes more complicated [159, 160].
Another class of adaptive controllers for regulation that attracted con-
siderable interest in the research community is based on search methods and
discontinuous adjustments of the controller gains [58, 137, 147, 148]. Of par-
ticular theoretical interest is the controller proposed in [137] referred to as
universal controller that is based on the rather weak assumption that only
the order nc of a stabilizing linear controller needs to be known for the sta-
bilization and regulation of the output of the unknown plant to zero. The
universal controller is based on an automated dense search throughout the
set of all possible nc-order linear controllers until it passes through a subset
of stabilizing controllers in a way that ensures asymptotic regulation and the
termination of the search. One of the drawbacks of the universal controller is
the possible presence of large overshoots as pointed out in [58] which limits
its practicality.
An interesting MRAC scheme that is also based on high gain feedback
and discontinuous adjustment of controller gains is given in [148]. In this
case, the MRC objective is modified to allow possible nonzero tracking errors
that can be forced to be less than a prespecified (arbitrarily small) constant
after an (arbitrarily short) prespecified period of time, with an (arbitrarily
small) prespecified upper bound on the amount of overshoot. The only
assumption made about the unknown plant is that it is minimum phase.
The adaptive controllers of [58, 137, 147, 148] where the controller gains
are switched from one constant value to another over intervals of times based
on some cost criteria and search methods are referred to in [163] as non-
identifier-based adaptive controllers to distinguish them from the class of
identifier-based ones that are studied in this book.
6.7.3
Assumption P3: Known Relative Degree n∗
The knowledge of the relative degree n∗ of the plant is used in the MRAC
schemes of the previous sections in order to develop control laws that are free
6.7. RELAXATION OF ASSUMPTIONS IN MRAC
415
of differentiators. This assumption may be relaxed at the expense of addi-
tional complexity in the control and adaptive laws. For the identifier-based
schemes, several approaches have been proposed that require the knowledge
of an upper bound n∗u for n∗ [157, 163, 217]. In the approach of [163], n∗u
parameterized controllers Ci, i = 1 , 2 , . . . , n∗u are constructed in a way that
Ci can meet the MRC objective for a reference model Mi of relative degree i
when the unknown plant has a relative degree i. A switching logic with hys-
teresis is then designed that switches from one controller to another based
on some error criteria. It is established in [165] that switching stops in finite
time and the MRC objective is met exactly.
In another approach given in [217], the knowledge of an upper bound n∗u
and lower bound n∗l of n∗ are used to construct a feedforward dynamic term
that replaces cor in the standard MRC law, i.e., up is chosen as
α( s)
α( s)
up = θ 1
+ θ
+ θ
Λ( s)
2 Λ( s)
3 yp + θ 4 b 1( s) n 1( s) Wm( s) r
where b 1( s) = 1 , s, . . . , s¯ n∗
, ¯
n∗ = n∗u − n∗l , θ∗ 4 ∈ R¯ n∗+1 and n 1( s) is an
arbitrary monic Hurwitz polynomial of degree n∗l. The relative degree of
the transfer function Wm( s) of the reference model is chosen to be equal to
n∗u. It can be shown that for some constant vectors θ∗ 1 , θ∗ 2 , θ∗ 3 , θ∗ 4 the MRC
objective is achieved exactly, provided θ∗ 4 is chosen so that kpθ∗ 4 b 1( s) is a
monic Hurwitz polynomial of degree n∗ − n∗l ≤ ¯ n∗, which implies that the
last ¯
n∗ − n∗ − n∗l elements of θ∗ are equal to zero.
The on-line estimate θi of θ∗i, i = 1 , . . . , 4 is generated by an adaptive law
designed by following the procedure of the previous sections. The adaptive
law for θ 4, however, is modified using projection so that θ 4 is constrained
to be inside a convex set C which guarantees that kpθ 4 b 1( s) is a monic
Hurwitz polynomial at each time t. The development of such set is trivial
when the uncertainty in the relative degree, i.e., ¯
n∗ is less or equal to 2. For
uncertainties greater than 2, the calculation of C, however, is quite involved.
6.7.4
Tunability
The concept of tunability introduced by Morse in [161] is a convenient tool for
analyzing both identifier and nonidentifier-based adaptive controllers and for
discussing the various questions that arise in conjunction with assumptions
416
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
P1 to P4 in MRAC. Most of the adaptive control schemes may be represented
by the equations
˙ x = A( θ) x + B ( θ) r
1
= C( θ) x
(6.7.1)
where θ : R+ → Rn is the estimated parameter vector and 1 is the estima-
tion or tuning error.
Definition 6.7.1 [161, 163] The system (6.7.1) is said to be tunable on a
subset S ⊂ Rn if for each θ ∈ S and each bounded input r, every possible
trajectory of the system for which 1( t) = 0 , t ∈ [0 , ∞) is bounded.
Lemma 6.7.1 The system (6.7.1) is tunable on S if and only if {C( θ) ,A( θ) }
is detectable for each θ ∈ S.
If {C( θ) , A( θ) } is not detectable, then it follows that the state x may grow unbounded even when adaptation is successful in driving 1 to zero by adjusting
θ. One scheme that may exhibit such a behavior is a MRAC of the type
considered in previous sections that is designed for a nonminimum-phase
plant. In this case, it can be established that the corresponding system of
equations is not detectable and therefore not tunable.
The concept of tunability may be used to analyze the stability properties
of MRAC schemes by following a different approach than those we discussed
in the previous sections. The details of this approach are given in [161, 163]
where it is used to analyze a wide class of adaptive control algorithms. The
analysis is based on deriving (6.7.1) and establishing that {C( θ) , A( θ) } is
detectable, which implies tunability. Detectability guarantees the existence
of a matrix H( θ) such that for each fixed θ ∈ S the matrix Ac( θ) = A( θ) −
H( θ) C( θ) is stable. Therefore, (6.7.1) may be written as
˙ x = [ A ( θ) − H( θ) C( θ)] x + B( θ) r + H( θ) 1
(6.7.2)
by using the so called output injection. Now from the properties of the adap-
tive law that guarantees ˙ θ, 1 ∈ L
m
2 where m = 1 + ( C 0 x)2 for some vector
C 0, we can establish that the homogeneous part of (6.7.2) is u.a.s., which,
together with the B-G Lemma, guarantees that x ∈ L∞. The boundedness
of x can then be used in a similar manner as in the previous sections to es-
tablish the boundedness of all signals in the closed loop