IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000
883
A New Linear Adaptive Controller: Design, Analysis
and Performance
Youping Zhang and Petros A. Ioannou
Abstract—The certainty equivalence and polynomial approach,
widely used for designing adaptive controllers, leads to “simple”
adaptive control designs that guarantee stability, asymptotic error
convergence, and robustness, but not necessarily good transient
performance. Backstepping and tuning functions techniques,
on the other hand, are used to design adaptive controllers that
guarantee stability and good transient performance at the expense
of a highly nonlinear controller. In this paper, we use elements
from both design approaches to develop a new certainty equivalence based adaptive controller by combining backstepping
based control law with a normalized adaptive law. The new
adaptive controller guarantees stability and performance, as well
as parametric robustness for the nonadaptive controller, that are
comparable with the tuning functions scheme, without the use of
higher order nonlinearities.
I. INTRODUCTION
T
HE “certainty equivalence” adaptive linear controllers
with normalized adaptive laws have received considerable
attention during the last two decades [5], [8], [15]. These
controllers are obtained by independently designing a control
law that meets the control objective assuming knowledge of all
parameters, along with an adaptive law that generates on-line
parameter estimates that are used to replace the unknowns
in the control law. The normalized adaptive law could be a
gradient or least squares algorithm [5], [8], [15]. The control
law is usually based on polynomial equalities resulting from a
model reference or pole assignment objective based on linear
systems theory. An important feature of this class of adaptive
controllers is the use of error normalization, which allows the
complete separation of the adaptive and control laws design.
Since the “normalizing signal” depends on closed-loop signals,
the performance bounds are hard to quantify a priori, and there
is no systematic way of improving them. In fact, it was shown
in [19] that the performance of these controllers can be quite
poor. With asymptotic performance characterized in the form
or mean-squared-error (MSE) bound, the undesirable
of an
phenomena such as bursting [6] cannot be prevented. Efforts
to improve performance include the use of fixed compensators
[18], [1] to depress the effect of estimation error, and the
introduction of “dynamic certainty equivalence” which avoids
Manuscript received April 7, 1998; revised March 24, 1999 and June 16,
1999. Recommended by Associate Editor, M. Krstic. This work was supported
in part by NASA under Grant NAGW-4103 and in part by NSF under Grant
ECS-9877193.
Y. Zhang is with Numerical Technologies, Inc., San Jose, CA 95134-2134
USA (e-mail: yzhang@numeritech.com).
P. A. Ioannou is with the Department of Electrical Engineering-Systems, University of Southern California, Los Angeles, CA 90089-2562 USA (e-mail:
ioannou@rcf.usc.edu).
Publisher Item Identifier S 0018-9286(00)04158-1.
direct error normalization by imposing a passive identifier
with “higher order tuners” [13], [16], [2]. All these controllers,
however, involve an increase of controller order, especially
the “dynamic certainty equivalence” controller. Moreover,
all these controllers are “direct adaptive controllers” which
directly estimate the controller parameters. This may increase
the number of parameters to be estimated relative to the number
of actual unknowns as the controller usually contains more
parameters than the plant for higher relative degree plants, and
a priori knowledge about the plant parameters is generally not
readily usable.
In [10] and [9], an adaptive control approach that deviates
from the traditional certainty equivalence ones was introduced
by employing integrator backstepping, nonlinear damping, and
tuning functions, which results in a relatively complex controller
structure. The design procedure is performed recursively, and
the adaptive law is determined without error normalization. The
resulting controller guarantees good transient performance in
the ideal case, and the performance bounds can be computed
and reduced systematically [12]. The adaptive law estimates the
unknown plant parameters directly, allowing full utilization of a
priori knowledge and eliminating the possible overparameterization introduced by traditional “direct” approach [8], [9]. Another remarkable feature of the new controller is the parametric
robustness when adaptation is switched off [10]. These properties are achieved, however, at the expense of a highly nonlinear
controller, even though the plant is linear. Higher order nonlinear terms can potentially lead to difficulty in proving robustness at the presence of high-frequency unmodeled dynamics,
especially those that lead to fast nonminimum phase zeros and
change in high-frequency gain sign [22]. In such a case, the
tuning function’s design can only guarantee local stability when
these “undesirable” unmodeled dynamics are present [7]. Moreover, since the tuning functions design is Lyapunov-based, it excludes the flexibility of choosing among different types of adaptive law.
In this paper, we use elements from both design approaches
to develop a new certainty equivalence adaptive controller by
combining a backstepping control law with a normalized adaptive law. All conditions equal, the new controller guarantees performance that is comparable with the tuning functions adaptive
backstepping controllers of [10] and [9], without introducing
higher order nonlinear terms, extra parameter estimates, or increasing controller order. Despite the presence of a normalizing
and
performance bounds are computable and
signal, the
can be reduced systematically via appropriate choice of design
parameters. Similar parametric robustness and performance of
the nonadaptive controller as shown in [12] are also achieved.
0018–9286/00$10.00 © 2000 IEEE
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000
The paper is organized as follows: In Section II, we introduce the new design procedure that gives a class of adaptive
controllers with various control and adaptive laws and prove
stability of the resulting adaptive control system. In Section III,
we establish performance bounds and parameter convergence
properties and explain how performance can be systematically
improved. Section IV shows the parametric robustness and performance when adaptation is switched off. Before concluding
the paper in Section VI, we present some simulation results to
demonstrate the properties of the new controller in Section V.
, respectively, as the
Notations: We denote
vector Euclidean norm and matrix Frobenius norm, i.e.,
For a time funcdenotes the
norm;
tion
denote the
norms, respectively;
and
denote
and
induced norm, respectively, of a stable proper
the
Unless specified otherwise, denotes
transfer function
denotes the estimation error.
the estimate of and
as an
identity matrix,
In addition, we denote
as an
zero matrix,
as the
zero vector. Finally,
we will use the symbol to denote interchangeably either the
operator
Laplace variable in the frequency domain, or the
in the time domain.
II. ADAPTIVE CONTROL DESIGN
The plant under consideration is the following singleinput/single-output (SISO) linear time invariant (LTI) system:
(1)
with the following standard assumption.
Assumption: The plant parameters
are unknown constants; an upper bound for the nominal plant
order the plant relative degree
and the sign of the
are known; the plant is minimum
high frequency gain
is Hurwitz.
phase, i.e.,
The control objective is to design an output feedback control
law, such that all the closed-loop signals are uniformly bounded,
asymptotically tracks a given reference
and the plant output
with known, bounded derivatives.
signal
The adaptive controller is designed based on the certainty
equivalence principle. The control law is designed by employing
backstepping, while the adaptive law is chosen by using a certain plant parameterization independent of the controller parameters.
A. Filter Design and Plant Parameterization
We introduce the standard filters as used in traditional adaptive controllers [8], [15]:
(2)
where
is a Hurwitz matrix defined by
(3)
Based on the plant input–output relation (1), we can get a
representation of the tracking error in terms of the filter signals
as follows:
(4)
where
(5)
(6)
$
(7)
is the initial condition of the state of the plant (1) in its observer canonical form, and $ represents the inverse Laplace
transform.
Equation (4) gives a standard bilinear parameterization in
terms of the unknown parameters for the tracking error The
signals, compared with
filter structure (2) involves a total of
signals in the Kresselmeier observer filters
the
-filters and
used in [10] and [9], which involves
-filters. Though the overall orders are the same for both filter
structures
our filter design greatly reduces the complexity
of the filter structure and subsequently the controller structure
by reducing the number of signals involved considerably.
B. Backstepping Control Law Design
To demonstrate the use of backstepping, we ignore the trivial
case and assume that
relative degree
1) Design Procedure—Initialization: A unique feature of
our backstepping design relative to that of [9] is that instead
of targeting at the tracking error directly from the first step,
as is mostly done for backstepping design [11], we change
our control objective to achieve convergence to zero of the
estimated tracking error based on (4)
(8)
are the estimates of
In subsequent design
where
steps, we continue to replace wherever it appears with its estimate
Since this is a certainty equivalence design, we treat
as
if they were the true parameters and consequently do not attempt
through “tuning
to eliminate the terms containing
functions.” Nor do we attempt to apply “nonlinear damping” to
achieve independent input-to-state-stability which may lead to
higher order nonlinear terms.
were the control input, we would choose
In view of (8), if
so that
which achieves the tracking obor
Since
is not the control input,
jective
ZHANG AND IOANNOU: A NEW LINEAR ADAPTIVE CONTROLLER: DESIGN, ANALYSIS AND PERFORMANCE
becomes the first error signal. The output mismatch
and the tracking error are represented in terms of the parameter error as
(9)
(10)
defined, we can proceed
With the first error signal
to Step 1. For convenience, we introduce the variable
Step 1: Using (6), (8), (2), and
compute as follows:
we
(11)
where
885
to the term as the “stabilizing damping” which places the
dynamics to a desired one. On the other hand, the parameter
terms, which we refer to as the “normalizing
dependent
damping,” is used to normalize the effect of the uncanceled
and
terms. We will explain what this normalization
means later in the section. The two types of damping carry different missions and will be used over and over again in the following steps.
is not the control input, the design does not stop
As
becomes the second error
here. The difference
signal, and the error dynamic equation for is
(20)
follow a unified proceThe subsequent steps 2 through
dure which can be performed recursively as follows.
: The following design steps will be
Step 2:
we
described in a recursive manner. After finishing step
stabilizing functions
will have obtained
(21)
(12)
(13)
Differentiating
and substituting in (11), we get
(14)
where
is an
vector of continuously
where
The stabilizing functions are so
differentiable functions of
chosen that the dynamics of the th error system is
(22)
are continuously
where
differentiable with respect to
To proceed to step we differentiate
and substitute in (21), (22), and (11). Following some straightforward arrangements, we arrive at
(23)
where
(15)
In the known parameter and nonadaptive case, would be
zero and would be exponentially decaying. So we would treat
as the “virtual” control and design the first stabilizing
function to be
(16)
(24)
(25)
where
(17)
(26)
(18)
(19)
bitrary
are design constants, and
is an arconstant matrix. We refer
(27)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000
and
are again design constants. From (23), we see
that to stabilize the sub-system, we can select the th stabiwhich results in
lizing function in the form of (21) with
the error dynamic equation (22) for
Now we have finished step and have inductively proven (21),
We can proceed to the next step.
(22) for
Step : In step , the control appears after differentiating
By following the derivation of step we can easily reach
where
is a design constant, which we refer to as the normalizing coefficient, that determines the strength of normalizais described by
tion. The normalizing property of
(28)
As the value of approaches zero,
where
diminishes.
the normalizing property of
The advantage of the adaptive law (32) based on bilinear
parameterization is that we circumvent the requirement of inverting an estimated parameter by estimating the parameter and
its inverse simultaneously. In return, it has the drawback of estimating an extra parameter, and it excludes a least square (LS)
based adaptive law.
2) Adaptive Law with Linear Parameterization: By way of
and at
a linear parameterization, we can avoid estimating
the same time, which is typically done in most standard “indirect” adaptive control schemes. Specifically, we estimate only
one of the two unknowns and obtain the other one by inversion.
However, it requires such a parameter estimate to be bounded
away from zero, which is achieved with parameter projection if
we have additional knowledge on the bound of the true parameter. In particular, we assume the following.
Assumption: An upper bound of the value of the high-freis known.
quency gain,
Alternatively, we can also assume knowledge of a lower
which was considered in [20]. The knowledge of
bound of
means that
Hence we can use parameter projection
to ensure the estimate of is bounded from below by
Then we can invert to obtain
The resulting paramis
eter error for
This way we are able to obtain a linear parameterization from
(9) as follows:
are functions of defined the same way
where
Since the control input is now availas (24)–(27) for
able for design, we can choose the same way we chose the
stabilizing functions to cancel the undesirable terms, i.e.,
(29)
that leads to
(30)
An alternative control law is to consider the fact that this is
the last design step, the control law will be fixed herein and
no further differentiation will occur. Hence we are allowed to
and
as
choose the control to cancel
well, both of which are available. Specifically, we choose the
following control law:
(31)
where the “normalizing damping” term, due to the elimination
and
becomes unnecessary, so that
of
This finishes the control law design.
we can set
We observe that during the intermediate design steps, the sigare absent from the stabilizing functions and
nals
their positions have been completely replaced by the new variables which is different from the tuning functions design [9].
This is important for the performance results of Section III to be
valid.
(34)
(35)
where
C. Normalized Adaptive Law
1) Adaptive Law with Bilinear Parameterization: Equation
(4) is the standard bilinear equation error model that appears
in many adaptive control systems [8], [15], based on which a
gradient based adaptive scheme is available that takes the form
of [8], [15]
(36)
we can choose the following update
Considering that
law for based on the linear parameterization (35):
(32)
(37)
are constant adaptive gains, and
is the normalizing signal, which is used to guarantee
A typical choice is
which has been considered in [20]. Here we employ a slightly
different version of the normalizing signal
where
(33)
is the
adapwhere
tation gain matrix. In the case of a gradient scheme,
is constant. In the case of a LS scheme, is determined by the
following dynamical equation:
(38)
ZHANG AND IOANNOU: A NEW LINEAR ADAPTIVE CONTROLLER: DESIGN, ANALYSIS AND PERFORMANCE
887
TABLE I
where
is a large constant. The projection operator
projects the estimate of from the boundary of
and
is to set
when projection of is in effect
the operator
in the LS scheme. Based on certainty equivalence, our controller
design does not involve implementation of in the stabilizing
functions. Therefore is not further differentiated, and unlike
the tuning functions scheme, differentiability of the projection
operator is not an issue.
is chosen to be the same as in the
The normalizing signal
is ensured by parameter projection,
bilinear case. Since
is described by
the normalizing property of
(39)
3) The Known High Frequency Gain Case: Going one step
further, if we have complete knowledge of the high-frequency
then the adaptive law is further simplified. That is, we
gain
and from the list of unknown parameters. The
drop both
control law is again totally unchanged, except we set
whereas the adaptive law is simplified to
(40)
Proof: We first consider the adaptive law with bilinear parameterization. Consider the Lyapunov function
(44)
and its derivative along the solution of (37)
(45)
where the last inequality is a result of completion of squares.
is nonincreasing, i.e.,
Therefore
which implies
from which (41) follows. Integrating (45) leads to
as a direct consequence of error
(42). Moreover,
together with and
normalization. Finally,
imply
The proof for the adaptive law with linear parameterization is the same by adopting the Lyapunov function
for the unknown high-frein
quency gain case and
the known high-frequency gain case.
D. Error System Property
The error system equation resulted from the control law (31)
can be written in the following compact form:
(46)
is the
adaptation gain
where
matrix similar to that of the linear parameterization case.
The known high-frequency gain case is of particular interest
as many performance, passivity, or parametric robustness results
are obtained under this assumption [1], [12], [9], [11].
4) Stability and Performance of Adaptation System: The
above adaptive laws (32), (37), (40) guarantee the following
stability and performance properties regardless of the stability
of the closed-loop control system.
Lemma 1 (Performance of Adaptive Law): The adaptive
law (32), (37), or (40) for the equation error model (4) guaranSpecifically, the
tees that
following performance bounds hold:
where
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
(47)
and
(bilinear only)
(41)
(48)
(42)
in the case of control law (29), or
where
(43)
and
are defined in Table I.
(49)
in the case of control law (31). The homogenous part of the error
state equation (46) is exponentially stable due to the uniformly
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000
negative definite (or screw symmetric)
The terms
and
due to the output mismatch and the adaptation, now act
as perturbations to the -system.
with completion of
Using (46), and the definition of
squares, we evaluate the derivative of
Essentially, the terms perform similar tasks as the terms.
we are able to eliminate the deBy appropriately choosing
with respect to the design parameters and the
pendency of
parameter estimates. We defer the discussion on this to the performance analysis section.
Proceeding from (50), we have
(56)
where
(50)
(57)
in the case of control law (31) and
in
where
the case of control law (29).
We now consider the last term of (50) for different adaptive
laws.
The Bilinear Parameterization Case: The adaptive law (32)
can be expressed in the form of
Integrating (56) gives
(51)
from which we get
(58)
Further integrating the first inequality of (58) gives
(52)
where in obtaining the constant
we have used the inequalities
(53)
(59)
and
and
Inequalities (58) and (59) directly relate the
norms of the backstepping error with the
norm of the
through a simple relation
parameter estimation error
and
where the effect of design parameters is clear: if
is independent of the design parameters and the parameter
estimates, then there exist upper bounds of the quantities
that are computable a priori and
only decrease with increasing
Linear Parameterization Case: Notice that the equivalent
can be derived as
adaptive law for
The adaptive law (37) can be rewritten as
E. Stability Analysis
(54)
Therefore (52) holds true for the linear parameterization case as
well.
Known High frequency Gain Case: This is only a special
in
case of the above two cases. Here we simply set
So we have (52) with
(55)
is independent of the filter paIt is interesting to notice that
rameter vector as compared to that defined in (53).
We are now in the position to state and prove the stability
properties of the proposed adaptive controller.
Theorem 2 (Stability and Asymptotic Tracking): Consider
the LTI plant (1). The adaptive controller with either control
law (31) or (29), and any of the adaptive law (32), (37), or (40)
guarantees uniform boundedness of all closed-loop signals and
asymptotic convergence of the tracking error to zero.
Proof: We prove this theorem by first showing uniform
The idea is to conboundedness of the normalizing signal
namely and and establish
sider the two components of
involving the normalized estimation error
an inequality for
that is suitable for applying the Bellman–Gronwall lemma.
ZHANG AND IOANNOU: A NEW LINEAR ADAPTIVE CONTROLLER: DESIGN, ANALYSIS AND PERFORMANCE
The partial regressor can be written in the following transfer
function form:
where
889
are defined by
(68)
(60)
is the strictly proper transfer function matrix from
where
to Substituting in (10) and applying Lemma 6
(61)
In view of (61), (59), and (58), we have
(62)
where
(63)
(64)
Therefore the normalizing signal
equality
satisfies the integral in-
Therefore the inequalities similar to (58) and (59) still hold.
do not provide much insight, beHowever, the quantities
cause they now depend on the closed-loop behavior, including
the parameter estimates as well as all the design parameters, in a
complex manner, and we are no longer able to predict what will
as the parameter estimates
happen to
are updated or as the design parameters are changed.
Remark 2: Using the “normalizing damping” instead of
“nonlinear damping,” we manage to preserve the linearity of
control law (31) or (29). Our adaptive controller design guaranregardless of the boundedness of
tees
the closed-loop signals. Hence all the terms in (31) or (29) are
[in the sense that
], which con“linear” in
tains all the closed-loop states, including and This assures
that the control signal can be bounded by the normalizing
signal
i.e.,
Such preservation of linearity is
crucial for proving stability when error normalization is used
as it allows us to arrive at an integral inequality in terms of
that is suitable for applying the Bellman–Gronwall lemma.
Such “linearity” of control law is also crucial for robustness
with respect to high-frequency unmodeled dynamics [8], [22].
III. PERFORMANCE AND PARAMETER CONVERGENCE
(65)
Applying the Bellman–Gronwall lemma, we obtain
In this section, we study the performance of the new adaptive
control system with or without persistent excitation (PE).
A. Transient Performance
(66)
so
Since all adaptive laws guarantee
Boundedness of
implies by (66)
Bounded
together with bounded and its derivatives imply bounded
and therefore bounded
which together with
Finally, inspecting (10), (31), or
bounded implies
(29), we easily see that all components are uniformly bounded,
and
Therefore all the
so
also
closed-loop signals are uniformly bounded. Bounded
In view of (59), we immediately
implies that
which together with
imply
Since
have
by Barbalat’s lemma, we
or
as
which completes
conclude that
the proof.
Remark 1: The stability analysis depends heavily on (56) in
It is, however, not necessary for
which we assume
to be nonzero to guarantee stability. Suppose we set
then instead of (56), we have an equivalent inequality
For traditional adaptive linear controllers with normalized
adaptive law, the unnormalized performance bounds are usually
proportional to the size of the normalizing signal which may
grow as design parameters are tuned toward improving performance of the normalized performance bounds. Consequently,
these adaptive controllers lack computable performance that
can be systematically improved. For the proposed adaptive
controller, however, an upper bound of the normalizing signal
can be determined that does not grow as the design parameters
tune to reduce the normalized performance bounds.
1) Filter Polynomial Design: Let
where
is a free constant that determine the role of
in
is defined
the performance, and the Hurwitz polynomial
by
(69)
(70)
Hence
(67)
(71)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000
The relationship between
and
applying the Parseval’s theorem
can be determined by
Bilinear Parameterization Case: Consider
defined in
(53). Notice that is a constant matrix, so if we choose
then
(77)
(72)
which
Using (72) in (43), we obtain
depends only on the plant and the initial conditions, provided
to be a constant. Moreover,
is
we maintain
That is, inmonotonically decreasing with respect to
leads to decrease in
creasing without changing
On the other hand, assume we have the decomposition of the biproper transfer function
where is a constant,
is a polynomial
both independent of Then
of degree
$
$
$
$
which is independent of and only decreases as increases or
decreases.
Linear Parameterization Case: In the linear parameterization case, the matrix defined by (54) is not constant, therefore
We note that
we cannot directly choose
(78)
Let
matrix
and consider the
(79)
(73)
represents time domain convolution. As
is
where
independent of
Hurwitz, there exists constant
such that $
Let
then
Since
and the determinant of
is
(80)
the Hermitian matrix
is nonnegative definite. Therefore
(81)
(74)
For the gradient adaptive law,
so that
choose
are constant, so we can
are independent of (72), (74) imply
Since the constants
that the effect of nonzero initial conditions on the plant in the
and
norms can be arbitrarily reduced
sense of both the
by increasing
Next we decomposed
to
(82)
(75)
because the coFor the LS scheme,
variance matrix does not increase. Hence,
which suggests the choice of
In either case, we have
where
is now biproper. The
are bounded by
induced norms of
(83)
(76)
which monotonically decrease to zero as goes to since is
independent of
2) Normalizing Damping: In the controller design, we state
that the matrix can be arbitrary. While we may arbitrarily
choose without losing stability, appropriate choice of bears
critical importance for performance. In particular, we need to
does not increase as or
choose such that the size of
increases or decreases.
which is again independent of and strictly decreasing with
Practically, since the adaptation gain can be
respect to or
very large, the resulting can also be very large. Therefore
should be chosen small to avoid extremely high gain.
Known High-Frequency Gain Case: In this special case,
defined in (55) does not involve the filter coefficients
the
hence we can choose
where the last two rows and
and are
columns of associated with the adaptation of
zeros, and
From the above analysis, the choice of the damping coefficients, namely
do not depend explicitly on any state
variables. Therefore these “normalizing damping” terms are not
ZHANG AND IOANNOU: A NEW LINEAR ADAPTIVE CONTROLLER: DESIGN, ANALYSIS AND PERFORMANCE
aimed at, as the “nonlinear damping”s of [11] are, damping out
the effect of these perturbation terms by way of nonlinear gains.
In contrast, the purpose of these coefficients is to normalize the
terms to eliminate the parameter depencoefficients of the
normalizes the
dency of these coefficients. The coefficient
term, and
normalizes the parameter
coefficients of the
term.
dependencies of the
3) Performance: Lemma 1 gives the performance bounds
on the normalized estimation error These normalized performance bounds are translated to their unnormalized versions via
From (66), we obtain
the bound of the normalizing signal
(84)
then from (63), (64), we can see that
is a decreasing function of
is a nonincreasing function of
Therefore
is noninand independent of
creasing with respect to
Using
we have the following performance properties.
Theorem 3 (Performance): Consider the LTI plant (1). The
adaptive controller with control law (31) or (29), and adapnorm of the
tive law (32), (37), or (40), guarantees that the
tracking error satisfies the following inequality:
If
(85)
in the case of adaptive law (32),
in the case of adaptive law (37), and
in the case of adaptive law (40). If in addition the plant highis known, then the adaptive controller with
frequency gain
control law (31) or (29), and adaptive law (40) guarantees the
bound for the tracking error
following
where
(86)
Proof: The proof of (85) follows by substituting (42), (59),
(58) into the following inequality:
(87)
performance (86) is
and plugging in the bound of (42). The
shown by applying (74) and (76) to the following inequalities:
(88)
and plugging in the bounds (41) and (42) developed in Lemma 1,
and the inequality
which is a direct
.
consequence of the normalizing property of
891
The
performance is only available when the high-frequency gain is known. We recall that the tuning functions
scheme has the same limitation [12].
Let us now discuss the roles of various design parameters in
performance improvement.
1) Reference signal and filter initialization: As we see from
the definitions of the constants involved in the performay
mance bounds, the nonzero initial condition of
However, by
be an increasing function of
simply zeroing the initial condition of the reference model
then
regardless of the initial
so that
condition of the plant. This greatly simplifies the complex filter initialization process that is required for the
tuning functions design [9]. Alternatively, for nonzero
we can choose
which also leads
a similar process as the tuning functions deto
sign. The effect of however, cannot be eliminated due
to the unknown plant parameters. This is one drawback
compared with the tuning functions design.
2) Adaptation gain : Large adaptive gain which attenuates
the effect of initial parameter error is vital for good
performance. This means that fast adaptation is needed to
minimize the energy of the tracking error caused by the
initial parameter error. Meanwhile, the condition number
needs to be kept small
of the adaptation gain matrix
(e.g., a constant adaptation gain,
3) Filter parameter : A large is critical for both the
and
performance in two senses. First, for nonzero
perforplant initial condition, in order to guarantee
so that when we
mance, we must fix the ratio of
we proportionally increase Otherwise,
increase
due
increasing with fixed could lead to increasing
to nonzero initial condition, which in turn results in inThis is, however, not necessary when the
creasing
Second,
plant initial condition is zero such that
but increase alone. Though the
suppose we fix
is limited, it has a deterministic effect on
effect on
performance shown in (86).
the improvement of the
correParticularly when we increase the value of
can be arbitrarily reduced.
spondingly, the size of
: The role of is to atten4) Damping coefficients
or
tracking error due to the backstepping
uate the
are not necessary for
error We have shown that
appear in the denominator in
stability. However, as
are
all the performance bounds, it is essential that
chosen nonzero to achieve the performance bounds (85),
can lead to improve(86). Moreover, increasing
performance, as
ment in performance, especially the
evident from (86).
5) Normalizing coefficient : Decreasing leads to weaker
normalization, which renders smaller normalizing signals
and hence improved performance. As
which is the unnormalized case.
We point out that if we had not used the normalizing damping
then we would not have been able to
terms
that is as
find an upper bound for the normalizing signal
straightforward as the one shown in Lemma 2. Not only are
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000
we unable to predict the possible effect on
as we tune the
design parameters, but we are also unable to establish perfordue
mance bounds even for the normalized tracking error
to the complex relation of the backstepping error and the pahence large
rameter estimation error We notice that
or small implies stronger “normalizing damping,” and consequently, large feedback gain.
It is interesting to see from the first term of (86) that though
may result in good
performance, it is not required.
good
In essence, large leads to a fast filter, which increases the controller bandwidth and feedback gain. A large will result in
a small estimation error for all time, therefore undermines the
adaptation process, leading to slower adaptation. So fast adaptation can be counteracted by high gain feedback. This point is
manifested in the next section where adaptation is completely
switched off.
We also observe that the tracking error differs from the error
signal used for backstepping by the estimation error This mismatch, however, does not prevent us from establishing performance for the tracking error as the effect of can be reduced by
employing fast adaptation and fast filter.
4) Performance Comparison with Tuning Functions Design: To compare the performance of the new controller
with the tuning functions controller, we consider the performance bounds derived in [12] with the conditions
shown as follows:
(89)
(90)
are constants determined by the chosen
where
and the initial conditions, which, however, cannot
filter
be systematically decreased with the filter design. We observe
and
that increasing can lead to improvement in both
performance, and increasing
can result in further improveperformance only. Increasing the adaptation gain
ment in
may help to further improve performance through, for example,
Detailed discussion on the role of adaplimited reduction in
tation can be found in [12]. However, it is unclear how the filter
design can play a role in further performance improvement.
We point out that the new controller can achieve the same
level of performance as promised by the tuning functions design, though the way they achieve such performance is different,
i.e., the parameters used to shape the performance are different.
Table II compares the two approaches. In the table, the word
“primary” and “secondary” refers to irreplaceable and replaceable role, respectively, in achieving arbitrary reduction in performance bounds, whereas “limited” refers to limited reduction
in performance bounds.
Another difference is the process of filter initialization. The
tuning function’s scheme employs a complex signal tracking
process for filter initialization to achieve complete elimination
of the effect of nonzero initial conditions. With our controller,
is automatically set to zero if we
the initial condition of
use zero initial condition for all filters and the reference model.
However, we are unable to initialize the filters so that the effect
COMPARISON
OF
TABLE II
NEW ADAPTIVE CONTROL DESIGN AND
FUNCTIONS DESIGN
THE
TUNING
of nonzero initial conditions of the plant can be completely eliminated from the tracking error. This is caused by the fact that the
tuning functions design starts with the tracking error, while the
new controller design starts with the “estimated” tracking error.
This effect, though, is exponentially decaying to zero which can
be made to die arbitrarily fast with large initial control effort.
When plant has nonzero initial condition, our controller may
experience high initial control effort due to high gain, whereas
the tuning functions design may not require large control effort
due to “nonlinear damping.”
B. Parameter Convergence
Finally, we establish the parameter convergence property of
the new adaptive controller.
Theorem 4 (Parameter Convergence): Assume that the two
and
in (1) are coprime. Consider
plant polynomials
the adaptive controller with the control law (31) or (29), we have
the following parameter convergence properties.
1) Known high-frequency gain with adaptive law (40). If the
reference signal is sufficiently rich (SR) of order
then the parameter error vector
as well as the error
converge to zero exponentially.
signals
2) Known upper bound of high frequency gain with adaptive
law (37). If the reference signal is SR of order
then the parameter error vector as well as the error
converge to zero exponentially.
signals
3) Unknown high-frequency gain with adaptive law (32).
converges asymptotically to a
The parameter estimate
(not necessarily equal to the true value
fixed constant
If in addition the reference signal is SR of order
then the parameter error converges asymptotthen for sufficiently
ically to Furthermore, if
the convergence rate for
is exponential.
large
Proof: We prove only the last case. The first two cases
follow closely.
hence
We have shown that
which implies that
converges to a fixed constant
not
necessarily equal to
Next we consider the regressor vector which we express as
(91)
transfer functions
It is straightforward to verify that the
are linearly independent. Hence when
in
the first term
the reference signal is SR of order
of in (91) is PE. The second term is an LTI operator applied
signal, which is again
So is PE [8]. With this PE
to an
ZHANG AND IOANNOU: A NEW LINEAR ADAPTIVE CONTROLLER: DESIGN, ANALYSIS AND PERFORMANCE
condition, we first show asymptotic convergence of the parameter estimates.
From the adaptive law (32), we have
(92)
then
Hence when
Let
is PE, so is
So the homogeneous system of (92) is
exponentially stable [15], [8], i.e.,
is an exponentially stable time varying matrix. Since the last
therefore
term in (92) converges to zero due to
as
Now let us first consider the equation. We redefine the conwhere
stant to be
is a constant to be determined. Also for convenience of notaThen instead
tion, we introduce
of (56), we have
893
Since neither
nor
nor
is a growing function of
therefore (97) can always be satisfied by design. This means
that the closed loop is exponentially stable, which is stronger
than the asymptotic stability that is obtained in traditional overparameterized adaptive system.
As a consequence of the backstepping design, the new adaptive controller also possesses stronger parameter convergence
property as it can guarantee exponential convergence, as compared with traditional linear adaptive schemes with bilinear parameterization, which has at most asymptotic convergence.
IV. PARAMETRIC ROBUSTNESS AND NONADAPTIVE
PERFORMANCE
For the nonadaptive controller,
Hence (46) becomes
(98)
which defines an LTI relationship between
and
(99)
(93)
where we have used the fact that
From (92), we get
is an
vector,
$
is the exponentially decaying to zero term due to
According to (58),
possible nonzero initial condition of
and satisfy
(59),
where
(100)
(94)
The quantity (99) can be rewritten as
(101)
Let be a positive constant such that
where
is an arbitrary constant, then
Next let us consider the quantity
Define
(102)
(95)
Since
there exists
such that
If satisfies
then
is a stable proper LTI transfer function which
Obviously
Noticing that
is independent of the design parameters
we have
and
(103)
where we have used the fact that
tuting (101) and (103) into (10) results in
Substi-
(96)
exponentially, (96) implies that
exponenSince
tially, which in turn implies that
exponentially fast. Since is arbitrary, the condition on is satisfied if we choose to satisfy
(104)
Solving (104) for
(97)
we get
(105)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000
the high frequency gain itself, then by choosing
that
where
and such
(111)
(106)
(112)
(107)
(109) will be satisfied, and closed-loop stability will be guaranteed. As a special case, when the high-frequency gain is
then (109) degenerates to
known precisely so that
which can be satisfied simply by choosing
the filter to be sufficiently fast.
then
we are only able to
Note that when
obtain the MSE performance defined as
(113)
(108)
are the closed-loop sensitivity functions with respect to
respectively. Since
are stable,
according to the Small Gain theorem [3], the closed-loop system
described by (105) is stable if
(109)
This means that the tracking error will not converge to zero
then
and
asymptotically. However, if
asymptotic tracking can be achieved. These nonadaptive properties are summarized by the following theorem.
Theorem 5 (Parametric Robustness): The nonadaptive controller (31) or (29) applied to the LTI plant (1) guarantees that
for any given parameter error except for the high-frequency gain
if
satisfy the inequalities
which satisfies
(111) and (112), then all the closed-loop signals are uniformly
bounded, and the tracking error satisfies the following performance bounds, as shown in (114)–(116) at the bottom of this
$
is an exponenpage, where
tially decaying to zero term.
Proof: The results can be shown by applying Lemma 6
and the following inequalities to (105):
where
(110)
and in reaching (109) we have used the fact that
(117)
It is now obvious that if
i.e., the uncertainty
on the high frequency gain is strictly smaller than the value of
(118)
(114)
(115)
(116)
ZHANG AND IOANNOU: A NEW LINEAR ADAPTIVE CONTROLLER: DESIGN, ANALYSIS AND PERFORMANCE
895
(119)
and using (100) and
As in the adaptive case, the performance bounds given in
(114) and (115) or (116) can be improved by increasing the
, or
But the achievable performance is limvalue of
is. The case where the
ited by how small the parameter error
is known exactly so that
is a
high-frequency gain
special case considered in [10], [12], in which parametric robustness and nonadaptive performance are established. Under
such assumption, and with zero initial condition, the performance bounds can be further simplified that allow systematic
improvement. This can be easily seen from (114) and (115) by
Specifically, both the MSE (or
and the
setting
performance can be arbitrarily improved by increasing the value
of i.e., using faster filters.
V. NUMERICAL EXAMPLE
In this section, we use a numerical example to illustrate the
new adaptive controller and the effect of different parameters
on the closed loop performance. Consider the following plant
transfer function:
(120)
is known but
are considwhere
ered unknown. The control objective is to make the closed loop
follow the reference model
(121)
All the initial conditions are set to zero, and we consider only
the control law (29) with the gradient adaptive law where
The filter polynomial is chosen to be
We note that since the reference signal
is SR of order and
we are estimating parameters. Therefore we expect parameter
estimates convergence to their true values.
The baseline set of parameters are chosen to be
The results are shown
in Fig. 1. A large transient is observed, and the tracking error
settles in about 20 s when the parameter estimates settle.
For comparison of performance, we consider different parameter settings. In the first case, reducing the strength of normalization ten times improves the transient overshoot by more than
10 times (Fig. 2). In the second case, increasing the adaptation
gain ten times results in even more dramatic reduction in transient overshoot (more than 25 times), and the speed of parameter
convergence is also faster. In the mean time, the tracking error
perforsettles in about 15 s, showing an improvement of
mance. In the third case, we retain the original adaptation gain
and normalization coefficient but increase the value of
to 10, 20, 1, respectively, then from Fig. 2 we can see that the
transient overshoot is restrained to be nearly 0.007 more than 50
times as good as the previous case. However, the convergence is
Fig. 1. Controller performance for baseline parameter settings ( = 1;
1; c
=
c
= 1; d
=
d
= 0:1; % = 1):
=
very slow—continued simulation shows that it takes over 3000 s
to achieve a ten times lower magnitude of the tracking error. This
verifies that faster filter and larger damping coefficients undermine the role of adaptation dramatically. If we switch off the
adaptation, then the first two cases result in instability, whereas
with the third case, stability is maintained, but performance is
not as good as with the adaptive case (Fig. 2). Moreover, the
tracking error does not tend to zero anymore. This demonstrates
the parametric robustness and nonadaptive performance of the
new controller.
Finally, we simulate a case where we estimate and
by applying a bilinear parameter estimator. For no particular
reason, the parameters are chosen to be
As Fig. 3 shows, both the tracking error
and the parameter errors converge quickly after a brief transient
performance is not guaranteed, we cannot
overshoot. Since
get rid of the spike. However, by increasing the adaptation gain
we can reduce the settling time which represents partially the
performance.
VI. CONCLUSION
In this paper, we propose a new linear adaptive design approach. The new controller combines a backstepping control
law with a traditional equation error model based adaptive law
896
Fig. 2. Systematic performance improvement via parameter settings.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000
Fig. 3.
(
via certainty equivalence. The resulting adaptive controller possesses nice properties of both the traditional certainty equivalence adaptive controllers and the tuning functions adaptive
backstepping controller: linear, separation of control and adaptive law design and freedom of choice of adaptive law, direct estimation of plant parameters, use of error normalization, computable performance bounds, parametric robustness, and least
possible controller order.
A significant contribution of this paper is that we can establish performance while preserving linearity of the control law
and error normalization in the adaptive law. The use of error
normalization has been responsible for the difficulty in quantifying performance for traditional certainty equivalence adaptive linear controllers. Consequently, performance improvement
are mostly obtained by way of eliminating normalization [16],
[2], [12]. For the new controller, with the use of “normalizing
damping” in the backstepping design, we eliminate the dependency of the size of the normalizing signal on the design constants or parameter estimates, leading to an upper bound of the
normalizing signal that can be computed a priori. This ensures
the performance of the unnormalized error signals once performance of the normalized error signals is established. This is
accomplished without introducing additional dynamic compensator like [18] and [1].
Performance of adaptive controller with unknown high frequency gain
= 10; = 2; c =;
d
= 0:01; % = 0:5):
Error normalization was initially introduced in [4] to obtain
a stable parameter estimator independent of the closed-loop
stability to achieve controller–estimator separation. Later it
was shown that such “static” error normalization contributes to
the robustness of the resulting adaptive control system [14]. A
stronger form of normalization is the “dynamic” normalization
introduced in [17] and later developed in [8], which further
strengthens the robustness of the adaptive controller. Since the
new controller preserves the two key properties—linearity and
error normalization—we expect the same level of robustness
for the new controller as the traditional certainty equivalence
adaptive controller as described by [8], the ability to guarantee
global signal boundedness in the presence of input unmodeled
dynamics that is only required to be small in the low frequency
range. In [21], such robustness property was established for a
similar adaptive controller using a simple modification to the
adaptive law with parameter projection and dynamic normalization. Following [14], it is also possible to show robustness
of the new controller with parameter projection only. With
the tuning functions design, however, due to nonlinearity, it
is unclear how to modify the controller design to achieve this
level of robustness, or even it is possible, as controllers with
higher order nonlinear terms are generally unable to guarantee
ZHANG AND IOANNOU: A NEW LINEAR ADAPTIVE CONTROLLER: DESIGN, ANALYSIS AND PERFORMANCE
robustness with high frequency unmodeled dynamics in the
global sense [22].
APPENDIX
INPUT–OUTPUT STABILITY
Lemma 6 [3], [8]: Let
where
is an th
order stable proper transfer function. Then the following inequalities hold:
(122)
If in addition,
is strictly proper, then
897
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Youping Zhang received the Bachelor’s degree in
1992 from the University of Science and Technology
of China, Hefei, Anhui, PRC, and the M.S. and Ph.D.
degrees in electrical engineering from the University
of Southern California, Los Angeles, CA, in 1994 and
1996, respectively. From 1996 to 1999, he was an Associate Research Engineer with the United Technologies Research Center in East Hartford, CT.
He joined Numerical Technologies, Inc. in San
Jose, CA, in July, 1999, where he is currently a
Senior Engineer. His research interests are in the
areas of intelligent control systems, optimizations, and computer aided designs.
Dr. Zhang is a member of IEEE, Phi Kappa Phi, and Eta Kappa Nu.
Petros A. Ioannou received the B.Sc. degree with
First Class Honors from University College, London,
England, in 1978 and the M.S. and Ph.D. degrees
from the University of Illinois, Urbana, IL, in 1980
and 1982, respectively.
Dr. Ioannou joined the Department of Electrical
Engineering-Systems, University of Southern California, Los Angeles, CA in 1982. He is currently a
Professor in the same Department and the Director
of the Center of Advanced Transportation Technologies. He held positions as
a Professor at the Department of Mathematics and Statistics and Dean of the
school of Pure and Applied Sciences at the University of Cyprus and as Visiting Professor at the Technical University of Crete. His research interests are in
the areas of dynamics and control, adaptive control, neural networks, intelligent
transportation systems, computer networks and control applications. In 1984,
he was a recipient of the Outstanding Transactions Paper Award for his paper,
An Asymptotic Error Analysis of Identifiers and Adaptive Observers in the Presence of Parasitics, which appeared in the IEEE TRANSACTIONS ON AUTOMATIC
CONTROL in August 1982. Dr. Ioannou is also the recipient of a 1985 Presidential
Young Investigator Award for his research in Adaptive Control. He has been an
Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, the International Journal of Control and Automatica. Currently he is an Associate Editor at Large of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and an Associate Editor of the IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION
SYSTEMS.
Dr. Ioannou is a fellow of IEEE, a Member of the AVCS Committee of ITS
America and a Control System Society Member on IEEE TAB Committee on
ITS. He is the author/co-author of 1 patent, 5 books, and more than 300 research
papers in the area of controls, transportation, and neural networks. He served as
a Technical Consultant with Lockheed, Ford Motor Company, Rockwell International, General Motors, ACORN Technologies and other companies dealing
with design, implementation and testing of control systems.