Chapter 4
Model Reference Adaptive
Control
1
Table of Contents
1. Introduction
2. Simple MRAC Schemes
3. MRC for SISO Plants
4. Direct MRAC
5. Direct MRAC
6. Indirect MRAC
7. Robust MRAC
8. Case Study
2
Introduction
In this chapter, we design and analyze a wide class of
adaptive control schemes based on model reference
control
(MRC)
referred
to
as
model
reference
adaptive control (MRAC).
In MRC, the desired plant behavior is described by a
reference model
and is driven by a reference
input. The control law
is then developed so that
the closed-loop plant has a transfer function equal
to
.This transfer function matching guarantees
that the plant will behave like the reference model for
any reference input signal.
3
Introduction
Model reference control
4
Introduction
Goal:
This goal, implies that
to satisfy certain
assumptions. These assumptions enable the calculation
of the controller parameter vector as
The above goal guarantees that the tracking error
converges to zero for any given reference
input signal r.
If
is known, then
and the controller
can be calculated using
can be implemented.
5
Introduction
When
is unknown the use of certainty equivalence
(CE) approach, where the unknown parameters are
replaced with their estimates, leads to the adaptive
control scheme referred to as indirect MRAC.
6
Introduction
Another way of designing MRAC schemes is to
parameterize the plant transfer function in terms of the
desired controller parameter vector . The structure of
the MRC law is such that we can write
In this case, the controller parameter
is updated
directly without any intermediate calculations, and for
this reason the scheme is called direct MRAC.
7
Introduction
Direct MRAC
8
Introduction
Various classes of MRAC
9
Simple MRAC Schemes
Scalar Example: Adaptive Regulation
Consider the scalar plant
The control objective is to determine a bounded
function
such that the state
is
bounded and converges to zero for any .
A possible procedure to follow in the unknown
parameter case is to use the same control law but with
k* replaced by its estimate k(t) i.e., we use
and search for an adaptive law to update k(t).
10
Simple MRAC Schemes
Scalar Example: Adaptive Regulation
11
Simple MRAC Schemes
Scalar Example: Adaptive Regulation
Barbalat's lemma
12
Simple MRAC Schemes
Scalar Example: Adaptive Regulation
We have shown that the combination of the control
law
with the adaptive law
meets the
control objective in the sense that it guarantees
boundedness for all signals and forces the plant state
to converge to zero. It is worth mentioning that, as in
the parameter identification problems, we cannot
establish that k(t) converges to k*. The lack of
parameter convergence is less crucial in adaptive
control than in PI, because in most cases the control
objective can be achieved without requiring the
parameters to converge to their true values.
13
Simple MRAC Schemes
Scalar Example: Direct MRAC without Normalization
Consider the following first-order plant:
where a, b are unknown parameters but the sign of b is
known. The control objective is to choose an
appropriate control law u such that all signals in the
closed-loop plant are bounded and x tracks the state of
the reference model given by
or
We propose the control law:
14
Simple MRAC Schemes
Scalar Example: Direct MRAC without Normalization
are calculated so that the closed-loop transfer
function from r to x is equal to that of the reference
model, i.e.,
plant is controllable
parameters a, b are unknown
control law
15
Simple MRAC Schemes
Scalar Example: Direct MRAC without Normalization
where
are the estimates of
, respectively,
and search for an adaptive law to generate
.
16
Simple MRAC Schemes
Scalar Example: Direct MRAC without Normalization
where
adaptive laws
are the parameter errors.
k   1ex sgn(b )
l   2er sgn(b )
17
Simple MRAC Schemes
Scalar Example: Direct MRAC without Normalization
18
Simple MRAC Schemes
Scalar Example: Indirect MRAC without Normalization
Consider the same problem in last example
where
are generated by an
adaptive law that we design.
SSPM
19
Simple MRAC Schemes
Scalar Example: Indirect MRAC without Normalization
20
Simple MRAC Schemes
Scalar Example: Indirect MRAC without Normalization
The boundedness of depend on
and then
.
The requirement that
be bounded away from zero is
a controllability condition for the estimated plant. One
method for preventing
from going through zero is to
modify the adaptive law as below. Such a modification
is achieved using the following a priori knowledge:
21
Simple MRAC Schemes
Scalar Example: Indirect MRAC without Normalization
Using the same arguments
If the reference input signal
is sufficiently rich of
order 2, then
and therefore
converge to zero
exponentially fast.
22
Simple MRAC Schemes
Scalar Example: Direct MRAC with Normalization
Consider the same first-order plant:
Control law:
or
as before
or
:Reference model
B-SPM
23
Simple MRAC Schemes
Scalar Example: Direct MRAC with Normalization
Using the PI techniques, the adaptive law is given by
normalizing signal:
Independent of the boundedness of
adaptive law guarantees that:
, the above
24
Simple MRAC Schemes
Scalar Example: Direct MRAC with Normalization
We can use properties (i)-(ii) of the adaptive law to
first
establish
signal
boundedness
and
then
convergence of the tracking error e to zero.
It can be follow in ref.
25
Simple MRAC Schemes
Scalar Example: Indirect MRAC with Normalization
Consider the same first-order plant:
SPM
the gradient algorithm
As last, the above adaptive law guarantees that
26
Simple MRAC Schemes
Scalar Example: Indirect MRAC with Normalization
Due to division by , the gradient algorithm has to
guarantee that does not become equal to zero.
Therefore, instead of above adaptive law we use
27
Simple MRAC Schemes
Scalar Example: Indirect MRAC with Normalization
As shown in last chapter, the above adaptive law
guarantees that
By applying some lemma and theorem, it can be
conclude that:
28
Simple MRAC Schemes
Vector Case: Full-State Measurement
Consider the nth-order plant
where
and
is controllable.
are unknown matrices
The control objective is to choose the input vector
such that all signals in the closed-loop plant are
bounded and the plant state follows the state
of a reference model:
where
is a stable matrix,
, and
is a bounded reference input vector. The reference
model and input r are chosen so that
represents a
desired trajectory that has to follow.
29
Simple MRAC Schemes
Vector Case: Full-State Measurement
Control Law: If the matrices A, B were known, we
could apply the control law
Comparison with
closed-loop plant
(*)
Matching condition
30
Simple MRAC Schemes
Vector Case: Full-State Measurement
In general, no
may exist to satisfy the matching
condition (*), indicating that the above control law may
not have enough structural flexibility to meet the
control objective.
In some cases, if the structure of
is known,
,
may be designed so that (*) has a solution for
.
Let us assume that
in (*) exist, and propose
the following control law to be generated by an
appropriate adaptive law.
31
Simple MRAC Schemes
Vector Case: Full-State Measurement
±
32
Simple MRAC Schemes
Vector Case: Full-State Measurement
It depends on the unknown matrix B. In the scalar case
we manage to get away with the unknown B by
assuming that its sign is known. An extension of the
scalar assumption to the vector case is as follows:
Let us assume that L* is either positive definite or
negative definite and
where
if L* is
positive definite and
if L* is negative definite.
Then
and above error dynamic becomes
33
Simple MRAC Schemes
Vector Case: Full-State Measurement
We propose the following Lyapunov function candidate:
where
satisfies the Lyapunov equation
Then
are bounded and
Note that The assumption
may not be realistic.
34
MRC for SISO Plants
In the general case, the design of the control law is not
as straightforward as it appears in last examples.
At firs we formulate the MRC problem for a general
class of LTI SISO plants and solve it for the case where
the plant parameters are known exactly.
The significance of the existence of a control law that
solves the MRC problem is twofold:
1) It demonstrates that given a set of assumptions
about the plant and reference model, there is
enough structural flexibility to meet the control
objective
2) It provides the form of the control law that is to be
combined with an adaptive law to form MRAC
35
schemes in the case of unknown plant parameters.
MRC for SISO Plants
Problem Statement
Consider the SISO LTI plant
where
are monic polynomials and
is a constant
referred to as the high-frequency gain. The reference
model, selected by the designer to describe the
desired characteristics of the plant, is described by:
where
constant.
are monic polynomials and
is a
36
MRC for SISO Plants
Problem Statement
The MRC objective is to determine the plant input
so
that all signals are bounded and the plant output,
,
tracks the reference model output
as close as
possible for any given reference input
. We refer to
the problem of finding the desired
, to meet the
control objective as the MRC problem.
In order to meet the MRC objective with a control law
that is free of differentiators and uses only measurable
signals, we assume that the plant and reference models
satisfy the following assumptions.
37
MRC for SISO Plants
Plant assumptions:
Reference model assumptions:
38
MRC for SISO Plants
Remark:
Assumption P1 requires that the plant be minimum
phase and no assumptions about the location of the
poles of plant; i.e., the plant is allowed to have
unstable poles.
Note that we allow the plant to be uncontrollable or
unobservable, since, by assumption P1, all the plant
zeros are in LHP, any zero-pole cancellation can occur
only in LHP, which implies that the plant is both
stabilizable and detectable.
Assumption P1 is a consequence of the control
objective which is met by designing an MRC control law
that cancels the zeros of the plant and replaces them
with those of the reference model. For stability, such
cancellations should occur in LHP, which implies the
39
assumption P1.
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
Let us consider the feedback control law as:
Structure of the MRC scheme
40
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
control law
41
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
The closed-loop plant should be equal with reference:
Choosing
and using
or
matching equation
42
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
Equating the coefficients of the powers of s on both
sides, we can express it in terms of the algebraic
equation
43
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
A state-space realization of the above control law:
44
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
45
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
We obtain the state-space representation of the overall
closed-loop plant by augmenting the state of the plant
with the states of the controller, i.e.,
46
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
reference model realization
47
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
state error:
output tracking error:
48
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
Example: Let us consider the second-order plant
and the reference model
choosing
and the control input
49
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
Computing the closed-loop transfer function
and the matching equation
50
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
Computing the closed-loop transfer function
and the matching equation
51
MRC for SISO Plants
MRC Schemes: Known Plant Parameters
and can be implemented as
52
Direct MRAC with Unnormalized Adaptive Laws
In this section, we extend the last scalar example
to the general class of plants where only the output
is available for measurement. The complexity of the
schemes increases with the relative degree n* of
the plant. The simplest cases are the ones where n*
= 1 and 2. Because of their simplicity, they are still
quite popular in the literature of continuous-time
MRAC and are presented in separate sections.
53
Direct MRAC with Unnormalized Adaptive Laws
Relative Degree n* = 1
Plant:
Model:
is chosen to have the same relative degree, and both
and
satisfy assumptions P1-P4 and M1 and
M2, respectively. In addition
is designed to be
SPR.
We have shown that the control law
meets the MRC objective
54
Direct MRAC with Unnormalized Adaptive Laws
Relative Degree n* = 1
estimate of
composite state-space of the plant and controller:
55
Direct MRAC with Unnormalized Adaptive Laws
Relative Degree n* = 1
Add and subtract the desired input
where Ac is as before.
56
Direct MRAC with Unnormalized Adaptive Laws
Relative Degree n* = 1
B-SSPM form
where
, which is in the form of the bilinear
parametric model
57
Direct MRAC with Unnormalized Adaptive Laws
Relative Degree n* = 1
The MRAC scheme is summarized by the equations
Theorem: The above MRAC scheme has the following properties:
(i) All signals in the closed-loop plant are bounded, and the tracking
error converges to zero asymptotically with time for any reference
input
.
(ii) If is sufficiently rich of order 2n,
relatively coprime, then the parameter error
tracking error converge to zero exponentially fast.
are
and the
58
Direct MRAC with Unnormalized Adaptive Laws
Example: Let us consider the second-order plant
reference model:
The control law is designed as
59
Direct MRAC with Unnormalized Adaptive Laws
The adaptive law is given by
For parameter convergence, we choose r to be
sufficiently rich of order 4. As an example, we select
We may not always choosing r to be sufficiently rich.
For example, if r = constant in order to follow a
constant set point at steady state, then the use of a
sufficiently rich input r of order 4 will destroy the
60
desired tracking properties of the closed-loop plant.
Direct MRAC with Unnormalized Adaptive Laws
Relative Degree n* = 2
Let us again consider the parameterization of the plant
in terms of , developed in the previous section, i.e.,
With n* = 2,
can no longer be designed to be
SPR, and therefore the last procedure fails to apply.
Instead, let us use the identity
61
Direct MRAC with Unnormalized Adaptive Laws
Relative Degree n* = 2
62
Direct MRAC with Unnormalized Adaptive Laws
Relative Degree n* = 2
using the transformation
63
Direct MRAC with Unnormalized Adaptive Laws
Relative Degree n* = 2
As before
Adaptive law
64
Direct MRAC with Unnormalized Adaptive Laws
Relative Degree n* = 2
The overall MRAC scheme is summarized as
65
Direct MRAC with Unnormalized Adaptive Laws
Relative Degree n* = 2
Theorem: The above MRAC scheme guarantees that:
(i) All signals in the closed-loop plant are bounded, and the
tracking error converges to zero asymptotically.
(ii) If
are coprime and r is sufficiently rich of order 2n,
then the parameter error
and the tracking error
, converge to zero exponentially fast.
66
Direct MRAC with Unnormalized Adaptive Laws
Example: Let us consider the second-order plant
reference model:
The control law is designed as
67
Direct MRAC with Unnormalized Adaptive Laws
where
The adaptive law is given by
For parameter convergence, the reference input r is
chosen as
68
Direct MRAC with Normalized Adaptive Laws
Let us use the MRC law
69
Direct MRAC with Normalized Adaptive Laws
whose state-space realization is given by
and search for an adaptive law to generate
DPM
70
Direct MRAC with Normalized Adaptive Laws
B-SPM
Using the results of PI
71
Direct MRAC with Normalized Adaptive Laws
Theorem: The above MRAC scheme guarantees that:
(i) All signals are uniformly bounded.
(ii) The tracking error
converges to zero.
(iii) If the reference input signal r is sufficiently rich of order 2n,
are coprime, the tracking error
parameter error
and
, converge to zero exponentially
fast.
72
Indirect MRAC with Unnormalized Adaptive Laws
As in the direct MRAC case with unnormalized adaptive
laws, the complexity of the control law increases with
the value of the relative degree n* of the plant. In this
section we demonstrate the case for n* = 1.
The same methodology is applicable to the case of
n*>2 at the expense of additional algebraic
manipulations.
We propose the same control law as in the direct MRAC
case,
73
Indirect MRAC with Unnormalized Adaptive Laws
where
plant parameters:
Controller parameters:
matching equations:
74
Indirect MRAC with Unnormalized Adaptive Laws
75
Indirect MRAC with Unnormalized Adaptive Laws
Considering the parametric model
where
As in the direct case, since n* = 1 we can choose
to be SPR.
76
Indirect MRAC with Unnormalized Adaptive Laws
state-space
77
Indirect MRAC with Unnormalized Adaptive Laws
All signals are bounded and
78
Indirect MRAC with Normalized Adaptive Law
Starting with the plant equation
where
is the high-frequency gain, we obtain
the following plant parametric model:
is a Hurwitz polynomial
79
Indirect MRAC with Normalized Adaptive Law
control law:
The controller parameter vectors
is
calculated using the mapping
.
The mapping
is obtained by using the matching
equations
where
is the quotient of
and
80
Indirect MRAC with Normalized Adaptive Law
If
are the estimated values of the
polynomials
, respectively, at each
time t, then
are obtained as solutions to
the polynomial equations
are evaluated from the estimate
of
81
Indirect MRAC with Normalized Adaptive Law
Summary:
Plant:
Model:
Plant Parameter Estimation
Controller:
Control Parameter Calculation:
82
Robust MRAC
In this section we consider MRAC schemes that are
designed for a simplified model of the plant but are
applied to a higher-order plant. We assume that the
plant is of the form
where,
is the modeled part of the plant
is an unknown multiplicative perturbation with stable poles
is a bounded input disturbance
We assume that the overall plant and modeled part
are strictly proper.
We design the MRAC scheme assuming that the plant
83
model satisfies assumptions P1-P4.
Robust Direct MRAC
We first use an example to illustrate the design and
stability analysis of a robust MRAC scheme with a
normalized adaptive law. Then extend the results to a
general SISO plant with unmodeled dynamics and
bounded disturbances.
Example Consider the SISO plant
with a strictly proper transfer function, where
is
unknown and
is a multiplicative plant uncertainty.
Let us consider the nonrobust adaptive control law
84
Robust Direct MRAC
where
is the desired closed-loop pole and is the
estimate of
designed for the plant model
and applied to the actual plant. The plant uncertainty
introduces a disturbance term in the adaptive law that
may easily cause
to drift to infinity and certain
signals to become unbounded. The above adaptive
control law is, therefore, not robust with respect to the
85
plant uncertainty
.
Robust Direct MRAC
This adaptive control scheme, however, can be made
robust if we it with a robust one developed by
following the procedure of last Chapter as:
then we can verify that the signal
guarantees that
generated as
86
Robust Direct MRAC
Hence, we can combine normalization with any
modification, such as leakage, dead zone, or
projection, to form a robust adaptive law. Let us
consider the switching σ-modification, i.e.,
where
is as defined in last chapter. According to the
results in last chapter, the above robust adaptive law
guarantees that
87
Robust Direct MRAC
Using the properties of the
norm, we have
For analyzing the stability properties follow the ref,
which implies that
That is, the regulation error is of the order of the
modeling error in m.s.s. where,
88
Robust Direct MRAC
The conditions of
summarized as follows:
to satisfy robust stability are
The constant
is arbitrary and chosen to
satisfy the above inequalities for small
89
Robust Direct MRAC
Simulation results of the MRAC scheme of Example for
different values of mu.
90
Robust Direct MRAC
General case
Let us now consider the SISO plant given by
Satisfying assumptions P1-P4, and the overall transfer
function of the plant is strictly proper. The
multiplicative uncertainty
satisfies the following
assumptions:
91
Robust Direct MRAC
Assumptions SI and S2 imply that
defined as
are finite constants which for robustness purposes will
be required to satisfy certain upper bounds. We should
note that the strict properness of the overall plant
transfer function and of
imply that
is a
strictly proper transfer function.
The control objective is to choose
so that all signals
in the closed-loop plant are bounded and the output
tracks, as closely as possible, the output of the
reference model
given by
92
Robust Direct MRAC
The transfer function
of the reference model
satisfies assumptions Ml and M2.
We start with the control law
It can be shown that the parametric model for * is
given by
93
Robust Direct MRAC
We can use a wide class of robust adaptive laws. For
example gradient algorithm as:
The constant
are analytic in
is chosen so that
This implies that
94
Robust Direct MRAC
The above adaptive law guarantees that
where
is the upper bound of
The control law
together with the mentioned
robust adaptive law form the robust direct MRAC
scheme whose properties are described by the
following theorem.
95
Robust Direct MRAC
Theorem Consider the MRAC scheme
designed for the plant model
the plant
With plant uncertainties
disturbance
. If
but applied to
and bounded input
where
96
Robust Direct MRAC
is such that
is analytic in
is an arbitrary constant, and
denotes
finite constants, then all the signals in the closed-loop
plant are bounded and the tracking error satisfies
for any T > 0, where
is an upper bound for
and
If, in addition, the reference signal r is dominantly rich
of order 2n and Zp, Rp are coprime, then the
parameter error and tracking error converge
exponentially to the residual set
97
Robust Direct MRAC
Remark1: it was shown that the projection
modification or switching σ-modification alone is
sufficient to obtain the same qualitative results as
those of Theorem. In simulations, adaptive laws using
dynamic normalization often lead to better transient
behavior than those using static normalization.
Remark2: The calculation of the constants
is tedious but possible. Because the constants,
depend
on
unknown
transfer
functions
and
parameters, the conditions for robust stability are
quite difficult to check for a given plant. The
importance of the robustness bounds is therefore
more qualitative than quantitative.
98
Case Study:
Adaptive Attitude Control of a Spacecraft
We consider the control problem associated with the
descending of a spacecraft onto a landing site such as
Mars. The attitude of the spacecraft needs to be
controlled in order to avoid tipping or crash. Due to
the consumption of fuel during the terminal landing,
the moments of inertia Ix, Iy, and Iz are changing
with time in an uncertain manner. In order to handle
this parametric uncertainty we consider an adaptive
control design.
In next figure,
are the body frame axes of
the spacecraft; X, Y, and Z are the inertial reference
frame axes of Mars; and O and C are the centers of
mass of the spacecraft and Mars, respectively.
99
Case Study:
Body frames of spacecraft and Mars.
100
Case Study:
The dynamics of the spacecraft are described by the
following equations:
where
are the input torques;
are
the moments of inertia; and
are the angular
velocities with respect to the inertial frame
Axes. Define the coordinates
where
are the unit vectors along the axis of
rotation and φ is the angle of rotation with
are the quaternion angles of rotation.
101
Case Study:
By assuming a small angle of rotation, i.e.,
we will have
, and the
attitude spacecraft dynamics will be described as
102
Case Study:
Let the performance requirements are to have settling
time less than 0.6 s. and overshoot less than 5%.
These performance requirements are used to choose
the reference model as a second-order system with two
complex poles corresponding to a damping ratio
and natural frequency
rad/sec, i.e.,
103
Case Study:
The control objective is to choose the input torques
so that each
follows the output of
the reference model
for any unknown moment of
inertia
The form of the MRC law is
104
Case Study:
where the controller parameters
are such that the closed loop transfer function in each
axis is equal to that of the reference model. It can be
shown that this matching is achieved if
Substituting these desired controller parameters into
the MRC law, we can express the overall MRC law in
the compact form
where,
105
Case Study:
I is the identity 3x3 matrix, and the design constant λ
is taken to be equal to 1. The relationship between the
desired controller parameters and unknown plant
parameters shows that if we use the direct MRAC
approach, we will have to estimate 12 parameters,
which is the number of the unknown controller
parameters, whereas if we use the indirect MRAC
approach, we estimate only 3 parameters, namely the
unknown inertias. For this reason the indirect MRAC
approach is more desirable.
Using the CE approach, the control law is given as
where
is the estimate of
to be generated by an
106
online adaptive law as follows.
Case Study:
We consider the plant equations
Filtering by
Define:
107
Case Study:
we obtain the SPMs
Using the a priori information that
we design the following adaptive laws:
108
Case Study:
where
adaptive gains:
The adaptive laws
together with the control law
form the indirect MRAC scheme
109
Case Study:
Next figure shows the tracking of the output qm of the
reference model by the three coordinates q1,q2,q3, as
well as the way the inertias change due to fuel
reduction and the corresponding estimates generated
by the adaptive law. It is clear that the control
objective is met despite the fact that the estimated
parameters converge to the wrong values due to lack
of persistence of excitation.
110
Case Study:
Simulation results of indirect MRAC.
111
THE END
112