chapter 3 1 parameter identification. table of contents o ne-parameter case tt wo parameters pp...
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Chapter 3
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Parameter Identification
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Table of ContentsTable of Contents One-Parameter Case Two Parameters Persistence of Excitation and Sufficiently Rich Inputs Gradient Algorithms Based on the Linear Model Least-Squares Algorithms Parameter Identification Based on DPM Parameter Identification Based on B-SPM Parameter Projection Robust Parameter Identification Robust Adaptive Laws State-Space Identifiers Adaptive Observers 22
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IntroductionIntroduction
33
The purpose of this chapter is to present the design, analysis, and simulation of algorithms that can be used for online parameter identification. This involves three steps:Step 1 (Parametric model ). Express the form of the parametric model SPM, DPM, B-SPM, or B-DPM.
Step 2 (Parameter Identification Algorithm). The estimation error is used to drive the adaptive law that generates online. The adaptive law is a differential equation of the form
( ) ( )t H t
( )H twhere is a time-varying gain vector that depends on measured signals.
Step 3 (Stability and Parameter Convergence). Establish conditions that guarantee *( )t
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Example: One-Parameter CaseExample: One-Parameter Case
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Consider the first-order plant model
Step 1: Parametric Model
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Example: One-Parameter CaseExample: One-Parameter Case
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Step 2: Parameter Identification Algorithm
parameter
errorAdaptive Law The simplest adaptive law for In scalar form may be introduced as
provided . In practice the effect of noise especially when is close to zero, may lead to erroneous parameter estimates.
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Example: One-Parameter CaseExample: One-Parameter Case
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Step 2: Parameter Identification Algorithm
Another approach is to update in a direction that minimizes a certain cost of the estimation error. As an example, consider the cost criterion:
where is a scaling constant or step size which we refer to as the adaptive gain and where is the gradient of J with respect to . We ill have
0, (0) adaptive law
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The adaptive law should guarantee that:parameter estimate and speed of adaptation are bounded andestimation error gets smaller and smaller with time.
Example: One-Parameter CaseExample: One-Parameter Case
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Step 3: Stability and Parameter Convergence
( )t
Note that these conditions still do not imply that unless some conditions on the vector referred to as the regressor vector.
*( )t ( )t
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Example: One-Parameter CaseExample: One-Parameter Case
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Step 3: Stability and Parameter Convergence
Analysis
1. Solving
2. Lyapunov
Solving
( ) 0t *( )t
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and are bounded
Example: One-Parameter CaseExample: One-Parameter Case
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Step 3: Stability and Parameter Convergence
is always bounded for any( )t ( )t
is bounded( ) ( )t t
is bounded
( )t ( )t
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Analysis by Lyapunov
Example: One-Parameter CaseExample: One-Parameter Case
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Step 3: Stability and Parameter Convergence
or
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Example: One-Parameter CaseExample: One-Parameter Case
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is uniformly stable (u.s.)
is uniformly bounded (u.b.)
asymptotic stability
So, we need to obtain additional properties for asymptotic stability
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Example: One-Parameter CaseExample: One-Parameter Case
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Example: One-Parameter CaseExample: One-Parameter Case
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adaptive law
summary(i)
(ii)
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Example: One-Parameter CaseExample: One-Parameter Case
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The PE property of is guaranteed by choosing the input u appropriately.
Appropriate choices of u:
and any bounded input u that is not vanishing with time.
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Example: One-Parameter CaseExample: One-Parameter Case
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Summary
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Example: Two-Parameter CaseExample: Two-Parameter Case
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Consider the first-order plant model
Step 1: Parametric Model
Step 2: Parameter Identification Algorithm
Estimation Model: Estimation Error:
A straightforward choice:
where is the normalizing signal such that
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Example: Two-Parameter CaseExample: Two-Parameter Case
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Adaptive Law: Use the gradient method to minimize the cost,
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Example: Two-Parameter CaseExample: Two-Parameter Case
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Step 3: Stability and Parameter Convergence
Stability of the equilibrium will very much
depend on the properties of the time-varying matrix
, which in turn depends on the properties of .
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Example: Two-Parameter CaseExample: Two-Parameter Case
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For simplicity let us assume that the plant is stable, i.e., . If we choose
at steady state
2 20 1
0( )
( )A
c c
is only marginally stable0e
is bounded but does not necessarily converge to 0. constant input does not guarantee exponential stability.
e
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Persistence of Excitation and Persistence of Excitation and Sufficiently Rich InputsSufficiently Rich Inputs
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Definition
Since is always positive semi-definite, the PE condition requires that its integral over any interval of time of length is a positive definite matrix.
Definition
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Persistence of Excitation and Persistence of Excitation and Sufficiently Rich InputsSufficiently Rich Inputs
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Let us consider the signal vector generated as
where and is a vector whose elements are strictly proper transfer functions with stable poles.Theorem
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Example: in the last example we had:
Persistence of Excitation and Persistence of Excitation and Sufficiently Rich InputsSufficiently Rich Inputs
In this case n = 2 and
is nonsingular
For , is PE.
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Example:
Persistence of Excitation and Persistence of Excitation and Sufficiently Rich InputsSufficiently Rich Inputs
Possible u
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Example: Vector CaseExample: Vector Case
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Consider the first-order plant model
Parametric Model
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Example: Vector CaseExample: Vector Case
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Filtering with
where,
a monic Hurwitz polynomial
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Example: Vector CaseExample: Vector Case
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If is Hurwitz, a bilinear model can be obtained as follows:Consider the polynomials
which satisfy the Diophantine equation
where is a monic Hurwitz polynomial of order 2n-m-1.y
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Example: Vector CaseExample: Vector Case
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Filtering by
B-SPM model
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Example: Vector CaseExample: Vector Case
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Note that in this case contains not the coefficients of
the
plant transfer function but the coefficients of the
polynomials
. In certain adaptive control systems such as
MRAC, the coefficients of are the controller
parameters, and the above parameterizations allow the
direct estimation of the controller parameters.
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Example: Vector CaseExample: Vector Case
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If some of the coefficients of the plant transfer function are known, then the dimension of the vector can be reduced. For example, if are known, then we have:
where,
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Gradient Algorithms Based on the Linear ModelGradient Algorithms Based on the Linear Model
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Different choices for cost function lead to different algorithms.As before we have:
Instantaneous Cost Function
referred to as the adaptive gain.
2
( )T
s
zJ
m
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Theorem
Gradient Algorithms Based on the Linear ModelGradient Algorithms Based on the Linear Model
Instantaneous Cost Function
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Integral Cost Function
where is a design constant acting as a forgetting factor
Gradient Algorithms Based on the Linear ModelGradient Algorithms Based on the Linear Model
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Integral Cost Function
Gradient Algorithms Based on the Linear ModelGradient Algorithms Based on the Linear Model
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Theorem:
Integral Cost Function
Gradient Algorithms Based on the Linear ModelGradient Algorithms Based on the Linear Model
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Least-Squares AlgorithmsLeast-Squares Algorithms
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LS problem: Minimize the cost:
Let us now extend this problem
Now we present different versions of the LS algorithm, which correspond to different choices of the LS cost function.
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Least-Squares AlgorithmsLeast-Squares Algorithms
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Recursive LS Algorithm with Forgetting Factor
where are design constants and
is the initial parameter estimate.
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Least-Squares AlgorithmsLeast-Squares Algorithms
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covariance matrix
is covariance matrix
where
Non-recursive LS algorithm
Recursive LS Algorithm with Forgetting Factor
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Using the identity
recursive LS algorithm with forgetting factor
Theorem:
Least-Squares AlgorithmsLeast-Squares Algorithms
Recursive LS Algorithm with Forgetting Factor
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When the above algorithm reduces to:
which is referred to as the pure LS algorithm.
Theorem
Least-Squares AlgorithmsLeast-Squares Algorithms
Pure LS Algorithm
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The pure LS algorithm guarantees that
without any restriction on the regressor .
If , however, is PE, then .
Convergence of the estimated parameters to constant
values is a unique property of the pure LS
algorithm.
2sm
*
Least-Squares AlgorithmsLeast-Squares Algorithms
Pure LS Algorithm
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One of the drawbacks of the pure LS algorithm is
that the covariance matrix P may become arbitrarily
small and slow down adaptation in some directions.
This is due to the fact that1 0P or P
This is the so-called covariance wind-up problem.
Another drawback of the pure LS algorithm is that
parameter convergence cannot be guaranteed to be
exponential.
Least-Squares AlgorithmsLeast-Squares Algorithms
Pure LS Algorithm
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Modified LS Algorithms
One way to avoid the covariance wind-up problem is
using covariance resetting modification to obtain
Least-Squares AlgorithmsLeast-Squares Algorithms
where is the time at which
and are some design scalars.
Due to covariance resetting,
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Modified LS Algorithms
Therefore, P is guaranteed to be positive definite for all t > 0. In fact, the pure LS algorithm with covariance resetting can be viewed as a gradient algorithm with time-varying adaptive gain P, and its properties are very similar to those of a gradient algorithm.
Least-Squares AlgorithmsLeast-Squares Algorithms
modified LS algorithm with forgetting factor
Where is a constant that serves as an upper bound for .
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Modified LS Algorithms
Theorem:
Least-Squares AlgorithmsLeast-Squares Algorithms
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Parameter Identification Based on DPMParameter Identification Based on DPM
Consider the DPM , it may be written as:
Where is chosen so that is a
proper stable transfer function, and is a proper
strictly positive real (SPR) transfer function.Normalizing error
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Parameter Identification Based on DPMParameter Identification Based on DPM
state-space representation
where
there exist matrices
such that:
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Parameter Identification Based on DPMParameter Identification Based on DPM
Theorem:
The adaptive law is referred to as the adaptive law based on the SPR-Lyapunov synthesis approach. It has the same form as the gradient algorithm.
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Parameter Identification Based on B-SPMParameter Identification Based on B-SPM
Consider the B-SPM . The estimation error is generated as
Let us consider the cost
where is available for measurement.
where are the adaptive gain.
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Parameter Identification Based on B-SPMParameter Identification Based on B-SPM
Since is unknown, this adaptive law cannot be implemented.
We bypass this problem by employing the equality
where . Since is arbitrary any can be selected without having to know .Therefore, the adaptive laws may be written as
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Parameter Identification Based on B-SPMParameter Identification Based on B-SPM
Theorem:
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Parameter ProjectionParameter Projection
In many practical problems, we may have some a priori knowledge of where is located in . This knowledge usually comes in terms of upper and/or lower bounds for the elements of or in terms of location in a convex subset of . If such a priori information is available, we want to constrain the online estimation to be within the set where the unknown parameters are located. For this purpose we modify the gradient algorithms based on the unconstrained minimization of certain costs using the gradient projection method.
where is a convex subset of with smooth boundary
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Parameter ProjectionParameter Projection
The adaptive laws based on the gradient method can be modified to guarantee that by solving the constrained optimization problem given above to obtain
where
denote the boundary and the interior, respectively, of
and
is the projection operator.
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Parameter ProjectionParameter Projection
The gradient algorithm based on the
instantaneous cost function with projection is
obtained by substituting
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Parameter ProjectionParameter Projection
The pure LS algorithm with projection becomes:
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5555
Parameter ProjectionParameter Projection
Theorem: The gradient adaptive laws and the LS
adaptive laws with the projection modifications
respectively, retain all the properties that are
established in the absence of projection and in
addition guarantee that provided
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Parameter ProjectionParameter Projection
Example: Consider the plant model
where a, b are unknown constants that satisfy some known bounds, e.g., b ≥ 1 and 20 ≥a ≥ -2.
SPM
SPM
The gradient adaptive law in unconstrained case is:
Now, apply the projection method by defining:
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Parameter ProjectionParameter Projection
applying the projection algorithm for each set, we obtain the following adaptive laws:
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Parameter ProjectionParameter Projection
Example: Let us consider the gradient adaptive law
SPM
with the a priori knowledge that for some known bound . In most applications, we may have such a priori information. We define
use projection method with to obtain the adaptive law
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5959
Robust Parameter IdentificationRobust Parameter Identification
In the previous sections we designed and analyzed a wide class of PI algorithms based on the parametric models that are assumed to be free of disturbances, noise, unmodeled dynamics, time delays, and other frequently encountered uncertainties. In the presence of plant uncertainties we are no longer able to express the unknown parameter vector in the form of the SPM or DPM where all signals are measured andis the only unknown term. In this case, the SPM or DPM takes the form
where is an unknown function that represents the modeling error terms.The following examples are used to show how above form arises for different plant uncertainties.
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Robust Parameter IdentificationRobust Parameter Identification
Example: Consider a system with a small input delay
Actual plant Nominal plant
where
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Robust Parameter IdentificationRobust Parameter Identification
Instability Example
Consider the scalar constant gain system
where d is a bounded unknown disturbance and . The adaptive law for estimating derived for d = 0 is given by
where and the normalizing signal is taken to be 1.
Parameter error equation
Now consider d ≠ 0, we have
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Robust Parameter IdentificationRobust Parameter Identification
Instability ExampleIn this case we cannot guarantee that the parameter estimate is bounded for any bounded input u and disturbance d. For example for:
i.e., the estimated parameter drifts to infinity even though the disturbance disappears with time. This instability phenomenon is known as parameter drift. It is mainly due to the pure integral action of the adaptive law, which, in addition to integrating the "good" signals, integrates the disturbance term as well, leading to the parameter drift phenomenon.
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Robust Adaptive LawsRobust Adaptive Laws
Consider the general plant
where is the dominant part, are strictly proper with stable poles and d is a bounded disturbance.
where
SPM
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Robust Adaptive LawsRobust Adaptive Laws
For robustness, we need to use the following modifications:
Design the normalizing signal to bound the
modeling error in addition to bounding the
regressor vector .
Modify the "pure" integral action of the adaptive
laws to prevent parameter drift.
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Robust Adaptive LawsRobust Adaptive Laws
Dynamic Normalization
Assume that are analytic in for some known .
1)2)3)
,s s
Lm m
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Robust Adaptive LawsRobust Adaptive Laws
σ-Modification
A class of robust modifications involves the use of a small feedback around the "pure“ integrator in the adaptive law, leading to the adaptive law structure
where is a small design parameter and is the adaptive gain, which in the case of LS is equal to the covariance matrix P. The above modification is referred to asthe σ -modification or as leakage.Different choices of lead to different robust adaptive laws with different properties.
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Robust Adaptive LawsRobust Adaptive Laws
Fixed σ -Modification
where σ is a small positive design constant. The gradient adaptive law takes the form
If some a priori estimate is available, then the term may be replaced with so that the leakage term becomes larger for larger deviations of from rather than from zero.
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Robust Adaptive LawsRobust Adaptive Laws
Fixed σ -Modification
Theorem:
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Robust Adaptive LawsRobust Adaptive Laws
Fixed σ -Modification
Main drawback:
If the modeling error is removed, i.e., , it will not
guarantee the ideal properties of the adaptive law
since it introduces a disturbance of the order of the
design constant σ.
Advantage:
No assumption about bounds or location of the
unknown is made.
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Robust Adaptive LawsRobust Adaptive Laws
Switching σ -Modification
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Robust Adaptive LawsRobust Adaptive Laws
Switching σ -Modification
Theorem:
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Robust Adaptive LawsRobust Adaptive Laws
Switching σ -Modification
Theorem:
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Robust Adaptive LawsRobust Adaptive Laws
ε -Modification
Another class of σ -modification involves leakage that
depends on the estimation error ε , i.e.,
where is a design constant. This modification
is referred to as the ε –modification and has
properties similar to those of the fixed σ -modification
in the sense that it cannot guarantee the ideal
properties of the adaptive law in the absence of
modeling errors.
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For parametric model in order to avoid parameter drift, we constrain to lie inside a bounded convex set that contains . As an example, consider the set
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Robust Adaptive LawsRobust Adaptive Laws
Parameter Projection
where is chosen so that .Following the last discussion, we obtain
where is chosen so that and
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Robust Adaptive LawsRobust Adaptive Laws
Parameter Projection
Theorem:
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Robust Adaptive LawsRobust Adaptive Laws
Parameter Projection
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Robust Adaptive LawsRobust Adaptive Laws
Parameter Projection
The parameter projection has properties identical to
switching
σ -modification, as both modifications aim at keeping
.
In the case of the switching σ -modification, may
exceed but remain bounded, whereas in the case of
projection provided .
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Robust Adaptive LawsRobust Adaptive Laws
Dead Zone
The principal idea behind the dead zone is to monitor the size of the estimation error and adapt only when the estimation error is large relative to the modeling error .
where is a known upper bound of the normalized modeling error . In other words, we move in the direction of the steepest descent only when the estimation error is large relative to the modeling error, i.e., when
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Robust Adaptive LawsRobust Adaptive Laws
Dead Zone
To the discontinuity in, the dead zone function is made continuous as :
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Robust Adaptive LawsRobust Adaptive Laws
Dead Zone
Normalized dead zone function
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Robust Adaptive LawsRobust Adaptive Laws
Dead Zone
Theorem:
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Robust Adaptive LawsRobust Adaptive Laws
Dead Zone
The dead zone modification guarantees that the estimated parameters always converge to a constant.
As in the case of the fixed σ-modification, the ideal properties of the adaptive law are destroyed in an effort to achieve robustness.
The robust modifications that include leakage, projection, and dead zone are analyzed for the case of the gradient algorithm for the SPM with modeling error. The same modifications can be used in the case of LS and DPM, B-SPM, and B-DPM with modeling errors.
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State-Space IdentifiersState-Space Identifiers
Consider the state-space plant model
SSPM
The above estimation model has been referred to as the series-parallel model in the literature. The estimation error vector is defined as
A straightforward choice for
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State-Space IdentifiersState-Space Identifiers
estimation error dynamics
where are the parameter errors.
where are constant scalars.
Adaptive laws:
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State-Space IdentifiersState-Space Identifiers
Theorem:
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Adaptive ObserversAdaptive Observers
Consider the LTI SISO plant
where . Assume that u is a piecewise continuous bounded function of time and that A is a stable matrix. In addition, we assume that the plant is completely controllable and completely observable. The problem is to construct a scheme that estimates both the plant parameters, i.e., A, B, C, as well as the state vector x using only I/O measurements.We refer to such a scheme as the adaptive observer.
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Adaptive ObserversAdaptive Observers
A good starting point for designing an adaptive observer is the Luenberger observer used in the case where A, B, C are known. The Luenberger observer is of the form:
Where K is chosen so that is a stable matrix, and guarantees that exponentially fast for any initial condition and any input u. For to be stable, the existence of K is guaranteed by the observability of (A, C).
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Adaptive ObserversAdaptive Observers
A straightforward procedure for choosing the structure of the adaptive observer is to use the same equation as the Luenberger observer , but replace the unknown parameters A, B, C with their estimates, generated by some adaptive law. But the problem we face with this procedure is the inability to estimate uniquely the n^2+2n parameters of A, B, C from the I/O data. The best we can do in this case is to estimate the parameters of the plant transfer function and use them to calculate . These calculations, however, are not always possible because the mapping of the 2n estimated parameters of the transfer function to the n^2 + 2n parameters of is not unique unless (A,B,C) satisfies certain structural constraints. One such constraint is that (A, B, C) is in the observer form, i.e., the plant is represented as:
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Adaptive ObserversAdaptive Observers
where
We can use the techniques presented in the previous sections.
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Adaptive ObserversAdaptive Observers
The disadvantage is that in a practical situation x may represent some physical variables of interest, whereas may be an artificial state vector.
However, the adaptive observer motivated from the Luenberger observer structure and is given by
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Adaptive ObserversAdaptive Observers
is a stable matrix that contains the eigenvalues of the observer.
A wide class of adaptive laws may be used to generate online. As in last Chapter , develop the parametric model
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THE ENDTHE END