chapter 3 1 parameter identification. table of contents o ne-parameter case tt wo parameters pp...

Chapter 3

11

Parameter Identification

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

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

Example: One-Parameter CaseExample: One-Parameter Case

44

Consider the first-order plant model

Step 1: Parametric Model


55

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.


66


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

The adaptive law should guarantee that:parameter estimate and speed of adaptation are bounded andestimation error gets smaller and smaller with time.


77

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


88


Analysis

1. Solving

2. Lyapunov

Solving

( ) 0t *( )t

and are bounded


99


is always bounded for any( )t ( )t

is bounded( ) ( )t t

is bounded

( )t ( )t

Analysis by Lyapunov


1010


or


1111

is uniformly stable (u.s.)

is uniformly bounded (u.b.)

asymptotic stability

So, we need to obtain additional properties for asymptotic stability


1212


1313

adaptive law

summary(i)

(ii)


1414

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.


1515

Summary

Example: Two-Parameter CaseExample: Two-Parameter Case

1616


Step 1: Parametric Model


Estimation Model: Estimation Error:

A straightforward choice:

where is the normalizing signal such that


1717

Adaptive Law: Use the gradient method to minimize the cost,


1818


Stability of the equilibrium will very much

depend on the properties of the time-varying matrix

, which in turn depends on the properties of .


1919

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

Persistence of Excitation and Persistence of Excitation and Sufficiently Rich InputsSufficiently Rich Inputs

2020

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


2121

Let us consider the signal vector generated as

where and is a vector whose elements are strictly proper transfer functions with stable poles.Theorem

2222

Example: in the last example we had:


In this case n = 2 and

is nonsingular

For , is PE.

2323

Example:


Possible u

Example: Vector CaseExample: Vector Case

2424


Parametric Model


2525

Filtering with

where,

a monic Hurwitz polynomial


2626

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


2727

Filtering by

B-SPM model


2828

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.


2929

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,

Gradient Algorithms Based on the Linear ModelGradient Algorithms Based on the Linear Model

3030

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

3131

Theorem


Instantaneous Cost Function

3232

Integral Cost Function

where is a design constant acting as a forgetting factor


3333



3434

Theorem:



Least-Squares AlgorithmsLeast-Squares Algorithms

3535

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.


3636

Recursive LS Algorithm with Forgetting Factor

where are design constants and

is the initial parameter estimate.


3737

covariance matrix

is covariance matrix

where

Non-recursive LS algorithm


3838

Using the identity

recursive LS algorithm with forgetting factor

Theorem:



3939

When the above algorithm reduces to:

which is referred to as the pure LS algorithm.

Theorem


Pure LS Algorithm

4040

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

*


Pure LS Algorithm

4141

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.


Pure LS Algorithm

4242

Modified LS Algorithms

One way to avoid the covariance wind-up problem is

using covariance resetting modification to obtain


where is the time at which

and are some design scalars.

Due to covariance resetting,

4343


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.


modified LS algorithm with forgetting factor

Where is a constant that serves as an upper bound for .

4444


Theorem:


4545

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

4646


state-space representation

where

there exist matrices

such that:

4747


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.

4848

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.

4949


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

5050


Theorem:

5151

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

5252


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.

5353


The gradient algorithm based on the

instantaneous cost function with projection is

obtained by substituting

5454


The pure LS algorithm with projection becomes:

5555


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

5656


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:

5757


applying the projection algorithm for each set, we obtain the following adaptive laws:

5858


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

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.

6060


Example: Consider a system with a small input delay

Actual plant Nominal plant

where

6161


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

6262


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.

6363

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

6464


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.

6565


Dynamic Normalization

Assume that are analytic in for some known .

1)2)3)

,s s

Lm m

6666


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.

6767


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.

6868



Theorem:

6969



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.

7070


Switching σ -Modification

7171



Theorem:

7272



Theorem:

7373


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.

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

7474


Parameter Projection

where is chosen so that .Following the last discussion, we obtain

where is chosen so that and

7575



Theorem:

7676



7777



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 .

7878


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

7979


Dead Zone

To the discontinuity in, the dead zone function is made continuous as :

8080


Dead Zone

Normalized dead zone function

8181


Dead Zone

Theorem:

8282


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.

8383

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

8484


estimation error dynamics

where are the parameter errors.

where are constant scalars.

Adaptive laws:

8585


Theorem:

8686

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.

8787


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).

8888


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:

8989


where

We can use the techniques presented in the previous sections.

9090


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

9191


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

9292

THE ENDTHE END

chapter 3 1 parameter identification. table of contents o ne-parameter case tt wo parameters pp...

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