I think that such parameters should not be referred to as " unidentifiable" but , simply , as "unknown".

I think taht your question is if a least - squares method or any of its generalizations or any other batch method for identification can be used also recursively.

Te answer is yes. You can re-arrange a batch least-squares procedure ( all the parameters identified with the whole set of input-output data)  in a recursive way. Both methods are asymptotically equivalent  for arbitrary initialization of the recursive version and exactly equivalent if the recursive method is initialized  after a previous minimum number of processed data with the batch algorithm ( depending on the number of parameters to identify). The first strategy ( asymptotic equivalence under arbitrary initialization) is suffiecient in practice.

For a good identification , the input should be sufficiently rich in frequency , for instance, adding a very weak in amplitude signal with a number a frequencies to the relevant input in the process, in the event that the relevant input  is not rich in frequency.

If you have also a tracking objective ( the output has to follow a certain reference signal) or a model-following  one ( in addition, to set asymptotically the closed-loop parameterizarion to a prefixed stable model) then the above strategy for recursive identification can be combined with an adaptive controller design with  the adaptive  time-varying controller parameters being generated from the recursive plant identification.

thanks for your answer. I am familiar with LS and its derivatives and I am using RLS in my resesrches. It seems that my question is not clear yet so I will clear it by specific application which I am working on. I am working on identification of mass parameters (e.g. mass, center of mass and inertia matrix) of elements of robots.

According to the literatuer although the dynamic equations of robots are linear w.r.t mass parameters but all of them are not identifiable which means that they can not be identified through identification. I am not sure but it may be related to measueablity of the system  I should note that baded on linear model most researches used RLS for identificatin purpise. So one problem is that how should we determine the parameters which cant be identifed. To answer this quedtion, these researhes used SVD to determine which parameters canot be identified.  

I am not expert in SVD but what which I found is that the whole time history of the regressors is required to determine which parameter is identifiable. It makes my computer code too slow so I want to find a better way to find unidentifiable parameters. I assumed that a simple way to avoid storing all regressos is to calculatr SVD recursively. it is like what whicg RLS is doing to avoid storing all data during the identification. But I found no method in literature to handle this problem yet. So I asked about the possibilty of calculation of SVD in recursive manner. 

I do not understand the question precisely.

For instance , it is clear that if a combination of parameters have a common regreesor , only teh combinsation and not each one can be identified.

For instance, from the equation:

dy(t)/dt= ( a.b +c) y(t) + du(t)

only d and (a.b+c) can be identified , but not separately , a , b an c.

 It is better to group the whole equation you have as

output=( coninitation of parameters 1)* regressor 1+ ( combination 2) *regressor 2+ 

Then, identify combination 1( all together), combination 2 ( all together ) etc. May be that combination 1 and combination 2 have common parameters. For instance:

a+b,  a+b+c

Then you can identify a+b, a+b+c and  also c as the second one minus the first one. But in this case you cannot identify a and b  individually if you do not have extra regressors to separate them.

That is , the idea is to identify one parameter or a combination of them ( which would be "a parameter" itself) per regressor component. Once this is done, one can try to use new operations ( as in the simple example) just to know  that some parameter(s) coupled to several regressors can be identified in this  second step.

Thanks Professor De La Sen.

We have a dynamic system with linear equations in the form of:

A(t).X=T(t)

which A(t) is time varying regressor matrix at each time instant, X is unknown time invariant parameters and T(t) is the excitation of the system.

When the analytical form of regressor matrix is available by determining the columns of A(t) which are linearly dependent (at each time instant) we can determine the identifiable parameters. It is in accordance with your comment. In the case of your example if a, b, c ad d are unknowns then

A(t) = [y(t)   y(t)]  u(t)]

X = [ab   c    d]^T

T(t) = dy(t)/dt

Because that the first and second columns are linearly dependent, ab and c can not be determined independently.

In robotic applications the problem is complex because that:

1-the dimensions of the regressor matrix is relatively high (For example for a robot with 3 arms in planar motion A(t) is a 5*8 Matrix to identify 8 unknown parameters(mass and inertia term of components of the robot)

2- A(t) have no clear mathematical equation (like above) and just can be calculated numerically at each time instant

What which I found in literature of robotics is that they used SVD technique to identify the unknown parameters.

Based on your comment it seems that the approach for solving such these problems should be determined first. So it seems that we first answer another question: How can we determine dependent parameters when the regressor matrix has not a clear analytical form?

I hope that I was able to describe the problem clearly.

Thanks again

I see .

Since A(t) is not full column rank then A(t)transpose*A(t) is singular and least- squares estimation Xhat(t)=[inverse ( Atranspose(t)*A(t))]*Atranpose(t)*T(t) cannot be applied  ( the n eeded  inverse does not exist)-Also , the recursive version cannot be used either since this needs that the batch version be well- posed.

Is  the rank of A(t) constant r for all time ( r < n , n being the order of A(t))?.

In this case C(t) and D(t) exist  so taht A(t)=C(t)*D(t) , C(t) being rxn of full row rank (i.e. rank equal to r ) and D(t), nxr of full column rank equal also to r. May be a batch ( least-squares like) estimation can be performed using  the Moore-Penrose pseudoinverse of A(t) [there are standard formulas for  such an  inverse]. The best approximation solution got  in this way minimizes ( in norm) the estimation error  output - estimated output .

The problem is to find  the factorization A(t)=C(t)*D(t) and  to guarantee that the rank of A(t) is constant for all time. But even if the rank changes ( r(t)

If you want to identify a completely unknown system, then you can identify only the part of system that is observable and controllable. 

Your question, if I understand correctly, is: If the parameters are time-varying and unidentifiable, then can they be identified by recursive SVD? 

If the parameters correspond to unobservable and uncontrollable part of the system for all times, then they cannot be identified. But if at some point of time they correspond to observable and controllable part of them system, and if you are performing some recursive identification technique, then you should be able to identify the parameters at that time. 

There are several recursive identification techniques, but mostly they tend to avoid SVD as using SVD recursively becomes computationally heavy. There exist many proposals that use different methods in place of SVD to identify the system recursively. 

Hope this helps. 

This is an interesting question for structural modeling. In 1998 I wrote a paper for SAE / AIAA on the inverse correlation of structural dynamics models to measured data sets that employed SVD for inversion of what appears from the tread of this post to be the same inverse equation / problem:

Ulmer, A., Structural Integrity Assessment  Assessment through Eigensolution, SAE Technical Paper 985584, 1998.

Also later published in SAE 1998 Transactions - Journal of Aerospace - V107 -1

Let me know how it goes:

email : [email protected]