Supposing there is a vector of system dynamics, called F, consisting of differential states, x, and algebraic states, y, and in addition to the manipulated control signals, u, there are some unmanipulated inputs, called w. There is also a vector of algebraic constraints, called G, that provides constraints on the differential and algebraic states; however, there are some unmanipulated inputs in these constraints as well. Since these latter signals are not fixed and change the algebraic constraints, is the system still a DAE?
Dear Arash,
first of all, the term "unmanipulated control signal" does not make any sense. variables w are, as far as I understood from the description you have provided, "unmanipulated input signals", more commonly denoted as disturbances or uncertainties. I am mentioning this only because using of such (more common) terms is crucial when you search the literature for some answers or similar problems.
Second thing is, I did not get if you were trying to simulate such system or optimize it. In either way, both optimization and simulation of such system is not going to be an easy task.
Well, to the point of your question. I would say that the system you are dealing with should be still classified as DAE, since there are differential and algebraic equations involved; but since there are these uncertainities involved, the actual classification should better change to "uncertain DAEs".
Thanks Radoslav. You are absolutely right about my mistake. I corrected it.
What I mean is that when there are such inputs, by my understanding, the jacobian of F with respect to the first momentum could be non-singular in some points means the violation of the condition of being a DAE.
Dear Arash,
in case you know how the uncertainties enter the system, what magnitude are they of or how they are bounded, you can definitely say more about the system. But still, in general, the system will be DAE I would say. I did not mention it previously, but, in general, there can exist cases where actual realization of the uncertainty will change the entire structure or index of DAE.
I am not sure if I understand what do you mean by "the jacobian of F with respect to the first momentum could be non-singular". General DAE can be written as
F(\dot x, x, y, u, w)=0
I am not familiar with the term "first momentum" in this context but I guess it signifies "\dot x". If so, then you should be worried about the case when Jacobian is singular (instead of non-singular) which will mean that differential equations of your system are transformed to algebraic ones and some of your variables are reaching equilibrium state. Did I get it correctly?
As you mentioned, by the first momentum I meant \dot x.
"actual realization of the uncertainty will change the entire structure or index of DAE ... the case when Jacobian is singular": you are absolutely right and it is what I meant. I was wondering that when there is unmanipulated inputs, I supposed that their values at some points may cause the jacobian to be non-singular and so the index to be zero and the algebraic constraints to be disappeared. I found my answer that though it may happen, we can still call it DAE or in more specific "uncertain DAE". Would you please correct me if I am wrong?
By the way, may I please ask you about any available numerical tools to calculate the index of a system represented by DAEs?
Hello Arash,
As far as I know, the fact that the Jacobian of F is singular does not mean that your equations of motion are no longer DAEs. On the contrary, it means that you cannot transform them to ODEs. If I understood correctly, in your particular case this would mean that 'some of the x variables would become y variables' (sorry, it's hard to express this properly without math).
By the way, I'm working with some collegues on a similar problem (the one where disturbances are present in the algebraic equations), from a purely theoretical point of view. Are you working on a particular application? (We would like apply our results to something concrete). Could you please provide more details on your problem statement? That is, how come the perturbations enter in the algebraic equations?
Thanks,
Fernando.
Thanks a lot for your answer Fernando. I am doing my research in the field of optimal energy management for renewable energy sites which specifically can be assumed as the application of NMPC to a system represented by DAEs. As a matter of fact, I have not considered the disturbances in my controller; however, there are large perturbations in algebraic equations. Physically, there are some components which are algebraic as well as dependent on the perturbations. Although these perturbations are random variables, at this stage I assumed them as deterministic but variable profiles.
Thanks for replying Arash. Could you please give me a reference to one of your papers (where the perturbations, though random, enter the algebraic equations)? Thanks again.
I am waiting for the reviewers comments on my paper, however, the paper entitled "ACAUSAL MODELING AND SIMULATION OF THE STANDALONE SOLAR POWER SYSTEMS AS HYBRID DAEs", which you can find it in my research gate page, indicates the point partially by presenting the algebraic constraint introduced by the PV array and the battery. Please note that the S, i.e. the solar irradiance, is the perturbation. Please check the other paper entitled "An Assessment of Solar Irradiance Stochastic Model for the UK", available with the link in my LINKEDIN page, that presents a stochastic model for the solar irradiance.
Hi arash and all,
I have a brief comment that, the comparison of your work which is a modeling problem (a far as my understanding) with work on modeling of under-actuated mechanical systems can lead to further insights. As lot of work has been done in this area with lots of evil constraints, it might help :)
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