Attaching a file, but essentially we want variables that allow an nth order DE to be reduce to n first order DEs which can be put in matrix form and are easier to solve...

See  http://www.roboticslab.ca/mass-spring-damper/ as well as Laplace transforms that replace the nth order DE with an nth order algebraic equation that again are easier to solve.

Thanks for your answer Mr. Nikos Boretos, I do understand the concept of it, but my question is, as state variables need not be physically measurable or observable quanities , so anything related to the system can be taken as the state variable like heat of engine, brakes friction etc..., but to have complete controllability and observability of the system it is better to choose only the most important variable which gives the complete information of the system and they should be linearly independent else, we loose controllability or observability of the system (some valuable info is lost).

taking this into consideration choosing position and velocity as our state variables that is x and xdot , though they are normally have some relation as derivative of x is velocity, is this doesnot violate the linearly independency of x and xdot.

I believe linear independence applies to the variables as a whole, e.g. a,v and x, not just between a subset of the variables.

then that would clear my doubt. Thanks for answering it.

Dear Prof. Nerella,

                                 In terms of the Copenhagen conception and the Heisenberg Principle these constants also act as variables as they are related amongst themselves by the uncertainty principle. So you have a point. But in terms of Schroedinger's wave conception of quantum mechanics and in terms of De Broglie number you need to consider these as state variables otherwise quantum calculations will become difficult if at all possible. I hope that helps. SKM QC

Hello Mr. Soumitra K Mallick

At first I am very touched for giving me Prof. Title, because my goal is to be so, but unfortunately I am still a masters student but thanks for my future.

Coming to your answer, as you told x and xdot (position and velocity) have to be considered as state variables while considering in quantum levels.

When I use this state variable concept for mechanical electrical systems , I don't think we are considering quantum level Sir still aren't they dependent

Thanks and Regards

Udaysimha Nerella

The main point is: what does "dependence" mean? Of course, position, velocity and acceleration are related to each other by definition. But this is not an algebraic dependence. An algebraic dependence would be like "tell me your position and I shall tell you how fast you move". In this sense, position and velocity are independent. To evaluate the velocity v1 = v(t1) at a certain time t1 you need more than just x1 = x(t1), you need the position x = x(t) as a function of time in a whole interval, even a very small one, around the time instant t1. In the opposite way, you need the whole history v(t) up to the time instant t1 to evaluate the single value x1 = x(t1). 

If you had a facility that couples the velocity to the position in such a way that at any position x you have a unique velocity v = f(x), then you could omit v as an independent state variable, since it is uniquely determined by x, even if this function f is nonlinear and complicated. Usually there is no such coupling. Therefore x and v have to be kept as independent state variables. The relation xdot = v that holds between them is already part of the dynamical description of the system. 

Hello Mr. Manfred Braun ,

Thankyou for your very detailed explanation, I think I understood it almost, there are few things I would like to get clarified on.

when we do this state space analysis in computer, to my knowledge  we use ode to get numerical solution so for velocity aren't we using a finite difference approximation, doen't this show algebraic relation between position and velocity?

also can you kindly explain me in more detail the following satement, "

To evaluate the velocity v1 = v(t1) at a certain time t1 you need more than just x1 = x(t1), you need the position x = x(t) as a function of time in a whole interval, even a very small one, around the time instant t1."

but the opposite way I understood it as we need to take integral over the time.

Thanks in advance

Of course, numerical simulation boils everything down to addition and multiplication of numbers, which is algebraic. But even then you don't get v at a certain time only from x at that time. Even in the simplest finite-difference approximation you need the positions at least for two neighboring time instances, and this is the discretized version of my previous statement that you need the function in an interval to get the derivative. So even in the discretized version there is no algebraic relation between x and v taken at the same time instant.

To get the derivative of x at a certain time instant t1, you need the function x(t) in a neighborhood of t1, otherwise you cannot form the derivative. From the single value x(t1) alone you don't get it. In the finite-difference approximation: you need at least x(t1) and x(t1 - h) to approximate xdot at t1.

If I were at our university, I would invite you to discuss it personally in my office. However, my answer comes from Tallinn, Estonia, where I am living.

Thank you very much sir for your time and detailed explanation, now I got the point.  yes it would be great to discuss in person, but I am happy even though people like you are not reachable in person for newbies, platforms like these bring us on to one table.

Thank you