Paper on passive reduced-order macromodeling

The text is written in August 2008 for the http://groups.google.com/group/mor4ansys group.

I have just read a paper

Rumi Zhang, Wei Wang, Anestis Dounavis, Graham A. Jullien
Passive Reduced-Order Macromodeling Algorithm for Microelectromechanical Systems
Journal of Microelectromechanical Systems, v. 17, N 3, 678-687, 2008.
http://dx.doi.org/10.1109/JMEMS.2008.921673

and would like to make some comments.

First, I like the paper and advise to read it. The authors have proved that a particular transformation from the second order dynamic system to the first one possesses very special properties. In the case when the output matrix is equal to the input matrix, it is possible to prove that the transfer matrix will be positive-real. It is very interesting indeed.

Some problem what I see if it is possible to apply the passivity criterion from the electrical circuits (the beginning of section 3.C) to the MEMS blindly. In order to understand this point, we should remember where these requirements have been obtained from. The starting point for the passivity is the statement that the system cannot generate energy and the equation for the energy as follows

E = i_1*o_1 + … i_n*o_n

where i_1 is the first input and so on, and o_1 is the first outputs and so on. Then starting from this they come to positive definiteness of the transfer function.

The equation above implies two things that are correct for an electrical circuit

1) There are conjugated inputs and outputs. This actually contradicts a bit to the original assumption of the control theory that one can define outputs arbitrarily. Well, for an electrical circuit to define a port is quite natural (still this implies some constraints on a dynamic systems). The outputs must be connected with inputs correctly, presumably based on physical considerations. If the order of inputs and outputs in the equation above is changed, the system may become not passive.

2) The product i_1*o_1 is energy. This is the case for a circuit indeed.

Both things may not be applicable for a dynamic system that describes MEMS. Let us take as an example a thermal system that is also considered in the paper.

In the case of a thermal system the product of input (power) by an output (temperature) does not lead to the energy (W multiplied by K is definitely not energy). This means that the second assumption is just wrong and this makes the whole proof actually meaningless for the thermal system. I mean it does not make sense to talk about not producing energy anymore as the original equation is not the energy.

Then in electrothermal simulation, the temperature is actually considered in the sense of information. There is nothing like a wire in a circuit that actually couples the input and output. A transistor dissipates power and then the temperature distribution changes the properties of the transistor.

Actually, a passive system is a special case of a dissipative system. See the two fundamental papers of Prof Willems

J.C. Willems, Dissipative dynamical systems. Part 1: general theory. Archive for Rational Mechanics and Anylisis 45(5) (1972) 321-351

J.C. Willems, Dissipative dynamical systems. Part II: linear systems with quadratic supply rates. Archive for Rational Mechanics and Anylisis 45(5) (1972) 352-393

http://homes.esat.kuleuven.be/~jwillems/

So, in my view, to discuss passivity/dissipativity of MEMS it would be good to start from the beginning. Just using the criterion from an electrical circuit, as it has been done in the paper, could be dangerous.

P.S. I have asked about the passivity for electrothermal simulation at sci.engr.control. The discussion could be found at