Controllable canonical form method

There are other methods in pole-placement design. One of them is the Ackermann s formula. The derivation of Eq. (9-21) predicates that we have put (9-13) in the controllable canonical form. Ackermann s formula only requires that the system (9-13) be completely state controllable. If so, we can evaluate the state feedback gain as 1... 

Because Newton s equations of motion are conservative, the natural ensemble is NVE (micro-canonical), Aat is one in which the internal energy rather than the temperature is held constant. This is inconvenient if one wishes to compare with experiment where it is the temperature that is generally controlled. In order to perform molecular dynamics in the canonical ensemble, a thermostat must be applied to the system. This is accomplished by constructing a pseudo-Lagrangian. Many forms for temperature-conserving Lagrangians have been proposed, most of which can be written in a form that adds a frictional (velocity-dependent) term to the equations of motion (Allen and Tildesley 1989). Physically, the thermostat can be thought of as a heat bath to which the system is coupled. In the NPT ensemble, in which the pressure is held constant, the cell size and shape fluctuates. The choice of dynamical variables is critical. If the lattice parameters are chosen as in the method of Parrinello and Rahman (1981), the time evolution may depend on the chosen size or shape of the supercell. This difficulty is... 

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