A physical quantity is a physical property or quality that can be quantified by measurement. A physical quantity is defined by the product of a numerical value and a unit of measurement. In this section we present a structured approach in order to obtain the dimensionless numbers of any dimensional relation. The approach is based on matrix computations, so it is very convenient for computer implementation.

In this chapter we present the conditions for establishing dimensional similarity between systems represented in transfer function form and in state space form. It is easier to establish dimensional similarity between transfer functions than between state space systems representation. The transfer function framework only requires two scaling factors between model and prototype, the gain scaling factor and the time scaling factor. On the contrary, in the state space framework, there are as many scaling factors as physical variables involved in the matrices representation, thus the scaling factors are problem dependent. The second half of the chapter is devoted to dimensional similarity between discrete time systems, considering also the transfer function and the state space representation. With discrete time systems it is necessary to stress between purely discrete time systems and the discrete time systems obtained from continuous time systems by discretization. For purely discrete time systems there is no time, and dimensional similarity is reduced to model equality up to a gain factor. For sampled data systems, the time concept is recovered and for preserving continuous time dimensional similarity the sampling times used must also be scaled properly.

In the present chapter we demonstrate that tuning rules derived from minimization of indexes depending on closed loop transfer functions yield homogeneous and nonhomogeneous tuning rules, depending on the closed loop functions considered in the minimization index.

The analysis and design of control systems in the transfer function framework are pervasive in control applications because normally the system state is not fully measurable and dynamical controllers are required. In this case, for continuous time dimensionally similar systems controlled by means of digital controllers, a similar performing control level can be achieved by modifying the controller in two straightforward ways: (i) by adjusting the controller gain and (ii) by adjusting the controller sampling time. Moreover, the adjusting factors are directly given by the system scaling factors.