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## Dynamical systems: dimensional similarity

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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.

Chapter Contents:

• 4.1 Introduction
• 4.2 Continuous time dynamical systems similarity
• 4.2.1 Transfer function dimensional similarity
• 4.2.2 State space dimensional similarity
• 4.3 Discrete time dynamical system similarity
• 4.3.1 Discrete time transfer function similarity
• 4.3.2 Sampled-data transfer function similarity
• 4.3.3 Discrete state space similarity
• 4.4 Exercises
• References

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