Transfer function matrix

It is a particularly useful construction for linear time-invariant (LTI) systems because it can be expressed in terms of the s-plane.

In some systems, especially ones consisting entirely of passive components, it can be ambiguous which variables are inputs and which are outputs.

If the matrix is to properly model energy flows in the system, compatible variables must be chosen to allow this.

Each entry in the matrix is in the form of a transfer function relating an output to an input.

For example, for a three-input, two-output system, one might write, where the un are the inputs, the ym are the outputs, and the gmn are the transfer functions.

For discrete time systems s is replaced by z from the z-transform, but this makes no difference to subsequent analysis.

[3] In electrical systems it is often the case that the distinction between input and output variables is ambiguous.

In such cases the concept of port (a place where energy is transferred from one system to another) can be more useful than input and output.

It is customary to define two variables for each port (p): the voltage across it (Vp) and the current entering it (Ip).

For instance, the transfer matrix of a two-port network can be defined as follows, where the zmn are called the impedance parameters, or z-parameters.

There are six basic matrices that relate voltages and currents each with advantages for particular system network topologies.

To correctly predict the behaviour of the circuit, the currents entering or leaving the ports must also be taken into account, which is what the transfer matrix does.

[6] The impedance matrix for the voltage divider circuit is, which fully describes its behaviour under all input and output conditions.

[7] At microwave frequencies, none of the transfer matrices based on port voltages and currents are convenient to use in practice.

These are the powers transmitted into, and reflected from a port which are readily measured in the transmission line technology used in distributed-element circuits in the microwave band.

In this example the effort and flow variables are torque T and angular velocity ω respectively.

Other sensors in the system may be transducers converting yet other energy domains into electrical signals, such as optical, audio, thermal, fluid flow and chemical.

[12] Acoustic systems are a subset of fluid dynamics, and in both fields the primary input and output variables are pressure, P, and volumetric flow rate, Q, except in the case of sound travelling through solid components.

An example of a two-port acoustic component is a filter such as a muffler on an exhaust system.

The component can be just as easily described by the z-parameters, but transmission parameters have a mathematical advantage when dealing with a system of two-ports that are connected in a cascade of the output of one into the input port of another.

[18] The first use of transfer matrices to represent a MIMO control system was by Boksenbom and Hood in 1950, but only for the particular case of the gas turbine engines they were studying for the National Advisory Committee for Aeronautics.