Control Systems/System Representations

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System Representations[edit]

This is a table of times when it is appropriate to use each different type of system representation:

Properties State-Space
Linear, Distributed no no no
Linear, Lumped yes no no
Linear, Time-Invariant, Distributed no yes no
Linear, Time-Invariant, Lumped yes yes yes

General Description[edit]

These are the general external system descriptions. y is the system output, h is the system response characteristic, and x is the system input. In the time-variant cases, the general description is also known as the convolution description.

General Description
Time-Invariant, Non-causal
Time-Invariant, Causal
Time-Variant, Non-Causal
Time-Variant, Causal

State-Space Equations[edit]

These are the state-space representations for a system. y is the system output, x is the internal system state, and u is the system input. The matrices A, B, C, and D are coefficient matrices.

[Analog State Equations]

State-Space Equations


These are the digital versions of the equations listed above. All the variables have the same meanings, except that the systems are digital.

[Digital State Equations]

State-Space Equations


Transfer Functions[edit]

These are the transfer function descriptions, obtained by using the Laplace Transform or the Z-Transform on the general system descriptions listed above. Y is the system output, H is the system transfer function, and X is the system input.

[Analog Transfer Function]

Transfer Function

[Digital Transfer Function]

Transfer Function

Transfer Matrix[edit]

This is the transfer matrix system description. This representation can be obtained by taking the Laplace or Z transforms of the state-space equations. In the SISO case, these equations reduce to the transfer function representations listed above. In the MIMO case, Y is the vector of system outputs, X is the vector of system inputs, and H is the transfer matrix that relates each input X to each output Y.

[Analog Transfer Matrix]

Transfer Matrix

[Digital Transfer Matrix]

Transfer Matrix

Control Systems