Control Systems/System Representations
System Representations[edit]
This is a table of times when it is appropriate to use each different type of system representation:
Properties  StateSpace Equations 
Transfer Function 
Transfer Matrix 

Linear, Distributed  no  no  no 
Linear, Lumped  yes  no  no 
Linear, TimeInvariant, Distributed  no  yes  no 
Linear, TimeInvariant, 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 timevariant cases, the general description is also known as the convolution description.
General Description  

TimeInvariant, Noncausal  
TimeInvariant, Causal  
TimeVariant, NonCausal  
TimeVariant, Causal 
StateSpace Equations[edit]
These are the statespace 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]
StateSpace Equations  

TimeInvariant 

TimeVariant 

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]
StateSpace Equations  

TimeInvariant 

TimeVariant 

Transfer Functions[edit]
These are the transfer function descriptions, obtained by using the Laplace Transform or the ZTransform 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 statespace 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  
