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System Representations
This is a table of times when it is appropriate to use each different type of system representation:
| Properties | State-Space Equations | Transfer Function | Transfer Matrix |
|---|---|---|---|
| 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
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 | y ( t ) = {∫}_{− ∞}^{∞} h ( t − r ) x ( r ) d r |
| Time-Invariant, Causal | y ( t ) = {∫}_{0}^{t} h ( t − r ) x ( r ) d r |
| Time-Variant, Non-Causal | y ( t ) = {∫}_{− ∞}^{∞} h ( t , r ) x ( r ) d r |
| Time-Variant, Causal | y ( t ) = {∫}_{0}^{t} h ( t , r ) x ( r ) d r |
State-Space Equations
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 |
|---|
y ( t ) = C x ( t ) + D u ( t ) |
| Time-Variant | {x}^{′} ( t ) = A ( t ) x ( t ) + B ( t ) u ( t )y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) |
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 |
|---|
y [ t ] = C x [ t ] + D u [ t ] |
| Time-Variant | {x}^{′} [ t ] = A [ t ] x [ t ] + B [ t ] u [ t ]y [ t ] = C [ t ] x [ t ] + D [ t ] u [ t ] |
Transfer Functions
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 |
|---|
Y ( s ) = H ( s ) X ( s ) |
[Digital Transfer Function]
| Transfer Function |
|---|
Y ( z ) = H ( z ) X ( z ) |
Transfer Matrix
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 |
|---|
𝐘 ( s ) = 𝐇 ( s ) 𝐗 ( s ) |
[Digital Transfer Matrix]
| Transfer Matrix |
|---|
𝐘 ( z ) = 𝐇 ( z ) 𝐗 ( z ) |
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