# Control Systems/System Representations

## 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 $y(t) = \int_{-\infty}^\infty h(t - r)x(r)dr$
Time-Invariant, Causal $y(t) = \int_{0}^t h(t - r)x(r)dr$
Time-Variant, Non-Causal $y(t) = \int_{-\infty}^\infty h(t, r)x(r)dr$
Time-Variant, Causal $y(t) = \int_{0}^t h(t, r)x(r)dr$

## 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
Time-Invariant $x'(t) = Ax(t) + Bu(t)$

$y(t) = Cx(t) + Du(t)$

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
Time-Invariant $x'[t] = Ax[t] + Bu[t]$

$y[t] = Cx[t] + Du[t]$

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
$\bold{Y}(s) = \bold{H}(s)\bold{X}(s)$

[Digital Transfer Matrix]

Transfer Matrix
$\bold{Y}(z) = \bold{H}(z)\bold{X}(z)$