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	$y(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}h(t-r)x(r)dr$
Time-Invariant, Causal	$y(t)=\underset{0}{\overset{t}{\int }}h(t-r)x(r)dr$
Time-Variant, Non-Causal	$y(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}h(t,r)x(r)dr$
Time-Variant, Causal	$y(t)=\underset{0}{\overset{t}{\int }}h(t,r)x(r)dr$

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.

State-Space Equations
Time-Invariant	$x^{\prime }(t)=Ax(t)+Bu(t)$ $y(t)=Cx(t)+Du(t)$
Time-Variant	$x^{\prime }(t)=A(t)x(t)+B(t)u(t)$ $y(t)=C(t)x(t)+D(t)u(t)$

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

State-Space Equations
Time-Invariant	$x^{\prime }[t]=Ax[t]+Bu[t]$ $y[t]=Cx[t]+Du[t]$
Time-Variant	$x^{\prime }[t]=A[t]x[t]+B[t]u[t]$ $y[t]=C[t]x[t]+D[t]u[t]$

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.

Transfer Function
$Y(s)=H(s)X(s)$

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.

Transfer Matrix
$𝐘(s)=𝐇(s)𝐗(s)$

Transfer Matrix
$𝐘(z)=𝐇(z)𝐗(z)$