Simulink Control Design™ - MathWorks

δx(t) =x(t)−x 0

δu(t) =u(t)−u 0

δy(t) =y(t)−y 0

The output of the system at the operating point is y(t 0 )=g(x 0 ,u 0 ,t 0 )=y 0.

The linearized state-space equations in terms of δx(t), δu(t), and δy(t) are:

δx ̇(t) =Aδx(t)+Bδu(t)

δy(t) =Cδx(t)+Dδu(t)

where A, B, C, and D are constant coefficient matrices. These matrices are the Jacobians
of the system, evaluated at the operating point:

A=

∂f

∂xt 0 ,x 0 ,u 0

B=

∂f

∂ut 0 ,x 0 ,u 0

C=

∂g

∂xt 0 ,x 0 ,u 0

D=

∂g

∂ut 0 ,x 0 ,u 0

This linear time-invariant approximation to the nonlinear system is valid in a region
around the operating point at t=t 0 , x(t 0 )=x 0 , and u(t 0 )=u 0. In other words, if the values of
the system states, x(t), and inputs, u(t), are close enough to the operating point, the
system behaves approximately linearly.

The transfer function of the linearized model is the ratio of the Laplace transform of δy(t)
and the Laplace transform of δu(t):

Plin(s) =

δY(s)

δU(s)

Multirate Models

Simulink Control Design software lets you linearize multirate nonlinear systems. The
resulting linearized model is in state-space form.

Multirate models include states with different sampling rates. In multirate models, the
state variables change values at different times and with different frequencies. Some of
the variables might change continuously.

The general state-space equations of a nonlinear, multirate system are:

Exact Linearization Algorithm........................ 2-