Simulink Control Design™ - MathWorks

x ̇t = fxt,x 1 k 1 ,...,xmkm,ut,t

x 1 (k 1 +1) =f 1 xt,x 1 k 1 ,...,xmkm,ut,t

⋮ ⋮

xm(km+1) =fixt,x 1 k 1 ,...,xmkm,ut,t

yt =gxt,x 1 k 1 ,...,xmkm,ut,t

where k 1 ,..., km are integer values and tk 1 ,...,tkm are discrete times.

The linearized equations that approximate this nonlinear system as a single-rate discrete

model are:

δxk+1≈Aδxk+Bδuk

δyk ≈Cδxk+Dδuk

The rate of the linearized model is typically the least common multiple of the sample

times, which is usually the slowest sample time.

For more information, see “Linearization of Multirate Models” on page 2-183.

Perturbation of Individual Blocks

Simulink Control Design software linearizes blocks that do not have a preprogrammed

linearization using numerical perturbation. The software computes the block linearization

by numerically perturbing the states and inputs of the block about the operating point of

the block.

The block perturbation algorithm introduces a small perturbation to the nonlinear block

and measures the response to this perturbation. The default difference between the

perturbed value and the operating point value is 10 −^5 1+ x , where x is the operating

point value. The software uses this perturbation and the resulting response to compute

the linear state-space of this block. For information on how to change perturbation levels

for individual blocks, see “Change Perturbation Level of Blocks Perturbed During

Linearization” on page 2-191.

In general, a continuous-time nonlinear Simulink block in state-space form is given by:

x ̇(t) =fx(t),u(t),t

y(t) =gx(t),u(t),t.

2 Linearization