Simulink Control Design™ - MathWorks

Plant Models for Gain-Scheduled Controller Tuning

Gain scheduling is a control approach for controlling a nonlinear plant. To tune a gain-

scheduled control system, you need a collection of linear models that approximate the

nonlinear dynamics near selected design points. Generally, the dynamics of the plant are

described by nonlinear differential equations of the form:

x ̇=fx,u,σ

y=gx,u,σ.

Here, x is the state vector, u is the plant input, and y is the plant output. These nonlinear

differential equations can be known explicitly for a particular system. More commonly,

they are specified implicitly, such as by a Simulink model.

You can convert these nonlinear dynamics into a family of linear models that describe the

local behavior of the plant around a family of operating points (x(σ),u(σ)), parameterized

by the scheduling variables, σ. Deviations from the nominal operating condition are

defined as:

δx=x−xσ, δu=u−uσ.

These deviations are governed, to first order, by linear parameter-varying dynamics:

δ ̇x=Aσδx+Bσδu,δy=Cσδx+Dσδu,

Aσ =∂f

∂x

xσ,uσ Bσ =∂f

∂u

xσ,uσ

Cσ =

∂g

∂x

xσ,uσ Dσ =

∂g

∂u

xσ,uσ.

This continuum of linear approximations to the nonlinear dynamics is called a linear

parameter-varying (LPV) model:

dx

dt

=Aσx+Bσu

y=Cσx+Dσu.

The LPV model describes how the linearized plant dynamics vary with time, operating

condition, or any other scheduling variable. For example, the pitch axis dynamics of an

aircraft can be approximated by an LPV model that depends on incidence angle, α, air

speed, V, and altitude, h.

11 Gain-Scheduled Controllers