Parameterize Gain Schedules
Typically, gain-scheduled control systems in Simulink use lookup tables or MATLAB
Function blocks to specify gain values as a function of the scheduling variables. For
tuning, you replace these blocks by parametric gain surfaces. A parametric gain surface
is a basis-function expansion whose coefficients are tunable. For example, you can model
a time-varying gain k(t) as a cubic polynomial in t:
k(t) = k 0 + k 1 t + k 2 t^2 + k 3 t^3.Here, k 0 ,...,k 3 are tunable coefficients. When you parameterize scheduled gains in this
way, systune can tune the gain-surface coefficients to meet your control objectives at a
representative set of operating conditions. For applications where gains vary smoothly
with the scheduling variables, this approach provides explicit formulas for the gains,
which the software can write directly to MATLAB Function blocks. When you use lookup
tables, this approach lets you tune a few coefficients rather than many individual lookup-
table entries, drastically reducing the number of parameters and ensuring smooth
transitions between operating points.Basis Function Parameterization
In a gain-scheduled controller, the scheduled gains are functions of the scheduling
variables, σ. For example, a gain-scheduled PI controller has the form:Cs,σ =Kpσ +Kiσ
s.
Tuning this controller requires determining the functional forms Kp(σ) and Ki(σ) that yield
the best system performance over the operating range of σ values. However, tuning
arbitrary functions is difficult. Therefore, it is necessary either to consider the function
values at only a finite set of points, or restrict the generality of the functions themselves.In the first approach, you choose a collection of design points, σ, and tune the gains Kp
and Ki independently at each design point. The resulting set of gain values is stored in a
lookup table driven by the scheduling variables, σ. A drawback of this approach is that
tuning might yield substantially different values for neighboring design points, causing
undesirable jumps when transitioning from one operating point to another.Alternatively, you can model the gains as smooth functions of σ, but restrict the generality
of such functions by using specific basis function expansions. For example, suppose σ is a
scalar variable. You can model Kp(σ) as a quadratic function of σ:11 Gain-Scheduled Controllers