Synthesis of gain-scheduled H∞ controllers
[gopt,pdK,R,S] = hinfgs(pdP,r,gmin,tol,tolred)
Given an affine parameter-dependent plant
where the time-varying parameter vector p(t) ranges in a box and is measured in real time,
hinfgs seeks an affine parameter-dependent controller
scheduled by the measurements of p(t) and such that
K stabilizes the closed-loop system
for all admissible parameter trajectories p(t)
K minimizes the closed-loop quadratic H∞ performance from w to z.
pdP of the parameter-dependent plant P is specified with
psys and the vector
r gives the number of controller inputs and outputs (set
r=[p2,m2] if y ∊ Rp2 and u ∊ Rm2). Note that
hinfgs also accepts the polytopic model of P returned, e.g., by
hinfgs returns the optimal closed-loop quadratic performance
gopt and a polytopic description of the gain-scheduled controller
pdK. To test if a closed-loop quadratic performance γ is achievable, set the third input
gmin to γ. The arguments
tolred control the required relative accuracy on
gopt and the threshold for order reduction. Finally,
hinfgs also returns solutions R, S of the characteristic LMI system.
The gain-scheduled controller
pdK is parametrized by p(t) and characterized by the values KΠj of at the corners ³j of the parameter box. The command
Kj = psinfo(pdK,'sys',j)
returns the j-th vertex controller KΠj while
pv = psinfo(pdP,'par') vertx = polydec(pv) Pj = vertx(:,j)
gives the corresponding corner ³j of the parameter box (
pv is the parameter vector description).
The controller scheduling should be performed as follows. Given the measurements p(t) of the parameters at time t,
Express p(t) as a convex combination of the ³j:
This convex decomposition is computed by
Compute the controller state-space matrices at time t as the convex combination of the vertex controllers KΠj:
Use AK(t), BK(t), CK(t), DK(t) to update the controller state-space equations.
Apkarian, P., P. Gahinet, and G. Becker, “Self-Scheduled H∞ Control of Linear Parameter-Varying Systems,” Automatica, 31 (1995), pp. 1251–1261.
Becker, G., Packard, P., “Robust Performance of Linear-Parametrically Varying Systems Using Parametrically-Dependent Linear Feedback,” Systems and Control Letters, 23 (1994), pp. 205–215.
Packard, A., “Gain Scheduling via Linear Fractional Transformations,” Syst. Contr. Letters, 22 (1994), pp. 79–92.