Robust Control Toolbox™

Synthesis of gain-scheduled H controllers

Syntax

• [gopt,pdK,R,S] = hinfgs(pdP,r,gmin,tol,tolred) 
  

Description

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.

The description 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 aff2pol.

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 is achievable, set the third input gmin to . The arguments tol and 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.

Controller Implementation

The gain-scheduled controller pdK is parametrized by p(t) and characterized by the values Kj of

at the corners ³j of the parameter box. The command

• Kj = psinfo(pdK,'sys',j)
  

returns the j-th vertex controller Kj 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,

1. Express p(t) as a convex combination of the ³j:

p(t) = 1³1 + . . .+ N³N, j 0,

1. This convex decomposition is computed by polydec.

2. Compute the controller state-space matrices at time t as the convex combination of the vertex controllers Kj:

3. Use AK(t), BK(t), CK(t), DK(t) to update the controller state-space equations.

Reference

Apkarian, P., P. Gahinet, and G. Becker, "Self-Scheduled H Control of Linear Parameter-Varying Systems," submitted to Automatica, October 1995.

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.

See Also
psys        Specification of uncertain state-space models

pvec        Quantification of uncertainty on physical parameters

pdsimul     Time response of a parameter-dependent system along a given parameter trajectory

polydec     Compute polytopic coordinates wrt. box corners

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