hinfgs

Synthesis of gain-scheduled H_∞ controllers

Syntax

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

Description

Given an affine parameter-dependent plant

$$P\{\begin{array}{c}\dot{x}=A(p)x+B_1(p)w+B_{2}u\\ z=C_1(p)x+D_{11}(p)w+D_{12}u\\ y=C_{2}x+D_{21}w+D_{22}u\end{array}$$

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

$$K\{\begin{array}{c}\dot{\zeta }=A_K(p)\zeta +B_K(p)y\\ u=C_K(p)\zeta +D_K(P)y\end{array}$$

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 ∊ R^p2 and u ∊ R^m2). 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 γ is achievable, set the third input gmin to γ. 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 K_Πj of $$\left(\begin{array}{cc}{A}_K(p)& B_K(p)\\ C_K(p)& D_K(p)\end{array}\right)$$ at the corners ³_j of the parameter box. The command

Kj = psinfo(pdK,'sys',j)

returns the j-th vertex controller K_Πj while

pv = psinfo(pdP,'par') 
vertx = polydec(pv) 
Pj = vertx(:,j)

gives the corresponding corner ³_j 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 ³_j:

   $$p(t)={\alpha }_1{{}^3}_1+\dots +{\alpha }_N{{}^3}_N,{\alpha }_j\ge 0,{\displaystyle \sum _{i=1}^N{\alpha }_j=1}$$

   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 K_Πj:

   $$\left(\begin{array}{cc}{A}_K(t)& B_K(t)\\ C_K(t)& D_K(t)\end{array}\right)={\displaystyle \sum _{i=1}^N{\alpha }_j}{K}_{\Pi }{}_{{}_{\iota }}.$$

3. Use A_K(t), B_K(t), C_K(t), D_K(t) to update the controller state-space equations.

References

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.

Version History

Introduced before R2006a