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Mixed H_{2}/H_{∞} synthesis with pole placement constraints
[gopt,h2opt,K,R,S] = hinfmix(P,r,obj,region,dkbnd,tol)
h2hinfyn performs multi-objective output-feedback synthesis. The control problem is sketched in this figure.
If T_{∞}(s) and T_{2}(s) denote the closed-loop transfer functions from w to z_{∞} and z_{2}, respectively, hinfmix computes a suboptimal solution of the following synthesis problem:
Design an LTI controller K(s) that minimizes the mixed H_{2}/H_{∞} criterion
$$\alpha {\Vert {T}_{\infty}\Vert}_{\infty}^{2}+\beta {\Vert {T}_{2}\Vert}_{2}^{2}$$
subject to
∥T_{∞}∥_{[[BULLET]]} < γ_{0}
∥T_{2}∥_{2} < ν_{0}
The closed-loop poles lie in some prescribed LMI region D.
Recall that ∥.∥∞ and ∥.∥_{2} denote the H_{∞} norm (RMS gain) and H_{2} norm of transfer functions.
P is any SS, TF, or ZPK LTI representation of the plant P(s), and r is a three-entry vector listing the lengths of z_{2}, y, and u. Note that z_{∞} and/or z_{2} can be empty. The four-entry vector obj = [γ_{0}, ν_{0}, α, β] specifies the H_{2}/H_{∞} constraints and trade-off criterion, and the remaining input arguments are optional:
region specifies the LMI region for pole placement (the default region = [] is the open left-half plane). Use lmireg to interactively build the LMI region description region
dkbnd is a user-specified bound on the norm of the controller feedthrough matrix D_{K}. The default value is 100. To make the controller K(s) strictly proper, set dkbnd = 0.
tol is the required relative accuracy on the optimal value of the trade-off criterion (the default is 10^{–2}).
The function h2hinfsyn returns guaranteed H_{∞} and H_{2} performances gopt and h2opt as well as the SYSTEM matrix K of the LMI-optimal controller. You can also access the optimal values of the LMI variables R, S via the extra output arguments R and S.
A variety of mixed and unmixed problems can be solved with hinfmix. In particular, you can use hinfmix to perform pure pole placement by setting obj = [0 0 0 0]. Note that both z_{∞} and z_{2} can be empty in such case.