# h2hinfsyn

Mixed H2/H synthesis with pole placement constraints

## Syntax

```[gopt,h2opt,K,R,S] = hinfmix(P,r,obj,region,dkbnd,tol)
```

## Description

`h2hinfyn` performs multi-objective output-feedback synthesis. The control problem is sketched in this figure.

If T(s) and T2(s) denote the closed-loop transfer functions from w to z and z2, respectively, `hinfmix` computes a suboptimal solution of the following synthesis problem:

Design an LTI controller K(s) that minimizes the mixed H2/H criterion

$\alpha {‖{T}_{\infty }‖}_{\infty }^{2}+\beta {‖{T}_{2}‖}_{2}^{2}$

subject to

• T[[BULLET]] < γ0

• T22 < ν0

• The closed-loop poles lie in some prescribed LMI region D.

Recall that ∥.∥∞ and ∥.∥2 denote the H norm (RMS gain) and H2 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 z2, y, and u. Note that z and/or z2 can be empty. The four-entry vector `obj` = [γ0, ν0, α, β] specifies the H2/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 DK. 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 H2 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 z2 can be empty in such case.

## References

Chilali, M., and P. Gahinet, "H Design with Pole Placement Constraints: An LMI Approach," IEEE Trans. Aut. Contr., 41 (1995), pp. 358–367.

Scherer, C., "Mixed H2/H-infinity Control," Trends in Control: A European Perspective, Springer-Verlag (1995), pp.173–216.