# msfsyn

Multi-model/multi-objective state-feedback synthesis

## Syntax

```[gopt,h2opt,K,Pcl,X] = msfsyn(P,r,obj,region,tol)
```

## Description

Given an LTI plant `P` with state-space equations

`$\left\{\begin{array}{c}\stackrel{˙}{x}=Ax+{B}_{1}w+{B}_{2}u\\ {z}_{\infty }={C}_{1}x+{D}_{11}w+{D}_{12}u\\ {z}_{2}={C}_{2}x+{D}_{22}u\end{array}$`

`msfsyn` computes a state-feedback control u = Kx that

• Maintains the RMS gain (H norm) of the closed-loop transfer function T from w to z below some prescribed value γ0 > 0

• Maintains the H2 norm of the closed-loop transfer function T2 from w to z2 below some prescribed value υ0 > 0

• Minimizes an H2/H trade-off criterion of the form

`$\alpha {‖{T}_{\infty }‖}_{\infty }^{2}+\beta {‖{T}_{2}‖}_{2}^{2}$`
• Places the closed-loop poles inside the LMI region specified by `region` (see `lmireg` for the specification of such regions). The default is the open left-half plane.

Set `r = size(d22)` and `obj =` [γ0, ν0, α, β] to specify the problem dimensions and the design parameters γ0, ν0, α, and β. You can perform pure pole placement by setting `obj = [0 0 0 0]`. Note also that z or z2 can be empty.

On output, `gopt` and `h2opt` are the guaranteed H and H2 performances, `K` is the optimal state-feedback gain, `Pcl` the closed-loop transfer function from w to $\left(\begin{array}{c}{z}_{\infty }\\ {z}_{2}\end{array}\right)$, and `X` the corresponding Lyapunov matrix.

The function `msfsyn` is also applicable to multi-model problems where `P` is a polytopic model of the plant:

`$\left\{\begin{array}{c}\stackrel{˙}{x}=A\left(t\right)x+{B}_{1}\left(t\right)w+{B}_{2}\left(t\right)u\\ {z}_{\infty }={C}_{1}\left(t\right)x+{D}_{11}\left(t\right)w+{D}_{12}\left(t\right)u\\ {z}_{2}={C}_{2}\left(t\right)x+{D}_{22}\left(t\right)u\end{array}$`

with time-varying state-space matrices ranging in the polytope

In this context, `msfsyn` seeks a state-feedback gain that robustly enforces the specifications over the entire polytope of plants. Note that polytopic plants should be defined with `psys` and that the closed-loop system `Pcl` is itself polytopic in such problems. Affine parameter-dependent plants are also accepted and automatically converted to polytopic models. 