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Multi-model/multi-objective state-feedback synthesis

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

Given an LTI plant `P`

with state-space equations

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

`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*γ*_{0}> 0Maintains the

*H*_{2}norm of the closed-loop transfer function*T*_{2}from*w*to*z*_{2}below some prescribed value υ_{0}> 0Minimizes an

*H*_{2}/*H*_{∞}trade-off criterion of the form$$\alpha {\Vert {T}_{\infty}\Vert}_{\infty}^{2}+\beta {\Vert {T}_{2}\Vert}_{2}^{2}$$

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 =`

[*γ*_{0}, ν_{0}, α, β] to specify the problem dimensions and the design parameters *γ*_{0}, ν_{0}, α, and β. You can perform pure pole placement by setting `obj = [0 0 0 0]`

. Note also that *z*_{∞} or *z*_{2} can be empty.

On output, `gopt`

and `h2opt`

are the guaranteed *H*_{∞} and *H*_{2} 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:

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

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

$$\left(\begin{array}{ccc}A(t)& {B}_{1}(t)& {B}_{2}(t)\\ {C}_{1}(t)& {D}_{11}(t)& {D}_{12}(t)\\ {C}_{2}(t)& 0& {D}_{22}(t)\end{array}\right)\in \text{Co}\left\{\left(\begin{array}{ccc}{A}_{k}& {B}_{k}& {C}_{k}\\ {C}_{1k}& {D}_{11k}& {D}_{12k}\\ {C}_{2k}& 0& {D}_{22k}\end{array}\right):k=1,\mathrm{...},K\right\}$$

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.