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

${\begin{matrix} \dot{x} = A x + B_{1} w + B_{2} u \\ z_{\infty} = C_{1} x + D_{11} w + D_{12} u \\ z_{2} = C_{2} x + D_{22} u \end{matrix}$

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_∞ tradeoff criterion of the form
$α {‖ T_{\infty} ‖}_{\infty}^{2} + β {‖ T_{2} ‖}_{2}^{2}$
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 $(\begin{matrix} z_{\infty} \\ z_{2} \end{matrix})$ , 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:

${\begin{matrix} \dot{x} = A (t) x + B_{1} (t) w + B_{2} (t) u \\ z_{\infty} = C_{1} (t) x + D_{11} (t) w + D_{12} (t) u \\ z_{2} = C_{2} (t) x + D_{22} (t) u \end{matrix}$

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

$(\begin{matrix} A (t) & B_{1} (t) & B_{2} (t) \\ C_{1} (t) & D_{11} (t) & D_{12} (t) \\ C_{2} (t) & 0 & D_{22} (t) \end{matrix}) \in Co {(\begin{matrix} A_{k} & B_{k} & C_{k} \\ C_{1 k} & D_{11 k} & D_{12 k} \\ C_{2 k} & 0 & D_{22 k} \end{matrix}) : k = 1, ..., K}$

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.

Version History

Introduced before R2006a

msfsyn

Syntax

Description

Version History

See Also