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

$$\{\begin{array}{c}\dot{x}=Ax+B_{1}w+B_{2}u\\ z_{\infty }=C_{1}x+D_{11}w+D_{12}u\\ z_2=C_{2}x+D_{22}u\end{array}$$

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 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

  $$\alpha {\Vert T_{\infty }\Vert }_{\infty }^2+\beta {\Vert T_2\Vert }_2^2$$

• 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 = [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:

$$\{\begin{array}{c}\dot{x}=A(t)x+B_1(t)w+B_2(t)u\\ z_{\infty }=C_1(t)x+D_{11}(t)w+D_{12}(t)u\\ z_2=C_2(t)x+D_{22}(t)u\end{array}$$

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.

Version History

Introduced before R2006a