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LQG loop transfer-function recovery (LTR) control synthesis

[K,SVL,W1] = ltrsyn(G,F,XI,THETA,RHO) [K,SVL,W1] = ltrsyn(G,F,XI,THETA,RHO,W) [K,SVL,W1] = ltrsyn(G,F,XI,THETA,RHO,OPT) [K,SVL,W1] = ltrsyn(G,F,XI,THETA,RHO,W,OPT)

`[K,SVL,W1] = ltrsyn(G,F,XI,TH,RHO)`
computes a reconstructed-state output-feedback controller K for LTI
plant `G` so that `K*G` asymptotically
recovers plant-input full-state feedback loop transfer function *L*(*s*)
= *F(Is–A)*^{–1}*B+D;* that
is, at any frequency `w>0,` `max(sigma(K*G-L,
w))`→0 as ρ→ ∞, where` L`= `ss(A,B,F,D)` is
the LTI full-state feedback loop transfer function.

`[K,SVL,W1] = ltrsyn(G,F1,Q,R,RHO,'OUTPUT')`
computes the solution to the `dual' problem of filter loop recovery
for LTI plant `G` where `F` is a
Kalman filter gain matrix. In this case, the recovery is at the plant
output, and `max(sigma(G*K-L, w))`→0 as ρ→∞,
where `L1` denotes the LTI filter loop feedback loop
transfer function `L1`= `ss(A,F,C,D)`.

Only the LTI controller` K` for the final value `RHO(end)`is
returned.

s=tf('s');G=ss(1e4/((s+1)*(s+10)*(s+100)));[A,B,C,D]=ssdata(G); F=lqr(A,B,C'*C,eye(size(B,2))); L=ss(A,B,F,0*F*B); XI=100*C'*C; THETA=eye(size(C,1)); RHO=[1e3,1e6,1e9,1e12];W=logspace(-2,2); nyquist(L,'k-.');hold; [K,SVL,W1]=ltrsyn(G,F,XI,THETA,RHO,W);

See also `ltrdemo`

The `ltrsyn` procedure may fail for non-minimum
phase plants. For full-state LTR (default `OPT='`* INPUT*'),
the plant should not have fewer outputs than inputs. Conversely for
filter LTR (when

[1] Doyle, J., and G. Stein, "Multivariable
Feedback Design: Concepts for a Classical/Modern Synthesis," *IEEE
Trans. on Automat. Contr*., AC-26, pp. 4-16, 1981.

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