[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.

Inputs

G

LTI plant

F

LQ full-state-feedback gain matrix

XI

plant noise intensity,

or, if OPT='OUTPUT' state-cost
matrix XI=Q,

THETA

sensor noise intensity

or, if OPT='OUTPUT' control-cost
matrix THETA=R,

RHO

vector containing a set of recovery gains

W

(optional) vector of frequencies (to be used for plots);
if input W is not supplied, then a reasonable default
is used

Outputs

K

K(s) — LTI
LTR (loop-transfer-recovery) output-feedback, for the last element
of RHO (i.e., RHO(end))

SVL

sigma plot data for the recovered loop transfer function
if G is MIMO or, for SISO G only,
Nyquist loci SVL = [re(1:nr) im(1:nr)]

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 OPT='OUTPUT'),
the plant should not have fewer inputs than outputs. The plant must
be strictly proper, i.e., the D-matrix of the plant
should be all zeros. ltrsyn is only for continuous
time plants (Ts==0)

where K_{c} = F and K_{f} = lqr(A',C',XI+RHO(i)*B*B',THETA).
The "fictitious noise" term RHO(i)*B*B' results
in loop-transfer recovery as RHO(i) → ∞.
The Kalman filter gain is $${K}_{f}=\sum {C}^{T}{\Theta}^{-1}$$ where
Σ satisfies the Kalman filter Riccati equation $$0=\sum {A}^{T}+A\sum -\sum {C}^{T}{\Theta}^{-1}C\sum +\Xi +\rho B{B}^{T}$$. See [1] for
further details.

Similarly for the 'dual' problem of filter loop recovery case, [K,SVL,W1]
= ltrsyn(G,F,Q,R,RHO,'OUTPUT') computes a filter loop recovery
controller of the same form, but with K_{f} = F is
being the input filter gain matrix and the control gain matrix K_{c} computed
as K_{c} = lqr(A,B,Q+RHO(i)*C'*C,R).

Example of LQG/LTR at Plant Output.

References

[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.