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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*(

`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.

| |
---|---|

`G` | LTI plant |

`F` | LQ full-state-feedback gain matrix |

`XI` | plant noise intensity, or, if |

`THETA` | sensor noise intensity or, if |

`RHO` | vector containing a set of recovery gains |

`W` | (optional) vector of frequencies (to be used for plots);
if input |

| |
---|---|

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

`SVL` | sigma plot data for the recovered loop transfer function
if |

`W1` | frequencies for SVL plots, same as |

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

`OPT='`

`OUTPUT`

`ltrsyn`

is only for continuous
time plants (`Ts==0`

)For each value in the vector `RHO`

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

computes the full-state-feedback
(default `OPT='`

`INPUT`

`'`

)
LTR controller

$$K(s)=\left[{K}_{c}{(Is-A+B{K}_{c}+{K}_{f}C-{K}_{f}D{K}_{c})}^{-1}{K}_{f}\right]$$

where * K_{c}* =

`F`

and `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 `lqr(A,B,Q+RHO(i)*C'*C,R)`

.**Example of LQG/LTR at Plant Output.**

[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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