Controllo lineare quadratico gaussiano

Il controllo lineare quadratico gaussiano (Linear Quadratic Gaussian, LQG) è un compensatore dinamico ottimo capace di recuperare la stessa funzione di trasferimento di un sistema di controllo osservabile per un sistema non osservabile. Tale sistema di controllo ottimo è basato su un controllore ottimo e un filtro ottimo, il primo sintetizzato tramite regolatore lineare quadratico (LQR) il secondo tramite loop transfer recovery (LTR).

Il problema

Con riferimento al controllo LQR si può avere una robustezza intrinseca forte, che garantisce una specifica di prestazione sulla sensibilità del controllo che scende di 20db/dec in alta frequenza. Ciò vale se il sistema è osservabile, cioè se lo stato del sistema è tutto fornito in uscita: in tal caso la matrice di trasferimento del sistema vista tagliando tra il regolatore e il processo risulta:

 $\mathbf {U_{0}=K_{opt}} (j\omega \mathbf {I-A)^{-1}B}$

dove j rappresenta l'unità immaginaria, ω è la pulsazione del sistema, I è la matrice identica, il -1 denota l'inversione di matrice, A e B sono le matrici che descrivono il sistema dinamico lineare stazionario e Kopt è la matrice ricavata dall'algoritmo del controllo LQR.

Se invece lo stato è solo rilevabile allora occorre inserire un osservatore dello stato:

 ${\frac {d\mathbf {\tilde {x}} }{dt}}=\mathbf {(A+VC+BK_{opt}){\tilde {x}}+Vy\,}$

dove $\mathbf {\tilde {x}}$ è lo stato stimato, y il vettore delle uscite e C la matrice che lega y allo stato. Quindi si modifica la funzione di trasferimento divenendo:

 $\mathbf {U_{0}=K_{opt}} (j\omega \mathbf {I-A-VC-BK_{opt})^{-1}VC} (j\omega \mathbf {I-A)^{-1}B\,}$

sfruttando LTR si riesce ad ottenere la stessa prestazione del caso osservabile, recuperando così la robustezza intrinseca garantita per il sistema osservabile controllato in LQR.

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