Design of optimal observers with specified eigenvalues via shifted Legendre polynomials

Optimal design of reduced-order observers with specified eigenvalues and performance measurement of minimizing estimation errors using evolutionary optimization

“…Although the asymptotic principle can be applied to improve the convergence of the estimated states by choosing the eigenvalues of the observer with a sufficiently negative real part, the design result is not satisfactory due to the fact that a large estimation error may occur during the transient period of observation. [13][14][15] Several previous studies have been devoted to coping with the design issue of reducing the large estimation error occurring during the transient period of observation. [12][13][14][15][16][17] The reduced-order observer reduces the order of the observer by using the sensed outputs.…”

“…[13][14][15] Several previous studies have been devoted to coping with the design issue of reducing the large estimation error occurring during the transient period of observation. [12][13][14][15][16][17] The reduced-order observer reduces the order of the observer by using the sensed outputs. To the authors' best knowledge, to date, only Kung and Yeh 13 as well as Horng and Chou 14,15 have studied the problem of transient estimation performance improvement for "reduced-order observers.…”

“…18,19 One design method requires the dynamical equation in an observable form, while the other design approach does not require so. Based on the dynamical system in an observable form, Kung and Yeh 13 as well as Horng and Chou 14,15 studied the problem of transient estimation performance improvement for the "reduced-order observer of an Observable-Form-Based dynamical system (OFB-reduced-order observer)." However, the methods proposed by Kung and Yeh 13 as well as Horng and Chou 14,15 cannot be applied to deal with the design issue of the "reduced-order observer of a Non-Observable-Form-Based dynamical system (NOFB-reduced-order observer)."…”

“…However, the methods proposed by Kung and Yeh 13 as well as Horng and Chou 14,15 cannot be applied to deal with the design issue of the "reduced-order observer of a Non-Observable-Form-Based dynamical system (NOFB-reduced-order observer)." On the other hand, it merits attention that when the number of outputs is greater than the number of unmeasurable states, the approaches presented by Kung and Yeh 13 as well as Horng and Chou 14,15 cannot be utilized to uniquely determine the observer gain matrix of a linear optimal OFB-reduced-order observer. The reason is that after the eigenvalues are assigned, in the works of Kung and Yeh 13 as well as Horng and Chou, 14,15 the degree of design freedom remaining is not fully utilized.…”

“…On the other hand, it merits attention that when the number of outputs is greater than the number of unmeasurable states, the approaches presented by Kung and Yeh 13 as well as Horng and Chou 14,15 cannot be utilized to uniquely determine the observer gain matrix of a linear optimal OFB-reduced-order observer. The reason is that after the eigenvalues are assigned, in the works of Kung and Yeh 13 as well as Horng and Chou, 14,15 the degree of design freedom remaining is not fully utilized.…”