## A Modified Functional Observer-Based EID Estimator for Unknown Continuous-Time Singular Systems  
[PERSON] 1, [PERSON] 1, [PERSON] 2,3,*, [PERSON] 1, [PERSON] 4, [PERSON] 5 and [PERSON] 6  
Department of Electrical Engineering, National Cheng-Kung University, Tainan 701, Taiwan; [EMAIL_ADDRESS] (J.S.-H.T.); [EMAIL_ADDRESS] (Y.-F.C.); [EMAIL_ADDRESS] (C.-Y.C.) Department of Electronic Engineering, National Kaohsiung University of Science and Technology, Kaohsiung 800-852, Taiwan  
Graduate Institute of Clinical Medicine, Kaohsiung Medical University, Kaohsiung 800-852, Taiwan  
Department of Computer Science and Information Engineering, National Cheng-Kung University, Tainan 701, Taiwan; [EMAIL_ADDRESS] Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204-4005, USA; [EMAIL_ADDRESS] Electrical Engineering School, Universidad del Zulia, Maracaibo 4005, Venezuela; [EMAIL_ADDRESS] Correspondence: [EMAIL_ADDRESS]  
Received: 26 February 2020; Accepted: 24 March 2020; Published: 28 March 2020  
This paper presents the design of a linear quadratic analog tracker (LQAT) based on the observer-Kalman-filter identification (OKID) method and the design of a modified functional observer-based equivalent input disturbance (EID) estimator for unknown square-non-square singular analog systems with unknown input and output disturbances. First, an equivalent mathematical model of the singular analog system is presented to simulate the time response of continuous-time linear singular analog systems to arbitrary inputs via the model conversion method. Then, for the unknown singular analog system, it constructs a linear quadratic analog tracker with state feedback and feed-forward gains based on the off-line OKID method. Furthermore, it extends the design methodology of the EID estimator for strictly proper regular systems with unknown matched-mismatched input and output disturbances to proper regular systems. It is important to mention that the newly developed modified functional observer for proper systems is used to estimate the unknown EID of singular analog systems and that the constraints on the dimensions of unknown disturbances can be eliminated by using the newly proposed EID estimation method. The contributions of this paper can be listed as follows: (1) based on both the OKID method and the discrete-to-continuous model conversion, the simulation of the time responses of the continuous-time linear singular models (which are not feasible using existing MATLAB toolboxes) become feasible; (2) for effective control of the unknown singular analog system, an off-line OKID method is proposed to design an LQAT with state feedback and feed-forward gains; and (3) based on the newly developed modified functional observer for the reduced-order proper regular system, the original EID estimator in the literature is newly extended to estimate the EID from the unknown strictly proper singular analog system, without the original dimensional constraints of the disturbances. It is important to mention that the disturbances of interest can be unknown matched-mismatched input and output disturbances.  
singular systems; generalized Riccati equation; input-output direct feed-through term; functional observer; observer-Kalman-filter identification; equivalent input disturbance (EID); disturbance estimator