A method of robust multi-rate state estimation Neeraj Zambare a , Masoud Soroush a, *, Babatunde A. Ogunnaike b a Department of Chemical Engineering, Drexel University, Philadelphia, PA 19104, USA b DuPont Central Science and Engineering, Experimental Station, E1/104, Wilmington, DE 19880, USA Received 4 March 2001; received in revised form 18 December 2001; accepted 16 May 2002 Abstract The inadequacy of the standard notions of detectability and observability to ascertain robust state estimation is shown. The notion of robust state estimation is defined, and for a class of processes the conditions under which the robust state estimation is possible, are given. A method of robust, nonlinear, multi-rate, state estimator design is presented. It can be used to improve robustness in an existing estimator or design a new robust estimator. Estimator tuning guidelines that ensure the asymptotic sta- bility of the estimator error dynamics are given. To ensure that estimation error does not exceed a desired limit, the sampling period of infrequent measurements should be less than an upper bound that depends on factors such as the size of the process dominant time constant, the magnitude of measurement noise, and the level of process–model mismatch. An expression that can be used to calculate the upper bound on the sampling period of infrequent measurements, is presented. The upper bound is the latest time at which the next infrequent measurements should arrive to ensure that estimation error does not exceed a desired limit. The expres- sion also allows one to calculate the highest quality of estimation achievable in a given process. A binary distillation flash tank and a free-radical polymerization reactor are considered to show the application and performance of the estimator. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Robust state estimation; Multi-rate state estimation; Integral observability; Integral detectability; Maximum sampling period; Binary distillation flash tank; Polymerization reactor 1. Introduction Effective control and monitoring of a process require sufficient frequent information on the essential variables of the process. In many processes, however, the essential variables are not measured or are measured infre- quently, and a state estimator can be used to obtain frequent estimates of the variables. In particular, in processes where measurements are available at different frequencies, multi-rate estimators can provide frequent estimates of the variables. Infrequent measurements are usually related to key process variables, and thus their use in estimation leads to more reliable estimates (espe- cially in the presence of model–plant mismatch and measurement noise) [2,29,31]. Since the 1960s state estimators have been used to estimate state variables from readily available measure- ments [2,6,7–9,12,13,15,16,19–24,28,29,31]. The need for multi-rate estimators has been well recognized in the process industry [2,7,12,16,20,22,24,29,31]. Most of the earlier studies have focused on extended Kalman Filters (EKFs). For example, Ellis and coworkers {6,7] used a multi-rate EKF to estimate unmeasurable process states continuously from frequent measurements of tempera- ture and density and infrequent and delayed measure- ments of average molecular weights. Gudi et al. [12] implemented a two time-scale EKF on a biochemical reactor. Yet another estimator based on EKF by Myers et al. [21] utilized noisy measurements of oxygen con- centration and oxygen uptake rate with infrequent and time-delayed measurements of biomass and substrate concentrations to estimate the state of a semi-batch biochemical reactor system. Mutha et al. [19] presented a fixed-lag smoothing-based extended Kalman-filter algorithm. They applied it to a complex, emulsion copolymerization reactor and showed estimation robustness against state and measurement noise. This multirate estimation algorithm was shown to be super- ior to the basic extended Kalman filter. They also showed the performance of the estimator by applying it 0959-1524/03/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0959-1524(02)00027-6 Journal of Process Control 13 (2003) 337–355 www.elsevier.com/locate/jprocont * Corresponding author. Tel.: +215-895-1710; fax: +215-895- 5837. E-mail address: masoud.soroush@coe.derxel.edu (M. Soroush)