Observer Luenberger

In Figure 8.8, sinee the observer dynamies will never exaetly equal the system dynamies, this open-loop arrangement means that x and x will gradually diverge. If however, an output veetor y is estimated and subtraeted from the aetual output veetor y, the differenee ean be used, in a elosed-loop sense, to modify the dynamies of the observer so that the output error (y — y) is minimized. This arrangement, sometimes ealled a Luenberger observer (1964), is shown in Figure 8.9. 

An observer for the states 6x,lo) is then constructed as a Luenberger observer of the form... 

Lemma 2. The dynamics of the states zi,ri can be reconstructed from measurements of the output y = z = St by means of the following high-gain Luenberger observer... 

Although there is a close relationship among the various quantitative model-based techniques, observer-based approaches have become very important and diffused, especially within the automatic control community. Luenberger observers [1,45, 53], unknown input observers [44], and Extended Kalman Filters [21] have been mostly used in fault detection and identification for chemical processes and plants. Reviews of several model-based techniques for FD can be found in [8, 13, 35, 50] and, as for the observer-based methods, in [1, 36,44],... 

The literature focused on model-based FD presents a few applications of observers to chemical plants. In [10] an unknown input observer is adopted for a CSTR, while in [7] and [21] an Extended Kalman Filter is used in [9] and [28] Extended Kalman Filters are used for a distillation column and a CSTR, respectively in [45] a generalized Luenberger observer is presented in [24] a geometric approach for a class of nonlinear systems is presented and applied to a polymerization process in [38] a robust observer is used for sensor faults detection and isolation in chemical batch reactors, while in [37] the robust approach is compared with an adaptive observer for actuator fault diagnosis. 

R. Tarantino, F. Szigeti, and E. Colina-Morles. Generalized Luenberger observer-based fault detection filter design An industrial application. Control Engineering Practice, 8 665-671, 2000. 

Extensions of Kalman filters and Luenberger observers [131 Solution polymerizations (conversion and molecular weight estimation) with and without on-line measurements for A4w [102, 113, 133, 134] Emulsion polymerization (monomer concentration in the particles with parameter estimation or not (n)) [45, 139[ Heat of reaction and heat transfer coefficient in polymerization reactors [135, 141, 142] Computationally fast, reiterative and constrained algorithms are more robust, multi-rate (having fast/ frequent and slow measurements can be handled)/Trial and error required for tuning the process and observation model covariance errors, model linearization required The number of industrial applications is scarce A critical article by Wilson eta/. [143] reviews the industrial implementation and shows their experiences at Ciba. Their main conclusion is that the superior performance of state estimation techniques over open-loop observers cannot be guaranteed. 

Modem system theory [43] offers mathematical methods, which—on the basis of these variables together with thermodynamics, chemical kinetics and a reactor model—enable the estimation of additional state variables (e.g. Kalman-Bucy filter, Luenberger observer). The more accurate the process model which is assumed as a base, and the more numerous and meaningful the starting measured variables are, the greater is the number and accuracy of the additional, estimated state variables. Consequently, variables which are usually available (e.g. temperature, pressure) should, if possible, be supplemented by further variables which are more meaningful. 

Luenberger, D.G. (1964) Observing the State of a Linear System, IEEE Trans. Military Electronics, MIL-8, pp. 74-80. 

High-gain Luenberger iike observers or non-iinear state observers [132, 144] Solution polymerizations (concentration and molecular weight) with A4w measurement [145] Emulsion polymerization (conversion, composition and parameter estimation, n [146 147]) Heat of reaction and heat transfer coefficient in polymerization reactors [148-150] Computationally fast, multi-rate schemes also reported, less tuning parameters/Tuning required No industrial applications reported so far... 

Consideration of the equations of a faulty LTI system and of a Luenberger state variable observer reveals that any faults affecting the system have an affect on the observer output error which, after transients have settled, can be used as a fault indicator ([34], Sect. 5.2.2). Assume that thedynamics of a system may be represented by the linear time-invariant state space model... 

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