Model-Based Nonlinear Observer

Therefore, an estimate of aq can be easily computed via (5.22) from the estimates of the reactants concentrations 

The convergence properties of both the state estimation error x = x — x and the parameter estimation error 60 = 9 - 60 are stated in the following theorem. 

Theorem 5.1 If the rate constants are bounded as in (2.32) and (2.33), then, there exists a set of observer gains such that the state estimation error x is globally uniformly convergent lo 0 as / - oc and the parameter estimation error 90 is bounded for every t. 

The proof is based on a Lyapunov-like argument and is reported in Appendix A. 1. 

Remark 5.1 As usual, in direct adaptive estimation and/or control schemes, the convergence to zero of the parameter estimation error 0O is not guaranteed, unless the persistency of excitation condition is fulfilled [5, 35], In detail, if there exist three scalars A.i 0, A.2 0, and r 0 such that 

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In order to tackle the problem of uncertainties in the available model, nonlinear robust and adaptive strategies have been developed, while, in the absence of full state measurements, output-feedback control schemes can be adopted, where the unmeasurable state variables can be estimated by resorting to state observers. The development of model-based nonlinear strategies has been fostered by the development of efficient experimental identification methods for nonlinear models and by significantly improved capabilities of computer-control hardware and software. 

The first approach adopts a nonlinear model-based adaptive observer to estimate the reactant concentrations (i.e., the state variables x, ..., xjvc), while the heat transfer coefficient is estimated via an adaptive update law. Then, the term aq is reconstructed from the estimated concentrations. 

When an accurate model of the reaction kinetics is not available (e.g., due to the lack of reliable data for identification), the previously developed approach may be ineffective and model-free strategies for the estimation of the effect of the heat released by the reaction, aq, must be adopted. In detail, the approach in [27] can be considered, where aq is estimated, together with the heat transfer coefficient, via a suitably designed nonlinear observer [24], Other model-free approaches can be adopted, e.g., those based on the adoption of universal interpolators (neural networks, polynomials) for the direct online estimation of the heat [16] and purely neural approaches [11], Approaches based on the combination of neural and model-based paradigms [2] or on tendency models [25] can be considered as well. 

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. 

Magnetic resonance imaging permitted direct observation of the liquid hold-up in monolith channels in a noninvasive manner. As shown in Fig. 8.14, the film thickness - and therefore the wetting of the channel wall and the liquid hold-up -increase nonlinearly with the flow rate. This is in agreement with a hydrodynamic model, based on the Navier-Stokes equations for laminar flow and full-slip assumption at the gas-liquid interface. Even at superficial velocities of 4 cm s-1, the liquid occupies not more than 15 % of the free channel cross-sectional area. This relates to about 10 % of the total reactor volume. Van Baten, Ellenberger and Krishna [21] measured the liquid hold-up of katapak-S . Due to the capillary forces, the liquid almost completely fills the volume between the catalyst particles in the tea bags (about 20 % of the total reactor volume) even at liquid flow rates of 0.2 cm s-1 (Fig. 8.15). The formation of films and rivulets in the open channels of the structure cause the further slight increase of the hold-up. 

In these studies the rate of the mass and contact diameter of water and -octane drops placed on glass and Teflon surfaces were investigated. It was found that the evaporation occurred with a constant spherical cap geometry of the liquid drop. The experimental data supporting this were obtained by direct measurement of the variation of the mass of droplets with time, as well as by the observation of contact angles. A model based an the diffusion of vapor across the boundary of a spherical drop has been considered to explain the data. Further studies were reported, where the contact angle of the system was 9 < 99°. In these systems, the evaporation rates were found to be linear and the contact radius constant. In the latter case, with 9 > 99°, the evaporation rate was nonlinear, the contact radius decreased and the contact angle remained constant. 

Nonlinear waves are very useful for a qualitative understanding of the concentration and temperature dynamics in an RD column. So far, only an incomplete understanding of the relation between the physicochemical complexity of the mixture, the design and operation of the column, and the observed spatiotemporal patterns is available. Much research is required to resolve the open issues. In addition, the phenomenon of a propagating wave can also be exploited to derive a simplified quantitative description of the column dynamics in the non-reactive [27, 67] as well as in the reactive [4, 51, 52] case. These reduced nonlinear models are most suitable to design and implement advanced model based control systems as discussed in the next section. 

Fig. 10.30 gives a comparison between open loop dynamics and closed loop dynamics for linear multi-input multi-output PI controllers and an advanced nonlinear controller. The nonlinear controller is based on asymptotic etsact input/out-put linearization as proposed by Gilles and coworkers [29]. This is a model based controller, where the concentration profiles inside the column are reconstructed online from the two temperature measurements by means of a nonlinear state observer [21, 33]. 

The observation of non-ohmic conductivity, below the temperature which corresponds to the onset of three-dimensional SOW order, strongly suggests that the nonlinearity is associated with the appearance of SDW. Indeed, it is difficult to explain the low threshold fields, at which deviations from Ohm s law are observed by models based on a single-particle picture. First, we note that the threshold fields are far too low, for Zener breakdown to be relevant. 

Taking into account that the target variable, the moisture content of the product, is not available for direct measurement, a NN based state observer is proposed for its estimation. The data provided by the NN state observer is used for feedback nonlinear model predictive control of the product moisture content, as shown in figure 4. 

Model based control schemes such as model predictive control are highly related to the accuracy of the process model. A regional-knowledge index is proposed in this study and applied in the analysis of dynamic artificial neural network models in process control. To tackle the extrapolation problem and assure stability of the control system, we propose to run a neural adaptive controller in parallel with a model predictive control. A coordinator weights the outputs of these two controllers to make the final control decision. The proposed analysis method and the modified model predictive control architecture have been applied to a neutralization process and excellent control performance is observed in this highly nonlinear system. 

This controller, with high gain setpoint observer, represents the limiting behavior attainable with any exact model-based cascade nonlinear SF controller. 

Tobushi et al. (2001) presented an empirical model based on the observation of the thermo-mechanical behavior of shape memoiy polymers (SMPs). In the model, the modulus of the SMPs was considered to linearly decrease with the temperature increase in a narrow region around T (T - T < T < T + T ). The stress vs strain profiles in shape memorization were predicted based on a nonlinear visco-elastic equation. This model mainly took account of the modulus variation above and below T, but noted no further microscopic changes in the shape memorization (Tobushi et al., 2001). 

In Chapter 4, it was noted that linear viscoelastic behavior is observed only in deformations that are very small or very slow. The response of a polymer to large, rapid deformations is nonlinear, which means that the stress depends on the magnitude, the rate and the kinematics of the deformation. Thus, the Boltzmann superposition principle is no longer valid, and nonlinear viscoelastic behavior cannot be predicted from linear properties. There exists no general model, i.e., no universal constitutive equation or rheological equation of state that describes all nonlinear behavior. The constitutive equations that have been developed are of two basic types empirical continuum models, and those based on a molecular theory. We will briefly describe several examples of each type in this chapter, but since our primary objective is to relate rheological behavior to molecular structure, we will be most interested in models based on molecular phenomena. The most successful molecular models to date are those based on the concept of a molecule in a tube, which was introduced in Chapter 6. We therefore begin this chapter with a brief exposition of how nonlinear phenomena are represented in tube models. A much more complete discussion of these models will be provided in Chapter 11. 

Thus, models based on multisite binding equilibria successfully describe the dependence of D on Cx at a fixed micelle concentration when the electroactive probe is almost totally bound to the micelles. The electrode reaction of the probe must not be accompanied by adsorption or chemical reaction. Nonlinear regression analysis of the data enables an assessment of the importance of polydispersity and an accurate estimate of diffusion coefficients of the micelles. Parameters proportional to binding constants are also obtained, and these can be converted to apparent binding constants if the micelle concentrations are known. The models also quantitatively predict the observed z decrease in measured diffusion coefficient with concentration of micelles. This work shows that the micellar system we have used for electrocatalysis are probably polydisperse. 

The determination of piezoelectric constants from current pulses is based on interpretation of wave shapes in the weak-coupling approximation. It is of interest to use the wave shapes to evaluate the degree of approximation involved in the various models of piezoelectric response. Such an evaluation is shown in Fig. 4.5, in which normalized current-time wave forms calculated from various models are shown for x-cut quartz and z-cut lithium niobate. In both cases the differences between the fully coupled and weakly coupled solutions are observed to be about 1%, which is within the accuracy limits of the calculations. Hence, for both quartz and lithium niobate, weakly coupled solutions appear adequate for interpretation of observed current-time waveforms. On the other hand, the adequacy of the uncoupled solution is significantly different for the two materials. For x-cut quartz the maximum error of about 1%-1.5% for the nonlinear-uncoupled solution is suitable for all but the most precise interpretation. For z-cut lithium niobate the maximum error of about 8% for the nonlinear-uncoupled solution is greater than that considered acceptable for most cases. The linear-uncoupled solution is seriously in error in each case as it neglects both strain and coupling. 

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