Nonlinear feedback

Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

Nonlinearity of a system also arises from the interaction of an element being a product as well as a controlling factor of any one of the reaction steps preceding this product. If a nonlinearity results, this type of nonlinearity is called a feedback nonlinearity. 

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. 

Another important reaction supporting nonlinear behaviour is the so-called FIS system, which involves a modification of the iodate-sulfite (Landolt) system by addition of ferrocyanide ion. The Landolt system alone supports bistability in a CSTR the addition of an extra feedback chaimel leads to an oscillatory system in a flow reactor. (This is a general and powerfiil technique, exploiting a feature known as the cross-shaped diagram , that has led to the design of the majority of known solution-phase oscillatory systems in flow... 

The term collectivism has sometimes been used to distinguish this AL philosophy from the more traditional top down and bottom up philosophies. Collectivism embodies the belief that in order to properly understand complex systems, such systems must be viewed as coherent wholes whose open-ended evolution is continuously fueled by nonlinear feedback between their macroscopic states and microscopic constituents. It is neither completely reductionist (which seeks only to decompose a system into its primitive components), nor completely synthesist (which seeks to synthesize the system out of its constituent parts but neglects the feedback between emerging levels). 

It is this nonlinear feedback between the information describing individual species (or the system s microscopic level) and the global ecology (or the system s macro-... 

Collectivism is thus distinct from both the top-down reductionist approach traditionally favored by most physicists (system as a simple edifice of its microscopic parts), and the more recent neural-net-like bottom-up approach favored by connec-tionists (system as a synthesis of its constituent parts). The nonlinear inter-level feedback loop that makes up the collective is what makes a traditional linear analysis of such systems difficult, if not impossible. 

A control algorithm has been derived that has improved the dynamic decoupling of the two outputs MW and S while maintaining a minimum "cost of operation" at the steady state. This algorithm combines precompensation on the flow rate to the reactor with state variable feedback to improve the overall speed of response. Although based on the linearized model, the algorithm has been demonstrated to work well for the nonlinear reactor model. 

The ar tide is organized as follows. We will begin with a discussion of the various possibilities of dynamical description, clarify what is meant by nonlinear quantum dynamics , discuss its connection to nonlinear classical dynamics, and then study two experimentally relevant examples of quantum nonlinearity - (i) the existence of chaos in quantum dynamical systems far from the classical regime, and (ii) real-time quantum feedback control. 

To illustrate an application of nonlinear quantum dynamics, we now consider real-time control of quantum dynamical systems. Feedback control is essential for the operation of complex engineered systems, such as aircraft and industrial plants. As active manipulation and engineering of quantum systems becomes routine, quantum feedback control is expected to play a key role in applications such as precision measurement and quantum information processing. The primary difference between the quantum and classical situations, aside from dynamical differences, is the active nature of quantum measurements. As an example, in classical theory the more information one extracts from a system, the better one is potentially able to control it, but, due to backaction, this no longer holds true quantum mechanically. 

In a nonlinear system the addition of a feedforward controller often permits tighter tuning of the feedback controller because the magnitude of the dis turbances that the feedback controller must cope with is reduced. 

It should be noted that Eqs. (14.38) and (14.39) give approximate values for 0) and X. because the relay feedback introduces a nonlinearity into the system. However, for most systems, the approximation is close enough for engineering purposes. 

State estimators are basically just mathematical models of the system that are solved on-line. These models usually assume linear DDEs, but nonlinear equations can be incorporated. The actual measured inputs to the process (manipulated variables) are fed into the model equations, and the model equations are integrated. Then the available measured output variables are compared with the predictions of the model. The differences between the actual measured output variables and the predictions of the model for these same variables are used to change the model estimates through some sort of feedback. As these differences between the predicted and measured variables are driven to zero, the model predictions of all the state variables are changed. 

There are several control problems in chemical reactors. One of the most commonly studied is the temperature stabilization in exothermic monomolec-ular irreversible reaction A B in a cooled continuous-stirred tank reactor, CSTR. Main theoretical questions in control of chemical reactors address the design of control functions such that, for instance (i) feedback compensates the nonlinear nature of the chemical process to induce linear stable behavior (ii) stabilization is attained in spite of constrains in input control (e.g., bounded control or anti-reset windup) (iii) temperature is regulated in spite of uncertain kinetic model (parametric or kinetics type) or (iv) stabilization is achieved in presence of recycle streams. In addition, reactor stabilization should be achieved for set of physically realizable initial conditions, (i.e., global... 

Here, a control law for chemical reactors had been proposed. The controller was designed from compensation/estimation of the heat reaction in exothermic reactor. In particular, the paper is focused on the isoparafhn/olefin alkylation in STRATCO reactors. It should be noted that control design from heat compensation leads to controllers with same structure than nonlinear feedback. This fact can allow to exploit formal mathematical tools from nonlinear control theory. Moreover, the estimation scheme yields in a linear controller. Thus, the interpretation for heat compensation/estimation is simple in the context of process control. 

The following example illustrate the calculations involved in the construction of an error feedback regulator for nonlinear systems. 

D.D. Bruns and J.E. Bailey. Process operation near an unstable steady state using nonlinear feedback control. Chem. Eng. Sci., 30 755-762, 1975. 

S. Monaco and D. Normand-Cyrot. Minimum phase nonlinear discrete-time systems and feedback stabilization. In IEEE Conf. Decision and Control (CDC), pages 979-986, Los Angeles, USA, 2002. 

K. Otawara and L.T. Fan. Increasing the yield from a chemical reactor with spontaneously oscillatory chemical reactions by a nonlinear feedback mechanism. Comput. Chem. Eng., 25 333-335, 2001. 

W.T. Baumaim and W.J. Rugh. Feedback control of nonlinear systems byex-tended linearization. IEEE Trans. Automat. Contr., 31(1) -, 1986. 

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