Nonlinear feedback model

A Nonlinear Feedback Model of the HPA Axis with Circadian Cortisol Peaks... 

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. 

Both competing reductions consume the cofactor nicotinamide adenine dinucleotide (NADH) and thereby interfere with the redox balance of the cell and feedback on glycolysis where NADH is regenerated on the one hand, while on the other hand NAD+ is required to keep the glycolytic pathway running. The nonlinear dynamical model combines the network of glycolysis and the additional pathways of the xenobiotics to predict the asymmetric yield (enantiomeric excess, ee) of L-versus D-carbinol for different environmental conditions (Fig. 3.4). Here, the enantiomeric excess of fluxes vy and i>d is defined as... 

To simulate the disease episodes and subthreshold oscillations we again refer to our neuronal modeling approaches (Fig. 7.2b). The algorithms have been implemented with a simple but physiologically plausible approach, i.e. with two nonlinear feedback loops, one positive and one negative. Depending on the parameter setting, such a system can attain stable dynamics but also can develop oscillations. 

The present paper (just as previous ones, references 2-6) considers the thermal mechanism as being responsible for the formation of DSs. In other words, the factor of nonlinearity is here the exponential dependence of the reaction heat generation intensity on temperature, which is the commonest in chemistry, and the concentration-velocity relation corresponds to the linear case of a first-order reaction. Consideration of the chemically simplest case aims at forming a basis of the theory of DS in heterogeneous catalysis and its further development by consistently complicating the kinetic law of a reaction and introducing into the model nonlinearities (feedbacks) of both... 

With regard to the role of nonlinear feedback control, periodic behaviour was illustrated in fig. 11.7 for a repression function characterized by a Hill coefficient of 4 however, a value of 2 or 1 for that cooperativity coefficient can also give rise to sustained oscillations. The cooperativity of repression provides a major source of nonlinearity required for sustained oscillatory behaviour. This is the reason why steep thresholds due to zero-order ultrasensitivity are not required here to generate limit cycle oscillations (see the relatively large values of the reduced Michaelis constants K, used in fig. 11.7), in contrast to the situation encountered for the phosphorylation-dephosphorylation cascade model analysed for the mitotic oscillator. 

This section discusses strategies for including Q—parametrization within an optimization framework of the type described earlier. Since Q is infinite dimensional, a useful first step is to approximate Q using a finite number of parameters. Both continuous and discrete-time approximations will be shown. The integration of the approximation into the optimization framework can be done in different ways. Two different approaches for doing this will be described - direct inclusion of the closed-loop transfer function, and inclusion of the individual components of the feedback system with appropriate interconnections between them. The latter approach admits nonlinear plant models. 

In view of the dramatic decreases in the ratio of computer cost to performance in recent years, it can be argued that physically based, nonlinear process models should be used in the set-point calculations, instead of approximate linear models. However, linear models are still widely used in MPC applications for three reasons First, linear models are reasonably accurate for small changes in u and d and can easily be updated based on current data or a physically based model. Second, some model inaccuracy can be tolerated, because the calculations are repeated on a frequent basis and they include output feedback from the measurements. Third, the computational effort required for constrained, nonlinear optimization is still relatively large, but is decreasing. 

Distinctive kernel relationships (e.g., between first-order and second-order kernels) exist for each type of cascade model, which can be used for validation of the chosen modular model on the basis of the kernel estimates obtained from the data—a task that is often referred to as structural identification. The combined use of modular and nonparametric approaches may yield considerable benefits. This idea can be extended to more complicated modular structures entailing multiple parallel and cascaded branches (Korenberg 1991 Marmarelis 1997,2004). The case of modular models of nonlinear feedback systems attracts considerable interest, because of its critical role in physiologic control and autoregulation (MarmareUs 1991,2004). Mention should also be made of a rather complex model of parallel L-N-L cascades (usually called the model) that covers abroad class of nonlinear systems (Rugh 1981). 

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... 

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. 

In this section we consider a model of interactions between the Kerr oscillators applied by J. Fiurasek et al. [139] and Perinova and Karska [140]. Each Kerr oscillator is externally pumped and damped. If the Kerr nonlinearity is turned off, the system is linear. This enables us to perform a simple comparison of the linear and nonlinear dynamics of the system, and we have found a specific nonlinear version of linear filtering. We study numerically the possibility of synchronization of chaotic signals generated by the Kerr oscillators by employing different feedback methods. 

The most general approach to model-based nonlinear control is the so-called Feedback Linearization (FL) [35], In fact, FL control approaches use the model of the plant to achieve a global linearization of the closed-loop systems, so as well-established linear controllers can be adopted for the globally linearized model. In... 

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. 

In order to briefly introduce the basic principle of the feedback linearizing control, consider the following Single Input Single Output (SISO) nonlinear model ... 

---