Feedback steady state systems

A mechanical system, typified by a pendulum, can oscillate around a position of final equilibrium. Chemical systems cannot do so, because of the fundamental law of thermodynamics that at all times AG > 0 when the system is not at equilibrium. There is nonetheless the occasional chemical system in which intermediates oscillate in concentration during the course of the reaction. Products, too, are formed at oscillating rates. This striking phenomenon of oscillatory behavior can be shown to occur when there are dual sets of solutions to the steady-state equations. The full mathematical treatment of this phenomenon and of instability will not be given, but a simplified version will be presented. With two sets of steady-state concentrations for the intermediates, no sooner is one set established than the consequent other changes cause the system to pass quickly to the other set, and vice versa. In effect, this establishes a chemical feedback loop. 

There is nothing in Equations 1-8 which is an all-or-none situation. There are no positive feedback loops which might cause some kind of flip-flop of states of operation of the system. There are some possibilities for saturation phenomena but all relationships are graded. Overall, transient or steady-state, the changes of concentration of P-myosin are continuous, monotonic functions of the intracellular Ca ion concentration. On this basis it is more appropriate to say that smooth muscle contraction is modulated rather than triggered by Ca ion. 

Example 14.1 shows how an isothermal CSTR with first-order reaction responds to an abrupt change in inlet concentration. The outlet concentration moves from an initial steady state to a final steady state in a gradual fashion. If the inlet concentration is returned to its original value, the outlet concentration returns to its original value. If the time period for an input disturbance is small, the outlet response is small. The magnitude of the outlet disturbance will never be larger than the magnitude of the inlet disturbance. The system is stable. Indeed, it is open-loop stable, which means that steady-state operation can be achieved without resort to a feedback control system. This is the usual but not inevitable case for isothermal reactors. 

Do the time response simulation in Example 7.5B. We found that the state space system has a steady state error. Implement integral control and find the new state feedback gain vector. Perform a time response simulation to confirm the result. 

In Chapter 3 the steady-state hydrodynamic aspects of two-phase flow were discussed and reference was made to their potential for instabilities. The instability of a system may be either static or dynamic. A flow is subject to a static instability if, when the flow conditions change by a small step from the original steady-state ones, another steady state is not possible in the vicinity of the original state. The cause of the phenomenon lies in the steady-state laws hence, the threshold of the instability can be predicted only by using steady-state laws. A static instability can lead either to a different steady-state condition or to a periodic behavior (Boure et al., 1973). A flow is subject to a dynamic instability when the inertia and other feedback effects have an essential part in the process. The system behaves like a servomechanism, and knowledge of the steady-state laws is not sufficient even for the threshold prediction. The steady-state may be a solution of the equations of the system, but is not the only solution. The above-mentioned fluctuations in a steady flow may be sufficient to start the instability. Three conditions are required for a system to possess a potential for oscillating instabilities ... 

In summary, the stationary-states at C3 and C3 are stable, and the state at Q is unstable, and cannot be achieved in steady-state operation. It can, however, be stabilized by a control system involving a feedback loop. For a simple system, as considered here, there... 

Based on a previously constructed model, two crucial questions with respect to dynamic properties were investigated [296] (i) What is the role of feedback regulation in maintaining the stability of the steady state. In particular, does the feedback regulation contribute to a stabilization of the in vivo metabolic state (ii) Is it possible to distinguish between different metabolic states That is, does knowledge of the metabolic state, characterized by the flux distribution and metabolite concentrations, indeed constrain the dynamic properties of the system ... 

Based on the linearized models around the equilibrium point, different local controllers can be implemented. In the discussion above a simple proportional controller was assumed (unity feedback and variable gain). To deal with multivariable systems two basic control strategies are considered centralized and decentralized control. In the second case, each manipulated variable is computed based on one controlled variable or a subset of them. The rest of manipulated variables are considered as disturbances and can be used in a feedforward strategy to compensate, at least in steady-state, their effects. For that purpose, it is t3q)ical to use PID controllers. The multi-loop decoupling is not always the best strategy as an extra control effort is required to decouple the loops. 

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 the course of time open systems that exchange matter and energy with then-environment generally reach a stable steady state. However, as shown by Glansdorff and Prigogine, once the system operates sufficiently far from equilibrium and when its kinetics acquire a nonlinear nature, the steady state may become unstable [15, 18]. Feedback regulatory processes and cooperativity are two major sources of nonlinearity that favor the occurrence of instabilities in biological systems. 

The analysis of feedback control systems can often be facilitated by conversion to an equivalent unity feedback system, i.e. a feedback loop in which the feedback path is represented by a steady-state gain of unity. There are two principal cases to consider. 

As shown notably by the thermodynamic school of Brussels,9,10 systems maintained far from equilibrium and endowed with appropriate non-linearities and feedback interactions may display such nontrivial behaviors as sustained oscillations and multiple steady states (see papers by I. Prigogine and G. Nicolis in this volume). Even though the structures studied are much simpler than biological systems, this type of work provides a firm fundamental basis, not sufficient but absolutely necessary, for the future understanding of such processes as cell differentiation. Clearly, the sustained oscillations and multiple steady states displayed by simpler systems are related, respectively, with the two types of biological regulation described above. 

Systems that display such complexities as sustained oscillations or multiple steady states always comprise feedback loop(s) in their logical structure. In our context this may mean, for instance, that substance a exerts a control on the rate of synthesis of (3, which exerts a control on the rate of synthesis of -y, which in turn exerts a control on the rate of a synthesis. More generally, there are elements whose rate of production is directly or indirectly influenced by their own concentration. 

The maximum productivity of the desired product B usually occurs at the middle unstable saddle-type steady state. In order to stabilize the unstable steady state, a simple proportional-feedback-controlled system can be used, and we shall analyze such a controller now. A simple feedback-controlled bubbling fluidized bed is shown in Figure 4.25. 

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