Takagi-Sugeno (TS) fuzzy models (1985, 1992) can provide an effective representation of complex nonlinear systems in terms of fuzzy sets and fuzzy reasoning applied to a set of linear input/output (I/O) submodels. In this paper, the TS fuzzy model approach is extended to the stability analysis and control design for both continuous and discrete-time nonlinear systems with time delay. The TS fuzzy models with time delay are presented and the stability conditions are derived using Lyapunov-Krasovskii approach. We also present a stabilization approach for nonlinear time-delay systems through fuzzy state feedback and fuzzy observer-based controller. Sufficient conditions for the existence of fuzzy state feedback gain and fuzzy observer gain are derived through the numerical solution of a set of coupled linear matrix inequalities. An illustrative example based on the CSTR model is given to design a fuzzy controller.

Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach

In this chapter, some recent stability and robust stability results on linear time-delay systems are outlined The goal of this guided tour is to give (without entering the details) a wide overview of the state of the art of the techniques encountered in time-delay system stability problems In particular, two specific stability problems with respect to delay (delay-independent and respectively delay-dependent) are analyzed and some references where the reader can find more details and proofs are pointed out The references list is not intended to give a complete literature survey, but rather to be a source for a more complete bibliography In order to simplify the presentation several examples have been considered

Stability and robust stability of time-delay systems: A guided tour

Publisher Summary This chapter focuses on two idealized patterns (of all possible flow patterns), plug flow and back-mix flow. Plug flow assumes that fluid moves through the vessel “in single file” with no overtaking or mixing with earlier or later entering fluid. Back-mix flow assumes that the fluid in the equipment is perfectly mixed and uniform in composition throughout the vessel. All patterns of flow other than plug and back-mix flow may be called nonideal flow patterns because for these the design methods are not nearly as straightforward as those for the two ideal flow patterns. The treatment of nonideal flow patterns discussed in the chapter divides naturally into two parts: flow of single fluids and flow of two fluids. The main emphasis falls on the nonideal flow of single fluids through process equipment. Conversion in a reactor with nonideal flow can be determined either directly from tracer information or by use of flow models. The distribution of residence times gives information on how long various elements of fluid spend in the reactor but not on the detailed exchange of matter within and among the elements. The exact nature of the surrounding molecules is of no importance. Thus, the distribution of residence times yields sufficient information for the prediction of the average concentration in the reactor effluent. Because of a lack of applications, conversion expressions for two region models have not been developed for homogeneous systems. For heterogeneous systems, the appropriate expressions can be found in the works of the individual investigators.

Patterns of Flow in Chemical Process Vessels

Publisher Summary This chapter describes the relations between the amounts of the various species during the course of the reaction and to relate the concentration changes to a minimal number of concentration independent parameters that characterize the reaction system. Reaction kinetics provides an important part of the understanding of highly coupled systems and provide the method for predicting their behavior. The monomolecular reaction systems of chemical kinetics are examples of linear coupled systems. Because, linear coupled systems are the simplest systems with many degrees of freedom, their importance extends far beyond chemical kinetics. The linear coupled systems may be characterized, in general terms, as arising from stochastic or Markov processes that are continuous in time and discrete in an appropriate space. In addition, the principle of detailed balancing is observed and the total amount of material in the system is conserved. The system is characterized by discrete compartments or states, and material passes between these compartments by first order processes. Such linear systems are good models for a large number of processes.

The Structure and Analysis of Complex Reaction Systems

Publisher Summary This chapter focuses on the unstructured models of microbial cell population, distributed structured models, segregated structured models, and growth of single microbial cells. It offers quantitative analyses of some simpler biological phenomena and suggests experimental work that may profitably be carried out. The chapter provides a summary of attempts of applying engineering analysis for elucidating growth and replication phenomena in very simple systems: cultures of unicellular organisms reproducing by binary fission. The study of biological systems so as to apply the knowledge gained to nonbiological systems is familiar to electrical engineers. State in an unstructured model can only refer to population density or to concentration of protoplasm, because, by definition, a model is unstructured if only one state variable appears. The basic supposition in distributed models is that protoplasm is composed of two structural components, whose interaction with each other and with the surrounding medium produces growth. The gross metabolic rates of the culture represent averages over the distribution of cell structures. One of the principal problems is then to establish the factors, which determine the distribution.

Dynamics of Microbial Cell Populations

The purpose of this paper is to consider the well agitated continuous reactor from the standpoint of stability of the steady state. It has been shown in the past that chemical-reaction systems may be unstable in the sense that on slight perturbation they tend to move to a more stable state or that they are stable in their steady states, small perturbations being self-correcting so that the system possesses autoregulation. In this paper methods of developing criteria for the quantitative determination of stability or instability or presented and applied to some simple problems. In order that the effect of large perturbations on the system may be determined, complete solutions of the rigorous equations are obtained on the analogue computer (R.E.A.C.). A complete plot of reaction paths in the concentration-temperature plane may be obtained in this manner. Because of the nonlinearity of the system one cannot predict with certainty what steady state will be approached after a given large perturbation, multiple steady states being assumed possible. From the phase plot of reaction paths the regions in the plane which lead to certain steady states are delineated. Also it is shown that the natural behavior of a reactor is not to approach an unstable state. So far as the reactor is concenrned, the unstable state does not exist. The stability of the system is important to the engineer, as control will be easy or difficult and product quality will be satisfactory or not depending upon the relative stability of the steady state. An unstable state would require more elaborate control than a stable state.

Chemical reactor stability and sensitivity

In this paper extensive calculations on the quasi isothermal tubular reactor are presented. Temperature and concentration profiles were obtained on an analogue computer (R.E.A.C.). The calculations tend to show that there are regions of operation in which the reactor effluent is very sensitive to operating conditions. For example, it is shown that in some regions of operation a small change in the heat transfer coefficient at the reactor wall or a small dilution of the feed will produce large changes in the effluent. In such cases the reactor is said to exhibit parametric sensitivity. It is shown analytically that this sensitivity may be predicted by analyzing the frequency response or transient response of the reactor approximated by a local linearization. This linearization requires complete solutions of the steady state problem. Semiquantitative results are then obtained for the regulation required from a given specification of product limits. The frequency-response analysis should be useful in connection with the control problem.



If the reactor is fed partially with a recycle stream, then experience with electrical systems indicates that the possibility of instability exists. It is shown that at least theoretically these instabilities do exist, and a method based on the transfer function is developed for derivation of criteria of stability or instability.

Chemical reactor stability and sensitivity: II. Effect of parameters on sensitivity of empty tubular reactors

Optimum temperature gradients in tubular reactors—I: General theory and methods

Optimum temperature gradients in tubular reactors—II: Numerical study☆

The theory and design of continuous stirred tank reactors are extended to the complex but industrially important field of continuous-heterogeneous-reaction systems. The present investigation is concerned with the elementary process occurring when solid particles and a liquid flow into and from a single reactor or a chain of these reactors. The equations are developed for dissolution processes, and the theoretical size distributions and specific areas are confirmed experimentally. Performance is also related to operating conditions and reactor design.

Olegh Bilous

Papers

Chemical reactor stability and sensitivity

Chemical reactor stability and sensitivity: II. Effect of parameters on sensitivity of empty tubular reactors

Optimum temperature gradients in tubular reactors—I: General theory and methods

Optimum temperature gradients in tubular reactors—II: Numerical study☆

Continuous‐flow stirred tank reactors: Solid‐liquid systems