Nonlinear control

Since many of our chemical engineering processes are nonlinear, it would seem intuitively advantageous to use nonlinear controllers in some systems. The idea is to modify the controller action and/or settings in some way to compensate for the nonlinearity of the process. 

For example, wc could use a variable gain controller in which the gain varies with the magnitude of the error. 

Another advantage of this kind of nonlinear controller is that the low gain at the setpoint reduces the effects of noise. 

Another type of nonlinear control can be achieved by using nonlinear transfonnations of the controlled variables. For example, in chemical reactor control the rate of reaction can be controller instead of the temperature. The two are, of course, related through the exponential temperature relationship. In high-purity distillation columns, a transformation of the type shown below can sometimes be useful to linearize the composition signal and produce improved control while still using a conventional linear controller. 

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Atherton, D.P. (1982) Nonlinear Control Engineering, Van Nostrand Reinhold, London. 

Farrell, R.J. and Yen-Cheng Tsai, 1994. Nonlinear controller for batch crystallization Development and experimental demonstration. In American Institute of Chemical Engineers National meeting. Atlanta, Paper 89e. 

These two basic advantages of the second method make it valuable for applications, particularly the difficult ones of stability of nonlinear control systems.13... 

A. M. Letov, Stability of Nonlinear Control System (in Russian), Moscow, 1955 English translation, Princeton University Press, Princeton, N.J., 1961. 

Linear control theory will be of limited use for operational transitions from one batch regime to the next and for the control of batch plants. Too many of the processes are unstable and exhibit nonlinear behavior, such as multiple steady states or limit cycles. Such problems often arise in the batch production of polymers. The feasibility of precisely controlling many batch processes will depend on the development of an appropriate nonlinear control theory with a high level of robustness. 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

For example, it is usually impossible to prove that a given algorithm will find the global minimum of a nonlinear programming problem unless the problem is convex. For nonconvex problems, however, many such algorithms find at least a local minimum. Convexity thus plays a role much like that of linearity in the study of dynamic systems. For example, many results derived from linear theory are used in the design of nonlinear control systems. 

These openloop-adaptive controllers are really just another form of nonlinear control. They have been quite successfully used in many industrial processes, particularly in batch processes where operating conditions can vary widely. 

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. 

A. Isidori. Nonlinear Control Systems. Springer Verlag, third edition, 1995. 

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 order to test the aforementioned nonlinear controller in the face of parameter uncertainty, we introduced changes on the Damkholer parameters during the course of the reactions. These changes are reported in Table 1. 

Consider again the CSTR where the consecutive reactions A B C take place. The required discrete nonlinear control law is found by obtaining a discrete version of the immersion developed in Example 3 and its associated exponential holder which has the form He, and in this particular example is a 1 X 5 dimension vector, where and H are presented in matrices (38). The exponential holder can be calculated by using the definition of an exponential matrix,... 

C.l. Byrnes and A. Isidori. Output regulation for nonlinear systems An overview. Int. J. Robust Nonlinear Control, 10 323-337, 2000. 

Robust Nonlinear Control of a Pilot-Scale Anaerobic Digester... 

Keywords Nonlinear Control, Total Organic Carbon, Volatile Fatty Acids, Anaerobic Digestion. 

Now, three types of differential operations involving real-valued functions, vector and co-vectors fields are described. These operations are frequently used in the analysis and design of nonlinear control systems (for more details see [15, 26]). 

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