Nonlinear Internal Model Control

Several process control design methods, such as the Generic Model Control (GMC) [41], the Globally Linearizing Control (GLC) [37], the Internal Decoupling Control (IDC) [7], the reference system synthesis [8], and the Nonlinear Internal Model Control (NIMC) [29], are based on input-output linearization. 

It is therefore necessary to develop control-relevant techniques for characterizing nonlinearity. Through use of the Optimal Control Structure (OCS) approach [5], Stack and Doyle have shown that measures, such as Eq. (1), may still be applied but to a controlrelevant system structure. In the OCS approach, the necessary conditions for an optimal control trajectory given a process and performance objective are analyzed as an independent system. The nonlinearity of these equations determine the control-relevant nonlinearity. The OCS has been used to determine the control-relevance of certain commonly-exhibited nonlinear behaviors [6]. Using nonlinear internal model control (IMC) structures, similar analysis has been performed on Hammerstein and Wiener systems with polynomial nonlinearities to examine the role of performance objectives on the controlrelevant nonlinearity [7]. Though not applied to the examples in section 5, these controlrelevant analysis techniques have been shown to be beneficial and remain an active research area. 

The first two points are valid for open-loop process nonlinearity measures as well. The third point is new in control-relevant nonlinearity quantification. In a more general context, one has not only to consider the performance criterion but additionally mention the controller design method. Following the idea of Ref 24, optimal control theory with an integral performance criterion will be used here as it represents a benchmark for any achievable performance. Considering nonlinear internal model control with different filter time constants is also possible, see for example Ref 23. 

M.A. Henson and D.E. Seborg. An internal model control strategy for nonlinear systems. AIChE Journal, 37 1065-1081, 1991. 

The combination of the nonlinear estimator (28a,b) with the nonlinear controller (Eq. 17) yields the measurement-driven controller in internal model control (IMC) form ... 

Henson, M. A., and D. E. Seborg, An Internal Model Control Strategy for Nonlinear Systems, AIChE J., 37,1065 (1991). 

ABSTRACT In this paper the Internal Model Control (IMC) approach for marine autopilot system is presented. The inversion by feedback techniques are employed for reahzation of inversion such nonlinear characteristics as saturation of rudder angle and rudder rate. The extension of the model and inverse model to a nonlinear form enabled to achieve a significant improvement in the control performance. 

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. 

Another important consideration is the development of an efficient control strategy, because this minimizes the costs by maintaining the process in its optimal conditions. Costa et al. (5) determined the best control structures and studied the control of the process proposed by Silva et al. (3) using a linear predictive controller. Later, a nonlinear predictive controller using FLNs as the internal model was developed and implemented with the same process with promising results (6). 

J.B. Balchen, B. Lie, and I. Solberg. Internal decoupling in nonlinear process control. Modeling Identification and Control, 9 137-148, 1988. 

S. Griiner, S. Schwarzkopf, I. Uslu, et ah, Nonlinear model predictive control of multicomponent distillation columns using wave models. Proceedings, jth International Symposium on Advanced Control of Chemical Processes. Vol. 1,... 

Once the large internal flow rates have been set via appropriate control laws, the index of the DAE system (7.21) is well defined, and a state-space realization (ODE representation) of the slow subsystem can be derived. This representation of the slow dynamics of the column can be used for the derivation of a model-based nonlinear controller to govern the input-output behavior of the column, namely to address the control of the product purity and of the overall material balance. To this end, the small distillate and bottoms flow rates as well as the setpoints of the level controllers are available as manipulated inputs. 

Mayne, D. Q., Nonlinear model predictive control An assessment, in Fifth International Conference on Chemical Process Control (Kantor, J. C., Garcia, C. E., and Carnahan, B., Eds.), AIChE Symposium Series, Vol. 93, pp. 217-231 (1997). 

Marquard, W, Nonlinear Model Reduction for Optimization Based Control of Transient Chemical Processes, Proceedings of the 6 International Conference of Chemical Process Control, AlChe Symp. Ser. 326, Vol. 98 (12), 2002. 

Three methods which do not require solution of the nonlinear partial differential equation are presented for estimating extractor performance. The choice of method depends on the value of the dimensionless outlet solute concentration, oj. If (oj + 1)/oJ is close to 1, the reaction is effectively irreversible and the pseudosteady-state solution of the advancing front model satisfactorily predicts performance after normalization to include solute solubility in the globule. If (oj + 1)/oJ is not close to 1, the advancing front results will still apply, provided that the amount of solute extracted by reaction is small and membrane solubility controls. When oj is small enough so that (oj + 1) is close to 1, then the reversible reaction model can be reduced to a linear equation with an analytical solution. Otherwise, for oj values when neither (oj + 1) nor (oj + 1)/o is nearly 1, a reasonable first approximation is made by adjusting the actual concentration of internal reagent to an effective concentration which equals the amount consumed to reach equilibrium. 

For an efficient numerical solution, problem (1.2) is first reformulated as a nonlinear programming (NLP) problem by parameterization of the controls and discretization of the model equations. Then, an infeasible-path optimization method is applied to solve the discretized problem. By this approach, simulation and optimization proceed simulianeously i.e., the model equations are satisfied only at the final solution. The necessary gradients can be calculated efficiently and reliably by internal numerical differentiation. In addition, reduced-space strategies allow to considerably reduce the number of gradient evaluations. 

Lakner, R., Hangos, K.M., Cameron, I.T, 1999, An Assumption-Driven Case-Specific Model Editor. Computers and Chemical Engineering, 23, S695-S698. Szederk6nyi, G., Kovacs, M., Hangos, K.M, 2002, Reachability of Nonlinear Fed-batch Fermentation Processes. International Journal of Robust and Nonlinear Control, 12, in print. 

Aoustin, Y., Chedmail, P., Glumineau, A., "Some simulation results on the robustness of a flexible arm nonlinear control law". International Journal of Modelling and Simulation, vol. 11, Issue 4, 1991. 

A digital computer by its very nature deals internally with discrete-time data or numerical values of functions at equally spaced intervals determined by the sampling period. Thus, discrete-time models such as difference equations are widely used in computer control applications. One way a continuous-time dynamic model can be converted to discrete-time form is by employing a finite difference approximation (Chapra and Canale, 2010). Consider a nonlinear differential equation. 

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