Nonlinear reference control

Figure 5.16 Nonlinear reference control (Servo control). Bringing the reactor from an arbitrary initial condition to the three open-loop steady states k =0.6). 1, upper steady state (stable) 2, intermediate steady state (unstable) 3, lower steady state (stable) (from [38]).

The nonlinear reference controller provides zero offset on setpoint change, even without explicit integral action. Unfortunately, the same is not true for disturbance rejection. Integral action may be incorporated into the controller in order to provide disturbance rejection capabilities. The addition of integral action to the controller is achieved by state augmentation. The integral action is adjoined to the proportional reference controller input u in the following manner ... 

Figure 5.17 Nonlinear reference control (disturbance rejection). 10% reduction is heat transfer coefficient. Curves 1,2 and 3 are the open loop, closed loop and set point, respectively (from [38]).

Most polymerizations are nonlinear. The applicability of nonlinear reference control schemes for solution polymerization reactors has been demon-... 

Adebekun, A. K. and Schork, F. J. (1989b) Continuous solution polymerization reactor control. 2. Estimation and nonlinear reference control during methylmethacrylate polymerization, Ind. Eng. Chem. Res. 28, 1846-61. 

Chapter 21. Chapter 7 in Shinskey [Ref. 3] is again an excellent reference for the practical considerations guiding the design of feedforward and ratio control systems. It also discusses the use of feedforward schemes for optimizing control of processing systems. Good tutorial references are the books by Smith [Ref. 2], Murrill [Ref. 8], and Luyben [Ref. 9]. The last one has a simple but instructive example on the nonlinear feedforward control of a CSTR. 

In Figure 5.17, a 10% decrease in the heat transfer coefficient is introduced. This disturbance enters the temperature loop directly and no other loop, and is assumed to be unmeasured. Hence, the controls are computed based on the old value of heat transfer coefficient. The controller parameters are fe = 2 and an integral gain of 1 on the temperature loop (X2). Because of the nature of the reference controller, one would expect the monomer (x,) and initiator (xj) loops to eventually return to the desired operating point after some dynamics even without integral action. Furthermore, because of the integral action in the temperature loop, at equilibrium, no offset exists in this loop. Also, note that the monomer and initiator loops are virtually unaffected by this disturbance. This follows from the nonlinear nature of the controller. Note that ultimately, there is no offset in the solvent loop (x J. In effect, the controller has perfectly rejected the disturbance in the entire Subsystem I (and hence, Subsystem II, in the absence of a disturbance to Subsystem II directly). It is clear that before the disturbance enters the temperature loop, the system is at equilibrium. The... 

The pioneering use of wavepackets for describing absorption, photodissociation and resonance Raman spectra is due to Heller [12, 13,14,15 and 16]- The application to pulsed excitation, coherent control and nonlinear spectroscopy was initiated by Taimor and Rice ([17] and references therein). 

As mentioned above, the backbone of the controller is the identified LTI part of Wiener model and the inverse of static nonlinear part just plays the role of converting the original output and reference of process to their linear counterpart. By doing so, the designed controller will try to make the linear counterpart of output follow that of reference. What should be advanced is, therefore, to obtain the linear input/output data-based prediction model, which is obtained by subspace identification. Let us consider the following state space model that can describe a general linear time invariant system ... 

Forced oscillation is a well-known technique for the characterization of linear systems and is referred to as a frequency response method in the process control field. By contrast, the response of nonlinear systems to forcing is much more diverse and not yet fully understood. In nonlinear systems, the forced response can be periodic with a period that is some integer multiple of the forcing period (a subharmonic response), or quasi-periodic (characterized by more than one frequency) or even chaotic, when the time series of the response appears to be random. In addition, abrupt transitions or bifurcations can occur between any of these responses as one or more of the parameters is varied and there can be more than one possible response for a given set of parameters depending on the initial conditions or recent history of the system. 

R.D. Bartusiak, C. Georgakis, and M.J. Reilly. Nonlinear feedforward/feedback control structures designed by reference synthesis. Chemical Engineering Science, 44 1837-1851, 1989. 

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. 

Figure 13.4 illustrates three aspects of the basal ganglia network. First, a reciprocal control exists between GPe and STN. Second the SNc acts not only on the striatum but also on the cortex, the STN and the GPi. Finally the location of the STN at the intersection between vertical and horizontal feedback loops is crucial. Reference [50] concludes that the BG can no longer be considered as a unidirectional linear system that transfers information based solely on a firing-rate code and must rather be seen as a highly organized network with operational characteristics that simulate a nonlinear dynamical system (Fig. 13.4). 

The analysis of ordinary differential equation (ODE) systems with small parameters e (with 0 < generally referred to as perturbation analysis or perturbation theory. Perturbation theory has been the subject of many fundamental research contributions (Fenichel 1979, Ladde and Siljak 1983), finding applications in many areas, including linear and nonlinear control systems, fluid mechanics, and reaction engineering (see, e.g., Kokotovic et al. 1986, Kevorkian and Cole 1996, Verhulst 2005). The main concepts of perturbation theory are presented below, following closely the developments in (Kokotovic et al. 1986). 

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