Feedback-optimized control

A robust experimental concept potentially apphcable to all time domains is the feedback-optimized control [21] which requires no knowledge of the underlying mechanism in effecting control. Optimal control theory [187] provides theoretical and computational description of the process to help understand the control mechanisms in effect. 

Equation (10.24) shows that, because of the recycle loop, the process is non-self-regulating. Consequently an integral controller cannot be used to regulate composition. This rules out any kind of feedback-optimizing control system. But because of the lack of self-regulation, end-point... 

A feedback optimizing controller must be used to hold minimum conductivity, but the process must be made self-regulating first. This can be accomplished by feeding solvent from one storage tank while flowing into another, switching when the feedtank is empty. 

Such a model can be developed to a new design to get a feedback (FB) and build up a quality control system for materials. This scheme also includes smart block (SB) for optimal control and generation of a feedback function (Figure 1). 

Assion A, Baumert T, Bergt M, Brixner T, Kiefer B, Seyfried V, Strehle M and Gerber G 1998 Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses Sc/e/ ce 282 919... 

Rajamani and Herbst (loc. cit.) compared control of an experimental pilot-mill circuit using feedback and optimal control. Feedback control resulted in oscillatory behavior. Optimal control settled rapidly to the final value, although there was more noise in the results. A more complete model should give even better results. 

B. Jakubczyk and W. Respondek, Geometry of Feedback and Optimal Control (1998)... 

M. Okamoto and K. Hayashi, Optimal control mode of a biochemical feedback system, BioSystems., 16, 315-321 (1984). 

Hornung, T., Meier, R., Zeidler, D., Kompa, K. L., Proch, D., and Motzkus, M. 2000. Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback. Appl. Phys. B-Lasers Opt. 71(3) 277-84. 

All of the analyses described above are used in a predictive mode. That is, given the molecular Hamiltonian, the sources of the external fields, the constraints, and the disturbances, the focus has been on designing an optimal control field for a particular quantum dynamical transformation. Given the imperfections in our knowledge and the unavoidable external disturbances, it is desirable to devise a control scheme that has feedback that can be used to correct the evolution of the system in real time. A schematic outline of the feedback scheme starts with a proposed control field, applies that field to the molecular system that is to be controlled, measures the success of the application, and then uses the difference between the achieved and desired final state to design a change that improves the control field. Two issues must be addressed. First, does a feedback mechanism of the type suggested exist Second, which features of the overall control process are most efficiently subject to feedback control ... 

Based on these developments, the experimental implementation of automated feedback optimized laser control has been achieved [23]. This type of control at the molecular level can be much more selective than traditional methods of control where only macroscopic parameters like the temperature can be varied. 

The primary interest in the pole placement literature recently has been in finding an analytical solution for the feedback matrix so that the closed loop system has a set of prescribed eigenvalues. In this context pole placement is often regarded as a simpler alternative than optimal control or frequency response methods. For a single control (r=l), the pole placement problem yields an analytical solution for full state feedback (e.g., (38), (39)). The more difficult case of output feedback pole placement for MIMO systems has not yet been fully solved(40). 

The LQP is the only general optimal control problem for which there exists an analytical representation for the optimal control in closed-loop or feedback form. For the LQP, the optimal controller gain matrix K becomes a constant matrix for tf>°°. K is independent of the initial conditions, so it can be used for any initial condition displacement, except those which, due to model nonlinearities, invalidate the computed state matrices. 

The equivalence of tuned PID controllers and optimal controllers can be demonstrated by augmentation of the state vector and judicious selection of the objective function (47), (48) ordinarily an optimal feedback controller contains higher order derivative terms, yielding significant phase advance (which can cause noise amplification and controller saturation). 

As shown in the above works, an optimal feedback/feedforward controller can be derived as an analytical function of the numerator and denominator polynomials of Gp(B) and Gn(B). No iteration or integration is required to generate the feedback law, as a consequence of the one step ahead criterion. Shinnar and Palmor (52) have also clearly demonstrated how dead time compensation (discrete time Smith predictor) arises naturally out of the minimum variance controller. These minimum variance techniques can also be extended to multi-variable systems, as shown by MacGregor (51). 

This second-level modeling of the feedback mechanisms leads to nonlinear models for processes, which, under some experimental conditions, may exhibit chaotic behavior. The previous equation is termed bilinear because of the presence of the b [y (/,)] r (I,) term and it is the general formalism for models in biology, ecology, industrial applications, and socioeconomic processes [601]. Bilinear mathematical models are useful to real-world dynamic behavior because of their variable structure. It has been shown that processes described by bilinear models are generally more controllable and offer better performance in control than linear systems. We emphasize that the unstable inherent character of chaotic systems fits exactly within the complete controllability principle discussed for bilinear mathematical models [601] additive control may be used to steer the system to new equilibrium points, and multiplicative control, either to stabilize a chaotic behavior or to enlarge the attainable space. Then, bilinear systems are of extreme importance in the design and use of optimal control for chaotic behaviors. We can now understand the butterfly effect, i.e., the extreme sensitivity of chaotic systems to tiny perturbations described in Chapter 3. 

Therefore, the controller is a linear time-invariant controller, and no online optimization is needed. Linear control theory, for which there is a vast literature, can equivalently be used in the analysis or design of unconstrained MPC (Garcia and Morari, 1982). A similar result can be obtained for several MPC variants, as long as the objective function in Eq. (4). remains a quadratic function of Uoptfe+ -iife and the process model in Eq. (22) remains linear in Uoptfe+f-ife. Incidentally, notice that the appearance of the measured process output y[ ] in Eq. (22) introduces the measurement information needed for MPC to be a feedback controller. This is in the spirit of classical hnear optimal control theory, in which the controlled... 

This last class of methods provides a way of avoiding the repeated optimization of a process model by transforming it into a feedback control problem that directly manipulates the input variables. This is motivated by the fact that practitioners like to use feedback control of selected variables as a way to cormteract plant-model mismatch and plant disturbances, due to its simphcity and reliability compared to on-line optimization. The challenge is to find functions of the measured variables which, when held constant by adjusting the input variables, enforce optimal plant performance [19,21]. Said differently, the goal of the control structure is to achieve a similar steady-state performance as would be realized by an (fictitious) on-line optimizing controller. 

The state estimation technique can also be incorporated into the design of optimal batch polymerization control system. For example, a batch reaction time is divided into several control intervals, and the optimal control trajectory is updated online using the molecular weight estimates generated by a model/state state estimator. Of course, if batch reaction time is short, such feedback control of polymer properties would be practically difficult to implement. Nevertheless, the online stochastic estimation techniques and the model predictive control techniques offer promising new directions for the improved control of batch polymerization reactors. 

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