Error feedback

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. 

Fig. 2. Control scheme for error feedback linear regulation...

Likewise, the error-feedback regulation problem consists of finding a controller of the form (5) such that the following conditions hold... 

Q g j — ( j(G0) s stable, then the error feedback regulator problem is also solvable by the controller... 

The following example illustrate the calculations involved in the construction of both the state and error feedback regulators. 

The results of applying these strategies under the influence of initial error conditions are shown in Figure 6. As can be observed, both linear regulators ensure the asymptotic tracking of the desired oscillatory temperature profile. Both regulators performance is different because the error feedback regulator behavior depends on the initial states of zi and Z2. 

Ji-Chang L, Chien-Hsing Y (1999) A heuristic error-feedback learning algorithm for fuzzy modeling. IEEE Trans Syst Man Cybernet Part A Syst Humans 29(6) 686-691... 

Another important research topic concerns the particular focus of explanations. Even when explanations are constructed and are therefore constrained to refer to the knowledge used by the problem solver, there are many choices in the type of information to include in error feedback. For example, the explanation could focus on the problem solving goal, the part of the student step that is correct, the location of the error, the reason why the step will not work, a description of the type of step that would be better, or the precise nature of the fix to the current step. Empirical work is necessary to investigate the type of information effective for error feedback, and the contingencies governing the selection of diis information. 

Optimality or near-optimality in computational performance is the result of error feedback leading to modification of output [15]. The process is described by an alphabet Q, I, Z, d, w where Qj is the structure s internal states, I represents the environmental inputs, Z is its output values, d is a next-state mapping function such that d I X Q, a Qj and w is an output function such that w Ij X Qj a Z,. Error E may then be expressed as the sum of accumulating environmental mispredictions... 

The "feedback loop in the analytical approach is maintained by a quality assurance program (Figure 15.1), whose objective is to control systematic and random sources of error.The underlying assumption of a quality assurance program is that results obtained when an analytical system is in statistical control are free of bias and are characterized by well-defined confidence intervals. When used properly, a quality assurance program identifies the practices necessary to bring a system into statistical control, allows us to determine if the system remains in statistical control, and suggests a course of corrective action when the system has fallen out of statistical control. 

Some of the inherent advantages of the feedback control strategy are as follows regardless of the source or nature of the disturbance, the manipulated variable(s) adjusts to correct for the deviation from the setpoint when the deviation is detected the proper values of the manipulated variables are continually sought to balance the system by a trial-and-error approach no mathematical model of the process is required and the most often used feedback control algorithm (some form of proportional—integral—derivative control) is both robust and versatile. 

The Smith dead-time compensator is designed to aUow the controUer to be tuned as tightly as it would be if there were no dead time, without the concern for cycling and stabUity. Therefore, the controUer can exert more reactive control. The dead-time compensator utilizes a two-part model of the process, ie, Gp, which models the portion of the process without dead time, and exp — sTp,pj ), which models the dead time. As seen from Figure 18b, the feedback signal is composed of the sum of the model (without dead time) and the error in the overaU model Gpj exp — sTppj )), ie, C —. Using... 

Feedback Control In a feedback control loop, the controlled variable is compared to the set point R, with the difference, deviation, or error e acted upon by the controller to move m in such a way as to minimize the error. This ac tion is specifically negative feedback, in that an increase in deviation moves m so as to decrease the deviation. (Positive feedback would cause the deviation to expand rather than diminish and therefore does not regulate.) The action of the controller is selectable to allow use on process gains of both signs. 

Feedforward Control If the process exhibits slow dynamic response and disturbances are frequent, then the apphcation of feedforward control may be advantageous. Feedforward (FF) control differs from feedback (FB) control in that the primary disturbance or load (L) is measured via a sensor and the manipulated variable (m) is adjusted so that deviations in the controlled variable from the set point are minimized or eliminated (see Fig. 8-29). By taking control action based on measured disturbances rather than controlled variable error, the controller can reject disturbances before they affec t the controlled variable c. In order to determine the appropriate settings for the manipulated variable, one must develop mathematical models that relate ... 

Other Considerations in Feedforward Control The tuning of feedforward and feedback control systems can be performed independently. In analyzing the block diagram in Fig. 8-32, note that Gy is chosen to cancel out the effects of the disturbance Us) as long as there are no model errors. For the feedback loop, therefore, the effects of L. s) can also be ignored, which for the sei vo case is ... 

Eodt (13-174) where V and are initial values, Kc and T are respectively feed-back-controller gain and feedback-reset time for integr action, and E is the error or deviation from the set point as given by ... 

This completes the design of the feedback loop compensation elements, and the error amplifier curves and the overall plots are also included in Figure 3-66. This also completes the design of the major portions of the switching power supply. The schematic is shown in Figure 3-67. 

Then find the value of the feedback capacitor C. The designer knows the value of the input resistor (R). It is the upper resistor in the voltage divider responsible for the voltage feedback to the error amplifier. One then performs Equation B.15. 

Fig. 4.1 Block diagram of a closed-loop control system. R s) = Laplace transform of reference input r(t) C(s) = Laplace transform of controlled output c(t) B s) = Primary feedback signal, of value H(s)C(s) E s) = Actuating or error signal, of value R s) - B s), G s) = Product of all transfer functions along the forward path H s) = Product of all transfer functions along the feedback path G s)H s) = Open-loop transfer function = summing point symbol, used to denote algebraic summation = Signal take-off point Direction of information flow.

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