Output response

Fig. 7. Residence time distributions where U = velocity, V = reactor volume, t = time, = UtjV, Cj = tracer concentration to initial concentration and Q = reactor volume (a) output responses to step changes (b) output responses to pulse inputs.

A reference trajectoiy is used to represent the desired output response over the prediction horizon. 

In principle, the step-response coefficients can be determined from the output response to a step change in the input. A typical response to a unit step change in input u is shown in Fig. 8-43. The step response coefficients are simply the values of the output variable at the samphng instants, after the initial value y(0) has been subtracted. Theoretically, they can be determined from a single-step response, but, in practice, a number of bump tests are required to compensate for unanticipated disturbances, process nonhnearities, and noisy measurements. 

A double trial-and-error procedure is needed to determine uq and Tq. If done only once, this is probably best done by hand. This is the approach used in the sample program. Simultaneous satisfaction of the boundary conditions for concentration and temperature was aided by using an output response that combined the two errors. If repeated evaluations are necessary, a two-dimensional Newton s method can be used. Dehne... 

Chapter 14 and Section 15.2 used a unsteady-state model of a system to calculate the output response to an inlet disturbance. Equations (15.45) and (15.46) show that a dynamic model is unnecessary if the entering compound is inert or disappears according to first-order kinetics. The only needed information is the residence time distribution, and it can be determined experimentally. 

A washout experiment is performed on a CSTR to measure its mean residence time. What is the effect of starting the experiment before the outlet concentration has fully reached Co Assume that the normalized output response is based on the outlet concentration measured at I = 0 so that the experimental washout function starts at 1.0. 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

Special considerations are required in estimating paraimeters from experimental measurements when the relationship between output responses, input variables and paraimeters is given by a Monte Carlo simulation. These considerations, discussed in our first paper 1), relate to the stochastic nature of the solution and to the fact that the Monte Carlo solution is numerical rather than functional. The motivation for using Monte Carlo methods to model polymer systems stems from the fact that often the solution... 

The method for estimating parameters from Monte Carlo simulation, described in mathematical detail by Reilly and Duever (in preparation), uses a Bayesian approach to establish the posterior distribution for the parameters based on a Monte Carlo model. The numerical nature of the solution requires that the posterior distribution be handled in discretised form as an array in computer storage using the method of Reilly 2). The stochastic nature of Monte Carlo methods implies that output responses are predicted by the model with some amount of uncertainty for which the term "shimmer" as suggested by Andres (D.B. Chambers, SENES Consultants Limited, personal communication, 1985) has been adopted. The model for the uth of n experiments can be expressed by... 

The aim of dynamic simulation is to be able to relate the dynamic output response of a system to the form of the input disturbance, in such a way that an improved knowledge and understanding of the dynamic characteristics of the system are obtained. Fig. 2.1 depicts the relation of a process input disturbance to a process output response. 

Consider a step input, x(t) = Mu(t) with X(s) = M/s, and the different cases with respect to the value of . We can derive the output response y(t) for the different cases. We rarely use these results. They are provided for reference. In the case of the underdamped solution, it is used to derive the characteristic features in the next section. 

We also see another common definition—bounded input bounded output (BIBO) stability A system is BIBO stable if the output response is bounded for any bounded input. One illustration of this definition is to consider a hypothetical situation with a closed-loop pole at the origin. In such a case, we know that if we apply an impulse input or a rectangular pulse input, the response remains bounded. However, if we apply a step input, which is bounded, the response is a ramp, which has no upper bound. For this reason, we cannot accept any control system that has closed-loop poles lying on the imaginary axis. They must be in the LHP. 1... 

Figure 11P.1 can be used to determine the dimensionless dispersion parmeter ( l/uL) for a system of interest. Use the transfer function method to evaluate the mean residence time and QjJuL) for a system subjected to the arbitrary input shown in the figure. Note that the output response has been shifted 62.5 sec to the left. Response values for the input and output streams were as follows. 

P5.01.18. INPUT OR OUTPUT RESPONSE CURVES, RAMP SHAPE... 

The response curve (RC) represents the calibrated output response of a sensor as a function of the measurand/s applied to its input. For instance, in the case of a chemical sensor based on conductivity (G), it is recommended to use one of the following notations [1] for the output response ... 

In the case of a sensor whose output is a frequency, the representations of the output responses may be as follows ... 

Residence time distributions can be determined in practice by injecting a non-reactive tracer material into the input flow to the reactor and measuring the output response characteristics in a similar manner to that described previously in Section 2.1.1. 

The DMC method uses the same statistical mathematics that are used in a standard least-squares procedure for determining the best values of parameters of an equation to fit a number of data points. In the DMC approach, we would like to have NP future output responses match some optimum trajectory by finding the best values of NC future changes in the manipulated variables. This is exactly the concept of a least-squares problem of fitting NP data points with an equation with NC coefficients. This is a valid least-squares problem as long as NP is greater than NC. 

Pulse testing also has problems in situations where load disturbances occur at the same time as the pulse is being performed. These other disturbances can effect the shape of the output response and produce poor results. The output of the prt)cess may not return to its original value because of load disturbances. 

To prevent rippling we will modify our desired output response to give the system two sampling periods to come up to the setpoint. The value of X( at the first sampling period, the Xi shown in Fig. 20.2d, is unspecified at this point. The output is now... 

This is the specified output response for a step change in setpoint. 

Fig. 6. Contour plot of the CMP down force vs the platen speed. The output responses are polish nonuniformity and rate.

Figure 8. Operational configurations and typical light output responses of (A) quadratic (B) linear, and (C) memory PLZT materials the heavy accented portions of the response curves indicate the usable range.

Fio. 2.2. Tracer measurements types of input signals and output responses (a) Step input—F-curve (A) Pulse input—C-curve (c) Sinusoidal input... 

Sarvomechanisms. Servomechanisms are part of a broad class of systems that operate on the principle of feedback. In a feedback control system, the output (response) signal is made to conform with the input (command) signal by feeding back to the input a signal that is a function of the output for the purpose of comparison. Should an error exist, a corrective action is automatically initiated to reduce the error toward zero. Thus, thru feedback, output and input signals are made to conform essentially with each other... 

Critical process variables should be set within their operating ranges and should not exceed their upper and lower control limits during process operation. Output responses should be well within finished product specifications. 

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