Step response coefficients

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. 

Laplace transform variable Search direction Step response coefficient Time... 

Figure 2 Step response coefficients of the extract and raffinate purities with respect to the flow rates (from top to bottom eluent, recycle, extract and solid ).

Substitution of (10) into the expression for the closed-loop map, (7), then gives the following expression for the step response coefficients from the sth component of w to the rth component of z ... 

Equation (5.4) gives an explicit relationship between the parameters of the FSF model and the step response coefficients. The step response coefficients, evaluated using a reduced order FSF model, can be obtained from... 

Now the question arises as to how neglecting these high frequency parar meters aifects the accuracy of the step response coefficients. To study this problem, we define... 

G(e °). The next most significant term corresponds to Z = 2. The weighting functions for Z = 7 indicate that as Z increases, the contributions from the corresponding higher frequencies parameters of the FSF model to the step response coefficients decrease. 

Now let us rewrite Equation (5.4), which converts the FSF parameters into step response coefficients, into a similar partitioned form... 

We have also been able to obtain confidence bounds for the step response model derived from the FSF model using Equation (5.4). The basic idea is to represent the step response coefficients as a linear transformation of the estimated FSF parameters, and then map the covariance matrix from the frequency domain to the time domain. 

FVom assumptions A 5.1-A 5.3, we know that d is an unbiased and normally distributed estimate of 6. Therefore, gm is an unbiased and normally distributed estimate of gm- The variance of the estimated step response coefficient at the sampling instant m is given by... 

The quality of the estimated model is measured here by the amount of departure from the true impulse and step response coefficients. More precisely, we use the sum of the squared deviations from the true impulse response coefficients Eimpuise) and the sum of squared deviations from the... 

In these equations, are the step response coefficients obtained by model identification. 

One can use the dynamic model of the emulsion polymerization to apply perturbations in the inputs u and record the model outputs y. The step response coefficients are then calculated as follows ... 

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