Low order models

Galerkin s method low-order model since a smooth function is... 

Even after linearization, the state-space model often contains too many dependent variables for controller design or for implementation as part of the actual control system. Low-order models are thus required for on-line implementation of multivariable control strategies. In this section, we study the reduction in size, or order, of the linearized model. 

The basic strategy of modal reduction approaches is to retain only certain modes of the high-order model in the low-order model. Wilson et al. (1974) summarized these techniques and showed that many of the published modal approaches are equivalent since they produce identical reduced models. Bonvin (1980) also provides a comparison of the various modal techniques with respect to their steady-state and dynamic accuracies as well as to the dependence of the reduced models on the retained state variables. 

Bird and co-workers have recently examined a number of low-order models which mimic finite extensibility (102, 338). The rigid dumbbell consists of two... 

Fig. 5.16. Comparison of steady-state concentration profiles in a coupled column system predicted by a rigorous model from first principles (dashed line) and a low-order model based on the wave approach (solid line).

Graham, W. R., J.P. Peraire, and K. Y. Tang. 1999. Optimal control of shedding using low-order models, part ii-model based control. Int. J. Numerical Methods Engineering 44 973-90. 

The order of the postulated model is a very important factor. For a well-known process such as a stirred-tank heater, it is not a problem. On the other hand, it is not obvious what order of dynamics we should assume for a fluid catalytic cracker (see Example 4.15). Also, it is not obvious what type of low-order model we should use to approximate the high-order models of even simple distillation columns (see Example 4.16). As a general starting point one could employ first- or second-order models with or without dead time. There exist a surprisingly large number of processes which could be effectively described by such low-order models. 

Rathnasingham and Breuer [4] developed the first low-order model of a synthetic jet using a control volume model for the flow and an empirical model for the stmctural dynamics of the diaphragm. A lumped element model of a piezoelectric-driven synthetic jet actuator was derived in [5]. 

Sanchez-Gasca J.J., Chow J.H., 1997. Computation of power system low-order models Irom time domain simulation using a Hankel matrix, IEEE Trans. Power Systems, Vol. 12, No. 4,pp. 1461-1467. 

Table 3.2 presents a summary of our findings. Prom this table, it can be seen that the PRESS statistic selects a model order which is close to the best order obtained using the complete data set (M = 1000) starting with M — 400. However, the FPE criterion does not select a model order close to the best order until M = 800. In addition, it is interesting to note that with the smallest data set examined (M = 200), the PRESS statistic also manages to select the best low order model ni = 3), according to our earlier analysis, but this was not the case for the FPE criterion. These results show that, for this example, the PRESS statistic provides a consistent order estimate and is more robust than the FPE criterion in terms of sensitivity to data length effects. 

Betlem, B.H.L. (2000) Batch distillation column low-order models for quality control. Chemical Engineering Science, 55, 3187-94. 

By selecting Estimate, Parametric models again in the ID-GUI we could easily select another model order. Through trial and error one could try to find the best model, although for controller design a low order model is preferred. 

In Chapter 5 we discussed the dynamics of relatively simple processes, those that can be modeled as either first- or second-order transfer functions or as an integrator. Now we consider more complex transfer function models that include additional time constants in the denominator and/or functions of s in the numerator. We show that the forms of the numerator and denominator of the transfer function model influence the dynamic behavior of the process. We also introduce a very important concept, the time delay, and consider the approximation of comphcated transfer function models by simpler, low-order models. Additional topics in this chapter include interacting processes, state-space models, and processes with multiple inputs and outputs. 

In this section, we present a general approach for approximating higher-order transfer function models with lower-order models that have similar dynamic and steady-state characteristics. The low-order models are more convenient for control system design and analysis, as discussed in Chapter 12. 

When should a step response or impulse response model be selected First, this type of model is useful when the actual model order or time delay is unknown, because this information is not required for step response models. The model parameters can be calculated directly using linear regression. Second, step or impulse response models are appropriate for processes that exhibit unusual dynamic behavior that cannot be described by standard low-order models. We consider such an example next. 

For processes with higher-order dynamics and/or time delay, the model can first be approximated by a low-order model, or the frequency response methods described in Chapter 14 can be employed to design controllers. First, the inner loop frequency response for a set-point change is calculated from (16-7), and a suitable value of Kc2 is determined. The offset is checked to determine whether PI control is required. After Kc2 is specified, the outer loop frequency response can be calculated, as in conventional feedback controller design. The open-loop transfer function used in this part of the calculation is... 

Varanasi KK, Nayfeh SA (2004) The dynamics of lead-screw drives low-order modeling and experiments. J Dyn Syst Meas Control 126 388-3%... 

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