We now derive the time-domain solutions of first- and second-order differential equations. It is not that we want to do the inverse transform, but comparing the time-domain solution with its Laplace transform helps our learning process. What we hope to establish is a better feel between pole positions and dynamic characteristics. We also want to see how different parameters affect the time-domain solution. The results are useful in control analysis and in measuring model parameters. At the end of the chapter, dead time, the reduced-order model, and the effect of zeros are discussed.

What Are We Up to?

• Even as we speak of a time-domain analysis, we invariably still work with a Laplace transform. Time domain and Laplace domain are inseparable in classical control.

• In establishing the relationship between time domain and Laplace domain, we use only first- and second-order differential equations. That is because we are working strictly with linearized systems. As we have seen in partial-fraction expansion, any function can be “broken up” into first-order terms. Terms of complex roots can be combined together to form a second-order term.

• Repeated roots (of multicapacity processes) lead to a sluggish response. Tanks-in-series is a good example in this respect.

• With higher-order models, we can construct approximate reduced-order models based on the identification of dominant poles. This approach is used in empirical controller tuning relations in Chap. 6.

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