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expressed in the frequency- and time-domain for both continuous- and discrete-time systems was
discussed. For additional details about the concepts mentioned in this section and those not covered, it
is recommended that the reader consider the attached references for further reading.

References

1. Franklin, G. F., et al., Feedback Control of Dynamic Systems, 3rd ed., Addison-Wesley, New York, 1994.
2. Hanselman, D., and Littlefield, B., The Student Edition of Matlab, Version 5, User’s Guide, Prentice-
Hall, Upper Saddle River, NJ, 1997.
3. Croft, D., et al., Creep, hysteresis, and vibration compensation for piezoactuators: atomic force
microscopy application, ASME J. Dyn. Syst., Meas., Control, 123, 35, 2001.
4. Dorny, C. N., Understanding Dynamic Systems—Approaches to Modeling, Analysis, and Design, Prentice-
Hall, Englewood Cliffs, NJ, 1993.
5. Locatelli, M., et al., Easy method to characterize a piezoelectric ceramic tube as a displacer, Rev. Sci.
Instrum., 59, 4, 1988.
6. Dorf, R. C., and Bishop, R. H., Modern Control Systems, 9th ed., Prentice-Hall, Upper Saddle River,
NJ, 2001.
7. Chen, T. C., Linear System Theory and Design, Oxford University Press, New York, 1999.
8. Friedland, B., Control System Design: An Introduction to State-Space Methods, McGraw-Hill, New
York, 1986.
9. Gillis, J. T., State space, in The Control Handbook, Levine, W. S., CRC Press, Salem, MA, 1996, Chap. 5.
10. Khalil, H. K., Nonlinear Systems, 2nd ed., Prentice-Hall, Upper Saddle River, NJ, 1996.
11. Ogata, K., Discrete-Time Control Systems, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1995.
12. Franklin, G. F., et al., Digital Control of Dynamic Systems, 3rd ed., Addison-Wesley, Menlo Park, 1998.

23.4 Transfer Functions and Laplace Transforms

C. Nelson Dorny
We perceive a system primarily through its behavior. Therefore, our mental image of a system usually
includes representative response signals. The step response, the behavior when we suddenly turn on the
system, is such a system-characterizing signal. We should view the step response as a description of the
system. The impulse response is another description of the system. For a system represented by linear
differential equations, the unit-step response is the integral of the unit-impulse response.
Let us represent time differentiation (d/dt) by the time-derivative operator, p. Then we can denote the
2
time derivative of a signal y by py, its second derivative by p y, its integral with respect to time by (1/p)y,
and so on. This operator notation simplifies the expressions for differential equations. We shall use the
expression system equations to mean a set of differential equations that determines fully the behaviors of
the dependent variables that appear in those equations. We can reduce a linear set of system equations
to a single input–output system equation by eliminating all but one dependent variable from the set.
The transfer function associated with that dependent variable is a mathematical expression that contains
all the essential information embodied in the system differential equation.
The Laplace transformation converts signals (functions of time) to functions of a complex-frequency
variable, s = σ + jω. There is a one-to-one correspondence between a signal and its Laplace transform.
We can retrieve the time function by inverse transformation. Laplace transformation produces images
that have some properties that are more convenient than those of the original signals. In particular, time
differentiating a signal corresponds to multiplying its Laplace transform by the complex-frequency
variable s. Hence, the transformation converts linear constant-coefficient differential equations to linear
algebraic equations. Such simplifications of time-domain operations make Laplace transformation useful.

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