Up to this point, we have developed the necessary tools to generate mathematical models of dynamic systems. The discussions in this area have covered physical systems comprising mechanical, electrical, thermal, and fluidic components, and, further, we have shown how some fundamental multi-domain problems may be addressed by the mechanical modeling approach. Under this paradigm, all the tools and techniques we have discussed inevitably lead to mathematical representations in the form of coupled ODEs of motion. Through these discussions, we have also demonstrated that the mechanical modeling approach does not necessarily produce a linear governing equation and may often result in mathematical models that have nonlinear terms. However, since the solution techniques for linear and nonlinear systems differ, all the tools we will discuss in Chs. 6–10 assume that the equations of motion are linear.

In Ch. 5, we introduced the concept of state-space models as an alternative to traditional systems of nth-order governing equations. Unlike governing equations derived from first principles, such as those discussed in Chs. 2–4, state-space models always comprise first-order ODEs and can be analyzed and solved using linear algebra rather than higher-order ODEs.

Following our discussions in Ch. 5 on developing analytical and numerical solutions of system responses, Ch. 6 is focused on critical features of these solutions, known as , for systems with responses that do not tend to infinity when excited by a pulse input. The primary characteristics of these responses include the settling time, the percent overshoot, and the frequency of oscillations (i.e., if they exist), all of which can be obtained analytically, numerically, or experimentally using simple test inputs – such as steps or sinusoids. Once the response characteristics are established for a test input, the implications for more complex excitations can be understood by applying the concept of superposition (see Sec. 1.3) for linear systems.

Through our discussions in Chs. 5–10, we developed many tools and techniques and are now equipped to analyze, design, and simulate linear feedback control systems. Up to this point, we have focused heavily on idealized mathematical representations and their numerical simulation. To put this knowledge into practice, we must now address several practical considerations and non-idealities that arise when implementing feedback control systems. By doing so, we aim to fortify our intuition for the behavior of real-world dynamic systems, improve our ability to quickly develop feasible designs, and develop an understanding of the important challenges and choices that arise in the implementation of feedback control systems prior to moving forward with our discussions of advanced techniques in Part IV of this text.

In this chapter we will discuss techniques for developing mathematical models of mechanical systems. In doing so, we will take a direct approach to derive governing equations for planar rigid bodies and point masses by first reviewing the equations that describe the behavior of individual mechanical elements and later applying well-known interconnection laws to combine these elements into meaningful representations of physical dynamics. Through this approach, known as mechanical modeling, we will develop mathematical and physical models of mechanical systems with translational, rotational, levered, and geared elements.

Continuing our discussions on the methods used to analyze dynamic systems and the characteristics of their responses, we will now focus our efforts on a special kind of Laplace-domain representation known as theand will develop a set of tools and techniques relevant to transfer function analysis. By analyzing transfer functions we will be able to identify properties of dynamic systems – rather than properties of their responses – and, later, specify desirable system design characteristics in terms of transfer function properties. In turn, transfer functions will enable our eventual transition from analysis tasks to design tasks and will be pivotal in moving forward. Accordingly, we will begin by demonstrating how a linear system can be put into the form of a transfer function, and we will review some basic properties of these representations.

As we wrap up our introduction to control engineering, it is worth reflecting on the techniques covered thus far. Beginning in Ch. 8, the concept of feedback has been pivotal in designing closed-loop systems that achieve transient and steady-state performance for continuous-time plants such as the mechanical, electrical, thermal, fluidic, and other systems modeled in Chs. 1–4. Using the Laplace transform and linear-algebra-based state-space methods, we have reviewed numerous techniques for placing the poles of the closed-loop system, tracking reference inputs, and rejecting disturbances such as impulses, steps, ramps, and sine waves. These topics form the foundation of control engineering practice and are essential for understanding and applying advanced techniques. In fact, with some additional mathematics, we can extend many of these concepts and results to a broader range of dynamics and greatly improve the performance, robustness, and operating regimes of our control systems.

Thus far, we have discussed the dynamics and response characteristics of various physical systems. In doing so, we have developed many useful modeling and analysis techniques that help us to understand the behavior of first-, second-, and higher-order dynamic systems. One important application of these tools is to design systems that meet certain requirements or criteria in the responses of their output variables, such as the settling time and final value of a motor’s velocity in response to a step input.