Following our discussion of translational and rotational mechanical systems, we will now extend our mechanical modeling approach (via the system decomposition technique) to electrical circuits. As in Ch. 2, we begin by considering the fundamental elements from which many electrical systems are built, and we review the corresponding mathematical element laws and electrical circuit diagrams. Then, moving forward, the focus shifts toward the rules that govern the interconnection of these elements as complete circuits and subsystems of series and parallel elements. Through these discussions, we will find that, just as Newton’s second law guided our modeling approach for mechanical systems, our treatment of electrical systems will follow Kirchhoff’s voltage and current laws.

To motivate our interest in the broad field of , let us begin by first dissociating the descriptor “dynamic” and focus solely on the objects of our interest, the “systems.” Throughout this text, we will considerof many kinds. To understand the true scope of potential applications for the tools that we will develop, it is advantageous to establish a clear vision of what qualifies as a system and why. To ensure suitability across many domains, we will prefer the flexibility of a loose conceptual definition, by which the term system will refer to any collection of elements (i.e., physical or mathematical) that have cause-and-effect relationships. Then, reintroducing “dynamic” as a reference to changes that occur over time, the field of dynamic systems encompasses the study of cause-and-effect relationships that propagate changes over time.

In this chapter, we continue our discussion of feedback control systems from the perspective of frequency-domain system properties, which were briefly introduced in Ch. 6. Accordingly, we begin this chapter with a comprehensive review of frequency-domain analysis techniques relating to the response of systems to inputs, or excitations, of the form.

Recall from our discussions in Part II that dynamic systems can be represented through various mathematical modeling strategies, including systems of ordinary differential equations, systems of Laplace-domain equations, transfer functions, functional block diagrams, and physical models. Through our review of the Laplace domain, we found that the zero-state input–output behavior of dynamic systems could be represented, conveniently, as the ratio of the system’s Laplace-domain output and its Laplace-domain input. These ratios, or transfer functions, simplified our analysis of interconnected systems and enabled the design of feedback control systems in Part III. However, while the Laplace-domain and transfer functions are undoubtedly powerful and useful tools for systems with a single input and a single output, known as(SISO) systems, the implementation of transfer functions can become cumbersome and complex for systems with multiple inputs and multiple outputs, known as(MIMO) systems.

In this chapter we will discuss an important graphical tool in feedback system design known as the . As the name suggests, this technique allows us to graph the closed-loop pole locations (i.e., the roots of the characteristic equation) on the complex plane as a function of a selected parameter (e.g., as the value of a resistor is varied or as controller gains are changed). Thus, not only does this technique allow us to extend our concept of stability analysis from the more mathematical Routh–Hurwitz approach in Ch. 8 but it also allows us to visualize the stability of a system and select its parameters to achieve desired system characteristics.

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.