This book is a tutorial about the Laplace transform and two cornerstone methods for the design and analysis of feedback control systems: (1) the Root Locus method; and, (2) the Frequency Response method. Further, the book is divided into two parts: (1) theory; and, (2) computer programming. In the book, the theory is proven using examples. Part one: In calculus, we learned how to solve differential equations by integration. We learned that the fundamental behavior of the solution is determined by the roots of the characteristic equation. That’s why the characteristic equation is so-named. About 250 years ago, Laplace invented his transform that turns a differential equation into a ratio of polynomials. The denominator polynomial contains the factors of the characteristic equation. Using algebra, this ratio can then be transformed back into the solution of the differential equation. Now let’s talk about feedback control theory using Laplace notation. Suppose the characteristic equation is: a s2 + b s + c = 0 where s = p + j q Let’s rearrange and rewrite it using a ratio. c / (a s2 + b s) = -1 As the value of c varies, the roots of the characteristic equation vary, forming a locus of roots . About 80 years ago, Evans developed Root Locus Rules . These rules describe this locus in the complex s-plane. A feedback control system designer uses the Root Locus Rules to relocate the roots of the characteristic equation. Because the characteristic equation can be arranged in many ways, the designer can use many different root locus plots. Now, in the ratio of the characteristic equation, let’s use s = j q. The ratio becomes: c / (- a q2 + j b q) As the value of q varies, the value of this ratio varies on a path that is called the frequency response. About 100 years ago, Nyquist developed a criterion for this path. If the criterion is satisfied, the solution of the differential equation will converge. A feedback control system designer shapes the frequency response to achieve a desired solution. Part two: Learning about feedback control systems usually starts in school. But real learning starts by doing. And a great way to do the theory, is to write computer programs about it. The second part of this book discusses the programs that generated all the data in the book. This discussion includes: the theory used in the programs; and, complete code listings. These programs are written in the computer language called Google Apps Script which is free and universally available in Google Sheets . The discussion also includes a Google Apps Script user manual and examples on how to use the programs.