1066-033X/21©2021IEEE » BOOKSHELF 80 IEEE CONTROL SYSTEMS » FEBRUARY 2021 ANALYTICAL DESIGN OF PID CONTROLLERS by IVÁN D. DÍAZ-RODRÍGUEZ, SANGJIN HAN, and SHANKAR P. BHATTACHARYYA Reviewed by Lee H. Keel T he proportional-integral- derivative (PID) controller dominates the control in- dustry and, by some estimates, accounts for more than 95% of the controllers in use world- wide [1], [2]. Its working is based on the following simple fact: if a stable dynamic system is subject to constant refer- ence and disturbance inputs, all signals in the system tend asymptotically to constant values and the input to each integrator tends to zero. This applies to continuous-time; discrete-time; linear; nonlinear; single-input, single-output (SISO); and multivariable, multiple-input, multiple-output systems with and without time delays. This immediately suggests that tracking and disturbance rejection of con- stant inputs can be accomplished by generating the track- ing error and using it to drive integrators that generate the control signal. This architecture is inherently robust relative to state feedback, observer-based, high-order op- timal systems [3]. This monograph is focused exclusively on the PID con- troller. It is timely and contributes substantially to the art and science of control system engineering. » Timely, because PID controllers are increasingly being used in dynamically changing environments, such as those encountered in driverless cars, unmanned aerial vehicles, and distributed robotics. In these applications, there is an urgent need for design meth- ods and algorithms that can quickly and accurately update controller gains. » Art, because the book proposes a multiobjective approach to design, allowing the designer to cre- atively blend various performance measures using classical and modern approaches. » Science, because rigorous analytical approaches to achieve such designs are also presented. This monograph is the third in a series (the others are [4] and [5]) focused on the development of PID controller design theory based on the computation of the stabilizing set. CONTENTS The book is organized into three parts. Part I lays the foun- dation for the rest of the book. It describes methods for the computation of the complete set of PID controllers stabi- lizing a given SISO linear time-invariant plant. The plant description can be in the form of a transfer function, or it may be given as frequency-response data for a continuous-time [6] or discrete-time system [7]. Clearly, every design must reside in this stabilizing set. It is shown that the stabilizing set can be computed by solving a set of linear programming problems with a scalar sweeping parameter varied over a specifiable range. The linear programming formulation results from the novel application of a root counting for- mula that generalizes the classical Hermite–Bieler theorem. This applies to continuous-time systems with time delay and discrete-time systems. In the latter case, the frequency response of the system is represented using Tchebyshev polynomials, which simplify the computations. An interest- ing by-product of these results is the computation of all sta- bilizing PID controllers for the Ziegler–Nichols plant (that is, first order with time delay), generalizing this classical 1942 result [8]. Another interesting result is the computation of all PID controllers that shift all the closed-loop poles to the left of a line at , s v =- resulting in the determination of the maximum achievable v for the given plant [9]. This number is inversely proportional to the settling time of the step response. Using the stabilizing set, Part II proceeds to search for achievable performance for the closed-loop system under I EEE Control Systems welcomes suggestions for books to be reviewed in this column. Please contact either Scott R. Ploen, Hong Yue, or Thomas Schön, associate editors for book reviews. Springer, 2019, ISBN, 978-3-030—18227-4, 314 pages, US$149.99. Digital Object Identifier 10.1109/MCS.2020.3032803 Date of current version: 18 January 2021