**1. Introduction**

The automatic control systems that are applied on the industrial process or on any production plant are previously analyzed, before they can be physically implemented. This analysis consists of obtaining some properties of interest such as qualitative and quantitative that are very helpful to achieve good performance in the closed-loop control system. The analysis is realized with the intention to predict the behavior that the control system will have when this is implement‐ ed. However, most of the time, this prediction lacks precision such that analysis is strongly based

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on the mathematical model of the physical process, and this does not represent in an exact way its dynamic behavior. This problem has been of great interest for scientific researchers in the last years, because it is desirable to get the best results when the control system is implement‐ ed but depends on these properties. A way of solving this problem is through the considera‐ tion of uncertainty on the mathematical model of the physical process. This uncertainty can be dynamic (see references [1–3]) or parametric (see references [4–6]). This chapter describes the qualitative property analysis of robust stability of a system. A case of study of an internal combustion engine is analyzed considering parametric uncertainty in the mathematical model. This case of study process is taken from references [5, 7], where conditions are obtained to verify the robust stability of the control system.

It is important to mention that in these research works several simplifications of the mathe‐ matical model were made, which may affect the real behavior of the system; one simplification performed by the authors is the cancellation of time delay in a part of the mathematical model, which has an influence that affects the stability property. The main contribution presented here is the consideration of the time delay on the mathematical model of an internal combustion engine, which is taken into account to obtain the property of robust stability. The methodology used is based on the application of the value-set concept to the particular case of the internal combustion engine (see reference [5]); to be more precise, it consists of the characterization of the value set of the resulting characteristic equation of the closed-loop system when the controller is connected. This controller assigns the poles in a position previously defined. Using this characteristic and applying the zero exclusion principle [8], it is possible to obtain robust stability conditions through a visual inspection of a graphic in the complex plane.

This chapter is organized as follows: Section 2 presents the elements used to verify the robust stability, Section 3 provides an abstract regarding obtaining the mathematical model of the internal combustion motor, in Section 4 the problem statement is set, in Section 5 the robust stability tools are used in simulation to verify robustness margin, and finally in Section 6 the conclusions of this work are presented.
