**6. Discussion and conclusion of results**

**Figure 8.** Transient response for unstable behavior.

160 Robust Control - Theoretical Models and Case Studies

*q*2 = [−19.72, −16.13], *q*3 = −29.25, −8.08], *q*4 = [11.01,17.35].

**Figure 9.** Value set with uncertainty in the parameters.

with time delay:

Finally, considering an uncertainty of 10% in each of the parameters, the following character‐ istic equation is obtained, where an overestimation was made to obtain interval polynomials

> ( ) ( ) 43 2 3 2 1 2 34 1 2 3 4 ,,, *T s <sup>d</sup> <sup>d</sup> <sup>s</sup> p s q r e s rs r s r s r qs q s q s q e* - -

where *r*1 = [11.60,11.99],, *r*2 = [50.95, 54.90], *r*3 = [61.80,75.53], *r*4 = [8.84,10.80], *q*1 = −1.98, −1.62],

Once again, using the lemma, the value set of the characteristic equation is obtained now with

uncertainty in the coefficients; This can be appreciated in the graphic of the **figure 9**.

= + + + ++ + + +

t

A robust stability verification methodology based on a graphic method was mentioned in this chapter. First, the mathematical model with nominal approximated parameters must be obtained; then, a control algorithm connected in feedback with the plant is analyzed when in the plant of the closed-loop system, uncertainty is introduced in both parameters and time delay. The main tools used are the value-set graph and the zero exclusion principle, then the analytical problem to verify the robust stability is transformed into a graphical problem, which consist of checking if the value-set plot for frequency interval does not include the origin. The case of study of an internal combustion motor was presented. The time delay appears when modeling the corresponding section of the engine combustion chamber of the motor. It is important to note that by considering the uncertainty in the parameters of the mathematical model, the time delay influences more in the control stability system; so, it can become significant even when it is very small and therefore the inclusion in the analysis is important. Then, with the tools presented here, a robustness margin for parameters and time delay can be obtained.
