1. Control Systems and Adaptive Process. Design, and control methods and strategies 1

2. Reliability and stability Reliability Reliability is one of the main characteristics that define any system (electronic, mechanical, etc.). Reliability is defined as the probability of continuing to function over time under certain conditions. Reliability is a fundamental aspect of the quality of any device, therefore, its quantification is essential, so that estimations can be made on the lifetime of any product. 2

3. Reliability and stability Reliability Instantaneous failure rate (λ): is the probability that, in a system that has worked up to time t, a failure occurs during the interval (t, t+Δt). Mean time between failures (MTBF): is the average time between two successive failures of a system. MTBF = 1 / λ. Mean time to repair (MTTR): is the average time it takes to repair a system. Instantaneous repair rate (µ): is defined as 1 / MTTR. Availability (A): is defined as A = MTBF / (MTBF + MTTR). 3

4. Reliability and stability Reliability Reliability diagram: is a block diagram that provides an answer about which element of a system must operate in a manner that ensures normal operation and which may be faulty. The concept of redundancy is thus introduced. Error-masking: is the way to mask any possibility of error in order to prevent uncontrolled direct influence on the development of the required function. 4

5. Reliability and stability Reliability Fault tolerance: defined as fault-tolerant such system that ensures required task development also when one of the parts of the system produces a fault. The defect tolerance is achieved by introducing redundancy or error masking (in the hardware or software). A fault tolerant system is composed essentially of the following elements: -Fault-detection: sensors and sensing circuits. -Fault-isolation: isolation of the defective part without affecting the rest of the system. -Defect Diagnosis: Diagnosis for identifying the cause of the defect. -Elimination of fault: replace the faulty module without interrupting the function performed by the system. -Return to operation: the return to normal operating condition should be performed without interrupting the function being performed. 5

6. Reliability and stability Reliability Redundancy: presupposes the existence in the system of various alternative possibilities to ensure the development of the desired function. Degree of redundancy: if from a set of n elements arranged in parallel k are redundant, that is, n-k elements can support the function that is performed, it can be said that the system possesses a degree of redundancy n-k over n. 6

7. Reliability and stability Reliability 7 When you have a set of N identical components, of which up to time t have failed N f and N s continue to operate, the probability of survival of a component at time t is and the probability that has failed at that instant is It is evident that

8. Reliability and stability Reliability Failure rate is defined as the number of failed components per unit time in relation to the number of components that survive. Operating and integrating we have which represents the reliability of a component, defined as the probability of survival at time t assuming operation at time 0. 8

9. Reliability and stability Reliability 9 In homogeneous populations of identical components is experimentally observed that the failure rate changes according to the known bathtub curve, in which three zones are distinguished. The first, the area of infant mortality, with a high rate of failure due to manufacturing defects, etc.., is represented by a downward curve. The second, called life zone is one in which the lower failure rate occurs and is represented by a substantially flat curve. The third zone, called the zone of aging, is one in which the life of the components is coming to an end, so the failure rate is represented by a curve with increasing slope.

10. Reliability and stability Reliability The reliability of a system depends on the reliability of each and every one of its components, hence the recourse to redundancy to increase reliability of a system. Therefore, in more or less complex systems, it is interesting to know the reliability of the system itself, not its components. Open loop systems are very sensitive to perturbations. 10

11. Reliability and stability Reliability Non-repairable system: suppose a non-repairable system. The probability that the system does not stop working is the product of the probabilities that does not fail any of its components, which is expressed mathematically. 11

12. Reliability and stability Reliability 12 Non-repairable system: Consider now N components placed in parallel so it is necessary that all fail for a system failure, the probability of system failure is the product of the probabilities of failure of each of the elements. Since the probability of failure of component i is For the system

13. Reliability and stability Reliability Repairable system: now suppose a repairable system in which the rate of repair is µ, and MTTR is defined by its inverse r = 1/ µ. The probability that an element is functioning at time t + Δt is equal to the probability that, while running on t, not to malfunction to Δt, plus the probability that, being damaged in t be repaired in Δt. We can conclude that, for a system composed of N components functionally in series, the error rate is the mean time to repair is, whenever λ i is small compared to unit 13

14. Reliability and stability Reliability Repairable system: For a system with functionally parallel- connected elements with the condition that λ i r i has a small value compared to 1, and 14

15. Reliability and stability Stability 15 The stability of a feedback system is closely related to the position of the roots of the characteristic equation of the transfer function of the system. A system is stable when all the poles of its transfer function are in the left half-plane s. The relative stability of a system is analyzed by studying the relative locations of the poles. According to R. Dorf, a stable system is a dynamic system with a bounded response to a bounded input.

16. Reliability and stability Stability The response of a dynamic system with an input is decreasing, increasing or neutral. By the definition of stability follows that a system is stable if, and only if, the value of its impulse response g(t) integrated over a finite interval is a finite value. The location of the poles in the left half-plane s ensures a decreasing response for perturbation inputs. The planes located on the jω axis result in a neutral response and those in the right half-plane s a growing response (and hence unstable). 16

17. Reliability and stability Stability Routh-Hurwitz stability criterion : This algebraic method is based on the characteristic equation of the system, which is usually written in the form The following array is constructed: the sign changes in the first column are the roots with positive real parts 17

18. Reliability and stability Stability Routh-Hurwitz stability criterion: If in a row the first element is zero, it will not be possible to continue calculating elements. This can be solved in three ways: 1. Making a change of variable s =1/x achieving a polynomial in x whose roots are the inverse of the polynomial in s. If I apply the method to the polynomial in x the solution is obtained. 2. Multiplying the polynomial by a factor (s+a) where a > 0, increasing in by l the degree of the polynomial, so will have a root in s = -a. 3. Substitute the zero by an infinitesimal ϵ, still building the table, taking ϵ as constant. In the end ϵ is made to tend to zero from the right. 18

19. Reliability and stability Stability Routh-Hurwitz stability criterion: The necessary and sufficient condition for all roots of the equation to be in the negative half-plane s is that all terms of the equation must be positive and all terms in the first column must be positive. The Routh stability criterion only considers the absolute stability of the system but not the relative stability. One of the methods used to determine the relative stability is to change the position of the axis of the plane s by making s= ŝ-k. From here, the equation in ŝ is constructed and the process is repeated to yield the number of roots located to the right of the new axis in s= -k. 19

20. Reliability and stability Stability Nyquist stability criterion: The Nyquist stability criterion relates the open-loop frequency response G(jω)H(jω) with the number of zeros and poles of 1+ G(s)H(s) located on the right- half s plane. This system is useful because it allows you to graphically determine the absolute stability in closed loop from the frequency response curves in open loop, without the need to determine the closed-loop poles. It is particularly useful in cases in which the mathematical expressions of a system are unknown and only its frequency response is known. 20

21. Reliability and stability Stability 21 Argument principle: It's called F(s) a quotient of polynomials of variable s, where P is the number of poles and z the number of zeros of the function, which are within a given contour in the s plane, considering the multiplicity (double pole). If the function does not pass through any singular point or zeros, then the representation of this function in the image plane will have a N number of bypasses to the origin, equal to the difference between the zeros and poles enclosed by the contour. The shape of the paths or of the contours does not affect the performance of the theorem.

22. Reliability and stability Stability 22 Argument principle: consider a system Operating we have The zeros of 1+GH(s) will give the system stability. They match the poles of A(s). The poles of 1+GH(s) coincide with the poles of GH(s).

23. Reliability and stability Stability Argument principle: For the system to be stable, the zeros of 1+GH(s) must be in the left half plane. Consider a closed path that encloses the entire right path, called Nyquist path If I make the representation of 1 + GH(s) of the closed path, applying the argument principle, we have that ; to make the system stable the number of zeros (Z) must be zero. 23

24. Reliability and stability Stability Measurement of relative stability by varying the Nyquist path: Relative stability indicates the extent to which a system is stable. The closer the poles are to the imaginary axis, the less stable the system. If the roots of this factor are calculated we have As increases, the imaginary part decreases and the real part becomes more negative. If we will have a double pole on the real axis. If we will have conjugate poles on the imaginary axis. 24

25. Reliability and stability Stability We need to know if there exist damping coefficients smaller than the given ones. With this angle we obtain a new Nyquist path. Then the Nyquist criterion is applied over the new path. 25

26. Reliability and stability Stability Stability of series systems: if there have been no cancellations, that of denominator of G(s) is given by the product of G 1 (s)G 2 (s), and the poles of G(s) is the set of poles of the two subsystems. We conclude that the system G(s) is stable if and only if the two subsystems are stable individually. If there have been cancellations there is a hidden part that does not guarantee stability under this criterion. 26

27. Reliability and stability Stability Stability of parallel systems: equally, if not cancellations occur, the denominator of G(s) corresponds to the product of the denominators of G 1 (s) and G 2 (s) and the poles of G(s) is the set of poles of both subsystems. Similarly we conclude that the system is asymptotically stable if and only if both subsystems are. However, when D 1 (s) and D 2 (s) have common factors, cancellations occur, so you have to keep the same caution in the previous case. 27

28. Reliability and stability Stability Stability of systems with state variables: The stability of a system modeled by a flow diagram of state variables can be determined without special complications. - If the transfer function is expressed in the domain of s, then the stability will be represented by the roots of the denominator. As already mentioned, this denominator is referred to as the characteristic equation. therefore, to analyze the stability the characteristic equation is analyzed using the Routh-Hurwitz method. - If the system is evaluated by a signal flow diagram, the characteristic equation is obtained by evaluating the determinant of the flow graph. - If the system is represented by block diagrams, the characteristic equation is obtained using the methods of simplification of blocks, with the precautions previously described. 28

29. Bibliography K. Ogata, Modern Control Engineering. R. Dorf, R. Bishop: Modern control systems. B. Kuo, F. Golnaraghi: Automatic Control Systems. P. Bolzern: Fundamentos de control automático. S. Martínez: Electrónica de potencia. Componentes, topologías y equipos Interesting links http://www.uoc.edu/in3/e-math/docs/Fiab_1.pdf http://laurel.datsi.fi.upm.es/~ssoo/STR/Fiabilidad.pdf http://it.aut.uah.es/danihc/DHC_files/menus_data/SCTR/ToleranciaFiabilidad.pdf http://isa.uniovi.es/docencia/TiempoReal/Recursos/temas/Fiabilidad.pdf http://web.usal.es/~sebas/TEORIA/TEMA7-REGULACION.pdf http://www.ing.unlp.edu.ar/controlm/electricista/archivos/apuntes/cap4.pdf http://www.ie.itcr.ac.cr/gaby/Licenciatura/Analisis_Sistemas_Lineales/Presentaciones/07_Est abilidadv2008s02.pdf http://www.herrera.unt.edu.ar/controldeprocesos/tema_4/Tp4B.pdf 29