Control Systems and Adaptive
Process. Design, and control
methods and strategies
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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.
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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).
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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.
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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.
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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.
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Reliability and stability
Reliability
• When you have a set of N identical components, of which
up to time t have failed Nf and Ns continue to operate, the
probability of survival of a component at time t is
S (t ) 
Ns
N
and the probability that has failed at that instant is
F (t ) 
Nf
N
It is evident that
S (t )  F (t ) 
NsN f
N
1
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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
1 dN f
survive.
 (t ) 
N s dt
• Operating and integrating we have
N 1 dN f
N dN f
1 dF (t )
1 d (1  S (t )  1 dS (t )
 (t ) 




N N s dt
N s Ndt S (t ) dt
S (t )
dt
S (t ) dt
t
S (t )  e

  ( t ) dt
0
which represents the reliability of a component, defined as
the probability of survival at time t assuming operation at
time 0.
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Reliability and stability
Reliability
• 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.
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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.
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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.
N
N
i 1
i 1
S s   S i   e  i t  e  ( 1  2   N )  e s t
N
s  1  2     N   i
i 1
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Reliability and stability
Reliability
• 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
Fi (t )  1  Si (t )
For the system
N
N
i 1
i 1
Fs (t )   Fi (t )   1  S i (t )
N
S s (t )  1   1  S i (t )
i 1
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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
N
functionally in series, the error rate is
s   i
i 1
the mean time to repair is, whenever λi is small compared to unit
N
rs 
 r
i 1
N
i i

i 1
i
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Reliability and stability
Reliability
• Repairable system: For a system with functionally parallelconnected elements
 N
 N 1 
s    i ri   
 i 1
 i 1 ri 
with the condition that λiri has a small value compared to 1,
and
1
rs  N
1

i 1 ri
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Reliability and stability
Stability
• 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.
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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).
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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
q( s )  a 0 s n  a1 s n 1    a n 1 s  a n  0
• The following array is constructed: the sign changes in the first
column are the roots with positive real parts
sn
an
an 2
an 4
s n 1 a n 1
s n  2 bn 1
an 3
a n 5
bn  3
bn 5
s n  3 cn 1


cn  3
cn 5


s 0 hn 1
a a  a0 a3
b1  1 2
a1
a a  a0 a5
b2  1 4
a1
a a  a0 a7
b3  1 6
a1
b1a 3  a1b2
b1
b a a b
c2  1 5 1 3
b1
b a  a1b4
c3  1 7
b1
c1 
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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.
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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.
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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 righthalf 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.
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Reliability and stability
Stability
• 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.
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Reliability and stability
Stability
• Argument principle: consider a system
A( s ) 
C ( s)
G( s)

;
R( s ) 1  G ( s ) H ( s )
G( s) 
P1 ( s )
P ( s)
; H ( s)  2
Q1 ( s )
Q2 ( s )
Operating we have
A( s ) 
P1 ( s ) P2 ( s )
P1 ( s ) P2 ( s )  Q1 ( s )Q2 ( s )
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).
1  GH ( s )  1 
P1 ( s ) P2 ( s )
Q1 ( s )Q2 ( s )
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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
I)
II )
III )
0     ; s  j
s  90,90º
     0 ; s  j
If I make the representation of 1 + GH(s) of the closed path, applying the
argument principle, we have that N  Z  P  Z  N  P ; to make the
system stable the number of zeros (Z) must be zero.
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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
s  n 
4 2n2  4n2
2
 n  jn 1   2
As  increases, the imaginary part decreases and the real part
becomes more negative. If   1 we will have a double pole on
the real axis. If   0 we will have conjugate poles on the
imaginary axis.
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Reliability and stability
Stability
• We need to know if there exist damping coefficients smaller
than the given ones.
tg 
n
n 1  
2
   sen
• With this angle we obtain a new Nyquist path. Then the
Nyquist criterion is applied over the new path.
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Reliability and stability
Stability
• Stability of series systems:
G ( s )  G1 ( s )G2 ( s ) 
N1 ( s) N 2 ( s)
D1 ( s ) D2 ( s )
if there have been no cancellations, that of denominator of G(s) is
given by the product of G1(s)G2(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.
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Reliability and stability
Stability
• Stability of parallel systems:
G ( s )  G1 ( s )  G2 ( s ) 
N 1 ( s ) D2 ( s )  N 2 ( s ) D1 ( s )
D1 ( s ) D2 ( s )
equally, if not cancellations occur, the denominator of G(s)
corresponds to the product of the denominators of G1(s) and G2(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 D1(s) and D2(s) have
common factors, cancellations occur, so you have to keep the same
caution in the previous case.
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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.
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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
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