Adaptive control and process systems. Design and methods and control strategies 1

Sensitivity and precision For a control system to carry out the mission for which it is designed, it should be sensitive to changes that occur so that it can actuate, if the variation is at the input, or correct it if it is at the output. At the same time a system has to be precise and accurate in its response. 2

Sensitivity and precision Definitions Sensitivity: we define sensitivity as the ability of response to very small excitations, stimuli or causes. E. Dorf defines the sensitivity of a system as the ratio of change in the transfer function of the system with respect to the change in the transfer function of the process, for a small incremental change. Precision: it is defined as the ability to produce accurate results. This concept is linked to the ability to produce the same result under the same conditions, that is, for the control case, the ability to generate the same output in different occasions at the same input. 3

Sensitivity and precision Definitions Accuracy: it is defined as the accuracy on executing something. This concept would be more related to the ability to get the output to be as close as possible in value to the desired value. Reliability: it is the quality of reliable, probability of good functioning of something. S. Martinez takes it up as the concept of reliability of equipment, in its current technical sense, as the probability of continuity of operation under certain conditions. 4

Sensitivity and precision Sensitivity to parameter variations Every process is subject to changing conditions in their environment (imbalances, aging, etc.). – Open loop system: very sensitive to variations. – Closed loop system: less sensitive/insensitive to parameter variations. 5

Sensitivity and precision Sensitivity to parameter variations Suppose a change in the process so that the new transfer function is G(s)+ΔG(s). Then, the output will change according to the type of system – Open loop: – Closed loop Si G(s)H(s)>>ΔG(s)H(s) less variation 6

Sensitivity and precision Sensitivity to parameter variations 7 According to R. Dorf, the sensitivity of the system is defined as the ratio of the percentage change in the transfer function of the system compared to the percentage change in the transfer function of the process. If the transfer function of the system is The sensitivity of the system is defined as

Sensitivity and precision Sensitivity to parameter variations Variations in direct loop: – Open loop systems: sensitivity = 1 – Closed loop systems: The sensitivity in a system in open loop is reduced if the factor G(s)H(s) increases 8

Sensitivity and precision Sensitivity to parameter variations 9 Variations in the feedback loop: – Closed loop: The changes in H(s) directly affect the output. It is important that the components used in the feedback do not vary with environmental changes and remain constant.

Sensitivity and precision Disturbances A disturbance is a signal that can affect the value of a system output. – Internal: it is generated within the system. – External: It is generated outside the system. It should be considered as another entry. – In both cases they are uncontrolled. Open loop systems are very sensitive to disturbances. Closed loop systems are less sensitive since they affect the output and the system tends to correct them. 10

Sensitivity and precision Disturbances Disturbances should be considered as inputs of the system. The superposition principle is applied in order to work with them. It can appear at any point of the system. The main problem is its variability. 11

Sensitivity and precision Disturbances 12 Disturbance in direct loop: If D(s) is zero the system behaves normal If R(s) = 0, the process is readjusted as

Sensitivity and precision Disturbances Disturbance in direct loop: by the superposition principle The sensitivity of the system to G 2 and the effect of the disturbance with R(s)=0 are 13

Sensitivity and precision Disturbances Disturbance in the feedback loop: which is equivalent to: 14

Sensitivity and precision Errors 15 The errors in the control systems are attributable to many factors. The variations in the input reference cause inevitable deviations during the transient periods and even in steady state. The imperfections of the components, the deterioration of the elements of the systems (sensors and actuators), the friction between mechanical parts, the rolling, the temperature drift of the electronic components, the aging, etc., cause the system to deviate from the expected results. Other types of errors are those that occur when the systems are not able to follow certain types of inputs. Any control system has in steady state an error in response to certain types of inputs.

Sensitivity and precision Errors Errors in steady state: in a system with a unity feedback we have The error will be 16

Sensitivity and precision Errors Error in steady state for a step input: The static position error constant: The error in steady state will be: 17

Sensitivity and precision Errors Error in steady state for ramp input: The static position error constant: The error in steady state will be: 18

Sensitivity and precision Errors Error in steady state for a parabolic input: the error in steady state will be 19

Sensitivity and precision Errors Errors on a system with open loop: K c = 1/K The error will be: as changes in environmental variables or aging of the components are produced G(0) will stop being the unity, provocking that the error will no longer be zero. 20

Sensitivity and precision Errors 21 Errors on the system in closed loop: K p K>>1 The error will be: The error on the system in closed loop will be, if we establish K p =100/K,

Sensitivity and precision Precision 22 S. Marcos defines the precision as the accuracy of a system in the tracking of an input signal. The precision is represented by the error in steady state. The designer always wants the system not to present any error, however, each system has a certain inability to follow certain types of inputs. It can be concluded that a system is more precise the smaller the errors in steady state are.

Sensitivity and precision Reliability Reliability: It is a measure of its compliance with a correct specification of its behavior. Fault: it is a deviation of the behavior of a system with respect to its specification. Error: it is the cause of the faults. Failure: it is the mechanical or algorithmic cause of an error. The presence of a fault does not need to cause an error. 23

Sensitivity and precision Reliability Types of failures: – Steady state: they are present until they are repaired. They generally cause the systems to stop. – Transients: they disappear by themselves, so they have less serious consequences. – Intermittents: they appear and disappear from time to time or with a certain periodicity 24

Sensitivity and precision Reliability Prevention and tolerance to failures: – Reduce the failures: Avoid them. Try them not to be produced during the design and construction phases. Eliminate them when they appear. Techniques to avoid failures: – Based on hardware – Based on software – Checks – Tests 25

Sensitivity and precision Reliability Redundancy: By this method additional components are used that perform simultaneously the same function or are able to detect incorrect behavior and restore functionality, it is what is called as masking of errors. However, the introduction of additional elements increases the possibility of failure. – static redundancy: with this type of redundancy, redundant components are always active. – Dynamic Redundancy: Redundant components are activated only when a fault occurs. 26

Bibliography K. Ogata, Modern Control Engineering. R. Dorf, R. Bishop: Sistemas de control moderno. B. Kuo, F. Golnaraghi: Automatic Control Systems. P. Bolzern: Fundamentos de control automático. S. Martínez: Electrónica de potencia. Componentes, topologías y equipos Links of interest %20Clase/Tema%2001%20-%20Introducci%F3n%20a%20los%20Sistemas%20de%20Control.pdf upo1/archivos/Fiabilidad%20presentacion.pdf 27