Automatic

The automatic belongs to the engineerings. This milked discipline of modeling, the analysis, the order and, of the Regulation of the dynamic systems. It has as theoretical bases the Mathématiques, the Théorie of the signal and the theoretical Informatique. The automatic allows the automation of tasks by machines functioning without human intervention. One speaks then about linked system or controlled.

The experts automatically or automatism name control engineers.

A simple example, is that of the speed regulator of a car, it makes it possible to maintain the vehicle at a constant speed, speed-instruction predetermined by the driver.

General information, concepts

One wishes to control the temperature of a furnace. The first task consists in analyzing the system " four" and with the to model in the form of equations. One will be able thus precisely to connect the entry of the sytème (a tension ordering the temperature) to the exit (the desired temperature of the furnace). This relation can be done in the form of a differential equation or a transfer transfer function. One determines also the conditions of stability of the system (it is not wanted that the furnace starts to increase the temperature without arréter). One then will synthesize a new system, the " régulateur" , this one will have for entries the instruction (i.e. the temperature desired inside the furnace) as well as the real temperature of the furnace provided by a sensor and as an exit the ordering of the furnace. The two systems " régulateur" and " four" are put in cascade. The unit forms what is called a linked system . This one must answer a certain number of requirements:

• stability (the regulator should not make the system unstable),
• continuation (the temperature of the furnace must reach the temperature in instruction, one can specify in the schedule of conditions if there are constraints of speed or going beyond),
• the rejection of the disturbances (one opens the door of the furnace, the temperature goes down, the temperature must join the desired temperature).

The " régulateur" can then be realized in analogical form (Electrical circuit with resistances, condensers…) or numerical.

In connection with the systems

A system is a modeling of a process under operation. It has one or more entries, and one or more exits.

The entries of the system are called variable exogenic, which gather the handled disturbances and variables, regulating orders or sizes. They are often represented in a generic way by the letter U or E . They are connected to the process as such by an actuator.

The exits of the system are called variable controlled, regulated measurements or sizes. They are often represented in a generic way by the letter there . The process is connected to the exit of the system by a sensor.

In the case of a sampled system, the entries and exit are at discrete time, but the system in him even remains at continuous time. The system thus included a digital-to-analog converter in entry, an analog-to-digital converter at exit and a clock allowing to fix the sampling rate.

There exists an infinity of examples of systems: mechanical systems, electric systems or chemical processes. The representation of the system could then be done only with good knowledge in the physical field corresponding.

Various systems

The systems can be classified in several categories.

Systems at continuous time, discrete time

• Systems at continuous times: In fact the systems exist naturally.

• Systems at discrete times: these are systems of which time at discretized summer. These systems do not exist in a natural state (the majority of the natural physical systems are of type at continuous time), but since the majority of the controllers used automatically are calculated by numerical processors, it is sometimes interesting to model the system ordered like a system at discrete time.

• Systems with discrete events: systems whose operation can be modelled by discrete events. Generally, these systems are modelled by Petri networks, or by the Algèbres of dioïdes. Examples are the railway networks, or the operation of an assembly line.

• hybrid Systems: Systems whose modeling requires the use of the techniques related on the continuous systems and the systems with discrete events, for example: a Gear box of car.

Monovariable systems, multivariable systems

Four possibilities exist:

• the system has an entry and an exit, it is a monovariable system or SISO (Individual Intput Single Outpout),
• the system has several entries and several exits, it is a multivariable system or MIMO (Multiple Intput Multiple Outpout),
• the system has an entry and several exits, system SIMO,
• the system has several entries and an exit, system MISO.

System invariant in time

These are systems whose behavior does not vary according to the moment when the entry signal is sent.

Or not linear linear systems

See also: linear System

It is said that a system is linear if the exit is linaire compared to the entry.

No system is strictly linear, would be this only by saturations (physical thrusts, for example) which it comprises or by the phenomena of Hystérésis.

Conversely, a non-linear system can sometimes be regarded as linear in a certain beach of use. It is necessary always to keep in mind that the system on which one can work is only a Mathematical model of the reality, and which consequently there is a loss of information at the time of the passage to the model. Of course, it falls on the engineer to judge the relevance of its model with respect to the laid down objectives.

Representation of the linear systems invariants

The control engineers are accustomed to graphically representing a linked system by the use of Schéma-bloc.

Differential equation and transfer transfer function

See also: Transfer function transfer

A physical system generally describes with differential equations (basic principle of dynamics, characteristic of a condensing or a winds…). The transformed of Laplace then makes it possible to pass from the differential equation to a transfer transfer function.

For a system at discrete time one uses the transformed into Z.

This function will make it possible to deduce the behavior input-output from the system.

Temporal representations

See also: Impulse response, indicielle Answer

One can be interested in the behavior of the system when one subjects it to certain signals like a impulse of Dirac or a level. One can deduct from it a certain number of characteristics of the system.

Frequential representations

See also: Diagram of Nyquist, Diagram of Bode

The diagram of Bode represents, on separated graphs, the profit and the phase according to the frequency.

The place of Nyquist represents the imaginary part of the transfer transfer function according to the real part.

Finally the diagram of Black represents the profit according to the phase.

Representation of state

See also: Representation of state

The representation of state is a matric representation of the system. One is interested in internal variables with the systems, called variables of state. One then represents the derivative of the variables of state according to themselves and the entry, as well as the exit according to the variables of state and of the entry.

From this representation one can deduce the behavior input-output from the system (One can deduce from the representation of state the transfer transfer function) but also a certain number of other information like the commandability or the observability.

The representation of state can also represent a non-linear system or a variable system in time.

Stability

In the case of the linear systems represented by a Transfer function transfer, the analysis of the poles makes it possible to conclude on the stability of the system. One points out that the poles of a transfer transfer function are the complexes p0, p1… which cancel the denominator.

• Danslecasde a transfer function transfer continues using the Transformée of Laplace, all the poles must be with strictly negative real part so that the system is stable.
• In the case of a discrete transfer function transfer using the Transformed into Z, all the poles must have a module lower than 1 so that the system is stable.

Attention, automatically, the stability term must be defined precisely because there exists ten kinds of different stabilities. In general one refers to a asymptotic Stabilité.

In the case of the systems non-linéraires, the stability is generally studied using the theory of Lyapunov.

Control

See also: Control (automatic)

Buckled system

The technique of the most widespread automation is control in closed loop. A system is known as in closed loop when the exit of the process is taken into account to calculate the entry. Generally the controller carries out an action according to the error between measurement and the desired instruction. The traditional diagram of a linear system equipped with a linear regulator in closed loop is the following:

The open loop of the system is made up of the process and the corrector. The transfer transfer function of this loop system open is thus:

$H\_\; \{BO\}\; (S)\; =H\; (S)\; \backslash \; cdot\; C\; (S)$

With this architecture one can recompute a new transfer transfer function of the system: the transfer transfer function in loop closed using the relations between the various variables:

$there\; (S)\; =H\; (S)\; \backslash \; cdot\; U\; (S)$
$U\; (S)\; =C\; (S)\; \backslash \; cdot\; E\; (S)$
$E\; (S)\; =r\; (S)\; -\; there\; (S)$

One obtains then: $y\; (S)\; =\; \backslash \; left\; (\backslash \; frac\; \{H\; (S)\; C\; (S)\}\{1\; +\; H\; (S)\; C\; (S)\}\; \backslash \; right)\; R\; (S)$

The function $H\_\; \{BF\}\; (S)\; =\; \backslash \; frac\; \{H\; (S)\; C\; (S)\}\{1\; +\; H\; (S)\; C\; (S)\}$ represents the transfer transfer function in closed loop. One can notice that $H\_\; \{BF\}\; (S)\; =\; \backslash \; frac\; \{H\_\; \{BO\}\; (S)\}\{1+\; H\_\; \{BO\}\; (S)\}$: it is the formula of Black which makes it possible to pass from a transfer transfer function in loop open to a transfer transfer function in closed loop.

Note:

• the loop of return is the way which leaves the exit and which returns to the comparator with the sign " moins". In this loop, there is generally a block representing, in the greatest majority of the cases, a sensor. If this block has like transfer function " transfer; 1" (what is equivalent to a abscence of block because the multiplication by 1 does not change anything), it is said that the block diagram is on unit return. The formula previously stated is valid only if the block diagram is on unit return.

• Whatever the block diagram (unit or not, with or without disturbance,…), the denominator of the transfer transfer function in closed loop is always: $1+\; H\_\; \{BO\}\; (S)$ with $H\_\; \{BO\}\; (S)$ being the transfer transfer function in open loop i.e. the product of all the blocks of the loop, including those of the loop of return.

The study of this transfer transfer function in closed loop allows the frequential and temporal analysis general system with the controller.

Example of loop of regulation

Let us take again the example of the automobile engine.

One orders it by choosing the opening of the butterfly of gas integrated into the system of injection of the engine. The opening is directly related to the force applied to the piston thus to the acceleration of the vehicle. Let us say that they are proportional (one neglects the losses and the resistance of the air on the vehicle).

One wants to maintain a certain speed, 90 km/h for example. 90 km/h are the instruction, it should be compared at the real speed given by a tachometer.
La difference gives the variation from speed to be carried out. One from of deduced acceleration to be requested from the vehicle.
Connaissant the relationship between the acceleration and the opening of the butterfly, one calculates the opening to be given to the butterfly to approach the speed of instruction. The speedometer then takes the new value speed to reiterate the operation. In this manner, when one approaches wanted speed, acceleration decreases until being cancelled without brutality.
On thus obtains this diagram.

Actually, for of the losses, it is necessary to maintain a certain acceleration inter alia fighting against the resistance of the air.

Various techniques

There exist various techniques to synthesize the regulators. The industrial technique most largely used is the Régulateur PID which calculates a Proportionnelle action, Intégrale and Derived according to the error/measurement consigns. This technique makes it possible to satisfy the regulation of more than 90% of the industrial processes. Nevertheless, of many techniques of orders known as “advanced” can be used for the regulation of more complex systems when the Régulateur PID is insufficient:

• the Commande with model interns
• the Commande by return of state
• the predictive Commande being based on the use of a dynamic model of the system to anticipate its future behavior.
• the robust Order allowing to guarantee stability compared to the disturbances and with the errors of model.
• the adaptive Order which carries out an identification in real-time to bring up to date the model of the system.
• the fuzzy Logical using a Network of neurons or a Expert system.
• nonlinear controllers using the theory of Aleksandr Lyapunov, like the linearizing Orders or the Order by slipping modes, more robust.
• the Order by differential flatness, which allows the inversion of model without passing by the integration of the differential equations, and thus to calculate the signals necessary on the entries to guarantee the trajectories wanted at exit.

Notes and references of the article

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