Control Systems and Adaptive Process . Regulators and Communication

1. Control Systems and Adaptive Process . Regulators and Communication

2. PID controllers • The PID controllers represent an efficient solution for many control problems. Currently around 95% of controllers are of the type PID, having survived to most innovative technological elements. • In this kind of controllers, the control action is built as an addition of three type of actions: proportional, integral and derivative. The control signal c(t) could be expressed by the following equation, where Kp is the value of the proportional gain, Ti is the integral time constant and Td is the differential time constant, applied to the error signal e(t).

3. PID controllers • This same signal, in the Laplace domain, would have the following expression: • Its main advantages are that the proportional action produces a control signal proportional to the error signal, so it introduces a major correction if the error is greater, the derivative action provides some advance on the response of the system and the integral term allows to eliminate the steady state error.

4. PID controllers • Kp will determine the value of the proportional action. If Kp is small also the proportional action will be and vice versa. This action is easy to tune and only depends on a parameter, and since the correction is proportional to the error made can reduce this, although not eliminate on steady state. • If there is only proportional action: • This implies that there is always error, which decreases as Kp increases, but if Kp increases greatly over impulse or instability can appear.

5. PID controllers • Ti is the time required for that the integral action contribute to output of the controller in an amount equal to the proportional action. If Ti is small, the integral action will be great. This action compensates for disturbances and maintains the controlled variable around the point consigned. Being an integral action eliminates stationary errors. In contrast, if Ti decreases much can destabilize the system. • Td is the time required for that the proportional action contribute to the output of the controller in an amount equal to the derivative action. If Td is small the derivative action will be small. The derivative error anticipates the effect of the proportional action estimating the error that will occur later, stabilizing quickly the controlled variable after any disturbance.

6. PID controllers comparative representation of the response to a step signal of a plant for different types of controllers (P, PI, PD, PID).

7. PID controllers The code in Scilab for this example would be: //Examples of controllers P, PD, PI, PID // clear s=poly(0,'s'); //time vector t=0:0.05:50; // //transference function of the plan (order 3) gp=1/((4*s+1)*(3*s+1)*(s+1)); // //controllers parameters Kc=3;Ti=8;Td=1.5; // //window xset('window',1) xname('Control System with several controllers PID') //--------------------------------- //P control gc=Kc; Mr=gc*gp/(1+gc*gp); Mrs=syslin('c',Mr); yp=csim('step',t,Mrs); //

8. PID controllers //PI control gc=Kc*(1+1/(Ti*s)); Mr=gc*gp/(1+gc*gp); Mrs=syslin('c',Mr); ypi=csim('step',t,Mrs); // //PD control gc=Kc*(1+Td*s); Mr=gc*gp/(1+gc*gp); Mrs=syslin('c',Mr); ypd=csim('step',t,Mrs); // //PID control gc=Kc*(1+1/(Ti*s)+Td*s); Mr=gc*gp/(1+gc*gp); Mrs=syslin('c',Mr); ypid=csim('step',t,Mrs); // //step response //graphic subplot(1,2,1);xset('font',2,3) plot2d(t',[yp; ypi; ypd; ypid]',style=[2,3,5,6]),xgrid(4) //títulos y leyendas xtitle('Step response','time','y(t)'); legends(['P control';'PIcontrol';'PDcontrol';'PID control'],[2,3,5,6],opt=1) xstring(22,0.6,'Gp=1/((4*s+1)*(3*s+1)*(s+1)), Kc= 3, Ti= 8, Td= 1.5') //---------------------------------

9. PID controllersTuning rules • PID controllers are usually adjusted in situ, in order to encompass all the features of the plant. There are methods of adjustment or tuning both analytical and experimental, even automatic. Analytical methods require the knowledge of the desired transfer function, so they are commonly used experimental methods. • When designing a controller may use a controller of which complexity coincide with the complexity of the process being controlled.However, for obvious reasons, it is necessary to using a controller whose complexity is more restricted, in which case you may either simplify the process model to approximate it to a PID controller or design a controller for a complex model and approximate it by a PID controller . In both cases it is necessary a tuning or adjustment of the controller to achieve to fulfil its mission in the most satisfactory mode. This need gave rise to the appearance of different methods for tuning these controllers. • The variation of any parameter may affect the operation of the controller, so it is necessary to pursue a particular adjustment process. In the following figure can be seen the effects of the decrease (left) or increase (right) of the independent variables, keeping unchanged the other.

10. PID controllersTuning rules Example: effect of the decrease (left) and increase (right) of the Ti value between 4 and 12 (the correct value for Ti could be 8).

11. PID controllersTuning rules • There are several methods for tuning, each best suited to each type of plant, the type of controller and/or specifications. Some of them are: • Step response Ziegler-Nichols method • Frequency response Ziegler-Nichols method • Chien, Hrones, and Reswick method (CHR) • Cohen-Coon method • Empirical tuning based on rules

12. PID controllersTuning rules • Step response Ziegler-Nichols method: is based on a registered information of the open loop step response of the process. The step response only need two parameters (figure). First we need to determine the point where the slope of the tangent straight line to unitary step response has the maximum value. The intersections between the tangent and the axis give the values for a and L. • Ziegler-Nichols have given the parameters for the controller in function of the values of a and L as shown in the next table:

13. PID controllersTuning rules • Frequency response Ziegler-Nichols method: is based in the knowing of the point where the Nyquist plot of the process intersects the negative real axis (this point is characterized by parameters Ku and Tu). This method represent that a point of the Nyquist plot can be moved by changing the PID controller parameters. The process is as shown: • To carry out the adjustment is necessary first to set the integral time constant Ti to its maximum value (∞) and differential time constant Td to minimum (0). Then, starting with a small value, would adjust the value of Kp until the process starts to oscillate, which occurs for a value of Kp = Ku being the oscillation period Tu. The values calculates by this method are the indicated in the next table:

14. PID controllersTuning rules • Experimentally, this method can be carried out as follows: 1. Proportional gain setting: To perform the adjustment first it’s necessary to adjust the integral time constant Ti to his maximum value and the differential time constant Td to his minimum value. Following Kp will be adjusted until to obtain the desired response values. 2. Integral action setting: After the proportional gain has been adjusted would proceed to reduce the integral time constant Ti to reduce the steady state error (objective of this action). An important swaying could appear. It would reduce the gain slightly and repeat the process until to obtain the desired response values​​. 3. Derivative or differential action setting: Keeping the previously set values ​​would increase the value of Td until to obtain a faster response. If necessary would increase slightly the gain value.

15. PID controllersTuning rules • Chien, Hrones, and Reswick method (CHR): for tuning a PID controller by this method, the parameters a and L will be determined in the same mode than Ziegler-Nichols method, and the controller parameters will be obtained in function of these values. • This method determines various settings in function of the percentage of overshoot and according if are obtained from the response to load disturbances or variations in the set point. The values of the controller parameters ​​obtained from the response to load disturbances are the indicated in the following table:

16. PID controllersTuning rules • The controller parameters obtained from the response to variations in the set point are the indicated in the following table:

17. PID controllersTuning rules • Cohen-Coon method: this method is based on a model of the process such that: The main design criterion is the rejection of load disturbances, and also provides a table of values ​​based on analytical and numerical calculations. considering:

18. PID controllersTuning rules • Empirical tuning based on rules: the methods described above are approximate methods that require a posterior manual tuning. This manual tuning is done on the closed loop response, introducing a disturbance (change in the set point, change in the control variable, etc..), analyzing and correcting the response of the controller parameters. These adjustments are based on simple rules, developed from a process of experimentation. These rules are: - Increasing the proportional gain decreases the stability. - The error decays faster if we decrease the integration time constant. - Decreasing the integration time constant decreases the stability. - Increasing the derivative time constant increases the stability. • As can be appreciated, changing a parameter in one direction or another affects the tuning differently. It is therefore common to use tuning maps which objective is to show, intuitively, how the changes in the controller parameters affect to the behaviour of the closed loop system. In this mode may be defined boundaries within which can move parameter values or limit values for which the system becomes unstable. These tuning rules were implemented in automatic tuning process.

19. PID controllersControllers design • Basic design: for a basic design, first we will choose the system topology, that is, where to locate the controller: serial, parallel. After selecting the controller settings, we must choose a type of controller that meets the required specifications, the most widely used is the PID controller. Thus the complexity of the controller is restricted. You can also use different compensators (lead, lag or lag-lead).

20. PID controllersControllers design • After selecting the controller, and according to the required specifications, we must to determine its parameters, for which we will choose the analysis method most appropriate to be used, according to specifications: root locus approach or frequency response analysis. • Finally we should check that the designed system meets the required function and, if necessary, adjust the parameters to make it so.

21. PID controllersControllers design • Improved design: sometimes it is desirable to move some of the actions of the controller to the feedback loop. For example, if a PID controller has a step input, the derivative action makes that appears a pulse in the control. Therefore other possible configurations are used to avoid the problems that can occur.

22. PID controllersControllers design • PI-D Controller: avoids the phenomenon of the set point reaction. Are avoided aggressive control actions that may damage the actuators. The action becomes slower and reduces the over shoot.

23. PID controllersControllers design • I-PD controller: the proportional and derivative actions are only on the feedback loop. With this kind of controller, with a step signal do not produce an abrupt change in the control signal, which may not be suitable for some types of actuators.

24. PID controllersControllers design • Integral control with state feedback configuration: allows a finest control if increases the order of the plant. • PI-PD, PID-PD controllers: The PI-PD controller provides an excellent four-parameters controller for control of integrating, unstable and resonant processes. • Feed-forward control: allows to measure the disturbances and perform a corrective action as soon as the disturbance appears.

25. PID controllersControllers design • With PID controllers can appear some problems, such as: - The tuning: the choice of values ​​of the parameters Kp, Ti and Td can become complicated, in fact, it was observed that according to the tuning method chosen these values ​​may be different. - Integrator windup effect: this effect appears when the control action increases so much that produce the saturation in the actuators. This problem breaks the control loop because the actuator will remain at its limit value regardless of the control signal. The control will not be effective until the control signal falls below the saturation level. To eliminate this effect there are many controller configurations that allow to override the problem.

26. PID controllersControllers design • Robust design: the first step in designing a control system is to design an equivalent mathematical model of the physical plant. Sometimes this model may not be linear or too complex. A complex model is not very useful because complicates excessively the design process. For this reason normally the design is done by simple models but which reflect the intrinsic characteristics of the physical system. Obviously this solution generates an uncertainty whether the designed control will be suitable for the required function. Uncertainty can be of different types depending on the features that are deemed (stability, gain margin, phase margin, etc..). Uncertainty is added to the nominal model and may be treated in different ways.

27. PID controllersControllers design • In robust control theory we expect to approximate the plant model by a linear model with constant coefficients, knowing it will be incurred an error that is intended to be bounded. In this way we can design control techniques valid for multivariable systems, to ensure at least the system stability. • In a robust control the system must to be stable for the entire plant situations and the performance must meet the design specifications for all possible situations. that is, in the presence of uncertainty.

28. PID controllersControllers design • It is therefore essential to determine the type of uncertainty (parametric, structured, unstructured) to determine its size and importance, and to enclose it. To estimate the uncertainty we can observe the data of the sensors and actuators, to experiment with the system at different operating points, etc. • For multivariable systems should choose the closest description to the effect that causes uncertainty. Robustness margin is related to the specific type of model uncertainty. • The control with two degrees of freedom: Allows to adjust the closed loop and feedback loop characteristics independently, which improves system response.

29. PID controllersControllers design • Design by pole placement method: many of the properties of a system can be expressed by its poles. With this we expect to design a controller so that the closed loop system has poles at the desired location. The method requires a complete model of the process. Is possible to find a controller that gives the closed-loop poles desired, provided that the controller is sufficiently complex. For a PID control is necessary to restrict the complexity of the model using approximate methods, therefore, the selected poles must be chosen to ensure that the model is valid. • The design process will depend on the type of controller (PI, PID) and the characteristics of the system (number of poles, order, oscillatory system, etc.)

30. PID controllersControllers design • Dominant poles design: is a simplification of the above method. Sometimes is difficult to specify all the closed loop poles, so that the dominant poles are used to characterize the system. • Lambda tuning: is a special case of the poles placement method. It is a simple method that can give good results in certain circumstances if the design parameter is chosen appropriately. The basic method cancels a pole of the process, which will result in a poor response of the load to disturbances in process dominated by time constant.

31. PID controllersControllers design • Algebraic design: is a process in which the transfer function of the controller is obtained from the specifications by a direct algebraic method. There are several methods, all of them related to the allocation of poles. - Standard forms: It begins by determining a transfer function in a certain way, calculating her parameters in mode to minimize the error criterion chosen. - Haalman method: for systems with a delay L, Haalman proposed to select a loop transfer function of the form depending on the transfer function of the process and applying this method will be easy to determine the parameters and the type of controller.

32. PID controllersControllers design • Internal model control (IMC): its name comes from the fact that the controller contains internally a model of the process. This model ( Pm(s) ) is connected in parallel with the process and applies the inverse of Pim(s) model and a filter (Gf). It is considered that all disturbances affecting the process are reduced to an equivalent disturbance d. The controller obtained can be represented by the following function, from which is deduced that this type of serial controller cancels the poles and zeros of the process.

33. PID controllersControllers design • Design for disturbance rejection: in the above methods we have only taken into account the characterization of the dynamics of the process, without considering the disturbances directly. For the study we have used a step disturbance and analyzed the response of the system, but not all the disturbances have this form. Furthermore we have not taken into account the amplification of the measurement noise in the feedback. With a design oriented to interference rejection we try to obtain a compromise solution between the attenuation of load disturbances and the amplification of measurement noise due to the feedback.

34. PID controllersOperation conditions • There are many requirements that are demanded to a control system. About 95% of the installed controllers are PID type which shows that these controllers work properly if the requirements are not too strictest. Usually most stable process can be controlled using a PI controller, in fact, it is frequent that the derivative action is not used. This occurs in the majority of first-order processes. • The PID controller is usually sufficient for second order processes and processes with a delayin where the derivative action may be sufficient to accelerate the system response. • Not will be sufficient for higher-order processes, in systems with large delays or in systems with oscillatory modes. For these cases it is advisable to use more sophisticated control systems that the PID.

35. References Bibliography • Karl J. Åström. Control PID Avanzado.  • Karl J. Åström. PID Controllers: Theory, Design and Tunning. Interesting links • http://www.dia.uned.es/~fmorilla/MaterialDidactico/El%20controlador%20PID.pdf • http://www.robolabo.etsit.upm.es/asignaturas/sctr/apuntes/transparencias/PID.pdf • http://www.imac.unavarra.es/~jorge.elso/IA/apuntes/Diapositivas%20tema%208.pdf • http://www.imac.unavarra.es/~jorge.elso/ICR/apuntes/Diapositivas%20tema%202.pdf • http://www.isa.cie.uva.es/~fernando/papers/doctorado_parte1.pdf • http://www.siam.org/books/dc14/DC14Sample.pdf • http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction&section=ControlPID • http://link.springer.com/article/10.1007%2Fs12555-009-0203-y#page-1