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PID Controllers in Control Systems Assistant Professor Dr. Khalaf S Gaeid Electrical Engineering Department/Tikrit University khalafgaeid@tu.edu.iq gaeidkhalaf@gmail.com +9647703057076 April 2018
Contents 1.Introduction 2. Requirements of a Good Control System 3.Controller Modes 4. The Characteristics of P, I, and D Controllers 5. Mathematical form 6. Electronic analogue controllers 7.Limitations of PID control 8.The tuning parameters essentially determine 9.Electronic PID Controllers 10. PID Tuning 11. PID Controllers and Compensation 12.Conclusions
1.Introduction A controller is a device that generates an output signal based on the input signal it receives. The input signal is actually an error signal, which is the difference between the measured variable and the desired value as can be shown in feedback control system ( Fig.1). Fig.1.Feedback control system
A sensor measures and transmits the current value of the process variable(PV) back to the controller. ▪ Controller error(e(t)) at current time t is computed as set-point(SP) minus measured process variable as in (1). e(t) = SP – PV (1) ▪ The controller uses this e(t) in a control algorithm to compute a new controller output signal. ▪ The controller output signal is sent to the final control element (e.g. valve, pump,heater, fan) causing it to change. ▪ The change in the final control element causes a change in a manipulated variable ▪ The change in the manipulated variable (e.g. flow rate of liquid or gas) causes a change in the PV
1.Introduction A proportional-integral-derivative controller (PID controller or three term controller) is a control loop feedback mechanism widely used in industrial control systems and a variety of other applications requiring continuously modulated control. A PID controller continuously calculates an error value e(t) as the difference between a desired set point (SP) and a measured process variable (PV) and applies a correction based on proportional, integral, and derivative terms (denoted P, I, and D respectively) which give the controller its name.
The first theoretical analysis and practical application was in the field of automatic steering systems for ships, developed from the early 1920s onwards. It was then used for automatic process control in manufacturing industry, where it was widely implemented in pneumatic, and then electronic, controllers. Today there is universal use of the PID concept in applications requiring accurate and optimized automatic control. PID control is widely used in all areas where control is applied(solves(90%of all control problems).
2. Requirements of a Good Control System The essential requirements of a good Control System can be listed as follows: 1) Accuracy: Accuracy must be very high as error arising should be corrected. Accuracy can be improved by the use of feedback element. 2) Sensitivity: A good control system senses quick changes in the output due to an environment, parametric changes, internal and external disturbances. 3) Noise: Noise is a unwanted signal and a good control system should be sensitive to these type of disturbances.
2. Requirements of a Good Control System 4) Stability: The stable systems has bounded input and bounded output. A good control system should response to the undesirable changes in the stability. 5) Bandwidth: To obtain a good frequency response, bandwidth of a system should be large. 6) Speed: A good control system should have high speed that is the output of the system should be fast as possible. 7) Oscillation: For a good control system oscillation in the output should be constant or at least has small oscillation.
3.Controller Modes In industry there are many control modes as follows: 1. ON-OFF controller/two position controller as temperature controller used for domestic heating system. 2. Three-position controller 3. Proportional Action Control 4. Integral/Reset Action Control 5. Derivative/Rate Action Control 6. P+I Control 7. P+D Control 8. P+I+D Control
P_Controller • P depends on the present error • I on the accumulation of past errors I _Controller •D is a prediction of future errors, based on current rate of change D_Controller So the importance of PID comes from the above
4. The Characteristics of P, I, and D Controllers A proportional controller (Kp ) will have the effect of reducing the rise time and will reduce but never eliminate the steady state error. An integral control (Ki ) will have the effect of eliminating the steady-state error for a constant or step input, but it may make the transient response slower. A derivative control (Kd ) will have the effect of increasing the stability of the system, reducing the overshoot, and improving the transient response.
In fact, changing one of these variables can change the effect of the other two. With the PID controller we can set the P+I+D values so that we will not have any Over or undershoot and reach set point directly. PID controller has all the necessary dynamics: fast reaction on change of the controller input (D mode), increase in control signal to lead error towards zero (I mode) and suitable action inside control error area to eliminate oscillations (P mode). This combination of{Present + Past + Future} makes it possible to control the application very well.
Proportional Controller In a proportional controller the output (also called the actuating/control signal) is directly proportional to the error signal. The position of Kp can be as shown in Fig.2. Fig.2. Proportional controller Control signal = Kp * e(t) (2) If the error signal is a voltage, and the control signal is also a voltage, then a proportional controller is just an amplifier.
Properties of proportional controller: In a proportional controller, steady state error tends to depend inversely upon the proportional gain, so if the gain is made larger the error goes down as in (3). (3) Where SSE is the steady state error Proportional controller helps in reducing the steady state error, thus makes the system more stable. Slow response of the over damped system can be made faster with the help of these controllers. P controller has the advantage of reducing down the steady state error of the system , but along with that it also has some serious disadvantages. These properties can be shown in Fig.3.
Fig.3.Response of PV to step change of SP vs time, for three values of Kp (Ki and Kd held constant)
Disadvantages of P Controller 1. Due to presence of these controllers we have some offsets in the system. 2. Proportional controllers also increase the maximum overshoot of the system. 3. It directly amplifies process noise. To avoid these errors and to make the controller more accurate and practical, we use the advanced and modified version of it known as the Proportional Integral Controllers (PI) and Proportional Derivative Controllers (PD).
Integral An integral term increases action in relation not only to the error but also the time for which it has persisted. So, if applied force is not enough to bring the error to zero, this force will be increased as time passes. A pure "I" controller could bring the error to zero, however, it would be both slow reacting at the start (because action would be small at the beginning, needing time to get significant), brutal (the action increases as long as the error is positive, even if the error has started to approach zero), and slow to end (when the error switches sides, this for some time will only reduce the strength of the action from "I", not make it switch sides as well), prompting overshoot and oscillations (see Fig.4). Moreover, it could even move the system out of zero error: remembering that the system had been in error, it could prompt an action when not needed. An alternative formulation of integral action is to change the electric current in small persistent steps that are proportional to the current error. Over time the steps accumulate and add up dependent on past errors; this is the discrete-time equivalent to integration.
Fig.4. Response of PV to step change of SP vs time, for three values of Ki (Kp and Kd held constant)
Derivative A derivative term does not consider the error (meaning it cannot bring it to zero: a pure D controller cannot bring the system to its set-point), but the rate of change of error, trying to bring this rate to zero. It aims at flattening the error trajectory into a horizontal line, damping the force applied, and so reduces overshoot (error on the other side because too great applied force). Applying too much impetus when the error is small and is reducing will lead to overshoot. After overshooting, if the controller were to apply a large correction in the opposite direction and repeatedly overshoot the desired position, the output would oscillate around the set-point in either a constant, growing, or decaying sinusoid. If the amplitude of the oscillations increase with time, the system is unstable. If they decrease, the system is stable. If the oscillations remain at a constant magnitude, the system is marginally stable. This can be illustrated in Fig.5.
Fig.5. Response of PV to step change of SP vs time, for three values of Kd (Kp and Ki held constant)
5. Mathematical form The overall control function can be expressed mathematically as in(4) (4) where Kp , Ki , and Kd , all non-negative, denote the coefficients for the proportional, integral, and derivative terms respectively (sometimes denoted P, I, and D). In the standard form of the equation (see later in article), Ki and Kd are respectively replaced by Kp/Ti and Kd*Td ; the advantage of this being that Ti and Td have some understandable physical meaning, as they represent the integration time and the derivative time respectively the above relationship is obtained from parallel configuration of PID controller as illustrated in Fig.6.
Although a PID controller has three control terms, some applications use only one or two terms to provide the appropriate control. This is achieved by setting the unused parameters to zero and is called a PI, PD, P or I controller in the absence of the other control actions. PI controllers are fairly common, since derivative action is sensitive to measurement noise, whereas the absence of an integral term may prevent the system from reaching its target value. Fig.6. Parallel configuration of PID controller
6. Electronic analogue controllers Electronic analog PID control loops were often found within more complex electronic systems, for example, the head positioning of a disk drive, the power conditioning of a power supply, or even the movement-detection circuit of a modern seismometer. Discrete electronic analogue controllers have been largely replaced by digital controllers using microcontrollers or FPGAs, to implement PID algorithms. However, discrete analog PID controllers are still used in niche applications requiring high-bandwidth and low-noise performance, such as laser-diode controllers.
7.Limitations of PID control While PID controllers are applicable to many control problems, and often perform satisfactorily without any improvements or only coarse tuning, they can perform poorly in some applications, and do not in general provide optimal control. The fundamental difficulty with PID control is that it is a feedback control system, with constant parameters, and no direct knowledge of the process, and thus overall performance is reactive and a compromise. While PID control is the best controller in an observer without a model of the process, better performance can be obtained by overtly modeling the actor of the process without resorting to an observer. PID controllers, when used alone, can give poor performance when the PID loop gains must be reduced so that the control system does not overshoot, oscillate or hunt about the control set point value.
7.Limitations of PID control They also have difficulties in the presence of non-linearities, may trade-off regulation versus response time, do not react to changing process behavior (say, the process changes after it has warmed up), and have lag in responding to large disturbances. The most significant improvement is to incorporate feed-forward control with knowledge about the system, and using the PID only to control error. Alternatively, PIDs can be modified in more minor ways, such as by changing the parameters (either gain scheduling in different use cases or adaptively modifying them based on performance), improving measurement (higher sampling rate, precision, and accuracy, and low-pass filtering if necessary), or cascading multiple PID controllers.
Closed-loop Response performance depends on the effects of PID parameters as can be shown in the following table when the parameters are increased as can be shown in table1. Table1. Increasing PID controller parameters with the performance of the closed loop system • Note that these correlations may not be exactly accurate, because P, I and D gains are dependent of each other.
8. The tuning parameters essentially determine: How much correction should be made? The magnitude of the correction (change in controller output) is determined by the proportional mode of the controller. How long the correction should be applied? The duration of the adjustment to the controller output is determined by the integral mode of the controller. How fast should the correction be applied? The speed at which a correction is made is determined by the derivative mode of the controller.
8. PID Tuning Algorithm Typical PID tuning objectives include: Closed-loop stability: The closed-loop system output remains bounded for bounded input. Adequate performance: The closed-loop system tracks reference changes and suppresses disturbances as rapidly as possible. The larger the loop bandwidth (the frequency of unity open-loop gain), the faster the controller responds to changes in the reference or disturbances in the loop. Adequate robustness: The loop design has enough gain margin and phase margin to allow for modeling errors or variations in system dynamics.
9.Electronic PID Controllers Electronic PID controllers can be obtained using operational amplifiers and passive components like resistors and capacitors. A typical scheme is shown in Fig.7. Fig. 7. Electronic PID controller
It is evident from Fig. 5, the proportional gain Kp is decided by the ratio R2/R1 of the first amplifier; the integral action is decided by R3 and C1 and the derivative action by R5 and C2. The final output however comes out with a negative sign, compared to eqn.(1) (though the positive sign can also be obtained by using a noninverting amplifier at the input stage, instead of the inverting amplifier). The op. amps. shown in the circuits are assumed to be ideal.
10. PID Tuning Users of control systems are frequently faced with the task of adjusting the controller parameters to obtain a desired behavior. There are many different ways to do this. One approach is to go through the conventional steps of modeling and control design as described in the previous section. Since the PID controller has so few parameters, a number of special empirical methods have also been developed for direct adjustment of the controller parameters. The first tuning rules were developed by Ziegler and Nichols. Their idea was to perform a simple experiment, extract some features of process dynamics from the experiment and determine the controller parameters from the features
Ziegler–Nichols tuning rules (Table2). (a) The step response methods give the parameters in terms of the intercept a and the apparent time delay τ. (b) The frequency response method gives controller parameters in terms of critical gain kc and critical period Tc. Table2. Ziegler–Nichols tuning rules (a) Step response method (b) Frequency response method by extensive simulation of a range of representative processes. A controller was tuned manually for each process, and an attempt was then made to correlate the controller parameters with a and τ
11. PID Controllers and Compensation The series configuration of PID control consists of a proportional plus derivative (PD) compensator cascaded with a proportional plus integral (PI) compensator. The purpose of the PD compensator is to improve the transient response while maintaining the stability. The purpose of the PI compensator is to improve the steady state accuracy of the system without degrading the stability. Since speed of response, accuracy, and stability are what is needed for satisfactory response, cascading PD and PI will suffice.
11. PID Controllers and Compensation Lead/Lag compensation is very similar to PD/PI, or PID control. The lead compensator plays the same role as the PD controller, reshaping the root locus to improve the transient response. Lag and PI compensation are similar and have the same response: to improve the steady state accuracy of the closed-loop system. Both PID and lead/lag compensation can be used successfully, and can be combined.
12.Conclusions • • Proportional action gives an output signal proportional to the size of the error. Increasing the proportional feedback gain reduces steady-state errors, but high gains almost always destabilize the system. • • Integral action gives a signal which magnitude depends on the time the error has been there. Integral control provides robust reduction in steady-state errors, but often makes the system less stable. • • Derivative action gives a signal proportional to the change in the Error. It gives sort of “anticipatory” control .Derivative control usually increases damping and improves stability, but has almost no effect on the steady state error • • These three kinds of control combined from the classical PID controller. • PID can be implemented in Hardware and software. • The PI controller can be considered as Lag compensator, The PD controller can be considered as lead compensator and PID same as Lag-Lead compensator works to improve transient and steady state region.
12.Conclusions • The tuning of the controller is one of the limitations of PID controller. • Proportional and integral control modes are essential for most control loops, while derivative is useful only in some cases. • Designing and tuning a PID controller appears to be conceptually intuitive, but can be hard in practice, if multiple (and often conflicting) objectives such as short transient and high stability are to be achieved. • Control engineers usually prefer P-I controllers to control first order plants. On the other hand, P-I-D control is vastly used to control two or higher order plants. • The major reasons behind the popularity of P-I-D controller are its simplicity in structure and the appilicability to variety of processes. Moreover the controller can be tuned for a process, even without detailed mathematical model of the process.
12.Conclusions • The choice of P-D, P-I or P-I-D structure de pends on the type of the process we intend to control. • There are few more issues those need to be addressed while using P-I controller. The most important among them is the anti-windup control.
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