2. PID tuning by classical method

9.3: PID tuning by the classical method

original

introduce

Currently, more than half of the controllers used in industry are PID controllers. In the past, many of these controllers were analog; however, many of today's controllers use digital signals and computers. When a mathematical model of the system is available, the parameters of the controller can be explicitly determined. However, when mathematical models are not available, parameters must be determined experimentally. Controller tuning is the process of determining the controller parameters that produce the desired output. Controller tuning allows to optimize the process and minimize the error between the process variable and its set point.

Types of controller tuning methods include trial and error and process response curve methods. The most common classical controller tuning methods are the Ziegler-Nichols and Cohen-Coon methods. These methods are typically used when a mathematical model of the system is not available. The Ziegler-Nichols method can be used for both closed-loop and open-loop systems, while Cohen-Coon is usually used for open-loop systems . A closed loop control system is a system that uses feedback control. In an open loop system, the output is not compared with the input.

The formula below shows the PID algorithm discussed in the previous PID Control section.

$$u(t)=K_c\left(ϵ\left(t\right)+\frac{1}{t_i}{\int }_0^tϵ\left(t^{\prime }\right)d{t}^{\prime }+t_d\frac{dϵ(t)}{dt}\right)+b$$

where

• u is the control signal
• ε is the difference between the current value and the set value.
• $K_c$is the gain of the proportional controller.
• $T_i$is the parameter of the scaled integral controller.
• $T_d$is a parameter of the scaling derived controller.
• t is the time taken for the error measurement.
• b is the setpoint value of the signal, also known as bias or offset.

The experimentally obtained controller gain (providing a stable and consistent oscillation for the closed-loop system) or the final gain is defined as $K_u$。$K_c$is the controller gain that has been corrected by the Ziegler-Nichols or Cohn-Kuhn method and can be entered into the above equation. $K_u$It is found through experiments that from $K_c$and adjust upwards until a consistent oscillation is obtained, as shown below. If the gain is too low, the output signal will be damped and will eventually reach equilibrium after interference occurs, as shown below.

On the other hand, if the gain is too high, the oscillations become unstable and become larger and larger over time, as shown below.

The Process Response Curve Methods section shows the parameters required for open-loop system calculations. The Ziegler-Nichols method section shows, for both open-loop and closed-loop systems, how to find $K_c$,$T_i$and * $T_d$*, the Cohen-Coon section shows another way to find $K_c$,$T_i$and $T_d$.

Open-loop systems typically use a quarter damping ratio (QDR) for oscillation damping. This means that the ratio of the magnitude of the first overshoot to the magnitude of the second overshoot is 4:1.

trial

Trial and error tuning methods are based on guess and check. In this approach, proportional action is the primary control, while integral and calculus actions refine it. Controller gain, $K_c$, making adjustments, keeping the integral and derivative actions to a minimum, until the desired output is obtained.

The following are some common values ​​of $K_c$,$T_i$and $T_d$Used to control flow, level, pressure or temperature for trial and error calculations.

flow

P or PI control can be used at low controller gains. Gain greater accuracy with highly integrated activities using PI control. Derivative control was not considered due to the fast fluctuations and large amount of noise in the flow dynamics.

$K_c$= 0.4-0.65

$T_i$= 6s

level

Either P or PI control can be used, but PI control is more common due to the inaccuracy caused by the offset in P-only control. Derivative control was not considered due to the fast fluctuations and large amount of noise in the flow dynamics.

Only the P setting below is such that the control valve is fully open when the vessel is 75% full, fully closed when the vessel is 25% full, and half open when the vessel is 50% full.

$K_c$= 2

bias deviation b = 50%

set point = 50%

For PI control:

$K_c$= 2-20

$T_i$= 1-5 minutes

pressure

The tuning here has a large range of possible values $K_c$Sum $T_i$For PI control, depending on whether the pressure measurement is liquid or gas phase.

Liquid
$K_c$= 0.5-2

$T_i$= 6-15 seconds

Gas
$K_c$= 2-10

$T_i$= 2-10 minutes

temperature

Since the temperature sensor responds relatively slowly to dynamic temperature changes, a PID controller is used.

$K_c$= 2-10

$T_i$= 2-10 minutes

$T_d$= 0-5 minutes

Treatment Response Curve

In this approach, the variables to be measured are those of an already existing system. A disturbance is introduced into the system and data can then be obtained from this curve. The system is first allowed to reach steady state, followed by disturbances, $X_o$, is introduced to it. Percent disturbances to the system can be introduced by changes in set points or process variables. For example, if you have a thermometer that you can only turn up or down 10 degrees, then raising the temperature by 1 degree will cause a 10% disturbance to the system. These types of curves are obtained in open-loop systems without system control, allowing disturbances to be recorded. The process response curve method typically yields a response to a step function change and can measure several parameters including: transport lag or dead time, $T_{d}ead$ , the time τ of the response change and the limit value of the response in the steady state,Mu.

$T_{d}ead$ = Transmission Lag or Dead Time: Time elapsed from the moment a disturbance is introduced to the first sign of change in the output signal

τ = time for the response to occur

$X_o$= size of step change

Mu = value at which the system responds when it returns to steady state

$$R=\frac{t_{\text{dead}}}{t}$$

$$K_o=\frac{X_o}{M_u}\frac{}{t_{\text{dead}}}$$

An example of these parameters determining a typical process response curve to a step change is shown below.
To find $T_{d}For\; the\; values\; of\; ead$ and T, draw a line at the inflection point tangent to the response curve, and find these values ​​from the graph.

To map these parameters to the P, I, and D control constants, see Tables 2 and 3 in the ZN and Cohen Coon sections below.

Ziegler-Nichols method

In the 1940s, Ziegler and Nichols devised two empirical methods for obtaining controller parameters. Their method is used for non-first-order plus dead-time cases and involves intensive manual calculations. With improvements in optimization software, most of these manual methods are no longer used. However, even with computer aids, the following two methods are still used today and are considered to be among the most common:

Ziegler-Nichols closed-loop tuning method

The Ziegler-Nichols closed-loop tuning method allows you to use the limiting gain value, $K_u$, and the final period of oscillation, Pu , to calculate $K_c$. This is an easy way to tune a PID controller, which can be improved to provide a better approximation of the controller. You can get the controller constant $K_c$,$T_i$and * $T_d$*In systems with feedback. The Ziegler-Nichols closed-loop tuning method is limited to tuning procedures that cannot be run in an open-loop environment.

Determine the final gain value, $K_u$, by finding the value of the proportional-only gain that causes the control loop to oscillate infinitely in steady state. This means that the gains of the I and D controllers are set to zero in order to determine the effect of P. It tests $K_c$value to optimize it for the controller. Another important value associated with this proportional-only control tuning method is the terminal period ( Pu ). The terminal period is the time required to complete one full oscillation when the system is in steady state. These two parameters, $K_u$and Pu to find the loop tuning constants of the controller (P, PI or PID). To find the values ​​of these parameters and calculate tuning constants, follow these steps:

Closed Loop (Feedback Loop)

1. Remove integral and derivative actions. Set the integration time ( $T_i$) to 999 or its maximum value, and set the derivative controller ( $T_d$) to zero.
2. Create small disturbances in the loop by changing the set point. Adjust the gain proportionally, increasing and/or decreasing the gain until the oscillations have a constant amplitude.
3. Record the gain value ( $K_u$) and oscillation period ( Pu ).

Figure 1. System tuning using the Ziegler-Nichols closed-loop tuning method

4. Substitute these values ​​into the Ziegler-Nichols closed-loop equation and determine the necessary settings for the controller.

Table 1. Closed loop calculation $K_c$,$T_i$,$T_d$

Advantage

1. Simple experiment; only P controller needs to be replaced
2. Including the dynamics of the entire process allows for a more accurate understanding of how the system behaves

Disadvantages

1. experimentation can be time consuming
2. It is possible to venture into unstable regions when testing the P controller, which may cause the system to go out of control

Ziegler-Nichols open-loop tuning method or process response method:

This method remains a common technique for tuning controllers that use proportional, integral, and derived operations. The Ziegler-Nichols loop-opening method is also known as the process response method because it tests the change in the loop-opening response of the process to the output of the controlled variable. This basic test requires recording the system's response, preferably with a plotter or computer. Once certain process response values ​​are found, they can be substituted into the Ziegler-Nichols equation with specific multiplier constants to obtain the gain of a controller with P, PI or PID action.

Open loop (feedforward loop)

To use the Ziegler-Nichols open-loop tuning method, the following steps must be performed:

1. Performing an Open Loop Step Test

2. Determine the transport lag or dead time from the process response curve, $T_{d}ead$ , the time constant or time τ of response change, and the extreme value reached by the response in steady state,Mu, for$X_o$step change.
$$K_o=\frac{X_o}{M_u}\frac{T}{T_{d}ead}$$

3. Determine the cycle tuning constant. Substitute the reaction rate and dead time values ​​into the Ziegler-Nichols open-loop tuning equation for the corresponding controller (P, PI, or PID) to calculate the controller constants. Use the table below.

Table 2. Open loop calculation $K_c$,$T_i$,$T_d$

Advantage

1. Faster and easier to use than other methods
2. This is a powerful and popular method
3. Of the two techniques, the process reaction method is the easiest to implement and the least disruptive.

Disadvantages

1. It relies on purely proportional measurements to estimate I and D controllers.
2. Approximate $K_c$,$T_i$and * $T_d$*Values ​​may not be completely accurate for different systems.
3. It is not available for I, D and PD controllers

Cohen-Coon method

The Cohen-Coon method of controller tuning corrects for the slow steady-state response given by the Ziegler-Nichols method when there is a large dead time (process delay) relative to the open-loop time constant; in order for this Approaches that are practical require large process delays that would otherwise predict unreasonably large controller gains. This method is only used for first-order models with time delays because the controller does not respond instantaneously to disturbances (step disturbances are asymptotic rather than instantaneous).

The Cohen-Coon method is classified as an "offline" tuning method, meaning that a step change can be introduced to the input once it is in steady state. The output can then be measured as a function of the time constant and time delay, and this response can be used to estimate initial control parameters.

For the Cohen-Coon method, there is a predetermined set of settings to obtain a minimum offset and a standard attenuation ratio of 1/4 (QDR). The 1/4 (QDR) decay ratio refers to the response in which the oscillations are reduced such that the amplitude of the second oscillation will be 1/4 of the amplitude of the first oscillation. These settings are shown in Table 3.

Table 3. Standard Recommendation Equations for Optimizing Cohn-Kuhn Forecasts

The variables P, N and L are defined as follows.

Or, $K_0$Can be substituted for (P/NL). $K_0$、τ 和 $T_{d}ead$ are defined in the Process Response Curve section. An example of using these parameters is shown here [1].

The process of the Kohn-Kuhn turning method is as follows:

1. Wait until the process reaches a steady state.
2. Introduces a step change in the input.
3. From the output, an approximate first-order process of the time constant τ delay τ is obtained when the input step is introduced into the unit .

The values ​​of τ and τ can be obtained by first recording the following time instances:

$t_0$= time $t_2$= time to half o'clock $t_3$= time to reach the 63.2% point

4. use $t_0$，$t_2$，$t_3$, measured values ​​at A and B, evaluate process parameter τ, $T_{d}ead$ sumKo.
5. According to τ, τ, find the controller parameters and Ko .

Advantage

1. For systems with time delay.
2. Faster closed loop response time.

Drawbacks and Limitations

1. Unstable closed-loop system.
2. Can only be used for first-order models that include large process delays.
3. offline method.
4. Approximate value $K_c$0 $X_i$and $T_d$Values ​​may not be completely accurate for different systems.

Other methods

These are other common methods used, but they can be complex and are not considered classical, so they are only briefly discussed.

internal model control

The Internal Model Control (IMC) method was developed with robustness in mind. Ziegler-Nichols open-loop and Cohen-Kuhn method controllers have large gain and short integration time, which are not conducive to chemical engineering applications. The IMC method involves closed-loop control without overshoot or oscillatory behavior. However, for systems with first-order dead times, the IMC method is very complicated.

Autotune changes

The automatic tuning variation (ATV) technique is also a closed-loop method for determining two important system constants ( Pu and $K_u$For example). These values ​​can be determined without disturbing the system, and the tuning value of the PID can be obtained from them. The ATV method is only suitable for systems with a large amount of dead time or the resulting period, Pu , will be equal to the sampling period.

Example 9.3.1

You're a controls engineer working at Perfect Design when your best controller fails. As a backup, you think that by using a cursory knowledge of classical methods, you can sustain the development of your product. After adjusting the gain to a set of data obtained from the controller, you will find that the final gain is 4.3289.

From the tuning graph below, determine the type of cycle this graph represents; then, calculate $K_c$,$T_i$and * $T_d$*Applies to all three types of controllers.

the solution

From the fact that this graph oscillates rather than a step function, we see that this is a closed loop. Therefore, the values ​​will be calculated accordingly.

We get the final yield, $K_u$= 4.3289. From the figure below, we see that the final period of this gain is Pu = 6.28

From this, we can calculate $K_c$,$T_i$and * $T_d$*Applies to all three types of controllers. The results are shown in the table below. (Results are calculated from the Ziegler-Nichols closed-loop equation.

Example 9.3.29.3.2

Your partner finds another set of data after a controller failure and decides to use the Cohen-Coon method because of the slow response time of the system. They also noticed that the control dial from 0 to 8 was set to 3 instead of 1. Fortunately, the response curve was obtained earlier, as shown below. From these data, he wants to calculate $K_c$,$T_i$and * $T_d$*. Help him determine these values. Note that the y-axis is the percent change in the process variable.

the solution

In order to solve $K_c$,$T_i$and * $T_d$*, L, ΔCp and T must first be determined. All these values ​​can be calculated from a given reaction curve.

From the process response curve, we can find:
L=3
T=11
\(Δ C_p= 0.55\, (55%)\)

Now that these three values ​​have been found, N and R can be calculated using the equations below.
$$N=\frac{C{\_}_p}{T}$$

$$R=\frac{L}{T}=\frac{NL}{C{\_}_p}$$

Using these equations, you find that
N = .05
R = 0.27
We also know that there is a 200% change due to the controller moving from 1 to 3.
P = 2.00

We use these values ​​to calculate $K_c$,$T_i$and * $T_d$*, for the three types of controllers based on the formulas in Table 3.

Tuning the parameters of the PID controller on the real system

There are several ways to tune the parameters of a PID controller. They involve the following procedures. For each, name the procedure and explain how the given measurement information is used to select the parameters of the PID controller.

Example 9.3.19.3.1

1. The controller is only set to P and the system operates in "closed loop", which means the controller is connected and working. Gain is tuned until resonance is obtained. The amplitude and frequency of this resonance are measured.
2. The system is kept in "open loop" mode, step function changes are manually made to the system (either by disturbance or by the controller itself). The resulting response of the system is recorded as a function of time.

the solution

one. We will use the Ziegler-Nichols method.

$K_i$=0.5$K_u$
$K_u$is the final gain when the system starts to oscillate.

2. We will use the Cohen-Coon method.

We can locate the inflection point of the step function and draw the tangent. $T_{d}ead$ lies at the intersection of this tangent with t and Τ lies at the intersection of the tangent of M(t)

Exercise 9.3.1

Which of the following do you document in the Ziegler-Nichols method?

1. $K_c$
2. $T_i$
3. $K_o$
4. $T_d$

• Answer : C

Exercise 9.3.2

For the Ziegler-Nichols method, it is important that:

1. Find the gain that produces ringing
2. Set P and I controllers to zero
3. record oscillation period
4. Calculate Tc

• Answer : A, C