1. PID control

introduce

Process control is necessary to design safe and efficient plants. Various process controls are used to operate a process, but the simplest and often most effective is the PID controller . The controller attempts to correct the error between the measured variable and the desired setpoint by calculating the difference, and then performs corrective actions to adjust the process accordingly. A PID controller controls a process through three parameters: proportional (P), integral (I), and derivative (D). These parameters can be weighted or adjusted to adjust their impact on the process.

PID controllers are more practical than typical on/off controllers, allowing for better tuning in the system. While this is true, there are some advantages to using on/off controllers, including that they are (1) relatively simple to design and implement, and (2) binary sensors and actuators (like on/off controllers) are generally more reliable and less expensive .

Despite some advantages, there are significant disadvantages to using an on/off controller scheme. They are (1) inefficient (using this control is like driving with full throttle and full rest), (2) noisy when seeking stability (may be significantly over or under set point), and ( 3) Physical wear and tear on valves and switches (continuously opening and fully closing valves/switches will cause them to wear out faster).

For better control and fine-tuning, most industrial processes use a PID controller scheme. This section will give a brief introduction to PID controllers and methods for modeling the controlled system in Excel.

Process gain ( $K$ ) is the ratio of the change in the output variable (the response variable) to the change in the input variable (the forcing function). It specifically defines the sensitivity of the output variable to a given change in the input variable.

$$K=\frac{\Delta \text{ Output }}{\Delta \text{input}}$$

Gain can only be described as a steady-state parameter, with no knowledge of the dynamics of the process, and independent of design and operating variables. Gain has three components including sign, value, and unit. This symbol indicates how the output responds to the process input. A positive sign means that the output variable increases with the increase of the input variable, and a negative sign means that the output variable decreases with the increase of the input variable. The units depend on the process considered, and the process depends on the variables mentioned.

For example, the pressure increases from 21 psi to 29 psi. This change changes the valve position from 30% vp to 22% vp.

$$K=\frac{(29-21)psi}{(22-30)\%vp}=-1.0psi/(\%vp)$$

Dead time ( $t_0$) is the value between the change in the input variable and the time the output variable starts. Dead time is important because it affects the controllability of the control system. Due to this parameter, changes in the setpoint do not happen immediately. Dead time must be considered during tuning and modeling.

control type

Process controls are instruments used to control parameters such as temperature, level, and pressure. PID controllers are a type of continuous controller because they continuously adjust the output versus on/off the controller while looking at feedforward or feedback conditions. An example of a temperature controller is shown in Figure 9.2.1

Figure 9.2.1: Temperature controllers in continuous reactors

As shown in Figure 1, the temperature controller controls the fluid temperature in the CSTR (Continuous Stirred Tank Reactor). The temperature sensor first measures the temperature of the fluid. This measurement produces a measurement signal. The measured signal is then compared to the controller's setpoint or desired temperature setting. The difference between the measured signal and the set point is the error. Based on this error, the controller sends an actuating signal to the heating coil, which adjusts the temperature accordingly. This type of process control is known as error-based control because the drive signal is determined by the error between the actual and desired settings. Different types of error-based control differ in the mathematical way they convert errors into drive signals, the most common of which is the PID controller. Also, understanding feedforward and feedback control is critical before exploring P, I, and D control.

I, D, PI, PD, PID control

As mentioned earlier, controllers vary in the way they relate the controller input (error) to the controller output (drive signal). The most commonly used controller is the proportional-integral-derivative (PID) controller . A PID controller relates an error to a drive signal in a proportional (P), integral (I) or derivative (D) manner. A PID controller can also use a combination of these controls to relate an error to a drive signal.

Proportional § Control

One type of action used in PID controllers is proportional control. Proportional control is a form of feedback control. It is the simplest form of continuous control and can be used in closed loop systems. P-only control minimizes fluctuations in the process variable , but does not always bring the system to the desired set point . It provides a faster response than most other controllers, initially making the P-only controller a few seconds faster. However, as the system becomes more complex (i.e. more complex algorithms), the response time difference may accumulate, making the P controller respond even minutes faster. While a P-only controller does have the advantage of a faster response time, it creates deviations from the set point . This deviation is called offset . The presence of an offset means that the system cannot maintain the desired set point in steady state. It is similar to systematic error in a calibration curve, where there is always a constant error to prevent the line from passing through the origin. Offset can be minimized by combining P-only control with another form of control such as I or D-control. However, it is important to note that it is not possible to completely remove the offset, which is implicit in every equation.

P-control linearly relates the controller output (drive signal) to error (difference between measured signal and set point). This P-control behavior is described mathematically in the formula.

$$c(t)=K_{c}e(t)+b\dots \dots \dots \dots \mathrm{9.2.1}$$

in

• c(t) = controller output
• $K_c$ = controller gain
• e(t) = error
• b = bias（

In this formula, bias (b) and controller gain $K_c$are constants specific to each controller. When the error is zero, the bias is just the controller output. Controller gain is the change in controller output every time the controller input changes. In a PID controller, the signal is usually transmitted electronically, and the controller gain relates a change in output voltage to a change in input voltage. These voltage changes are then directly related to the properties that are changing (i.e. temperature, pressure, liquid level, etc.). Therefore, gain is ultimately related to changes in input and output characteristics. If the output changes more than the input, Kc will be greater than 1. If the change in the input is greater than the change in the output, Kc will be less than 1. Ideally, if $K_c$Equal to infinity, the error can be reduced to zero. However, $K_c$This infinitesimal property increases loop instability, since zero error means that the measured signal is exactly equal to the set point. As mentioned in the lecture, exact equality is never achieved in control logic; instead, in control logic, errors are allowed to vary within a certain range. Therefore, for $K_c$There are limits to the size, which are defined by the system. A graphical representation of the impact of these variables on the system is shown in PID tuning by classical methods.

It can be seen from the above equation that only P control provides a linear relationship between the system error and the output of the system controller. This type of control provides a response based on a signal that adjusts the system so that any oscillations are eliminated and the system returns to a steady state. The inputs to the controller are setpoint, signal and bias. The controller calculates the difference between the set point and the signal, known as the error, and sends this value to the algorithm. Combined with the bias, the algorithm determines what action the controller should take. The graphical representation of the P controller output when the input step increases at time t0 is shown in Figure 2 below. This diagram is exactly the same as the step input diagram itself.

Figure 2. P controller output for step input.

To illustrate this linear P-control relationship, consider P-only control controlling the fluid level in a tank. Initially, the flow into the tank is equal to the flow out of the tank. However, if the flow out of the tank decreases, the fluid level in the tank will increase because more fluid is entering than leaving. The P only control system will adjust the flow out of the tank so that it is again equal to the flow into the tank and the level will again remain constant. However, this level is no longer equal to the initial level in the tank. The system is in steady state, but there is a difference between the initial set point and the current position in the tank. This difference is the P control offset.

Integral (I) Control

Another action used in PID controllers is integral control. Integral control is the second form of feedback control. It is often used because it removes possible biases . Thus, the system returns to the steady state and original settings. A negative error will result in a reduced signal to the system, while a positive error will result in an increased signal. However, I-only controllers have much slower response times than P-only controllers because they depend on more parameters. If there must be no drift in the system, an I-only controller should be used, but it requires a slower response time. This slower response time can be reduced by combining I-only control with another form such as P or PD control. I-only control is typically used when the measured variable needs to be held within a very narrow range and fine-tuned control is required. I controls affect the system by responding to accumulated past errors. The philosophy behind integral control is that deviations will be affected proportionally to the cumulative sum of their magnitudes . The main advantage of adding I control to the controller is that it will eliminate offset. The downside is that it destabilizes the controller, and there is integral windup, which increases the time it takes the controller to make changes.

I-control relates the controller output to the error integral. The integral of the error is with respect to time. It is the total error associated over the specified time. This I control behavior is mathematically illustrated in Equation 9.2.2 (Schek et al .).

$$c(t)=\frac{1}{T_i}\int e(t)dt+c\left(t_0\right)\dots \dots \dots \dots \mathrm{9.2.2}$$
where

• c(t) is the controller output
• $T_i$is the integral time
• e(t)is the error
• c($t_0$) is the controller output before integration

In Equation 9.2.2, the integral time is the time it takes for the controller to change its output equal to the error value. The controller output before integration is equal to the initial output at time t=0, or the controller output at the previous step measured. A graphical representation of the impact of these variables on the system is shown in PID tuning by classical methods.

The rate of change of the I-only control controller output is determined by a number of parameters. While only the P controller is determined by e, only the rate of change of I depends on e and $T_i$. Since c(t) and $T_i$Inversely proportional relationship, which reduces the rate of change of the I-only controller.

An I-only controller operates in much the same way as a P-only controller. The inputs are again setpoint, signal and bias. Calculate the error again, and send this value to the algorithm. However, instead of using only a linear relationship to calculate the response, the algorithm now uses an integral to determine which response should be taken. Once the points are evaluated, a response is sent and the system adjusts accordingly. Due to dependence $T_i$, the algorithm takes longer to determine the correct response. A graphical representation of the I controller output at time t0 for an input step increase is shown in Figure 3 below. As expected, this plot represents the area below the stepping input plot.

Figure 3. I controller output for step input.

Derivative (D) Control

Another type of action used in PID controllers is derivative control. Unlike P-only and I-only control, D control is a form of feedforward control. D-control predicts process conditions by analyzing error changes. Its function is to minimize erroneous changes so that the system maintains a consistent setup. The main advantage of a D controller is its resistance to system changes, the most important of which is oscillation . The control output is calculated from the rate of change of the error over time. The greater the rate of error change, the more pronounced the controller response.

Unlike proportional and integral controllers, derivative controllers do not steer the system to a steady state. Due to this property, a D controller must be coupled with a P, I, or PI controller to properly control the system.

D-control relates the controller output to the derivative of the error. The derivative of the error is taken with respect to time. It is a false change relative to a time change. This D control behavior is mathematically illustrated in Equation 9.2.3.
$$c(t)=T_d\frac{d}{dt}\dots \dots \dots \dots \mathrm{9.2.3}$$
Among them

• c(t) is the controller output
• Td is the derivative time constant
• de is the differential change in error
• dt is the differential change in time

A graphical representation of the impact of these variables on the system is shown in PID tuning by classical methods.

Mathematically, derivative control is the opposite of integral control. Although I-only control exists, D-only control does not exist. The D control only measures the change in error. D-control doesn't know where the setpoint is, so it's often used in conjunction with another control method, such as P-only or PI-combined control. D-controls are typically used for processes where the process output changes rapidly . However, like I control, D control is mathematically more complex than P control. Adding a D control slows down the controller's response time because the computer algorithm takes longer to compute the derivative or integral than simply linearly correlating the input and output variables. Graphical representation of D controller output with input at $t_0$Time gradually increases, as shown in Figure 4. As expected, this graph represents the derivative of the stepped input graph.

Figure 4. D controller output for step input.

The controller's impact on the system

The diagram below is intended to provide a visual representation of how the P, I and D controllers will affect the system.

describe

Figure 5. Stable data sample.

Figure 6. Data interference.

Figure 7. Effect of the P controller on the data.

Figure 8. Effect of the I-controller on the data.

Figure 9. Effect of the D controller on the data.

Read on to see the results of combining controllers.

Proportional Integral (PI) Control

One combination is PI control, which lacks the D control of a PID system. PI control is a form of feedback control. It provides a faster response time than an I-only control due to the added proportional action. PI control stops system fluctuations and is also capable of returning the system to its set point. While the response time of PI control is faster than I-only control, it is still 50% slower than P-only control. Therefore, to increase response time, PI control is often combined with D-only control.

PI control relates the controller output to the error and the error integral. This PI control behavior is mathematically explained in Equation 9.2.4.

$$c(t)=K_c\left(e(t)+\frac{1}{Ti}\int e(t)dt\right)+C\dots \dots \dots \dots \mathrm{9.2.4}$$
q where

• c(t) is the controller output,
• Kc is the controller gain,
• Ti is the integration time,
• e(t) is false, and
• C is the initial value of the controller

In this equation, the integral time is the time required for the I-only part of the controller to match the control provided by the P-only part of the controller.

This equation shows that a PI controller works like a simplified PID controller with zero derivative terms. Alternatively, a PI controller can also be viewed as a combination of P-only and I-only control equations. The bias term in the P-only control is equal to the integral action of the I-only control. P-only control only works when the system is not reaching the set point. When the system is at the set point, the error is equal to zero and the first term disappears from the equation. The system is then controlled by only the I part of the controller. If the system deviates from the setpoint again, P-only control will be implemented. A graphical representation of the PI controller output at time t0 for an input step increase is shown in Figure 5 below. As expected, this plot resembles a qualitative combination of a P-only plot and an I-only plot.

Figure 10. PI controller output for step input.

Effect of Kc and Ti

When using a PI control system, the controller activity varies with Kc and $T_i$decrease, but they can act alone on the aggressiveness of the controller's response. Consider Figure 11 below, the center plot is the base case for a linear second order system.

Figure 11. $K_c$Sum $T_i$Influence

The image depicts $T_i$and $K_c$How both affect the performance of the system, whether they both affect the system or each affects it independently. Increasing controller gain (moving from bottom to top on the graph) increases controller activity regardless of integration time. Likewise, decreasing the integration time (moving from right to left on the graph) will increase controller activity independent of controller gain. As expected, increasing $K_c$and reduce $T_i$Will increase sensitivity and create the most aggressive controller scenarios.

Since there are only two interacting parameters in a PI control system, similar performance diagrams can still cause confusion. For example, plots A and B in the figure both look very similar, although the parameters of each plot are affected. This can cause further problems and create a very aggressive system if the wrong parameters are corrected. While trial and error might work for PI systems, in PID it becomes cumbersome as a third parameter is introduced and the plots become increasingly similar.

Another notable observation is the plot $K_c$and double $T_i$. The plot depicts how the proportional term is useful, but the integral doesn't get enough weight initially, causing slight oscillations before the integral term eventually catches up and helps the system reach the set point.

Proportional derivative (PD) control

Another control combination is PD control, which lacks the I control of PID systems. PD control is a combination of feedforward and feedback control as it operates on current and predicted process conditions. In PD control, the control output is a linear combination of the error signal and its derivative. PD control includes proportional control for damping fluctuations and differential control for process error prediction .

As mentioned earlier, PD control relates the controller output to the error and the derivative of the error. This PD control behavior is mathematically illustrated in Equation 5.

$$c(t)=K_c\left(e(t)+T_d\frac{d}{dt}\right)+C\dots \dots \mathrm{9.2.5}$$

where

• c(t) = controller output
• Kc = proportional gain
• e = error
• C = initial value of the controller

This equation shows that the PD controller works like a simplified PID controller with zero integral term. Alternatively, the PD controller can also be viewed as a combination of P-only and D-only control equations. In this control, only the purpose of the D control is to predict the error in order to improve the stability of the closed-loop system. PD control is not commonly used due to the lack of an integral term. Errors in steady state operation are not minimized without the integral term. PD control is typically used in batch pH control loops where there is no need to minimize errors in steady state operation. In this application, the error is related to the actuation signal through a proportional term and a derivative term. A graphical representation of the PD controller output at time t0 for an input step increase is shown in Figure 6 below. Also, as expected, this plot is a combination of a P-only plot and a D-only plot.

Figure 12. PD controller output for step input.

Proportional-integral-derivative (PID) control

Proportional-integral-derivative control is a combination of all three control methods. PID control is the most commonly used because it combines the advantages of each type of control. This includes faster response times due to P-only control, and offset reduction/zero for combined derivative and integral controllers. This offset was eliminated by additional use of the I control. The addition of D-control greatly improves the response of the controller when used in combination because it predicts disturbances to the system by measuring changes in error. Conversely, as mentioned earlier, when used alone, it has a slower response time than the faster P-only controls. However, while the PID controller seems to be the most suitable controller, it is also the most expensive. Therefore, it is not used unless the process requires the precision and stability that a PID controller provides.

PID control relates the controller output to error, error integral and error derivative. This PID control behavior is mathematically illustrated in Equation 6 (Scrcek* et al.)*.

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where

• c(t) = controller output
• Kc = controller gain
• e(t) = error
• T i = integration time
• Td = derivative time constant
• C = initial value of the controller

As shown in the above equation, PID control is a combination of all three types of control. In this equation, the gain is multiplied by the integral and derivative terms as well as the proportional term, because in PID combined control, the gain also affects the I and D actions. Due to the use of derivative control, PID control cannot be used for very noisy processes because the noise interferes with the feed-forward aspect of prediction. However, PID control is used when the process does not require offset and fast response time. A graphical representation of the PID controller output at time t0 for an input step increase is shown in Figure 7 below. This plot resembles a qualitative combination of P-only, I-only, and D-only plots.

Figure 7. PID controller output for step input.

In addition to PID control, P, I and D control can also be combined in other ways. These alternative combinations are simplifications of PID control.

Note: the order of e(t)

The order of elements in e(t) may vary from case to case. It can be fixed elements minus variable elements, or vice versa. To better illustrate this concept, let's look at an example. Suppose you are creating a PID control to control the liquid level in a storage tank by manipulating an outlet valve. When the fluid level in the tank exceeds your set point, you will want the valve to open more to allow more flow out of the tank. You are looking for a positive response. So your e(t) should give a positive value when the liquid level is above the set point. In this case your e(t) will be (V-Vset). The same logic can be used in other systems to determine what e(t) should be in PID control.

Modeling a PID Controller Using Euler in Excel

Like many engineering systems, PID controllers can be modeled in Excel by numerical methods such as Euler's method. First start with the initial values ​​of the given parameters. The variation of this parameter at a time step is determined by summing the three controllers P, I, and D for that step, which are listed above in the P, I, D, PI, PD, PID control section The equation is found. Make this change, multiply it by the selected time step, and add it to the previous value of the parameter of interest. For more details, see Numerical Ordinary Differential Equation Solving in Excel. An example of a chemical engineering problem using this approach can be seen in Example 4 below.

Troubleshooting PID Modeling in Excel

When setting up an Excel spreadsheet to model a PID controller, you may receive an error message stating that you have created a circular reference. Suppose you are controlling one reactant (B�) to a reactor (one by one) that depends on the concentration of another reactant already inside the reactor. Your PID equation looks like this:

$$FB=1+K_c\left[\left(A-A_{\mathrm{set}}\right)+\frac{1}{t_i}\int \left(A-A_{set}\right)dt+t_d\frac{d\left(A-A_{set}\right)}{dt}\right]$$

和
$$x_i=\frac{d\left(A-A_{set}\right)}{dt}$$

After setting columns A − A Set $A-A_{set}$,$d(A-A_{set})/dt$，$x_i$As well as the cells for the parameters, such as Kc, τi, and τd, you need to set the PID column in the PID column, which contains the PID equation. After entering a formula into the first cell of a PID column, you may receive a circular reference error message when attempting to drag the formula down to another cell in the column.

You can do two things:

1. You may need to start the PID equation in the second or third cell of the PID column. Enter reasonable values ​​in the first few cells before starting the PID equation, and you will find that these values ​​do not affect the final result of the controller.
2. You can also try reducing the step size ( $\Delta t$).

Summary Table

Table 1 shows a summary of the pros and cons of the three controls.

**Table 1.** Advantages and disadvantages of controls

Table 2 shows typical usage guidelines for the various controllers.

**Table 2.** Typical uses of P, I, D, PI and PID controllers

Table 3 shows a summary of definitions of terms and symbols.

**Table 3.** Definitions of terms and symbols.

Example 9.2.19.2.1

Suppose an industry has just put you in charge of one of their batch reactors. Your task is to find a way to maintain the set point level inside the reactor. Your boss wants to use some type of regulator controller, but he's not quite sure which one to use. Help your boss find the right type of controller. It is very important that the liquid level in the reactor is at the set point. Large fluctuations and errors cannot be tolerated.

the solution

You may wish to use a PID controller. Because of the P control, the system will respond to changes very quickly. Due to the action of the I control, the system is able to return to the setpoint. Finally, since it is critical that the system stays at a constant set point, D control will measure changes in error and help adjust the system accordingly.

Example 9.2.29.2.2

You go back to your high school and you notice an oven in your old chemistry class. An oven is used to remove moisture from the solution. Using the knowledge you've gained from ChE 466, you're starting to wonder what type of controller your oven uses to maintain its set temperature. You notice that some high school students are eager to learn, and you decide to share your knowledge with them, hoping to inspire them to become chemical engineers. Explain to them the type of control most likely to be inside the oven, and how that control works.

the solution

Since the oven is only used to remove moisture from the solution, fluctuations, errors and lags between the set point and actual temperature are all acceptable. Therefore, the easiest and easiest controller to use is the switch controller. When the temperature in the oven is lower than the set temperature, the switch controller turns on the heating mechanism. If the oven temperature rises above the set temperature, the controller will turn off the heating mechanism.

Example 9.2.39.2.3

After accepting your suggestion, you are assuming that the owner of an industry decides to install a PID controller to control the liquid level in a batch reactor. When starting the reactor for the first time, the controller initially receives a step input. As the reactor reaches steady state, the liquid level in the reactor tends to fluctuate, sending a pulsed input to the controller. For pulse inputs, a graphical representation of the output of the PID controller is provided.

Figure 8. Pulse input.

the solution

The PID controller output will be a combination of P-only, I-only, and D-only controller outputs. Similar to the P-controller output for a step input, the P-controller output for a pulse input is exactly like the input.

Figure 9. P controller output for pulse input.

The I controller output represents the area below the input graph. Unlike a step input, the area under the pulse input graph drops to zero once the pulse passes through. Therefore, the I-controller output graph eventually does not increase continuously, but tends to level off.

Figure 10. I controller output for pulse input.

The D controller output represents the derivative of the input graph. The derivative at the first discontinuity of the graph will be positive infinity. The derivative of the second downward discontinuity is negative infinity.

Figure 11. D controller output for pulse input.

Combining the qualitative properties of all three graphs, we can determine the output of the PID controller for the pulsed input.

Figure 12. PID controller output for pulse input.

Example 9.2.39.2.3

Different types of disturbances can arise when determining the robustness of a PID controller. These different disturbances are used to simulate possible changes within the system. For the CSTR reactor, you decide to heat the system to handle the cold outside weather. The perturbation of the input temperature is a ramp perturbation, as shown in Fig. What will be the output of the PID controller if the controller responds to the input temperature?

Figure 13. Ramp input.

the solution

Using only a controller with only p controllers, we will see a proportional change in the controller output corresponding to a change in the input variable. See Figure 14 below

Figure 14. P controller output for ramp input.

With the I-only controller, we will see the controller corresponding to the area below the graph, which in this case appears to grow exponentially with the slope geometry.

Figure 15. I controller output for ramp input.

With a D-only controller, we will see a step response to ramp disturbances. This is because the D component corresponds to the derivative, and the ramp input exhibits a constant slope (positive in this case) that differs from the starting condition slope (usually zero). See Figure 16.

Figure 16. D controller output for ramp input.

With a PID controller, all three components play a role in the controller output. As we would expect, the result will be just a simple addition of the three separate component graphs.

Figure 17. D controller output for ramp input.

Example 9.2.69.2.6

Below is the P&ID for process A+B–>C.

Figure 18. P&ID of the reaction process.

What is the PID controller expression on the V3 to control the volume in the TK001 to a setpoint of 50 liters? NOTE: The PID controller uses LC1 to measure volume.

the solution

The general formula for a PID controller is:

$$c(t)=K_c\left[e(t)+\frac{1}{T_i}\int e(t)dt+T_d\frac{d}{dt}\right]+C$$

where

• c(t)�(�)= controller output
• Kc��= controller gain
• e(t)�(�) = error
• T i i = integration time
• Td��= derivative time constant
• C�= initial value of the controller

So for this example the solution is:

$$F_{\mathrm{out}}=offset+K_c\left[\left(V_1-V_{\mathrm{set}}\right)+\frac{1}{T_i}\int \left(V_1-V_{\mathrm{set}}\right)dt+T_d\frac{d\left(V_1-V_{\mathrm{vet}}\right)}{dt}\right].$$

Example 9.2.69.2.6

In this problem, the differential equations describing a particular first order system with a PID controller will be determined. This aims to show how well-defined systems can be modeled or explained mathematically. As a complement to this question, visit Building a Block Diagram . Note that this is an example of a solution using the integral-differential operator instead of the Laplace transform. Here, we set Kp=1��=1.

Consider a general first-order process:

$$\tau p{Y}^{\prime }(t)+Y(t)=X(t)$$

where Y ( t ) is the output of the system and X ( t ) is the input. Add a PID controller to the system and solve a single simple differential equation. The operator or formula of the PID controller is as follows. Assume there is no dead time in the measurement.

$$G=K_c\left[1+\frac{1}{t_I}{\int }_0^{t}d\tau +t_D\frac{d}{dt}\right]$$

Solution: Use the construction block diagram as a reference when troubleshooting this issue.
Equations define the system,

• Process: τ pY '( t ) + Y (t) = X ( t )
• Controller: X ( t ) = Gε ( t )
• Comparator: ε(t) = R (t) − M ( t )
• Measure: M ( t ) = Y ( t )
• Process: τpY’(t) + Y(t) = X(t)
• Controller: X(t) = Gε(t)
• Comparator: ε(t) = R(t) − M(t)
• Measurement: M(t) = Y(t)

When these equations are combined into one equation, the following formula results. This is accomplished by adding the comparator's measurement to the controller's process equation.
$$t_p{Y}^{\prime }(t)+Y(t)=G(R(t)-Y(t))$$

Replace the controller operator, then evaluate yield:

$$t_p{Y}^{\prime }\left(t\right)+Y\left(t\right)=K_c\left[1+\frac{1}{t_I}{\int }_0^{t}d\tau +t_D\frac{d}{dt}(R(t)-Y(t))\right]$$

Because there is an integral in the differential equation, it is necessary to take the derivative with respect to time.

To put this into the standard notation for solving second order differential equations, the Y(t)�(�) needs to be on one side, and the R(t)�(�) term needs to be on the opposite side. Also, the coefficient Y(t)�(�) term needs to be one.

$$\frac{t_I\left(t_p-K_c{t}_D\right)}{K_c}{Y}^{\prime \prime }(t)+\frac{t_I\left(1+K_c\right)}{K_c}{Y}^{\prime }(t)-Y(t)=t_I{t}_D{R}^{\prime \prime }\left(t\right)+t_I{R}^{\prime }\left(t\right)+R(t)$$

The above equations can then be solved by hand or using a program such as Mathematica. If a computer program is used, different values ​​of the control parameters Kc , τI , τD can be selected and the response to system changes can be evaluated graphically.

Exercise 9.2.19.2.1

What type of controller does the equation below show?

$$c(t)=K_c\left[e(t)+\frac{1}{T_i}\int e(t)dt\right]$$

1. Feedforward
2. PID
3. Derivative
4. proportional integral

• answer

exercise 9.2.29.2.2

Which type of controller improves the stability of the system by keeping it at a consistent setting?

1. Derivative
2. proportional
3. switch
4. complete

• answer

exercise 9.2.39.2.3

Which type of controller would provide increased responsiveness to reach the desired set point as quickly as possible while eliminating drift?

1. switch
2. proportional
3. complete
4. proportional integral

• answer

A microbiology lab has discovered a deadly new strain of bacteria called P. Woolfi in the city's water supply. In order to sterilize water, bacteria must be heat-killed at a temperature of 105 degrees Celsius. However, this is above the boiling point of water at 1 atmosphere, and bacteria are only heated in liquids due to spore formation in gases or vapors. To accomplish this sterilization, it is recommended to use an autoclave to keep the water in a liquid state by keeping it at a pressure of 30 atmospheres while heating for 5 seconds. The autoclave can only handle pressures up to 7 atmospheres before exploding, so to ensure the process operates to the desired specifications, a PID controller model must be created in Excel. See Figure 18 for a visual view of the system.

Figure 18. Automatic kettle with temperature and pressure PID control

Click this link for the formulated Excel solution

explain:

In order to simulate the real situation of pressure changes in the system, column B invokes an equation to generate random fluctuations in pressure. Psets are just the required specifications. Error is the difference between the set pressure and the measured signal. du/dt is the sum of the P, I and D terms. The equations used to calculate each can be found in the article, which take into account the errors associated with each time step. dU/dt is the parameter that is varied to correct for the difference between the measured pressure and the desired pressure.

quote

• Astrom, Karl J., Hagglund, Tore., "Advanced PID Control," Society for Instrumentation, Systems, and Automation.
• Cooper, Douglas J. "The Electronic Textbook of Practical Process Control" http://www.controlguru.com
• Scrcek, William Y., Mahoney, Donald P., Young, Brent R. "A Real-Time Approach to Process Control", 2nd ed. John Wiley & Sons, Ltd.