8
PID Controller Design for Complex Systems

8.1 Introduction

The PID controller design methods discussed in the previous chapters are either model based approaches or rule based approaches. It is clear that when using a model based approach, a first order model yields a PI controller and a second order model yields a PID controller. In some applications, the first order and second order models are basically an approximation to the actual physical systems. In other applications, the underlying physical systems are complex and are of higher order. This chapter studies how to design PID controllers for higher order systems directly using frequency response data.

8.2 PI Controller Design via Gain and Phase Margins

This section presents PID controller design via specification of gain margin and phase margin.

8.2.1 PI Controller Design Using Gain Margin and Phase Margin Specifications

The starting point is to assume that the frequency response of a desired open-loop transfer function at a specific frequency point is available. It is also assumed that the frequency response of the system is available at .

At the frequency , the actual open-loop frequency response with a PI controller is

Letting the actual open-loop frequency response equal its desired counterpart leads to

(8.1)

which is

(8.2)

Comparing the left-hand side with the right-hand side of (8.2) gives

(8.3)

(8.4)

From images and images , the PI controller parameters are calculated using the following relationships

One of the choices for the frequency response of a desired open- loop transfer function images is to specify the gain margin for the PI control system. Say, if one wishes to have a gain margin of 2, then images . However, the frequency images still needs to be determined, where images in this specification is the cross-over frequency for the desired closed- loop system. Had a proportional controller images been used in the design, images could be chosen as the cross-over frequency for the images . Following the same practice, because of the images phase-lag introduced by the integrator in the controller, a reasonable practice is to select images in the vicinity of the frequency images where images for the first time crosses the imaginary axis. Note that images because at images , images , consequently resulting in images . A practice is to set images .

Similar to the specification of the desired gain margin, the phase margin can also be used to specify the desired open-loop frequency response images . It is known that images at the frequency (say images ) that defines the phase margin. Hence, by denoting the phase margin as images , then

8.2.2 Design Examples

The following example demonstrates the performance of the PI controller when using gain margin and phase margin specifications.

Example 8.1

Consider a system with the following transfer function,

(8.5)

Choose the frequency .

Selecting gain margin and respectively, design a PI controller with these two gain margin specifications and evaluate the closed-loop performance.
Selecting phase margin , design a PI controller with the phase margin and simulate the closed-loop response.

Solution. We will first determine the frequency by presenting the real part of as shown in Figure 8.1(a). By inspection of Figure 8.1(a), the parameter is found to be . With this frequency parameter, .

Image described by caption and surrounding text. — **Figure 8.1** Illustration of the design process and closed-loop responses (Example 8.1) when using the gain margin. (a) Determining . (b) Output signal. (c) Control signal. Key: line (1) with a specified gain margin of ; line (2) with a gain margin of .

images — **Figure 8.1** Illustration of the design process and closed-loop responses (Example 8.1) when using the gain margin. (a) Determining . (b) Output signal. (c) Control signal. Key: line (1) with a specified gain margin of ; line (2) with a gain margin of .

Design using the gain margin. The specification of a gain margin of 2 () gives

which leads to and . Hence, the proportional controller gain and the integral time constant .

Repeating the above computation with the specification of a gain margin of 3 (), it can verified that

Here has been reduced while stays the same.

Closed-loop simulation studies have been performed for both controllers where a unit step reference signal is introduced at time and an input step disturbance with magnitude is injected at (s). Figures 8.1(b)–(c) show the closed-loop output responses and the control signal responses. It is seen that with a larger gain margin, the closed-loop system response is slower and less oscillatory.

Design using phase margin. By choosing the desired phase margin , the desired open-loop frequency response at is

Using this information together with the information of and , we calculate

yielding

The PI controller parameters are and . Apparently, the design using phase margin specification gives a larger proportional gain and integral time constant. Closed-loop simulation studies have been performed for this PI control system for evaluation of reference following and input disturbance rejection. Figure 8.2(a) shows the output response and Figure 8.2(b) shows the control signal response. It is seen that the closed-loop response speed is faster than the closed-loop systems designed using the gain margin specification in the first part of this example.

The following example is to show how the PI controller is used to control a higher order complex system with time delay, which is very difficult if other design methods are used for this case. The closed-loop performance when using the specification of gain margin is left as an exercise (see Problem 8.1).

Example 8.2

The transfer function of a complex system is given by

(8.6)

Design a PI controller for this system by specifying the desired phase margin with and evaluate the closed- loop performance.

Two graphs depicting Time (sec) on the horizontal axis, output and Control on the vertical axis, and curves plotted for Closed-loop responses when using phase margin. — **Figure 8.2** Closed-loop responses (Example 8.1) when using the phase margin. (a) Output signal. (b) Control signal.

Solution. The frequency response is shown in Figure 8.3(a) and its real part is shown in Figure 8.3(b), from which the frequency is chosen, leading to . With the given desired phase margin , the desired open-loop frequency response at is specified as

Then the following quantity is calculated:

This gives and . Finally, and .

To evaluate the closed-loop performance of the PI control system, simulation studies are performed using unit step reference input and a step disturbance with magnitude of entering the system at (s). The proportional control is implemented on the output only. Figures 8.3(c) and (d) show the closed-loop control results. The results show that even though the process is higher order and complex, the PI controller produces stable and satisfactory results. However, the response to the input disturbance is quite slow.

It must be emphasized that the PI controller design using the specification of either gain margin or phase margin does not work for severely underdamped systems and unstable systems. This is because for these classes of systems, the desired open-loop frequency response images must be calculated in a more sophisticated way to reflect the undesired process characteristics. Problem 8.1 is left for the verification of these statements.

8.2.3 Food for Thought

If we are to fit the frequency response of a higher order transfer function with a first order plus delay model or a second order plus delay model, how would you propose to estimate the time delay parameter ?
Would you say that the stability of the closed-loop system is not guaranteed when a PID controller is designed using an approximate model?
Is it correct that the closed-loop stability is checked against the original higher order system?
If the amplitude of the frequency response of the original system, , asymptotically decays as increases, do you think that the PI controller design methods using the specification of gain margin or phase margin will guarantee closed-loop stability for the original system?
Will the amplitude of the frequency response of the original system containing a pair of underdamped modes asymptotically decay as increases?

8.3 PID Controller Design using Two Frequency Points

This section discusses an intuitive and simple approach to PID controller design from the perspective of curve fitting of the frequency response of the loop transfer function.

8.3.1 Finding the PID Controller Parameters

As we know from the PI controller design, the process frequency response information at one frequency images is sufficient to find the two parameters ( images and images ). Because there are three parameters contained in a PID controller, naturally it requires the process frequency response information at two frequencies images and images ( images ) to uniquely determine the three parameters ( images , images and images ).

The starting point is to assume that the desired open-loop frequency response images is specified at images and images , and the plant frequency response images is also known at images and images . Furthermore, we assume that images . The specifications of the images , and the frequency points images and images will be discussed later.

For a PID control system, the actual open-loop frequency response at images is

and at images is

Thus, by letting

(8.7)

(8.8)

and comparing their real and imaginary components, the following linear equations hold:

(8.9)

(8.10)

(8.11)

where for notational simplicity,

(8.12)

(8.13)

From (8.9), the coefficient images is calculated as

(8.14)

From (8.10) and (8.11), the coefficients images and images are calculated by solving the two linear equations, giving

(8.15)

(8.16)

Finally, the PID controller parameters are given in relation to images as

Because there are two frequencies images and images used in the design, more thought is required in the selection of not only images at images and images , but also the frequencies images and images themselves. Naturally, one assumes that the desired gain margin and phase margin are good candidates for the selection of images . However, the challenge is to find the suitable values for the desired gain margin and phase margin together with images and images .

The following example is used to demonstrate the difficulties in specifying images .

Example 8.3

Consider PID controller design for the higher order system used in Example 8.2 [see the transfer function given by (8.6)]. Choosing , (cross-over frequency of ), and desired phase margin , design a PID controller by varying the desired gain margin.

Solution. It can be easily verified that for any choice of desired gain margin less than , the derivative gain is negative. As in practice, a derivative filter is often used in conjunction with the derivative gain , which has the form . Then with the negative , the filter becomes unstable. This definitely renders the controller better off without the derivative term.

By increasing the desired gain margin to , together with , , and the desired phase margin , the PID controller parameters are calculated as

In comparison to the PI controller designed in Example 8.2, the proportional gain is the same, however the integral time constant is reduced. Closed-loop system simulation is performed with proportional control implemented on the output only. The derivative filter with time constant is used in the implementation of derivative control. Figure 8.4 compares the closed-loop responses of the PID control system with those of the PI control system from Example 8.2. It is seen that the performance improvement using the PID controller is very small in comparison to the one with the PI controller. Again, the closed-loop response to input disturbance is oscillatory and slow.

8.3.2 Desired Closed-loop Performance Specification using Two Frequency Points

It is apparent that the desired closed-loop performance specification via choice of images plays an important role in the design of a PID controller using the frequency response. The parameters such as gain margin and phase margin are relatively easy to specify in terms of closed-loop stability; however, it is difficult to relate them to the actual closed-loop response performance for reference following and disturbance rejection.

Going back to the drawing board, it is necessary to find a systematic and yet a simple way to specify images such that the closed-loop response performances for reference following and disturbance rejection are met. Another aspect in PID controller design apart from the performance specification is that, because of the limited complexity of the controller structure, there is a difference between what is desired and what is achievable. In other words, what we ask for in a PID control system is not necessarily achievable.

One of the effective ways to specify the desired open-loop frequency response images is via the specification of the desired frequency response of the complementary sensitivity function images , where

Hence, if images is specified, then images is calculated as

(8.17)

The properties of images are directly related to reference following and noise attenuation, as well as indirectly to disturbance rejection via the frequency response of the desired sensitivity function

What are the key characteristics of a complementary sensitivity function? There are four basic characteristics listed as below.

The desired complementary sensitivity function must have all poles on the left-hand side of the complex plane.
With a PID controller in the feedback control, the complementary sensitivity must be equal to unity at .
The plant unstable zeros contained in will be in the presence of because the plant unstable zeros cannot be changed through feedback control.
The plant time delay will appear in the desired complementary sensitivity function because the plant time delay can not be changed through feedback control.

All the characteristics can be easily verified with closed-loop transfer function calculations, which is left as an exercise.

In view of these characteristics of the desired complementary sensitivity function, without a complete knowledge about the system transfer function images , it could be a difficult task to choose a suitable images in its own right. This task could become even more difficult when the plant frequency information images is given at one or two frequency points.

The specification of images is proposed as follows so that the PID controller design using frequency response data remains effective while maintaining the original simplicity. This specification was originally proposed in Wang et al. (1995b) and was described in more detail in Wang and Cluett (2000).

We assume that the plant transfer function images is stable with all poles on the left-half complex plane and the system has no severely underdamped poles. With these assumptions, the behaviour of a control signal to a step reference signal in an over-damped closed-loop control system can be approximated by a first order response. This behaviour is then described by the desired control sensitivity function images with the first order transfer function:

(8.18)

where images is the desired closed-loop time constant for the control signal, the parameter images is selected so that images is approximately equal to the dominant time constant of the system, images is the steady-state gain of the system. When the dominant time constant of the system is unknown, which is the case for using the plant frequency response data in the design, the parameter images is a tuning parameter.

The desired complementary sensitivity function follows from the desired control sensitivity function in the form:

(8.19)

Clearly, images is stable as images and the transfer function images is assumed to be stable; images at steady-state ( images ) is equal to unity because of the factor images , where images is the steady-state gain of images ; and the time delay or zeros in images are contained in images . Therefore, all four characteristics of images have been included in this simple specification.

If the dominant time constant of the plant is estimated (or known) as images , then the desired closed-loop time constant images is chosen to be images , where images .

8.3.3 Design Examples

Example 8.4

Consider PID controller design for the higher order system used in Example 8.2 (see the transfer function given by (8.6)). Choosing , (cross-over frequency of ), which are the same as those used in Example 8.3. Fixing , investigate the effect of on the closed-loop performance in terms of reference following and disturbance rejection.

Solution. Because the system transfer function is known, the desired complementary sensitivity function is seen to be, with the specification of ,

Here, the choice of adds a small time constant to the desired closed-loop system, which essentially does not change much the desired closed-loop response speed. However, the specification of will add a lead element to , which will have an impact on the desired closed-loop response speed.

We will first consider the selection of . This leads to pole-zero cancelation for the pole corresponding to the largest time constant . With this specification, the desired loop frequency responses at and are calculated using (8.17),

The calculation of PID controller coefficients follows from (8.14)–(8.16) based on which the PID controller parameters are obtained as,

(8.20)

For comparison purpose, we investigate the effect of on the closed-loop response. With the choice of , the PID controller parameters are calculated as

Further decrease of to leads to the PID controller parameters as

Clearly, with the decrease of , the proportional control gain has increased in magnitude, and integral time constant has reduced and the derivative gain has increased. The parameters of the PID controller indicate that the closed-loop responses become faster as decreases. Figure 8.5 compares the closed-loop simulation results for the three cases for a unit step response and an input disturbance rejection where the disturbance magnitude is 2 entering the system at (sec). In comparison with the closed-loop response when using the gain and phase margin specifications (see Example 8.3 and Figure 8.4), it is clearly seen with this specification that the closed-loop response to the input disturbance has been significantly improved.

Note that a derivative filter with time constant has been used in the simulation, and both proportional control and derivation control are implemented on the output only.

8.3.4 MATLAB Tutorial on PID Controller Design Using two Frequency Points

The objectives of the following two tutorials are to produce a MATLAB program for PID controller design using two frequency response points (see Tutorial 8.1) and to test this program using a simulation example (see Tutorial 8.2).

Tutorial 8.1

We need plant frequency information at two frequencies that are denoted as 'w1' and 'w2'. The frequency response at 'w1' is denoted by 'Gjw1' and at 'w2' is denoted by 'Gjw2'. As in PI controller design, K is the estimated steady-state gain, 'beta' and 'taucl' are specified according to the control sensitivity function

Step by Step

Create a new file called FR4PID.m for the MATLAB function.
Define the input and output variables for the MATLAB function. Enter the following program into the file:
```
function [Kc,tauI,tauD]=FR4PID(beta,taucl,w1,w2,Gjw1,Gjw2,K) 
```
Calculate the frequency response of the control sensitivity function at and . Continue entering the following program into the file:
```
j=sqrt(-1);
Sujw1=(j*w1*taucl/beta+1)/(K*(j*w1*taucl+1));
Sujw2=(j*w2*taucl/beta+1)/(K*(j*w2*taucl+1)); 
```
Calculate the frequency response of the complementary sensitivity function and the desired open-loop frequency response at and . Continue entering the following program into the file:
```
Tjw1=Sujw1*Gjw1;
Ljw1=Tjw1/(1-Tjw1);
Tjw2=Sujw2*Gjw2;
Ljw2=Tjw2/(1-Tjw2); 
```
Calculate and using (8.12) and (8.13). Continue entering the following program into the file:
```
Xjw1=j*w1*Ljw1/Gjw1;
Xjw2=j*w2*Ljw2/Gjw2; 
```
Calculate the controller coefficients , and using and . Continue entering the following program into the file:
```
c1=imag(Xjw1)/w1;
c2=-(real(Xjw2)-real(Xjw1))/(w2^2-w1^2);
c0=c2*w1^2+real(Xjw1); 
```
Finally, calculate the PID controller parameters, , and . Continue entering the following program into the file:
```
Kc=c1;
tauI=c1/c0;
tauD=c2/c1;
```

The program needs to be tested so that we can use it for applications.

Tutorial 8.2

We select a system with dominant time delay that has the following transfer function,

Step by Step

We create a new file called ‘test4FR2PID.m’.

Define the system under test. Enter the following program into the file:

delay=20;
K=0.4;
num=K;
tau1=2;
tau2=2;
tau3=1;
den1=conv([tau1 1],[tau2 1]);
den=conv(den1,[tau3 1]);
w=0.001:0.01:1;
Gjw0=freqs(num,den,w).*exp(-j*w*delay);

plot the real and imaginary parts of 'Gjw0' as before to find and .
These two frequencies will be used to find the PID controller parameters. We need to add time delay to the frequency calculation. Continue entering the following program into the file:
```
w1=[0.061 0.131];
Gjw=freqs(num,den,w1).*exp(-j*w1*delay); 
```
We choose and , then use the function 'FR4PID.m' to calculate the PID controller parameters. Continue entering the following program into the file:
```
taucl=5;
beta=5;
[Kc,tauI,tauD]=FR4PID(beta, taucl,w1(1),w1(2),Gjw(1),Gjw(2),K); 
```
The function gives , and .
If one wishes to have a slower closed-loop response, the desired closed-loop time constant could be increased. When , , , and .

Closed-loop simulation of the PID control systems are performed using a derivative filter with the filter time constant being images . Both the proportional control term and derivative control term are implemented on the output only. Figure 8.6 shows the closed-loop responses for reference following of a unit step signal and disturbance rejection. The input step disturbance with amplitude of 2 enters the system at images (s). It is seen that by increasing images , the closed-loop response speed is reduced, however, the slight oscillation with the smaller images is overcome.

Note that the MATLAB program FR4PID.m will be used for auto-tuner design in Chapter 9 where the plant frequency information at images and images will be found by the relay feedback experiments. Additionally, the PID controller will degrade to a PI controller if the derivative gain images is either negative or is too small. In the case of the PI controller, the proportional gain images and images remain unchanged from the calculation of the FR4PID.m program.

8.3.5 PID Controller Design for Beer Filtration Process

In the work by Lees and Wang (2015), two transfer function models were estimated for a beer filtration process at different operational conditions.

For the first operational condition, step response experiments were conducted to obtain the estimated transfer function:

For the second operational condition, the estimated transfer function is

where the time unit for the transfer functions is minute, instead of second. The filtration process is clearly a nonlinear system, in which the system dynamics change with respect to operating conditions.

In order to design a single PID controller for the system, the frequency responses of the transfer function models are then averaged point-by-point. Figure 8.7 shows the frequency response of images , images and the averaged frequency response images . To obtain the two frequency response points images and images , the real and imaginary parts of images are examined, where images is identified as the point when the real part of images changes sign from negative to positive and images is identified as the point where the imaginary part changes sign from positive to negative. The corresponding frequency response images at images is images and at images is images .

The desired closed-loop performance is specified at images and images through the following relationship:

(8.21)

where images , images and images . Note that we have selected images , which corresponds to the dominant time constant of images .

Using the MATLAB function FR4PID.m produced in Tutorial 8.1, we calculate the PID controller parameters:

Figure 8.8 shows the frequency response images and images , from which we can estimate that the closed-loop control systems have the minimum gain margin of 2 and phase margin images .

In the closed-loop simulation, the PID controller is discretized and a derivative filter with time constant images is added to the derivative term to avoid amplification of measurement noise. The discretized control signal ready for implementation is calculated using ((4.40)) in Chapter 4. Additionally, the control signal is computed with quantization for the possible implementation of the control system by a plant operator. The control signal with quantization is chosen to be a multiple of 0.01, which corresponds to 1 percent change in the control signal as the basis unit. Also, if the calculated control signal change images is less than 1 percent, then the control signal remains constant.

The control objective is to maintain a constant output images , and due to the filtration operation, it drops with respect to time. The closed-loop control system is simulated with an output disturbance added to the system while maintaining a constant reference response. The typical case of the disturbance mimics the situation where the images reduces in a series of step changes. Because of the nonlinearity, the same PID controller is used to control both images and images in the simulation studies. Figure 8.9 shows the control signal response and output response to the output disturbance in a series of steps. It is seen that the closed-loop PID control has maintained the constant output value despite of the disturbance. Note that with the same filter, but at different operational time, images has a smaller steady-state gain, corresponding to the filter condition deteriorating. As a result, a larger steady-state control signal is required to maintain the same operational conditions. This is evident from comparing the control signals in Figure 8.9 (a).

Four graphs depicting Time on the horizontal axes, u (a) and y (b) on the vertical axes, and curves plotted for closed-loop control simulation for output stair case disturbance rejection. — **Figure 8.9** Closed-loop control simulation for output stair case disturbance rejection. (a) Control response (top figure: results from using , bottom figure: results from using ). (b) Output response (top figure: results from using , bottom figure: results from using ).

8.3.6 Food for Thought

In the PID controller parameter solutions (see (8.14)–(8.16)), is it correct to say that the proportional controller gain only uses the information from the first frequency point , however, and use the information from both first and second frequency points?
Upon finding the controller parameters, , and , using the information from the two frequency points, we have the options to test different combinations of controllers without changing their parameters. First instance, we can use the proportional controller with , PI controller with and , PID controller with , and or PD controller with and . Why do you think that the frequency domain design can lead to such a result?
We have chosen and for easy implementation. Can we choose other frequency points as and as long as they are in the medium frequency range with cross-over frequency contained? Why is that?
In the specification of desired closed-loop transfer function (see 8.19), if the parameter , the closed-loop dominant time constant is specified to be equal to the open-loop dominant time constant but with unity steady-state gain. Would you consider this as a default choice?

8.4 PID Controller Design for Integrating Systems

PID control of integrating systems has become increasingly important in control engineering applications. A large number of electro-mechanical systems can be classified as integrating plus time delay systems. For instance, the angular position control of a robot is the control of integrating system, and the quadrotor control is also related to control of integrating systems.

The most widely encountered integrating systems have time delay in addition to first order or higher order dynamics. Because the integral action is expressed as a pole on the origin of the complex plane, which essentially is the dominant dynamics for an integrating system, it may not be necessarily to capture the first order or higher dynamics in the design of PID controllers. Instead, these stable dynamics are approximated using an equivalent time delay to describe the effect of their phase lag in the PID control system design.

8.4.1 The Approximate Model

The approximate model of an integrating system is assumed to be of the following form:

(8.22)

where images is the gain of the integrating system and images is its time delay. For most physical systems, there are more or less approximations involved in obtaining the integrating plus time delay model. An easy way to find the parameters in (8.22) is through frequency response analysis.

Assume that the frequency response images is available at the frequency images . This frequency information images is estimated using the relay experiments in many applications as shown in the next chapter.

Now, letting the frequency response of the integrating plus delay model (8.22) be equal to the measured images leads to

(8.23)

Equating the magnitudes on both side of (8.23) gives

(8.24)

where images . Additionally, from (8.23), the following relationship holds:

This gives the estimate of time delay as

It is seen here that if the system is truly integrating with time delay, the plant information at a single frequency is sufficient to determine the plant gain and time delay.

8.4.2 Selection of Desired Closed-loop Performance

Because the transfer function for the time-delay images is irrational, approximation is often needed when using the model based designs (see Chapter 3). An effective way to avoid the approximation is to derive the PID controller parameters using the frequency response analysis.

Similar to the PID controller design introduced in the previous section, we will first introduce the specification of desired closed-loop performance. Considering the PID controller structure

together with the integrating plus delay model

it is clear that the loop transfer function

(8.25)

contains a double integrator. Therefore, this characteristic should be reflected in the selection of the desired closed-loop performance. Additionally, the four characteristics of the desired complementary sensitivity function specified in Section 8.3.2 should be satisfied. It is simpler to choose the desired control sensitivity function to this effect.

A candidate for such a choice is the control sensitivity to have the following form:

(8.26)

where images is the desired closed-loop time constant and images is the damping coefficient typically chosen as 0.707 or 1. A larger images corresponds to a slower closed-loop response speed.

The desired complementary sensitivity function images is composed of the control sensitivity images and the model images given by

(8.27)

where the steady-state gain images and the factor images have been cancelled to obtain (8.27).

It is seen from (8.27) that the desired complementary sensitivity function images has all poles on the left-hand side of the complex plane. Additionally, the complementary sensitivity images is equal to unity at images and the plant time-delay images appears in the numerator of images . Therefore, all the characteristic requirements discussed in Section 8.3.2 are satisfied for the integrating with time delay system by PID control.

Furthermore, a stable zero at images is introduced in the desired complementary sensitivity. The introduction of this stable zero is to ensure that at images the desired loop transfer function images will have the structure of a double integrator, which matches that of the actual loop transfer function images in (8.25). This claim can be verified through the following calculation:

(8.28)

Writing the irrational transfer function images in Taylor series expansion gives

(8.29)

where images denotes the higher order terms in the Taylor series. Then, the denominator of images is expressed as

(8.30)

Since from (8.30) the higher order images contains a factor images , then it is clearly seen that the denominator images contains the factor images .

It is emphasized that the choice of images given by (8.27) ensures the desired loop transfer function images has the feature of a double integrator at images matching that of the actual loop transfer function images . This means that at the lower frequency region, the PID controller parameters will automatically lead to the low frequency requirement of the sensitivity functions. This choice of desired complementary sensitivity function reduces the errors in the frequency curve fitting for computation of the PID controller parameters.

8.4.3 Normalization of the Parameters and Empirical Rules

To normalize the process parameters for derivation of the PID controller parameters in empirical rules, the actual loop transfer function images from (8.25) is re-written as

(8.31)

where images , images , images and images . For convenience in computation, (8.31) is expressed as

(8.32)

where the parameters are defined as

Note that the loop transfer function images is free of the process gain images and the time delay images . Similarly, the desired loop transfer function images given by (8.28) is required to be normalized. To this end, the desired closed-loop time constant images is selected as the function of the time delay images :

(8.33)

where images is the desired closed-loop performance parameter used in the design. This leads to the re-writing of (8.28) in the following form:

(8.34)

Note that the desired loop transfer function images is also free of the time delay parameter images .

The solution of the PID controller parameters follows from the frequency domain solution proposed in Section 8.3, but with different choices of the frequency points images and images (see Wang and Cluett (2000)).

Because the PID controller parameters are normalized, there are only the desired closed-loop time constant images and the damping coefficient adjustable. Thus, we can find the normalized PID controller parameters numerically with respect to the parameter images and form empirical rules. There are two sets of empirical rules obtained below through polynomial fitting of the normalized PID controller parameters, together with gain and phase margins.

Selecting 100 images values from images to images with increment of 0.1, together with a damping coefficient images , there are 100 sets of normalized PID controller parameters calculated. By using the polynomial fitting tool in MATLAB to find the calculated PID parameters, the following empirical rules for the normalized parameters are obtained as shown in Tables 8.1–8.2. With the normalized PID controller parameters calculated, the actual PID controller parameters are then obtained with the scaling parameters images and images , as

(8.35)

The polynomial functions in both Tables have provided quite accurate descriptions to the original data (see Figure 8.10 as an illustration). Therefore, when an integrator with delay is given, the PID controller parameters will be calculated simply using the polynomial equations presented in the tables.

Two graphs depicting beta on the horizontal axes, Kc on the vertical axes, and curves plotted marked 1 and 2. — **Figure 8.10** Calculated normalized proportional controller gain. (a) , . (b) , . Key: line (1) data; line (2) using Tables 8.1 and 8.2.

8.4.4 Gain and Phase Margins

Because the original PID controller parameters are calculated using two frequency response data points, the PID controller parameters can be used in combination to obtain PID controller, PI controller and PD controller.

The gain and phase margins for the PID controllers are calculated using the empirical forms, which are also function of the parameter images and are shown in Figure 8.11. Additionally, the gain and phase margins for PI and PD controllers are calculated shown in Figures 8.12 and 8.13. These gain and phase margins are useful in measuring closed-loop performance and quantify robustness of the PID control system designed. It also provides some guidance on the choice of controller structures.

8.4.5 Simulation Examples

Example 8.5

Consider the integrator plus time delay system described by the transfer function

In the first part of the example, we will evaluate the performance of the PID, PI, and PD controllers designed. In the second part of the example, we will show that the closed-loop performance is similar to a PID controller using the pole assignment design method. However, there is no comparable PI or PD controller when using model based design. The damping coefficient is chosen to be .

Two graphs depicting beta on the horizontal axes, gain and phase on the vertical axes, and curves plotted marked 1 and 2 for PID controllers. — **Figure 8.11** Calculated gain and phase margins for PID controllers. (a) Gain margin. (b) Phase margin. Key: line (1) using Table 8.1 (); line (2) using Table 8.2 ().

Two graphs depicting beta on the horizontal axes, gain and phase on the vertical axes, and curves plotted marked 1 and 2 for PI controllers. — **Figure 8.12** Calculated gain and phase margins for PI controllers. (a) Gain margin. (b) Phase margin. Key: line (1) using Table 8.1 (); line (2) using Table 8.2 ().

Two graphs depicting beta on the horizontal axes, gain and phase on the vertical axes, and curves plotted marked 1 and 2 for PD controllers. — **Figure 8.13** Calculated gain and phase margins for PD controllers. (a) Gain margin. (b) Phase margin. Key: line (1) using Table 8.1 (); line (2) using Table 8.2 ().

Solution. From Figures 8.11–8.13, when , we can see that the phase margin for the PID control system is about , for the PI control system about , and for the PD control system about . All gain margins exceeded . Thus, it is expected that the closed-loop systems are stable for all three types of controller. There is a closed-loop oscillation when using the PI controller because its phase margin is too small. We usually need to have a phase margin larger than to avoid closed-loop oscillation.

From Table 8.1, the normalized PID controller parameters are calculated, and from (8.35) the actual PID controller parameters are calculated to give

The closed-loop systems are simulated for a step reference change at and a step input disturbance entering system at (s). The PI controller takes the form and PD controller takes the form . All derivative terms are implemented on output only. Figure 8.14 compares the control signals and the output signals for the three control systems. It is seen that the PID control system shows satisfactory performance, the PI control system has oscillatory responses due to the small phase margin, and PD controller has a better performance in reference tracking; however, it is not capable of rejecting the step input disturbance, leading to steady-state errors in the disturbance rejection.

The second part of this example is to design a PID controller for the integrator plus delay model using the pole assignment controller design method. Here the Padé approximation is used to approximate the time delay to yield

(8.36)

Two graphs depicting Time on the horizontal axes, u and y on the vertical axes, and curves plotted marked 1 to 3. — **Figure 8.14** Comparison of closed-loop performance for three types of controllers (Example 8.5). (a) Control signal. (b) Output. Key: line (1) PID control response; line (2) PI control response; line (3) PD control response.

Two graphs depicting Time on the horizontal axes, u and y on the vertical axes, and curves plotted marked 1 to 2 for comparison of closed-loop PID control performance between the model based design and the tuning rules. — **Figure 8.15** Comparison of closed-loop PID control performance between the model based design and the tuning rules (Example 8.5). (a) Control signal. (b) Output signal. Key: line (1) model based design; line (2) using tuning rules.

Choose the desired closed-loop polynomial as

where , which is equivalent to the selection of the dominant closed-loop time constant . With this selection, the dominant closed-loop time constant is equal to the previous case. Using the pole-assignment design method, the PID controller parameters are found as

Although the two design approaches have selected an equivalent desired closed-loop time constant, the model based design method leads to the closed-loop control system with oscillation [see Figure 8.15]. This is likely caused by the modeling error between the approximated transfer function and the true integrator with time delay model [see (8.36)].

Interestingly, for the model-based design, the direct combination of proportional control and integral control is unstable for this design method.

8.4.6 Food for Thought

In the derivation of the PID controller rules for integrating with delay system, why is the desired closed-loop control sensitivity function specified with a stable zero?
Do you expect an overshoot in closed-loop response to the step reference signal by observing the desired complementary sensitivity function in (8.27)? If you wish to eliminate such an overshoot, which reference filter should you choose in a two-degrees of freedom controller implementation?
To obtain an approximate integrating with time delay model, a frequency is required. Which region on the Nyquist curve of a system is a good candidate? Why?

8.5 Summary

This chapter has discussed several approaches to PID controller design using frequency domain information. PID controllers can be designed using gain margin and phase margin as their performance specifications in the frequency domain. One drawback with respect to the gain margin and phase margin specification is that these parameters are not related to the closed-loop response speed in a simple and intuitive way.

The other important aspects in the chapter are summarized as follows.

PID controller parameters can be found analytically in a manner related to curve fitting of the open-loop frequency response in two frequency points. In this approach, the closed-loop performance specification is the desired dominant time constant via complementary sensitivity function.
In order to produce the best fit possible with the limited number of controller parameters, the complementary sensitivity function contains the zeros of the plant as well as the time delay of the plant.
A special case of the approach is the PID controller for integrator with delay system. With a normalized delay parameter, the PID controller parameters are simply expressed in empirical forms analytically with achieved gain margin and phase margin.

8.6 Further Reading

The PID controller design methods using two points of frequency response data were originally introduced in Wang et al. (1995b), Wang and Cluett (1997b) and Wang and Cluett (2000).
More PID controller design techniques using gain margin and phase margin specifications can be found in Ho et al. (1995), Ho et al. (1996), Ho and Xu (1998), Ho et al. (1998), Ho et al. (2000).
Second order with delay model was obtained for PID controller design by using two frequency response points together with nonlinear optimization in Wang et al. (1999).

Problems

8.1 The transfer function of a complex system is given by
(8.37)
1. Design a PI controller for this system by specifying the desired gain margin of .
2. Determine the phase margin and delay margin using Nyquist diagram.
8.2 The transfer function of a complex, underdamped system is given by
(8.38)

where .
1. Design a PI controller for this system by specifying the desired phase margin of .
2. Determine the gain margin and delay margin using Nyquist diagram.
3. Show using simulation studies that the closed-loop response to a step input disturbance is oscillatory no matter what you do.
4. Explain your results by examining the sensitivity function between the step input disturbance and output.
8.3 The mathematical model for the eighth reactor in a copolymerization reactor train is described by the following transfer function model (Madhuranthakam and Penlidis (2016)):
(8.39)

where the input is the flow rate of the Chain Transfer Agent (CTA) to the first reactor in the reactor train and the output is the weight-based average molecular weight (). The parameters in the transfer function for the reactor are given as , and . Design PID controller for this polymer reactor using two frequency response points (see Section 8.3).
1. Choosing , which is case that the desired closed-loop transfer function equal to the open-loop transfer function, find the PID controller parameters.
2. Choosing and , find the PID controller parameters.
3. Choosing and , find the PID controller parameters.
4. Compare the Nyquist plots for the PID control systems and find their gain margin, phase margin and delay margin.
5. Simulate the closed-loop PID control systems with unit step reference signal and a unit step disturbance entering the systems at half of the simulation time. The sampling interval is 1 (), and the simulation time is 3000 () because it is a very slow process. In the simulation, both proportional control and derivative control are implemented on the output only to reduce overshoot in the reference response. The derivative filter time constant is selected as . For the simulations studies, use the MATLAB real-time function PIDV.slx created in Tutorial 4.1.
6. What are the observations when we compare the three PID control systems? What are the observations when we compare the three PID control systems with the results obtained in Example 2.7?
7. What are the reasons behind the closed-loop performance improvement when using the frequency domain based design technique for this particular system?