# Design and implementation of high-order PID for second-order processes with time delay

^{1}School of Control and Computer Engineering, North China Electric Power University, Beijing, China^{2}School of Power Engineering, Mongolian University of Science and Technology, Ulaanbaatar, Mongolia

In this study, a state-space pole placement approach is first proposed to design high-order PID controllers for high-order processes. The method makes use of a single parameter to determine the locations of closed-loop poles; thus, a high-order PID controller can be tuned with this parameter. To implement the high-order PID controller in practice, an observer-based PID structure is proposed. The structure utilizes a model-free observer to estimate the plant output and its derivatives, thus retaining the high-order PID structure but can filter the measurement noise and make the high-order derivatives of the plant output available for control. The proposed method is applied to design high-order PID controllers for second-order processes with time delay. Simulation results show that high-order PID can indeed improve the performance of conventional PID controllers for second-order processes with time delay in disturbance rejection and robustness.

## 1 Introduction

Proportional–integral-derivative (PID) control is the most common feedback control mechanism in the industrial process (Åström and Hägglund, 2001; Chen and Seborg, 2002; Åström and Hägglund, 2006). The reason is that the three parameters of the PID controller have a clear relationship with the performance of the system, so it is easy to tune online. However, with the increase in the complexity of industrial processes and the uncertainties of the controlled plant, the traditional PID control may not meet the requirements of control performance due to its structural characteristics. Some limitations of the PID controller are illustrated in Oliveira et al. (2009). Some techniques are proposed to improve the PID control, including adding integral feedback, feedforward, or delay compensation to the traditional PID loop, which makes the PID structure complicated and requires extra parameters (Duan et al., 2008). On the other hand, control scholars put forward many advanced control methods to replace PID control, such as LQG (Garrido-Moctezuma et al., 1997), H infinity (Tan et al., 2006), and adaptive control (Mocci et al., 2020). Although these advanced control methods improve the control performance, they are rarely used in practice due to implementation, tuning, and maintenance issues. Therefore, more than 90% of the loops in the industrial process still use the PID structure (Dash et al., 2015; Latif et al., 2020).

It is well-known that PI control is suitable for first-order processes and PID control is suitable for second-order processes. For second-order processes with time delays, will high-order PID with a second-order derivative

Vrančić and Huba, (2021) show that the high-order PID controllers can significantly improve the control performance of various process models. Lin et al. (2021) illustrate that high-order linear active disturbance rejection controllers (LADRCs) can be interpreted as high-order PID with low-pass filters for speed servo systems and position servo systems. Huba et al. (2021) find that high-order PID control enables faster transients by simultaneously reducing the negative effects of measurement noise and increasing the closed-loop robustness. Huba and Vrani (2018) extend the 2DOF PI and PID controller design for the first-order time-delayed plant by the multiple real dominant pole method to the 2DOF

In practice, PI controllers are used more often than PID controllers, since the latter significantly increases the controller output noise (Ediga and Ambati, 2021). One reason is that the derivative term is very sensitive to the measurement noise, so an appropriate filter can significantly improve the closed-loop performance (Segovia et al., 2014). Naturally, with higher degrees of controllers, the problem becomes aggravated. Therefore, the appropriate high-order filter is inevitable in practical applications. Furthermore, obtaining derivative and high-order derivative signals may amplify sensor noises. Therefore, the filter is usually considered an integral part of the PID design, and the filter constants need to be selected as a compromise between robustness and performance, so PID tuning is essentially a four-parameter design procedure. However, noise is not considered in high-order PID at present. In view of this, considering the form of the high-order PID controller and ensuring the selection of its adjustable poles should be an improvement way that can be considered. Therefore, a high-order PID design method and noise suppression method are proposed in this study.

In this study, an observer is used to estimate the output and derivative of the system, and a high-order PID control structure is proposed, which does not depend on the controlled plant model and retains the traditional PID control structure but increases the controller’s disturbance rejection ability. Usually, a time delay is an approximation of high-order dynamics of the controlled plant except the pure transport delay. So, a process with time delay is in fact a high-order system; thus, a high-order PID controller can be used to achieve better control performance. The goal of the study was to propose a practical control structure that can utilize high-order derivatives of the process to improve the control performance, as compared with the conventional three-term PID controller, and we tried to design the controller via the modern control method. The structure and the design method can build a link between the modern control theory and classical control theory. The main contributions of this study are as follows: 1) an observer-based PID structure is proposed to improve the performance of conventional PID by providing high-order derivatives. The proposed observer-based PID structure is an extension of the conventional PID + filter structure in which high-order derivatives can be incorporated into the structure and the measurement noise can be handled through the extended state observer. 2) A state-space pole placement method is proposed to directly design PID gains, thus linking the modern control theory with the classical control theory. It retains the essential terms of conventional PID. To obtain a PID controller gain, the trick is to transform the model into the canonical observable form. The method is applied to second-order processes with time delay, and the simulation results show that the structure can achieve better performance than other variations of PID, and the design is in the usual modern control theory framework.

The structural arrangement of this study is as follows: in Section 2, the high-order PID controller is proposed to design through the pole placement method; in Section 3, an observer-based PID is proposed to implement high-order PID in practice; the structure and tuning of high-order PID for a variety of second-order timed-delayed systems are tested in Section 4; finally, conclusions are given in Section 5.

## 2 Design of high-order PID controllers *via* pole placement

There are many methods to get the parameters of an ideal PID. It is possible to apply the state feedback design in the PID design which is a convenient and useful control design method in the modern control theory. It is shown that the available plant model information used in the high-order PID design can guarantee the stability of the actual plant. Suppose the controlled plant has a minimal state-space realization as follows:

In this study, we will propose a method to design a high-order PID controller. Our idea is to obtain a state feedback control law *x* is not necessarily composed of *y* and its derivatives,

where

where

Since

Define a new variable

Here, *y*, and the extended state-space model of the controlled plant is

Suppose a state-feedback control law

To obtain a static state feedback gain with only derivatives and integral of y, by ignoring the derivatives of u, then we have

Here, *u* is the combination of the derivatives and integrals of *y*. It is in the form of a high-order PID control, with gain.

For the second-order system (*m* = 2), (10) is the conventional PID control, with

For the third-order system (*m* = 3), (10) is

The state-feedback law (9) can be designed by the well-known pole-placement method. It is well-known that the dominant poles determine the rise time and damping of the closed-loop system, so they can be used to tune the parameters of high-order PID. To design different-order controllers, different numbers of poles are selected as suggested in (13). The two poles ^{2}), four poles (a pair of dominant poles + two (same) negative real poles) can be used. For simplicity, the poles can be chosen as:

From extensive simulations, in order to have good damping and fast settling time in the disturbance rejection response, the damping ratio

## 3 Practical implementation of high-order PID controllers

In the previous section, ideal PID and

In this study, we try to estimate the derivatives of the output via an observer instead of using filters. Consider the ideal

It can be treated as the state-feedback gain

where the state vector is defined as

and the controller gain is

and the state-space matrices are

The state vector

with the observer gain

For simplicity, the observer gain

when the observer gain is chosen properly,

Combining the state feedback control and the observer, a state-space realization (*m*th-order) of

Note that (23) is a standard state-feedback-observer form; however, the novelty is that the state-feedback gain can be interpreted as the (high-order) PID gain due to the special state-space coefficients, and thus it builds a link between modern control and classical control theories. Not every state-feedback-observer form can be interpreted as this. It is a practical implementation of the ideal

The structure of

It is noted that the output

instead of using the true model. Thus, the model-independent control structure of PID is retained.

^{2}) will be obtained. So the order can be chosen by the designer. In this study, we are interested in the PIDD^{2} controller, so a second-order model with time delay (the third-order model) is used in the design. For example, when

when

## 4 Simulation

This section verifies the high-order PID design for a variety of second-order processes with time delay. To use the state-space pole placement method, the delay can be approximated using the first-order Pade approximation. As an application of the high-order PID control, we consider disturbance rejection and robustness analysis of different control methods. To measure the robustness of the designed system, the peak of the sensitivity function *S* usually occurs at low- and mid-frequencies, so *u*(t) for different controllers are also used in this study. They are compromised with pole positions, which are determined by

where *n* = 20) if the references have not given the practical PID form. Also, since the initial response of a PID controller is large without setpoint weighting, while the proposed OB-PID has setpoint weighting, the control signals of the simulations are plotted within a suitable limit to distinguish them from the load disturbance rejection response. The performance indexes, ITAE and TV, are computed only for the load disturbance rejection response.

**Example 1. **Considering the following second-order plus time delay (SOPTD) process

a high-order PID control can be tuned with

The observer bandwidth

where a setpoint weight *n* = 20) for the ideal PID as follows (Wang et al., 2016):

The performance indexes for the two controllers are listed in Table 1. It clearly shows that the observer-based PID (29) has better disturbance rejection performance and robustness than the practical PID (30). Figure 2A shows the responses of the closed-loop system, where at t = 0 s, a unit step reference is inserted and a step disturbance input of magnitude 1 is inserted at t = 40 s. The red dotted dashed red line and black solid line represent the responses of the practical PID form (30) and observer-based PID (29), respectively.It can be clearly seen from Figure 2A that the observer-based PID has better disturbance rejection, faster response speed, and smaller overshoot than the practical PID as shown in Wang et al. (2016). The Bode plots (Figure 2C) show that the observer-based PID (29) and the practical PID (30) roll off at high frequencies, thus attenuating measurement noise as expected. The observer-based PID has a larger roll-off rate; thus, the observer-based PID has better performance against high-frequency noise. In Figure 2D, the system with a white noise with the power of 0.0001 and a sampling period of 0.1 s is introduced into the process output. It shows that the proposed observer-based PID has better sensor noise rejection performance than the practical PID (30). In the actual situation, there always exists a model mismatch. In order to evaluate the robustness of the controller, a 20% increase in the process gain and time delay and a 20% decrease in the two-time constants are considered, i.e., the perturbed plant is *M*_{s}, and TV of the control input u(t) for the perturbed system are also listed in Table 1. The TV of the observer-based PID is smaller and thus has better control effort. The simulation results show that the proposed observer-based PID gives better control performances in the set-point tracking, disturbance rejection performance, and robustness than the practical PID as given in Wang et al. (2016).

**FIGURE 2**. Responses for Example 1. **(A)** Nominal plant. **(B)** 20% variations in model parameters. **(C)** Bode plots of controllers. **(D)** Measurement noise.

**Example 2. **Considering the unstable process with a large time delay (USOPDT):

To tune a high-order PID for this plant, ^{2} with the following gains:

The extended observer gain can be tuned with the observer bandwidth

Consider the controller designed by Shamsuzzoha and Lee (2009). A practical PID (*n* = 20) is implemented in practice for the controller:

The responses of the processes under the observer-based PID (33) proposed in this study and practical PID in Shamsuzzoha and Lee, (2009) are shown in Figure 3. Compared with the practical PID, the observer-based PID can achieve better tracking and disturbance rejection performance. The performance indexes for different controllers are listed in Table 2. It clearly shows that the observer-based PID (33) has better disturbance rejection performance and better control effort with smaller TV than the practical PID (34). Figure 3A shows the responses of closed-loop systems, where at t = 0 s, a unit step reference is inserted and a step disturbance input of magnitude 0.1 is inserted at t = 100 s. The red dotted dashed red line and black solid line represent the response of the practical PID form (34) and observer-based PID (33), respectively.It can be clearly seen from Figure 3A that the observer-based PID has better disturbance rejection and faster response speed than the practical PID in Shamsuzzoha and Lee, (2009). The Bode plots (Figure 3C) show that the observer-based PID (33) and the practical PID (34) roll off at high frequencies, thus attenuating measurement noise as expected. The observer-based PID has a larger roll-off rate; thus, the observer-based PID has better performance against high-frequency noise. In Figure 3D, the system with a white noise with a power of 0.00001, and a sampling period of 0.1 s is introduced into the process output. It shows that the proposed observer-based PID has better noise rejection performance than the practical PID (34). In the actual situation, there always exists a model mismatch. In order to evaluate the robustness of the controller, a 5% increase in the process gain and time delay and a 5% decrease in two-time constants are considered, i.e., the perturbed plant is *Ms*, and TV of the control input u(t) for the system are also listed in Table 2. The TV of the observer-based PID is smaller and thus has better control effort. The simulation results show that the proposed observer-based PID gives better control performances both in the setpoint tracking and disturbance rejection performance than the practical PID as given in Shamsuzzoha and Lee, (2009).

**FIGURE 3**. Responses for Example 2. **(A)** Nominal plant (b) 5% variations in model parameters. **(C)** Bode plots of controllers. **(D)** Measurement noise.

**Example 3. **The following second-order dead time process with two unstable poles process (SODPTUP) is considered:

A two-degree-of-freedom control structure with a disturbance estimator in the form of the practical PID (*n* = 20) was proposed by Shamsuzzoha and Lee (2009) for this process:

To tune a high-order PID for this plant, choose ^{2} with the following gains:

The extended observer gain can be tuned with the observer bandwidth

The performance indexes for different controllers are listed in Table 3. It clearly shows that the observer-based PID (38) has better robustness and control effort than the practical PID (36). Figure 4A shows the responses of the closed-loop system, where at t = 0 s, a unit step reference is inserted and a step disturbance input of magnitude 1 is inserted at t = 20 s. The red dotted dashed red line and black solid line represent the response of the practical PID form (36) and observer-based PID (38), respectively. The Bode plots (Figure 4C) show that the observer-based PID (38) and the practical PID (36) roll off at high frequencies, thus attenuating measurement noise as expected. The observer-based PID has a larger roll-off rate; thus, the observer-based PID has better performance against high-frequency noise. In Figure 4D, the system with a white noise with a power of 0.00001 and a sampling period of 0.1 s is introduced into the process output. In order to evaluate the robustness of the controller, a 20% increase in the process gain and time delay and a 20% decrease in two-time constant are considered, i.e., the perturbed plant is *M*_{s}, and TV of the control input u(t) for the system are also listed in Table 3. The two controllers have almost the same TV, thus having almost the same control effort when there is noise. The simulation results show that the proposed observer-based PID gives good control performances and good robustness and almost the same noise attenuation compared with the practical PID given in Shamsuzzoha and Lee, (2009).

**FIGURE 4**. Responses for Example 3. **(A)** Nominal plant. **(B)** 5% variations in model parameters. **(C)** Bode plots of controllers. **(D)** Measurement noise.

**Example 4. **Considering the unstable process with an integrator:

To tune a high-order PID for this plant, choose ^{2} with the following gains:

With

For comparison, consider the two-degree-of-freedom control structure with a disturbance estimator in the form of the practical PID (*n* = 20) proposed by Shamsuzzoha and Lee, (2009) for this process.

The performance indexes for different controllers are listed in Table 4. It clearly shows that the observer-based PID (41) has better control effort than the practical PID (42) but disturbance rejection performance and robustness similar to the practical PID. Figure 5A shows the responses of the closed-loop system, where at t = 0 s, a unit step reference is inserted and a step disturbance input of magnitude 1 is inserted at t = 15 s. The red dotted dashed red line and black solid line represent the response of the practical PID form (42) and observer-based PID (41), respectively.It can be clearly seen from Figure 5A that the observer-based PID has almost the same disturbance rejection performance as the practical PID as shown in Shamsuzzoha and Lee, (2009). The Bode plots (Figure 5C) show that the observer-based PID and the practical PID roll off at high frequencies, thus attenuating measurement noise as expected. The observer-based PID has a larger roll-off rate; thus, the observer-based PID has better performance against high-frequency noise. In Figure 5D, the system with a white noise with a power of 0.0001 and a sampling period of 0.1 s is introduced into the process output. It shows that the proposed observer-based PID has better noise rejection performance than the practical PID. In order to evaluate the robustness of the controller, a 10% increase in the process gain and time delay and a 10% decrease in the time constant are considered, i.e., the perturbed plant is *M*_{s}, and TV of the control input u(t) for the system are also listed in Table 4. Simulation results show that the proposed observer-based PID has almost the same performance in disturbance rejection and robustness but better noise attenuation than the practical PID as shown inShamsuzzoha and Lee, (2009).

**FIGURE 5**. Responses for Example 4. **(A)** Nominal plant. **(B)** 10% variations in model parameters. **(C)** Bode plots of controllers. **(D)** Measurement noise.

## 5 Conclusion

A state feedback pole-placement method was proposed to design high-order PID controllers, and an observer-based PID structure was proposed to implement high-order PID controllers. The observer-based PID can effectively suppress noise by estimating the derivatives of the plant output through a model-independent observer. It was shown that the observer-based PID can be tuned with two parameters,

It is observed that if a cost function is used in the state-feedback design and observer design, then it can be directly related to an optimal controller; thus, an observer-based PID can also be designed via the well-known LQG method, which will be investigated in the future.

## Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

## Author contributions

WH substantially contributed to the conception and design of the work and the acquisition of data for the work. QH analyzed and interpreted data for the work. UD drew the picture and checked the work. WT revised the manuscript critically for important intellectual content.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: observer-based PID, high-order PID, second-order processes with time delay, parameter tuning, pole placement

Citation: Han W, Hu X, Damiran U and Tan W (2022) Design and implementation of high-order PID for second-order processes with time delay. *Front. Control. Eng.* 3:953477. doi: 10.3389/fcteg.2022.953477

Received: 26 May 2022; Accepted: 27 July 2022;

Published: 29 August 2022.

Edited by:

Tao Liu, Dalian University of Technology, ChinaReviewed by:

Ramon Vilanova, Universitat Autònoma de Barcelona, SpainShengquan Li, Yangzhou University, China

Copyright © 2022 Han, Hu, Damiran and Tan. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Wenjie Han, 13253151532@163.com