A Relay Feedback Method for the Tuning of Linear Active Disturbance Rejection Controllers

Active disturbance rejection control (ADRC) is a new control method that estimates the total disturbance by an extended state observer and compensates it by a state feedback law, and good disturbance rejection performance is expected compared to traditional PID control method. However, the parameters of ADRC are hard to tune, even for linear ADRC (LADRC). This paper proposes a new tuning method for second-order LADRC based on relay feedback, which is simple to implement without knowing the information of the controlled plant. Simulation examples show that the proposed method can achieve good performance in disturbance rejection and robust performance. The tuning rule is also tested for the control of a steam condenser. Compared with PID, the proposed method has smaller overshoot and shorter settling time.


I. INTRODUCTION
Tuning of controllers for plants with uncertainties are important and challenging for control systems design. Lots of efforts have been made to deal with uncertainties, e.g., traditional robust control methods for internal uncertainties [1]- [3], and disturbance accommodation control (DAC) for external uncertainties [4], [5], etc. PID, as the most commonly used control method in the current industrial process, has a long history in the industrial field because of its simple structure and easy tuning [6]. However, PID control cannot achieve good effect when the controlled system has a large delay, nonlinearity, or uncertainty. To solve this problem, Han proposed a new control method called active disturbance rejection control (ADRC) in [7], [8]. The core idea of ADRC is to estimate the disturbance that is composed of the internal uncertainties and the external disturbances in real time through an extended state observer, and to compensate it with a simple state-feedback control law. However, ADRC is limited in practical applications due to its multiple parameters The associate editor coordinating the review of this manuscript and approving it for publication was Jun Hu . and complex structure. Later, Gao improved ADRC and implemented all the nonlinear functions in ADRC with linear forms to obtain linear ADRC (LADRC) [9], [10]. Since then lots of engineering applications [11]- [14] and theoretical verifications [15]- [18] have been done for LADRC. Up to now, substantial efforts have been focused on tuning the parameters of LADRC. Reference [9] presented the bandwidth conception and reduced the parameters to two; [19] presented a method that needs only one parameter to tune when the desired settling time is given; [20] suggested that any strictly proper controller which has integration function can be converted to a general ADRC; [21], [22] proposed an idea that uses the extra dynamic information to tune LADRC [21], [22]; [23] gave an explicit tuning formula for second-order LADRC for FOPDT; [24], [25] showed that PID parameters can be converted into the parameters of LADRC, which make it possible to tune LADRC via the tuning formula of PID controllers.
Though there are many methods proposed to tune the parameters of LADRC, there are some problems to be solved. For example, how to get the initial values of the tuning parameters as soon as possible without knowing the model information of the plant has become an urgent problem, which is a critical step in the application of LADRC. In the 1980s, Astrom proposed a method to obtain the ultimate point of the plant based on the relay feedback [26]. A relay is used in the feedback loop connected to the process and after that the parameters of PID are obtained by Ziegler-Nichols tuning rules. Luyben extended this method in [27], and showed that it is a good way to obtain plant information and to design controllers in process control by the method of relay feedback [28]- [31]. This motivates us to utilize the relay feedback method to obtain the initial parameters of LADRC without identifying the plant model.
A tuning formula is proposed in this paper for second-order LADRC based on relay feedback. The method is based on the LADRC tuning rule proposed in [23], thus it not only includes almost all the advantages of the literature [23], but also combines them with the relay feedback to obtain the relationship between the critical point and LADRC tuning parameters. The tuning formula is tested for a series of process models like FOPDT and high-order systems, and a steam condenser with a Simulink model. Compared with the previous methods, the performances obtained by using relay-tuning method are better.
The rest of the paper is arranged as follows: The basic theory of second-order LADRC and relay feedback is introduced in Section 2; Section 3 reviews [23] briefly and the tuning formula is derived; In Section 4, some benchmark systems are used to verify the applicability of the proposed method; In Section 5, the tuning formula is applied to a close-to-real steam condenser. And conclusions are given in Section 6.

A. SECOND-ORDER LADRC
In general, consider a second-order nonlinear system [32]: where y(t), u(t), d(t) and b are the output, the input, the external disturbance and the high-frequency gain of the system, respectively. Besides, f (y, u, d) is treated as the total disturbance. In order to estimate f (y, u, d) accurately, it is represented as a new state variable in the system. Let Assuming f is differentiable, the derivative of f is regarded as a new state variable. And then the system in (1) can be described as: where For process (3), the following LESO can be designeḋ where L o is the observer gain: When L o is appropriate, z 1 , z 2 and z 3 will approximate y,ẏ and f . The following control law can be used to compensate the total disturbance where K o is the controller gain: In conclusion, the structure of LADRC can be represented as in FIGURE 1. The state space form of LADRC can be expresses as Both L o and K o are needed to tune. To simplify the difficulty of parameter tuning, Gao [9] suggested that the tuning of L o and K o can be simplified to tune two parameters by the method of pole placement:ω c and ω o . Thus K o can be calculated from and L o can be calculated from In process control, the plants are usually of high-order, thus the high-frequency gain b of the model (1) is unknown or hard to identify, so in practice b is also treated as another tuning parameter. In sum, there are three tuning parameters for (2nd-order) LADRC: process gain b, observer bandwidth ω o and controller bandwidth ω c . VOLUME 8, 2020

B. RELAY FEEDBACK
In 1984, Åströmn and Hägglund [26] put forward a relay feedback method to obtain the information of the ultimate point of the controlled plant. Steps of the relay feedback experiment are very simple and summarized as follows.
As shown in FIGURE 2(a), this method only requires the addition of a relay to replace the controller in the original control loop. Figure 2(b) shows the working principle of the relay feedback, the controlled plant input u is increased by h. Since the controlled process has a time delay τ , only after the delay time the output begins to increase. At this time, the relay turns to the reverse position, and then the output lags the input by −π radians, and the system will produce continuous oscillation. And the frequency of the oscillation can be calculated from where P u is the period of oscillation. The input of the relay is regarded as a sinusoidal signal, the output of the relay is expanded in Fourier series form and the harmonic analysis is carried out. If only the influence of the main harmonic is considered, the ultimate gain can be described as [26], [27] where h is the height of the relay and a is the amplitude of oscillation. K u and P u can be used to calculate PID controller settings, e.g., with the original Ziegler-Nichols tuning rule, as shown in Table 1.

III. TUNING OF SECOND-ORDER LADRC VIA REALY FEEDBACK
Consider the FOPDT plant with a normalized delay: An existing second-order LADRC tuning formula for such plant is shown below [23]: For the process (14), it is found that the relationship between the tuning parametersb,ω c ,ω o in (15) and the ultimate frequencyω u of the process (14) is shown in FIGURE 3 asτ increases from 0.05 to 5 with a step 0.05. FIGURE 3 shows that there is an approximate linear relationship between the ultimate frequency and the tuning parameters in (15). It is not difficult to get the linear relation by the curve fitting method as: However, most controlled plants are not in the form of the normalized delay, so it is necessary to extend this formula to the following general FOPDT system: Lets then (17) can be written as: Assume that plant (17) can be stabilized with K c (s), then the controllerK will stabilize plant (19) since G(s)K c (s) =Ḡ(s)K c (s). Taking (20) into (15), for plant (17), the tuning parameters can be calculated from the following formulas: i.e., a tuning rule for the general FOPDT process (17) It is easy to show that the ultimate frequency ω u for the general FOPDT (17) is proportional to the ultimate frequencȳ ω u for the FOPDT (14) as then (22) In order to make the tuning formula (24) easy and only relate to relay parameters. We need to know the steady-state gain K and the time constant T besides the ultimate frequency ω u .
By performing a step response experiment, the parameter K which represents the steady-state gain of the system can be obtained, thus it can be supposed to be known. However, the time constant T is unknown. Fortunately, it is related to relay feedback parameters. For FOPDT process (17), it is not difficult to obtain [31] where K u is the ultimate gain of the general FOPDT process (17). Substituting (25) into (24), a tuning formula for second-order LADRC for general FOPDT process (17) is given as follows: where λ is an adjustable factor [23]. The smaller λ is, the faster the system response will be, and the overshoot will increase accordingly.  1) Obtain the steady-state gain K of the process by a step change of the reference.
2) Switch to the relay controller. Obtain the critical parameters of the process, K u and ω u , from the relay feedback experiment.
3) Obtain the parameters of a 2 nd -order LADRC using tuning formula (26). 4) Switch to the 2 nd -order LADRC for closed-loop control. Note: In general, the adjustable factor λ in equation (26) is set to 1. If the dynamic performance is not very good, λ can be adjusted appropriately.
Tuning rule (26) connects the parameters of LADRC with the ultimate parameters of process, so that we can obtain the initial value of parameters of LADRC by relay experiment. Like the tuning rule in [23], though it is derived for the FOPDT models, it can also be applied to a wide range of process models, as will be shown in the next section.

IV. SIMULATIONS
Some benchmark systems in [33] and high-order systems in [34] will be used to verify the applicability of the proposed method in this section. In all the examples a step setpoint of magnitude 1 is conducted at t = 0 s, and a step disturbance is inserted at some appropriate time at the input of the system when the system reaches steady state.
Example 1: Consider the following FOPDT models. They represent delay-dominated (T = 0.2), balanced (T = 2), and lag-dominated (T = 10) systems, respectively [23]. Table 2 shows the parameters and performance indices of the proposed second-order LADRC tuning method based on relay feedback and the method in [23] for the three processes. The performance indices include the integral of time square error (ITSE) caused by the load disturbance as well as the robustness measure discussed in [23]. It is observed that there are some errors in converting the tuning rule in [23] into formula (26) as the relation of the normalized delay and the ultimate frequency is approximated by the linear equations in (16) for delay-and lag-dominated FOPDT models. However, for balanced FOPDT models, the approximation is very close. Thus the proposed tuning rule can be directly applied VOLUME 8, 2020   to balanced FOPDT models. For delay-and lag-dominated FOPDT models, the tuning parameters need to be re-tuned to improve the performance. It can also be concluded that the tuning formula (26) can achieve smaller ITSE for suppressing the load disturbance than the tuning rule in [23], with some sacrifice on the robustness ( Table 2). The time-domain responses shown in FIGURE 5 verify the observation.
Though this tuning rule is obtained from FOPDT models, as a lot of processes can be approximated as FOPDT models, the tuning rule is expected to work for high-order systems, as shown in Example 2.
Example 2: Consider the following high-order systems in [33], [34]: G 3 (s) = e −3s (s 2 + 10s + 1)(s + 1) 2 (29)  G 5 (s) = e −2s (9s 2 + 2.4s + 1)(s + 1) Table 3 shows the parameters and performance indices of the proposed second-order LADRC tuning method based on relay feedback and the method in [23], [34] for the high-order systems. It can be concluded that the performances obtained in this paper are better than LADRC tuned for G 2 in [23], and the PID tuned in [34] for G 3 and G 4 , but worse for G 5 , an oscillatory process that cannot be well approximated by a FOPDT model. It is noted that even though G 4 is a slightly oscillatory process with a damping ratio 0.7, the proposed relay-tuning method can still achieve satisfactory performance.
The response for the high-order systems under the control of the tuned controllers are shown in FIGURE 6, and it is easy to find that the LADRCs tuned by the proposed method achieve satisfactory disturbance rejection performance with small overshoot for high-order non-oscillatory or slightly oscillatory systems.
As we all know, non-minimum phase systems are hard to control because of the right-half-plane (RHP) zeros. The following example shows that the proposed relay-tuning method can work well for high-order non-minimum phase system despite it is derived from FOPDT models.
Example 3: Consider the following high-order nonminimum phase systems: G 7 (s) = (−0.5s + 1)e −s (2s + 1)(s + 1) 2 (33) Table 4 shows the parameters and performance indices of the proposed relay-tuning method for second-order LADRC and the method in [23], [34] applied to the non-minimum phase systems. It can be concluded that proposed method can make ITSE smaller than the LADRC tuned in [23] and the PID tuned in [34], except for a process with RHP zeros close to the imaginary axis. The responses of the non-minimum phase system under the tuned controllers illustrated in FIGURE 7 verify the observation. Thus the proposed method is also applicable to high-order non-minimum phase systems.

V. CLOSE-TO-REAL SIMULATION
To further verify the effectiveness of the proposed method, we will use a steam condenser model to illustrate the advantage of relay-tuning method which can obtain the initial value of parameters of LADRC quickly without knowing the information of plant in this section. This dynamic model of the steam condenser described in [35] is based on energy balance and cooling water mass balance as shown in FIGURE 8, where the controlled variable is the condenser pressure and the manipulated variable is the mass flowrate of the cooling water.
The Simulink model is treated as a real system to test the proposed method to tune a second-order LADRC. Relay feedback experiment is done first, and then read off the relay parameters. Finally, tuning parameters can be obtained by substituting the obtained critical point into formula (26).
FIGURE 9(b) shows that the responses of the steam condenser under the tuned LADRC controller and the PID controller tuned in [35]. It can be concluded that the condenser pressure converges to the setpoint smoothly, while the PID tuned in [35] is quite oscillatory.

VI. CONCLUSION
In this paper, a new tuning formula is proposed for LADRC via the relay feedback. The advantage of this method is that the initial values of the tuning parameters can be obtained quickly by a relay experiment without identifying the model of the controlled plant. This method was tested for a range of processes and a close-to-real steam condenser Simulink model. The result shows that systems under the control of the tuned controllers by us can obtain satisfactory performance.
It should be noted that this new tuning method for LADRC uses a relay feedback experiment to estimate the ultimate point of the controlled plant. Once the controlled plant does not have an ultimate point, the method will not work. Furthermore, because the method is derived from FOPDT models, if the controlled plant shows different dynamics from an FOPDT model, e.g., an oscillatory process or a process with RHP zeros close to the imaginary axis, the proposed method may not achieve satisfactory performance. These issues will be investigated in the future research.