The calculated areas for the processes (20) without and with the controller filter.
The main tasks of control in various industries are either tracking the setpoint changes or rejecting the process disturbances. While both aim at maintaining the process output at the desired setpoint, the controller parameters optimised for setpoint tracking are generally not suitable for optimal disturbance rejection. The overall control performance can be improved to some extent by using simpler 2-DOF PID controllers. Such a controller structure allows the disturbance rejection to be optimised, while it also improves the setpoint tracking performance with additional controller parameters (usually through the setpoint weighting factors). Since such 2-DOF structures are usually relatively simple, the optimization of tracking performance is usually limited to the reduction of process overshoots instead of achieving an optimal (fast) tracking response. In this chapter, an alternative approach is presented in which the parameters of the PID controller are optimised for reference tracking, while the performance of the disturbance rejection is substantially increased by introducing a simple disturbance estimator approach. The mentioned estimator requires adding two simple blocks to the PID controller. The blocks are the second-order transfer functions whose parameters, including the PID controller parameters, can be calculated analytically from the process characteristic areas (also called process moments). The advantage of such an approach is that the mentioned areas can be analytically calculated directly from the process transfer function (of any order with time delay) or from the time response of the process when the steady state of the process is changed. Both of the above calculations are absolutely equivalent. Moreover, the output noise of the controller is under control as it is considered in the design of the controller and compensator. The closed loop results on several process models show that the proposed method with disturbance estimator has excellent tracking and disturbance rejection performance. The proposed controller structure and tuning method also compare favourably with some existing methods based on non-parametric description of the process.
- tuning method
- disturbance rejection
- disturbance estimator
- multi-objective design
The control of industrial processes requires efficient control loops. A majority of the control loops in various industries are implemented by the Proportional-Integrative-Derivative (PID) control algorithms. For efficient control, the PID controllers require proper tuning of the PID controller parameters. The parameters can be calculated to optimise various performance criteria such as integral of error (IE), integral of absolute error (IAE), integral of squared error (ISE) and similar [1, 2, 3, 4]. However, the most important decision that should be made in advance is the choice of the main purpose of the closed-loop system. Namely, the user should choose between the optimal closed-loop responses to reference changes (so-called tracking responses) or the optimal response to process disturbances. While there are many industrial processes that require optimal reference tracking responses, such as robot manipulation, welding, and batch processes, the majority of industrial processes require optimal disturbance rejection.
The history of tuning rules is long, originating in the 1940s with the famous Ziegler-Nichols tuning rules. In the following decades, many other tuning rules have been developed [1, 2, 4, 5, 6, 7, 8, 9, 10]. The rules can be generally categorised according to the required data of the process. The process can be described either in parametric form, e.g., as a process model (transfer function), or in nonparametric form, e.g., as a process time-response.
A relatively new tuning method that optimises either closed-loop tracking or disturbance rejection is the Magnitude-Optimum-Multiple-Integration (MOMI) method [7, 9, 11, 12]. The MOMI method is based on the Magnitude Optimum method, which aims to optimise the frequency response of the closed loop to achieve fast and stable closed loop time response [10, 13, 14, 15]. An interesting feature of the MOMI method is that it works either on the process given by its transfer function (of arbitrary order with time delay) or directly on the time response of the process during the steady state change. It is worth noting that both the parametric and non-parametric process data give exactly the same PID tuning results.
Many tuning methods for PID controllers provide different sets of controller parameters for tracking and disturbance rejection response. Similarly, the MOMI method primarily optimises the tracking response, while its modification, the Disturbance-Rejection-Magnitude-Optimum (DRMO) method, aims at optimising the disturbance rejection response. The latter significantly improves the disturbance rejection response, while the tracking response slows down due to the implemented reference-weighting gain or reference signal filter [9, 16, 17].
The main approach presented in this chapter is the alternative approach. First, the parameters of the PID controller are optimised for tracking performance. Then, a simple disturbance estimator is introduced to significantly increase the disturbance rejection performance [18, 19]. The advantages of the above approach are twofold. First, the disturbance rejection performance can significantly outperform that obtained by the DRMO method. Second, the parameters of the disturbance estimator can also be obtained directly from the non-parametric process data in the time domain. Therefore, the proposed approach can still be applied to the process data which is either in parametric or non-parametric form.
However, in practice, the process output noise is always present. If the controller or estimator gains are too high, the process input signals may be too noisy for practical applications. Therefore, noise attenuation should already be taken into account when calculating the controller and estimator parameters. This chapter shows how to achieve the best trade-off between performance and noise attenuation.
2. Process and controller description
The classic 1-degree-of-freedom (1-DOF) control loop configuration of the process and the controller is shown in Figure 1, where the signals
A process model (1) can be described by the following process transfer function:
The PID controller is described by the following expression:
The closed-loop transfer function
Since the structure of a 1-DOF PID controller does not provide optimal tracking and disturbance rejection at the same time, the 2-degrees-of-freedom (2-DOF) controller can be used instead [1, 2, 4, 8, 16, 20], where
as shown in Figure 2, where parameters
3. MOMI and DRMO tuning methods
The MOMI and DRMO methods, as mentioned earlier, are based on the Magnitude Optimum (MO) method, which goes back to Whitley in 1946 . The MO method shapes the closed-loop amplitude frequency response equal to one in a wide frequency range [6, 7, 10, 12, 13, 14, 21]. Such a closed-loop frequency response is usually “mirrored” into a fast and stable closed-loop time response.
The calculation of controller parameters has been simplified when using the MO method by determining the process characteristic areas or moments, which can be measured directly from the time responses during the change of the process steady-state [12, 15, 21, 22]. The mentioned areas or moments (
The controller parameters, for a given filter time constant
where the modified areas A0* to A5* are:
by using expression (7) . The aforementioned modification of the method, referred to as the MOMI method, allowed the controller parameters to be computed directly from the process time response [12, 21] or from the process transfer function.
Since the MOMI method aims at optimising the tracking performance, the disturbance rejection performance may be degraded for some types of processes.
To improve the disturbance-rejection performance, the optimisation criteria of the MOMI method were modified accordingly. The new method, referred to as the DRMO (Disturbance-Rejection-Magnitude-Optimum) method, achieved significantly improved disturbance rejection performance [9, 16, 17].
Similar to the MOMI method, the controller parameters in the DRMO method are also based on characteristic areas or moments. Therefore, the controller parameters can be calculated either from the process time-response or from the process transfer function.
and the derivative gain
The DRMO tuning method significantly improved the disturbance rejection performance, especially for the lower-order processes. However, the reference tracking becomes slower due to the reference-weighting factors
4. DE-MOMI tuning method
In order to improve the disturbance rejection response, while retaining the tracking response obtained by the MOMI method, a disturbance estimator has been added to the PID controller
The disturbance estimator consists of the process model
the estimated disturbance
In this case the ideal disturbance compensation is achieved. However, in practice, model mismatch may occur (due to changing process characteristics in time or working point, lower-order process model or the process non-linearity), and the inverse of the process usually cannot be obtained, since majority of the actual processes are either strictly proper or they have time delays. Therefore, another strategy is required.
For practical applications, the solution has to be as simple as possible. In this manner we decided to use the following process model, the inverse process model and the disturbance estimator filter:
The remaining question is how to obtain the process model if the actual process is of the higher order or if the actual process is not known (e.g. the areas (moments) were calculated directly from the process time-response)? Fortunately, the process model can be calculated directly from the obtained areas (5), as derived in :
Now, all the model parameters are known and the disturbance filter
Derivation of disturbance filter parameters depends mainly on desired disturbance rejection performance. It is natural that the disturbance signal reconstruction (
With sufficiently small time constant (
One remaining parameter of the disturbance filter
It means that, by applying
Figure 4 shows an example on delayed second-order process, when applying the step-wise external process input disturbance signal
The remaining question is how to find the most appropriate filter gain
should be optimised according to the modified MO criterion [9, 16]. Note that expression (17) holds when the process and the model transfer functions are equivalent. Since the disturbance filter time constant is defined, and all of the controller and the model parameters are calculated, the only optimisation parameter is the gain
For the given controller filter (
Illustrative example 1
To illustrate the proposed design of DE-MOMI method, according to control structure in Figure 5, let us calculate the controller, model and disturbance filter parameters for the following processes:
The a-priori chosen filter time constants were:
Next, the PID controller parameters are calculated from (6) and from (9), since we are going to compare the proposed DE-MOMI method with MOMI and DRMO methods. The calculated controller parameters are given in Table 2.
|MOMI controller for ||1.81||0.89||0.93|
|DRMO controller for ||2.25||1.49||0.93|
|MOMI controller for ||1.61||0.64||1.08|
|DRMO controller for ||1.93||0.98||1.08|
The process models
Finally, the disturbance filter gain
Therefore, the complete inverse of the models with accompanying disturbance filters (see Figure 3) are the following:
The closed-loop responses, obtained with the calculated controller, model and filter parameters, for the MOMI, DRMO and the proposed DE-MOMI method, are given in Figures 6 and 7. At
The disturbance rejection performance of the DE-MOMI method can be increased by decreasing the disturbance filter time constant
Calculating the controller and DE parameters is a relatively simple process. However, to simplify it even further, all Matlab/Octave scripts are available on the OctaveOnline Bucket website . The layout of the website is shown in Figure 8. To calculate the controller and DE parameters, the user must 1) change the process and filter parameters, 2) press the “Save” button, and 3) press the “Run” button. The script will be executed and on the right side of the web screen all calculated parameters will be displayed. Note that users can change the content of the script only temporarily.
5. Noise attenuation of DE-MOMI method
As already mentioned in the previous sub-chapter, the output noise of disturbance estimator (
In practice, it is important to keep the controller output noise within some limits. Namely, if the controller’s and the estimator’s filter time constants are too low, the DE-MOMI controller output noise can be so high that the controller would be useless in practice.
The controller noise is mainly caused by the process output noise
In practice, on the other side, it is enough to keep the noise sufficiently low at some sufficiently high frequency. The definition of “high frequency” is arguable. In discrete-realisation of the controller, the sampling frequency is
As already mentioned above, the source of controller noise is the process output noise
In practical applications of the DE-MOMI method, the noise specifications (limitations) should be given in as simple form as possible for the user (operator). We decided that the actual parameters, given by the user should be the high-frequency gains of the controller (
The actual gain of the PID controller around the chosen high frequency
The controller filter time constant can then be calculated as:
Since the PID controller parameters depend on the filter time constant
The calculation of the disturbance filter high-frequency gain
In a similar manner, the disturbance filter time constant can be derived as:
Illustrative example 2
Let us illustrate the calculation procedure for the following processes:
Note that other process models were chosen as in the previous case (20) in order to test different types of processes. The chosen high-frequency gains of the PID controller and the disturbance filter are
The initially chosen filter time constants were (the values are not critical):
The characteristic areas are calculated from (5). For the given high-frequency gain
Note that indexes 3 and 4 in above filter time constants stand for the processes
|MOMI controller for ||2.35||1.88||0.48|
|DRMO controller for ||2.91||3.83||0.48|
|MOMI controller for ||0.84||0.26||0.77|
|DRMO controller for ||0.94||0.32||0.77|
The process models
According to the chosen high-frequency gain
Therefore, the complete inverse of the models with accompanying disturbance filters (see Figure 3) are the following:
The closed-loop responses for the MOMI, DRMO and the proposed DE-MOMI method, are given in Figures 11 and 12. Again, the disturbance rejection performance of the DE-MOMI method is the best (note that the unity-step process input disturbance signal was applied at the half of experiment time). The level of controller output (
The disturbance rejection performance of the DE-MOMI method can be additionally improved by increasing the high-frequency gain
The computation of the controller and the DE parameters can be performed similarly as before on another OctaveOnline Bucket website . The calculation of the parameters can be performed similarly as shown in Figure 8, with the difference that the name of the script is now Octave_Calc_GC_GF_Noise.m. To calculate the controller and DE parameters, the user must 1) change the process and noise gain parameters, 2) press the “Save” button, and then 3) press the “Run” button. The script will run and the right side of the web screen will display all the calculated parameters. Note that users can only temporarily change the contents of the script.
6. Comparison to some other methods
In this sub-chapter the proposed method will be compared to some other tuning methods based on non-parametric description of the process. Besides the already introduced MOMI and DRMO methods, the DE-MOMI method will be compared to Åström and Hägglund’s tuning method  (denoted as “AH”) and to ADRC method .
The AH method  is based on the calculation of the maximum sensitivity index
The method does not require the process transfer function. However, few user-defined parameters, like the observer speed, the desired settling time and the main controller gain
Since ADRC method depends on three user-defined parameters, which, in great extent, determine the closed-loop performance, we were limited to the set of processes tested in . Someone would argue that, by limiting our choice to the mentioned processes, we are favouring the ADRC method. However, it should be noted that in , the ADRC method was tested on 8 different processes, so the choice of processes was actually not significantly limited. In this regard, the following two processes have been selected:
The PID controller parameters for the MOMI, DRMO, DE-MOMI and AH methods are given in Tables 5 and 6. The ADRC controller parameters are given in Table 7. The chosen high frequency gains for the PID controller and disturbance estimator are
The sampling time for
It can be seen that the proposed DE-MOMI method, when compared to some other methods, gives quite good responses. The AH method for process
For more objective comparison between the methods, the integral of absolute error (IAE) measure is used. The IAE value has been measured on tracking response (unity step-change of the reference
The DE-MOMI method, therefore, compares favourably with few other methods, based on the non-parametric description of the process.
The process closed-loop responses for all the process models tested in this chapter (
In the chapter, it was shown that the disturbance rejection performance of the PID controller can be improved by adding a simple disturbance estimator (DE). The disturbance estimator consists of the process model and the inverse process model with DE filter. The advantage of the proposed approach is that the DE parameters can also be obtained directly from the nonparametric process data (time response of the process) without prior process identification. The same is true for the PID controller parameters, which are obtained using the MOMI tuning method. Of course, all PID and DE parameters can also be calculated from the process transfer function if it is known.
The proposed solution, called DE-MOMI method, has been tested on several different process models. It was shown that the control performance of the DE-MOMI method was significantly improved compared to similar MOMI and DRMO methods, especially for lower order processes with smaller time delays. In contrast, the improvements were noticeable but not as significant for higher order processes or processes with larger time delays. The additional advantage of the proposed method was that the tracking performance remained similar to that of the MOMI method.
The controller noise was controlled by the high frequency noise factors KPIDn and KDEn. The advantage of using these factors is that they can be easily understood and defined by the user.
The DE-MOMI method was also compared with some other non-parametric disturbance-rejection methods including the ADRC method. The results showed that the DE-MOMI method has either comparable or better control and tracking performance than the other tested methods. Nevertheless, it should be mentioned that the ADRC method uses a somewhat simpler control structure.
Future research activities could therefore focus on combining the advantages of the DE-MOMI and ADRC methods.
The authors gratefully acknowledge the contribution of the Ministry of Higher Education, Science and Technology of the Republic of Slovenia, Grant No. P2-0001 as well as the support by the grants APVV SK-IL-RD-18-0008 Platoon Modelling and Control for mixed autonomous and conventional vehicles: a laboratory experimental analysis and VEGA 1/0745/19 Control and modelling of mechatronic systems in emobility.