Improving Disturbance-Rejection by Using Disturbance Estimator

Abstract

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.

1. Introduction

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 r, e, u, d and y represent the reference, the control error, the controller output, the process input disturbance, and the process output, respectively.

Figure 1.

The 1-DOF PID controller and the process in the closed-loop configuration.

A process model (1) can be described by the following process transfer function:

$$G_P\left(s\right)=\frac{K_{\mathit{PR}}\left(1+b_{1}s+b_2{s}^2+\cdots +b_m{s}^m\right)}{1+a_{1}s+a_2{s}^2+\cdots +a_n{s}^n}{e}^{-s{T}_{\mathit{del}}}$$E1

where a₁ to a_n are the denominator coefficients, b₁ to b_m are the numerator coefficients, K_PR is the process gain, and T_del is the process time delay. Note that n > m represents a strictly proper process transfer function and that the process is stable.

The PID controller is described by the following expression:

$$G_C\left(s\right)=\frac{K_I+K_{P}s+K_D{s}^2}{s\left(1+s{T}_F\right)}$$E2

where K_I is the integrating gain, K_P is the proportional gain, and K_D is the derivative gain. Note that all three controller terms are filtered by the first-order filter with time constant T_F.

The closed-loop transfer function G_CL between the reference (r) and the process output (y) is as follows:

$$G_{\mathit{CL}}=\frac{G_C{G}_P}{1+G_C{G}_P}$$E3

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 G_CR and G_CY denote the controller transfer function from the reference and the process output, respectively:

$$u=G_{\mathit{CR}}\left(s\right)r-G_{\mathit{CY}}\left(s\right)y$$

$$G_{\mathit{CR}}=\frac{K_I+b{K}_{P}s+{\mathit{cK}}_D{s}^2}{s\left(1+s{T}_F\right)}$$

$$G_{\mathit{CY}}=\frac{K_I+K_{P}s+K_D{s}^2}{s\left(1+s{T}_F\right)},$$E4

as shown in Figure 2, where parameters b and c are reference-weighting parameters for the proportional and derivative terms, respectively.

Figure 2.

The 2-DOF PID controller and process in the closed-loop configuration.

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 [10]. 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 (A₁ to A_k) can also be calculated from the process model:

$$A_0=K_{\mathit{PR}}$$

$$A_1=K_{\mathit{PR}}\left(a_1-b_1+T_{\mathit{del}}\right)$$

$$A_2=K_{\mathit{PR}}\left[b_2-a_2-b_1{T}_{\mathit{del}}+\frac{T_{\mathit{del}}^2}{2!}\right]+A_1{a}_1$$

$$⋮$$

$$A_k=K_{\mathit{PR}}\left[{\left(-1\right)}^{k+1}\left(a_k-b_k\right)+\sum _{i=1}^k{\left(-1\right)}^{k+i}\frac{T_{\mathit{del}}^i{b}_{k-i}}{i!}\right]+\sum _{i=1}^{k-1}{\left(-1\right)}^{k+i-1}{A}_i{a}_{k-i}$$E5

The controller parameters, for a given filter time constant T_F, are then calculated as follows:

$$\left[\begin{array}{c}{K}_I\\ K_P\\ K_D\end{array}\right]={\left[\begin{array}{ccc}-A_1^{\ast }& A_0^{\ast }& 0\\ -A_3^{\ast }& A_2^{\ast }& -A_1^{\ast }\\ -A_5^{\ast }& A_4^{\ast }& -A_3^{\ast }\end{array}\right]}^{-1}\left[\begin{array}{c}-0.5\\ 0\\ 0\end{array}\right],$$E6

where the modified areas A₀* to A₅* are:

$$A_0^{\ast }=A_0$$

$$A_1^{\ast }=A_1+A_0{T}_F$$

$$A_2^{\ast }=A_2+A_1{T}_F+A_0{T}_F^2$$

$$⋮$$E7

The reference-weighting factors are b = c = 1. Note that the areas (moments) in expression (6) apply areas of the process including the controller filter G_F with time constant T_F(4):

$$G_F=\frac{1}{\left(1+s{T}_F\right)}$$E8

by using expression (7) [9]. 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.

The PID controller parameters are calculated according to the following expressions when using the DRMO method [9, 16, 17]:

$$K_P=\frac{\beta -\sqrt{{\beta }^2-\mathit{\alpha \gamma }}}{\alpha }$$

$$K_I=\frac{{\left(1+K_P{A}_0^{\ast }\right)}^2}{2\left(K_D{A}_0^{\ast 2}+A_1^{\ast }\right)}$$E9

where

$$\alpha =A_1^{\ast 3}+A_0^{\ast 2}{A}_3^{\ast }-2A_0^{\ast }{A}_1^{\ast }{A}_2^{\ast }$$

$$\beta =A_1^{\ast }{A}_2^{\ast }-A_0^{\ast }{A}_3^{\ast }+2K_D\left(A_0^{\ast }{A}_1^{\ast 2}-A_0^{\ast 2}{A}_2^{\ast }\right)$$

$$\gamma =K_D^3{A}_0^{\ast 4}+3K_D^2{A}_0^{\ast 2}{A}_1^{\ast }+K_D\left(2A_0^{\ast }{A}_2^{\ast }+A_1^{\ast 2}\right)+A_3^{\ast }$$E10

and the derivative gain K_D is calculated directly from expression (6). The reference-weighting factors are b = c = 0.

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 b = c = 0 in the 2-DOF control structure (4). The problem can be circumvented by including a simple disturbance estimator in the control scheme. Such a solution is denoted as DE-MOMI method.

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 G_C(s) (2), as depicted in Figure 3.

Figure 3.

The PID controller with disturbance estimator.

The disturbance estimator consists of the process model G_M, the inverse process model G_MI and the filter G_FD. In hypothetical case, when the process model is ideal representation of the bi-proper process without time-delay, and the filter G_FD = 1:

$$G_M=G_P,G_{\mathit{MI}}=G_M^{-1},G_{\mathit{FD}}=1,$$E11

the estimated disturbance d_f equals to the actual disturbance d:

$$d_f=d.$$E12

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:

$$G_M=\frac{K_{\mathit{PRM}}{e}^{-s{T}_{\mathit{\text{delm}}}}}{1+a_{1m}s+a_{2m}{s}^2}$$

$$G_{\mathit{MI}}=\frac{1+a_{1m}s+a_{2m}{s}^2}{K_{\mathit{PRM}}}$$

$$G_{\mathit{FD}}=\frac{K_{\mathit{FD}}}{{\left(1+s{T}_{\mathit{FD}}\right)}^3}$$E13

where K_PRM and T_delm are the process model gain and time delay and a_1m and a_2m are the process model dynamic parameters. Parameters K_FD and T_FD are the disturbance filter gain and time constant, respectively.

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 [23]:

$$\frac{T_{\mathit{\text{delm}}}^3}{6}-\frac{A_1}{2A_0}{T}_{\mathit{\text{delm}}}^2+\left(\frac{A_1^2}{A_0^2}-\frac{A_2}{A_0}\right)T_{\mathit{\text{delm}}}+\frac{2A_1{A}_2}{A_0^2}-\frac{A_3}{A_0}-\frac{A_1^3}{A_0^3}=0$$

$$K_{\mathit{PRM}}=A_0$$

$$a_{1m}=\frac{A_1}{A_0}-T_{\mathit{\text{delm}}}$$

$$a_{2m}=-\frac{A_2}{A_0}+\frac{a_{1m}{A}_1}{A_0}+\frac{T_{\mathit{\text{delm}}}^2}{2}$$E14

The process model delay T_delm is calculated from the third-order equation in (14). The solution is the smallest real positive result [23].

Now, all the model parameters are known and the disturbance filter G_FD parameters should be derived. Before continuing the derivation we should be aware of the fact that G_MI(s) is not proper, so it cannot be realised in practice without the accompanied filter G_FD(s). Multiplication of both is strictly proper, so the entire block can be easily implemented inside the controller.

Derivation of disturbance filter parameters depends mainly on desired disturbance rejection performance. It is natural that the disturbance signal reconstruction (d_f) is faster if the filter time constant T_FD is smaller. In hypothetical case, if G_M = G_P, the reconstructed disturbance signal d_f becomes:

$$d_f=K_{\mathit{FD}}\frac{e^{-s{T}_{\mathit{\text{delm}}}}}{{\left(1+s{T}_{\mathit{FD}}\right)}^3}d$$E15

With sufficiently small time constant (T_FD → 0), where the disturbance filter gain K_FD = 1, and there is no process time delay, df ≈ d. In this case the reconstructed disturbance signal d_f perfectly compensates the disturbance d. On the other hand, smaller disturbance filter time constant significantly increases the process output noise present in the measurements and forwards it to the controller output. Therefore, the T_FD should be selected according to the tolerated noise gain of the disturbance estimator, as will be discussed in detail in the next sub-chapter.

One remaining parameter of the disturbance filter G_FD(13) is the gain K_FD. One would, naturally, expect that the most optimal value should be K_FD = 1, since only in this case, after some time, d_f becomes the same to d(15). Therefore, the process input disturbance d is eliminated by the reconstructed disturbance d_f. However, as will be shown below, the optimal disturbance response is obtained with lower values of the gain K_FD. Namely, due to the disturbance compensator, the external process input signal d generates the delayed reconstructed disturbance signal d_f (15). Combined together, the actual process input u, due to disturbance d, is d-d_f. The step-like signal d, therefore, generates pulse-like actual process input disturbance signal d-d_f. Since the PID controller is present in the loop, and it contains the integrating term, the process output (y) deviation in one direction (e.g. above the reference) should be compensated by the process output deviation in the opposite direction (e.g. below the reference). Namely, when using K_FD = 1, the integral of the control error must be:

$${\int }_{t=0}^{\infty }e\left(t\right)\mathit{dt}=0.$$E16

It means that, by applying K_FD = 1, the additional process undershoot, after the initial process overshoot due to the disturbance d, is inevitable.

Figure 4 shows an example on delayed second-order process, when applying the step-wise external process input disturbance signal d, and when using K_FD = 1 (upper figure) and K_FD = 0.44 (lower figure). The process output undershoot in the upper figure is clearly seen. By appropriately reducing the filter gain to K_FD = 0.44, the disturbance rejection response is improved (lower figure).

Figure 4.

The closed-loop signals when applying step-wise external process input disturbance signal with disturbance filter gains K_FD = 1 (upper figure) and K_FD = 0.44 (lower figure).

The remaining question is how to find the most appropriate filter gain K_FD. Certainly, the K_FD should be chosen so as that the disturbance rejection is optimised. Here we can use the same optimisation criteria as in the DRMO tuning method. Therefore, the transfer function G_CLD(s) between the external disturbance (d) and the process output (y):

$$G_{\mathit{CLD}}\left(s\right)=\frac{Y\left(s\right)}{D\left(s\right)}=\frac{G_M\left(1-G_{\mathit{FD}}{e}^{-s{T}_{\mathit{\text{delm}}}}\right)}{1+G_M{G}_P}$$E17

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 K_FD. By using similar derivation as in [9, 16], the optimal filter gain K_FD is calculated as:

$$K_{\mathit{FD}}=\frac{-b_0+\sqrt{b_0^2-4a_0{c}_0}}{2a_0}$$E18

where

$$a_0=-d_0+2f_0{K}_{\mathit{IP}}+K_{\mathit{IP}}^2\left(T_F^2-3T_{\mathit{FD}}^2\right)$$

$$b_0=2d_0-4f_0{K}_{\mathit{IP}}+K_{\mathit{IP}}^2\left(-2T_F^2+12T_{\mathit{FD}}^2+6T_{\mathit{FD}}{T}_{\mathit{\text{delm}}}+T_{\mathit{\text{delm}}}^2\right)$$

$$c_0=-d_0+2f_0{K}_{\mathit{IP}}+K_{\mathit{IP}}^2{T}_F^2$$

$$K_{\mathit{IP}}=K_I{K}_{\mathit{PRM}}$$

$$K_{\mathit{PP}}=K_P{K}_{\mathit{PRM}}$$

$$K_{\mathit{DP}}=K_D{K}_{\mathit{PRM}}$$

$$d_0={\left(1+K_{\mathit{PP}}\right)}^2$$

$$f_0=a_{1m}+K_{\mathit{DP}}+T_F+T_{\mathit{\text{delm}}}$$E19

For the given controller filter (T_F) and the disturbance filter (T_FD) time constants (note that the calculation of both time constants, according to the desired level of noise, will be derived in the next sub-chapter), the calculation of remaining controller, model and disturbance filter parameters proceeds as given in Figure 5.

Figure 5.

Calculation of the controller, model and filter parameters.

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:

$$G_{P1}=\frac{e^{-0.5s}}{{\left(1+s\right)}^2}$$

$$G_{P2}=\frac{e^{-0.2s}}{{\left(1+s\right)}^3}$$E20

The a-priori chosen filter time constants were:

$$T_F=T_{\mathit{FD}}=0.1$$E21

The characteristic areas, calculated from (5) and (7), are given in Table 1.

	A0	A1	A2	A3	A4	A5
Areas GP1	1	2.50	4.13	5.77	7.42	9.07
Areas GP1 with controller filter	1	2.60	4.39	6.21	8.04	9.87
Areas GP2	1	3.20	6.62	11.26	17.12	24.21
Areas GP2 with controller filter	1	3.30	6.95	11.96	18.32	26.04

Table 1.

The calculated areas for the processes (20) without and with the controller filter.

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.

Controller parameters	KP	KI	KD
MOMI controller for GP1	1.81	0.89	0.93
DRMO controller for GP1	2.25	1.49	0.93
MOMI controller for GP2	1.61	0.64	1.08
DRMO controller for GP2	1.93	0.98	1.08

Table 2.

The calculated controller parameters for the processes (20) for MOMI (6) and DRMO (9) method, taking into account the chosen controller filter T_F = 0.1.

The process models G_M and inverse process models G_MI are then calculated from (14), where G_MI is the inverse of G_M without time-delay:

$$G_{M1}=\frac{e^{-0.5s}}{1+2s+s^2}$$

$$G_{\mathit{MI}1}=1+2s+s^2$$

$$G_{M2}=\frac{e^{-0.616s}}{1+2.58s+1.84s^2}$$E22

$$G_{\mathit{MI}2}=1+2.58s+1.84s^2$$

Finally, the disturbance filter gain K_FD, when taking into account the chosen T_FD = 0.1, is then calculated from (18):

$$K_{\mathit{FD}1}=0.57$$

$$K_{\mathit{FD}2}=0.59$$E23

Therefore, the complete inverse of the models with accompanying disturbance filters (see Figure 3) are the following:

$$G_{\mathit{MI}1}{G}_{\mathit{FD}1}=\frac{0.57\left(1+2s+s^2\right)}{{\left(1+0.1s\right)}^3}$$

$$G_{\mathit{MI}2}{G}_{\mathit{FD}2}=\frac{0.59\left(1+2.58s+1.84s^2\right)}{{\left(1+0.1s\right)}^3}$$E24

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 t = 0 s, the reference value (r) was changed from 0 to 1 and at half of experiment time the process input disturbance (d) was changed from 0 to 1. It is obvious that the disturbance rejection performance of the DE-MOMI method is the best. Note that when applying the DE-MOMI method, due to the difference between the actual process and the process model in the second example (G_P2), the process input signal, during the reference change, is not smooth. This is expected, since the inverse process model with filter is amplifying the difference between the actual process and the process model. In this case, the response can be made smoother by increasing the disturbance filter time constant (T_FD). Note that a possible limitation of the control signal can also help to smooth out the oscillations after the reference step [24].

Figure 6.

The closed-loop responses on the process G_P1, when using the MOMI, DRMO and DE-MOMI method.

Figure 7.

The closed-loop responses on the process G_P2, when using the MOMI, DRMO and DE-MOMI method.

The disturbance rejection performance of the DE-MOMI method can be increased by decreasing the disturbance filter time constant T_FD. However, as already mentioned above, the process input signal can become oscillatory when the actual process and the process model differ. In this case, too small T_FD can even render the closed-loop system unstable. Besides that, the process noise (signal n in Figure 3) is also amplified via block G_MIG_FD, so small T_FD can cause excessive noise of signal d_F. The selection of T_FD is, therefore, important in practical realisation of the DE-MOMI method.

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 [25]. 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.

Figure 8.

The website layout for the calculation of the controller and the DE parameters.

5. Noise attenuation of DE-MOMI method

As already mentioned in the previous sub-chapter, the output noise of disturbance estimator (d_F) depends on the selection of disturbance filter T_FD. However, according to Figure 3, some noise is also present at the output of the PID controller block (signal u_C). In this sub-chapter we will give some guidelines regarding the noise attenuation in practical realisation of DE-MOMI controller.

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 n (see Figure 3). The noise power at the controller output (u) depends on the power of measurement noise n and the frequency properties of noise, PID controller and disturbance estimator. The relation between the filters (T_F and T_FD) time constants and the controller output noise is rather complex, but can be calculated according to Parseval’s theorem if the measurement noise frequency characteristics are known. However, this relation is higher-order and non-linear. Therefore, the search for adequate filter time constants T_F and T_FD would require optimisation procedure, which would significantly complicate the otherwise simple method.

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

$$f_S=\frac{1}{T_S}$$E25

where T_S is the controller’s sampling time. The highest signal, which may be sent to discrete function is, due to Shannon’s theorem, f_S/2. Therefore, any frequency close to f_S/2 can be considered as high frequency. In this research we have arbitrarily decided that the “high frequency” f_HF is the quarter of controller’s sampling frequency f_S:

$$f_{\mathit{HF}}=0.25f_S$$

$${\omega }_{\mathit{HF}}=2\pi f_{\mathit{HF}}=\frac{0.5\pi }{T_S}$$E26

As already mentioned above, the source of controller noise is the process output noise n (Figure 3). In DE-MOMI controller, the overall high-frequency control noise consists of the PID controller (u_PIDn) and the disturbance estimator (u_DEn) high-frequency noise:

$$u_{\mathit{\text{PIDn}}}\left({\omega }_{\mathit{HF}}\right)=K_{\mathit{\text{PIDn}}}n\left({\omega }_{\mathit{HF}}\right)$$

$$u_{\mathit{DEn}}\left({\omega }_{\mathit{HF}}\right)=K_{\mathit{DEn}}n\left({\omega }_{\mathit{HF}}\right),$$E27

where K_PIDn and K_DEn are the high-frequency gains (around frequency ω_HF) of the PID controller and the disturbance estimator, respectively.

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 (K_PIDn) and the disturbance estimator (K_DEn). Therefore, in practice, by selecting the mentioned two gains, the user would limit the amount of controller noise at high frequencies.

The actual gain of the PID controller around the chosen high frequency ω_HF can be calculated from the controller transfer function (2):

$$K_{\mathit{\text{PIDn}}}=\frac{\sqrt{{\left(K_I-K_D{\omega }_{\mathit{HF}}^2\right)}^2+K_P^2{\omega }_{\mathit{HF}}^2}}{{\omega }_{\mathit{HF}}\sqrt{1+T_F^2{\omega }_{\mathit{HF}}^2}}$$E28

The controller filter time constant can then be calculated as:

$$T_F=\frac{\sqrt{{\omega }_{\mathit{HF}}^4{K}_D^2+{\omega }_{\mathit{HF}}^2\left(K_P^2-2K_I{K}_D-K_{\mathit{\text{PIDn}}}^2\right)+K_I^2}}{{\omega }_{\mathit{HF}}^2{K}_{\mathit{\text{PIDn}}}}$$E29

Since the PID controller parameters depend on the filter time constant T_F, the T_F should be calculated by an iterative procedure given in Figure 9.

Figure 9.

Calculation of the filter and controller parameters according to the desired controller high-frequency gain K_PIDn.

The calculation of the disturbance filter high-frequency gain K_DEn is similar as for the PID controller:

$$K_{\mathit{DEn}}=\sqrt{\frac{K_{\mathit{FD}}^2\left(1+{\omega }_{\mathit{HF}}^2\left(a_{1m}^2-2a_{2m}\right)+{\omega }_{\mathit{HF}}^4{a}_{2m}^2\right)}{K_{\mathit{PRM}}^2{\left(1+{\omega }_{\mathit{HF}}^2{T}_{\mathit{FD}}^2\right)}^3}}$$E30

In a similar manner, the disturbance filter time constant can be derived as:

$$T_{\mathit{FD}}=\frac{1}{{\omega }_{\mathit{HF}}}\sqrt{\sqrt[3]{\frac{K_{\mathit{FD}}^2\left(1+{\omega }_{\mathit{HF}}^2\left(a_{1m}^2-2a_{2m}\right)+{\omega }_{\mathit{HF}}^4{a}_{2m}^2\right)}{K_{\mathit{PRM}}^2{K}_{\mathit{DEn}}^2}}-1}$$E31

Since the calculated filter gain K_FD depends on the filter time constant T_FD (see expression (18)), the calculation of expression (31) is iterative as well, as given in Figure 10.

Figure 10.

Calculation of the disturbance filter parameters according to the desired disturbance filter high-frequency gain K_DEn.

Illustrative example 2

Let us illustrate the calculation procedure for the following processes:

$$G_{P3}=\frac{e^{-0.2s}}{\left(1+0.2s\right)\left(1+s\right)}$$

$$G_{P4}=\frac{e^{-s}}{{\left(1+s\right)}^4}$$E32

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 K_PIDn = 4 and K_DEn = 4, respectively. The chosen closed-loop sampling time was T_S = 0.01 s. Therefore, the chosen high-frequency is:

$${\omega }_{\mathit{HF}}=\frac{0.5\pi }{T_S}=157.1s^{-1}$$E33

The initially chosen filter time constants were (the values are not critical):

$$T_F=T_{\mathit{FD}}=0.1s$$E34

The characteristic areas are calculated from (5). For the given high-frequency gain K_PIDn = 4, the filter and controller parameters are calculated according to procedure given in Figure 9. The calculated filter time constants (after 2 iterations) were

$$T_{F3}=0.119s$$

$$T_{F4}=0.192s$$E35

Note that indexes 3 and 4 in above filter time constants stand for the processes G_P3 and G_P4, respectively.

The areas are given in Table 3 and the controller parameters are given in Table 4.

	A0	A1	A2	A3	A4	A5
Areas GP3	1	1.40	1.50	1.52	1.53	1.53
Areas GP3 with controller filter	1	1.52	1.68	1.72	1.73	1.73
Areas GP4	1	5.00	14.50	32.17	60.71	102.8
Areas GP4 with controller filter	1	5.19	15.50	35.14	67.45	115.8

Table 3.

The calculated areas for the processes (32) without and with the controller filter.

Controller parameters	KP	KI	KD
MOMI controller for GP3	2.35	1.88	0.48
DRMO controller for GP3	2.91	3.83	0.48
MOMI controller for GP4	0.84	0.26	0.77
DRMO controller for GP4	0.94	0.32	0.77

Table 4.

The calculated controller parameters for the processes (20) for MOMI (6) and DRMO (9) method, taking into account the calculated controller filters.

The process models G_M and inversed process models G_MI are then calculated from (14):

$$G_{M3}=\frac{e^{-0.2s}}{1+1.2s+0.2s^2}$$

$$G_{\mathit{MI}3}=1+1.2s+0.2s^2$$

$$G_{M4}=\frac{e^{-1.94s}}{1+3.06s+2.69s^2}$$

$$G_{\mathit{MI}4}=1+3.06s+2.69s^2$$E36

According to the chosen high-frequency gain K_DEn = 4, the T_FD and K_FD were calculated according to the procedure given in Figure 10 (2 iterations were sufficient):

$$T_{\mathit{FD}3}=0.06$$

$$T_{\mathit{FD}4}=0.116$$

$$K_{\mathit{FD}3}=0.69$$

$$K_{\mathit{FD}4}=0.36$$E37

Therefore, the complete inverse of the models with accompanying disturbance filters (see Figure 3) are the following:

$$G_{\mathit{MI}3}{G}_{\mathit{FD}3}=\frac{0.69\left(1+1.2s+0.2s^2\right)}{{\left(1+0.06s\right)}^3}$$

$$G_{\mathit{MI}4}{G}_{\mathit{FD}4}=\frac{0.36\left(1+3.06s+2.69s^2\right)}{{\left(1+0.116s\right)}^3}$$E38

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 (u) noise is close to the expected one taken into account that both, the PID controller (u_C) and the disturbance estimator output (d_F) noise should be 4-times higher than the measurement noise at high frequencies.

Figure 11.

The closed-loop responses on the process G_P3, when using the MOMI, DRMO and DE-MOMI method.

Figure 12.

The closed-loop responses on the process G_P4, when using the MOMI, DRMO and DE-MOMI method.

The disturbance rejection performance of the DE-MOMI method can be additionally improved by increasing the high-frequency gain K_DEn. However, increased gain is associated with higher controller output noise and decreased closed-loop stability if the actual process and the process model differ.

The computation of the controller and the DE parameters can be performed similarly as before on another OctaveOnline Bucket website [26]. 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 [1] (denoted as “AH”) and to ADRC method [27].

The AH method [1] is based on the calculation of the maximum sensitivity index M_S, which is the inverse of the smallest open-loop Nyquist curve distance to the critical point (−1,0i). The method was developed for values M_S = 1.4 and M_S = 2. In this comparison we will use M_S = 2, since it gives better disturbance-rejection performance. However, even though the process transfer function does not need to be derived, the method requires the identification of the process steady-state gain and the inflexion point along with maximum slope of the process output signal during the step-change of the process input signal. Note that those parameters usually require manual measurements and cannot be easily performed by using automatic calculation. The AH method is using the PID controller structure with adjustable reference-weighting factor b, and by fixing factor c = 0 (Figure 2).

The ADRC method [27, 28, 29, 30, 31] is based on a simple controller with three gains associated with extended state-observer (ESO), as shown in Figure 13.

Figure 13.

The ADRC control structure with the controller gains (up) and the extended state observer (down).

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 K_C, should be defined by the user before calculating the rest of ADRC parameters. As shown in Figure 13, the ADRC method is using control structure which consists of an extended state observer (ESO) with three gains (β₁, β₂ and β₃) and three controller gains (K_C, K_P and K_D) [27].

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 [27]. 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 [27], 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:

$$G_{P5}=\frac{1}{\left(1+s\right)\left(1+0.2s\right)\left(1+0.04s\right)\left(1+0.008s\right)}$$

$$G_{P6}=\frac{e^{-5s}}{{\left(1+s\right)}^3}$$E39

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 K_PIDn = K_DEn = 20 for G_P5 and K_PIDn = K_DEn = 4 for G_P6. The higher gains were chosen for G_P5, since the closed-loop tracking and control performance was substantially improved when using higher gains. Increasing the gains for G_P6 above 4 did not significantly improve the performance.

Process	Tuning method	KP	KI	KD	TF	b	c
GP5	MOMI	6.45	5.35	1.108	0.055	1	1
	DRMO	9.69	23.71	1.108	0.055	0	0
	DE-MOMI	6.45	5.35	1.108	0.055	1	1
	AH	21.35	53.05	2.22	0.055	0.24	0
GP6	MOMI	0.53	0.126	0.66	0.165	1	1
	DRMO	0.57	0.140	0.66	0.165	0	0
	DE-MOMI	0.53	0.126	0.66	0.165	1	1
	AH	0.52	0.136	0.52	0.165	0.36	0

Table 5.

The calculated controller parameters for the processes (39) for MOMI, DRMO, DE-MOMI and AH method.

Process	KPRM	a1m	a2m	Tdelm	TFD	KFD
GP5	1	1.205	0.205	0.043	0.018	0.909
GP6	1	2.58	1.84	5.42	0.077	0.159

Table 6.

The calculated disturbance estimator’s parameters for the processes (39) for DE-MOMI method.

Process	KC	KP	KD	β1	β2	β3
GP5	1/5	100	20	120	4800	19200
GP6	1/3	0.16	0.8	4.8	7.68	30.72

Table 7.

The calculated ADRC controller parameters for the processes (39).

The sampling time for G_P5 is chosen as T_S = 0.001 s and for G_P6 as T_S = 0.01 s.

The closed-loop process responses are given in Figures 14 and 15. In both experiments the unity-step process input disturbance signal was applied at the half of experiment time.

Figure 14.

The closed-loop responses on the process G_P5, when using the MOMI, DRMO and DE-MOMI method.

Figure 15.

The closed-loop responses on the process G_P6, when using the MOMI, DRMO and DE-MOMI method.

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 G_P5 gives somehow oscillatory response. For the same process, the ADRC method gives slightly oscillatory response during the reference change (see the process input signal). While DE_MOMI and MOMI methods clearly give the best tracking responses on process G_P6, all of the methods have similar disturbance-rejection performance. Only slightly oscillatory response can be observed for ADRC method.

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 r) and on disturbance rejection response (unity step-change of the process input disturbance d). The results are given in Table 8. It can be seen that the best values (marked with greyed colour) were obtained with DE-MOMI method.

Process	experiment	DE-MOMI	MOMI	DRMO	AH	ADRC
GP5	tracking	0.216	0.217	0.526	0.336	0.256
	DR	0.017	0.186	0.055	0.020	0.019
GP6	tracking	8.66	8.66	12.34	11.06	12.32
	DR	7.80	8.42	8.79	8.89	8.83

Table 8.

The calculated IAE values for tracking and disturbance rejection (DR) responses for the processes (39).

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 (G_P1 to G_P6) revealed that the proposed method can significantly improve the disturbance-rejection performance of the lower-order processes with smaller delays, while the improvement of the higher-order processes and/or processes with higher delays is not so significant. Therefore, the application of the method for lower-order processes with smaller delays might be beneficial in practice.

7. Conclusions

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.

Acknowledgments

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.