PID vs. Model-Based Control for the Double Integrator Plus Dead-Time Model: Noise Attenuation and Robustness Aspects

Abstract

One of the most important contributions of modern control theory from the 1960s was the separation of the dynamics of state-space controller design from the dynamics of state reconstruction. However, because modern control theory predates the mass spread of digital controllers and was predominantly focused on analog solutions that avoided modeling dead-time elements, it cannot effectively cover all aspects that emerged with the development of programmable devices and embedded systems. The same historical limitations also characterized the development of proportional-integral-derivative (PID) controllers, which began several decades earlier. Although they were used to control time-delayed systems, these solutions, which are most commonly used in practice today, can also be referred to as simplified disturbance observers that allow the avoidance of the the direct use of dead-time models. Using the example of controlling systems with a double integrator plus dead-time model, this article shows a novel controller design that significantly improves control performance compared to conventional PID controllers. The new control structure is a combination of a generalized state-space controller, interpreted as a higher-order derivative controller, and a predictive disturbance observer that uses the inversion of double integrator dynamics and dead-time models. It enables the elimination of the windup effect that is typical for PID control and extends the separation of the dynamics of setpoint tracking from the dynamics of state and disturbance reconstruction to time-delayed processes as well. The novelty of the presented solution offers several orders of magnitude lower amplification of measurement noise compared to traditional PID control. On the other hand, it offers high robustness and a stable transient response despite the unstable internal feedback of processes like the magnetic levitation system. The improvements achieved are so high that they call into question the classical solutions with PID controllers, at least for DIPDT models. In addition to the comparison with PID control, the relationship with traditional state space controllers, which today form the basis of active disturbance rejection control (ADRC), is also discussed and examined for processes including dead time.

1. Introduction

The term proportional-integral-derivative (PID) is commonly used to describe the most widely used automatic control solution in practice. However, its use is associated with several interpretative and historical inaccuracies, which also have serious practical consequences. First, it should be noted that the Taylor Fulscope controller, which was launched in 1940 and had three adjustable knobs for optimal adaptation to the process, was essentially equivalent to a stabilizing controller, including a simplified disturbance observer derived for the integrating process [1]. The stabilizing controller should ensure stable setpoint tracking based on the known model (transfer function) of the process. It can be complemented by the reconstruction and compensation of external and internal disturbances by using the process model (or its inversion) with a suitable disturbance observer (DOB) that does not affect the setpoint tracking. The manufacturer of the Fulscope controller did not bother with the theoretical background, but produced a “user manual”, which was published shortly afterwards [2]. Of the two possible approaches to controller tuning, one was based on a step response method in the time domain using a tangent drawn through the inflection point of the process response. This also enabled its application to time-delayed processes. However, since the analog technologies used at the time did not allow for the modeling of dead-time elements, there was no interest in more precise model-based interpretations. A simplified approach by tuning all three PID controller parameters together was chosen instead of designing the stabilizing controller and the disturbance rejection separately (using the DOB, including the dead-time model). This simplified PID controller was based on the equivalent transfer function of the original controller in the linear control region.

In the following decades, a large number of different textbooks, recommendations, and papers on PID control were published (see, e.g., [3,4,5,6]). Although the growing gap between theory and practice was regularly discussed in many forums [7,8,9], the essence of the original solutions was more or less forgotten. The later designation as series PID controllers did not help to emphasize their model-based nature. This certainly has not helped the further development of PID control. When setting the controller parameters, the differences between the solutions were not so significant. However, the lack of knowledge of the internal structure of the first controllers led to misinterpretations and incorrect solutions, and the inaccurate labeling had a negative impact on further modifications. It makes a big difference whether the new design of generalized controllers is based on a model-based structure with automatic reset or on an equivalent structure with an explicit integrator. When PID implementations started to use structures with an explicit integrator, this had serious consequences. The first was the windup effect, which occurred mainly in digital implementations of PID control. In the digital domain, integration can be implemented simply by summing small increments. However, in systems with explicit integrators and control constraints, undesired integration occurs after reaching the limits of the control signal. This phenomenon could make PID control unusable in practice. The anti-windup solutions developed in [3,10] compensate for this problem, but for generalized controllers with higher derivatives, they do not always provide optimal dynamics [11].

The usual interpretation of the three terms of the PID control, based on the present, previous, and future control error, and thus seemingly exploring all available alternatives, did not support the introduction of higher-order (HO) derivatives. The problem of HO derivatives was revisited only recently, when the origins of PID control were recalled and reviewed in a series of publications [12,13] with a model-based interpretation explaining a disturbance observer simplified for the integrator plus dead-time (IPDT) model. Generalized HO-PIDs are usually required for more complex processes that exhibit dynamics that are difficult to control [14,15,16,17,18,19,20,21,22,23,24,25]. By generalizing the solution with the parallel and series controllers for IPDT systems using HO derivatives, a new degree of freedom to modify the closed-loop performance could be gained [11]. With the increasing number of controller terms, traditional descriptions based on the present, past, and future of the control error are no longer efficient and new design paradigms are required. The number of papers dealing with generalized PID controllers using HO derivatives is rapidly increasing. Remarkably, none of the papers cited above attempt to extend the methodology developed for historically older and simpler solutions based on automatic reset. This situation is of course related to the fact that, for decades, no one interpreted industrial (series) PIDs as model-based controllers with a simplified DOB applied to ultra-local (integrating) models [1]. In other existing solutions, this is only the case with active disturbance rejection controllers (ADRCs). These also use ultra-local (integrating) models, but for the reconstruction of lumped disturbances based on extended state observers (ESO) and for state-space-based disturbance reconstruction. However, in modern control theory, the state-space-based approaches, as well as ADRCs, were developed before the advent of dead-time modeling and model-based approaches for series PID controllers. Therefore, the relationship between ADRCs and PIDs (understood as heuristic controllers with an explicit integrator) is sometimes completely contradictory, as for example in [9,26,27,28,29,30,31,32,33,34,35]. Without paying particular attention to the role of ultra-local linear models, the relationship between ADRCs and the state-space approach is also not clarified. Inadequacies are also noted in clarifying the relationship between PID control and intelligent (model-free) PIDs [36]. Although they use integrating process models, they use significantly different types of finite impulse response (FIR) filters [37,38].

Of course, the question arises whether the original Fulscope controller, whose concept can be interpreted as model-based, is also suitable for processes approximated with HO models when using HO derivatives and considering control signal constraints. However, from the point of view of the possible generalization of historical series PIDs to HO-PIDs, there is an important fact that the series HO-PIDs based on IPDT models cannot be directly as well as effectively extended to the systems described by the double integrator plus dead-time (DIPDT) models. The analytical design of series PID controllers using the MRDP (multiple real dominant pole) method for the DIPDT models is not feasible. The design of series PID controllers using the feedback from the constrained controller output [12], which was a typical analog solution, is only applicable for DIPDT systems using the performance portrait method (PPM) [39]. Although the PPM enabled the calculation of the controller parameters, the achieved series PID was still associated with significantly reduced closed-loop dynamics and robustness.

A more favorable situation exists with parallel PID controllers. An analysis of the controller design using delay equivalences has shown possible simplifications of the filters used in HO derivatives. In [39], an extension of the MRDP methodology to the proportional-integral-derivative-acceleration (PIDA) controller was proposed. However, parallel controllers require additional anti-windup protection for constrained control. Another serious problem associated with DIPDT models in controlling a nonlinear unstable magnetic levitation system in [40] is the high noise gain combined with slow closed-loop dynamics and low robustness. In comparison, the PD controllers designed using the same MRDP method but without the explicit integral action provide much faster closed-loop dynamics with a much lower noise gain.

One of the best-known traditional methods for studying the robustness of systems is Kharitonov’s theorem [41]. Numerous traditional approaches in the field of PID control use robust controller settings that are mainly derived from the frequency domain and are based on gain and phase margins or sensitivity functions [42,43,44,45,46,47,48]. However, these methods have the disadvantage that they are limited to stable processes. As discussed in [12], the recommended values of the sensitivity function are much higher for unstable systems. For example, in [14], controllers with $M_s=20$ are proposed, which is ten times the recommended maximum value for stable systems. From the point of view of utilization, robustness analysis in the time domain is far more suitable even for unstable processes. In this case, the performance measures based on deviations from the ideal input and output shapes [49] allow for formulating the conditions to achieve the invariance of the performance measures selected from variable circuit parameters. Thus, by providing responses with zero deviations from the ideal shapes, the MRDP controller is important from a robust control point of view.

All these findings motivate us to look for alternative solutions to DIPDT models that utilize the advantages of digital implementation. Digital-saturated model-based controllers with a predictive disturbance observer for DIPDT models have already been discussed in [50]. However, continuous-time design with a PDO for the implementation of simpler quasi-continuous solutions for DIPDT models is investigated in this paper [51]. HO derivatives are used to increase the closed-loop dynamics and smoothness of the responses. To compare the performance of several alternative solutions, a combined performance measure based on the integral of the absolute error ($IAE$) and the deviation of the process input signal from its ideal shape is used. A modified measure $T{V}_2\left(u\right)$ based on the total variation $TV$ introduced in [52] is used. This is because an ideal double integrator input signal for setpoint changes should consist of two pulses (with three monotonic intervals). The advantage of the model-based solution compared to PID control is that it reduces the cost function $J=IAE\ast T{V}_2\left(u\right)$ (which combines the control speed with the excessive controller effort) by more than a hundred times. Therefore, in this work, parallel PID (or PIDA) controllers are replaced by model-based solutions using HO derivatives, which is motivated by simple consideration of control signal constraints, time delays, achievable closed-loop dynamics, attenuation of measurement noise, and robustness.

The rest of the paper is structured as follows. Section 2 explains the design of a series PID controller for IPDT processes, originally called pre-act or hyper-reset, by simplifying model-based controllers with disturbance reconstruction using disturbance observers (DOBs). It is shown why this procedure cannot be repeated for DIPDT models. Section 3 compares the performance measures of processes obtained with a heuristic parallel PID analytically tuned by the MRDP method with a simple stabilizing P controller combined with a feedback controller using the mth-order derivative tuned by the same MRDP method. However, the stabilizing ${\mathrm{P}}^m$ controller is complemented by a PDO containing the dead-time model of the double integrator process model and its inverse. Section 4 demonstrates the MRDP-optimal PID controller through a simulation experiment and shows that it outperforms standard PID controllers for DIPDT processes in terms of speed of transient responses and measurement noise amplification. The high performance and robustness of the new ${\mathrm{P}}^m$-PDO controller are demonstrated in Section 5, where the attenuation of the measurement noise and the invariance of the process uncertainties are compared with the MRDP-based PID controller for unstable magnetic levitation processes approximated by the DIPDT model. Section 6 discusses the results obtained. The conclusions summarize the basic results from a historical point of view. The implications of the simplified interpretation of PID control for its further development are explained, taking into account the historically available technological solutions. The possibility of significant improvements points to the need for a paradigm shift in the development of DIPDT processes.

2. Disturbance Reconstruction and Compensation for IPDT and DIPDT Models

2.1. Reconstruction and Compensation Based on IPDT Models

For the integrator plus dead-time (IPDT) model:

$$S\left(s\right)={\displaystyle \frac{Y\left(s\right)}{U\left(s\right)}}=S_0\left(s\right)e^{-T_{dp}s};S_0\left(s\right)={\displaystyle \frac{K_{sp}}{s}}$$

(1)

with the gain $K_{sp}$ and the dead time $T_{dp}$ representing the simplest two-parameter model, which is often used for the design of controllers for processes with dominant first-order dynamics [53]. For model parameters corresponding to the nominal process parameters $K_s$ and $T_d$, the index “p” can be omitted to simplify the resulting formulas. The IPDT-based controller design, inspired by the work of Ziegler and Nichols [2], leads to an approximation of the process step response by a dashed line reminiscent of the step response of the IPDT model. This can be particularly interesting in situations where it is not possible to measure the open-loop responses over long periods of time up to a steady state. Similarly, it is also possible to approximate the initial part of open-loop responses by DIPDT models treated later [50].

Input disturbances of a process according to Figure 1 can be reconstructed as the difference between the current process input $u_a\left(t\right)$ and the controller output $u\left(t\right)$

$$d_i\left(t\right)=u_a\left(t\right)-u\left(t\right)$$

(2)

The actual process input $u_a\left(t\right)$ can be estimated from its output $y\left(t\right)$ by inversion of the process dynamics. However, since the dead time $T_d$ cannot be inverted in the causal system, the inversion according to $Y\left(s\right)/S_0\left(s\right)$ only results in a time-delayed process input $u_a(t-T_d)$. For a correct disturbance reconstruction, this value must be compared with the likewise delayed $u(t-T_d)$, which ultimately leads to a delayed reconstruction of the input disturbance $d_i(t-T_d)$, corresponding to

$$d_{id}\left(s\right)={\displaystyle \frac{1}{S_0\left(s\right)}}Y\left(s\right)-e^{-T_{d}s}U\left(s\right)$$

(3)

In order to obtain a feasible reconstruction, this operation introduced by (3) must be extended by filtering with a low-pass filter $Q_n\left(s\right)$

$$d_{if}\left(s\right)=Q_n\left(s\right)\left({\displaystyle \frac{1}{S_0\left(s\right)}}Y\left(s\right)-e^{-T_{d}s}U\left(s\right)\right);Q_n\left(s\right)={\displaystyle \frac{1}{{(1+T_{f}s)}^n}}={\displaystyle \frac{1}{P_n\left(s\right)}}$$

(4)

The chosen time constant $T_f$ cannot be arbitrarily small in relation to $T_d$ and $T_s$ [12,13], and the chosen filter degree n must be greater than or equal to the relative degree of $S_0\left(s\right)$ in order to guarantee proper transfer function $Q_n\left(s\right)/S_0\left(s\right)$. Thus, disturbance reconstruction can be expressed as follows:

$$d_{if}\left(s\right)=S_{yd}\left(s\right)Y\left(s\right)-S_{ud}\left(s\right)U\left(s\right);S_{yd}\left(s\right)={\displaystyle \frac{1}{S_0\left(s\right){(1+T_{f}s)}^n}};S_{ud}\left(s\right)={\displaystyle \frac{e^{-T_{d}s}}{{(1+T_{f}s)}^n}}$$

(5)

For the IPDT model (1), the minimum realizable filter degree value is $n=1$. With this filter, for a sufficiently long $T_f$, the contribution of the signal corresponding to $S_{yd}\left(s\right)Y\left(s\right)$ and generated by the output inversion can be neglected together with the dead-time model $e^{-T_{d}s}$ used in $S_{ud}\left(s\right)$. The simplified disturbance reconstruction is then

$$d_{if}\left(s\right)\approx {\displaystyle \frac{1}{1+T_{f}s}}Y\left(s\right)$$

(6)

Reconstructed in this way, the disturbance can be used as a feedforward control that counteracts the input disturbance $d_i$ by correcting the offset of the stabilizing controller, which is added to the output $u_s$ according to

$$u=u_s-d_{if}$$

(7)

In the proportional range of the controller, this corresponds to

$$U\left(s\right)={\displaystyle \frac{1+T_{f}s}{T_{f}s}}{U}_s\left(s\right)=\left(1+{\displaystyle \frac{1}{T_{f}s}}\right)U_s\left(s\right)$$

(8)

Obviously, this simplified disturbance reconstruction introduces a parallel integral (I) action with an integral time constant

$$T_i=T_f$$

(9)

In combination with the proportional-derivative (PD) stabilizing controller, which is used for calculation of $u_s$, this results in a series PID controller. However, due to the simplification of the nominal DOB, its dynamics will differ from the dynamics achieved by the stabilizing controller and its parameters usually need to be adjusted. Changing the closed-loop dynamics of the PD controller by introducing the I-component is one of the fundamental features of PID controller design. For example, the optimal analytical tuning of the PD controller

$$C_{PD}={\displaystyle \frac{U\left(s\right)}{E\left(s\right)}}=K_P(1+T_{D}s)$$

(10)

for the IPDT model using the MRDP method (derived by solving a system of equations constructed from the characteristic closed-loop polynomial and its first two derivatives in [54]; see also the next section) gives the dimensionless parameters

$$\kappa =0.5413;{\tau }_D=0.25;{\tau }_0=1/2$$

(11)

Here, $\kappa =K_p{K}_s{T}_d$ represents the dimensionless value corresponding to the proportional gain $K_p$, ${\tau }_D=T_D/T_d$ is the dimensionless derivative time constant, and ${\tau }_0=T_0/T_d$ corresponds to the triple closed-loop time constant achieved by this controller.

For the parallel PID controller

$$C_{PID}\left(s\right)={\displaystyle \frac{U\left(s\right)}{E\left(s\right)}}=K_p\left(1+{\displaystyle \frac{1}{s{T}_i}}+s{T}_D\right);K_i={\displaystyle \frac{K_p}{T_i}};K_d=K_p{T}_D$$

(12)

with ${\tau }_i=T_i/T_d$, the same MRDP methodology (solving system of equations constructed from the closed-loop characteristic polynomial and its first three derivatives in [54,55,56]) yields a quadruple real dominant pole ${\tau }_0$ and dimensionless controller parameters

$$\kappa =0.7836;{\tau }_D=0.2629;{\tau }_i=3.732;{\tau }_0=0.789$$

(13)

The comparison with the PD controller (10) now shows that the P-gain has increased by 1.45 times and the derivative time constant by 1.05 times. This leads to an increased noise amplification. At the same time, the dynamics of the circuit have slowed down considerably. Instead of the triple time constant ${\tau }_0=0.5$, it now has a quadruple time constant ${\tau }_0=0.789$ that is 1.58 times higher. The slowdown is best characterized by the $IA{E}_s$ values of the unit setpoint step response. The closed-loop transfer function of the IPDT system with PD controller is

$$F_{PD}\left(s\right)={\displaystyle \frac{K_s{K}_p(1+T_{D}s)}{s{e}^{T_{d}s}+K_s{K}_p(1+T_{D}s)}}$$

(14)

To avoid overshooting of the step responses, a setpoint prefilter

$$F_p={\displaystyle \frac{1}{1+T_{D}s}}$$

(15)

will be used. Using the optimum parameters (11), the circuit modified in this way gives the value of the $IA{E}_s$ calculated using the Laplace transformation of $I{E}_s$

$$IA{E}_s=2.097T_d.$$

(16)

The transfer function of IPDT system in the closed loop with PID controller is

$$F_{PID}\left(s\right)={\displaystyle \frac{K_s{K}_p(1+T_{i}s(1+T_{D}s))}{s^2{e}^{T_{d}s}+K_s{K}_p(1+T_{i}s(1+T_{D}s))}}$$

(17)

In order to avoid overshooting of the step responses, a setpoint prefilter

$$F_p\left(s\right)={\displaystyle \frac{1}{1+T_{i}s(1+T_{D}s)}}$$

(18)

will be used. Using the optimum parameters (13), the circuit modified in this way gives the value of the $IA{E}_s$ calculated using the Laplace transformation of $I{E}_s$

$$IA{E}_s=3.732T_d,$$

(19)

which is about 1.8 times more than with the PD control. In addition to the increased sensitivity to measurement noise, the slowing of transients in PID control means slower elimination of disturbances, which also results in lower robustness.

Remark 1

(Reconstruction of $d_i$ at $t\to \infty$). A comparison of (10) and the values of ${\tau }_i$ and ${\tau }_0$ in (13) confirms that for the MRDP design and (9) the condition for the simplification of the DOB (4) to (6), which can also be expressed as

$${\tau }_i>>{\tau }_0,$$

(20)

is sufficiently fulfilled. As a result, the DOBs implicitly included in PIDs reconstruct the input disturbances only close to steady states.

Remark 2 

(Advantages of the MRDP design). The multiple real dominant pole method is one of the oldest approaches to optimal controller tuning [57].

1.

It leads to setpoint responses with monotonic output, which allows for the calculation of the corresponding $IA{E}_s$ values using the Laplace transform as the value of the integral error $I{E}_s$. In the case of disturbance responses, the return to the desired setpoint is also monotonic without changing the sign of the control error, which in turn enables the calculation of $IA{E}_d$ by Laplace transformation from the value $I{E}_d$.

2.

The MRDP controller leads to transients with zero deviations from the ideal input and output shapes. According to [49], this is a condition for the invariance of the selected performance measures to variable circuit parameters.

3.

The advantage of the MRDP design also lies in the possibility of its extension to controllers with higher-order derivatives [11,12,54]. The standardized methodology offers the possibility to compare individual solutions.

The advantage of the DOB modification used in the derivation of the PID controller is that it does not include a dead-time model, which was essential in the implementation of the analog controller.

The advantage was also that no additional anti-windup had to be introduced when compensating for disturbances by introducing positive feedback from the limited output of the controller.

Remark 3

(Relation of series PIDs to model-based control approaches). When modern control theory (MCT) emerged around 1960, the model-based nature of industrial controllers with automatic reset (series PIDs) was overshadowed by their interpretation based on explicit I-action. As a result, MCT, as well as all subsequent postmodern approaches, such as internal model control (IMC), disturbance observer control (DOBC), and active disturbance rejection control (ADRC), emphasized their difference from heuristic PID control, but failed to recognize the essence of automatic reset controllers widely used in industry. However, automatic reset-based controllers were already using disturbance reconstruction and compensation long before the advent of these “modern” approaches [1]. They just did not publicize it anywhere and no one in the scientific control community noticed.

Remark 4

(Series PIDs for IPDT models and ADRC). The series PIDs use DOB, simplified for ultra-local IPDT models with a first-order low-pass filter ($n=1$), with a large time constant $T_f$. By using an ultra-local model, they are similar to ADRC, which, however, is usually based on an identical state observer for a process state extended by a constant disturbance model called the extended state observer (ESO). The latter is used for IPDT models of a higher order than $n=1$ in series PIDs [58], which led to their relationship being overlooked.

Remark 5

(Series HO-PIDs for IPDT models). The series HO-PIDs with higher-order derivatives bring improvements in the design of controllers with limited controller output. However, for controllers with even derivative degree m, the controller tuning designed using the multiple real dominant pole (MRDP) method must be modified to avoid complex zeros in the controller transfer function [12,13].

2.2. Pid Controllers Based on DIPDT Models

The excellent performance of series HO-PIDs in constrained control motivates us to consider a possible generalization of this concept. However, such a generalization of the concept derived for IPDT models to processes modeled by DIPDT systems, which needs to consider numerous aspects, is far from straightforward. As already mentioned, DIPDT models can also be used to approximate the initial part of the measured open-loop step responses of different processes [50]. The following applies to the double integrator plus dead-time (DIPDT) model

$$S\left(s\right)={\displaystyle \frac{Y\left(s\right)}{U\left(s\right)}}=S_0\left(s\right)e^{-T_{dp}s};S_0\left(s\right)={\displaystyle \frac{K_{sp}}{s^2}}$$

(21)

where the disturbance reconstruction according to (5) must be used. With a higher value of n, this model will not result in the series PID, even if other conditions for simplifying the DOB are met, i.e., neglecting $S_{yd}$ and $e^{-T_{d}s}$ in $S_{ud}$ for a relatively long $T_f$.

From this point of view, it is not surprising that the attempt to find the parameters of the series PID controller using the MRDP method did not provide a solution. This seemed to confirm the statement that parallel PIDs are more useful than series ones, as they can be optimally tuned for a broader class of systems. While parallel PIDs can also be optimally tuned for DIPDT systems using the MRDP method, their parameters cannot be converted to those of series PID controllers [39]. But is this really an advantage of parallel PIDs?

For a stabilizing PD controller (10) corresponding to a triple real dominant pole (TRDP) tuning found for DIPDT processes using the MRDP method in [36], the MRDP-optimal parameters $K_p$, $T_D$ and the triple dominant time constant $T_0$ are

$$\begin{array}{c}{K}_p={\displaystyle \frac{e^{\sqrt{2}-2}(10\sqrt{2}-14)}{K_s{T}_d^2}}={\displaystyle \frac{0.079}{K_s{T}_d^2}};T_D={\displaystyle \frac{\sqrt{2}-1}{5\sqrt{2}-7}}{T}_d=5.828T_d;T_0={\displaystyle \frac{T_d}{2-\sqrt{2}}}=1.71T_d\hfill \end{array}$$

(22)

The parallel PID controller (12), which corresponds to a quadruple real dominant pole (QRDP) tuning found using the MRDP method in [36], where $T_i$ is the integral time constant, has the MRDP-optimal dimensionless parameters

$$\begin{array}{c}\kappa =K_p{K}_s{T}_d^2=0.125;{\tau }_i=T_i/T_d=10.324;{\tau }_D=T_D/T_d=4.043;{\tau }_0=T_0/T_d=2.405.\hfill \end{array}$$

(23)

To avoid overshooting of the setpoint responses, a prefilter must be added to this controller

$$F_p\left(s\right)={\displaystyle \frac{{(1+T_{0}s)}^z}{1+T_{i}s(1+T_{D}s)}}$$

(24)

to make a two-degrees-of-freedom (2DoF) controller. Thus, the polynomial $N_p\left(s\right)={(1+T_{0}s)}^z;z\in [0,2]$ can be used to delete z dominant time constants $T_0=-1/s_o$ [56,59] and thus accelerate the setpoint responses.

Remark 6 

(Disadvantages of introducing the parallel I-term). With regard to the achieved performance, it can be seen that

• the PID gain is 1.58 times larger than the gain of the PD controller (22) and its derivative time constant $T_D$ is only 0.69 of the PD time constant, but the dominant time constant $T_0$ is 1.41 times larger than in the case of the PD controller. Together with the higher multiplication of the dominant pole, this leads to a significant slowdown of the transients.
• Similarly to the IPDT model, it would be possible to calculate the $IA{E}_s$ for the unit setpoint step responses to show that, using a PID controller, a DIPDT system, and the simplest prefilter (with $z=0$), the achieved values ($IA{E}_s=10.323T_d$) are about 1.8 times higher than for a PD controller with $IA{E}_s=5.828T_d$.

The need to recalculate the controller settings therefore leads to an increased sensitivity to measurement noise, which in turn is associated with a lower robustness of the circuit. When applied to the control of an unstable and highly nonlinear magnetic levitation system [40], all these modifications resulted in significantly lower performance compared to the dynamics achieved with the PD controller.

In terms of the design of constrained controllers, the analytical recalculation of the parameters of the parallel controller (23) derived with the MRDP method for series controllers, an equivalent transfer function of the controller, is impossible with real numbers. However, in the paper [39], some “optimal” series controllers were calculated using the performance portrait method (see also [49]) (PPM). The PPM represents a digitization of the “trial-and-error” method inspired by the behavior of practitioners who first check the loop performance for all relevant controller parameters by experimentation. However, the transients obtained showed significantly slower dynamics than the parallel PID controller. Thus, summarizing the results of the previous findings series and parallel PID controllers for the DIPDT models, the following conclusions can be drawn:

• For DIPDT processes, the design of stabilizing PD controllers can be performed analytically using the MRDP method.
• The advantage of controllers with parallel I-action for DIPDT processes is the possibility to perform an analytical design using the MRDP method, which also applies to controllers with higher-order derivatives.
• The disadvantage is that the controller setting must be recalculated, which leads to a considerable increase in sensitivity to measurement noise, a slowdown in transients, and a reduction in the robustness of the controller.
• There is no real solution for an analytical design of PIDs with automatic reset (series PIDs) using the MRDP method. It is possible to calculate their best possible setting using the PPM but, even then, the performance achieved is significantly lower than when using parallel PIDs.

Additionally, since the implementation of model-based disturbance reconstruction and compensation using dead-time models according to (5) no longer poses any problems today, one must ask oneself whether it still makes sense under the given circumstances to deal with the simplified solutions that were advantageous in the past but are no longer so interesting today. Thus, the search for the MRDP-optimal series PID setting is an inappropriately formulated task. By slowing down the transients compared to the simpler PD controller, the robustness is reduced at the same time the influence of the measurement noise increases. The parallel PID controller itself proves to be a much less suitable solution.

Hypothesis 1

(Separability of setpoint tracking and disturbance reconstruction and rejection in the state-space approach). In the state-space approach, the dynamics of setpoint tracking and state and disturbance reconstruction (at least for a nominal system) can be designed separately and without mutual interference. Therefore, it can be assumed that the dynamics of a properly designed $P^m$ controller for the setpoint tracking of a DIPDT system will not change even after the adding the disturbance reconstruction & compensation by PDO, at least in the nominal case and without the addition of measurement noise.

Therefore, in the next section, we deviate from the above problem of series PID control for DIPDT models. To approximate the methods used in state-space control, the dead-time element must be replaced by the Taylor series expansion with an increasing number of terms in the design of the stabilizing controller, resulting in an increase of the state (phase vector). This can also be interpreted as the use of controllers with an increasing degree of derivatives. A significant increase in control performance is also achieved with the predictive disturbance observer (PDO), which reconstructs disturbances according to (5) [51]. Both aspects will show the improvements compared to the solutions used in the past.

3. Stabilization and Compensation of DIPDT Processes Using Higher-Order Derivatives

This section generalizes stabilization of processes approximated by ultra-local DIPDT model enhanced by disturbance reconstruction and compensation via PDO. In order to avoid use of 2DoF controllers with a prefilter, in the interest of simplicity, let us consider a significant part of the controller dynamics in feedback. The reconstruction of a “less delayed” new output $Y^m\left(s\right)$ from the measured output $Y\left(s\right)$ by controller $C^m\left(s\right)$, including a low-pass filter term $P_n\left(s\right), n\ge m$, uses output derivatives up to mth-order

$$C^m\left(s\right)={\displaystyle \frac{Y^m\left(s\right)}{Y\left(s\right)}}={\displaystyle \frac{1+T_{1}s+...+T_m{s}^m}{P_n\left(s\right)}}.$$

(25)

$Y^m\left(s\right)$ will be fed back to a stabilizing proportional (P) controller that processes the control error $E^m\left(s\right)=W\left(s\right)-Y^m\left(s\right)$, given as

$$R_s\left(s\right)={\displaystyle \frac{U_s\left(s\right)}{W\left(s\right)-Y^m\left(s\right)}}=K_0.$$

(26)

For $P_n\left(s\right)=1$, such a control structure yields the closed-loop transfer function

$$F_w^m\left(s\right)={\displaystyle \frac{Y\left(s\right)}{W\left(s\right)}}={\displaystyle \frac{K_s{K}_0}{e^{T_{d}s}{s}^2+K_s{K}_0(1+T_{1}s+...+T_m{s}^m)}}$$

(27)

with the characteristic polynomial

$$A^m\left(s\right)=e^{T_{d}s}{s}^2+K_s{K}_0(1+T_{1}s+...+T_m{s}^m).$$

(28)

By introducing new variables

$$p=T_{d}s;{\kappa }_0=K_s{K}_0{T}_d^2;\mathcal{P}=e^p{p}^2;{\kappa }_1=K_s{K}_0{T}_d{T}_1;{\kappa }_2=K_s{K}_0{T}_2;...{\kappa }_m={\displaystyle \frac{K_s{K}_0{T}_m}{T_d^{m-2}}}$$

(29)

and with a simplified denotation $A^m\left(p\right)=A\left(p\right)$, (28) can be transformed to

$$A\left(p\right)=\mathcal{P}+{\kappa }_0+{\kappa }_{1}p+{\kappa }_2{p}^2+...+{\kappa }_m{p}^m$$

(30)

A successive differentiation of $A\left(p\right)$ then yields a sequence of equations

$$\begin{array}{c}{\displaystyle \frac{dA}{dp}}={\displaystyle \frac{d\mathcal{P}}{dp}}+{\kappa }_1+2{\kappa }_{2}p+...+m{\kappa }_m{p}^{m-1}\hfill \\ {\displaystyle \frac{d^{2}A}{d{p}^2}}={\displaystyle \frac{d^2\mathcal{P}}{d{p}^2}}+2{\kappa }_2+...+m(m-1){\kappa }_m{p}^{m-2}\hfill \\ ⋮\hfill \end{array}$$

(31)

that can finally be expressed in the form of a system with a triangular matrix and unknown values ${\kappa }_j,j\in [0,m]$ appearing in the first m rows

$$\begin{array}{c}\hfill \left[\begin{array}{c}{A}^{\left(0\right)}\\ A^{\left(1\right)}\\ A^{\left(2\right)}\\ ⋮\\ A^{\left(m\right)}\\ A^{(m+1)}\end{array}\right]-\left[\begin{array}{c}{\mathcal{P}}^{\left(0\right)}\\ {\mathcal{P}}^{\left(1\right)}\\ {\mathcal{P}}^{\left(2\right)}\\ ⋮\\ {\mathcal{P}}^{\left(m\right)}\\ {\mathcal{P}}^{(m+1)}\end{array}\right]=\\ =\left[\begin{array}{cccccc}1& p& p^2& \cdots & p^{m-1}& p^m\\ 0& 1& 2p& \cdots & (m-1)p^{m-2}& m{p}^{m-1}\\ 0& 0& 2& \cdots & (m-1)(m-2)p^{m-3}& m(m-1)p^{m-2}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & 0& m!\\ 0& 0& 0& \cdots & 0& 0\end{array}\right].\left[\begin{array}{c}{\kappa }_0\\ {\kappa }_1\\ {\kappa }_2\\ ⋮\\ {\kappa }_{m-1}\\ {\kappa }_m\end{array}\right],\end{array}$$

(32)

All the parameters ${\kappa }_j$ drop out of the difference $A^j-P^j$ firstly for $j=m+1$. Based on (32), the fastest possible non-oscillatory transients can be achieved by multiple real dominant pole (MRDP) tuning, fulfilling some $p_o<0$ requirements

$$\begin{array}{c}{\left[A\left(p\right);{\displaystyle \frac{dA\left(p\right)}{dp}};{\displaystyle \frac{d{A}^2\left(p\right)}{d{p}^2}};\cdots {\displaystyle \frac{d{A}^{m+1}\left(p\right)}{d{p}^{m+1}}}\right]}_{p=p_o}=\mathbf{0}\hfill \end{array}$$

(33)

From the equation $P^{(m+1)}\left(p_o\right)=0$, the dominant MRDP optimal pole $p_o$ and the corresponding time constant ${\tau }_0=-1/p_o$ can be determined as

$$\begin{array}{c}{p}_o=\sqrt{m+1}-(m+1)<0;T_0=1/\left((m+1)-\sqrt{m+1}\right);m=1,2,...\hfill \end{array}$$

(34)

Obviously, this result represents a generalization of (22) corresponding to $m=1$. Therefore, when solving the equation following from the last considered row in (32)

$$P^{(m+1)}=\left(p^2+2(m+1)p+m(m+1)\right)e^p=0$$

(35)

it is necessary to choose the pole located closer to the origin than the other possible solution $p_o=-\sqrt{m+1}-(m+1)$. The remaining dimensionless parameters ${\kappa }_j, j\in [0,m]$ are then calculated from $j=m$ up to $j=0$. Finally, the controller parameters $\kappa$ and ${\tau }_j, j\in [0,m]$ in

Table 1. Optimal ${\mathrm{PD}}^m$ parameters corresponding to DIPDT model (21) for $m\in [1,5]$.

	$\mathit{m}=1$	$\mathit{m}=2$	$\mathit{m}=3$	$\mathit{m}=4$	$\mathit{m}=5$
$-p_o$	0.5858	1.2679	2.0000	2.7639	3.5505
${\tau }_o$	1.7071	0.7887	0.5000	0.3618	0.2816
$\kappa$	0.0791	0.2100	0.3610	0.5237	0.6947
${\tau }_1$	5.8284	3.7321	3.0000	2.6180	2.3798
${\tau }_2$	0	0.9811	1.1250	1.1434	1.1310
${\tau }_3$	0	0	0.1250	0.1962	0.2373
${\tau }_4$	0	0	0	0.0124	0.0243
${\tau }_5$	0	0	0	0	0.0010

are enumerated by substituting for $s_o=p_o/T_d$ into (29).

3.1. Filter Design

To obtain proper controller activity, the filter term $P_n\left(s\right)$ must be designed with respect to closed-loop stability and required measurement noise attenuation. To keep the number of controller parameters to a minimum, a binomial filter will be chosen, specified by the order n and a time constant $T_f$

$$P_n\left(s\right)={\left(T_{f}s+1\right)}^n;n=1,2,..;n\ge m$$

(36)

Its effect can be respected by an equivalent dead time $T_e$ [60] added to the process delay $T_{dp}$, when the “total” closed-loop dead time becomes

$$T_d=T_{dp}+T_e$$

(37)

and a simple $T_e$ expression by the filter time constant $T_f$ can already be achieved by

$$T_e=n{T}_f.$$

(38)

Alternatively, filter dynamics can already be included in the process approximation (1) [61]. Although the minimum value of $n=m=1$ was considered when determining the series PID controller based on the IPDT model, it is generally recommended to choose $n>m$. Where applicable, for the sake of simplicity, the default value $n=m+2$ has been used in several recent works and this paper. The time constant $T_f$ cannot be arbitrarily small with respect to $T_d$ and $T_s$ [12,13]. Together with the values of m and n, they are the basic parameters that affect the speed of transient responses and noise attenuation.

3.2. Stabilizing Control with Disturbance Reconstruction and Compensation

The basic advantage of model-based disturbance reconstruction, according to (5), is that by adding disturbance compensation to a closed circuit with a stabilizing controller, its setting does not have to change in the nominal case.

Remark 7 (From automatic reset and pre-act to automatic offset controller (AOC)). When it is not necessary to know the actual values of the reconstructed disturbance $d_{if}$, the control scheme can be rearranged according to Figure 2 below, which allows a direct comparison with previously known controller schemes. Positive feedback from the output limitation of the controller can be considered a generalization of automatic reset and its modifications for systems with dead-time (see, e.g., Figure 4.18 Block diagram of PPI controller with $\lambda =1$ on pp. 159 in [3]). To clearly distinguish it from older solutions, it can be called automatic offset. When choosing the same filter $P_n\left(s\right)$ in both $C^m\left(s\right)$ and PDO, in the negative feedback from the output of the process given by $N_c\left(s\right)=N_c^m\left(s\right)$

$$N_c\left(s\right)={\displaystyle \frac{Y_p^m\left(s\right)}{Y\left(s\right)}}=1+T_{1}s+\left(T_2+{\displaystyle \frac{1}{K_s{K}_0}}\right)s^2+...+T_m{s}^m,$$

(39)

it is possible to combine the dynamic members from the controller $C^m\left(s\right)$ (25) and from the block $S_{yd}\left(s\right)$ of the disturbance reconstruction (5) into a single polynomial. The corresponding feedback controller

$$C_y^m\left(s\right)={\displaystyle \frac{N_c\left(s\right)}{P_n\left(s\right)}}={\displaystyle \frac{1+T_{1}s+\left(T_2+\frac{1}{K_s{K}_0}\right)s^2+...+T_m{s}^m}{{(1+T_{f}s)}^n}}$$

(40)

can be denoted as an output predictive controller and the whole structure as an automatic offset controller (AOC). Unlike the traditional interpretation of a PID controller with three terms responding to actual, past, and future control error values, an AOC works with a predicted value of the output $Y_p^m\left(s\right)$, which can be refined compared to the current value represented by $Y\left(s\right)$ by increasing the order of the maximum derivative used m. At the same time, it also takes into account the occurrence of disturbances $d_i$. The past, in AOC activity, is reflected in the output saturation values of the controller, which, in steady states, is equal to the acting disturbance $d_i=const$ and is fully formed as the offset of the controller, while the control error takes zero values.

Figure 2. A two-degrees-of-freedom PID controller (above) and a ${\mathrm{P}}^m$-PDO controller transformed to the structure with an automatic offset and a generalized predictive output (below) for a DIPD system with measurement noise generated in Matlab/Simulink by the uniform random number block.

Figure 2. A two-degrees-of-freedom PID controller (above) and a ${\mathrm{P}}^m$-PDO controller transformed to the structure with an automatic offset and a generalized predictive output (below) for a DIPD system with measurement noise generated in Matlab/Simulink by the uniform random number block.

Of course, in applications requiring the reconstructed disturbance $d_{if}$, the controller can still be used with the structure corresponding to Figure 1. This allows us to choose filters of $C^m\left(s\right)$ and $S_{yd}\left(s\right)$ separately.

4. Simulation Experiment—DIPDT System

The simulation evaluation of the derived controllers will be carried out in two steps: first, we will compare the MRDP-optimal PID controllers, with filter (4) taken into account according to (37) and (38), and prefilter (18) with other known alternative designs. In the second step, we compare the MRDP-optimal PID with different ${\mathrm{P}}^m$-PDO control options.

4.1. Comparison of MRDP-PID with Sensitivity-Constrained Optimal Design

The design of controllers for the DIPDT model using controllers of the PID type has been addressed by many authors (see, for example, works [52,53,62,63,64] and the references therein). However, when we look at these works from the point of view of noise attenuation, it must be noted that they did not pay special attention to the design of the filters that are absolutely necessary, at least for the implementation of the derivative component. Although [63] states the optimal setting of PI-PD controllers from the point of view of disturbance responses for five different cost functions (including IAE), it does not state how measurement noise is taken into account. In the next comparison, we will limit ourselves to the SIMC method. It was developed with ambition, becoming “probably the best simple PID tuning rules in the world” [52,65], and so far has enjoyed great popularity. The results of the comparison can thus be considered sufficiently representative. In [52] the implementation of a series PID controller was proposed, with an additional first-order filter with the time constant $T_f=T_D/N,N=100$ used only in the derivative action, when

$$C_{PID}\left(s\right)={\displaystyle \frac{U\left(s\right)}{E\left(s\right)}}=K_p\left(1+{\displaystyle \frac{1}{s{T}_i}}+{\displaystyle \frac{s{T}_D}{1+T_{D}s/N}}\right);K_i={\displaystyle \frac{K_p}{T_i}};K_d=K_p{T}_D$$

(41)

However, the author admitted that, especially for noisy processes, lower values of N that yield $T_f\in [T_D/10,T_D/5]$ can be required. The more recent work [64] does not mention the need for a low-pass filter in controller implementation at all. It notes, however, that the use of a parallel PID controller allows for improving the dynamics achieved with the series PID proposed in [52]. Ref. [64] presents the optimal design of the parameters of a parallel PID controller that minimizes the values of $IA{E}_d$ corresponding to the disturbance steps by considering several values of sensitivity constraints. The values of the dimensionless parameters of the controller are shown in

Table 2. Optimal dimensionless PID controller parameters for disturbance steps according to [64].

${\mathit{M}}_{\mathit{s}}/{\mathit{M}}_{\mathit{t}}$	$1.40/1.64$	$1.59/1.64$	$1.80/1.63$	$2.00/1.68$
$K_p{K}_s{T}_d^2$	0.0416	0.0694	0.0974	0.1215
$T_i/T_d$	16.39	13.39	11.98	11.28
$T_D/T_d$	7.17	5.76	5.09	4.68

for several values of sensitivity peaks.

In the next comparison, the DIPDT (21) parameters were initially chosen to be equal to the values achieved by our own approximation of the magnetic levitation system, which was based on the evaluation of the initial phase of open-loop step responses in [40] as follows:

$$K_{sp}=5000;T_{dp}=0.002$$

(42)

To illustrate effects of a noisy measurement, the amplitude of the measurement noise generated in Matlab/Simulink by the uniform random number block with the sampling period $T_s=0.01$ ms was set to

$${\Delta }_y=0.01$$

(43)

In the setpoint steps, the input signal takes two constant values

$$w_0=0.25;w_1=0.5$$

(44)

with a step change in the middle of the simulation time $T_{max}=0.12$ s. Thus, the input disturbance is $d_i=0$.

Similarly, in the disturbance steps, the variable input takes the two constant values

$$d_0=0.5;d_1=1.0$$

(45)

whereas $w=0$.

The speed of the changes at the process output after the step changes at its input is evaluated by the integral of the absolute error

$$IAE={\int }_0^{\infty }\left|e\left(t\right)\right|dt;e=w-y,$$

(46)

where w is the reference setpoint value, y is the process output, and e is the control error.

The excessive control effort at the process input (controller output) is measured in terms of deviations from the two-pulse (2P) signal, ideally consisting of three monotonic intervals. This requirement corresponds to an ideal situation of a delay-free double integrator. In evaluating the ideal input of DIPDT systems, the control signal could also have, besides the two extremes $u_{m1}$ and $u_{m2}$ lying out of the interval specified by the initial value $u_0$ and the final $u_{\infty }$, some additional extremes (see, e.g., the notion of dynamical classes of control in [66,67]). However, such a performance evaluation represents an already advanced topic. Here, the excessive input deviation from the 2P shape can be evaluated as follows:

$$\begin{array}{c}{TV}_2\left(u\right)=\sum _i\left(\left|u_{i+1}-u_i\right|\right)-\left|2u_{m1}-2u_{m2}+(u_{\infty }-u_0)sign(u_{m1}-u_{\infty })\right|.\hfill \end{array}$$

(47)

When formulating the design problem as a trade-off between the setpoint tracking ($IAE$) and measurement noise injection leading to an excessive controller activity (input usage $T{V}_2\left(u\right)$), a combined cost function can be specified as follows:

$$J=IAE\ast T{V}_2\left(u\right)$$

(48)

The sampling time is chosen as $T_s=0.01$ ms.

The setpoint and input disturbance step responses in Figure 3, together with the corresponding performance measures in Figure 4, show that the MRDP-optimal 2DoF PID controller using the second-order filter $Q_2\left(s\right)$ produces a significantly lower noise impact than the SIMC PID (41). When applying the derivative filter with value $N=100$ recommended in [52] to the SIMC PID (41), the performance measures $T{V}_2\left(u\right)$ and J of MRDP PID would be even more than 100 times lower with roughly equivalent IAE values.

4.2. Comparison of MRDP-Optimal PID and Model-Based Controllers Based on DIPDT Models

Evaluations of the influence of the growing value of the derivative degree m in several previous contributions have refuted the old dogmas that the design of derivative filters represents a tedious problem leading to a trivial decision to use only the simplest PI control [68], the derivative part is the most difficult to tune [69], the derivative action is not suitable for noisy and time-delayed processes [4], or that there exists an absence of reliable methods for derivative action tuning [4,69]. It turned out that the opposite is true, and that to meet needs for more effective filtering, the controllers must be supplemented with the use of higher-order derivatives. Otherwise, the filtering itself actually leads to the slowing down of transients and, subsequently, to a decrease in the robustness of the circuit. This is especially critical when dealing with unstable and marginally stable processes.

In [11], with the aim of showing a new degree of freedom created by using possibly higher-order derivatives $m\in [0,5]$, the design of PIDs for the IPDT model was applied by the multiple real dominant pole (MRDP) method to the whole family of higher-order (HO) PID controllers. In this paper, to investigate the performance of DIPDT systems under measurement noise impact, a new family of model-based proportional controllers extended by the use of higher-order derivatives and predictive disturbance observer (PDO) (denoted simply as ${\mathrm{P}}^m$-PDO) will be evaluated for the range of derivative degrees

$$m\in [1,5]$$

(49)

and some $n>m$. The parameters of $C^m\left(s\right)$ (40) derived for the DIPDT model (21) by the MRDP method are set according to Table 1. With the exception of the derivative degree $m=1$, when $n=m+2$, for the remaining derivative degrees (49), the filter degree was chosen as $n=m+1$. The vector of corresponding time constants $T_f=T_e/n$ is given by the vector of $T_e$ values related to the dead-time process $T_{dp}$

$$T_e/T_{dp}=[1/32345].$$

(50)

The 2DoF PID controller (12)–(23) with $m=1$ and $z=0$ in (24) was specified with

$$n=m+2=3;T_e/T_{dp}=1/5.$$

(51)

The transients corresponding to the chosen experiment setting are shown in Figure 5. Obviously, the dynamics of PID control are slower but still have a significant measurement noise impact. This observation can be enumerated by displaying the corresponding performance measures.

The performance measures of the setpoint step responses of ${\mathrm{P}}^m$-PDO controllers in Figure 6 show that, compared to the 2DoF PID controller, the transients are faster, but the excessive controller effort $T{V}_2=T{V}_2\left(u_s\right)$ and the values of the combined cost function $J_s=IA{E}_s\ast T{V}_2\left(u_s\right)$ in Figure 7 are significantly reduced.

If we divide the $J_s$ value for the ${\mathrm{P}}^m$-PDO controller $m\in [1,5]$ by the $J_s$ value for the 2DOF PID control, the possibility of a 120-fold improvement of this parameter is shown. Because this indicator maintains a growing trend, from this point of view, it might also make sense to consider values of the maximum derivative $m>5$.

The input disturbance step responses of the 2DoF PID controller (51) and the ${\mathrm{P}}^m$-PDO controllers in Figure 8 are due to a tight feedback on the noise level. The disturbance impact is visible just through the corresponding evaluation of the performance measures in Figure 9. It shows that for smaller values of m, the transients are a bit slower than for the 2DoF PID controller. However, excessive controller effort $T{V}_2=T{V}_2\left(u_d\right)$ (Figure 9) and the values of the combined cost function $J_d=IA{E}_d\ast T{V}_2\left(u_d\right)$ (Figure 10) are significantly reduced.

Regarding a detailed comparison of computational costs or complexity between PID and model-based controllers, in the simplest case, the design of the PID controller requires the use of a first-order filter, which can be implemented by a difference equation and the update of one state variable. When using the model-based controller design, the minimum order of the filter is n = 2, which corresponds to the use of two difference equations, the operation of shifting to emulate dead-time in the PDO, and the update of two state variables. Compared to other operations necessary for closing the feedback loop (loading the output, calculating a new control action value, and sending it to the input of the process), this is usually a relatively small increase in the complexity of the calculation. When trying to reduce the high sensitivity to the PID controller measurement noise by choosing a filter with n = 2, the differences in the complexity of both approaches can be further reduced.

In summary, simulation experiments performed with the control of the DIPDT system fully confirm the expectations regarding the increase in the smoothness of the waveforms by increasing m, which can be achieved even without slowing down the processes. At the same time, the achievable degree of improvement shows that the seemingly exceptional position of PIDs with regard to the possibilities of further development may not be certain at all.

5. Magnetic Levitation Control

The robustness of the design based on the use of DIPDT models can be best demonstrated by controlling unstable processes. With them, the imperfection of control with model uncertainties and the impact of measurement noise cannot be eliminated by slowing down transients. Its consequence would be a loss of stability. However, when controlling unstable systems, it is not possible to use even traditional methods to evaluate robustness based on peaks of sensitivity functions [11]. The recommended maximum sensitivity values are usually chosen from an interval $M_s\in [1.2,2]$. This is not suitable for unstable systems with sensitivity peaks reaching much higher figures of more than 20 [14]. Instead, robustness can be assessed by evaluating deviations from the ideal shapes of responses at the input and output of the process.

Here, the main advantages of the controllers derived for ultra-local (integral) models will further be demonstrated by comparison with the results achieved by the 2DoF PID and the ${\mathrm{P}}^m$-PDO controllers, shown in Figure 2, in controlling an unstable second-order magnetic levitation system (USO) from the work [40]. Its system transfer function derived by the producer

$$S\left(s\right)={\displaystyle \frac{K}{T_{00}^2{s}^2-1}};K=1.7;T_{00}=0.018$$

(52)

replaced the DIPDT model in control schemes according to Figure 2. When focused in the controller design on the fastest modes of transients, (52) can be approximated for $t\to 0$, i.e., $s\to \infty$, by a double integrator with the transfer function

$$S_0\left(s\right)={\displaystyle \frac{K_s}{s^2}};K_s={\displaystyle \frac{K}{T_{00}^2}};K=1.7;T_{00}=0.018;K_s=52.47$$

(53)

Evidently, this DIPDT model is significantly different from (42). Although it refers to the same device, the differences result from the different methods used for identification. In the end, the usefulness of both models can only be verified by a real experiment. However, even without such a real experiment, their comparison can be used to assess the robustness of the controller design method used.

Since the producer does not specify a dead time, it will be set to the minimum possible value of a quasi-continuous implementation represented by the sampling period, i.e., $T_{dp}=T_s$. Without appropriate compensation, short dead-time values in combination with nearly 100 times lower $K_s$ will result in high controller gains and increased noise amplitudes. To avoid these problems, $T_e$ value characterization in (37) and (38) the low-pass filter delay related to the dead-time process $T_{dp}=T_s$ have to be appropriately increased. They will be specified by the vector

$$T_e/T_{dp}=\left[100110130180240\right];n=m+2;T_f=T_e/n.$$

(54)

The performance achieved by ${\mathrm{P}}^m$-PDO controller is compared with the 2DoF PID controller (12)–(23). Its filter tuning was specified by

$$n=m+2=3;T_e/T_{dp}=10;T_f=T_e/3.$$

(55)

The transients corresponding to the chosen experiment setting are shown in Figure 11.

Remark 8

(Robustness of magnetic levitation control). For the setting (53)–(55), the dynamics of PID control are slower, albeit with a significant measurement noise impact and an overshooting caused by the different dynamics of the process (52) and its approximation by the DIPDT model (53). In the case of unstable systems, overshooting under PID control cannot simply be avoided by choosing longer $T_e$. This overregulation indicates a lower robustness of the PID control compared to all options of the ${\mathrm{P}}^m$-PDO control considered, in which the responses better keep the ideal shapes. The high closed-loop robustness is demonstrated by small differences in transients and performance measures obtained by applying derived controllers to the USO magnetic levitation system and the simplified DIPDT model.

Performance measures of ${\mathrm{P}}^m$-PDO controllers in Figure 12 show that, compared to the 2DOF PID controller, transients are faster, while excessive controller effort $T{V}_2=T{V}_2\left(u\right)$, together with values of the combined cost function $J=IAE\ast T{V}_2\left(u\right)$, is significantly reduced with increasing m in Figure 13.

When dividing the $J_s$ value for the ${\mathrm{P}}^m$-PDO controller, $m\in [1,5]$, by the $J_s$ value for the 2DoF PID control, the possibility of an almost 120-fold improvement of this parameter appears again. Because this ratio of $J_s$ values for $m\in [1,5]$ maintains a growing trend, it might again make sense to also consider derivative degrees with $m>5$.

6. Discussion

All conducted experiments indicate that by returning to model-based controller design using HO derivatives, the adverse impact of measurement noise can be substantially reduced without slowing down responses and increasing their sensitivity to unmodeled dynamics. Of course, the comparison made only with the PID controller having $m=1$ is not entirely correct because, even when using controllers with an explicit integrator, it is possible to improve filtering with a combination of low-pass filters and higher-order derivatives with $m>1$.

The effective compensation of disturbances results from a comparison of the actual values of the process signals with their expected values calculated using the model. This was the basis of industrial automatic reset controllers, but also state observers developed within the framework of modern control theory and other postmodern approaches such as IMC [70], DOBC [71], or ADRC [72,73]. However, if we look at the invited lectures of the recently held IFAC conference PID’2024 in Almeria, Spain, despite the clear role of controllers using automatic reset and other methods of reconstruction based on the comparison of actual and expected quantities, the pillar of process control is still considered to be the use of integral control [74]. Likewise, the theory of DOBC [75] has still not discovered their use in historical automatic reset controllers. The control community (with the exception of ADRC and model-free control [38]) also does not pay systematic attention to the important role of ultra-local (integral) models. They are often used without explanation or justification, as in [64], despite the fact that observations of their exceptional properties have appeared in the professional literature since their early origins. Sufficiently clearly formulated, they were mentioned more than a half century ago (see, e.g., [76]). One of the few areas where the current development of the issue surpasses the framework established in the period after the Second World War is the design of controllers with HO derivatives [20,23,24,77] and with fractional-order controllers [78,79]. With these, the use of higher-order controllers remains hidden behind fractional-order operators requiring implementation by HO integer-order approximations.

7. Conclusions and Future Work

The paper reviewed the design of PID controllers using IPDT and DIPDT models from the point of view of generalizing the standards set by modern control theory. It was shown that the extension of PID control to the reconstruction and compensation of disturbances (considered as a special case of state reconstruction) leads to the following outcomes:

• the redesign of the stabilizing controller;
• an increase in the controller gains and thus an increase in the negative effects of measurement noise;
• a reduction in the speed of transient responses.

Thus, the setpoint tracking and reconstruction and compensation of disturbances are not separated, as can be achieved by their state space design within MCT. The work refers to the historical context of the development of PID. It interprets the first widely used industrial controllers based on automatic reset (AR) as an analog counterpart to model-based solutions with a simplified disturbance observer. However, the introduction of the usual interpretation of AR controllers based on their equivalent transfer function also had negative effects that can be avoided by predictive disturbance observers. While the equivalent transfer functions of AR controllers in the linear domain led to the ambiguous designation of series PID controllers, which represent a subset of parallel PID controllers, model-based design also offers solutions, including dead-time models. In the linear domain, parallel PID controllers are more general than ARCs, which is why they prevail in most textbooks and publications. Later, especially “in the digital era”, the problem of redundant integration, also known as integrator windup, arises when the control signal is constrained. A formal advantage of parallel PIDs is that they can be analytically designed for processes based on the DIPDT model with higher-order dynamics. However, as has been shown, it is possible to achieve multiple improvements in control performance by consistently applying a model-based controller design that decouples setpoint tracking and disturbance reconstruction and compensation. With the use of programmable devices and embedded control, such a design is fully feasible and even profitable. Therefore, the only solid advantage of using PID controllers, apart from their long-standing tradition and popularity among control engineers, denoted as the control of processes with dominant second-order dynamics, disappears. In favor of the model-based approach are the possibilities of unified interpretation together with historical AR-based and modern AOC-based controllers, including MCT, ADRC, or intelligent PID controllers. Of course, there are opportunities for effective implementation through the use of various programmable devices and embedded systems, combined with a significant improvement in performance measures that have a direct impact on real-world applications. In addition to re-evaluating the design of PID controllers for systems approximated by DIPDT models, a recent publication [80] also showed the need to re-evaluate the use of PIDs in systems with dominant sensor dynamics. Here, we need monotonic responses of the control signal to step changes in the reference and process input disturbance. Furthermore, encouraged by the significant improvement in performance measurements when using PDO in combination with stabilizing controllers with HO derivatives and filters, we can also test them for the dominant first-order dynamic processes approximated by the IPDT model. This means a re-evaluation of series PIDs for systems with dominant first-order dynamics [12]. Here, we can take advantage of the technological advances offered by embedded controllers and programmable devices. Alternatively, the given method could be extended to processes approximated by triple integrator plus dead-time models.

When it comes to the saturation of control signals due to existing control constraints, which was one of the main reasons for using controllers with automatic reset (series PID controllers), the use of model-based solutions alone may not be sufficient for the DIPDT systems. Even if they allow for keeping the integrator windup under control, there is still the possibility of the so-called “plant windup” [81,82,83] and the need for nonlinear controller design [84]. This can also lead to transients with overshoot of the process output [85,86,87]. The first saturation controllers based on DIPDT models were realized in the discrete time domain [50]. However, the ${\mathrm{P}}^m$-PDO controllers, in combination with the reference model control [88], now enable a simple formulation of higher-order saturating controllers in the continuous time domain. They enable relatively short sampling periods, allowing them to compete with model predictive control (MPC) in single-input, single-output control [89,90,91,92,93]. Research in this direction will be one of our main goals for the future.