Practice-Oriented Controller Design for an Inverse-Response Process: Heuristic Optimization versus Model-Based Approach

Abstract

The proposed practice-oriented controller design (POCD) aims at stabilizing the system, reconstructing and compensating for disturbances while achieving fast and smooth step responses. This is achieved through a simple approach to process identification and controller tuning that takes into account control signal constraints and measurement noise. The proposed method utilizes POCD by eliminating the influence of the unstable zero dynamics of the inverse-response processes, which limits the achievable performance. It extends the previous work on PI and PID controllers to higher-order (HO) automatic reset controllers (ARCs) with low-pass filters. It is also extended according to POCD requirements while maintaining the simplified process model. The final result is an extremely simple design for a constrained controller that provides sufficiently smooth and robust responses to a wide family of HO-ARCs with odd derivatives, designed using integral plus dead time (IPDT) models and tuned by the multiple real dominant pole method (MRDP) and the circle criterion of absolute stability. The proposed design can be considered as a generalization of the Ziegler and Nichols step response method for inverse response processes and HO-ARCs.

1. Introduction

Some processes, such as distillation columns and chemical reactors [1], aircraft pitch control, power supply systems with hydro turbine [2], and boost converters [3], have an initial response to an input step change in the opposite direction to that of its new steady-state value. This represents a strong limitation for the use of robust control methods based on high-gain controllers [4]. The control of these processes is influenced by two parallel competing dynamics working against each other [5,6]. In the simple case with two counteracting first-order subsystems with gains $K_1,K_2$ and time constants $T_1,T_2$,

$$S_0\left(s\right)=\frac{K_1}{T_{1}s+1}-\frac{K_2}{T_{2}s+1}=\frac{(K_1{T}_2-K_2{T}_1)s+K_1-K_2}{(T_{1}s+1)(T_{2}s+1)}$$

(1)

the system $S_0\left(s\right)$ has a non-minimum phase (unstable) zero for $(K_1-K_2)(K_1{T}_2-K_2{T}_1)<0$. With $K=K_1-K_2$, $T_1=T,T_2=a{T}_1$ and $bT=-(K_1{T}_2-K_2{T}_1)/(K_1-K_2)$, it can be rewritten as

$$S_0\left(s\right)={\displaystyle \frac{K(-bTs+1)}{(Ts+1)(aTs+1)}}$$

(2)

If system (2) is extended by a dead time $T_d$, which represents the sum of the remaining shorter loop delays, the system becomes a second-order plus unstable zero (SOPUZ)

$$\begin{array}{c}S\left(s\right)={\displaystyle \frac{Y\left(s\right)}{U\left(s\right)}}=S_0\left(s\right)e^{-T_{d}s}\hfill \end{array}$$

(3)

The presence of an unstable zero causes problems in model-based control design approaches based on process inversion such as the disturbance observer-based (DOB) approach [7], or internal model control (IMC) [6].

The problems with unstable zeros also persist in the heuristic design of PI and PID controllers using optimization methods based on the values of the integral of the absolute error (IAE) and the total variation in the control signal (TV) for setpoint and disturbance step responses. For the inverse response processes with transfer function (3), with $a\in [0.1,1];b\in [0.1,3]$, controller tuning was proposed in [8]. The study was conducted at two robustness levels. It was shown that the proposed filtered PID controllers with two degrees of freedom (2DoF)

$$U\left(s\right)=\left[(\beta +\frac{1}{T_{i}s})W\left(s\right)-(1+\frac{1}{T_{i}s}+T_{D}s)Y\left(s\right)\right]\frac{K_p}{1+T_{f}s}$$

(4)

provide better performance (IAE) than 2DoF PI controllers tuned with the same approach at the same robustness level. Parameter $\beta$ represents the setpoint weighting, $T_i$ the integral time constant, $T_D$ the derivative time constant, $K_p$ the proportional gain and $T_f$ the filter time constant. The method provided responses with low undershoot and overshoot of the controlled variable, smooth control signal and low controller effort.

However, in real control applications, manipulation carried out solely with controller and process transfer functions usually does not lead to acceptable results. In practice, many other aspects of control need to be considered. Let us therefore introduce these aspects using the concept of practice-oriented controller design (POCD) as follows:

Definition 1

(Practice-Oriented Controller Design). Practice-oriented controller design ensures the stability of the closed loop together with the reconstruction and compensation of the acting disturbances. The aim is to achieve fast and smooth transients (characterized by the smallest number of monotonic sections of their responses to step changes in the inputs) using the simplest possible process model. At the same time, the specified attenuation of the measurement noise and the robustness of the closed loop should be achieved. The limitations of the control signal should be taken into account, while the resulting dynamics of the circuit should meet the technological requirements.

The consideration of all the above-mentioned POCD requirements is not a matter of course even for some well-known methods for designing controllers. For example, the most commonly used linear PI controllers have no additional filters and do not allow attenuation of high-frequency noise. This problem is even more pronounced with PID controllers. In addition, PI and PID controllers without adequate anti-windup protection, which prevents unwanted integration due to control limitations (windup phenomenon), are generally not useful in practice [9].

Compensation for constant disturbances is encountered in practically every control task and includes not only compensation for physical external influences but also internal disturbances resulting from simplified models of possibly time-varying processes. Machine or process health monitoring and fault diagnosis help to operate the systems safely and efficiently. In addition to compensation, this also requires explicit reconstruction of disturbances. There are many control approaches, such as simple PI and PID controllers, that are able to compensate for faults but do not provide explicit information about their values.

The design of a high-quality controller requires a sufficiently accurate method of process identification. While there are a number of approaches that directly provide the transfer function of the given system (3), such as Prony’s identification method [10], their practical application encounters a number of problems caused by measurement noise, non-linearities of process, quantization of signal levels, etc. Therefore, simpler methods can be useful as they can calculate the individual parameters of the transfer function step by step graphically and numerically [11]. However, the question remains whether the necessary identification of the process can be further simplified. For example, the process identified by the method of Ziegler and Nichols [12] can be approximated by an integral plus dead time (IPDT) model [13].

Another question is whether the design of a PID controller based on two different values for the maximum sensitivity, which was applied in [8], sufficiently fulfils the requirements of practice and allows fine tuning of the controller settings.

The numerous aspects of POCD have even led to the inclusion of inverse-response systems in interactive open-source software tools for control loop analysis [14]. The tools offer simulations under different conditions and can thus compensate for the missing aspects of the design methods under consideration.

PI and PID controllers are proclaimed as the most commonly used solutions in practice [15]. Although this statement is generally correct, it can be slightly misleading as the first controllers with the integrating and derivative terms were called automatic reset and hyper reset controllers, respectively, [16]. These control realizations do not contain an explicit integrator, but instead a filter in a positive feedback loop. It turns out that these controllers especially can easily fulfill POCD requirements.

Several recently published papers (see, e.g., [17,18]) have shown that the essential idea of these historical and still-existing in industry automatic reset controllers (ARCs) is the use of a positive feedback from the controller output (see Figure 1). The problem is that the implementation has been neglected for a long time. To explain the nature of AR, a model-based approach can be used, formulated for an integral process model describing an input–output behavior with a differential equation $\dot{y}=K_{s}u+d_i$, whereby $\dot{y}=dy/dt$. The reconstructed value of the constant input disturbance ${\widehat{d}}_i$ of such a system can be calculated as the difference between the estimated input of the process $\widehat{u}=\dot{y}/K_s$ and the output of the controller u as ${\widehat{d}}_i=\dot{y}/K_s-u$. To perform the entire calculation, the reconstructed disturbance must be supplemented with a low-pass filter (see Figure 1)

$$F_R\left(s\right)=\frac{1}{1+T_{i}s}$$

(5)

If you choose the time constant $T_i$ to be greater than the time constant of the closed-loop transients, the first term $\dot{y}/K_s$ can be neglected near to steady states and the disturbance value can simply be calculated as ${\widehat{d}}_i=-u$. The compensation of the reconstructed disturbance in the form of a change of the offset at the output of the stabilizing controller (for the integrating process, it is sufficient to use a simple gain) finally leads to a positive feedback of automatic reset and hyper reset controllers.

Due to the fact that the disturbance observer (DOB) used in the $P{D}_n^{m}R$ controller in Figure 1 is based on disturbance estimation from steady states (with an output derivative of zero), it was possible to reduce the number of its blocks from two (which is usually the case in the DOB [7], in the state space [19] or in the ADRC [20] based on the inverse process model [21], see also [22]) to one with $F_R\left(s\right)$, which is used in HO-ARCs.

Definition 2

(HO-AR Controller). The $P{D}_n^{m}R$ controller $m\in {\mathbb{Z}}^{+};n\in \mathbb{Z},n\ge m$ (see Figure 1) represents a combination of the $P{D}_n^m$ controller, saturation non-linearity, and automatic reset, modifying the $P{D}_n^m$ offset by a positive feedback with a low-pass filter $F_R$ (5); Thereby, the $P{D}_n^m$ controller consists of an ideal $P{D}^m$ controller (6) with a low-pass implementation filter $Q_n$ (8).

To compensate for the process dead time contained in the IPDT models, the corresponding exponential term $e^{-T_{d}s}$ can be approximated by the finite number of terms of its expansion into the Taylor series (see, e.g., [23]). The combination with the integrating process also increases the total dimension of the corresponding state vector. Its special case is the phase vector, which is formed by the output of the process and its derivatives. Thus, the automatic reset controller can be considered as a model-based solution derived for a specific IPDT model and extended to the whole family of controllers with higher-order (HO) derivatives. To simplify the controller design, only HO-ARCs with odd values of the derivative degree m are used in the proportional–derivative $P{D}^m$ controller (Figure 1).

$$\begin{array}{c}P{D}^m\left(s\right)=K_c\left(1+T_{1}s+\cdots +T_m{s}^m\right);m=1,3,5,7\hfill \end{array}$$

(6)

with a positive feedback from the limited controller output $F_R$ are considered in the study. This represents a special case of HO-ARC, which was considered in [17] for $m\in {\mathbb{Z}}^{+}$, $m\in [0,5]$. The separate treatment of even and odd values of m in the constrained control case is based on the requirement to achieve conditions for absolute stability by factorizing the zeros of the controller transfer function derived using the multiple real dominant pole (MRDP) method (see, e.g., [18], which deals with the case $m=2$, and [24], which presents a particular solution for $m=4$).

In the proportional zone of the controller, the HO-ARC structure yields the transfer function

$$\begin{array}{c}P{D}^{m}R\left(s\right)=K_c\left(1+T_{1}s+\cdots +T_m{s}^m\right){\displaystyle \frac{1+T_{i}s}{T_{i}s}}=\hfill \\ =K_p+{\displaystyle \frac{K_i}{s}}+K_{1}s+\dots +K_m{s}^m;K_i=K_c/T_i;K_p=K_c(1+T_1/T_i);\hfill \\ K_j=K_c(T_j+T_{j+1}/T_i;j=1,2,\dots ,m-1;K_m=K_c{T}_m.\hfill \end{array}$$

(7)

This means that this closed-loop transfer function can be considered as an alternative to the parallel HO-PID controller [25] with an explicit integrator $K_i/s$ in the vicinity of steady states. The use of HO derivatives is one way of increasing the allowable gain of the system. To investigate the practical limits of this concept in the context of POCD, we will try to answer the question raised in [8], in which the PID controller with derivative term achieved better results compared to simple PI control. Is it possible to generalize the results to processes with unstable zero dynamics controlled by HO-ARCs?

The implementation of the derivatives requires the inclusion of low-pass filters. In contrast to works that specify the necessary filter for each derivative [26], both HO-PID [25] and HO-AR [17] controllers use a much more efficient design of a single nth-order binomial filter

$$\begin{array}{c}{Q}_n\left(s\right)=Y_f\left(s\right)/Y\left(s\right)=1/{\left(T_{f}s+1\right)}^n=1/P_n\left(s\right)\end{array}$$

(8)

The above-mentioned joint filter can also be seen as a generalization of the approach used in [8]. The combination of $Q_n\left(s\right)$ with an ideal $P{D}^m\left(s\right)$ controller (6), for $n\ge m$, yields fully realizable $P{D}_n^m\left(s\right)$. Furthermore, in combination with AR (5) and saturation non-linearity, this yields the $P{D}_n^{m}R$ controller.

For a comparison, the approach in [26] requires the computation of three first-order differential equations for the implementation of PIDA controller filters. However, a joint filter (8) based on three equations provides access to the first, second and third derivatives of the output (see, e.g., [23]). If only the first two derivatives are used, the $P{D}_3^{2}R$ controller can contribute to more effective noise attenuation. The common filter (8) has the further advantage that all components contained in the control signal (7) are delayed in the same way. When using controllers with HO derivatives, the simplification by using only one filter becomes even more obvious.

When designing HO-ARCs, it is still sufficient to use an IPDT model with two unknown parameters as in simple PI or AR controllers. Moreover, the proposed method takes into account the POCD requirements and it simplifies the process identification as it uses a simpler process model than SOPUZ (see, e.g., [11]). The optimization phase of the controller is then reduced to the selection of a derivative degree m and a suitable low-pass filter required for the implementation of the controller and noise attenuation. The choice of the filter depends on the stability requirements, the closed-loop speed, the maximum overshoot and undershoot, the attenuation of the measurement noise, and the constraints of the control signal.

The rest of this article is organized as follows. Section 2, extended by Appendix A, briefly summarizes the MRDP rules for setting $P{D}_n^{m}R$ controllers for IPDT models. A sample of HO-ARCs based on a simple inverse process approximation without measurement noise is presented. Section 3 provides a more detailed analysis of process noise using the speed–effort (SE) and speed–wobbling (SW) characteristics. The practical limitations of linear and constrained controller designs are analyzed in Section 4. Section 5 and Section 6 discuss and summarize the lessons learned and make suggestions for future research.

2. MRDP Design of HO-ARCs for IPDT Model

For an integrator plus dead time (IPDT) process model with gain $K_{sp}$ and dead time $T_{dp}$

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

(9)

the tuning of the $PI{D}_n^m$ and $P{D}_n^{m}R$ controllers on the basis of MRDP has been considered in [17,25]. For a nominal tuning, in which the indices “p” can be omitted, the closed control loop with (7) and (9) yields

$$\begin{array}{c}{F}_{wy}\left(s\right)={\displaystyle \frac{Y\left(s\right)}{W\left(s\right)}}={\displaystyle \frac{K_c{K}_s(1+T_{i}s)(1+T_{1}s+\cdots +T_m{s}^m)}{T_i{s}^2{e}^{T_{d}s}+K_c{K}_s(1+s{T}_i)(1+T_{1}s+\cdots +T_m{s}^m)}}\hfill \\ F_{dy}\left(s\right)={\displaystyle \frac{Y\left(s\right)}{D_i\left(s\right)}}={\displaystyle \frac{K_s{T}_{i}s}{T_i{s}^2{e}^{T_{d}s}+K_c{K}_s(1+s{T}_i)(1+T_{1}s+\cdots +T_m{s}^m)}}\hfill \end{array}$$

(10)

The parameters of the optimal controller (7), including the reset time constant $T_i$, are listed for odd integers $m\in [1,7]$ in the Appendix A.

The filter dynamics can be included in the overall tuning of the controller in two ways:

• Implicitly—a filter with a sufficiently high relative degree $n-m\ge 0$ can be approximated together with the controlled process by the parameters of the IPDT model (9) (see, e.g., [13,23]). The design then starts with $T_d=T_{dp}$;
• Explicitly—the equivalent delay $T_e$ of the filter can be added to the delay of the identified process model $T_{dp}$ by setting the total dead time of the loop as follows

  $$T_d=T_{dp}+T_e;T_f=T_e/n.$$

  (11)

However, without the use of continuous-time simulation blocks in, e.g., Matlab/Simulink, quasi-continuous methods must be used for the numerical implementation of the filter. In this way, it is possible to avoid sampling zeros that occur in the discrete-time transfer functions of higher-order filters calculated with zero-order hold equivalents [27].

2.1. IPDT Approximation of Non-Minimum Phase Process

An even simpler identification method is presented here, based on the approximation of the non-minimum phase process by the IPDT model. This method was inspired by the fact that simple PI or PID controllers are sufficient to control the given process [8].

A direct specification of parameters of HO-PID controller (7) would allow control of a slightly broader class of processes than when they are specified by HO-ARCs. However, although the MRDP method [25] can be used for the direct specification of the parameters of parallel HO-PIDs, it is more often combined with heuristic optimization carried out for the given process transfer function. In each case, however, an additional design of suitable anti-windup measures is required. In contrast to this, as we will show in this article, for the model-based design of HO-ARCs, it is enough to use an analytical MRDP method based on the IPDT process model, which is supplemented with simple restrictions for the design of filter (8). Therefore, the first task is to find a suitable IPDT model approximation for the measured open-loop step response of a given non-minimum phase process.

The IPDT approximation of the non-minimum process can be derived by finding the time at which its step response characteristic crosses the initial (zero) value (see Figure 2). This value can be interpreted as the dead time of the IPDT model $T_{dp}$. The slope of the characteristic curve at this intersection point can then be used to determine the IPDT parameter $K_{sp}$. For example, the step response of the selected $S\left(s\right)$ in Figure 2 can be used to obtain its approximation

$$S\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=2\frac{-s+1}{{(2s+1)}^2};\mathrm{IPDT}:T_{dp}=1.526,K_{sp}=0.37$$

(12)

For the above transfer function $S\left(s\right)$ and the selected sensitivity value $M_s=2$, the alternative tuning method [8] results in the following filtered PID controller (4):

$$K_P=0.9497;T_i=4.8853;T_D=1.7785;T_f=1.8567;\beta =0.4050$$

(13)

2.2. Closed-Loop Responses Based on the IPDT Model of the SOPDTUZ Process

The setpoint and disturbance step responses corresponding to $P{D}_n^{m}R$ controllers (7) and (8) implemented according to Figure 1, with odd $m\in [1,7]$, are shown in Figure 3. The controller parameters are based on the IPDT model of process (12) using different values of tuning parameter $T_e$ (11) and setpoint prefilter (A7) with $p\in [0,2]$. For simplicity, these responses for $m>1$ do not cover the full set of possible prefilter options. To remain simple, the introductory analysis also avoids the control constraints, which were chosen to be sufficiently high ($u\in [-10,10]$). As a preliminary, the amplitude of the measurement noise was set to zero.

In addition to the output and input of the process, the reconstructed disturbance signal $d_{irec}$, which is given by the negative value of the controller feedback from $F_R$, is also recorded.

For the setpoint step responses with the unit step setpoint $w=1$ and the external disturbance $d_i=0$, the controller steady-state value corresponding to process (12) is $u(\infty )=-1/S\left(0\right)=-0.5$. Since the internal feedback of the SOPDTUZ process is not taken into account by the IPDT model, it forms the lumped disturbance together with the external disturbances (as with ADRC). As the steady-state value of the reconstructed disturbance $d_{irec}$ shows, it assumes a non-zero value during the setpoint change, although the actual external disturbance is zero. The values of the actual and the reconstructed disturbance are only equal ($d_{irec}=d_i=1$) near to steady state at $w=0$.

For a large $T_e=2T_{dp}$ (top, Figure 3), it can be seen that all processes are sufficiently smooth despite the extremely simple IPDT approximation of the process. The speed of the $P{D}_m^{m}R$ response increases and becomes slightly more oscillatory with increasing controller order m.

For $T_e=T_{dp}$ (center, Figure 3), the control signal of the $P{D}_7^{7}R$ controller shows some spikes thanks to the “aggressive” setting. This shows that the tuning parameter $T_{em}$ of the individual $P{D}_n^{m}R$ controller cannot be arbitrarily small with regard to the stability of closed loops (previously analyzed for an IPDT system and $m\in [0,2]$ in [28]).

For the even more aggressive setting with $T_e=0.5T_{dp}$ (bottom, Figure 3), the responses corresponding to $m=5$ and $m=7$ are already unstable.

The closed-loop results for all setpoint responses with the simplest prefilter numerator $p=0$ and $N_p=1$ are significantly slower compared to the filtered PID according to [8]. As the value of $p\in [0,2]$ increases, the speed of the setpoint step responses of the $P{D}_m^{m}R$ controllers increases. It must be mentioned that not all possible prefilters for $m>1$ are shown for reasons of clarity of the images. It should also be noted that all process outputs for setpoint step responses of $P{D}_m^{m}R$ controllers are strictly monotonic, while [8] provides slight overshoots. This overshoot also contributes to the fact that [8] provides faster setpoint closed-loop responses.

The $P{D}_n^{m}R$ controllers provide more efficient disturbance rejection at $T_e\le T_{dp}$ than [8]. For $T_e=2T_{dp}$, the comparison is not clear and requires a more detailed analysis.

Due to higher values of n and $T_e$, one can intuitively expect, even without a more detailed quantitative evaluation of the obtained waveforms, that controllers corresponding to higher values of m can filter the measurement noise more intensively. This is, of course, in contrast to the widely held belief that the use of controllers with a derivative component is not suitable for noisy systems [15,29]. The following evaluation will confirm this intuitive expectation and show that $m=1$ can lead to over-damped responses when the measurement signals need to be filtered more intensively. In other words, this article will extend and complement the conclusions in [8] regarding the comparison of PI and PID control with that of HO-ARCs. In the next section, a more detailed quantitative evaluation of each controller using appropriate performance measures and measurement noise will be shown.

3. Speed–Effort and Speed–Wobbling Characteristics

The design of HO-AR controllers offers several possibilities for improving the setpoint responses, depending on the number of poles p that are canceled by the prefilter numerator $N_p\left(s\right)$ (see Appendix A.1). For $m=7$, for example, the pre-filter denominator degree $deg\left(D_p\right)$ formally allows up to $p=8$ dominant poles to be canceled by the pre-filter numerator $N_p$ (A7). However, a consistent evaluation of all possible prefilters and controllers with odd $m\in [1,7]$ would take a lot of time and space. For the sake of brevity, we leave that to a separate paper and limit ourselves here to disturbance of measurement noise on controller tuning on disturbance responses. Note that the uncertainties in the models used for the controller design have a similar impact as external disturbances and good disturbance rejection is the first requirement for the robustness of the proposed controller design. Therefore, the following simulation analysis is based on the evaluation of disturbance responses as a function of the basic tuning parameters, i.e., the derivative degree m, the equivalent delay $T_e$, and the order n of the low-pass filter $Q_n\left(s\right)$.

The speed–effort (SE) and speed–wobbling (SW) characteristics introduced in [25,30] relate the speed of the closed-loop responses (expressed here in terms of $IA{E}_d$) to excessive control effort and excessive process output wobbling. They are used for detailed analysis of the tuning of higher-order $P{D}_n^{m}R$ tuning and for comparison with the filtered PID controller (4).

The SE and SW characteristics of the disturbance responses are evaluated in the $\xi ,\eta$ planes, which are labeled with the coordinates

$$\begin{array}{c}\mathrm{SE}:\xi =T{V}_1\left(u_d\right),\eta =IA{E}_d;\hfill \\ \mathrm{SW}:\xi =T{V}_1\left(y_d\right),\eta =IA{E}_d.\hfill \end{array}$$

(14)

The integral of the absolute error ($IA{E}_d$) is defined as

$$IA{E}_d={\int }_0^{\infty }\left|e_d\left(t\right)\right|dt;e_d=w_d-y_d,$$

(15)

where $w_d=0$ is the desired reference setpoint, $y_d$ is the process output, and $e_d$ is the control error.

$T{V}_1\left(y_d\right)$ and $T{V}_1\left(u_d\right)$ represent the deviations of the process output and input from the ideal disturbance step responses of the integral first-order plants with a one-pulse (1P) shape. Such responses consist of two monotonic intervals separated by an extreme point $y_m\notin (y_0,y_{\infty })$, or $u_m\notin (u_0,u_{\infty })$

$$\begin{array}{c}T{V}_1\left(y_d\right)=\sum _i\left|y_{i+1}-y_i\right|-\left|2y_m-y_{\infty }-y_0\right|;\hfill \\ T{V}_1\left(u_d\right)=\sum _i\left|u_{i+1}-u_i\right|-\left|2u_m-u_{\infty }-u_0\right|\hfill \end{array}$$

(16)

The evaluation of the responses was based on performance measurements obtained with the simulation time $t_{sim}=100$. The length of the simulation time was chosen with respect to the slowest responses corresponding to $m=1$ and $T_e=2$.

Stable (non-distorted) disturbance step responses are obtained by using the following parameters $T_e$ e:

$$\begin{array}{c}m=1,T_{e1}\in [0.1,2.0];\Delta T_{e1}=0.1\hfill \\ m=3,T_{e3}=T_{e1}+0.4;\hfill \\ m=5,T_{e5}=T_{e1}+0.9;\hfill \\ m=7,T_{e7}=T_{e1}+1.5.\hfill \end{array}$$

(17)

with three different filter degrees. In the following figures, $m=1$ corresponds to the black, $m=3$ to the blue, $m=5$ to the green, and $m=7$ to the red curves in combination with the filter orders $n=m$ (full), $n=m+1$ (dashed), and $n=m+2$ (dotted). The special range of values in (17) is explained in more detail in the following section.

For each controller order m, 20 different control loop responses with gradually increasing $T_e$ are measured. From the loop responses of the $P{D}_1^{1}R$ controller (equivalent to the PID controller) in the control loop where the particular values of $T_e$ are given by the vector $T_{e1}$ (17) (see Figure 4), it can be seen that the responses are accelerated by decreasing the parameter $T_e$. However, the oscillations of the process inputs and outputs increase. It is therefore not easy to find the optimum value of $T_{em}$ for individual values of m.

To illustrate the effects of noise in the controller design, we will first evaluate the SE and SW characteristics without measurement noise (Figure 5). It can be seen that as the filter degree n increases, the excessive increments of the process input and output signals increase. The reason for this is that the numerical integration used in computer simulations is never perfect. By choosing a sufficiently fine step and low permissible simulation errors, we try to reduce the numerical errors below the limit that would lead to a loss of significance of the observed transients. Increasing the derivative degree m increases the excessive control effort and the closed loop speed ($IA{E}_d$ decreases). However, increasing m also reduces the excessive wobbling in process output. The performance metrics of the filtered PID controller (4) and (13), marked with a “+” sign, show that the excessive control effort at the controller output is lower than when using $P{D}_n^{m}R$ controllers. This confirms the first visual impression from the responses in Figure 3.

Next, to illustrate the control problem in the POCD context with non-zero measurement noise, the SW and SE characteristics in Figure 6 are calculated at “smaller” ($\left|\delta \right|\le 0.05$) and “larger” ($\left|\delta \right|\le 0.5$) amplitudes of measurement noise. This noise was generated in Matlab/Simulink using the ”Uniform Random Number” block with a sampling period $T_s=0.001$.

The amplitudes of the applied noise were chosen with respect to the output amplitudes of the analyzed disturbance responses in Figure 4 and correspond to about 4% and 40% of the output amplitude. Although the adjectives “smaller” and “larger” have limited validity from the point of view of generality, the SE and SW characteristics obtained by considering non-zero amplitudes of the noise already point to generally valid conclusions: The excessive increments of the heuristically optimized PID controller are lower than in the case of equally fast $P{D}_m^{m}R$ control, but higher than in the case of $P{D}_n^{m}R$ controllers with $n>m$. The given conclusion would probably be valid until the level of external noise is reduced to the level of internal noise by numerical integration.

For both noise levels considered, the detailed quantitative evaluation of the responses obtained clearly shows the advantages of the new controller design method. The analysis shows that, compared to the PID controller (4), the measurement noise can be significantly reduced by using $P{D}_n^{m}R$ controllers with $n>m$, which enables a clear horizontal shift of the performance points to the left (by several decades of the logarithmic scale). Alternatively, by keeping a constant excessive amount, the performance point can be shifted downwards to lower $IA{E}_d$ values.

According to the SE and SW characteristics in Figure 5 (without measurement noise), the use of HO low-pass filters with $n>m$ leads to an increase in the excessive deviations of the process output from the 1P shape. However, in circuits with $m>1$, even with a relatively small measurement noise (Figure 6, top), the filters with an increased order n lead to a significant reduction in redundant increments at the process input and output. The positive effect of the increased filter order on the excess deviations is even clearer with larger noise amplitudes (Figure 6, bottom).

In the presence of the higher level of measurement noise, filtration made it possible to reduce the excess controller effort $T{V}_1\left(u_d\right)$ when using $P{D}_9^{7}R$ with the same value $IA{E}_d=6$ as in [8] more than 1000 times, or reduce $T{V}_1\left(u_d\right)$ approximately 400 times and at the same time, reduce the value of $IA{E}_d$ by approximately 1/3 third to the level of $IA{E}_d\approx 4.35$. For higher measurement noise, the reduction in excess controller effort $T{V}_1\left(u_d\right)$ due to filtering can be significant when using $P{D}_9^{7}R$ with the same value $IA{E}_d=6$ as in [8]. At the same time, the $T{V}_1\left(u_d\right)$ can be reduced by a factor of about 400 and the value of $IA{E}_d$ by a factor of about 3 (to $IA{E}_d\approx 4.35$). The adverse effects of measurement noise in practice are many—from the generation of heat losses as a result of excess activity of the actuator, through wear and tear of equipment to unwanted vibrations, acoustic noise disturbing the surroundings, violation of technological tolerances, etc. Oscillations caused by the noise could, for example, even lead to the destruction of valves used as one of the most common actuators in process control.

4. Linear and Constrained Controller Design

Stability and Absolute Stability of the Closed Loop with MRDP Controller

The selection of the controller’s low-pass filters is usually neglected when designing and tuning PID controllers. By default, such a filter is only present in the derivative term, which is usually realized by the following relation: $s{T}_D/(s{T}_D/N+1)$, where $N\approx$ 5–100 [31]. As mentioned above, the generalization of such an approach to the design of PIDA or higher-order controllers can be difficult. Designing a separate filter for each of the derivative components might be inefficient due to the complex computations involved. Another issue regarding the effectiveness of separate filters arises from noise attenuation, as filters with a higher-order denominator prove to be more effective. An important consideration for $m>0$ and $n\ge m$ is also the choice of $T_e$ (or $T_f=T_e/n$), which guarantees the stability of the closed loop.

The control signal constraints and anti-windup protection can have a significant impact on the closed-loop performance when the control signals are limited [25]. Therefore, consideration of the constraints deserves due attention when designing the controller. This can be done using the circle criterion of absolute stability [9,32,33]. Here, the $P{D}_n^{m}R$ controller from Figure 1 must be converted into the standard non-linear form, which expresses deviations from a required steady state and is composed of the saturation non-linearity and the associated linear loop blocks [17,18]

$$L_s\left(s\right)=P{D}_n^m\left(s\right)S\left(s\right)-{\displaystyle \frac{1}{1+T_{i}s}}$$

(18)

The use of HO-ARCs on IPDT models in [17,18] has shown that their MRDP design for odd m directly satisfies the conditions for absolute stability. However, for $m>1$, this is only true for suitably chosen $T_e$. Figure 7 shows the Nyquist curves $L_s\left(s\right)$ corresponding to $S\left(s\right)$ (9) drawn for the controller parameters corresponding to the IPDT model of the SOPDTUZ process (12) for the set of $T_{em}$ values

$$\begin{array}{c}m=1,T_{e1}=0.1;\hfill \\ m=3,T_{e3}=0.3;\hfill \\ m=5,T_{e5}=0.9;\hfill \\ m=7,T_{e7}=1.4.\hfill \end{array}$$

(19)

These curves do not cross the critical line representing the critical circle (CC) limit case with the center at minus infinity, so they guarantee the absolute stability of the considered circuit. A further reduction in some $T_{em}$ values in order to speed up the responses would lead to the intersection of the corresponding Nyquist curve with the vertical line with the real value −1 and thus to a deterioration of the control performance.

The Nyquist curves in Figure 8 have the calculated parameters of the $P{D}_n^{m}R$ controllers when considering the original SOPDTUZ process (12). They fulfil the absolute stability conditions for a slightly increased set of values $T_{em}$

$$\begin{array}{c}m=1,T_{e1min}=0.1;\hfill \\ m=3,T_{e3min}=0.5;\hfill \\ m=5,T_{e5min}=1.0;\hfill \\ m=7,T_{e7min}=1.6.\hfill \end{array}$$

(20)

which correspond to the minimum values considered in (17). In Figure 9, this set of $T_e$ values is illustrated by responses in the time domain. With the exception of $T_e=0.1$ for $m=1$, the significant difference between the actual SOPDTUZ process and the IPDT model used to design the $P{D}_n^{m}R$ controller is compensated by slowing down the transients (increasing the equivalent delay $T_{em}$).

The choice of the minimum value $T_e=0.1$ for $m=1$ was motivated by the analysis of the SE and SW characteristics in Figure 6, which correspond to a circuit with non-zero measurement noise. Although the $P{D}_1^{1}R$ controller guarantees absolute stability conditions also for lower values of $T_e$, it is not necessary to reduce them, since the SE and SW curves corresponding to equally low $IA{E}_d$ values and $m>1$ already give lower values of excess increments. It turns out that for $m>1$ the circle criterion, which is a generalization of the Nyquist criterion for circuits with a sector non-linearity and a linear part $L_s\left(s\right)$, can also be useful in the design of HO-AR controllers. According to the Nyquist curves in Figure 8, the circuit fulfils the stability conditions when the Nyquist curve passes the real axis on the right side of the critical point $(-1,0j)$. The conditions of absolute stability are fulfilled if the Nyquist curve lies on the right-hand side of the vertical line with the real value −1.

A non-linearity $u=F\left(\tilde{e}\right)$ is said to satisfy the sector conditions $[\alpha ,\beta ]$ if it satisfies the conditions $\alpha \le F\left(\tilde{e}\right)/\tilde{e}\le \beta$. While the position of the Nyquist curve is evaluated according to the critical point $(-1,0j)$, the circle criterion replaces this with a critical circle (CC), which is defined by the vertices $-1/\alpha ,-1/\beta$ (lying on the real axis and related to the sector $[\alpha ,\beta ]$). As already mentioned, the control loop is stable if the linear part of $L_s\left(s\right)$ crosses the real axis on the right-hand side of the critical point. To guarantee absolute stability, the same requirement applies to the critical circle and the Nyquist curve cannot even pass through CC. Absolute stability guarantees that there is a $c>0$ and $\gamma >0$, so that each transient of the system fulfils the requirement of a monotonic decrease in the equivalent deviation $\tilde{e}$

$$|\tilde{e}\left(t\right)|\le c{e}^{-\gamma t}\left|\tilde{e}\left(0\right)\right|,\forall t>0.$$

(21)

Such a monotonic decrease can also be used for tuning the loop performance, e.g., to determine the minimum values of the tuning parameter $T_{em}$ for unconstrained control.

The considered saturation non-linearity fulfills the sector conditions $[0,1]$. Such a sector corresponds to the critical circle defined by the points $-1/\alpha =-\infty$ and $-1/\beta =-1$ of the real axis, with a radius $R=|\alpha -\beta |/2\to \infty$. Such a limit-critical boundary is a vertical line that intersects the point $-1$ of the real axis (Figure 8).

For $m=3$ and $m=7$, the nominal Nyquist curves (solid lines) lie on the right-hand side of CC. For $m=5$, the Nyquist curve crosses it. Therefore, for $m=5$, the only permissible sector is $[ϵ,1]$, where $tan\left(ϵ\right)=1/{\tilde{e}}_0$ limits the maximum permissible value of the deviation ${\tilde{e}}_0$. The limit of absolute stability CC should be replaced by a smaller circle with the vertices $-1/ϵ$ and $-1$.

The conclusions regarding the absolute stability are relatively accurate if the smoothness of the process output and process input curves is taken into account. This is illustrated in Figure 9 by the responses obtained for $m=7$ and $T_e=1.6$. For $T_e=1.5$, the process input is already oscillating. The same can be observed for $m=5,T_e=1$. Since the corresponding Nyquist curve crosses the CC slightly, the corresponding process input signal already shows some higher harmonics. Thus, the results obtained by using the circle criterion of absolute stability can be considered as a useful tool in the search for the tuning parameter $T_e$ of HO-AR controllers for unconstrained control. Since distorted signals can also occur in other works (see, e.g., [34]), checking the settings of the proposed controllers using absolute stability could also be useful in other contexts.

Finally, Figure 10 shows a comparison of disturbance step responses of $P{D}_n^{m}R$ controllers to the constrained setpoint and the disturbance corresponding to the minimum values of the tuning parameter $T_e$. With the given quantization $\Delta T_e=0.1$, all controllers appear to be approximately equal for $m\ge 1$. Although prefilter tuning with $p=1$ (dotted line) improves the performance of the $P{D}_3^{1}R$ controllers in setpoint tracking, the HO controllers ($m>1$) showed significantly better performance in disturbance rejection. When analyzing the relationship between performance and complexity using the POCD method, the $P{D}_5^{3}R$ controller appears to reach the sweet spot. The distortion in the control signal due to higher harmonics is significantly lower than with linear control due to the limitation. The maximum amplitude of the output deviation is significantly lower with HO controllers.

5. Discussion

The evaluation of the filtered PID controller [8] and $P{D}_n^{m}R$ controllers depends on whether the measurement noise is taken into account. For example, the evaluation according to Figure 3 and Figure 5 (without measurement noise) would be completely different from the evaluation according to Figure 6 (with measurement noise). The latter results are closer to the POCD requirements as they correspond better to reality. Realistic evaluation is very important for controller design, so we briefly summarize the changes introduced by this article:

• Instead of parallel filtered PID controllers based on heuristic optimization and knowledge of the exact process model [8], the article proposes the use of HO-AR controllers with odd values of the derivative degree $m\in [1,7]$, whose parameters are tuned based on a simplified IPDT process model;
• Instead of $n=m=1$ in the original article [8], not allowing for modification of the noise attenuation, it is recommended that $n>m>1$ (ideally $m=3$ and if possible at least $n=m+2$) be used to better attenuate the measurement noise;
• A substantial increase in control performance (reduction in excessive controller effort in the range of ${10}^3$ and also reduction in the output wobbling) can be achieved by choosing $m\ge 3$;
• While [8] optimized the total variation $T{V}_u$ of the control signal ${\sum }_i\left|u_{i+1}-u_i\right|$ considering both the setpoint and disturbance responses, the present article only considers the excessive increments in the process input and output signals (16) during disturbance step changes;
• From the point of view of the POCD, no further optimization of the $P{D}_n^{m}R$ controller is required apart from the selection of the tuning parameters m, $T_{em}$, and n;
• The structure of the HO-AR controllers used is based on the IPDT process model [17,35]. Therefore, the step response of the given non-minimum phase process is directly approximated by the IPDT model, which greatly simplifies the experimental identification of the process;
• Thanks to the nature of such an IPDT approximation of the measured step response and the controller design used, the whole procedure can be described as a generalization of the Ziegler and Nichols method [12] for HO-ARCs and inverse response processes;
• By using ultra-local integral models, the presented controller design is similar to some alternative approaches, such as active disturbance rejection control (ADRC) [36,37,38,39,40,41,42] or model-free control (MFC) [43,44], which differ mainly in the type of disturbance observer used;
• An additional advantage of the new approach is that HO-AR controllers offer easier adaptation to technological requirements, allow reconstruction of input disturbance, ensure high-quality transients even in systems with limited control signal, and thus, better fulfil all POCD requirements.

6. Conclusions

In this paper, the control performance of the SOPDTUZ process using a PID controller design based on heuristic optimization [8] was compared with model-based HO-ARCs (with an odd number of derivative terms $m\in [1,7]$) using a simple IPDT process approximation, tuned by the MRDP design and the circle criterion of absolute stability. The comparison was based on the aspects of practice-oriented controller design (POCD) outlined in Definition 1.

The previous results based on the exact process model in [8] complemented by using generalized automatic reset controllers tuned to an IPDT process model showed that the performance can still be significantly increased by using $P{D}_n^{m}R$ controllers with derivative degree $m>1$ and filter order $n=m+2$. Of course, sufficient noise attenuation can only be achieved if the sampling period $T_s$ is chosen to be sufficiently smaller than the filter time constant $T_f$ ($T_f>>T_s$). A significant improvement was already achieved with $m=3$, while a further increase in m no longer brought any visible improvement. Similar results showing that increasing the controller and filter orders improve performance only up to a certain level have also been reported in [45]. Therefore, in future work, it will also be interesting to evaluate the performance when using controllers corresponding to the neighboring even values $m=2$ and $m=4$.

An evaluation of MRDP-tuned HO-ARCs by absolute-stability-based analysis showed that as the order of the derivatives m increases, the equivalent delay ($T_e$) must also be increased in order to maintain the stability of the circuit. This result, together with the relationship $n\ge m$, leads to an increase in the filter order and thus to an additional attenuation of the measurement noise. This refutes the long-held doctrine that the use of derivative controllers is unsuitable for controlling noisy processes.

In future work, we intend to focus on a more detailed evaluation of the setpoint step responses, the limitation of the process undershoot amplitude, and on controllers with even derivative degrees m.