A comparative study on time-delay estimation for time-delay nonlinear system control

1.   Introduction

In the domain of automation and teleoperation systems, time delays and uncertainties are significantly associated with inefficient operation of the overall control system ^[1,2,3]. This is mainly due to factors like time delay, hysteresis, parameter perturbation, external disturbances, and noise, which are inevitable in real-world system applications. Moreover, besides external output disturbances acting on a system, uncertainties like nonlinear backlash-like hysteresis, nonlinear components' interaction with their surroundings, and time delay persist in communication channels for a controlled object/plant, causing nonlinearity in a control system. Hence, this requires designing and improving robust disturbance rejection controllers with a time-delay estimation (TDE) mechanism to obtain stable and steady-state transient system responses.

Several model-based and model-free control approaches have been developed over time. The former is affected by the system's parameter uncertainty and unmodelled dynamics. In contrast, the latter considers a variety of system uncertainties, including internal and external disturbances, nonlinear hysteresis, sensor noise, and varying time delay ^[4,5,6].

In the literature, one of the most widely employed model-free controllers is an active disturbance rejection control (ADRC), whose model is updated continually to perform disturbance compensation in time-delay control ^[7,8]. Han first developed the ADRC in the mid-1990s ^[9], followed by its application by Gao ^[7]. The characteristic feature of the ADRC controller is its ability to measure internal and external disturbances actively, generally coined as "total disturbance", and then compensate for these total disturbance effects in real-time via the feedback loop. This is accomplished with an extended state observer (ESO), an integral feature of the ADRC controller ^[7].

Further, the time delay aspect is the most crucial aspect that greatly impairs the performance of the controlled system. Impairments include response overshooting with oscillations, reduced damping ratio, and moving the closed-loop system response to instability. Thus, controller design becomes more challenging due to the time delay and the different forms of disturbances to the system, which need to be controlled. Therefore, to mitigate the effects of different types of uncertainties acting on a concerned system, the ESO component has been amended over time to accommodate the ADRC operation in industrial applications; for example, in robotic systems ^[10], power plants ^[11], teleoperation systems ^[12], and piezoelectric-related medical fields ^[13]. Such modifications in the control law helped attain a decent trade-off between disturbance rejection and controller operation in a concerned time-delayed system.

Furthermore, another widely used and established industrial controller, called the proportional-integral-derivative (PID) controller, has evolved over time. The PID controller depicts decent accomplishment in overcoming various nonlinearities, such as the chattering effect, delay, and external disturbances present in robotic industries ^[14]. However, despite consistent efforts to conquer the time-delay problem, the adverse effects of delay are still difficult to overcome. Thus, time-delay estimation followed by compensation is still a work in progress for various disturbance rejection control methods ^[15,16]. Moreover, recent research has shown that the performance of an ADRC controller concerning estimation and attenuation of nonlinear backlash-like hysteresis under time delay and uncertainties is greatly improved in conjunction with TDE mechanisms ^[8,17]. Other controller methods commonly used in integration with the TDE mechanism for attenuation of system nonlinearities include, for example, the PID controller ^[18], adaptive sliding mode controller (SMC) ^[19], and fuzzy logic approach-based ^[20] control methods. These methods are reviewed in the following subsection.

1.1.   Literature survey

As an estimation technique, TDE mainly estimates the unknown dynamics of the controlled system by using the system's delayed dynamics. Usually, the TDE method is used in conjunction with other control strategies to attain the necessary uncertainty compensation and enhanced system transient steady-state performance. Some examples from the literature include TDE with PID control, ADRC-based control, SMC, and fuzzy-based control.

One of the TDE-PID approaches in ^[18] provides a unified TDE-based PID-nonsingular terminal SMC that takes advantage of both PID characteristics and the terminal SMC (TSMC) to manage cable-driven manipulators with matching disturbance. However, it was recommended that the mismatched disturbance be studied. Further, to estimate and counteract nonlinearity, such as friction and external system disturbance in a robotic manipulator, a modified TDE method was constructed by combining a fractional-order TSMC with PID ^[21]. It was recommended that future work address varying system time delays.

Furthermore, to supervise robotic manipulators impacted by hysteresis and external system disturbances, a robust controller was designed by combining the TDE with a nonsingular PID and a fast TSMC ^[22]. This approach used the fuzzy system to regulate PID gains. Future research will enhance the system's performance when dead-zone nonlinearity and applied measurement noise are present. Moreover, to counteract the adverse effects of complex lumped uncertainties and backlash hysteresis on system performance, an adaptive robust TSMC approach employing the TDE method has been developed for robotic manipulators ^[23], followed by multivariable fast TSMC for cable-driven manipulators ^[24]. Future research will examine chattering effects, dead zones, saturation, and noise reduction.

In another study ^[25], an ADRC predicted and attenuated the nonlinearities in an all-clamped plate system by applying the TDE definition to the ESO in the frequency domain. However, future research will analyze the external disturbances impacting the controlled object. Therefore, in most TDE-based methods, it has been observed that the stability and performance of the controller design are greatly impaired under the influence of external disturbances and varying system time-delay circumstances. Hence, in this paper, a comparative analysis is performed in terms of estimation and compensation of uncertainties based on experimental responses and performance measurement criteria on recently proposed control methods, namely the predictive TDE-based ADRC method ^[8] and a TDE-based PID control method ^[17], which will be employed to control a given system termed the controlled object.

2.   Methodologies used in the controller designs

This section describes the preliminary mathematical concepts used to design the complete TDE-based predictive ESO-based ADRC and the TDE-based PID controllers. These are experimentally analyzed under different case studies in Section 3.

2.1.   System or controlled object description

For simplicity and reliability, most controllers focus on lower-order, simpler models for analysis, as higher-order systems are frequently examined using lower-order ones (usually the second-order approximation) in control theory. The second-order system used in this analysis is represented by the state-space equation given by Equation (1).

where the system state vector is given by $x\left(t\right) = [{{x}_{1}\left(t\right), {x}_{2}\left(t\right)]}^{T}$, and the system output is $y\left(t\right)$. Matrix $A$ is the state transition, $B$ is the input, and $C$ is the output matrix of the concerned system/plant. The dimensions of these matrices are given by $Aϵ{\mathbb{R}}^{(n\times n)}$, $Bϵ{\mathbb{R}}^{(n\times 1)}$, and $Cϵ{\mathbb{R}}^{(1\times n)}$. $n = 2$ for the system shown in Equation (1). The system time delay is indicated by $\tau$, and the control signal to the system is given by $u\left(t\right)$. $f\left(x\left(t\right), t\right)$ is the total disturbance acting on the system, which includes input and output disturbances, nonlinear backlash-like hysteresis, and unknown system dynamics.

2.2.   Time-delay estimation mechanism

TDE is a method for estimating system disturbance that uses time-delayed input from the preceding sampling event. The sampling period, represented by ${t}_{s}$ in Equations (2) and (3), is a small delay that the TDE technique introduces when constructing a controller. Consequently, the TDE reduces the total effect of cumulative disturbances. Equation (3) presents the TDE law, as shown in Figure 1.

where ($t-{t}_{s}$) represents the time-delayed value, and $\widehat{\zeta }\left(t\right)$ indicates the estimate of all uncertainties. $\ddot{y}\left(t-{t}_{s}\right)$ represents the dynamics of the delayed system output. The parameter $\eta$, which is a positive nominal value, is adjusted to obtain the best feasible controller performance. Furthermore, when the sample period ${t}_{s}$ decreases, the estimate $\widehat{\zeta }\left(t\right)$ gets stronger as the TDE can operate effectively, according to Equation (3). Moreover, for robotic manipulators, $\eta$ is given by a diagonal matrix ^[26].

Hence, the proposed hypothesis in this study utilizes the TDE approach with an additional control method to achieve the required performance and steady-state transient system response. Thus, in this comparative study, it will be demonstrated that the robustness of the TDE mechanism is advantageous when combined with the predictive ESO-based ADRC and PID controllers individually by combating various system nonlinear uncertainties impacting the time-delay system and, therefore, will improve the controlled system's quality.

2.3.   Hysteresis model: Duhem backlash-like hysteresis model

The hysteresis effect is widely acknowledged to affect the precision of both closed-loop and open-loop control systems and lead to persistent oscillations. This highlights how a change in the direction of an input signal can alter the response of an output signal, thus introducing nonlinear behavior in control systems theory ^[27,28]. Therefore, in control systems, hysteresis can result in oscillation, instability, and incorrect positioning. Hence, it may prove challenging to develop controllers that can precisely follow the intended trajectory due to nonlinear behavior. The Duhem model given in ^[8,26] contributes to dynamic hysteresis in this study. Equations (4) and (5) are the foundation for the mathematical description of the Duhem backlash-like hysteresis framework. Constants $\alpha$, $\beta$, and $\gamma$ correspond to hysteresis constants and are greater than 0; also, $\alpha > \beta$.

wherein the initial conditions of the input $v$ and output $w$ are ${v}_{0}$ and ${w}_{0}$, respectively. For an input signal $v\left(t\right) = \rho \mathrm{sin}(2.5t)$, the response of the Duhem hysteresis curve in Figure 2 is generated for $\rho$ = 2 using the model parameter values $\gamma$ = 1, $\alpha$ = 1.16, and $\beta$ = 0.35. Also, the initial values ${w}_{0}$ and ${v}_{0}$ are set to zero.

2.4.   ADRC with a delayed input system

An ADRC consists of three primary subsystems: a tracking differentiator (TD), a nonlinear state error feedback (NLSEF) controller, and the extended state observer (ESO) ^[7]. The block diagram of a basic ADRC structure with delayed input is given in Figure 3 ^[29]. The ESO is a fundamental element of the ADRC that actively estimates and compensates for the overall disturbance $f\left(t\right)$ imposed on the controlled object given by Equation (1).

In Figure 3, the TD produces a softer signal with an estimated derivative signal. By extracting the difference signals ${e}_{1}$ and ${e}_{2}$ from the TD-generated signals ${v}_{1}\left(t\right)$ and ${v}_{2}\left(t\right)$ and the ESO-generated estimated system states ${z}_{1}$ and ${z}_{2}$, the NLSEF produces the control law denoted by ${u}_{0}$.

The TD and NLSEF equations for Figure 3 are given by Equations (6), (7), and (8).

where ${v}_{1}\left(t\right)$ and ${v}_{2}\left(t\right)$ are softened and differentiated signals obtained by supplying TD with the reference signal $v\left(t\right)$. Thus, TD improves the reference input reactivity while producing the intended transient profile. By setting ${r}_{0}$ to $r$ and applying the nonlinear function $\mathrm{f}\mathrm{h}\mathrm{a}\mathrm{n}({v}_{1}-r, {v}_{2}, {r}_{0}, h)$, which is defined by a series of relations in Equation (8), the fastest convergence from ${v}_{1}$ to $v$ with the least amount of overshoot is achieved ^[7]. The TD parameters denote the simulation step and the transition process speed ${r}_{0}$ and $h$, respectively. The nonlinear function $\mathrm{f}\mathrm{h}\mathrm{a}\mathrm{n}$ has equations in stages that are provided in ^[7,30], with the NLSEF's signal output ${u}_{o}$.

Equation (7) displays the NLSEF parameters, given by control gain $r$, simulation step $c$, and damping coefficient ${h}_{1}$. The optimal controller response is achieved by adjusting these NLSEF controller parameters.

For Figure 3, the ESO equations and the control law are stated in Equations (9) and (10), respectively ^[30].

where the ESO gains are given by ${\beta }_{01}$, ${\beta }_{02}$, and ${\beta }_{03}$. Parameters ${b}_{0}$ and ${g}_{0}$ are additional gain factors, whereas ${b}_{0\mathrm{D}\mathrm{c}}$ represents the disturbance compensation factor. The nonlinear feedback functions $\mathrm{f}\mathrm{e}$ and ${\mathrm{f}\mathrm{e}}_{1}$ are defined as $\mathrm{f}\mathrm{e} = \mathrm{f}\mathrm{a}\mathrm{l}(e, 0.5, {h}_{ESO})$ and ${\mathrm{f}\mathrm{e}}_{1} = \mathrm{f}\mathrm{a}\mathrm{l}\left(e, 0.25, {h}_{ESO}\right)$, respectively. For $\alpha$ < 1, the error ($e$) will decrease to zero more quickly. The simulation step size of ESO is denoted by ${h}_{ESO}$. The complete estimated disturbance signal is denoted by ${z}_{3}$. The controlled object (system) has ${u}_{a}$ as its control signal and $y$ as its output signal. Equation (11) provides the nonlinear function $\mathrm{f}\mathrm{a}\mathrm{l}$ value ^[30].

2.4.1.   Block diagram of TDE integration with predictive ESO-based ADRC controller (TDE-PESO-ADRC)

As seen in Figure 4, the predictive ESO-based ADRC (PESO-ADRC) mechanism is derived by combining the ADRC with the input that is delayed (shown in Figure 3) and the extended state predictor observer (ESPO) ^[31], to predict the unknown dynamics of the concerned system that were generated as a disturbance amid the delay. The estimated disturbance $\widehat{\delta }\left(t\right)$ is then removed from ${u}_{a}\left(t\right)$ to provide sufficient compensation for the total disturbance.

The ESPO is expressed in the literature by a series of mathematical equations ^[30,31]. An essential component of the ESPO is observer bandwidth (${\omega }_{e}$), which is adjusted to produce the required system response. The Hurwitz stability criterion regulates the selection of ${\omega }_{e}$; thus, the complete details of the ESPO used in this study are outlined in ^[30,31]. Moreover, as seen in Figure 4, the estimation error fed to the ESPO is defined as the difference between the system output $y\left(t\right)$ and the estimated system output $\widehat{y}\left(t\right)$. Hence, the TDE-based predictive ESO-based ADRC is a cascaded arrangement of the PESO-ADRC with the TDE mechanism. The complete structure is denoted as the TDE-PESO-ADRC controller method.

Equation (12) represents the overall disturbance ($\delta \left(x\left(t\right), t\right)$) affecting the controlled object,

where $f\left(x\left(t\right), t\right)$ is inclusive of the external output disturbance applied and parameter perturbation in the system under time-delay control. $w\left({v}_{m}\right)$ is the nonlinear Duhem backlash-like hysteresis given by Equations (4) and (5).

The TDE-PESO-ADRC controller's control law, $u\left(t\right)$, is given by Equation (13),

where $\widehat{\zeta }\left(t\right)$ is the TDE mechanism given by Equations (2) and (3). Also, ${u}_{p}\left(t\right)$, the control law for the PESO-ADRC, is given in Equation (14) ^[8].

2.4.2.   Block diagram of the TDE-based PID control method (TDE-PID)

The PID controller is one of the most widely used control processes in the field of industrial applications ^[32]. When nonlinearity or disturbance is added to a PID-controlled system, it drives the system to instability or inadequate performance. Thus, improving the conventional PID controller's operation is important, as it requires modifying its control law to achieve the desired tracking performance of the object to be controlled via optimal control operation.

In this study, the TDE-PID controller developed in ^[17] is analyzed. The integrated TDE-PID controller's structure is pictorially presented in Figure 5.

The time-delay system to be controlled is given by Equation (15).

where ${u}_{c}\left(t\right)$ is the control law of the conventional PID controller. Other variables in Equation (15) are defined in Equation (1).

The control law ${u}_{t}\left(t\right)$ of the cascaded TDE-PID controller structure is given in Equation (16).

where $r\left(t\right)$ is the input reference signal fed to the TDE-PID-controlled system. The standard PID control law ${u}_{c}\left(t\right)$ is the cumulative sum of the proportional (P), integral (I) and derivative (D) segments, along with their respective gains ${K}_{p}$, ${K}_{i}$, and ${K}_{d}$, respectively. $e\left(t\right)$ is the variation between the output signal $y\left(t\right)$ and the input signal $r\left(t\right)$. The gain factor $b$ is utilized to tune the contribution from TDE to the control variable ${u}_{t}\left(t\right)$. The TDE equation for the TDE-PID control method in Figure 5 is given by Equation (17).

For a given sampling period ${t}_{s}$ and constant $\eta$ in Equation (17), the delayed input and output dynamics are indicated by ${u}_{c}\left(t-{t}_{s}\right)$ and $\ddot{y}\left(t-{t}_{s}\right)$, respectively.

3.   Experimental results and discussions

The recent works in ^[8] and ^[17] presented control of a time-delay system by a cascaded TDE with a predictive ESO-based ADRC and the PID control method, respectively. Thus, this section performs a comparative analysis of the two TDE-based methods to discuss their performance measures and transient responses under different control scenarios. As specified in section 2.1, the controlled object considered in the experimental analysis is a second-order system with a time delay. Such a system is chosen as a benchmark system under testing for various industrial plants. The system concerned is represented in its transfer function form by Equation (18).

When Equation (18) is represented in its state-space form, seen in Equation (1), the state-space matrix values obtained are $A = \left[\begin{array}{cc}-3 & 1\\ 2 & 0\end{array}\right], B = \left[\begin{array}{c}0\\ 2\end{array}\right], C = \left[\begin{array}{cc}1 & 0\end{array}\right]$; and system delay $\tau$ = 0.1 s.

The ADRC controller parameters of the PESO-ADRC method were obtained by optimizing the delayed input ADRC structure shown in Figure 2 using the genetic algorithm (GA) tool of MATLAB software version R2020a in the Optimization tools window ^[30]. On optimization, the controller parameter values attained are $c$ = 0.7645, ${h}_{1}$ = 1.0831, $r$ = 56.0350, ${g}_{o}$ = 0.3097, ${b}_{0}$ = 3.1303, and ${b}_{0\mathrm{D}\mathrm{c}}$ = 1.0314. Further, the fixed parameter values of the TD component are ${r}_{0}$ = 10 and $h$ = 0.03; also, the remaining parameters of the ESO components are given by $[{\beta }_{01}$, ${\beta }_{02}$, ${\beta }_{03}$, ${h}_{ESO}$] = [100,300, 1000, 0.01] ^[33]. Furthermore, the observer bandwidth ${\omega }_{e}$ of the ESPO is tuned to obtain the desired system behavior, and its selection is based on the Hurwitz stability definition defined in ^[30,31]. Moreover, for the PID controller, the parameter values ${K}_{p}$, ${K}_{i}$, and ${K}_{d}$ were manually tuned to 1, 0.8, and 0.1, respectively. The gain factor $b$ was set to 0.2 ^[17]. Also, the input signal given to both controlled structures is a unit step.

In the experiments that follow, the robustness of the TDE-based controller designs is analyzed based on certain transient system response characteristics like rise time [RT (s)], response time $[{t}_{r}$ (s)], settling time [ST (s)], and overshoot [OV (%)] under the impact of different uncertainties. Further, the controller performance is evaluated using the system performance index measure termed integral of time-weighted absolute error (ITAE) and is also evaluated based on the improvement of performance percentage, ${P}_{i}$(%), of all the specified transient characteristics, given in Equation (19):

where ${P}_{1}$ and ${P}_{0}$ represent the response characteristic parameters for TDE-based and non-TDE-based control strategies, respectively.

3.1.   Experiment 1: Impact of TDE on control mechanisms in the absence of uncertainties

In this analysis, PID and PESO-ADRC control strategies are assessed in the presence and absence of the TDE method for the case of no uncertainties (disturbances and nonlinearity) acting on the time-delayed system in Equation (18). For TDE-based PID control, the TDE parameters are $\eta$ = 0.01 and ${t}_{s}$ = 0.001 s; for TDE-based PESO-ADRC control, the parameters are $\eta$ = 0.2 and ${t}_{s}$ = 0.002 s. The observer bandwidth is ${\omega }_{e}$ = 3 and $\tau$ = 0.1 s. Figure 6 illustrates the system responses obtained, and Table 1 tabulates the control performance criteria for both control strategies when tested with and without the TDE mechanism.

It was observed that with the TDE present, the controlled system rose faster for both control scenarios (TDE-PID and TDE-PESO-ADRC), i.e., for the PID controller RT = 1.5263 s with TDE, and RT = 1.8958 s without the TDE mechanism. A similar improvement in rise time of 23.81% is also obtained for the TDE-PESO-ADRC method. Further, the settling time, ITAE value, and overshoot are decreased by 62.07%, 52.26%, and 55.25%, thus showing a greater performance improvement for PESO-ADRC with TDE control. However, a significant improvement in performance measures was not obtained in the case of the TDE-PID control method in comparison to its non-TDE structure, as seen in Table 1. Hence, under this scenario, the integration of TDE with the ADRC strategy provided a better response than its respective non-TDE version and TDE-PID control approach.

3.2.   Experiment 2: Impact of TDE on control mechanisms in the presence of nonlinear hysteresis

In this study, the effectiveness of the control strategy of the TDE approach cascaded with the PID and PESO-ADRC controllers is investigated on a time-delay system, as given in Equation (18), under the influence of nonlinear backlash-like hysteresis represented by Equations (4) and (5). The term $w\left({v}_{m}\right)$ in Equation (12) depicts the dynamic hysteresis output. For the PID controller, the variables used for hysteresis are $\gamma$ = 2, $\alpha$ = 1.16, and $\beta$ = 0.35, whereas for the predictive ADRC controller, the variables are $\gamma$ = 0.1, $\alpha$ = 1.16, and $\beta$ = 0.35. Moreover, the TDE parameters for both PID and predictive ADRC methods are $\eta$ = 0.01, ${t}_{s}$ = 0.001 s, $\tau$ = 0.1 s, and $\eta$ = 0.001, ${t}_{s}$ = 0.001 s, and $\tau$ = 0.1 s, respectively. The ESPO observer bandwidth is ${\omega }_{e}$ = 4.

Figure 7 and Table 2 demonstrate the effectiveness of the TDE mechanism in counteracting the nonlinear nature of the hysteresis effect on the PESO-ADRC method. Meanwhile, the TDE mechanism improved performance for the PID controller regarding quicker settling time, smaller steady-state error, and smaller ITAE. However, as shown in Table 2, there is a minimal decrease in system overshoot in the TDE-PID control strategy and a longer rising time.

Further, the ITAE index measure with the TDE present is 2.1260 and 1.0130 for both PID and PESO-ADRC controllers, respectively, which shows an improvement of 64.74% and 81.23% as compared to without the TDE approach present. Furthermore, by integrating the TDE with the predictive ADRC method, the system response in Figure 7 shows robustness and thus less oscillatory behavior, reduced overshoot, and quick system recovery to uncertainties, contrary to the TDE-PID system response and performance.

3.3.   Experiment 3: Impact of TDE on control mechanisms in the presence of nonlinear hysteresis and perturbed system parameter

This section provides an experimental analysis for the case of nonlinear hysteresis and parameter perturbation present as uncertainties in the time-delayed system in Equation (18) under control. In this simulation, 25% of parameter perturbation is inserted at 20 s under hysteresis for both the system time constants and the system time delay. Hence, the perturbed system transfer function for Equation (18) is given by Equation (20).

where $\widetilde{A} = \left[\begin{array}{cc}-2.40 & 1\\ -1.28 & 0\end{array}\right]$, $\widetilde{B} = \left[\begin{array}{c}0\\ 1.28\end{array}\right]$ and $\widetilde{C} = \left[\begin{array}{cc}1 & 0\end{array}\right]$.

Perturbed system delay, $\widetilde{\tau }$ = 0.125 s. For PID with TDE control, the TDE parameters considered are $\eta$ = 0.01 and ${t}_{s}$ = 0.001 s. Further, for PESO-based ADRC with TDE control, the TDE parameters considered are $\eta$ = 0.001, ${t}_{s}$ = 0.001 s, $\tau$ = 0.125 s, and ${\omega }_{e}$ = 4. Furthermore, for hysteresis, the variables in PID control are $\gamma$ = 2, $\alpha$ = 1.16, and $\beta$ = 0.35; and for PESO-ADRC, the variables are $\gamma$ = 0.1, $\alpha$ = 1.16, and $\beta$ = 0.35.

The perturbed system's performance measures are displayed in Figure 8 and listed in Table 3. Table 3 and Figure 8 demonstrate that the systems responded more quickly and had a shorter rise when TDE was present. The RT was 1.0316 s with TDE vs 1.1677 s without TDE for the PID control scheme, and 0.4007 s with TDE compared to 0.7052 s without TDE for the PESO-ADRC approach. For the TDE-PESO-ADRC method, the initial system response's overshoot was greatly reduced by 45%; however, for the case of PID with TDE, the decrease in initial overshoot was not observed. However, from Table 3, under the effect of parameter perturbation at 20 s, the TDE system reached a faster settling time for the cases of both TDE with PID and PESO-ADRC control approaches. Hence, the percentage improvement ${P}_{i}$ obtained in ST, ITAE, and OV was significant for the predictive ADRC-TDE control (34.79%, 80.44%, and 42.64%, respectively), contrasted to the TDE-PID control method, thus showing greater robustness to the concerned system's perturbed parameters and nonlinear hysteresis acting under time-delay control.

3.4.   Experiment 4: Impact of TDE on control mechanisms in the presence of nonlinear hysteresis, perturbed system parameters, and added external disturbance

This experiment extends the one from the preceding subsection, which included hysteresis and a 25% incrementation in system parameters at 20 s. Thus, the cumulative applied uncertainty given by Equation (12) includes the nonlinear hysteresis, parameter perturbation, and input disturbance in the form of external disturbance added to the perturbed system in Equation (20). Two different forms of external disturbances are assessed in this section. Case 1 is a step input disturbance of magnitude 0.6 applied at 20 s, ${f}_{1}\left(x\left(t\right), t\right)$ stated in Equation (21), and the other case is with periodic forms of uncertainties ${f}_{2}\left(x\left(t\right), t\right)$ given by Equation (22).

For hysteresis, under all cases, the variables used are $\gamma$ = 2, $\alpha$ = 1.16, and $\beta$ = 0.35. Under case 1, for TDE-PID, $\eta$ = 0.01 and ${t}_{s}$ = 0.001 s, whereas for TDE-PESO-ADRC, $\eta$ = 0.001 and ${t}_{s}$ = 0.001 s. Other parameters for the ADRC and PID controllers are equal to those specified in the preceding section. Figures 9 and 10 illustrate the impact on system responses for both TDE-PID and TDE-PESO-ADRC controllers under the nonlinear effects of hysteresis, perturbed system parameters, and an added input external disturbance to the perturbed time-delay system. Table 4 lists the corresponding performance metrics for Figures 9 and 10.

As indicated in Table 4, Figures 9 and 10 demonstrate that both the controlled system with the TDE mechanism displayed a shorter rise (RT) when the TDE was introduced. In addition, the TDE mechanism in the TDE-PESO-ADRC showed a decrease in overshoot at the beginning of the transient response, contrary to the slight increase in response obtained in the TDE-PID controller between 0 and 5 s for the case of added disturbance ${f}_{1}\left(x\left(t\right), t\right)$ given by Equation (21). However, for the TDE-PID controller with added disturbance in Equation (21), the overshoot decreased by 14% between 20 and 25 seconds. For the PESO-ADRC controller, the system response was unstable at the onset of the external disturbance in the absence of TDE. Thus, the response could not be plotted.

Moreover, employing the TDE mechanism to control the perturbed system under the effect of periodic disturbance ${f}_{2}\left(x\left(t\right), t\right)$ given by Equation (22) is pictorially presented in Figure 10. The system's response characteristics are provided in Table 4. For both cases, with the TDE mechanism, the predictive-based ADRC's response shows system instability after adding an external disturbance to the perturbed time-delay system under nonlinear hysteresis. Thus, the system response for the same is not plotted in both Figures 9 and 10. Therefore, the improvement values [${P}_{i}$ (%)] could not be tabulated in Table 4 for the PESO-ADRC controller method. However, by incorporating the TDE mechanism in this circumstance, the TDE-PESO-ADRC could conquer this unsteady behavior and provide an acceptable response curve that demonstrated considerable resistance to system uncertainties (Figure 10 and Table 4).

Similarly, both Figure 10 and Table 4 do not include the system response's performance measures for the TDE-PID method because of the oscillatory character of the periodic external disturbance given in Equation (22) added to the system under test, seen in Equation (18). Hence, it is inferred that the TDE-PESO-ADRC control method is less susceptible to different forms of input disturbances like periodic, nonlinear, and step forms applied to the concerned controlled object, in contrast to the TDE-PID control mechanism. Therefore, this emphasizes how crucial the TDE control mechanism is for analyzing and mitigating all kinds of nonlinear phenomena subjected to a time-delayed system under control.

3.5.   Experiment 5: Impact of TDE on control mechanisms in the presence of nonlinear hysteresis, perturbed system parameter, added external disturbance, and varying system time delay

The robustness of the suggested TDE-PESO-ADRC and TDE-PID techniques is evaluated in this experiment over a range of system delay values when hysteresis, parameter perturbation, and input disturbance are added externally to a system. The delay ${\tau }_{s}$ in this experimental test is equivalent to the concerned system's time delay (in Figures 11 and 12).

For the ADRC, ${\omega }_{e}$ = 4, and the TDE parameters are $\eta$ = 0.001 and ${t}_{s}$ = 0.001 s. For hysteresis, the variables used are $\gamma$ = 0.1, $\alpha$ = 1.16, and $\beta$ = 0.35. For the PID controller, the TDE parameters considered are $\eta$ = 0.01 and ${t}_{s}$ = 0.001 s, and for hysteresis, the variables used are $\gamma$ = 2, $\alpha$ = 1.16, and $\beta$ = 0.35. In this analysis, the input disturbance added externally at 15 s to the perturbed system under control is given by ${f}_{1}\left(x\left(t\right), t\right)$ indicated by Equation (21). Further, 25% of parameter perturbation is injected at 20 s for the TDE-PESO-ADRC method (Figure 11). While assessing the resilience of the TDE-based controllers in this analysis, the delay block present in the input channel of the ESO and the ESPO subsystem, seen in Figure 4, has a value of 0.1 s. This delay is referred to as a delay design in this test. On the other hand, ${\tau }_{s}$ is varied through a range of known delay values in this experiment to evaluate for robustness. Their corresponding performance index measures are tabulated in Tables 5 and 6, and the system responses to study the impact of varying time delays are presented in Figures 11 and 12.

For the TDE-based PESO-ADRC controller in Figure 11, the experiments were performed on an increase in delay from ${\tau }_{s}$ = 0.12 and 0.14 to 0.16 s. It is seen that the system responses were therefore oscillatory for greater system latencies, but they could compensate for nonlinear uncertainties, demonstrating decent strength and steady performance for distinct delay scenarios. Besides the transient response features like RT, OV, and ITAE, Table 5 also lists the response time (${t}_{r}$), which provides the time needed for the system response to reach a steady state after overcoming the external disturbance applied.

Further, for the TDE-PID-based controller, the peak values of the oscillating response were decreased for different time-delay situations, such as ${\tau }_{s}$ = 0.2 s, 0.4 s, and 0.6 s, following the onset of the external disturbance given by Equation (21) and 25% parameter perturbation at 30 s (as seen in Figure 12). Furthermore, the performance criteria displayed in Table 6 for changes in the TDE-PID controlled system delay responses showed robustness and significant compensation for all forms of disturbances present. However, the response curve oscillates for delay values of 0.5 s and above, paving the way for further research. Therefore, it was observed that the TDE-PID controlled system could withstand greater delay values up to 0.6 s, unlike the TDE-PESO-ADRC control method, which could compensate for a second-order system having 0.16 s delay, subjected to various disturbances and nonlinear backlash-like hysteresis under time-delay control. Thus, this showcases the more robust and endurance behavior of the TDE-PID control method when affected by variations in system time delay.

3.6.   Experiment 6: Control signal for TDE-based controllers

In this section, the parameter values used for the TDE-based controllers are the same as those in the preceding section. The control signals fed to the system under control are provided in Figures 13(a) and 13(b) for TDE-PESO-ADRC and TDE-PID control, respectively. Both responses seen in Figure 13 are under the impact of nonlinear hysteresis, parameter perturbation, and an added external disturbance (${f}_{1}\left(x\left(t\right), t\right)$), under varying system time-delay scenarios.

For TDE-PESO-ADRC control, the controlled object could withstand smaller delay values (seen in Figure 11 and Table 5). Meanwhile, for TDE-PID control, the controlled object showed resilience to higher system delay values (shown in Figure 12 and Table 6). Hence, the control signals were plotted for each TDE-based control method as per the simulations performed in this section. Figure 13 (a) shows that the TDE-PESO-ADRC method reached a steady state much earlier for a range of smaller, varying time-delay values. However, the TDE-PID control method took longer to achieve a steady state, as seen in Figure 13 (b). It is to be noted that the TDE-PID is more aggressive at the beginning of the control (the control signal reaches 11), which might trigger hard constraints for some applications. Therefore, the controller operations are challenged by adding a hard constraint on the control signal to the system, with their respective plots shown in Figure 14. This is achieved by inserting a saturation block before the nonlinear hysteresis phenomenon, which leads to the controlled object shown in Figures 4 and 5. In this scenario, the overall system comprises hysteresis, the concerned system with time delay, and other uncertainties acting on it. The saturation block is set to an upper limit of 5 and a lower limit of -5. In Figures 14 and 15, ${\tau }_{s}$ = 0.12 s for the TDE-PESO-ADRC method, and with ${\tau }_{s}$ = 0.2 s for the TDE-PID control method.

As observed in Figure 14(a), for the TDE-PESO-ADRC method, the initial amplitude of the control signal is reduced from almost 10 to 5. Moreover, for the TDE-PID control method, the initial peak is reduced from 11 to 5. Therefore, the saturation block further limits the control signal fed to the overall system, making it less aggressive in nature, as seen in Figure 14. Furthermore, Figure 15 provides the corresponding system output responses for the case of saturation added in Figure 14. For both the TDE-PESO-ADRC and TDE-PID control methods, it is noted that the output response curves obtained with saturation present are in accordance with those obtained from the non-saturation case. Hence, an acceptable minimal difference is observed in the output responses by limiting the value of the control signal to the overall system.

3.7.   Experiment 7: Sensitivity analysis of the TDE parameters on control mechanisms in the presence of nonlinear hysteresis

This subsection conducts a sensitivity analysis to determine the choice of TDE parameters used for experimentation. Figure 16 illustrates the choice of TDE parameters for the TDE-PID controller operation, by first keeping the positive constant $\eta$ varying from values 0.001 to 0.2 (in Figure 16 (a)), when $b$ = 0.2. Similarly, in Figure 16 (b), the parameter $\eta$ is kept fixed at 0.001 for a varying gain factor $b$ value ranging from 0.001 to 0.2.

The significance of parameters $\eta$ and $b$ are related to the TDE-PID control law ${u}_{t}\left(t\right)$ given by Equation (16). Figure 16 (a) plots the sensitivity performance for $\eta$ values tuned to 0.001, 0.01, 0.1, and 0.2, and the gain factor $b$ constant at 0.2. It is noted that the transient response characteristics measured, such as the RT, ST, OV, and ITAE, remain consistent for all the abovementioned $\eta$ values. Therefore, this indicates that the TDE-PID controlled system is less sensitive to changes in $\eta$. Further, Figure 16 (b) illustrates the performance analysis for the case where the parameter $b$ is varied from 0.001, 0.01, 0.1, to 0.2; and for all $\eta$ = 0.001. Hence, it is seen that the performance criteria measures experience certain changes when $b$ of the TDE-PID controller is varied, but these variations observed are within acceptable limits, hence showcasing less sensitivity to varying $b$ parameter in a relatively small range.

Moreover, a sensitivity performance analysis was conducted on the choice of TDE parameters for the TDE-PESO-ADRC control method. It can be inferred from Figure 17 that for $\eta$ values tuned to 0.001, 0.01, 0.1, and 0.2, the performance appears to increase slightly for all transient response characteristics (RT, ST, OV, ITAE). Hence, the TDE-PESO-ADRC control method is less sensitive to $\eta$ values. It is important to note that the parameter $b$ is not included in the control law of TDE-PESO-ADRC, given by Equations (13) and (14); therefore, it has not been plotted in Figure 17.

4.   Conclusions

This paper performs a comparative analysis between two recent time-delay estimation-related controllers, namely the TDE-based predictive ESO-based ADRC method ^[8] and the TDE-based PID control method ^[17], to highlight the importance of TDE methods in predicting and overcoming the various uncertainties and nonlinearities manifested in the control system theory field.

The experiments performed in this paper were tested using the TDE-based controllers on a second-order time-delay system under different circumstances, such as the presence of nonlinear backlash-like hysteresis, followed by system parameter perturbation, then an external input disturbance, followed by varying the perturbed system's time delay. Under all these scenarios, the TDE-PESO-ADRC approach improved the estimation and mitigation of the uncertain prevailing nonlinearities compared to its non-TDE counterpart. Also, the TDE-PID controller showed good robustness to various uncertainties, with shorter rise time and settling time, stable system response, and decreased oscillations. The results obtained were validated using measures such as performance index metrics and transient response characteristics. Furthermore, a sensitivity analysis was conducted on the choice of specific TDE parameters for TDE-based controllers to validate their respective selected values.

However, it was demonstrated that the TDE-PESO-ADRC method could estimate and counteract the different forms of periodic and step forms of external disturbances added to the perturbed system to be controlled, unlike the TDE-PID controller, which showed an unstable response under the impact of a periodic kind of disturbance, hence directing the use of adaptive TDE gains under future scope. On the other hand, the TDE-PID control approach could sustain a satisfactory and stable response to an increasing system time delay, unlike the TDE-PESO-ADRC method, which showed endurance to a small range of carrying system time delay. Therefore, future work is required to obtain a decent trade-off by improving the TDE-based control mechanism for systems with unknown and higher delay values, along with considering noise and friction acting on a higher-order system under control. Currently, the setup of a hardware-based human-machine interface platform is underway, followed by the modeling of its digital twin setup. These time-delay estimation-based controllers will be combined with the modeled digital twin to minimize the error signal between the physical system output and the digital twin output. Hence, the control and time-delay compensation of such higher-order systems will contribute to real-world industrial applications.

Author contributions

Syeda Nadiah Fatima Nahri: conceptualization, methodology, software, investigation, validation, formal analysis, writing ‒ original draft preparation, writing ‒ review & editing; Shengzhi Du: conceptualization, methodology, software, validation, writing ‒ review & editing, supervision; Barend J. van Wyk: validation, writing ‒ review & editing; Oluwaseun Kayode Ajayi: writing ‒ review & editing. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We thank Tshwane University of Technology and the National Research Foundation for granting us the resources and assistance needed for this research study. Moreover, this work is based on the research supported in part by the National Research Foundation of South Africa (grant numbers PSTD2204143453 and SRUG2203291049).

Conflict of interest

All authors declare no conflicts of interest in this paper.