Tuning of PID/PIDD2 Controllers for Second-Order Oscillatory Systems with Time Delays

Abstract

PID control is the longest history and the most vital basic control mode that has been widely applied in the production process. The oscillatory dynamics of the process have various features, and parameter tuning is complicated. To reduce the complexity of the parameter tuning process and improve the performance of the system, in this research, we propose a new tuning method for the PID/PIDD2 controllers for second-order oscillatory systems with time delays under the constraint of certain robustness. In comparison to existing PID for second-order oscillatory systems with time delays, simulation findings demonstrate that the tuning method of the proposed PID/PIDD2 controllers trades off robustness and disturbance rejection performance.

1. Introduction

1.1. Background

Proportional-integral-derivative (PID) control is the most vital basic control mode that has been widely applied in the production process [1]. The principal reason is that it only needs three adjustable parameters, and does not require the exact information of the system [2]. Because of this, the majority of research in the field of process control has concentrated on PID control [3,4,5,6,7].

The underdamped systems are commonly found in practical applications, like bridge cranes control systems [8], pipeline pressure control systems [9,10], robot control systems [11], load frequency control systems [12,13], and vibration control systems [14,15]. However, the systems mentioned above require particular attention since their step response exhibits oscillatory behavior. Moreover, for the underdamped system, the conflict between rapidity and overshoot cannot be resolved [16,17]. Therefore, to obtain satisfactory control results, that is, the closed-loop response of the system without or with a relatively small overshoot [18] and the shortest possible stabilization time, we need to research the appropriate controller parameter tuning method for underdamped oscillatory systems.

1.2. Analysis of the Existing Literature

To the best of our knowledge, extensive research on oscillatory systems still adopts PID. Ho et al. [19] studied the relationship between the control parameters and specifications (the gain and phase margin) and obtained the PID controller parameter values through the method of specifying specifications, as it is difficult to select suitable gain and phase for different underdamped systems. Wang et al. [20] provided a simple calculation formula for the oscillatory system based on the traditional root locus analysis approach, the process of solving equations is complicated. Shen [21] adopted the fuzzy neural network approach to tune the PID controller for the underdamped oscillatory system. The training speed of the neural network is slow and the calculation amount is large, so it is not convenient to implement. Skogestad [22] modified the integral form of the internal model control (IMC)-PID tuning criterion and obtained the simple tuning formula, wherein the controller parameters met the requirements of IAE, TV, and robustness. The performance of this method does not show a significant advantage over currently used control methods. Huang et al. [23] presented an inverse-based synthesis PID controller design method, and the robustness of this method is analyzed via the gain and phase margin; this method also needs to select the appropriate gain and phase. Maclaurin performed the tuning of the PID controller based on [24] for the first-order process with a dead-time (FOPDT) and second-order process with a dead-time (SOPDT) process. Chen et al. [25] designed a PID controller based on direct synthesis and disturbance closed-loop transfer function to reject disturbance, and obtained the tuning formula; this method has both large and small robustness and ITAE. Ho et al. [26] used an orthogonal simulated annealing (OSA)-based approach optimal fuzzy neural network model to tune PID controller parameters to minimize the mean tracking error and satisfy a robustness constraint for the underdamped oscillatory system; this method also suffers from high computational complexity. Malwatkar et al. [27] derived the PID controller tuning procedure for higher-order oscillatory systems through the reduced order of the model. For this method for the moderate oscillatory system, the disturbance response is still oscillating and not eliminated. Eriş et al. [28] proposed PID tuning rules based on direct digital control for the SOPDT system and obtained a fast response from the closed loop system without overshooting. This method uses a discrete controller and is suitable for critically damped systems. Kurokawa et al. [29] discussed the relationship between set point tracking performance and robust stability, and provided an optimal PID design method for the SOPDT system. The method was designed using the maximum value of the sensitivity function ($M_s$) and the PID parameters were decided by IAE. However, no clear tuning formula is given. Other studies on oscillatory systems will not be listed here.

Consider the four methods (gain and phase margin-based PID [19], direct synthesis PID [23], closed-loop specified (CS)-PID [25], and IMC-PID [30]) mentioned above as examples; these four tuning methods are typical, so they are examined as a comparison. the closed-loop system robustness ($\mathsf{\epsilon }$) [31], disturbance rejection performance (the integral of time and absolute error (ITAE)), and the step response obtained for the underdamped SOPDT systems are shown in

Table 1. Performance indices of existing PID tuning methods for the underdamped SOPDT systems.

$\mathit{\tau }$		$\mathit{\xi }=0.1$	0.2	0.3	0.4	0.5	0.6
0.8	Ho [19]	$\epsilon$ = 2.6742	2.2971	2.1900	2.1387	2.1099	2.0913
		ITAE = 167.8861	147.0132	134.4632	126.8786	122.2789	119.6878
	Maclaurin [30]	$\epsilon$ = 10.2749	20.4934	8.2138	6.4983	5.4516	4.9848
		ITAE = 181.0100	152.0334	134.7830	123.8297	116.3050	111.0823
	Chen [25]	$\epsilon$ = 15.2057	21.0351	7.8210	5.2678	4.2224	3.6717
		ITAE = 163.0601	142.4455	128.8943	120.5432	115.5798	112.6540
	Huang [23]	$\epsilon$ = 3.2782	2.8444	2.6859	2.6020	2.5550	2.5224
		ITAE = 176.2622	153.3725	139.1954	130.1162	123.8912	119.4415
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
1.2	Ho [19]	$\epsilon$ = 2.1639	2.0346	2.0047	2.0090	2.0156	2.0187
		ITAE = 170.7906	151.0417	140.0866	134.2907	131.5404	130.7761
	Maclaurin [30]	$\epsilon$ = 4.3237	33.6121	11.5089	6.5598	5.4500	4.7244
		ITAE = 219.7949	189.0479	165.3703	148.7675	137.2231	129.2231
	Chen [25]	$\epsilon$ = 1.9443	3.6429	3.0389	2.3784	2.0517	1.9003
		ITAE = 152.5156	143.8841	140.3592	138.5792	137.1669	135.8013
	Huang [23]	$\epsilon$ = 2.9966	2.6885	2.5766	2.5183	2.4856	2.4641
		ITAE = 171.4444	154.4553	144.0193	137.7687	133.8287	131.1532
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
1.6	Ho [19]	$\epsilon$ = 1.9820	1.9861	1.9913	1.9962	2.0005	2.0042
		ITAE = 169.3500	151.5267	142.8030	139.5709	139.8092	140.5478
	Maclaurin [30]	$\epsilon$ = 3.4187	13.9663	10.0828	6.1866	4.8760	4.2371
		ITAE = 220.5395	193.8723	172.8015	157.2165	146.3086	138.7042
	Chen [25]	$\epsilon$ = 2.5808	2.2093	2.1180	2.0634	1.9639	1.9684
		ITAE = 160.6442	158.0982	156.4578	154.9632	153.5713	152.2662
	Huang [23]	$\epsilon$ = 2.8441	2.6061	2.5199	2.4760	2.4519	2.4356
		ITAE = 167.8376	152.6414	144.7046	141.1966	139.7917	138.8289
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
2.0	Ho [19]	$\epsilon$ = 1.9948	1.9964	1.9982	2.0001	2.0019	2.0036
		ITAE = 167.7916	152.2218	147.2653	147.2348	147.8758	148.5063
	Maclaurin [30]	$\epsilon$ = 5.6069	17.3869	6.7806	4.8391	4.0691	3.6961
		ITAE = 210.8966	188.2239	170.9862	158.5524	149.8659	143.8656
	Chen [25]	$\epsilon$ = 2.7157	2.5561	2.4301	2.3322	2.2574	2.2047
		ITAE = 171.0477	169.3779	167.9577	166.6392	165.3989	164.2274
	Huang [23]	$\epsilon$ = 2.7512	2.5563	2.4891	2.4548	2.4340	2.4202
		ITAE = 165.9403	151.9473	146.8995	145.7931	145.4290	145.2653

and Figure 1. Since the range of the magnitude of ITAE over the entire time horizon is large, it is represented in terms of dB (20lg(ITAE)) in Figure 1. From Table 1 and Figure 1, we observe:

(1)

The robustness value and ITAE of Maclaurin’s method [30] are apparently larger than the other three methods; thus, it is very sensitive to unmodeled dynamics. The step response of Maclaurin’s method is also seriously oscillatory.

(2)

Although the robustness value of Ho et al. [19] is smaller than Huang et al. [23], the ITAE’s of both are similar. Chen et al. [25] has both large and small robustness and ITAE.

(3)

Maclaurin’s and Huang’s methods are based on the model inverse; for the moderate oscillatory system (Figure 1c), the disturbance response is still oscillating and not eliminated. At the same time, Ho et al.’s approach can be included in addition to the above problems and also needs to select a suitable gain and phase for different underdamped systems, which is not conducive to practical engineering practice. The step response of Chen et al. eliminates the oscillation; however, from the point of view of both disturbance resistance and robustness, it is not optimal.

Remark 1. 

The process of calculating the performance indices ITAE and $\epsilon$ in Figure 1 is based on (4) and (7).

1.3. Motivation and Contribution of the Paper

From the previous discussion, it is necessary to find a new optimal PID controller design method that can trade-off between robustness and disturbance rejection performance for underdamped oscillatory systems, and this paper will first derive a PID tuning method for such systems.

As far as we know, proportional integral double derivative (PIDD2) can be used to improve the performance of the conventional PID [32,33]. The application of the PIDD2 controller in the voltage and frequency control of power systems has achieved satisfactory robustness [34,35]. Even though the parameter tuning of the PIDD2 controller has been studied in the literature [36,37,38], very little was found in the literature [39] on the question of PIDD2 controller for oscillatory systems. In this paper, the tuning formula of the PIDD2 controller for the underdamped oscillatory with time delays system will also be derived. The tuning formula of the PID/PIDD2 controller parameters are obtained by using the optimization algorithm while satisfying the robustness constraint and minimizing the ITAE.

To further illustrate the necessity of the PIDD2 control method proposed in this paper, here, we use the form of simulation to carry out a simple comparative analysis with the literature [39]. The detailed specifications and step responses of the process are shown in

Table 2. Parameters of the PIDD2 and IMC_PIDD2 controllers.

Methods	Controller Parameters				Disturbance Rejection		Robustness Index	ITAE Index
	${\mathit{K}}_{\mathit{p}}$	${\mathit{K}}_{\mathit{i}}$	${\mathit{K}}_{\mathit{d}}$	${\mathit{K}}_{\mathit{d}\mathit{d}}$	${\mathit{T}}_{\mathit{s}}/\mathit{s}$	$\mathit{\sigma }$%	$\mathit{\epsilon }$
PIDD2	0.0028	0.2218	0.5467	0.2217	10.5	60	2.5106	1.063 × 10³
IMC_PIDD2	−0.3954	0.1238	0.2618	−0.0187	14.7	97	3.1120	2.103 × 10³

and Figure 2. From the Figure 2 and Table 2, key findings emerge:

(1)

The two methods are different, and the tuning formulas in [39] need to be fitted twice, and the derivation process is complicated. In this paper, a simplified method is adopted to divide the oscillating system according to the damping ratio $\xi$, which greatly reduces the fitting process and is more convenient for practical engineering applications. In addition, both use different controllers and have different control structures: [39] uses state space PIDD2, and this paper uses PID and PIDD2.

(2)

The disturbance rejection performance and robustness of [39] are not as good as PIDD2.

The contributions of this paper include:

• The tuning formulas of PID/PIDD2 for the underdamped oscillatory plant are proposed.
• The parameters of the tuning formula are obtained by minimizing the integral of time and absolute error (ITAE) under a prescribed robustness through the optimization algorithm. The simulation results verify the applicability of the proposed formulas.

The remainder of the paper contains four chapters. Section 2 describes the system models and robustness design. Section 3 presents the parameter tuning of PID controllers. Section 4 presents the parameter tuning of PIDD2 controllers. Section 5 presents an application in power system control. Finally, Section 6 presents the conclusion.

2. System Models and Robustness Design

2.1. System Models

The general industrial process control system generally includes two parts: the controller and the controlled plant. Figure 3 is the simplified structure block diagram of the system.

As is well-known, the oscillatory system can be approximated in terms of the SOPDT model; thus, we can write the corresponding equivalent form as:

$$\begin{array}{c}P\left(s\right)=\frac{k}{T^2{s}^2+2T\xi s+1}{e}^{-\tau s}\end{array}$$

(1)

where $k,T, \tau$, and $\xi$ are the system gain, the time constant, the time delay, and the damping ratio, respectively.

The nominal form of the controller expressions are:

$$\begin{array}{c}{C}_{PID}\left(s\right)=K_p+\frac{K_i}{s}+K_{d}s\end{array}$$

(2)

$$\begin{array}{c}{C}_{PIDD2}\left(s\right)=K_p+\frac{K_i}{s}+K_{d}s+K_{dd}{s}^2\end{array}$$

(3)

where $K_p$,$K_i$, $K_d$, and $K_{dd}$ are the proportional, integral, derivative gain, and double derivative gain, respectively.

2.2. Robustness Design of Controller

To evaluate the performance of the controller, we take the integral of time and absolute error (ITAE) as the disturbance rejection capability of the closed-loop system:

$$\begin{array}{c}ITAE={{\displaystyle \int }}_0^{\infty }t\left|e\left(t\right)\right|dt\end{array}$$

(4)

where $e\left(t\right)=r\left(t\right)-y\left(t\right)$ is the difference between the reference input signal and the output signal of the system.

Since the system will be affected by various uncertainties in the process of operation, we need a robustness index to measure the quality of the controller design. There are two general robustness indices $M_s$ and $M_t$, where $M_s$ and $M_t$ are maximum sensitivities,$S\left(s\right)$ and $T\left(s\right)$ are sensitivity functions:

$$\begin{array}{c}{M}_s=\underset{\omega }{max}\left|S\left(i\omega \right)\right|,S\left(s\right)=\frac{1}{1+P\left(s\right)C\left(s\right)}\end{array}$$

(5)

$$\begin{array}{c}{M}_t=\underset{\omega }{max}\left|T\left(i\omega \right)\right|,T\left(s\right)=\frac{P\left(s\right)C\left(s\right)}{1+P\left(s\right)C\left(s\right)}\end{array}$$

(6)

According to [30,40], we can obtain a measure of the robustness of single-loop control systems:

$$\begin{array}{c}\epsilon :=\underset{\omega }{sup}\left(\left|S\left(i\omega \right)\right|+\left|T\left(i\omega \right)\right|\right)\end{array}$$

(7)

where $\epsilon$ represents the robustness of the system, and the larger $\epsilon$ is, the worse the robustness. To ensure that the system has appropriate robustness, we choose $\epsilon <2.5$.

3. Parameter Tuning of PID Controllers

Here, we consider the transfer function of the normalized second-order oscillatory system model, with the normalized delay time $\overline{\tau }=\tau /T$ [41]:

$$\begin{array}{c}P\left(s\right)=\frac{1}{s^2+2\xi s+1}{e}^{-\overline{\tau }s}\end{array}$$

(8)

Now that we have the model for the system in model (8), how to make full use of the known parameter information of the model to obtain the controller parameters is the focus of this section. It has been mentioned above that the value of damping $\xi$ is vital for the underdamped oscillatory system. In general, when $\xi <0.4$, it is a heavily oscillatory system; otherwise, it is a moderately oscillatory system. The basic requirements of the control system are moderate damping, fast response speed, and short settling time. The above two cases also have different demands for the controller. For simplicity, we study the problem of controller parameter tuning for the underdamped oscillatory systems under two cases: $\xi =0$ and $\xi =0.4$, i.e., we use model (8) with $\xi =0$ to represent the heavily oscillatory systems and derive the tuning formula for systems with $\xi <0.4$, and model (8) with $\xi =0.4$ to represent moderately oscillatory systems and derive the tuning formula for systems with $\xi \ge 0.4$. It is possible to derive tuning formulas for each $\xi$, however, this is quite complicated, as was previously performed in [16,17] for the tuning of active disturbance rejection controllers. We note that the PID/PIDD2 controllers are robust against the damping ratio, so we can simplify the tuning procedure into two cases.

Now, we can move on to the controller parameter optimization issue. The specific process of parameter optimization is:

$$\begin{array}{c}mi{n}_{{\overline{K}}_p,{\overline{K}}_i,{\overline{K}}_d} ITAE\\ s. t. \epsilon <2.5\end{array}$$

(9)

Remark 2. 

 

Step 1: Initialize the controller’s parameters $K_p$, $K_i$, and $K_d$ (or $K_p$, $K_i$, $K_d$, and $K_{dd}$).

Step 2: Parameters iteration: to minimize the load disturbance attenuation performance in the integral of time and absolute error (ITAE) sense, under the constraint of a specified robustness measure ($\epsilon <2.5$) for the second-order process with a dead-time (SOPDT) process model.

Step 3: If Step 2 meets the requirements, the controller parameters can be obtained, otherwise, the iteration continues until the requirements are met.

From the optimization process above, the parameters’ fitting curves for plant (8) are shown in Figure 4, and the tuning formulas are shown in (10) ($\xi <0.4$) and (11) ($\xi \ge 0.4$).

$$\begin{array}{c}\begin{array}{c}{\overline{K}}_p=0.02987{\overline{\tau }}^{-2.363}-0.186 \\ {\overline{K}}_i=0.06231{\overline{\tau }}^{-1.676}+0.1159\\ {\overline{K}}_d=1.41{\overline{\tau }}^{-0.4422}-1.027 \end{array}(\xi <0.4)\end{array}$$

(10)

$$\begin{array}{c}\begin{array}{c}{\overline{K}}_p=0.2455{\overline{\tau }}^{-1.496}+0.183 \\ {\overline{K}}_i=0.1691{\overline{\tau }}^{-1.619}+0.2083 \\ {\overline{K}}_d=0.4185{\overline{\tau }}^{-0.9354}+0.1609 \end{array}\left(\xi \ge 0.4\right)\end{array}$$

(11)

For the general SOPDT plant (1), the tuning formula (10) can be converted into (12), and (11) can be converted into (13) [41].

$$\begin{array}{c}\begin{array}{c}{K}_p=\frac{1}{k}\left(0.02987{(\frac{\tau }{T})}^{-2.363}-0.186\right) \\ K_i=\frac{1}{kT}\left(0.06231{\left(\frac{\tau }{T}\right)}^{-1.676}+0.1159\right)\\ K_d=\frac{T}{k}\left(1.41{\left(\frac{\tau }{T}\right)}^{-0.4422}-1.027\right) \end{array}(\xi <0.4)\end{array}$$

(12)

$$\begin{array}{c}\begin{array}{c}{K}_p=\frac{1}{k}\left(0.2455{\left(\frac{\tau }{T}\right)}^{-1.496}+0.183\right) \\ K_i=\frac{1}{kT}\left(0.1691{\left(\frac{\tau }{T}\right)}^{-1.619}+0.2083\right)\\ K_d=\frac{T}{k}\left(0.4185{\left(\frac{\tau }{T}\right)}^{-0.9354}+0.1609\right)\end{array}\left(\xi \ge 0.4\right)\end{array}$$

(13)

The responses of the normalized second-order oscillatory plants (8) with different damping ratios and time delays are shown in Figure 5, Figure 6 and Figure 7 (we focus on robustness and disturbance rejection performance; the tracking performance can be improved by setting a suitable weight factor to the controller’s proportional gain). The responses are for a step reference signal at $t=0 s$ and a step input disturbance signal is added to these systems at an appropriate time to test the disturbance rejection performance and robustness. As we can observe, for systems with small damping ($\xi <0.4$), the proposed tuning formula (12) is slightly slow in disturbance rejection, but has a very small oscillation, while the responses of the other two methods are quite oscillatory. For systems with large damping ($\xi \ge 0.4$), the proposed tuning formula (13) has a better control effect than the other two methods.

Consider the following oscillatory examples for ($G_1\left(s\right)-G_2\left(s\right)$ [20], $G_3\left(s\right)$ [27], $G_4\left(s\right)$ [29]):

$$\begin{array}{c} \begin{array}{c}{G}_1\left(s\right)=\frac{1}{\left(s^2+s+1\right){\left(s+2\right)}^2}{e}^{-0.1s},G_2\left(s\right)=\frac{1}{{\left(s^2+2s+3\right)}^3\left(s+3\right)}{e}^{-0.3s}\\ G_3\left(s\right)=\frac{1}{\left(9s^2+2.4s+1\right)\left(s+1\right)}{e}^{-2s},G_4\left(s\right)=\frac{1.5210}{s^2+0.3380s+1.6900}{e}^{-0.37s}\end{array}\end{array}$$

(14)

Dynamic responses of the plants are given in Figure 8. The controller parameters, system parameters., and controller performance indexes are shown in

Table 3. Parameters of the PID controllers.

System Parameters				Methods	Controller Parameters			Disturbance Rejection		Robustness Index	ITAE Index
	$\mathit{\xi }$	$\mathit{\tau }$	$\mathit{T}$		${\mathit{K}}_{\mathit{p}}$	${\mathit{K}}_{\mathit{i}}$	${\mathit{K}}_{\mathit{d}}$	${\mathit{T}}_{\mathit{s}}/\mathit{s}$	$\mathit{\sigma }$%	$\mathit{\epsilon }$
G1	0.4911	0.8370	1.1207	Proposed_PID	2.5314	1.9243	3.5821	8.5894	7.1	2.1349	121.0161
				Malwatkar_PID [27]	1.3039	1.3104	1.3351	8.328	10.18	1.8672	164.6253
				Wang_PID [20]	1.503	1.366	1.715	10.287	8.96	1.7707	155.7447
				Huang_PID [23]	2.9570	2.6864	3.3740	8.724	8.07	2.4594	116.8912
				Ho_PID [19]	2.147	1.484	0.777	14.756	9.96	2.7118	193.5172
G2	0.4927	1.8540	0.7927	Proposed_PID	20.7364	26.0712	22.8381	4.7	0	2.4471	13.1827
				Wang_PID [20]	17.562	22.485	14.13	4.86	0	2.3796	13.1787
				Huang_PID [23]	17.342	22.203	13.953	4.88	0	2.3956	13.5873
				Ho_PID [19]	13.712	17.051	10.715	4.82	0	2.0124	16.3049
G3	0.4154	2.3	3.2024	Proposed_PID	0.5858	0.1553	2.3418	43.7	40	3.0569	3.86 × 103
				Huang_PID [23]	0.5784	0.2174	2.2294	47.1925	36	3.0657	5.32 × 103
				Ho_PID [19]	0.4950	0.1669	1.7121	44.82	43.5	2.4928	5.14 × 103
G4	0.13	0.37	0.7692	Proposed_PID	−0.0196	0.4743	0.7879	10.0878	44.8	2.1503	393.2047
				Kurokawa_PID [29]	0.0932	0.2903	0.2311	17.6833	58.2	2.1007	939.5594
				Huang_PID [23]	0.3003	1.5015	0.8885	24.6275	39.1	2.9645	1.326 × 103
				Ho_PID [19]	0.3565	1.1531	0.6823	19.0112	39.2	2.5466	844.5659

. From Figure 8 and Table 3, we can see that the proposed PID controller exhibits a satisfactory control performance, especially when the $\xi <0.4$ (such as $G_4\left(s\right)$). The tracking overshoot of $G_2\left(s\right)$ can be reduced by setting the weight of the response to the controller’s proportional gain. It is worth noting that the control effect is good when enlarging the disturbance response of $G_2\left(s\right)$. Both $G_1\left(s\right)$ and $G_3\left(s\right)$ can also achieve good control effect. Overall, the PID tuning methods proposed in this paper trade-off between disturbance rejection performance and robustness.

.

4. Parameter Tuning of PIDD2 Controllers

PIDD2, as a variant of PID, has a new degree of freedom (differential action) in comparison to PID, which can accelerate the transient and closed-loop performance of the control system [42], and contributes to promoting the phase margin, steady-state accuracy and stability of the controlled plant [43]. As such, in this section, we adopt the PIDD2 controller for the control of the underdamped oscillatory systems with time delays.

Similar to the parameter optimization process in Section 3, we can obtain the PIDD2 controller parameters and the fitting curves for normalized plant (8) as shown in Figure 9, the tuning formulas for (8) as (15) and (16).

$$\begin{array}{c}\begin{array}{c}{\overline{K}}_p=0.05731{\overline{\tau }}^{-2.714}-0.05446\\ {\overline{K}}_i=0.071{\overline{\tau }}^{-2.386}+0.1508 \\ \begin{array}{c}{\overline{K}}_d=1.246{\overline{\tau }}^{-0.6861}-0.6993 \\ {\overline{K}}_{dd}=-0.1609{\overline{\tau }}^{0.5542}+0.3826 \end{array}\end{array}(\xi <0.4)\end{array}$$

(15)

$$\begin{array}{c}\begin{array}{c}{\overline{K}}_p=0.1237{\overline{\tau }}^{-2.487}+0.5163 \\ {\overline{K}}_i=0.1587{\overline{\tau }}^{-2.249}+0.2437 \\ \begin{array}{c}{\overline{K}}_d=0.3494{\overline{\tau }}^{-1.375}+0.4229 \\ {\overline{K}}_{dd}=0.01554{\overline{\tau }}^{0.9568}+0.2602\end{array}\end{array} \left(\xi \ge 0.4\right)\end{array}$$

(16)

The tuning formulas (15) and (16) for normalized plant (8) can be converted to the general form of (17) and (18) for general SOPDT plant (1) [41]:

$$\begin{array}{c}\begin{array}{c}{K}_p=\frac{1}{k}\left(0.05731{\left(\frac{\tau }{T}\right)}^{-2.714}-0.05446\right)\\ K_i=\frac{1}{kT}\left(0.071{\left(\frac{\tau }{T}\right)}^{-2.386}+0.1508\right) \\ \begin{array}{c}{K}_d=\frac{T}{k}\left(1.246{\left(\frac{\tau }{T}\right)}^{-0.6861}-0.6993\right) \\ K_{dd}=\frac{T^2}{k}\left(-0.1609{\left(\frac{\tau }{T}\right)}^{0.5542}+0.3826\right)\end{array}\end{array}(\xi <0.4)\end{array}$$

(17)

$$\begin{array}{c}\begin{array}{c}{K}_p=\frac{1}{k}\left(0.1237{\left(\frac{\tau }{T}\right)}^{-2.487}+0.5163\right) \\ K_i=\frac{1}{kT}\left(0.1587{\left(\frac{\tau }{T}\right)}^{-2.249}+0.2437\right) \\ \begin{array}{c}{K}_d=\frac{T}{k}\left(0.3494{\left(\frac{\tau }{T}\right)}^{-1.375}+0.4229\right) \\ K_{dd}=\frac{T^2}{k}\left(0.01554{\left(\frac{\tau }{T}\right)}^{0.9568}+0.2602\right)\end{array}\end{array}\left(\xi \ge 0.4\right)\end{array}$$

(18)

Figure 10, Figure 11 and Figure 12 show the responses of the normalized SOPDT system (8) under different damping $\xi =0.1, 0.4, 0.6$ with a step reference signal at $t=0 s$ and a step input disturbance signal is added to these systems at an appropriate time. It can be observed from the simulation results that:

(1)

The control performance (especially disturbance rejection performance) of the proposed PIDD2 is improved compared with that of PID.

(2)

The control performance of PIDD2 has no significant advantage over the proposed PID as the $\tau /T$ increases. Therefore, PIDD2 can improve the performance of PID for SOPDT systems with small delay; however, for large delay, the control improvement of PIDD2 is not noticeable.

Since PIDD2 controller can improve the performance of PID controller under certain conditions (as explained in the previous subsection), to verify the applicability of PIDD2, we conduct a simulation and a comparative analysis with the step responses of several typical complex oscillatory plants (($G_5\left(s\right),G_6\left(s\right),G_8\left(s\right)$ [21], $G_7\left(s\right)$ [28])).

$$\begin{array}{c} \begin{array}{c}{G}_5\left(s\right)=\frac{15}{(s^2+0.9s+5)(s+3)},G_6\left(s\right)=\frac{18}{(s^2+s+2){(s+3)}^2}\\ G_7\left(s\right)=\frac{15.4358}{(s^2+3.79436s+68.695)}{e}^{-0.4s},G_8\left(s\right)=\frac{20}{(s^2+2.4s+4)\left(s+5\right)}{e}^{-2s}\end{array}\end{array}$$

(19)

Dynamic responses of plants and parameters of the PIDD2 and PID are given in Figure 13 and

Table 4. Parameters of the PIDD2 and PID controllers for (19).

System Parameters				Methods	Controller Parameters				Disturbance Rejection		Robustness Index	ITAE Index
	$\mathit{\xi }$	$\mathit{\tau }$	$\mathit{T}$		${\mathit{K}}_{\mathit{p}}$	${\mathit{K}}_{\mathit{i}}$	${\mathit{K}}_{\mathit{d}}$	${\mathit{K}}_{\mathit{d}\mathit{d}}$	${\mathit{T}}_{\mathit{s}}/\mathit{s}$	$\mathit{\sigma }$%	$\mathit{\epsilon }$
G5	0.1999	0.2	0.4468	PIDD2 PID	0.4533 0.0136	1.4191 0.7958	0.6539 0.44	0.0558	3.11 6.72	23.8 40	1.7173 1.7548	102.1007 181.4656
G6	0.3650	0.48	0.8111	PIDD2 PID	0.1835 −0.083	0.4920 0.3380	0.8813 0.6093	0.1726	12.482 15.721	33.4 42.9	1.6918 1.9740	575.9491 888.3398
G7	0.2289	0.4	0.1207	PIDD2 PID	−0.232 −0.812	5.7124 4.5834	−0.081 −0.106	0.0045	3.5782 4.0958	16 16	3.5385 3.5748	16.3339 21.6371
G8	0.6	2.2	0.4775	PIDD2 PID	0.5191 0.2080	0.5211 0.4661	0.2224 0.1247	0.0746	32.006 36.163	53.9 53.9	5.5247 5.9557	2.36 × 103 3.96 × 103

. From $G_5$ and $G_6$, it can be seen that the control effect of PIDD2 is better than that of PID, both in terms of tracking speed and disturbance attenuation performance, which is further illustrated by combining the two indexes of settling time and overshoot in Table 4. While $G_7$ and $G_8$ are not obviously seen in Figure 13, a numerical analysis and comparison can also result in the same conclusion from Table 4. Based on the results, the main conclusion that can be drawn is that the tuning formula of the PIDD2 controller proposed in this paper can indeed improve the performance of the tuning formula of the PID controller.

5. Application in Power System Control

One of the most important roles in power system operation is to maintain a continuous energy power supply to the consumers while considering quality and security requirements. This objective is achieved by matching the total generation with the total load by using the well-known load frequency control (LFC). It is widely acknowledged that LFC systems are typical oscillatory SOPDT systems [12], which is a very important factor in the operation of power systems and provide fast load disturbance rejection and robust performance in the presence of uncertainties. However, research on frequency stability (balance between power supply and generation) faces serious challenges that constrain the development of power systems. Renewable energy has become a popular research topic in recent years. It is encouraging that the participation of renewable energy sources such as electric vehicles can effectively improve the frequency stability of the power system [44,45]. Therefore, the proposed controller is applied to the LFC system with electric vehicles in this section to test its feasibility.

To illustrate the issue, we take the benchmark system with electric vehicles as an example [46]. The transfer function model of the LFC system is shown in Figure 14. The specific parameter values of the transfer function are: $M=0.1667, D=0.0083, R=2.4, T_g=0.08, T_t=0.3, K_r=0.5, T_r=10, T_e=1, ag=0.9, ae=0.1$. The transfer function of each part is below.

$$\begin{array}{c}\begin{array}{c}\mathrm{Governor}:G_g\left(s\right)=\frac{1}{T_{g}s+1}\\ \mathrm{Turbine}:G_t\left(s\right)=\frac{1}{T_{t}s+1}\\ \begin{array}{c}\mathrm{Reheat} \mathrm{turbine}:G_r\left(s\right)=\frac{K_r{T}_r}{T_{r}s+1}\\ \begin{array}{c}\mathrm{Generator}:G_p\left(s\right)=\frac{1}{Ms+D}\\ \mathrm{Electric} \mathrm{vehicles}:G_e\left(s\right)=\frac{1}{T_{e}s+1}\end{array}\end{array}\end{array}\end{array}$$

(20)

The load frequency control system can be approximated as the SOPDT model [12]. Thus, the approximated SOPDT model parameters are $K=2.3547,T=0.6461,\xi =0.5285,\tau =0.0646$. The controller parameters can be obtained according to (13) and (18).

A step disturbance signal of $\mathsf{\Delta }{P}_d=0.01pu$ is added at $t=5 s$. The dynamic response of the frequency deviation is shown in Figure 15. From Figure 15, we can conclude that the proposed tuning formula has a faster response speed and better disturbance rejection performance. Nevertheless, the proposed tuning formulas have more advantages (disturbance rejection capability) in the load frequency control system.

6. Conclusions

The main goal of the current study was to put forward the tuning formula for PID/PIDD2 controllers for second-order oscillatory systems with time delays. The tuning formulas are obtained by the optimization algorithm to minimize the ITAE under a robustness constraint. The proposed formulas are used on a variety of plants; it should be noted that the PIDD2 controller can improve the performance of the PID under a small time delay, but for large time delay, its effect is not obvious. In contrast to the other PID tuning formulas, the advantage of our study is that the proposed PID/PIDD2 tuning method can provide good robustness and disturbance rejection performance.

The present study confirmed the findings about a new understanding of PID controllers. A further study should consider filter tuning for PID/PIDD2 controllers and PID/PIDD2 control for higher-order oscillatory systems with zeros.