Indirect response-based tuning of PID controller with adjustable robustness

Abstract

The response-based design approach is a straightforward method for proportional–integral–derivative (PID) controller tuning. In this method, an open-loop process response is used to tune the PID controller for fixed robustness. In this article, new response-based PID tuning rules are proposed with adjustable robustness and response. Instead of three PID variables, the proposed PID tuning method allows the user to adjust the process response with a single variable $(ψ)$ . The range of constant $(ψ)$ is provided for ensuring the stability of the process. Three MATLAB/SIMULINK examples and one temperature controller experiment are used to validate the proposed controller performance. The simulation result is also compared with three well-known response-based PID tuning rules Z-N, Cohen-Coon, Astrom et al., and with a model-based PID tuning proposed by Wang et al. Moreover, the proposed tuning provides lesser IAE, ITAE, ISE, and ITSE values than the existing methods.

Keywords

PID controller response-based tuning auto tuning indirect design approach robustness

Introduction

The simple structure, easy implementation, and satisfactory closed-loop performance of the proportional–integral–derivative (PID) controller make it the most extensively used controller in the industrial process. The proportional gain $(K_{c}),$ integral time constant $(T_{i}),$ and derivative time constant $(T_{d})$ are the three tunable parameters that determine the PID controller’s performance. In literature, several PID tuning methods have been developed, which can be classified into three design approaches: response-based design, model-based design, and stochastic optimization.

In the response-based tuning technique, the PID controller’s parameters are tuned based on the processes’ transient response. The response-based PID tuning approach was proposed by Ziegler and Nichols (1942). The Ziegler–Nichols (Z-N) open-loop PID tuning control method, also known as the process reaction curve method, is a widely used tuning method in the process industry. This tuning method is frequently used as the foundation for tuning procedures used by controller manufacturers and the process industry. The first-order plus time-delayed (FOPTD) parameters are used to calculate the PID controller parameters. The Z-N rules only apply to processes with a dead time that is less than half the length of the time constant. After that, Cohen and Coon (1953) made a significant improvement by defining process dynamics with three parameters. The Cohen and Coon (1953) tuning method is similar to the Z-N tuning method where both seek a quarter-delay amplitude response. The Cohen–Coon method, on the contrary, produces better results when the processes with a dead time are less than twice the length of the time constant. However, excessive overshoot problems exist with these tuning methods. The problem of excessive overshoot is solved by Hang et al. (1991) and Åström and Hägglund (1995) by incorporating a set-point weight with the tuned PID controller. Åström et al. updated the frequency-based tuning method of Z-N. Originally the frequency-based PID tuning method of Z-N uses the ultimate gain and an ultimate frequency of the process. whereas Åström et al. provided the PID tuning parameters in terms of system gain, dead time, and the time constant of the process. Zhuang and Atherton (1992) developed analytical tuning rules by using the curve-fitting approach. Response-based PID tuning is still used in industries due to the model-free design.

The limitation of model-based design approaches is that one must identify the parametric transfer function of the process first and then design a suitable controller for the obtained transfer function. Several model-based controller design methods can find in the literature, such as the Smith predictor (Astrom et al., 1994; C.C. Hang et.al., 1989; Mai et al., 2011; Matausek and Micic, 1996; Uma and Rao, 2016; Vranc and Strmc, 2001), internal model control (IMC) (Begum et al., 2017; Garcia and Morari, 1982; Rivera et.al., 1986; Wang et al., 2001; Shamsuzzoha and Lee, 2008; Verma and Padhy, 2019), and stability margin controller design (Åström and Hägglund, 1984; Ho et al., 1997; Khuen et al., 1995). Garcia and Morari (1982) introduced an IMC approach for controlling industrial processes. After that, Rivera et al. (1986) introduced the IMC filter and suggested a PID controller design approach based on the IMC scheme. Following that, literature suggests a variety of IMC filters (Begum et al., 2017; Shamsuzzoha and Lee, 2008; Wang et al., 2001). As the filter adds an extra phase lag to the process, the IMC filter reduces the controller’s performance and robustness. Verma and Padhy (2019) overcame this additional phase lag problem by introducing an IMC-PID indirect design technique that does not require an IMC filter.

Apart from that, the specification of phase and gain margin is another traditional approach of PID design, which involves determining the PID controller parameters using the process transfer function and phase and gain margin analysis. Åström and Hägglund (1984) presented a method for tuning a PID controller to meet specified gain and phase margin. Following that, as indicated by Khuen et al., (1995) and Ho et al. (1997), several phases and gain margin-based approaches are introduced in the literature. The robustness of the model-based PID controller design approach is subject to model accuracy.

In the stochastic optimization-based controller design approach, the control problem must be in a mathematical function called the objective function. After that, the objective function is minimized by using numerical optimization techniques such as the steepest descents method, genetic algorithm (Holland, 1992), particle swarm optimization (Gaing, 2004), or cuckoo search algorithms (Yang et al., 2009) to obtain the optimal value of controller settings. Various objective functions can be found throughout the literature. The majority of them are error-based objective functions like integral absolute error (IAE) (Taylor et al., 2007), integral square error (ISE) (Zhuang and Atherton, 1991a, 1991b), integral time absolute error (ITAE), and integral time square error (ITSE) (Visioli, 2001). However, IAE, ISE, ITAE, and ITSE have no control over the sensitivity and do not focus on the dead time region. To resolve this problem, sigmoidal weighted error performance criteria considering three regions, that is, the error during the dead time, the error during the transient time, and the steady-state error separately, and the exponential weighted error function introduced (Verma, 2017). However, this approach provides a good response, but implementing such a design approach is difficult and time-consuming.

As discussed earlier, to tune a PID controller by model-based and stochastic optimization-based design approach, a parametric transfer function of the process is required. And accurate parametric transfer function modeling of the process is difficult and time-consuming. In contrast, the response-based PID tuning method skips the step of transfer function modeling, which provides the fast tuning of the controller. But the drawback of the response-based controller is fixed robustness. So, it is required new response-based PID controller tuning rules where it is possible to adjust the robustness of the controller.

It is discussed that the existing response-based PID controllers are available for limited robustness and transient performance. These challenges are addressed in this article, and new response-based PID tuning rules are proposed for a wide range of robustness. The indirect design approach-2 (IDA-2) is exploited to derive the proposed tuning rules that offer adjustable robustness and transient performance with a single variable. The proposed tuning variable has a monotonic relation with the robustness of the controller and gives a better transient performance as compared to the existing response-based design methods. A higher phase and gain margin can be achieved by selecting a higher value of $ψ$ within the derived range and vice versa.

The paper is organized in the following manner. Section “IDA-2” explains the idea of an IDA, where a new PID controller parameter can be formulated in the form of constant parameter $ψ$ and pre-existing parameter of the PID controller. Section “Proposed indirect tunning rules and stability margin” explains the proposed tunning formula using Z-N tuning rules and IDA-2. Since the new PID controller parameters are dependent on constant parameter $ψ$ , a suitable range of $ψ$ is required for the desired response, which is also explained in section “Proposed indirect tunning rules and stability margin.” Section “Simulation results” is for validation of the proposed PID controller tuning formula in MATLAB/SIMULATION environment, and real-time validation is done in section “Experimental validation.” Finally, section “Conclusion” concludes the overall work of this paper.

IDA-2

If the open-loop transfer function $C (s) G (s)$ is translated into $C_{i} (s) C_{p} (s - ψ) G (s)$ , the gain margin and phase margin can be adjusted for any stable closed-loop system with open-loop transfer function $C (s) G (s)$ . $ψ$ determines the value of gain and phase margins; increasing the value of $ψ$ will increase the margins, and vice versa (Verma and Padhy, 2020).

Consider the PID controller of the following form

C (s) = K_{C} (1 + \frac{1}{T_{I} s} + T_{D} s)

(1)

where $K_{C}$ stands for proportional gain, $T_{I}$ for integral time constant, and $T_{D}$ for derivative time constant. Equation (1) can be factorized as follows

C (s) = C_{i} (s) C_{p} (s)

(2)

where $C_{i} (s) = K_{C} / s$ and $C_{p} (s) = (T_{D} s^{2} + s + 1 / T_{I})$ . Now by applying the IDA-2 in equation (2), the $ψ$ -dependent adjustable parameters of the controller can be obtained as follow

\begin{matrix} K_{C}^{'} = K_{C} (1 - 2 T_{D} ψ) \\ T_{I}^{'} = \frac{(1 - 2 T_{D} ψ)}{(T_{D} ψ^{2} - ψ + 1 / T_{I})} \\ T_{D}^{'} = \frac{T_{D}}{1 - 2 T_{D} ψ} \end{matrix}

(3)

Proposed indirect tunning rules and stability margin

Proposed tuning rules

The reaction curve parameters, such as time constant $T$ , time delay $θ$ , and gain $K$ , are used to develop the proposed tuning rules, which can be represented in a first order time delayed process as

G (s) = K \frac{e^{- θ s}}{Ts + 1}

The reaction curve and relation with parameters are shown in Figure 1. Consider the Z-N tuning rules for the same reaction curve parameters

K_{C} = \frac{1.2 T}{K θ}; T_{I} = 2 θ and T_{D} = 0.5 θ

It can be observed that the above tuning rules are designed for a fixed value of robustness. These tuning rules can be modified with the IDA-2, as mentioned in equation (3). Equation (3) can be rewritten by substituting the above expression of $K_{c}, T_{I}$ , and $T_{d}$ in equation (3) as

K_{C}^{'} = \frac{1.2 T (1 - θ ψ)}{K θ}

T'_{I} = \frac{2 θ (1 - θ ψ)}{θ^{2} ψ^{2} - 2 θ ψ + 1}

T'_{D} = \frac{0.5 θ}{1 - θ ψ}

where $ψ$ is the shifting variable. The user can change the process response using the shifting variable, which is a tunable parameter. A positive constant value of $ψ$ produces a more robust response, while a negative number produces a more aggressive response than the nominal. In the following section, the limit of the $ψ$ is derived and explained.

Selection of shifting constant parameter $ψ$

The choice of the constant parameter $ψ$ is crucial in fine-tuning the controller. The system will become unstable if the value of the $ψ$ is too high or too low. To avoid instability, a suitable range of $ψ$ needs to be decided.

Consider a PID controller of the following form with the assumption $K_{c} > 0, T_{i} > 0$ , and $T_{d} > 0$

C (s) = \frac{K_{c} (T_{i} T_{d} s^{2} + T_{i} s + 1)}{T_{i} s}

The above form of the PID controller can be realized if both the zeros are real that is

T_{i} \geq 4 T_{d}

(4)

To satisfy the condition in equation (4), assume that the minimum value of $ψ$ $(ψ_{\min})$ is $(- 1 / 2 θ)$ and the maximum value $(ψ_{\max})$ is $(1 / 2 θ)$ .

Substituting the value of $ψ_{\min}$ in the parameter of the proposed tuning rule

\begin{matrix} K_{c} |_{ψ = ψ_{\min}} = \frac{1.8 T}{K θ}, \\ T_{i} |_{ψ = ψ_{\min}} = 1.33 θ, \\ T_{d} |_{ψ = ψ_{\min}} = 0.33 θ \end{matrix}}

(5)

Substituting the obtained value of $T_{i} |_{ψ = ψ_{\min}}$ and $T_{d} |_{ψ = ψ_{\min}}$ in equation (4), it can be observed that the value of $ψ_{\min}$ is satisfying the condition given in equation (4).

Similarly, substituting the value of $ψ_{\max}$ in the parameter of the proposed tuning rule

\begin{matrix} K_{c} |_{ψ = ψ_{\min}} = \frac{0.6 T}{K θ}, \\ T_{i} |_{ψ = ψ_{\min}} = 4 θ, \\ T_{d} |_{ψ = ψ_{\min}} = θ \end{matrix}}

(6)

Substituting the obtained value of $T_{i} |_{ψ = ψ_{\min}}$ and $T_{d} |_{ψ = ψ_{\min}}$ in equation (4), it is observed that value of $ψ_{\min}$ is also satisfying the condition given in equation (4). Therefore, it can be concluded from the above proof that the value of $ψ$ should lie between the range $- 1 / 2 θ$ and $1 / 2 θ$ , that is, $ψ \in (- 1 / 2 θ, 1 / 2 θ)$ .

Simulation results

Proposed tuning rules are validated with the three simulation examples and one experiment on the temperature controller. Normalized dead time $(θ_{n})$ can characterize the process dynamics into three different categories and can be formulated as follows

θ_{n} = \frac{θ}{T + θ}; θ_{n} \in (0, 1)

(7)

when $θ_{n}$ approaches one, the process is known as a delay-dominant process, when $θ_{n}$ approaches zero, it is known as a lag-dominant process, and when $θ_{n}$ approaches 0.5, it is known as a balanced lag and delay process.

Example 1: FOPTD process

The proposed tuning rule is validated using the FOPTD transfer function of the following form

G (s) = \frac{K e^{- θ s}}{Ts + 1}

(8)

where $θ = 5$ is the time delay, $T = 2$ is the time constant, and $K = 1$ is the steady-state gain of the process. The proposed method is also compared with the Cohen–Coon (Cohen and Coon, 1953), Åström and Hägglund (1995), Z-N (Ziegler and Nichols, 1942) PID controller, and Wang et al. (2016).

The normalized dead time $(θ_{n})$ is calculated as 0.7093, that is, this is the case of a dominant delay process. For the comparison purpose, the value of $ψ$ is chosen to be $- 0.1$ which provides a phase margin of 85.30 and a maximum sensitivity of 2.55 (see Table 1). It also provides the IAE, ISE, ITAE, and ITSE of [13.56, 7.1932, 1010, and 172.22], respectively (see Table 2). As shown in Figure 2(a), the proposed method has a better response than the Z-N, Cohen–Coon, Astrom, and Wang et al. methods, and it is also suitable for disturbance handling. The step response is shown in Figure 2(b) for various values of $ψ$ , and it can be shown that the proposed method may change the maximum sensitivity and phase margin. The values of controller parameters, phase margin, and gain margin for different values of $ψ$ are shown in Table 3.

Table 1.

Response comparison of indirect PID tuning rule with Astrom, Cohen-Coon, Z-N, and Wang et al. tuning rule.

Process $(G (s))$	Method	$ψ$	$K_{c}$	$T_{i}$	$T_{d}$	$G_{m}$	$P_{m}$	$M_{s}$	Rise time	Settling time	Overshoot
$\frac{e^{- 5 s}}{2 s + 1}$	Cohen-Coon	–	0.8038	7.2787	1.2433	1.69	85.46	2.43	3.03	40.80	12.81
	Astrom et. al	–	0.3855	10	2.5	2.08	96.56	1.82	58.59	111.20	0
	Z-N	–	0.4921	10	2.5	1.63	98.37	2.30	45.95	89.42	0
	Wang et al.	–	0.4696	5.42	1.05	2.98	80.99	1.51	19.35	41.97	0
	Proposed	−0.1	0.7381	6.6667	1.6667	1.63	85.30	2.55	3.29	38.14	7.83
$\frac{e^{- 0.4 s}}{{(s + 1)}^{2}}$	Cohen and Coon	–	5.6337	1.5461	0.2402	1.3975	11.963	5.2776	0.40254	16.536	80.988
	Astrom et al.	–	3.7483	1.3634	0.3408	2.3613	31.019	2.3223	0.5144	5.8167	45.424
	Z-N	–	4.785	1.3634	0.3408	1.8507	24.681	2.9566	0.4061	6.1950	58.706
	Wang et al.	–	1.1037	1.9760	0.4939	7.1656	76.684	1.2166	2.9229	5.5227	0
	Proposed	0.21	4.1	1.5912	0.3978	2.1206	36.089	2.2578	0.4440	3.5947	40.045
	Proposed	0.73	2.4038	2.7140	0.6785	2.6550	74.2378	1.69	0.7078	5.950	0

Table 2.

Different performance indices of indirect PID tuning rule.

Process $(G (s))$	Method	$ψ$	IAE	ISE	ITAE	ITSE
$\frac{e^{- 5 s}}{2 s + 1}$	Cohen-Coon	–	13.75	7.12	1025	167.48
	Astrom et al.	–	38.90	15.21	3689	692.86
	Z-N	–	30.48	12.03	2716.9	479.58
	Wang et al.	–	17.36	8.65	1325	265.132
	Proposed	−0.1	13.56	7.19	1010	172.22
$\frac{e^{- 0.4 s}}{{(s + 1)}^{2}}$	Cohen-Coon	–	4.18	1.55	23.55	5.672
	Astrom et al.	–	2.51	1.31	10.57	2.774
	Z-N	–	2.58	1.34	11.26	2.919
	Wang et al.	–	2.87	1.67	12.83	3.877
	Proposed	0.21	2.03	1.13	7.54	2.083
	Proposed	0.73	2.05	1.15	8.24	1.896

Table 3.

Effect of $ψ$ on phase margin and maximum sensitivity.

Process	$ψ$	$K_{c}$	$T_{i}$	$T_{d}$	$P_{m}$	$M_{s}$
$\frac{e^{- 5 s}}{2 s + 1}$	−0.06	0.64	7.69	1.92	92.19	2.48
	−0.04	0.59	8.33	2.08	94.77	2.52
	−0.02	0.54	9.09	2.27	96.82	2.46
	0.01	0.47	10.53	2.63	98.95	2.47
$\frac{e^{- 0.4 s}}{{(s + 1)}^{2}}$	−0.147	5.26	1.30	0.30	18.3	3.8
	−0.047	4.94	1.39	0.33	23.1	3.1
	−0.053	4.6	1.48	0.35	28.1	2.7
	0.153	4.29	1.59	0.38	33.4	2.4
	0.253	3.96	1.71	0.41	39.11	2.2

Example2: Second-order plus time-delay process

The following form of the second-order plus time-delay (SOPTD) process is considered. The same process has also been considered by Hang et al. (1991)

G (s) = \frac{e^{- 0.4 s}}{{(s + 1)}^{2}}

The identified FOPTD process of the above transfer function from the reaction curve is as follows

G (s) = \frac{0.998 e^{- 0.6817 s}}{2.7128 s + 1}

The value of normalized dead time $(θ_{n})$ is calculated as 0.20, which is the case of a lag-dominant process. The suitable range of $ψ$ is (−0.7335, 0.7335), and for the comparison purpose, the value of $ψ$ is chosen to be 0.21 and 0.73, which provides the maximum sensitivity of (2.2578, 1.69) and a phase margin of (36.089, 74.2378), respectively (see Table 1). It also provides the value of IAE, ISE, ITAE, and ITSE (2.03, 1.13, 7.54, and 2.083) for $ψ = 0.21$ and (2.05, 1.15, 8.24, and 1.896) for $ψ = 0.73$ , respectively (see Table 2). It is seen from Figure 3(a), the response of the proposed method provides faster disturbance rejection than the Z-N, Cohen–Coon, Astrom et al., and Wang et al. Figure 3(b) depicts the step response for various values, whereas Table 3 lists the controller parameters, phase margin, and gain margin for different values of $ψ$ .

Example3: Higher-order plus time-delay process

The higher-order plus time-delay (HOPTD) process of the following form is considered.

G (s) = \frac{1}{{(s + 1)}^{10}} e^{- s}

Following is the identified FOPTD process of the above transfer function from the reaction curve

G (s) = \frac{0.9997}{7.5876 s + 1} e^{- 6.8685 s}

The value of normalized dead time $(θ_{n})$ is calculated as 0.4751, which is the case of a balanced lag and delay process. It is observed that Z-N and Cohen–Coon methods provide an unstable response, whereas Astrom et al. and proposed tuning rules provide a stable response. The value of the controller parameter is determined by setting the value of $ψ$ to 0.065 in the proposed tuning method. Figure 4 shows the step response for Z-N, Astrom et al., and the proposed method. The values of the controller parameter $[K_{c}, T_{i}, T_{d}$ ] for Cohen–Coon [1.7419, 12.9052, 2.1518], for Astrom et al. [ 1.0387, 13.7370, 3.4342], for Z-N [1.326, 13.737, 3.4342], and proposed tuning method [0.7340, 24.8162, 6.2040], respectively. The gain margin, phase margin, and maximum sensitivity are observed to be [1.25, 94.4132, 4.5386] for Astrom et al. and [1.125, 114.85, 5.5144] for the proposed tuning method.

Experimental validation

An experimental setup of the temperature control laboratory depicted in Figure 5(a) is used to validate the proposed method. The first-order time-delay transfer function model of the temperature control system is obtained using the reaction curve method. Equation (9) represents the temperature control system’s first-order transfer function model

G (s) = \frac{0.8221 e^{- 71.9 s}}{67.300 s + 1}

(9)

For the identified transfer function model, the PID controller is designed by using the proposed indirect PID tuning rule shown in Table 4. The designed Indirect PID controller parameter is given in Table 5. The suitable range for the $ψ$ is −0.007 to 0.007. For the comparison purpose, the value of $ψ$ is chosen to be −0.007 and −0.002. The performance of the designed controller is shown in Figure 5(b) with the set point of $40^{°} C$ . The Indirect PID tuning rules give IAE of [1106.3, 1518.6] and ISE of [6456.5, 9774.6] when the value of $ψ$ is chosen to be −0.007 and −0.002, respectively. For the same setpoint value, Astrom et al., Cohen-Coon, Z-N, and Wang et al. give IAE of [2869, 1323.8, 1412.6, 1138.3] and ISE of [16,757, 8646.6, 9294.4, 6771], respectively.

Table 4.

Indirect PID tuning rules.

$K_{c}$	$T_{i}$	$T_{d}$
$\frac{1.2 T (1 - θ ψ)}{K θ}$	$\frac{2 θ (1 - θ ψ)}{θ^{2} ψ^{2} - 2 θ ψ + 1}$	$\frac{0.5 θ}{1 - θ ψ}$

Table 5.

PID controller parameter for (9).

Method	$ψ$	$K_{P}$	$T_{i}$	$T_{d}$	IAE	ISE
Astrom et al.	–	1.070	143.8	35.95	2869	16,757
Cohen-Coon	–	1.841	130.221	21.920	1323.8	8646.6
Z-N	–	1.366	143.800	35.950	1412.6	9294.4
Wang et al.		2.3969	33.7153	8.1179	1138.3	6771.4
Proposed	−0.007	2.054	95.656	23.914	1106.3	6456.5
Proposed	−0.002	1.562	125.721	31.430	1518.6	9774.6

Conclusion

A new response-based PID controller tuning rule is proposed in this article. It becomes possible to adjust the phase margin and maximum sensitivity of the process without direct tuning the controller with the help of the IDA-2 method. A too-low and too-high value of $ψ$ may lead the process to instability. Therefore, it is derived that the suitable range of the constant parameter $(ψ)$ should vary between $- 1 / 2 θ$ and $1 / 2 θ$ to keep the process in a stable region. The simulation results, experiment, and obtained data show that the proposed method provides a low value of IAE, ISE, ITSE, and ITAE. The overall performance of the PID controller is enhanced. It is shown in simulation and experimental results that an aggressive response can be achieved by reducing the value of the proposed tuning variable within the given range and vice versa. Optimal controller responses exist within the given range of $ψ$ ; in the future, it can be studied in detail.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Bipin Singh

Bharat Verma

Sudeep Sharma

Prabin Kumar Padhy

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Indirect response-based tuning of PID controller with adjustable robustness

Abstract

Keywords

Introduction

IDA-2

Proposed indirect tunning rules and stability margin

Proposed tuning rules

Selection of shifting constant parameter ψ

Simulation results

Example 1: FOPTD process

Example2: Second-order plus time-delay process

Example3: Higher-order plus time-delay process

Experimental validation

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

Selection of shifting constant parameter $ψ$