Transfer function identification of a liquid flow-line pressure control system from simple relay auto-tuning

Abstract

Accurate modeling of liquid flow systems is necessary for the effective operation of the plant under different conditions. The plant dynamics may change with time, and due to this, the operation may not be satisfactory. In this article, relay autotuning of a liquid flow-line is proposed. Well-known describing function approximation is utilized for the estimation of gain of relay used in a feedback loop to excite the plant to be identified. The expressions are derived for the identification of a class of overdamped second-order plus dead-time and first-order plus dead-time plant dynamics. The identification method tested on different standard systems shows that the proposed method has good identification accuracy. Therefore, the proposed identification method is applied to a liquid flow line pressure control system. The experimental results show that an accurate dynamic model is extracted using the proposed method with a minimal set of data from an obtained limit cycle oscillation. The measured quantities of the limit cycle are utilized in the derived explicit expressions for the identification of unknown variables of the dynamic plant model. A distributed process control system is utilized here as a standard industrial platform to design virtual relay and conduct the proposed autotuning. The efficiency of the modeling method is demonstrated through the comparison of Nyquist and relay response plots with the transfer functions obtained from the MATLAB system identification toolbox.

Keywords

Dead-time transfer function modeling and identification plant dynamics pressure control system relay autotuning

Introduction

In various industrial sectors such as oil-gas, thermal power, food-drinks, and different bulk product plants, monitoring of ultimate process variables is an important and essential job to the controlled engineers. With the enhancement of intelligent control methods on model-free plants, conventional controllers also play a vital role more directly on plants with known dynamic models, which are generally identified from classical relay autotuning, step test, and pulse testing. Modeling can be done using different methods, but getting the correct model of the plant is very important as it helps in designing a controller based on the plant dynamics with which it is possible to achieve productive and fruitful control action. The plant dynamics may not remain the same all the time, and it may change with time due to environmental factors, load changes, and noise. As a result, the performance of the plant degrades irrespective of the controller design. Therefore, it is necessary to know the fundamental dynamic of a plant. In tutorial reviews in the last three to four decades, Hang et al. (2002) and Liu et al. (2013) have given various process identification schemes using step tests and relay autotuning experiments. In the literature based on input-output characteristics and relay configurations, the autotuning methods are categorized as symmetrical (Bajarangbali et al., 2014; Thyagarajan and Yu, 2003; Vivek and Chidambaram, 2005) and asymmetrical (Atherton and Majhi, 1998; Kaya and Atherton, 2001; Korbel and Prokop, 2015; Liu et al., 2008; Ramakrishnan and Chidambaram, 2003; Shen et al., 1996; Theorin and Berner, 2015) feedback tests. The process output is controlled under such relay configurations by varying relay amplitudes. Ziegler and Nichols (1942) proposed the universal tuning rules for the conventional proportional-integral-derivative (PID) controller in a simplified fashion. Later, the various difficulties of the Z-N method of controller tuning were overcome by Åström and Hägglund (1984) with the help of relay autotuning. An online PI controller tuning guideline based on half-limit cycle data from the closed-loop relay test was also presented by Mehta and Majhi (2012). Their proposed identification and controller tuning method works well on the process of electronic circuits as the electronic system always has lower values of time constants as well as dead-time or time delay. Recently, another online identification method was proposed by Ghorai et al. (2019) based on PD-controller in parallel with relay for industrial processes. In this research, the identification accuracy is high enough in comparison with off-line based modeling, but the presence of an additional controller and its tuning makes the method more complex and sophisticated for simple industrial applications. Further, Pandey and Majhi (2020) proposed online first-order plus dead-time (FOPDT) model identification based on state-space analysis on relay autotuning in parallel with a PI-controller, but the authors have not shown the guaranteed stability of limit cycles as the presence of I-controller may not allow the system to generate stable oscillation. Panda and Yu (2003) proposed an interesting and systematic set of analytical expressions for relay responses from the detailed study of the order of plant transfer function as well as the value of damping factor for second-order dynamics. Further, Berner et al. (2016) proposed an asymmetrical autotuning scheme with the adaptivity of relay heights. They have also discussed several issues on the choice of parameters, handling of disturbances, and other sources of error in a subtle way. However, from the complexity point of view, the proposed method is simpler than their proposed method. In fact, the proposed method is so simple that any control engineer can easily be adapted because of its straight forward application procedure and less computational effort and time.

The relay feedback based identification is a classical method and used extensively in most of the process industries. However, earlier methods reported in the literature suffer difficulty in the generation of sustained oscillation in real-time systems because of relay configuration. In most of the chemical industrial plants, it may not be possible for users to consider such parameters within the absolute full scale, such as a pressure control system. Though pressure becomes negative in the absolute scale, however, in maximum industrial processes, the selected working range of pressure is in the only positive zone. Again, the chemical processes comprise several parameters of unknown subsystems in their corresponding control loops, and in general they are approximated in terms of the lower-order transfer function (TF) model. In practice, it is not always possible to generate limit cycle output from all industrial processes while utilizing the classical relay autotuning with zero setpoints. Hence, the investigation is required for an alternative method, which helps in the generation of sustained oscillation through conducting an identification test using classical relay autotuning on the real-time plant. The relay needs to be configured around a reference value known as the setpoint to avoid the typical issues involved in the conventional relay autotuning. Moreover, several frequency and time domain methods (Panda, 2006; Pekař, 2013) have been reported to estimate the unknown plant models in terms of TF form. The state-space approaches proposed by Atherton and Majhi (1998); Ghorai et al. (2016); and Majhi (2007) yield more accurate process models as compared with the frequency domain method. This is because in frequency domain methods, relay gains are generally approximated by describing function (DF) approximation. However, the frequency domain approach provides a more straightforward set of expressions that are highly preferable to derive unknown model parameters.

The non-unity steady-state gain of plant dynamic is quite observable in most of the real-time cases, and due to this, the generation of limit cycles is asymmetrical. Again, sensor, measurement, and environmental noises are also the key factors to bring asymmetricity in limit cycles even though the relay configuration is symmetrical. Therefore, the symmetrical relay autotuning methods fails to identify plant dynamics if the obtained limit cycles are asymmetrical. This is because the expression for equivalent relay gain from DF approximation, proposed by Åström and Hägglund (1984), may not be applied directly to deduce analytical expressions for model parameters if the steady-state plant gain is other than unity. Therefore, the expression for equivalent gain of symmetrical relay configuration is extended for an asymmetrical setpoint weighted relay, which can be directly applied for autotuning and effectively used to derive the simple set of mathematical expressions.

The remaining portions of the paper are arranged as follows. In section ‘Dead-time plant dynamics’, models of dead-time plants are described followed by the development of analytical expressions from relay autotuning procedures and modeling plant dynamics in section ‘Autotuning and identification’. Then, simulation studies are included, followed by autotuning on a real-time pressure control system and unknown model parameters identification in section ‘Autotuning on real-time system’. Finally, the conclusion is drawn.

Dead-time plant dynamics

An unknown plant to be identified in the form of dead-time TF is kept under a setpoint weighted asymmetrical relay autotuning scheme, as shown in Figure 1. Here, attempts have been made for the estimation of plant dynamical model parameters. A dead-time overdamped second-order plus dead-time (SOPDT) plant TF can be written as

G (s) = \frac{k e^{- δ s}}{(τ_{1} s + 1) (τ_{2} s + 1)}

(1)

where $k$ is the plant gain, $τ_{1}$ and $τ_{2}$ are the two time-constants and $δ$ is dead-time of the considered plant model. When $τ_{2} = 0$ , the above model becomes stable FOPDT plant dynamics. Therefore, the frequency domain representation of the assumed stable SOPDT process TF model can be written as

G (j ω_{c}) = \frac{k e^{- j δ ω_{c}}}{(j ω_{c} τ_{1} + 1) (j ω_{c} τ_{2} + 1)}

(2)

where $ω_{c} = \frac{2 π}{τ_{p}}$ is the frequency of the obtained limit cycle oscillation from relay autotuning.

Autotuning and identification

In this relay autotuning experiment, a relay whose heights are asymmetrical with respect to the setpoint is placed in the closed-loop along with the plant to produce limit cycle output, as shown in Figure 2. The process input $u (t)$ , that is, relay switching is given as

u (t) = {\begin{matrix} \begin{matrix} (R + h_{1}^{'}) & if y (t) < setpoint (R) \end{matrix} \\ \begin{matrix} (R - h_{2}^{'}) & if y (t) > setpoint (R) \end{matrix} \end{matrix}

(3)

where, $R$ is the setpoint given to the plant, $(R + h'_{1})$ and $(R + h'_{2})$ are the relay heights at ON and OFF conditions, respectively. Since the relay heights are chosen with respect to the setpoint, the generation of sustained oscillations from the considered dead-time processes is possible only when the condition given in Ghorai et al. (2016) is satisfied. Thereafter, the measured quantities are obtained from a detailed analysis of sustained oscillatory output, as shown in Figure 2, and substituted in derived mathematical expressions for the evaluation of plant dynamic parameters. Hence, the overall TF of the closed-loop control shown in Figure 1 is written as

T_{cl} (s) = \frac{N (A) G (s)}{1 + N (A) G (s)}

(4)

where $N (A)$ is the approximate gain of the switching element known as the relay, considered in closed-loop control. Using DF approximation, the modified expression of gain of the setpoint weighted asymmetrical relay with hysteresis ( $ε$ ) is derived from Åström and Hägglund (1984) and given in equation (5). Though, we have considered $ε = 0$ throughout the manuscript to simplify the mathematical expressions. However, non-zero hysteresis value can be considered in a noisy environment to improve identification accuracy

N (A) = \frac{2 (h_{1}^{'} + h_{2}^{'}) [\sqrt{({A_{m}}^{2} (1 - {(\frac{B}{A_{m}})}^{2}) - ε^{2})} - j ε]}{π {A_{m}}^{2}}

(5)

where $h'_{1} = (h_{1} - R)$ and $h'_{2} = (R - h_{2})$ , are the operative relay amplitudes at ON and OFF conditions, respectively. $A_{u}$ and $A_{l}$ are the upward and downward peak amplitudes, respectively, in reference to the setpoint. $A_{m} = \frac{(A_{u} + A_{l})}{2}$ , is the average peak amplitude, and $B = | A_{u} - A_{m} | = | A_{l} - A_{m} |$ is the bias between upward and downward peak amplitudes. From close-loop TF (3), the characteristic equation becomes

N (A) G (j ω_{c}) = - 1

(6)

At the sustained oscillatory situation, the gain and phase margin conditions observed by Luyben (1987) are written as

| N (A) G (j ω_{c}) | = 1

(7)

and

∠ [N (A) G (j ω_{c})] = - π

(8)

respectively. In the next two subsections, the above two expressions are extended for the identification of two unknown model parameters of overdamped SOPDT and FOPDT plant dynamics.

Identification of overdamped SOPDT dynamics

In this subsection, a set of mathematical expressions for the estimation of a class of overdamped SOPDT processes using an ideal relay is deduced. These general expressions can be extended further for the identification of FOPDT process model parameters while substituting one of the eigenvalues equal to zero. Substituting (2) and (5) in the condition given by (6), the following expression is obtained

\frac{k e^{- j ω_{c} δ} [2 (h_{1}^{'} + h_{2}^{'}) \sqrt{1 - {(\frac{B}{A_{m}})}^{2}}]}{π {A_{m}}^{2} (j ω_{c} τ_{1} + 1) (j ω_{c} τ_{2} + 1)} = - 1

(9)

Assuming the unknown process yields sustained oscillations, then (9) can be written in more simplified form using magnitude condition given in (7) as

\frac{[2 k (h_{1}^{'} + h_{2}^{'}) \sqrt{1 - {(\frac{B}{A_{m}})}^{2}}]}{π A_{m} \sqrt{({(ω_{c} τ_{1})}^{2} + 1) ({(ω_{c} τ_{2})}^{2} + 1)}} = 1

(10)

and phase condition from (8) as

\tan^{- 1} (ω_{c} τ_{1}) + \tan^{- 1} (ω_{c} τ_{2}) = π - ω_{c} δ

(11)

Now, from (10)

τ_{1} + τ_{2} = \sqrt{\frac{1}{ω_{c}^{2}} ({(N (A))}^{2} - 1) + 2 τ_{1} τ_{2} - {(Ω_{c} τ_{1} τ_{2})}^{2}}

(12)

and from (11)

τ_{1} τ_{2} = \frac{1}{ω_{c}^{2}} (1 - \frac{ω_{c} (τ_{1} + τ_{2})}{\tan (π - ω_{c} δ)})

(13)

Hence, the above two equations can be written in terms of sum and product of time constants as

{ω_{c}}^{2} ({s_{τ}}^{2} - 2 p_{τ}) + {p_{τ}}^{2} {ω_{c}}^{4} + 1 = k^{2} {(N (A))}^{2}

(14)

and

p_{τ} = \frac{1}{{ω_{c}}^{2}} (1 - \frac{s_{τ} ω_{c}}{\tan (π - ω_{c} δ)})

(15)

where, $s_{τ} = (τ_{1} + τ_{2})$ and $p_{τ} = τ_{1} τ_{2}$ . Substituting $p_{τ}$ from (15) in (14) and simplification $s_{τ}$ becomes

s_{τ} = \pm \frac{\tan (π - ω_{c} δ)}{ω_{c}} \frac{k (N (A))}{\sqrt{\tan^{2} (π - ω_{c} δ) + 1}}

(16)

Again, substituting $s_{τ} = - \frac{\tan (π - ω_{c} δ)}{ω_{c}} \frac{kN (A)}{\sqrt{\tan^{2} (π - ω_{c} δ) + 1}}$ from (16) in (15), $p_{τ}$ becomes

p_{τ} = \frac{1}{{ω_{c}}^{2}} (\frac{kN (A)}{\sqrt{\tan^{2} (π - ω_{c} δ) + 1}} + 1)

(17)

Now, the difference $d_{τ} = (T_{1} - T_{2})$ is written in terms of $s_{τ}$ and $p_{τ}$ as

{d_{τ}}^{2} = {s_{τ}}^{2} - 4 p_{τ}

(18)

d_{τ} = \pm \sqrt{({s_{τ}}^{2} - 4 p_{τ})}

(19)

Considering the positive root of the above equation and substituting $s_{τ}$ and $p_{τ}$ from (16) and (17), respectively, in (19), we get

d_{τ} ω_{c} = \sqrt{\frac{k^{2} {(N (A))}^{2} \tan^{2} (π - ω_{c} δ)}{\tan^{2} (π - ω_{c} δ) + 1} - 4 [\sqrt{\frac{k^{2} {(N (A))}^{2}}{\tan^{2} (π - ω_{c} δ) + 1}} + 1]}

(20)

Using (16) and (20) the explicit expressions for two unknown parameters $τ_{1}$ and $τ_{2}$ can be derived as

τ_{1} = \frac{1}{2 ω_{c}} [\sqrt{\frac{k^{2} {(N (A))}^{2} \tan^{2} (π - ω_{c} δ)}{\tan^{2} (π - ω_{c} δ) + 1}} + \sqrt{\frac{k^{2} {(N (A))}^{2} \tan^{2} (π - ω_{c} δ)}{\tan^{2} (π - ω_{c} δ) + 1} - 4 \sqrt{\frac{k^{2} {(N (A))}^{2}}{\tan^{2} (π - ω_{c} δ) + 1}} - 4}]

(21)

τ_{2} = \frac{1}{2 ω_{c}} [\sqrt{\frac{k^{2} {(N (A))}^{2} \tan^{2} (π - ω_{c} δ)}{\tan^{2} (π - ω_{c} δ) + 1}} - \sqrt{\frac{k^{2} {(N (A))}^{2} \tan^{2} (π - ω_{c} δ)}{\tan^{2} (π - ω_{c} δ) + 1} - 4 \sqrt{\frac{k^{2} {(N (A))}^{2}}{\tan^{2} (π - ω_{c} δ) + 1}} - 4}]

(22)

An overdamped second-order plant dynamic given in (1) has four parameters $k$ , $δ$ , $τ_{1}$ and $τ_{2}$ , to be estimated for effective modeling. The plant gain $k$ is measured from integration-cum-area method as described in Atherton and Majhi (1998), where the areas under the plant input and output are written as

A'_{u} = \int_{0}^{τ_{p}} u (t) dt

(23)

A'_{y} = \int_{0}^{τ_{p}} y (t) dt

(24)

Therefore, the expression for steady-state gain, $k$ is derived from the ratio of (24) and (23) as

k = \frac{A_{y}^{'}}{A_{u}^{'}}

(25)

and dead-time can be measured from the derivative method, as suggested by Majhi (2007). This dead-time is responsible for the internal delay of the plant, signal or material(s) traveling path length, computational time, and many more. It has been proved by Majhi (2007), from extensive simulation studies as well as mathematically, that the dead-time or time delay parameter can be deduced for an $n^{th}$ -order plant with $n^{th}$ -order differentiation of its half-limit cycle output. From this phenomenon, it has been observed that the $2^{nd}$ derivative of the limit cycle output obtained from relay autotuning of a $2^{nd}$ order plant shows an abrupt change in slope after $δ$ unit of time from the beginning of the upper peak of limit cycle output. Once $k$ and $δ$ are estimated, two time-constants $τ_{1}$ and $τ_{2}$ can be determined by substituting the required measured quantities in the above explicit expressions.

Identification of FOPDT dynamics

The mathematical expressions are derived using an asymmetrical relay autotuing for stable FOPDT plant given in (1) considering $τ_{2} = 0$ . The necessary measured quantities for the FOPDT dynamic are obtained from limit cycle data. The process gain $k$ is obtained from the area method, as explained in the previous subsection. After that, $δ$ and $τ_{1}$ are derived from the marginal conditions of gain and phase of the characteristic equation presented in (6) with substitution of steady-state gain $k$ and required quantities. Now, substituting $G (j ω_{c})$ and $N (A)$ from (2) and (5), respectively, in (7) and (8) $τ_{1}$ and $δ$ are obtained as

τ_{1} = \frac{\sqrt{(k^{2} {(N (A))}^{2} - 1)}}{ω_{c}}

(26)

δ = \frac{1}{ω_{c}} [π - \overset{- 1}{\tan} (\sqrt{(k^{2} {(N (A))}^{2} - 1)})]

(27)

The effectiveness of the dynamic modeling of FOPDT and SOPDT is given in the next subsequent sections through simulation studies as well as the identification of a simple real-time single-input single-out (SISO) plant model.

Simulations

The proposed identification method is tested on two different processes. The identification error ( $I_{e}$ ) of the identified process is obtained using (28)

I_{e} = \int_{0}^{ω_{c}} | \frac{G_{a} (j ω) - G (j ω)}{G_{a} (j ω)} | d ω

(28)

where $G_{a} (j ω)$ is the actual process model. The identification of the process is carried out under the presence of load disturbance as well as in noisy environments, and the noisy signal is fitted through the utilization of the Fourier series based Curve Fitting Toolbox (2002) available in MATLAB. Thereafter, limit cycle parameters are measured from a fitted signal for further computation and plant modeling.

Example 1

Consider an overdamped SOPDT process $G_{1} (s) = \frac{e^{- 2 s}}{(10 s + 1) (s + 1)}$ . Three relay feedback tests have been performed by setting $R = 50$ , $h_{1} = 80$ and $h_{2} = 10$ . The first one is performed without any noise and load, then the second one is performed with a step load disturbance of magnitude 0.2, and the third one is performed with a noise of 15 dB (SNR). The detailed measurements from limit cycle output along with the relay configurations for each of the three cases are presented in Table 1. The measured quantities are substituted in derived expressions (26) and (27) to obtain two unknown parameters of the FOPDT plant model with the prior estimation of $k$ from (25) for each case, and the obtained models are presented in Table 2. Thereafter, three overdamped SOPDT models are derived from the corresponding set of same limit cycle data by the utilization of derived expressions (21) and (22) with prior estimations of dead-time $δ$ from the $2^{nd}$ derivative of limit cycle output and $k$ from (25) for each case. All the deduced models are shown in Table 2 with their calculated identification errors in comparison with the model obtained from the method by Ramakrishnan and Chidambaram (2003) as well as Ghorai et al. (2019). The proposed overdamped SOPDT model has an error $I_{e}$ of 0.0433, and for the FOPDT model, the error is 0.0204. For the same process, Ramakrishnan and Chidambaram (2003) have identified the model with an error of 0.0236. From the comparative table, it is observed that the calculated error for each of the proposed models is very less, almost of the order of $10^{- 2}$ , and sometimes lesser than the earlier proposed model. Another interesting observation is that for the considered plant dynamic, the model proposed by Ghorai et al. (2019) has a higher error value, which is not expected because their method is online, and it is obvious that the online method should give lower error than the off-line method. However, this might be the fact of model mismatch error as their method is applicable only for underdamped SOPDT dynamic, and in this case, the obtained model by utilizing their method is an overdamped SOPDT.

Table 1.

Parameters considered in simulation study and measured quantities.

Ex.	$u (t)^{§}$	SNR^★	${σ^{2}}^{*}$	$τ_{p}$	$A_{u}$	$A_{l}$	$A'_{y}$	$A'_{u}$
$1$	$R = 50; h_{1} = 80; h_{2} = 10$	–	–	$10.617$	$5.809$	$7.776$	$520$	$520$
		$L = 0.2$	–	$10.609$	$5.84$	$7.737$	$521$	$521$
		$15$ dB	$93.141$	$10.7126$	$5.264$	$7.527$	$514$	$520$
$2$	$R = 50; h_{1} = 80; h_{2} = 10$	–	–	$10.684$	$29.9157$	$39.864$	$491$	$491$
		$L = 0.2$	–	$10.683$	$30.114$	$39.665$	$496$	$493$
		$15$ dB	$918.696$	$10.696$	$34.332$	$44.924$	$503$	$492$

Relay configuration or plant input; ★ or load disturbance;* variance.

Table 2.

Identified dynamics from simulation study for the considered two different processes.

Actual process	Method of identification	Identified model	$I_{e}$
$\frac{1.0 e^{- 2 s}}{(10 s + 1) (s + 1)}$	Proposed SOPDT model without noise & load disturbance	$\frac{1.0 e^{- 2.0006 s}}{(9.0570 s + 1) (1.0882 s + 1)}$	$0.0433$
	Proposed FOPDT model without noise & load disturbance	$\frac{e^{- 2.9166 s}}{(10.8289 s + 1)}$	$0.0204$
	Model proposed by Ramakrishnan and Chidambaram (2003)	$\frac{1.05 e^{- 1.814 s}}{(9.766 s + 1) (1.271 s + 1)}$	$0.0236$
	Proposed SOPDT model with load disturbance	$\frac{0.9959 e^{- 2.0006 s}}{(9.0076 s + 1) (1.0938 s + 1)}$	$0.0456$
	Proposed FOPDT model with load disturbance	$\frac{0.9959 e^{- 2.92005 s}}{(2.92005 s + 1)}$	$0.0201$
	Proposed SOPDT model with 15 dB noise	$\frac{1.001 e^{- 2.229 s}}{(11.0125 s + 1) (0.7212 s + 1)}$	$0.0322$
	Proposed FOPDT model with 15 dB noise	$\frac{1.001 e^{- 2.8907 s}}{(11.9966 s + 1)}$	$0.0554$
	SOPDT model by Ghorai et al. (2019)	$\frac{1.0 e^{- 2.0008 s}}{(9.6742 s^{2} + 10.2272 s + 1)}$	$0.1495$
$\frac{1.0 e^{- 4 s}}{{(0.5 s + 1)}^{3}}$	Proposed FOPDT model without noise & load disturbance	$\frac{1.0 e^{- 4.2238 s}}{1.3132 s + 1}$	$0.0265$
	Model from ATV method proposed by Liu et al. (2008)	$\frac{1.0841 e^{- 4.4239 s}}{(1.6362 s + 1)}$	$0.0365$
	Proposed FOPDT model with load disturbance	$\frac{0.9956 e^{- 4.2359 s}}{(1.2948 s + 1)}$	$0.0278$
	Proposed FOPDT model with 15 dB noise	$\frac{0.9976 e^{- 4.566 s}}{(0.8203 s + 1)}$	$0.2152$
	SOPDT model by Ghorai et al. (2019)	$\frac{1.0 e^{- 4.0 s}}{(0.5514 s^{2} + 1.4739 s + 1)}$	$0.0497$

Example 2

Let us consider a higher-order process $G_{2} (s) = \frac{e^{- 4 s}}{{(0.5 s + 1)}^{3}}$ . The asymmetrical relay heights $h_{1}$ and $h_{2}$ are chosen as 80 and 10, while the setpoint input $R$ is set to 50 during the relay feedback test. Similar to Example 1, three relay feedback tests have been performed, and the measured quantities are tabulated in Table 1 with the above-mentioned relay settings, setpoint, and also with the same load disturbance and noise. Utilizing the measured quantities in the derived expressions (26) and (27) to obtain two unknown parameters of the FOPDT plant model with the prior estimation of $k$ from (25) for each case and three FOPDT models are obtained. All the deduced models are shown in Table 2 with their calculated identification errors in comparison with the model obtained from the method proposed by Liu et al. (2008) as well as Ghorai et al. (2019). The proposed FOPDT models without load disturbance and noise have an error of 0.0265 and with load 0.0278, whereas the enhanced autotune variation method proposed a FOPDT model has an error of 0.0365. On the other hand, the propped noisy model has an error of 0.2152, which is a little higher, and this might be the fact of higher-order of the actual plant. From the table, in comparison to the model proposed by Ghorai et al. (2019), it is observable that the identification error for the proposed method is almost same or lower than the online method. However, the online method is computationally complex and cost-effective as it needs an additional controller for model estimation.

Autotuning on real-time system

The implementation of the proposed identification technique on a real-time system is presented here. The proposed relay feedback test has been conducted on a simple real-time flow-line liquid pressure control system to show the practical implementation of autotuner and successive identification of plant dynamics.

Experimental set-up

A distributed control system (DCS), along with industry-standard field instruments, are utilized to conduct the relay feedback experiment. Yokogawa (2002) Field Control Station and CENTUM CS3000 are employed to build the setpoint weighted relay logic for the conduction of autotuning. A VLnet (control bus) and its controller are utilized to establish the real-time data communication between FCS and engineering station. The process and instrumentation diagram of the simple pressure control scheme along with a photograph of the experimental set-up is shown in Figure 3. The back-pressure of the pipe-line is considered here as a final controlled variable. A pump (P300) is used to supply liquid to the pipe-line from reservoir and the back-pressure of the line is controlled by an automatic pressure control valve (PCV). MV12 and MV13 are two manual control valves, used to connect a pressure transmitter (PT) for remote pressure signal transmission to FCS and a pressure gauge (PG) for local display, respectively. A third manual control valve (MV11) is used as an auxiliary pressure control element to bypass the line, which was in a fully closed position while conducting relay autotuning. The detail experimental results are well described in the next subsection.

Experimental results

During the experiment, a reference input is given as a pipe-line liquid desired pressure percentage scale through setpoint weighted relay feedback. An asymmetrical relay of height h₁ = 90.0 and h₂ = 20.0 produces an asymmetrical sustained limit cycle output around a setpoint of R = 55.0% as shown in Figure 4. From the starting point of the autotuning relay was in ON condition and water back pressure started to rise up to setpoint value, once the pressure crossed the setpoint relay started switching and turned OFF. Just after the moment when pipe-line pressure reaches below the setpoint, the relay turned ON. The process is continued, and as a result, the sustained oscillatory output is generated. The first and second derivatives of the limit cycle with respect to time are also plotted in Figure 4 and utilized to measure the plant dead-time (δ) during the estimation of parameters of overdamped SOPDT dynamics. From the yielded limit cycle output important parameters are measured as $A_{u}$ = 3.884, $k$ = 15.078, $τ_{p}$ = 10.0206, $A_{y}$ = 495.2165 and $A_{u}$ = 690. For more clarity, all the measured quantities, and relay heights are presented in Table 3 from raw limit cycles and fitted limit cycles with SNR and covariance. Then the measured quantities are substituted in derived expressions (26) and (27) to obtain two unknown parameters of the FOPDT plant model as τ₁ = 4.0355, δ = 3.1059 with the prior estimation of k = 0.7177 from (25). Another overdamped SOPDT model is derived from the same limit cycle data with dead-time δ = 2.5000 is measured from the 2^nd derivative of limit cycle output applying the method as mentioned in subsection ‘Identification of overdamped SOPDT dynamics’. Hence, two unknown τ₁ and τ₂ are derived with substituting the measured quantities in the derived expressions (21) and (22) as τ₁ = 3.6318 and τ₂ = 0.7071 respectively, with the help of k estimated as earlier. Thereafter, two dynamics analogous to the earlier proposed FOPDT and SOPDT TFs are derived utilizing the appropriate set of expressions using fitted limit cycle output obtained from MATLAB Curve Fitting Toolbox (2002) and presented in Table 4. All four models are presented in Table 4 in comparison with an underdamped second-order model proposed by Ghorai et al. (2019). Further, supplying the same raw data in MATLAB system identification toolbox (SIT), a stable first and second dynamics are derived and compared with the proposed models, as shown in the Table 4.

Table 3.

Relay configuration considered in experimental study and measured quantities.

Relay Config.	SNR	$σ^{2}$	$τ_{p}$	$A_{u}$	$A_{l}$	$A'_{y}$	$A'_{u}$
R = 55; h₁ = 90; h₂= 20	–	–	10.0206	3.884	15.078	495.2165	690.0
	37.9589 dB	43.5612	9.992	3.4977	14.9934	492.150	689.5615

Table 4.

Identified dynamics of the considered pressure control system.

Methods	Identified models
Proposed FOPDT dynamic^†	$\frac{0.7177 e^{- 3.1059 s}}{(4.0355 s + 1)}$
Proposed FOPDT dynamic^‡	$\frac{0.7137 e^{- 3.1036 s}}{(3.9755 s + 1)}$
Proposed SOPDT dynamic^†	$\frac{0.7177 e^{- 2.1 s}}{(2.4502 s + 1) (1.7486 s + 1)}$
Proposed SOPDT dynamic^‡	$\frac{0.7137 e^{- 2.2793 s}}{(3.1358 s + 1) (1.1048 s + 1)}$
SOPDT dynamic from Ghorai et al. (2019)	$\frac{0.7214 e^{- 1.5 s}}{(5.0997 s^{2} + 3.1527 s + 1)}$
FOPDT dynamic from Lennart (2007)^†	$\frac{0.7152 e^{- 2.598 s}}{(3.8326 s + 1)}$
SOPDT dynamic from Lennart (2007)^†	$\frac{0.7140 e^{- 1.55 s}}{(2.3921 s + 1) (1.9246 s + 1)}$

†

Using raw data; ^‡Using fitted data.

Validation

The dynamic models of the real-time flow-line pressure control system are obtained as stable FOPDT and overdamped SOPDT TFs from single relay autotuning. Therefore, to validate the experimental outcomes obtained from the above pressure control system, the Nyquist frequency response plots for all the four dead-time TF models derived from raw data are drawn and shown in Figures 5 and 6 separately, based on their order in comparison with the model proposed by Ghorai et al. (2019). Interestingly, it is clearly seen from the Table 4, and frequency response plots that the proposed first and second models are close to the dynamics obtained from MATLAB SIT as well as the model proposed by Ghorai et al. (2019), which is an online method and well accepted. Finally, all the proposed four obtained dynamics from the raw data of the real-time pressure control system in dead-time TF form are simulated in MATLAB along with two models obtained from MATLAB SIT and one model proposed by Ghorai et al. (2019), considering the identical setpoint weighted relay settings which have been chosen during the real-time experiment. Thereafter, simulated outputs of the proposed models and models obtained from MATLAB SIT as limit cycles are plotted in Figures 7 and 8 separately based on their order to compare with the actual limit cycles obtained from the experimental relay test on the pressure control system as well as to the model proposed by Ghorai et al. (2019). From the nature of the limit cycle oscillations, it is concluded that the dynamics of the above plant can be considered a SOPDT model since from the validation curves, it is observed that the actual experimental limit cycles matched more with the proposed second-order models than that of first-order models.

Conclusion

An autotuning relay test is proposed to derive the unknown industrial plant model parameters in terms of simple dead-time TF form in a real-time fashion. The unknown model parameters of plant dynamics such as steady-state gain, time constants, and dead-time are estimated explicitly from the measurement of a few important limit cycle information. Two well-known plant models have been simulated and identified to show the accuracy and simplified procedure of the proposed method. Though the accuracy of the proposed method is not so high with respect to MATLAB SIT because of relay gain approximation from DF analysis, however, the attention is on real-time implementation of simple relay autotuning towards plant dynamics identification and its application in real-time industries. The autotuning, based on relay feedback, is one of the simplest identification methods that can be easily applied to estimate plant dynamics in terms of proactive dead-time TF further helping in controller design. The method also showed that the application of a simple relay test identifies a diverse class of processes realistically. The main advantage of the proposed asymmetrical relay autotuner is that the identification is regardless of uncertainties of the symmetricity of the limit cycle output obtained from general relay autotuning. The uncertainty could be caused by either unknown noise, nonlinearities, or non-unity plant steady-state gain. Finally, the paper provides a way to implement a simple setpoint weighted asymmetrical relay autotuner for the identification of dynamics of unknown industrial plants.

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 iD

Prasenjit Ghorai

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