Perspective Chapter: Modeling and Identification of Two Tank System Using Relay Feedback – An Experimental Approach

Abstract

An experimental approach for modeling and identification of two tank system was considered. For the system that we considered modeling has been carried out with theoretical values. The practically obtained values from series of experiments the used to correlate the simulated values, the process we have considered is the MIMO process which is difficult being the interaction exists, the interactions can be eliminated for better control action using the existing the methods. Simulations with MATLAB has been carried out and the simulated results are used in the real time experimental set using LAB VIEW. There is great degree of the Accuracy between the estimated values and simulated which can be practically demonstrated with the experimental setup. In this relay feedback approach is being considered for conducting the autocuing test to estimate the parameters.

1. Introduction

Multivariable process systems are either time-lag dominated or dead-time dominated systems and can be modeled as First Order Plus Dead time (FOPDT) structures. The sequential identification and modified Ziegler-Nichols controller design method form the basic structure for the multivariable auto tuner. These methods consider desired response (output: for example, y₁ in case of 2x2 system) and input (u₁) for system identification and analysis. Also, they do not discuss about methods to reduce interactions nor do they give any exact analytical expressions for the relay responses that may help in analyzing the interaction behavior between input/output and may provide information regarding closed-loop parameters (PID using model based tuning rules) of the MIMO (Multi input and Multi output) system. Moreover, it is felt that exact model parameters and information on interactions can be better obtained/calculated from mathematical model of undesired relay responses for MIMO systems. As the off-diagonal closed loop transfer functions contain information on interactions, it is better to analyze the control system based on their time domain characteristics.

1.1 Motivation

Most of the chemical process are multivariable in nature.i.e. more than one input such as distillation column which are used for separation of the chemical components in the Industries. Identification and control of the process plays a crucial role in the Process industries to maintain the certain variable at set point. There are well defined Control strategies are there for identification and control of SISO [2014] (single input and single output process).

The control of MIMO (Multi Input and Multi out) is challenging due to presence of interaction exists among the manipulate variable and control variable. There are lot of methods available for identification and control of the process by reducing the interaction analysis.

Åström and Hägglund [1] were pioneered in the introducing the concept of auto-tuning to generate the sustained oscillations. Yu [2] has explored the relay feedback to investigate the shapes of various higher order systems. Tau and Liu [3] has presented the research gaps and further explored the further possible investigations. Shankar Prasad and Yaddanapudi Jaya [4] have proposed the new identifications for stable Process. Sujatha and Panda [5] has introduced the concept of estimating the parameters of MIMO systems. Ramesh and Panda [6] has explored the relay feedback and presented the report. Kalapana [7] has extended and developed the analytical method of identifying parameters. Chidambaram and Padma Sri [8] has discussed elaborately discussed the methods for unstable systems.

In this work an attempt is made by taking the two tank system (To control Liquid level) using the relay feedback (sequential tuning approach) and validation of the simulation results with the experimental results. The method used in this research work is sequential tuning approach which is widely acceptable for estimation and identification of the process parameters for proposed two tank (Liquid Level) system. Experimental transfer function is obtained from two tank system a simulation is carried out a comparison is made between experimental one and simulated one.

1.1.1 Two tank interacting system

The schematic diagram of coupled tank MIMO process is shown in Figure 1. The input flow for tank1 and tank2 are ${\mathrm{F}}_{\text{in}1}$ and ${\mathrm{F}}_{\text{in}2}$_. The controlled variables are level h₁ and h₂ in the tank1 and tank2. The mass balance and Bernoulli’s law yield.

Figure 1.

Block diagram of coupled tank system.

$$A_1\frac{{\mathit{dh}}_1}{\mathit{dt}}=k_1{u}_1-{\beta }_1{a}_1\sqrt{2{\mathit{gH}}_1}-{\beta }_x{a}_{12}\sqrt{2g\left(H_1-H_2\right)}$$E1

$$A_2\frac{{\mathit{dh}}_2}{\mathit{dt}}=k_2{u}_2+{\beta }_x{a}_{12}\sqrt{2g\left(H_1-H_2\right)}-{\beta }_2{a}_2\sqrt{2{\mathit{gH}}_2}$$E2

${\mathrm{A}}_{1,}{\mathrm{A}}_2$, cross sectional area of tank 1 and tank 2 $\left({\mathrm{cm}}^2\right)$;

${\mathrm{a}}_{1,}{\mathrm{a}}_1$, cross sectional area of output pipe in tank 1 and tank 2 $\left({\mathrm{cm}}^2\right)$;

$a_{12}$, cross sectional area of interaction pipe between tank 1 and tank 2 $\left({\mathrm{cm}}^2\right)$;

${\mathrm{h}}_1{\mathrm{h}}_2$, water level of tank 1 and tank 2 (cm),

${\mathrm{F}}_{\text{in}11}{\mathrm{F}}_{\text{in}22}$, inflow of tank 1 and tank 2 $\left({\mathrm{cm}}^3/\mathrm{s}\right)$;

${\mathrm{F}}_{\text{out}11},{\mathrm{F}}_{\text{out}22}$, outflow of tank 1 and tank 2 $\left({\mathrm{cm}}^3/\mathrm{s}\right)$;

${\mathrm{u}}_{1,}{\mathrm{u}}_2$, input voltage to pump 1 and pump 2 (V);

${\mathrm{\beta }}_{1,}{\mathrm{\beta }}_2$, valve ratio at the outlet of tank 1 and tank 2;

${\mathrm{\beta }}_{\mathrm{x}}$, valve ratio of jointed pipe between tank 1 and tank 2;

${\mathrm{k}}_1,{\mathrm{k}}_2$, gain of the pump 1 and pump 2 $\left({\mathrm{cm}}^3/\mathrm{v}-\mathrm{s}\right)$;

g, gravity $\left({\mathrm{cm}}^3/\mathrm{s}\right)$.

The parameter values of the coupled tank process are given in Table 1. The nominal operating conditions of the process are shown in Table 2.

A1, A2	a1, a1, a12	${\mathrm{\beta }}_1$	${\mathrm{\beta }}_2$	${\mathrm{\beta }}_{\mathrm{x}}$
154	0.5	0.7498	0.8040	0.2445

Table 1.

Parameters of coupled tank MIMO process.

u1	u2	h1	h2	k1	k2
2.5	2	24.6	14.4	33.336	25.002

Table 2.

Operating condition of coupled tank MIMO process.

The transfer function of coupled tank process is identified using system identification tool box. The levels in the tanks are initially maintained at the operating condition of 24.6cm and 14.4cm by giving the input voltage of 2.5 and 2 volts to the pump1 and pump2 respectively. Then the input to the pump1 is changed from 2.5 to 3 voltages by keeping pump2 input as constant and the level in tank1 and tank2 are recorded. The same procedure is repeated by changing the pump2 input from 2 to 2.5 volts by keeping the pump1 input as constant. The open loop response for the change in input1 and input2 are shown in Figures 2 and 3.

Figure 2.

Open loop response for input change in pump1.

Figure 3.

Open loop responses for input change in pump2.

The experimentally identified transfer function model is

$$\left[\begin{array}{c}{h}_1\\ h_2\end{array}\right]=\left[\begin{array}{ll}\underset{¯}{16.99e^{-12.89s}}& \underset{¯}{6.69e^{-72.57s}}\\ 214.7s+1& 204.93s+1\\ \underset{¯}{9.23e^{‐35.01s}}& \underset{¯}{11.38e^{‐25.04s}}\\ 256.44s+1& 169.15s+1\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]$$E3

1.2 Relay test for the g₁₂ interactive transfer function

The relay feedback test is conducted on 2X2 MIMO (Two tanks) system. The undesired relay response of ${\mathrm{g}}_{\mathrm{p}12}$ is obtained in the step2 of sequential auto tuning [9].

The relay feedback test output thus obtained is shown in Figure 4. It is ${\mathrm{g}}_{12}$ the interactive transfer function relay response and it has the interaction of ${\mathrm{g}}_{11}$ transfer function. The time intervals are taken to derive the analytical expression for the above relay response.

Figure 4.

Relay output response for g₁₂ (step2).

1.2.1 Relay test for the ${\mathbf{g}}_{\mathbf{21}}$ interactive transfer function

The relay feedback test is conducted on 2X2 MIMO (Two tank) system. The undesired relay response of ${\mathrm{g}}_{21}$ is obtained in the step3 of sequential auto tuning. The relay feedback test output thus obtained is shown in Figure 5. It is ${\mathrm{g}}_{21}$ interactive transfer function relay response and it has the interaction of ${\mathrm{g}}_{22}$ transfer function. The time intervals are taken to derive the analytical expression for the above relay response.

Figure 5.

Relay output response for ${\mathit{g}}_{\mathit{21}}$.

Under decentralized PI control, with known pairing (y1–u1) and (y2–u2) 2x2 MIMO system is employed. First Relay is placed between ${\mathrm{y}}_1$ and ${\mathrm{u}}_1$, while loop 2 is on manual mode. Following the relay-feedback test, a controller can be designed from the ultimate gain and ultimate frequency. Next performed relay feedback test between ${\mathrm{y}}_2$ and ${\mathrm{u}}_2$, while loop 2 is on automatic mode. A controller is designed from the ultimate gain and ultimate frequency for the loop 2. Third performed relay feedback test between ${\mathrm{y}}_1$ and ${\mathrm{u}}_1$, while loop 2 is on automatic mode. Generally, new set of tuning constants are founded for the controller in loop1. Based on the concept of sequential auto tuning method each controller is designed in sequence. The controller’s parameters are converged in 3–4 relay feedback test is shown in the below Table 3.

S. No	Iterations	Loops	$k_c$	$k_i$
1.	Iteration1	Loop1	0.5729	0.01236
		Loop2	0.2685	0.002496
2.	Iteration2	Loop1	0.5780	0.01275
		Loop2	0.2685	0.002492
3.	Iteration3	Loop1	0.5729	0.01245
		Loop2	0.2685	0.002494
4.	Iteration4	Loop1	0.5859	0.01258
		Loop2	0.2685	0.002498

Table 3.

Controller parameters computed from biased the relay feedback response of step 2 and step 3 by sequential autotuning.

1.3 Derivation of analytical expression for 2X2 MIMO system

The ideal relay output consists of a series of step changes in manipulated variables. Hence, the stabilized output is a sum of infinite terms of step responses due to those step changes. The process output converges to the stationary oscillation in one period and the limit cycle oscillation for ideal relay test is characterized by the deriving analytical expressions. Analytical expressions are mathematical expressions for the stabilized relay feedback output responses and are useful for back calculation of exact process model parameters

1.3.1 Analytical expression for the ${\mathbf{g}}_{\mathbf{12}}$ interactive transfer function

Analytical expression is derived for the undesired response obtained from the relay

Feedback test. The biased relay response is assumed to be formed by n number of small

Step changes. Let ${\mathrm{\mu }}_{+}={\mathrm{\mu }}_0+\mathrm{\mu }\text{and}{\mathrm{\mu }}_{‐}={\mathrm{\mu }}_0–\mathrm{\mu }$. The process input in the relay feedback test consists of a series of step changes with down amplitude, ${\mathrm{\mu }}_{‐}$ and up amplitude ${\mathrm{\mu }}_{+}$. At first interval (after synchronizing input with output by time shift), the response can be described.

as:

$$y_1=k_{12}{\mu }_0\left[1-e^{\frac{-t}{{\tau }_{12}}}\right]-\frac{k_{11}{k}_{c1}}{{\tau }_{i1}}{\mu }_{-}\left[t-\left({\tau }_{i1}+{\tau }_{11}\right)e^{\frac{-t}{{\tau }_{11}}}\right]y_1\left(t-1\right)$$E4

$$\begin{array}{l}{y}_2=k_{12}{\mu }_0\left[\left(1-e^{\frac{t+D_1}{n_2}}\right)-2\left(1-e^{\frac{-t}{n_2}}\right)\right]\\ -\frac{k_{11}{k}_{c1}}{{\tau }_{11}}{\mu }_{-}\left[\left(t+D_{11}\right)-{\left.\left({\tau }_{11}+{\tau }_{11}\left(1-e^{\frac{t+D_{11}}{n_{11}}}\right)\right]\right|}_{y_2\left(t-1\right)}\left[t-\left({\tau }_{11}+{\tau }_{11}\left(1-e^{\frac{-t}{{\tau }_{11}}}\right)\right]\mid \right.\right.\end{array}$$E5

Where ${\mathrm{D}}_{21}$ and ${\mathrm{D}}_{11}$ are time delays of the individual transfer functions of the system.

The above Eq. (5) can be simplified as follows:

$$\begin{array}{l}{y}_2=k_{12}{\mu }_0\left\{\left[1-2\right]-e^{\frac{-t}{{\tau }_{21}}}\left[e^{\frac{-D_{12}}{{\tau }_{12}}}-2\right]\right\}-\frac{k_{11}{k}_{c1}}{{\tau }_{i1}}\times \\ {\mu }_{-}\left\{t\left[1-2\right]+\left(\left({\tau }_{i1}-{\tau }_{11}\right)\left(1-e^{\frac{-t}{{\tau }_{11}}}\right)\left[e^{\frac{-D_{11}}{{\tau }_{11}}}\right]-2\right)\right\}{y}_2\left(t-1\right)\end{array}$$E6

Let ${\mathrm{p}}_{+}$ and ${\mathrm{p}}_{-}$ be the positive and negative half cycle’s periods. The third interval lags by an amount $D_{21}\square \mathit{pu}/2\text{and}{D}_{22}\square \mathit{pu}/2$ from input can be given a

$$-\frac{k_{11}{k}_{c1}}{{\tau }_{n1}}{\mu }_{-}\left\{\left[1-2+2\right]+\left(\left(r_n+{\tau }_{11}\right)\left(1-e^{\frac{-t}{{\tau }_{11}}}\right)\left[e^{\frac{-D_{11}+\frac{P_n}{2}}{{\tau }_{11}}}-2e^{\frac{\frac{p_u}{2}}{{\tau }_{11}}}+2\right]\right\}{y}_3\left(t-1\right)\right.$$E7

The Eq. (7) can easily be simplified (8)

The Eq. (8) for $y_3$ slowly forms a series, as time tends to infinity the response becomes stabilized and it can be described as in given in Eq. (9)

The RHS of Eq. (9) has 4 parts:

$$y_t=k_{12}{\mu }_0\left\{\left[1-2+2-\cdots \right]-e^{\frac{t}{T_{12}}}\left[e^{\frac{D_{12}+{\sum }_{k=1}^{n-1}\frac{p_1}{2}}{f_{12}}}-2e^{\frac{{\sum }_{k=1}^{n-1}\frac{p_1}{2}}{{\tau }_{12}}}++2e^{\frac{\frac{p_1}{2}}{{\tau }_{12}}}-2\right]\right\}$$E8

$$\frac{-k_{11}{k}_{c_1}}{{\tau }_{i1}}\mu -x\left[\left(1-2+2\dots \right]+\left[\left({\tau }_{i1}-{\tau }_{11}\right)\left(1-e\frac{-t}{{\tau }_{11}}\right)\left[e\frac{D_{11}+{\sum }_{n=1}^{n-1}\frac{P_0}{2}}{{\tau }_{11}}\right.\right.\right.\left.\left.\left.-2e^{\sum _{n=1}^{n-1}\frac{P_u}{2}}+\cdots +2e^{\frac{\rho }{{\tau }_{11}}}-2\right]\right]\right]$$

$$\left[y_1\left(t-1\right)+y_2\left(t-1\right)+y_3\left(t-1\right)+\cdots +y_n\left(t-1\right)\right]$$E9

The generalized analytical expression for ${\mathrm{g}}_{12}$ interactive transfer function of 2x2 MIMO process is given by: Similarly, the generalized analytical expression for ${\mathrm{G}}_{21}$ interactive transfer function of 2x2 MIMO process is given by:

$$y_m=\begin{array}{l}\left(k_{21}{\mu }_{+}-2k_{21}{\mu }_0{e}^{\frac{-r}{r_{23}}}\left(\frac{1-e^{\frac{-p_2}{r_{21}}}}{1+e^{\frac{-p_{21}}{2r_{2t}}}}\right)\right)-(\frac{k_{22}{k}_{e2}}{r_{12}}\left({\mu }_{-2}{t}_1+\mathit{\mu ur}\right)\\ -\frac{k_{22}{k}_{e2}}{r_{12}}\left(r_{12}+r_{22}\right))\times \left[{\mu }_{+}-2\mu e^{\frac{-t}{r_{2x}}}\right]\left(\frac{1-e^{\frac{-p_{-}}{r_{23}}}}{1+e^{\frac{-p_{-}}{2r_{31}}}}\right)y_n\left(t-1\right)\end{array}$$E10

The term ${\mathrm{y}}_{\mathrm{n}}\left(\mathrm{t}-1\right)$ in the above Eqs. (9) and (10) is one step ahead prediction of ${\mathrm{y}}_{\mathrm{n}}\left(\mathrm{t}\right)$.

1.4 Boundary conditions to estimate the model parameters of 2X2 mimo system (two tanks)

There are twelve parameters to be estimated in the 2x2 MIMO process. The four dead times ${\mathrm{D}}_{11},{\mathrm{D}}_{12,}{\mathrm{D}}_{21,}{\mathrm{D}}_{22}$ can be obtained straight away from the initial relay response. The remaining eight parameters ${\mathrm{K}}_{11,}{\mathrm{K}}_{12,}{\mathrm{K}}_{22,}{\mathrm{K}}_{21,}{\mathrm{\tau }}_{11,}{\mathrm{\tau }}_{12}{\mathrm{\tau }}_{22}$ and ${\mathrm{\tau }}_{21}$ can be estimated by applying four different boundary conditions in Eqs. (9) and (10) and solving them.

First, we have to measure ${\mathrm{y}}_1,{\mathrm{t}}_{1,}{\mathrm{y}}_2,{\mathrm{t}}_2{\mathrm{y}}_3,{\mathrm{t}}_{3,}{\mathrm{y}}_{\min }$ and ${\mathrm{t}}_{\min }$, as shown in Figures 6 and 7.

Figure 6.

Relay output response for ${\mathit{g}}_{\mathit{22}}$.

Figure 7.

Boundary conditions from the biased ${\mathbf{g}}_{\mathbf{21}}$ relay response.

The boundary conditions are as follows:

$$\mathrm{at}\mathrm{t}={\mathrm{t}}_1,\mathrm{y}={\mathrm{y}}_1$$E11

$$\mathrm{at}\mathrm{t}={\mathrm{t}}_2,\mathrm{y}={\mathrm{y}}_2$$E12

$$\mathrm{at}\mathrm{t}={\mathrm{t}}_3,\mathrm{y}={\mathrm{y}}_3$$E13

$$\mathrm{at}\mathrm{t}={\mathrm{t}}_{\min }$$E14

1.5 Procedure for parameter estimation using biased relay test

Ideal relay feedback test is conducted on the 2x2 MIMO systems. The model

Parameters are estimated as follows:

1. Relay feedback test is conducted on 2x2 MIMO process.

2. Dead times ${\mathrm{D}}_{12,}{\mathrm{D}}_{21}$ are obtained directly from the initial part of undesirable relay response of 2x2 MIMO process and the dead times ${\mathrm{D}}_{11,}{\mathrm{D}}_{22}$ are obtained directly

     From the initial part of desirable relay response of 2x2 MIMO process.

3. Record ${\mathrm{t}}_{1,}{\mathrm{t}}_2,{\mathrm{t}}_3{\mathrm{t}}_{min,}{\mathrm{y}}_{1,}{\mathrm{y}}_{2,}{\mathrm{y}}_3$ and ${\mathrm{y}}_{\min }$_.

4. Using the information obtained in step3, apply the boundary conditions given in Eqs. (11–14) in Eq. (10) and Eq. (11).

5. Use “fslove” to solve the equations obtained in step4 and estimate model Parameters K11, K12, K21, K22, τ11, τ12, τ21, and τ22.

1.5.1 Parameter estimation of two tank system

The dead times of the process is directly recorded from the responses. The information obtained during the stable oscillating condition for the process is given in Table 4.

	Process (Two tank system)
Measured values	${\mathbf{g}}_{\mathbf{12}}$ (loop1)	${\mathbf{g}}_{\mathbf{21}}$ (loop2)
t1	76.2	36.85
t2	80.1	38.7
t3	85.2	40.6
tmin	90	42.5
y1	−0.04	−0.02
y2	−0.08	−0.04
y3	−0.12	−0.06
ymin	−0.16	−0.08

Table 4.

Parameters computed from biased relay feedback responses.

Using this information, the process parameters are estimated by applying the boundary conditions given in Eqs. (10–13) in Eq. (10) and Eq. (11). The comparisons between the parameters and transfer function of true and estimated process are given Tables 5 and 6 respectively.

Parameters	Actual	Estimated	% Error
	Values	Values
${\mathrm{k}}_{11}$	16.99	16.91	−0.47086
${\mathrm{\tau }}_{11}$	214.03	213.95	−0.0373
${\mathrm{k}}_{12}$	6.69	6.59	−1.4947
${\mathrm{\tau }}_{12}$	204.93	204.1	−0.4050
${\mathrm{k}}_{21}$	9.23	9.200	−0.3250
${\mathrm{\tau }}_{21}$	256.44	256.39	−0.0194
${\mathrm{k}}_{22}$	11.38	11.32	−0.5272
${\mathrm{\tau }}_{22}$	169.15	169.11	−0.0236

Table 5.

Comparison of process parameters of true process with estimated process.

S.no	True process	Estimated process
1.	$\frac{16.99e^{-12.89s}}{214.03s+1}$	$\frac{16.9e^{-12.89s}}{213.9s+1}$
2.	$\frac{6.69e^{-72.57s}}{204.93s+1}$	$\frac{6.59e^{-72.57s}}{204.1s+1}$
3.	$\frac{9.23e^{-35.01s}}{256.44s+1}$	$\frac{9.2e^{-35.01s}}{256.39s+1}$
4.	$\frac{11.38e^{-25.04s}}{169.15s+1}$	$\frac{11.32e^{-25.04s}}{169.11s+1}$

Table 6.

Comparison of True process and estimated process.

It is found that the estimated model parameters are very close to the true process Parameters.

1.6 Interaction analysis using relative gain array (RGA)

• From the estimated model parameters information on interaction is obtained by using RGA as follows: The steady (gain) model is expressed as in given in equation

$$k=\mid 11\left[\begin{array}{cc}\begin{array}{l}k\\ \end{array}& \begin{array}{l}k\\ 12\end{array}\\ k& k\end{array}\right]=\mid \left[\begin{array}{ll}\begin{array}{l}16.900\\ \end{array}& \begin{array}{l}6.5900\\ \end{array}\\ 9.200& 11.32\end{array}\right]$$E15

• The relative gain array for a 2 × 2 MIMO (Two tank) system can be expressed as

$$\mathrm{\Lambda }=\left[\begin{array}{cc}1.4639& -0.4639\\ & \\ -0.4639& 1.4639\end{array}\right]$$E16

• Pair the controlled and manipulated variables so that corresponding relative gains

  Are positive and as close to one as possible. The input-output pairing of 2x2 MIMO systems is ${\mathrm{y}}_1$ and ${\mathrm{u}}_{1,}{\mathrm{y}}_2$ and ${\mathrm{u}}_2$_.

2. Real time implementation of two tank experimental setup using biased relay feedback test

2.1 Introduction

Multivariable process systems are either time-lag dominated or dead-time dominated systems and can be modeled as First Order Plus Dead time (FOPDT) structures. The sequential identification and modified Ziegler-Nichols controller design method form the basic structure for the multivariable auto tuner. This method considers desired response (output: for example, ${\mathrm{y}}_1$ in case of 2x2 system) and input (${\mathrm{u}}_1$) for system identification and analysis and they do not discuss about methods to reduce interactions nor do they give any exact analytical expressions for the relay responses that may help in analyzing the interaction behavior between input/output and may provide information regarding closed-loop parameters (PID using model based tuning rules) of the MIMO system. Moreover, it felt that exact model parameters and information on interactions can be better obtained / calculated from mathematical model of undesired relay responses for MIMO systems. Biased relay tests are carried out on 2 x 2 MIMO processes and undesirable responses are modeled.

2.2 Process description

The real time process considered for the sequential auto-tuning algorithm (Figure 8) is a multiple tank process in which two tanks are considered. The MIMO process considered here is the couple tank system, which consists of two cylindrical tanks connected in interacting fashion at two different heights. The system is configured as a MIMO system with two input and two output variables.

Figure 8.

Two tank interacting system.

The couple tank system, which consists of two cylindrical tanks connected in interacting fashion at two different heights. To measure the interaction packers’ valve are being used in the connecting lines between the two tanks. Water is pumped into the two tanks from the reservoir using the pumps. The levels of water in the two tanks are measured using differential pressure transmitters (DPT). Control valves are provided at the inflow lines of the two tanks in order to regulate the flow of water into the tanks. To measure the interaction, the valve is being used. Two I/P converters are used to change stem position of control valve and to measure the level in the tanks two DPT are used. All these sensors and actuators are connected through NI-6008 interface DAQ card; it has 8 analog inputs, 2 analog outputs, 4 digital inputs and 4 digital outputs compatible with LabVIEW software.

2.3 Biased relay feedback test

LabVIEW (short for Laboratory Virtual Instrumentation Engineering Workbench) is a system design platform and development environment for a programming language from National Instruments. LabVIEW is commonly used for Instrument control, and industrial automation. LabVIEW provides three key elements. They are Data acquisition tools, Data analysis tools and Data visualization tools. Data acquisition is the process of gathering or generating information in an automated fashion from analog and digital measurement sources such as sensors and devices under test. Here DAQ 6008 is used for Data Acquisition.

A key benefit of Lab VIEW over other development environments is the extensive support for accessing instrumentation hardware. Lab VIEW provides tight software-hardware integration. It has the ability to solve and execute complex algorithms in real time.

2.3.1 Sequential autotuning

The sequential identification and modified Ziegler-Nichols controller design method form the basic structure for the multivariable auto tuner. This method considers undesired response (output: for example, ${\mathrm{y}}_1$ in case of 2x2 system) and input (${\mathrm{u}}_1$) for system identification and analysis, they give exact analytical expressions for the relay responses that may help in analyzing the interaction behavior between input/output and may provide information regarding closed-loop parameters (PID using model based tuning rules) of the MIMO system.

2.3.2 Sequential autotuning—step 1

Based on the concept of sequential auto tuning method each controller is designed in sequence. Consider a 2 x 2 MIMO system with a known pairing (${\mathrm{y}}_1–{\mathrm{u}}_1$) and (${\mathrm{y}}_2{\mathrm{u}}_2$) under decentralized PI control. Relay is placed between ${\mathrm{y}}_1$ and ${\mathrm{u}}_1$, while loop 2 is on manual mode. Following the relay-feedback test, a controller can be designed from the ultimate gain and ultimate frequency. The block diagram shows in Figure 9 gives the sequential auto tuning step 1, the DAQ Assistant is used to acquire analog signals from the level transmitters which measure the level of the tank 1 and tank2. DAQ Assistant2 is used to generate analog signals to control the final control elements. The block also includes MATLAB script which performs the relay operation.

Figure 9.

Block diagram of biased relay feedback test (step - 1).

Relay is placed between ${\mathrm{y}}_1$ and ${\mathrm{u}}_1$, while loop 2 is on manual mode and a controller can be designed from the ultimate gain and ultimate frequency of the relay response (Figure 10).

Figure 10.

Biased relay feedback response (step 1) obtained from two tank experimental setup.

2.3.3 Sequential autotuning—step 2

• Perform the relay-feedback test between ${\mathrm{y}}_2$ and ${\mathrm{u}}_2$ while loop 1 is on automatic mode. A controller can also be designed for loop 2 following the relay-feedback test.

The block diagram shows in Figure 11 gives the sequential auto tuning step 2, the DAQ Assistant is used to acquire analog signals from the level transmitters which measure the level of the tank 1 and tank2. DAQ Assistant2 is used to generate analog signals to control the final control elements. The block also includes MATLAB script which performs the relay operation. Relay is placed between ${\mathrm{y}}_2$ and ${\mathrm{u}}_2$_, while loop 1 is on automatic mode. A controller is designed from the ultimate gain and ultimate frequency (Figure 12) for the loop 2.

Figure 11.

Block diagram of biased relay feedback test (step 2).

Figure 12.

Biased relay feedback response (step 2) obtained from two tank experimental setup.

2.3.4 Sequential autotunng—step 3

• Once the controller on the loop 2 is put on automatic mode, another relay-feedback experiment will be performed between ${\mathrm{y}}_1$ and ${\mathrm{u}}_1$. Generally, a new set of tuning constants will be found for the controller in loop 1.

• This procedure is repeated until the controller parameters converge. Typically, the controller parameters converge in 3–4 relay-feedback tests for 2 × 2 MIMO Systems.

The block diagram shows in Figure 13 gives the sequential auto tuning step 3, the DAQ Assistant is used to acquire analog signals from the level transmitters which measure the level of the tank 1 and tank 2. DAQ Assistant 2 is used to generate analog signals to control the final control element. The block also includes MATLAB script which performs the relay operation. Another relay-feedback experiment will be performed between ${\mathrm{y}}_1$ and ${\mathrm{u}}_1$ while loop 2 is on automatic mode and a new set of tuning constants will be found (Figure 14) for the controller in loop 1. The controller’s parameters are converged in 3–4 relay feedback tests [10, 11, 12].

Figure 13.

Block diagram of biased relay feedback test (step 3).

Figure 14.

Biased relay feedback response (step 3) obtained from two tank experimental setup.

Sequential auto tuning is conducted under decentralized PI control, with known pairing (y1–u1) and (y2–u2) 2x2 MIMO systems is employed. First relay is placed between ${\mathrm{y}}_1$ and ${\mathrm{u}}_1$, while loop 2 is on manual mode. Following the relay-feedback test, a controller can be designed from the ultimate gain and ultimate frequency. A relay feedback test is performed between ${\mathrm{y}}_2$ and ${\mathrm{u}}_2$, while loop 1 is on automatic mode. A controller is designed from the ultimate gain and ultimate frequency for the loop 2. Afterwards relay feedback test is performed between ${\mathrm{y}}_1$ and ${\mathrm{u}}_1$, while loop 2 is on automatic mode. Generally, new set of tuning constants are founded for the controller in loop1. Based on the concept of sequential auto tuning method each controller is designed in sequence. The controller parameters are converged in 3–4 relay feedback tests is shown in the Table 7.

S.no	Iterations	Loops	$k_c$	$k_i$
1.	Iteration1	Loop1	0.543	1.0427
		Loop2	0.910	0.630
2.	Iteration2	Loop1	0.6110	1.1305
		Loop2	0.925	0.6141
3.	Iteration3	Loop1	0.6175	1.2854
		Loop2	0.923	0.6161
4.	Iteration4	Loop1	0.6163	1.1311
		Loop2	0.924	0.6158

Table 7.

Controller Parameters computed from biased the relay feedback response of step 2 and step 3 by sequential auto tuning.

2.3.5 Derivation of analytical expression for two tank process

1. Analytical expressions are the mathematical expressions for the stabilized relay feedback output responses.

2. Analytical expressions are the time domain model equations useful for back calculation of exact process model parameters.

3. The shifted version of a typical relay feedback response provides the basis for the derivation.

   Analytical expression is derived for the relay response obtained from the relay feedback test when biased relay is used.

   The generalized analytical expression for the undesired relay response of tank 1 (2x2 MIMO process) is given by

$$y_n=\begin{array}{l}\left[k_{12}{\mu }_{+}^{-}\frac{-t}{2k12{\mu }_0{e}^{{\tau }_{12}}}\left[\frac{1-e^{\frac{-{\mathit{pu}}^{{\tau }_{12}}}{}}}{1+e^{\frac{-\mathit{pu}}{2{\tau }_{12}}}}\right]\right]-\left[\frac{k_{11}{k}_{e1}}{{\tau }_{i1}}\left(\mu -t_1+\mathit{\mu t}\right)-\frac{k_{11}{k}_{e1}}{{\tau }_{i1}}\left({\tau }_{i1}+{\tau }_{11}\right)\right]\\ \times \left[{\mu }_{+}-\frac{-t}{2\mu e^{{\tau }_{11}}}\right]\left[\frac{1-e^{\frac{-{\mathit{pu}}^{{\tau }_{11}}}{}}}{1+e^{\frac{-\mathit{pu}}{2{\tau }_{11}}}}\right]y_n\left(t-1\right)\end{array}$$E17

Similarly, the generalized analytical expression for the undesired relay response of tank 2 (2x2 MIMO process) is given by

$$y_n=\begin{array}{l}\left[k_{12}{\mu }_{+}^{-}\frac{-t}{2k21{\mu }_0{e}^{{\tau }_{12}}}\left[\frac{1-e^{\frac{-{\mathit{pu}}^{{\tau }_{21}}}{}}}{1+e^{\frac{-\mathit{pu}}{2{\tau }_{21}}}}\right]\right]-\left[\begin{array}{cc}\frac{k_{22}{k}_{e2}}{{\tau }_{i2}}\left(\mu -t_1+\mathit{\mu t}\right)-& \frac{k_{22}{k}_{e2}}{{\tau }_{i2}}\left({\tau }_{i1}+{\tau }_{11}\right)\end{array}\right]\\ \times \left[{\mu }_{+}-\frac{-t}{2\mu e^{{\tau }_{11}}}\right]\left[\frac{1-e^{\frac{-{\mathit{pu}}^{{\tau }_{22}}}{}}}{1+e^{\frac{-\mathit{pu}}{2{\tau }_{22}}}}\right]y_n\left(t-1\right)\end{array}$$E18

The term ${\mathrm{y}}_{\mathrm{n}}\left(\mathrm{t}‐1\right)$ in the above Eqs. (17) and (18) is one step ahead prediction of y_n(t).

3. Conclusion

Relay feedback test is conducted on both simulation and real time two tank experimental setup using biased relay and the responses are obtained. Analytical expressions are derived for the undesired relay responses in time domain using biased relay feedback test. The model parameters are estimated for the two tank system (simulation). The simulation results indicate that the estimation of Kp (Process gain) is more accurate when biased relay is used. Real time implementation of two tank experimental set up using biased relay feedback test is done.