Implementation of Simple Fuzzy PI Controller for Liquid Level Cascade Control

The control based on Fuzzy strategies was first proposed by Lotfi A. Zadeh in 1965. Since then, fuzzy logic has emerged as a powerful technique for complex industrial process control due to non linearities or time-varying responses, where it is difficult to model their dynamic behavior. Conventional PID controllers do not work well for these systems.

In such cases, fuzzy controllers, which are intrinsically non-linear, offer better performance. However, the traditional fuzzy controller uses many fuzzy sets for the input and output blocks, which increases the time taken to calculate the control value and limits their implementation on an industrial PLC. The simplest fuzzy controller, on the other hand, is one that uses a minimum number of fuzzy sets for the input and the output. The input-output relations of linear discrete-time tow input and tow output PI controller in speed form which are given by the study [11]:

$\Delta {{u}_{1}}(q)={{K}_{{{P}_{1}}}}\Delta {{e}_{1}}(q)+{{K}_{{{I}_{1}}}}\Delta {{e}_{1}}(q)+{{s}_{21}}\left[ {{K}_{{{P}_{2}}}}\Delta {{e}_{2}}(q)+{{K}_{{{I}_{2}}}}\Delta {{e}_{2}}(q) \right]{{u}_{1}}(q)={{u}_{1}}(q-1)+\Delta {{u}_{1}}(q)$                          (1)

$\Delta {{u}_{2}}(q)={{s}_{12}}\left[ {{K}_{{{P}_{1}}}}\Delta {{e}_{1}}(q)+{{K}_{{{I}_{1}}}}{{e}_{1}}(q) \right]+{{K}_{{{P}_{2}}}}\Delta {{e}_{2}}(q)+{{K}_{{{I}_{2}}}}{{e}_{2}}(q){{u}_{2}}(q)={{u}_{2}}(q-1)+\Delta {{u}_{2}}(q)$                          (2)

Equivalently, the linear discrete-time of proportional derivative controller in position form formulas are given by:

${{u}_{1}}(q)={{K}_{{{P}_{1}}}}{{e}_{1}}(q)+{{K}_{{{D}_{1}}}}\Delta {{e}_{1}}(q)+{{s}_{21}}\left[ {{K}_{{{P}_{2}}}}{{e}_{2}}(q)+{{K}_{{{D}_{2}}}}\Delta {{e}_{2}}(q) \right]$                     (3)

${{u}_{2}}(q)={{s}_{12}}\left[ {{K}_{{{P}_{1}}}}{{e}_{1}}(q)+{{K}_{{{D}_{1}}}}\Delta {{e}_{1}}(q) \right]+{{K}_{{{P}_{2}}}}{{e}_{2}}(q)+{{K}_{{{D}_{2}}}}\Delta {{e}_{2}}(q)$                           (4)        

where,

$\begin{matrix}  {{e}_{1}}(q)={{r}_{1}}(q)-{{y}_{1}}(q) \\  \Delta {{e}_{1}}(q)={{e}_{1}}(q)-{{e}_{1}}(q-1) \\\end{matrix}$                           (5)

$\begin{matrix}  {{e}_{2}}(q)={{r}_{2}}(q)-{{y}_{2}}(q) \\  \Delta {{e}_{2}}(q)={{e}_{2}}(q)-{{e}_{2}}(q-1) \\\end{matrix}$                          (6)

with

$K_{P_1}$ and $K_{P_2}$ are proportional factors;

$K_{I_1}$ and $K_{I_2}$ are integral factors;

r₁and r₂are reference commands;

y₁and y₂are outputs of TITO process;

u₁and u₂are outputs of TITO controller;

s₁₂and s₂₁(- or +) signs depend on the loops interactions.

Figure 1 indicates the typical block diagram of the TITO fuzzy PI-PD controller with:

S_e1, S_Δe1, S_e2, and S_Δe2 are input scaling factors;

$S_{\Delta u_1}^{-1}$ and $S_{\Delta u_2}^{-1}$ are output scaling factors for PI;

$S_{\Delta u_1}^{-1}$ and $S_{\Delta u_2}^{-1}$ are output scaling factors for PD;

e_1s(q), Δe_1s(q), e_2s(q), and Δe_2s are scaled input;

Δu_1s(q) and Δu_2s(q) are scaled outputs for PI;

u_1s(q) and u_2s(q) are scaled outputs for PD.

It must be noted that the interaction between the two loops determines the signs of S12 and S21. Proposed in the study [17] and finalised in the study [11], the type of input-output relationships of the simplest nonlinear fuzzy PI/PD controller is based on L-type and Γ-type member ship function to calculate the scaled input variables in the fuzzification part. The control contains five rules, obtained via Larsen product.

1.png

Figure 1. TITO fuzzy controller block diagram

The fuzzification part of this controller use L-type and Γ-type membership functions, as scaled input variables and the membership functions for the scaled outputs Δujs(q) for j=1, 2 are shown in Figure 2.

2.png

Figure 2. Input and output membership function

The control rule base consists of five rules in the following form:

$\begin{matrix}  f{{e}_{1s}}(q)=E_{1}^{_{{{I}_{1}}}}\text{and }\Delta {{e}_{s1}}(q)=\Delta E_{1}^{{{I}_{2}}}\text{ and } \\  {{e}_{2s}}(q)=E_{1}^{_{I3}}\Delta {{e}_{s2}}(q)=\Delta E_{1}^{{{I}_{4}}} \\\end{matrix}\\\text{then }{{\Delta }_{{{u}_{1s}}}}(q)=\Delta U_{1}^{{{J}_{1}}}\text{ and }{{\Delta }_{{{u}_{2s}}}}(q)=\Delta U_{2}^{{{J}_{2}}}$                    (7)

where I₁, I₂, I₃ and I₄ are the indices of the input fuzzy sets with values -1 or +1, J₁and J₂ are the indices of the output fuzzy sets having value (-2, -1, 0, +1 or +2) with membership functionsµ µ_ΔUj^Jn  for  j, n=1 and 2. The inference engine uses the membership functions of the modified fuzzy sets ΔŨ_j^-2, ΔŨ_j^-1, ΔŨ_j⁰, ΔŨ_j⁺¹ and ΔŨ_j⁺², obtained using Larsen product, which are represented by the hatched areas in Figure 3.

3.png

Figure 3. Modified output membership function

The region of scaled input plan detailed in the study [11] and given in Figure 4 shows the top view of the three-dimension plot with the axes e_s(q), ∆e_s(q) and µ. In this scaled input plan, all combinations of input variables are represented. The appropriate control law in each region of the scaled input plan is shown in Figure 3.

The calculation of the scaled output value in the defuzzification part uses the CoS method based on the study [18]. This scaled output value is given by:

\[\Delta u_{js}^{*}(q)=\frac{\sum\nolimits_{l=1}^{5}{A({{\mu }_{Rl}})C({{\mu }_{Rl}})}}{\sum\nolimits_{l=1}^{5}{A({{\mu }_{Rl}})}}\]

where, A(µ_Rl) and C(µ_Rl) are the area and centroid, respectively, of the l^th lth derived output membership function of the j^thoutput.

4.png

Figure 4. Scaled input plan

In this study, the PLC installed for control system is a highly modular Siemens S7-300 PLC (shown in Figure 3) programmed to monitor, control and analyse the sequential mode for safety operating. The implementation of the fuzzy system described in section 2 is by the using of STEP 7 ladder programming.

This PLC is fitted with 24V, 2A power supply, 314-2DP CPU module, an MPI/DP communication processor used to load the program in the programming station into PLC and HMI panel, 24 digital input channel, 16 digital output channel, analog I/O module which allows us to use 5 analog Input channel and 2 Output analog ones, 12-bit ADC and DAC converters with ±10V maximum input/output range. As input variables of the fuzzy block program, two bipolar voltages are considered on the PIW752 and the PIW752. The resulting outputs voltage is applied on analog Output module part at the address PQW752 and PQW754. In the main program, a simplified method of the fuzzy controller is developed, where at each acquisition cycle, the voltages are adapted by applying scaling gains, and then determine their membership degree to corresponding fuzzy sets. At the end, after the application of inference and defuzzification block, the output voltage is determined and transferred to the analog outputs addressed more exactly at PQW752, PQW754. The HMI virtual sliders are used as set-point variables. The level and flow responses measured respectively by level and flow sensors are set to the controller using two lanes of PLC’s analog input module. As detailed in Arnu and Mohan (2016), the mathematical relations between the inputs and outputs of the fuzzy logic regulator are obtained by means of the algebraic product as AND operator, bound sum for OR, feedback product for the inference and the CoS method for defuzzification according the study [5], the crisp value of the scaled output is expressed by the Eq. (8) previously presented.

The expression of the surfaces and centers of the inferred belonging functions of the outputs are:

$\begin{gathered}A\left(\mu_{-2}\right)=0.5\left(A_j+B_j\right) \mu_{-2} ; A\left(\mu_{+2}\right)=0.5\left(A_j+B_j\right) \mu_{+2} \\ A\left(\mu_{-1}\right)=\left(A_j+B_j\right) \mu_{-1} ; A\left(\mu_{+1}\right)=\left(A_j+B_j\right) \mu_{+1} \\ A\left(\mu_0\right)=\left(A_j+B_j\right) \mu_0\end{gathered}$

and

$\begin{gathered}C\left(\mu_{-2}\right)=\frac{-5\left(A_j+B_j\right)^2-A_j B_j}{3\left(A_j+B_j\right)} ; C\left(\mu_{+2}\right)=\frac{5\left(A_j+B_j\right)^2+A_j B_j}{3\left(A_j+B_j\right)} \\ C\left(\mu_{-1}\right)=-\left(A_j+B_j\right) ; C\left(\mu_{+1}\right)=\left(A_j+B_j\right) ; C\left(\mu_0\right)=0\end{gathered}$

with A_j, B_j are the linguistic outputs values of the modified output membership function shown in Figure 3.

The S7-SCL (Structured Control List) language is used to ensure a degree of flexibility in the implementation program. FB and FC blocks are programmed to perform the various calculations required. The first step in the controller implementation process is to calculate its input variables, consisting of four inputs: the first error e₁, the derivative of this error ∆e₁, the second error e₂ and the derivative of this error ∆e₂, which are programmed in block FB20. Depending on the value of each error and its derivative, the corresponding cell can take on an integer value between 1 and 9 based on the subspaces shown previously in Figure 4.

The assembly of the previously determined cells is programmed in function block FC22. This block merges the result of the two cells found from block FC21 to give an integer expressing the global cell ranging from 11 to 99. The FC23 block calculates the controller outputs, which are the control variations as a function of the global cell.

The numeric value of the output will be transformed to integer value (0-27648) using an unscale function block FC106 and loaded into the PQW752/ PQW754 output address device. The biggest advantage of this technique is its simplicity of implementation in a PLC’s environment.

This section presents PLC-based experimental results for the simplest classical PID controller and non-linear fuzzy PI controller. The experiments were done on an experimental Level/Flow control unit level for several level set-point and disturbance rejection. The total programme memory required for this simplest fuzzy controller implementation on the PLC, by choosing the design described above is 312 Bytes. The execution time of the algorithm developed is between 14 and 21 ms, achievable through the use of the OB35 timer interrupts or other dedicated PLC resources. The experimental results are as shown in Figure 8.

8.png

Figure 8. PID- PLC based cascade control tracking response

First experiment which has been done on the control system was oriented to test the PID mode controller, note that the static error is zero, rise time (Tr) of 16 seconds and disturbance rejection time is 9 seconds with a small overshoot during the tracking test, this overshoot being the result of the constraints imposed on the speed of the dynamic response.

9.png

Figure 9. Simulation results

The simplest fuzzy PI controller algorithm developed was originally implemented in MATLAB/Simulink where the simulation results are shown in Figure 9. The simulation results obtained shows that the mathematical model of the simplest nonlinear fuzzy PI controller model provides stability, steady state error is zero, rise time of 11 seconds with a small overshoot during the tracking test.

The central unit of our PLC begins by analysing the main programming block, called organisation block 1 (OB1), in which all the sub-programs used to process inputs/outputs, data conversions and normalisations are programmed in LD.

The control algorithms developed are programmed in 3 different Function Blocks. Firstly, the FB30 contain the fuzzy program controllers. Secondly, the function block FB20 is used for input calculation. Third, the FB21 has the role of controlling calculations using the instance data block (DB30, DB20, DB21). These blocks associated four Function (FC) where FC20 is used for input normalization, and FC21 and FC22 for cell calculation and assembly. The calculation of output signal control is realized by the FC23 block. The measurement input and output signals are normalized by Scale/Unscale preprogrammed function respectively FC105 and FC106. Finally, both the SCL and LAD are used for programming the control algorithm. The structure of the controller is shown in Figure 10.

10.png

Figure 10. PLC’s program structure for simplest nonlinear fuzzy PI controller

In the experimental tests we applied several set-point tracking. The results show that the simplest non-linear fuzzy PI controller we have implemented provides a good margin of stability, the steady-state error is zero, the rise time is 12 seconds and the disturbance rejection time is 7 seconds without overshoot during the tracking test which validates our implementation approach. However, the hysteresis effect of the proportional valve on the control is more apparent in this mode.

According to Figure 11 control signal have oscillation because the control law has different cells. In other words, the value is calculated based on the position of different input signals for each cell includes inside the called scaled input plan presented in Figure 4. The program memory capacity required to implement this simplest non-linear fuzzy controller on the PLC is 33206 bytes.

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Figure 11. Simplest nonlinear fuzzy PI experimental results

The supervisory control and data acquisition (SCADA) system of process developed (for real-time monitoring and control) using the SIMATIC HMI panel is composed of several windows. The HMI process window shown in Figure 12 allows us to interact with the control unit, navigate to other process windows, graphically display the data provided by the I/O server.

12.png

Figure 12. Controller HMI process window

The parameters of the simplest TISO PI fuzzy controller are obtained by the minimisation of the integral of the absolute error (IAE) criteria and are shown in Table 1.

Table 1. Parameters of TISO PI fuzzy controller

This work was supported by the Direction Générale de la Recherche Scientifique et du Développement Technologique (DG RSDT).