Design and Real Time Implementation of Fractional Order Proportional-Integral Controller ( PI λ ) in A Liquid Level System

This research article deals with the design and real time implementation of a fractional order Proportional-Integral controller (PI ? ) for a Liquid Level System (LLS). The system is approximated as a First Order Plus Time Delay (FOPTD) model. The equivalent transfer function of this system in polynomial format is considered here for controller design. Expressions for controller parameters (K P and K I ) in terms of frequency (?) and fractional order (?) are derived from the Fractional Order Characteristic Polynomial (FOCP) of the closed loop system. The global stability region based on K P and K I for each ? is constructed. Average values of K P and K I , for each ?, are taken. Among these values, the best fit of K P average and K I average and corresponding ? are identified by means of optimization techniques. The real time implementation of PI ? controller with the identified controller settings in LLS is done. The PI ? controller performances are analyzed in terms of ISE and IAE. A comparison of this control strategy with other conventional based controller techniques is made. PI ? controller outperforms the conventional PI controllers. In addition the load disturbance studies are also carried out and it justifies the supremacy of PI ? controller.


Introduction
PID controllers belong to the dominating form of feedback industrial controllers and there is a continuous effort to improve their quality and robustness.Design and tuning of PID controllers have been a large research horizon ever since Ziegler and Nichols presented their methods in 1942.Specifications, stability, design, applications and performance of the PID controller have been widely treated since then.In recent years, there is an increasing number of studies related to the application of fractional controllers in many areas of science and engineering.Fractional Order PID (FO-PID) controllers could benefit the industry significantly with a wide spread impact when FO-PID parameter tuning techniques have been well developed.This fact is due to a better understanding of the Fractional Calculus (FC) revealed by studies on viscoelasticity, damping, chaos, diffusion, wave propagation, percolation and irreversibility.
The FC concepts are adapted to frequency-based methods.The introduction of fractional order calculus idea to conventional controller design extends the opportunity of added performance improvement.The frequency response and the transient response of the non-integer integral and its application to control systems was introduced by Manabe (Manabe, S, 1960).Oustaloup (Oustalouo,A, 1990) studied the fractional order algorithms for the control of dynamic systems and demonstrated the superior performance of the CRONE (Commande Robuste d'Ordre Non Entier) method over the PID controller.
Podlubny (Igor.Podlubny, 1999) proposed a generalization of the PID controller, namely the PI λ D μ controller, involving an integrator of the order λ and a differentiator of the order μ.He also demonstrated that the response of this type of controller is better as compared to the classical PID controller.Research activities are now focused to develop new tuning rules for fractional controllers for real systems.Some of these techniques are based on an extension of the classical PID control theory.An optimal fractional order PID controller based on specified gain margin and phase margin with a minimum ISE criterion has been designed by using a optimization techniques.
In general, the transfer function Gc(s) of a PI λ controller is defined as Where E(s) is the error signal and U(s) is controller's output.The parameters (K P , K I and K D ) are the proportional, integral and derivative gains of the controller, respectively.The PI λ D µ algorithm is represented by a fractional integro-differential equation of type: Clearly, depending on the values of the orders λ and µ, we get an infinite number of choices for controller's type (defined through the (λ, µ)-plane).Conventional systems are derived from differential equations of integer order whereas fractional order systems are derived from fractional order differential equations.Since PID control is popular in many industry sections, PI λ D µ controller should provide additional potentials to achieve better performance.In this work, an attempt is made for the design and real time implementation of a PI λ controller of the form for the integer order Liquid Level System (LLS).Here, three parameters can be tuned in this control structure (K P , K I and λ ).This paper is organized as follows: In Section 2, PI λ controller design is explained.Experiments and analysis of real time implementation of PI λ controller for Liquid Level System (LLS) is discussed in Section 3. Results and discussions and concluding remarks are given in Section 4 & 5.

Design of PI λ controller
The transfer function of the process, after approximating the delay using Pade first order approximation, is expressed in polynomial format as The transfer function of the PI λ D µ controller is where (λ, µ > 0 ) (2) Here the differential element is not considered (i.e)K D =0, then the PI λ D µ controller becomes PI λ controller where (λ > 0) (3) The output of the feedback PI λ controller with a process (Ref: Figure 1) is given as The denominator of equation ( 4) represents the Fractional Order Characteristic Polynomial (FOCP) of the closed loop system.
(i.e) FOCP=P(s)=1+C(s)G(s)=0 (5) Substituting equations ( 1) and (3) in ( 5), FOCP is written as (Bhaba et al., 2007;Hamamci et al., 2008) Using mathematical identity, equation ( 7) is written as Equating Real and Imaginary parts of P(jω) to zero, Real part is given as follows: And Imaginary part is given as: Solving equations ( 9) and ( 10), the expressions for KP and KI ( PI λ controller parameters) are derived as ω changes from 0 to ω maximum , whose value is determined by substituting K I = 0 in equation ( 12).Using equations ( 11) and ( 12), K P and K I values are calculated for each value of λ (varying from 0.1 to 1.9, in steps of 0.1), by substituting ω from 0 to ω maximum .A stability curve in the K P -K I plane is constructed for each λ (Ref: Figure 2).All regions bounded in between stability curve and the stability line is represented as Global Stability Region.From the Global Stability Region, the average values of K P and K I corresponding to the each value of λ is obtained Among these values , the best fit of K P average and K I average and corresponding λ are identified by means of optimization techniques.

Experimental setup
The functional diagram of Liquid Level System is shown in Figure 3.The setup consists of process tank, collection tank, variable speed pump, RF capacitance level sensor and Interface card VMAT01.The variable speed pump is attached to the collection tank and speed of the pump is controlled by Thyristor Power Control (TPC) unit.The specifications of all the above said major hardware parts of the system are given in Table 1.Water in the collection tank is pumped to the process tank by means of a variable speed pump.The level in the process tank is measured by RF capacitance level sensor and it converts the physical quantity of level to current signal which in turn is converted into a voltage signal of 0 to 5 V by I / V converter.A newly designed VMAT01 interface board consisting of a multifunction, high speed, Analog to Digital Converter (ADC) and Digital to Analog Converter (DAC) is interfaced with the PC-AT Pentium 4. The special feature of VMAT01 is that it can run the real time control algorithms in simulink tool of MATLAB platform directly.

Model parameters identification
In open loop scheme, after the level in the tank reaches the steady state, a step magnitude of 5% DAC output to the variable speed pump is given .The level in the tank varies and this variation in level is recorded against time until a new steady state is reached.This recorded data is converted into fractional response and plotted against time to get the process reaction curve.The parameters for the model of LLS are estimated from this reaction curve using S-K (Sunderasan,K.R and Krishnaswamy,P.R, 1978) identification method.
Published by Canadian Center of Science and Education 191

Real time implementation of PI λ controller in LLS
For real time implementation of both the conventional and PI λ controller in LLS, their corresponding simulink blocks are used.The latter PI λ block is incorporated by using VALERIO's NINTEGER (nipid) (Valerio,D , 2005;Ivo Petras , 2009) MATLAB toolbox.
Real time runs are carried out in LLS with conventional control schemes and PI λ control scheme separately.Three different conventional tuning rules (Ziegler-Nichols, 1942), (Padmasree-Srinivas-Chidambaram, 2004) and (Hsiao Ping HUANG-Jyh Cheng Jeng-KUO Yuan Luo,2005) are used in this work for estimating conventional PI controller settings .Set point tracking of magnitudes (±5% & ±10%) at three nominal operating points 40%, 55%, and 70% of level for all the control scheme in this system are performed.In addition, Load Rejection test at two nominal operating points 40% and 55% of level are also carried out.Tracking responses in both cases are recorded.

Construction of Global stability region
The model parameters of LLS, as detailed in section 3.2, are identified as Process Gain Kp=4.31,Process Time constant τ p =22.8s and time delay L =6s.These model parameters are used to estimate the conventional PI controller parameters for three conventional tuning rules and they are listed in Table 2. From expressions ( 11) and ( 12) of section 2, K P and K I values for different λ (varying from 0.1 to 1.9 in steps of 0.1) and corresponding frequency ω (varying from 0 to ω maximum) are computed .The global stability region based on K P and K I for different λ are constructed.A sample construction of Global stability region for λ=0.5 is given in Figure 2. Average values of K P and K I , for each λ, are taken.Among these values, the best fit of K P average and K I average and corresponding λ are identified by means of optimization techniques.The identified K Paverage , K Iaverage and λ are listed as K Paverage =0.4639,K I average =0.2255, λ=0.5.

Performance of PI λ controller
With these values of conventional and PI λ controller parameters, real time runs are performed in LLS for set point tracking of ±5% & ±10% at the operating point 40% of level and load rejection test at the same operating point.The tracking responses are recorded in Figure 4 to Figure 5. From these Figures, controller performance indices such as ISE & IAE for each control schemes are estimated and values are tabulated in Table 3 & Table 4. From these tabulated values, it is observed that PI λ controller outperforms the conventional PI control techniques in both tracking cases.
To analyze the robustness of the proposed PI λ controller an experimental run at other operating points 55% & 70% of level in LLS are carried out .The results of set point tracking of ±5% & ±10% at these operating levels are recorded in Figure 6 to Figure 7. Load rejection test at 50% operating point with all controllers are carried out.The performance measures are given in Table 5 to 7 clearly indicates the supremacy of PI λ controller.

Conclusion
In this paper a fractional order PI (PI λ ) controller for an integer order liquid level system is proposed.
Expressions for two main parameters of PI λ controller (K P and K I ) are developed in terms of frequency ω and fractional order λ.

Figure 1 .
Figure 1.Block diagram representation of fractional order controller with process

Table 1 .
Global stability regions are constructed and best fit values of K P and K I are identified by adapting optimization technique.Real time runs with these controller parameters are carried out for various set point tracking and load rejection.Performance analysis of the proposed controller is done.In addition a comparison with other conventional tuning rules based PI controller is made.Result shows the supremacy of the proposed fractional controller PI λ .Specification of Hardware parts in LLS

Table 3 .
Performance measures of PI λ controller in terms of ISE and IAE at operating point 40% of level

Table 5 .
Comparison of performance measures (ISE) of PI λ controller with other conventional controllers

Table 6 .
Comparison of performance measures (IAE) of PI λ controller with other conventional controllers

Table 7 .
Load Rejection: Performance measures of Controllers at operating point 55% of level