Design a fractional filter based IMC controller for the nonlinear process

Abstract. Recently, fractional calculus found its applications in the field of process automation. Here the mathematical perspectives of fractional property are applied to control the process. This article presents a design of fractional filter based internal model controller (IMC) for the nonlinear process. The process chosen is pH process plays significant role in the industrial applications. Control of pH process is highly challenging due to its severe nonlinear characteristics and process uncertainty as echoed in the titration curve. An attempt has been made to design the fractional filter based IMC controller for pH process based on desired phase margin and cross over frequency using MATLAB-SIMULINK.Here Oustaloup approximation technique is used to approximate the Fractional filter. Servo and regulatory performance of the process are analyzed and reflects that better control actions are achieved by the fractional filter based IMC through simulation and it is found to be satisfied.

1 Introduction

In spite of tremendous development in the field of advanced control schemes in the process industries, use of conventional proportional integral derivative (PID) controller is remarkable because of the reduced number of parameters to be tuned and also they give satisfactory servo and regulatory performance for a wide range of operating conditions for different processes. The primary function of designed controller is to bring the process variable to setpoint by utilizing minimum energy optimally. Different tuning methods are available to determine the controller parameters (proportional, integral and derivative gain) based on the time and the frequency domain specifications as cited in [1].Now a days , the closed loop performance of the process has been improved by applying the concept of fractionalcalculusincontrollerdesign. These Fractional-order proportional-integral-derivative (FOPID) controllers more robust than the integer order (IO) controller due to the fractional powers of the integral and derivative s terms. Fractional order calculus is the extension of integer order calculus whose power are of non integer values. Literature survey says that several tuning rules are available for FO PID controller design for the IO process [1—6]. The optimization techniques such as genetic algorithm and the particle swarm optimization techniques are also used to obtain the FO PID parameters [7—8]. Real world processes are likely to be fractional, though the fractionality may be less [9]. Systems such as voltage-current relation of a semi-infinite lossy RC line, the diffusion of heat into a semi-infinite solid [9], heating furnace [10] and gas turbine [11] are of fractional order. For FOS, the FO controller parameters are tuned using gain margin and phase margin specifications [12—13]. Advanced FO controllers such as CRONE controller [14], fractional order lead compensator [15], self tuning regulator [16], model reference adaptive control [17], adaptive high gain controller [18], sliding mode controller [19], and iterative learning control [20] have been implemented to improve the performance and robustness in the closed loop control systems. It has been proved that, the stabilizing set of the fractional order PID controllers is wider than the integer order PID controller [21—22]. If the stabilizing set of the controller parameters is wider, then the controller is more robust i.e. it can accommodate more model uncertainties compared to integer order controller.

In the last decade, the internal model control (IMC) based PID controller design has gained widespread acceptance in the control community because the controller can be easily designed by taking inverse of the model with a single tuning parameter namely the IMC filter time constant. The optimal value of filter time constant is determined by a trade-off between speed of response (small value of time constant) and robustness (large value of time constant). An IMC based PID controller has been designed for the desired bandwidth specification [23]. Increase in the bandwidth provides less attenuation in reference signal and faster response. An IMC based FO PID controller has been designed for the IO first order plus time delay process [24] and a class of FOS [25]. Further a design of the internal model control (IMC) based single degree of freedom (SDF) fractional order (FO) PID controller with a desired bandwidth specification for a class of fractional order system (FOS) has been designed .And stability and robust performances of the Fractional order controller is proved through the simulation results. An IMC based integer PID controller cascaded with simple fractional order controller being implemented for the class of fractional order system. Advanced control schemes Such as nonlinear model predictive controller (MPC), Economic model predictive control of nonlinear process systems using empirical models are being developed to improve the closed loop performance in presence of process nonlinearity

In this paper, an attempt has been made to design fractional filter based IMC controller for the nonlinear process. Here the process chosen is pH process. Controlling pH is very important in many processes. The excessive non-linearity of the pH process makes control by a conventional linear PID controller difficult.

3 Process Descriptions

pH is a measurement of the concentration of hydronium ions ([H3O+] commonly abbreviated to [H+]) in aqueous solution. Hence pH control is really concentration control of a mixing process and consequently exhibits the characteristics of mixing processes, such as mixing dead time, residence time and dynamic gain. However, pH control also has its own unique attributes, the most distinctive of which is characterized by a neutralization or titration curve. pH is the negative logarithm of hydronium ion (or hydrogen ion) concentration, and this results in the familiar ‘S’ or ‘z’ shaped titration curve which defines the steady-state characteristic of a pH process. Because of the logarithmic nonlinearity, it is possible for the gain of a pH process to change by as much as a factor of 10 per pH unit which is given by the equation (4).

pH = – log10[H+] (4)

The process considered for the control design is a batch process. The acetic acid is added with sodium hydroxide to form sodium acetate in a continuously stirred tank as shown in the figure 1. Here volume is considered to be constant. Either the out let from the process tank should be at the top of the tank or there should be a level measurement and control unit to check the level and use an on/off control for the inlet flow. After the volume needed (1000 l) is reached. The content of the tank is mixed well using a stirrer.

Using a pH sensor the pH value of the solution is measured. Here, base flow rate is kept constant and the acid flow-rate is continuously varied in steps. Sodium hydroxide of constant flow-rate is continuously neutralized by Acetic acid which acts as a manipulated variable. The steady-state of process occurs when the concentrations of hydrogen and hydroxyl ions are equivalent, for different base flow rate steady state pH is obtained. pH of the mixture is measured continuously using pH sensor and transmitted to the pH transmitter.

Fig. 1pH Process

Based on mass balance equation, two equations have been derived to express the pH process. The derivation of these equations follows the general approach adopted by previous researchers in this field

The ph process considered is acetic acid (weak acid) added with sodium hydroxide (strong base) to form sodium acetate. The general equation for the pH process may be written as follows

The titration curve or the I/O characteristic curve should be obtained which gives the perfect dynamics of the process. The titration curve for pH process is shown in the fig.2 In this plot the curve is ‘z’ shaped because we neutralize base with the flow of acid. And the pH value stops increasing around 4 because of the buffering effect that is caused by the weak acid and its salt. Servo response of pH process with conventional PI controller at different operating regions is shown in fig.3.From the response it is in inferred that, performance of process with PI controller is not found to be satisfactory due to the process nonlinearity.

Table 1.Process Parameters

Process Variables / Parameters Nominal values

Acid dissociation constant(ka) 1.8*10-5

Water dissociation constant(Kw) 10-14

Acid inflow rate(l/min)(F1) 0-200

Base inflow rate(F2) 512

Acid inflow concentration(moles/l)(C1) 0.32

Base inflow concentration(moles/l)(C2) 0.05

Volume of the tank(l3) 1000

Fig. 2.Titration Curve

Fig. 3 Closed loop response of pH process with PI controller

4 Controller Design

A model-based design method, internal model control (IMC), was developed by Morari and coworkers. Now a days, design of PID controller based on internal model control (IMC) has played predominant role in the control community in which controller parameters depends on the inverse of process model parameters with IMC filter time constant as tuning parameters tuned based on trade off between speed of response and robustness [23].Fig.4 shows the general structure of IMC with a process model Gm(s) and the controller output ‘u’ are used to calculate the model output response ‘ym’. Difference of plant response and model response is used as the input signal to the IMC controller CIMC(s).

Fig.4 General structure of IMC controller

2 Fractional calculus The fractional-order differentiator is often denoted by a〖D^α〗_t , where a and t are respectively the lower- and upper bounds, and α is the order of derivative or integrals, which can be non-integers, or even complex numbers [25-29].The definition of fractional-order operator is:

(1)

where is the real part of α. The most commonly used definitions are the Grunwald-Letnikov (GL) definitions and the Riemann-Liouville (RL) definitions.

The RL definition is

(2) Where m-1< α < m, and is the well-known Euler Gamma function. The GL definition is

(3)

Fractional order system as contrary to the integer order system represented by powers of non integer values.

G(s)=1/(a_n s^(β_n )+a_(n-1) s^(β_(n-1) )+⋯+a_1 s^(β_ 1)+a_0 s^(β_0 ) )

Where β_(k,) (k=0,1,…,n) is a non integer numbers.

In other words, fractional order systems better describes the dynamic behavior of the process expressed as the fractional order differential equations given by the following expressions

a_n 〖D_t〗^(β_n ) y(t)+a_(n-1) 〖D_t〗^(β_(n-1) ) y(t)+⋯+a_1 〖D_t〗^(β_1 ) y(t)+b_0 〖D_t〗^(β_0 ) y(t) = u(t)

4.1 Fractional filter based IMC controller for integer order system:

In this section, internal model controller cascaded with filter having fractional characteristics has been designed for pH process. The primary objective of the design is to control pH process with desired closed loop specifications based on the tuning techniques as proposed by Bettayeb Maamar et al [30] method of designing fractional filter based internal model controller for a class of systems. General procedure to design of fractional filter based IMC controller is summarized as follows:

To design IMC controller, process model G_m (s), is factorized as non invertible and invertible part.

〖 G〗_m (s)=G_m^+ (s)G_m^- (s)

Here G_m^+ (s) is the invertible part of process model G_m (s) and G_m^- (s) is the non invertible part of process respectively.

General structure of fractional filter is given by f(s)=1/(1+〖ϔ_c〗^ s^α ) 0 <ð›,< 1

Time constant Ï”_c and non integer ð›, are selected based on desired phase margin 〖ϕ 〗_m and the cross over frequency w_c of the closed loop.

ð›, =( π-〖ϕ 〗_m)/(π/2)-1 and Ï”_c=1/(w_c^(α+1) )

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