7CHAPTER 2 LITERATURE SURVEY 2.1 INTEGER ORDER MODELING AND CONTROL The great majority of PID tuning rules assumes that FOPDT model of the process is available. This is motivated by the fact that many processes can be described effectively by this dynamics which needs a simple structure of a PID controller. Different methods have been proposed in the literature to estimate the three parameters of FOPDT model by performing simple experiment on the plant. They are typically based either on an open loop step response or on a closed loop relay feedback experiment. The tangent method, the area method, two points based method, least squares based method, optimization based method are some of the well known open loop methods. Standard relay (symmetrical relay, asymmetrical relay and relay with hysteresis) feedback method is a well known closed loop method. Some of the open loop and closed loop modeling methods and tuning of integer order controllers are briefly described in the following paragraphs. Ziegler Nichols (1942) proposed a time domain approach to identify the parameters of P, PI and PID controllers in terms of the parameters obtained from the process reaction curve (open loop response curve obtained for a step change in manipulated variable) and a frequency domain approach which is a closed loop method in which the controller parameters are calculated from the ultimate gain (which causes sustained oscillations in the He concluded that open loop step responses of all the processes were sigmoidal (‘s’ shaped) curves and also they can be approximated as FOPDT models. Then a low order model was derived by applying a model reduction algorithm. Then he used different performance criteria to obtain the tuning formulae for the tuning parameters of P.3% of its final steady state value respectively. Sundaresan and Krishnaswamy (1978) proposed a method based on the two time instants of the process reaction curve for estimation of dead time and time constant of FOPDT model. PI and PID controllers. PI and PID controllers in terms of process parameters of FOPDT model. A salient feature of this methodology is that it does not require any special input to the process but it can be applied in different operating conditions. and obtained the process reaction curve for a step change in the manipulated variable. Sung et al (1998) proposed a least squares method to model a system as a higher order transfer function. Cohen and Coon (1953) derived empirical formulae for finding the tuning parameters of P.8 closed loop step response) and ultimate period (which is the period of oscillation of sustained oscillations). The process gain was calculated in the same way as the previous method. It consists of determination of two time instants t1 and t2 when the process output attains 35. The dead time and the time constant of the model were calculated from the two time instants t1 and t2.3% and 85. A remarkable robustness with respect to measurement noise was achieved by considering the integrals of the input and output signals instead of their derivatives. . He found that the inclusion of the controller resulted in the oscillatory response and hence he removed the controller and opened the closed loop. In this method experimental step response and the model step response were compared and the model parameters were adjusted in order to minimize the difference between them. An internal model controller (IMC) based PID method was used for tuning the outer loop controller. The internal feedback has been used for stabilizing the process and the outer loop has been used for good set point tracking. The method has been tested successfully on many low order processes. where the control system led to the stability limit. The ultimate frequency u and hence the ultimate period Tu was calculated from the period of sustained oscillations. Astrom and Hagglund (1984) proposed a closed loop method in which relay feedback controller was used to obtain the non parametric model of the process namely ultimate gain Ku and ultimate frequency u. They suggested that auto tuning based on relay feedback or the Ziegler–Nichols method can be used for tuning an inner loop controller and the tuning parameter ( ) used to tune IMC-PID can also be used as a time constant of a set point filter which can reduce the peak overshoot. The proposed relay feedback experiment used the standard symmetrical relay to generate a sustained oscillatory response of the process output.9 Mitchell (1998) mentioned a genetic algorithm based optimization method to estimate the three parameters of FOPDT model. Vijayan and Panda (2012) proposed a double-feedback loop/method to achieve stability and better performance of the process. in analogy with the original idea of the ultimate sensitivity experiment of Ziegler Nichols (1942). The ultimate gain Ku was calculated from the relay amplitude and the amplitude of the sustained oscillations. The major drawback of this method is that it involves significant computational effort. . O’Dwyer (2006) listed many tuning rules that were developed based on these two values. . The step response and derived analytical formulae were used to calculate the steadystate gain. Nithya et al (2008) designed a GA based Fuzzy Logic Controller for liquid level control of spherical tank modeled as a FOPDT system and its performance is compared with Skogestad's PI method for the nonlinear process in terms of ISE and IAE. The parameters were estimated from the coordinates of the peaks of under damped closed loop response curve. Kavdia and Chidambaram (1996) proposed a method for unstable FOPDT models under proportional only controller where Pade approximation was used for the delay term.10 For estimation of the process parameters from a closed loop test applying a step change in the set point. Salim Ahmed et al (2007) proposed methods to estimate process model parameters from both open loop and closed loop step responses even if the step input was applied when the process was not at a steady state and developed estimation equations on industrial data without preprocessing. Padma Sree and Chidambaram (2006) proposed a simple method of identifying first order plus time delay transfer function model for unstable systems. Ananth and Chidambaram (1999) proposed a simple closed loop method with PID controller for identifying first order plus time delay transfer function for unstable systems. time delay and time constant of the unstable system. Panda and Yu (2003) proposed a systematic approach to derive the analytical expressions for relay feedback responses and explained how to identify unknown system parameters from the derived analytical expressions. The method is based on a single experiment on a closed-loop system with a PI or PID controller with a step change in the set point. 2009) have been proposed in the literature. Identifying a given system from the experimental data becomes more difficult. 1999) addressed fractional order modeling and control in various fields of engineering. system identification has become a standard tool. Thus. Many methods of tuning of integer order PI (Johan et al . Oldham and Spanier (1974). Tsao 1989. when fractional orders are present. once the maximum order of the system to be identified is chosen. the fractional order has significantly complicated the identification process (Bijoy 2009). Oustaloup (1981). Podlubny (1994. Ross (1975). Maia 1998). Several . For unknown systems. and finally the coefficients of the operators. These authors identified electrode-electrolyte polarization and mechanical damping behavior using frequency domain techniques for specifically chosen transfer function forms. For fractional-order systems. the parameters of the model can be optimized directly. 2002)and PID controllers for stable and unstable systems of different models (Panda.11 Madhavasarma and Sundaram (2008) modeled the spherical tank as FOPDT system from the experimental data with an error of less than five percent and compared the performances of model based Smith Predictor controller.2 FRACTIONAL ORDER MODELING AND CONTROL Application of fractional calculus in the field of modeling and control was described by many authors. Many methods have been proposed in the literature for identifying the integer order systems from the experimental data. identification requires the choice of the number of fractional operators. For integer-order systems. Previous work in this area has been limited (Sun 1984. IMC controller and IMC PID controller. the fractional power of the operators. Fractional order PID controllers have been increasingly used for fractional order systems over the last few years (Podlubny 1999). 2. Among the methods proposed based on frequency domain specifications Ziegler–Nichols type empirical rule developed by Valerio and Sa da Costa (2004). . 2012). Bacterial Foraging Optimization (BFO). ITAE). and tuning of FOPI/FO[PI] controllers for controlling fractional order systems by Ying Luo et al (2010) based on the afore mentioned first three constraints are some of the systematic methods. fractional PD (Zhang and Pi. (Padula and Visioli. tuning based on the specified phase margin ( m).e. PI Dµ controllers both in frequency domain and time domain (Biswas 2009) (Chen YQ 2006). The various time domain approaches are dominant pole placement tuning (Maiti Deepyaman 2008). flat phase curve around gc gc). high frequency noise rejection index(in dB) and sensitivity function(specified error in dB) using the NelderMead direct search simplex minimization method proposed by Monje et al (2004. 2012) based tuning of FOPID controllers are also presented in literature. GA (Saptarshi Dasa. Some of the methods involve both frequency as well as time domain criteria. 2008). iso-damping/robustness ). ISE. 2012). gain cross-over frequency ( criteria (i. (Indranil Pan 2011). PSO and BFO (Sanjoy Debbarma and Lalit Chandra Saikia. Chen YQ (2005). and hybrid optimization algorithms for fractional controller tuning. Generalization is done in recently developed heuristic algorithms such as Particle Swarm Optimization (PSO).12 methods have been proposed for tuning of fractional PI. tuning of FOPI/FOPD controllers for controlling integer order systems by Ho (1995). optimal tuning (Cao Jun-Yi 2005). tuning of FOPI controller by Bhaskaran et al (2007). Bettou and Charef (2009) have proposed a combination of frequency domain and time domain approach for tuning FOPID controllers. Applications of fractional controllers to different types of processes are addressed in literature (Chunna zhao 2008). 2012) tuning based on minimization of time domain integral performance indices (IAE. 13 A few methods of fractional order modeling. Igor Podlubny (1999) proposed the concept of fractional-order PID controller and derived explicit analytical expressions for the unit-step and unit-impulse response of a linear fractional order system with fractional order controller both for the open and closed loop. Petras (1999) explained the mathematical description and synthesis of fractional order controllers using modified root locus method. He compared the performances of fractional order controller on an integer order system and a fractional order system and proved the inadequacy of approximation of fractional order system with integer order system for controller design and the robustness of the fractional order controllers to the process parameter variations and controller parameter variations. On the other hand they considered a new class of dynamic systems (systems of an arbitrary real order) and new types of controllers (fractional order controllers). They proved that the fractional order PID controllers could significantly improve static and dynamic control system properties and could be used as robust controllers because of their less sensitivity to controlled system parameters and controller parameters variations. The most important limitation of the method presented in their work was that only linear systems with constant coefficients could be considered. Petráš et al (1998) designed the fractional order PID controllers for fractional order systems in the frequency domain for the determined stability and dumping level. tuning of fractional order controllers for integer order systems and tuning of fractional order controllers for fractional order systems are reviewed below. They presented synthesis of fractional PID controllers and analysis of their behavior in simulation. . They also pointed out the nonadequate approximation of non-integer systems by integer order models and differences in their closed loop behavior. fractional calculus fundamentals. They presented basic ideas and technical formulations of the four different types of fractional order controllers with some comparative comments. The four parameters of a non integer order PI D controller are validated by step by step extension of classical control theory. Here both differential and integral operator is of the same fractional order. models or representations of fractional order systems and fractional order controllers in time domain. PI D controller and fractional lead-lag compensator. Chunna Zhao et al (2005) designed FOPD and FOPID controller for a class of fractional order plants in frequency domain and compared with IOPD and IOPID controllers in simulation and proved the efficiency of fractional order controllers over integer order controllers by considering two different examples. controllability and observability and to find the error static coefficients. So there are four tuning parameters.14 Caponetto et al (2002) proposed a new frequency domain based technique for tuning the parameters of the fractional order PID controller. Laplace domain and Z domain to study their transient and steady state performances and to determine the conditions for stability. Xue and Chen (2002) compared the performances of four representative fractional-order controllers in the literature. CRONE controller (Controle Robuste d’Ordre Non Entier). and they used optimization . For finding p. frequency domain approach of analog and discrete approximations of fractional order operators. namely. Vinagre et al (2002) discussed historical introduction to fractional calculus. TID (Tilted Proportional and Integral) controller. He also designed optimal fractional controller for a class of commensurate fractional order systems based on Wiener – Hopf design method. In their proposed design they used required phase margin and gain margin values. c. c. They concluded that properly tuned fractional PID controllers with proper realization method for fractional powers outperform the integer PID controllers. Jun-Yi Cao and Bing-Gang Cao (2006) designed enhanced PSO the objective of which is the weighted combination of ITAE and control input based FOPID controllers for different order processes. To design the parameters of FOPID controllers.15 method based on some specified constraints and derived the equations for Kp. Jun-Yi Cao et al (2005) designed FOPID controllers for different order processes based on GA based optimization technique. The numerical realization of FOPID controllers used the methods of Tustin operator and continued fraction expansion. designed a fractional PID controller using minimization of ISE and ITAE by selecting the range of and between 0. Experimental results showed the effectiveness of the proposed design method in tuning the parameters of FOPID controllers and its performance was compared with GA based FOPID controllers and the efficiency of the enhanced PSO based FOPID controllers over GA based FOPID controllers were proved.5 randomly and proposed a modified approximated realization method for implementing fractional PID controller. which guaranteed the particle position inside the defined search spaces with momentum factor. Yi-Cheng Cheng and Chyi Hwang (2006) considered the problem of stabilizing unstable first-order time-delay (FOTD) processes using . Ki and Kd interms of p. and . the enhanced PSO algorithms was adopted. The numerical realization of FOPID controllers used the methods of Tustin operator and continued fraction expansion.5 and 1. the objective of which is the weighted combination of ITAE and control input. Dingyu Xue et al (2006) considered a bench mark problem of a DC motor with elastic shaft. The D-partition technique is used to characterize the boundary of the stability domain in the space of process and controller parameters. they tried to simultaneously maximize the jitter margin and ITAE performance (minimize ITAE performance index) for a set of hundred KLT systems having different time-constants and timedelay values. Varsha Bhambhani et al (2008) provided a detailed explanation of design of a robust-jitter controller called optimum fractional proportional integral controller (OFOPI) and compared its performance with OPID controller for systems with small value of . It is shown that for the same derivative gain. They investigated how the fractional derivative order in the range (0. Such a fractional-order PD controller can allow the use of higher derivative gain than an integer-order PD controller. Based on their previously proposed FOPI controller tuning rules using fractional Ms constrained integral gain optimization (F-MIGO). Further. Chunyang Wang et al (2009) proposed two new tuning methods of the FOPI and FO[PI] controllers based on constrained optimization for the .16 fractional-order proportional derivative (PD) controllers. 2) affects the stabilizability of unstable FOTD processes. They observed that the optimization results in enlarged jitter margin of all systems at expense of a slight decrease in ITAE performance of delay dominated systems. a fractional-order PD controller with derivative order less than unity has greater ability to stabilize unstable FOTD processes than an integer-order PD controller. The characterization of a stability boundary allows one to describe and compute the maximum stabilizable time delay as a function of derivative gain and/or proportional gain. Simulation results are presented to verify the proposed new tuning rules for best jitter margin and ITAE performance. the F-MIGO optimization based tuning rules were summarized by approximation of optimized gain parameters and fractional orders of the FOPI controller. so the closed loop system was robust to gain variations and the step response exhibited an iso-damping property. which are robust to high frequency model changes and applied in design of controllers for water distribution in a main irrigation canal pool and the results were compared with more complex control techniques as predictive control and robust H1 controllers.17 typical first-order velocity servo system in simulation. . Ying Luo and Yang Quan Chen (2009) developed a optimization based practical and systematic tuning procedure for the proposed FOPD and FO[PD] controller for a class of fractional order system and verified both in simulation and real time and were compared with IOPID controllers tuned based on the same constraints. special attention is paid to time delay changes. Feliu-Batlle et al (2009) proposed a new methodology to design fractional integral controllers combined with Smith predictors. In particular. the gain crossover frequency was zero at the gain crossover frequency. Yang Quan Chen et al (2009) discussed about various types of fractional order controllers and simulation of Fractional Order Transfer Function (FOTF) in MATLAB environment and stability checking of FOTF in both time domain and frequency domain. These controllers show also less sensitivity to high frequency measurement noise and disturbances than PI or PID controllers. The FOPI and FO[PI] controllers designed by the proposed tuning methods improved the performance and robustness of the first order velocity servo system.r. They also proved the better performance of FO[PI] controller over the FOPI controller among the two fractional control schemes. The given gain crossover frequency and the phase margin were achieved and the phase derivative w.t. They considered a first-order plant with time delay and a first-order plant with an integrator with time delay. Bettou and Charef (2009) proposed a new conception method for tuning of fractional order PI D controller by which enhancement of control quality was achieved. Illustrative examples were presented to show the effectiveness and the simplicity of the proposed method.18 Mohamad Adnan Al-Alaoui (2009) addressed about employment of direct and indirect discretization methods to obtain a rational discrete approximation of continuous time parallel fractional PID controllers. They also proved the robustness of the controller to model uncertainty. TI and TD and minimization of ISE criterion by Hall–Sartorius method was used for setting the fractional integration action order and the fractional differentiation action order . . In the proposed conception method can use any other classical parameters tuning method. The formulations of this new conception method have been derived using the rational function approximation of the fractional integrator and differentiator operators. in a given frequency band of practical interest. They concluded that their conception technique will be very suitable for already tuned PID controllers because in order to implement the fractional PI D controller the already existing classical PID controller can be used with given fractional order differentiator and fractional order integrator. Classical Ziegler– Nichols tuning rules were used for setting the parameters Kp. From the simulation results it was proved that the fractional PI D controllers have significantly improved the performance characteristics of the feedback control systems compared to the classical PID controllers. When the step response of the unity feedback system with analog approximation was compared with direct and indirect discretization approaches it was observed that direct discretization methods yield shorter rise time than the analog approximation but indirect discretization using bilinear CFE can approximate at best. Vale´rio and Sa`da Costa (2010) presented a study on the fundamentals of the theory of derivatives and integrals with arbitrary real or complex orders.2010) designed two fractional order proportional integral controllers. fractional transfer functions and their approximations. FOPI and FO[PI] based on a set of imposed tuning constraints for a class of fractional order systems and compared with IOPID controller designed following the same set of the imposed tuning constraints. whose open-loop transfer function is given by Bode’s ideal transfer function. The proposed technique appears to have promise for the control of fractional order systems instead of designing a integer order counterpart. identification of fractional transfer function models from experimental data. Ying Luo et al (2009.19 Bettou and Charef (2010) introduced a six parameter PI -PD controller for a first order plus integrator with time delay process to enhance the closed loop control performances and proved that the new controller is robust and well suited for models with noise compared with conventional integer order controllers. The robustness of the closed loop system to process gain variations and the isodamping property are proved. Anuj Narang et al (2010) proposed a design method of fractional order PI controller for fractional order models. The proposed strategy is based on a reference model. The parameters of the controller are estimated by formulating a constrained non-linear optimization problem. The performance of the fractional order PI controller designed based on the proposed method has been demonstrated through three fractional order dynamic models. Saptarshi Das et al (2011) modeled the nonlinear process dynamics of an operating Pressurized Heavy Water Reactor (PHWR) as several . has lesser robustness. a nice ability to suppress load disturbances and an inability to filter noise. the complete feasible region of the gain crossover frequency and phase margin were obtained and visualized in the plane. This feasible region for the FOPI controller was compared with that for the . With this region as the prior knowledge. The areas of these two feasible regions for the IOPID controller and the FOPI controller were compared. As a basic step. high probability of building integral windup. the potential advantages of one controller over the other in terms of achievable performances. a scheme for finding the stabilizing region of the FOPI/IOPID controller was presented first. Ying Luo et al (2011) presented a guideline for choosing feasible or achievable gain crossover frequency and phase margin specifications. around various operating points and reduced as NIOPTD-I and NIOPTD-II (non integer order plus time delay) models.20 linearized transfer function models from practical test-data with standard variants of LSE. the complete information about the feasible region of gain crossover frequency and phase margin was collected. Using this synthesis scheme. Thereafter. all combinations of the phase margin and gain crossover frequency were verified before the controller design. better capability of high frequency noise rejection. for the first time. lower value of control signal and hence reduced size of the actuator where as time domain optimal tuning methodology is faster. This area comparison revealed. and then a new scheme for designing a stabilizing FOPI/IOPID controller satisfying the given gain crossover frequency. and proposed a new FOPI/IOPID controller synthesis for all FOPTD systems. Saptarshi Das et al (2011) compared the performances of FOPID controllers tuned in time domain and frequency domain for reduced fractional order models and concluded that frequency domain approach gives better performance in terms of robustness (iso-damping). phase margin and flat phase constraint was proposed in details. Mehmet Önder Efe (2011) proposed a neural network based computationally simple PI D Control for a Quadrotor UAV. . fractional transfer functions and their approximations. In the first method Kp and Ki values are tuned using ZN method. This area comparison showed the advantage of the FOPI over the traditional IOPID clearly. In the second method five non linear equations are formed based on five constraints to ensure the robustness and an optimization technique is used to tune the five parameters of the PI D for a first order and FOPDT systems with modeling uncertainties.and second generation Crone controller. identification of fractional transfer function models from experimental data. The neural network is used to find the coefficients of a Finite Impulse Response (FIR) type approximator. He concluded that the response of the proposed scheme is highly similar to the response of the target PI D controller and the computational burden of the proposed scheme is very low. The results obtained showed that the neural network aided FIR type controller is very successful in driving the vehicle to prescribed trajectories accurately.21 traditional IOPID controller. Vale´rio and Sa`da Costa (2011) presented a study on the fundamentals of the theory of derivatives and integrals with arbitrary real or complex orders. Kd is tuned using Astrom–Hagglund method and . are tuned based on optimization technique for the required phase margin. third-generation Crone control and fractional proportional-integral-derivative control. Yeroglu and Tan (2011) proposed two different tuning methods. that approximates the response of a given analog PI D controller having time varying action coefficients and differintegration orders. first. Simulation illustration was presented to show the effectiveness and the performance of the designed FOPI controller comparing with the designed IOPID controller following the same synthesis. Salah Chenikher et al (2012) proposed a methodology based on optimization with constraints to minimize a cost function subject to H -norm for synthesis of a robust multi-variable fractional order PID controller.22 Luo et al (2011) designed PI and the (PI) controllers based on a set of imposed tuning constraints for improving the flight control performance of a small UAVs and compared with IOPI controller designed with modified ZN rules for the approximated FOPDT model of UAV. two-area reheat thermal System under deregulated . The obtained results showed the efficiency of the proposed method in time and frequency domains over a standard multi-variable PID controller. The FOPID controller was applied to MIMO plant with importantly multiple delays. Debbarma and Saikia (2012) designed a Bacterial Foraging Optimization based FOPID controller in automatic generation control (AGC) of an interconnected environment. Minimization of IAE is set as the objective function of optimization for both servo and regulatory performances. Macias and Sierociuk (2012) modeled the heating process as a fractional order system in frequency domain and validated in time domain. Padula and Visioli (2012) presented a set of optimal tuning rules for standard (integer-order) proportional-integral-derivative (PID) and fractional-order PID controllers for integral and unstable processes. Fractional order PID controller is tuned with ZN and minimization of ISE technique. They addressed that these performance specifications are possible only with a good choice of the weighting functions. The feedback control system with proposed controller guaranteed robustness and best performances. Also they proved that the iso-damped nature of the response allows design of extremely fast systems. . 2010) and fractional disturbance observer (Laurentz et al. MATLAB contains fractional system tool box and CRONE tool box developed by Oustaloup (2000) which are very useful in the analysis of fractional derivatives and integrals and has resulted rapid growth in this field.23 Besides modeling and control fractional calculus finds applications in some other area like fractional order phase shaper (Suman saha et al. Craig Yang Quan Chen (2011) designed a fractional order low pass filter (Q-filter) as a fractional order disturbance observer (FO-DOB) for run-of-mine (ROM) ore milling circuit to get an optimal set-point tracking and disturbance rejection performance and tested on a 3X3 linear time invariant MIMO plant model to evaluate the performance in simulation. Suman Saha et al (2010) proposed a method to enhance parametric robustness of any PID control loop with a FO phase shaper for process control applications where system gains tend to vary with time. Laurentz E. provided the actuator constraints can be met. etc. 2011). keeping overshoot constant. Olivier and Ian K.