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In recent years the industrial application of advanced control techniques for the process industries has become more demanding, mainly due to the increasing complexity of the processes themselves as well as to enhance requirements in terms of product quality and environmental factors. Therefore the process industries require more reliable, accurate, robust, efficient and flexible control systems for the operation of process plant. In order to fulfil the above requirements there is a continuing need for research on improved forms of control. There is also a need, for a variety of purposes including control system design, for improved process models to represent the types of plant commonly used in industry. Continuous Stirred Tank Reactor (CSTR)  is a nonlinear chemical reactor widely used in chemical industries which can be simplified as an affine nonlinear system. It's operation, however, is corrupted with various uncertainties. Some of them arise from varying or not exactly known parameters, such as reaction rate constants, heat transfer coefficients etc. In other cases, operating points of reactors may vary or the reactor dynamics gets affected by various changes of parameters and even causes instability of closed loop control systems. The CSTR reactor is usually used for liquid-phase or multiphase reactions that have fairly high reaction rates. Reactant streams are continuously fed into the vessel, and product streams are withdrawn. Cooling or heating is achieved by a number of different mechanisms like jacket surrounding the vessel or an internal coil. Perfect mixing of the liquid in the reactor is usually assumed, so the modelling of a CSTR involves ordinary differential equations. The control of a CSTR or a series of CSTRs is often a "regulator" problem, in which the temperature(s) and/or concentration(s) are held at the desired values in the face of disturbances. Of course, some continuous processes produce different grades of products at different times, so the transition from one mode of operation to another is a servo problem.
Certain types of chemical systems or processes have highly nonlinear characteristics due to the reaction kinetics involved and the associated thermodynamic relationships. In these circumstances, conventional linear controllers no longer provide adequate and achievable control performance over the whole operating range. Thus, designing a nonlinear controller which is robust in terms of its performance for different operating conditions is essential.
One of the most popular controllers both in the realm of the academic and industrial application is Proportional-Integral-Derivative (PID)  controller. Easy implementation of PID controller, made it more popular in system control applications. It tries to correct the error between the measured outputs and desired outputs of the process in order to improve the transient and steady state responses as much as possible. In one hand, PID controller appear to have an acceptable performance in most of the systems, but sometimes there are functional changes in system parameters that need an adaptive based method to achieve more accurate response. Several researches [3,4,5] are made available that combined the adaptive approaches on PID controller to increase its performance with respect to the system variations. On the other hand, although PID controller is used widely in the area of both academic and industrial control applications, its tuning  is still the controversial scope of investigation. There is also increasing interest in the potential of "intelligent" control methods for process applications. Intelligent control  can be described as a control approach or solution that tries to imitate important characteristics of the human way of thinking, especially in terms of decision making processes and uncertainty. Success of the fuzzy logic, which is based on the approximate reasoning instead of crisp modeling assumptions, remarks the robustness of this method in real environment applications. Fuzzy logic controller [8, 9] emulates the behavior of the experts in controlling the system. Not needing the precise mathematical modeling is a remarkable merit, which causes fuzzy controller more flexible in dealing with complex nonlinear problems. One of the remarkable issues in designing the fuzzy controllers is the strict dependence on the expert knowledge for making the rule bases. In both of the aforementioned controllers, PID and fuzzy, the challenge is fine design and tuning in order to achieve accurate and acceptable results.
Limitation of traditional approaches in dealing with constraints is one of the main reasons for the emergence of powerful and flexible methods. Bio- inspired intelligent computing has been successfully applied to solve complex problems in recent years. In PID tuning, optimization algorithms ,such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Ant Colony Optimization (ACO) are drastically used to find the optimum values of PID parameters. In addition, these bio-inspired algorithms help to design desired fuzzy controller. Along with these controllers, Gain Schedule Controller, Linear Model Predictive Controller and Sliding Mode Controllers are also the most popular control schemes that have been widely implemented throughout the chemical process industries for the past two decades.
The main contribution of this work is the development of suitable controller for the CSTR process. The introduction of powerful nonlinear control schemes and intelligent techniques to modify the above mentioned controllers according to the requirements is proposed in this work. In order to accommodate the nonlinearity and set point variations, different bio inspired techniques and intelligent techniques are utilized to tune the PID controller for this process. The Intelligent Sliding PID controller is developed; also the intelligent form of Gain Schedule Controller (GSC) is introduced. The strategy towards the optimum design in this work is employing a heuristic search hybrid algorithm, named Bacterial Forging-Particle Swarm Optimization (BF-PSO), which finally offers the control parameters so that the criteria of the control scheme would be optimized. In this research work, the highly nonlinear CSTR system is represented as a family of local linear models and different nonlinear controllers has been designed to control the process. The advanced control strategy developed can provide accurate, efficient and flexible operation of the CSTR plant around which the research was based. Besides that, the work involves investigation of issues such as robustness, stability, and overall performance optimization. The first aspect of the work described the development of a mathematical model of a specific CSTR process. The second stage of this research concerns with the development of tuning algorithms for various Controllers in the CSTR process and validate them through performance indices.
1.2 Motivation of this work
Due to its strong nonlinear behaviour, the problem of controlling of CSTR is always an attracting task for control system engineers. Conventional control techniques are suited for completely known and well understood Linear Time-Invariant systems. When the system exhibits parameter inaccuracy and time varying dynamics, conventional control methods encounters problem. Despite of the difficulty in achieving high control performance, the fine-tuning of controller parameters is a tedious task that requires expert knowledge both in control theory and process information. Therefore, the control of systems with nonlinear, complex, unknown and uncertain dynamics is a serious challenge to control community. Optimal control of systems is used to provide a desired control of the process. In order to accommodate the nonlinearity, a gain scheduled control scheme is investigated. The following explains the recent happening in the area of controller developments for CSTR process.
Continuous Stirred Tank Reactors (CSTRs) belong to a typical class of nonlinear dynamical systems described by Ordinary Differential Equations ODEs (Luyben (1990)) .They can present a complex behavior, for instance, they can be operated under multiplicity because of highly nonlinear constitutive relations (reaction kinetics...) (Viel etal. (1997); Favache and Dochain (2010)) . As a consequence, such CSTRs have been investigated with respect to control design for stabilization (Luyben (1990; Hoang et al. (2011, 2012),; Favache et al. (2011) ; Alvarez et al.(2011)) ; and state observer synthesis in a large number of studies (Gibon-Fargeot et al. (1994); Soroush (1997)  ; Alvarez-RamÂ´Ä±rez (1995)  ;Dochain et al. (2009)).The underlying motivation for controlling the CSTRs is that industrial chemical reactors may have to be operated at unstable operating conditions which correspond to some optimal process performances (Bruns and Bailey (1975)). Numerous strategies have been developed to control such non linear systems. For example: input/output feedback linearization (Viel et al. (1997)) for control under constraints, nonlinear PI control (Alvarez-RamÂ´Ä±rez and Morales (2000)), classical Lyapunov based control (Antonelli and Astolfi (2003))  and more recently thermo dynamical Lyapunov based control (Hoang et al. (2012)), (pseudo) Hamiltonian framework (Hangos et al.(2001); RamÂ´Ä±rez et al. (2009); DÂ¨orfler et al. (2009); Hoang et al. (2011); Alvarez et al. (2011)), power-shaping control (Favache and Dochain (2010)) and inventory control (Du et al. (2010)).Observation/estimation strategies have been developed for industrial applications since it is often the case that online measurements of reactant concentrations are difficult and/or very expensive to obtain. Usually, the reactor temperature is the only on-line available measurement. The missing state variables are then estimated (Gibon-Fargeot et al. (1994); Soroush (1997); Alvarez-RamÂ´Ä±rez (1995); Dochain et al. (1992, 2009))  and used in the control strategy. Results given in (Alvarez-RamÂ´Ä±rez (1995);Dochain et al. (1992, 2009)) are related to systems in which unfortunately, no feedback is imposed. It is required to construct a model which consists of all the possible variables as on line measurement variables in order to develop accurate control system. The nonlinear components should also be incorporated in the model. It is also necessary to design the control mechanism which can track the variations in plant parameters with a view to achieving invariant operation throughout the domain of operation of the plant. Adaptive controllers are one such approach, yet even these controllers do not always demonstrate satisfactory performance throughout the domain of operation of the plant and may, on occasion, lose control altogether. Robust controllers, another approach, also have their limitations since they must deal with system dynamics that vary over a wide range though using constant parameters only. Clearly this class of controllers can only operate satisfactorily over a limited domain.
The suitability of variable gain controllers for industrial processes is a matter of further studies. Recent developments in variable gain controllers are:
_ A self-organizing fuzzy controller (Kazemian, 2001)  based on a PID correction factor P obtained from fuzzy inference and pole placement. The strategy was implemented in a robot arm model with invariant parameters. No numerical comparison is made between this strategy and the standard PID or fuzzy controller.
_ Soft computing applications to PID variable tuning are illustrated in DC motor control (Ravichandran and Karray, 2001) . They present three strategies (PID, fuzzy variable tuning, and Artificial Neural Networks (ANN) variable tuning). Tests demonstrated that the neurofuzzy improved tuning leads to good closed-loop response.
_ A fuzzy gain scheduler where only the proportional gain is adjusted in the PID controller, and gain scheduling is performed based on a differential equation relating the fuzzy input and the output variables (Blanchett et al.,2000) . Results illustrate the behavior of this strategy in the temperature control of an aluminum block in a heating process.
_ A fuzzy supervisory predictive PID controller (Tsai andLu, 1998) , a fuzzy logic-based auto-tuner (Bandyopadhyay and Patranabis, 1998) , a fuzzy self-tuning PID controller (Lou et al., 1996) , and the a-fuzzy self-tuning controller (He et al., 1993 ).
These four strategies use either an additive or multiplicative term based on the error that modifies the tuning parameters.Some non-fuzzy based variable gain tuning methods are an Adaptive PI algorithm (Ali, 2000)  , a Self-Tuning PID by Adaptive Interaction (Lin et al., 2000) , an Adaptive/Self-Tuning PID by Frequency Loop-Shaping (Grassi et al.,2000) , a Self-Tuning PID using a Genetic Algorithm (Mitsukura et al., 1999) , and a Self-Tuning PID based on a Generalized Minimum Variance Control (GMVC) Law (Yamamoto et al., 1998) . Even though the previous methods demonstrate good performance in the published application it will be difficult to implement them in an industrial control problem. Their disadvantage is their main characteristic: continuously changing tuning parameters. It is important to note that the literature generally does not present the behavior of these strategies in time variant systems, such as industrial processes or models. Usually their performance is evaluated in linear or time invariant models. Therefore Considerable effort has gone into developing controllers that can track the variations in plant parameters with a view to achieving invariant operation throughout the domain of operation of the plant. Gain-scheduling is a well-known technique of industrial control and is used when a plant is subject to large changes in its operating state, a situation that is typical in industry. Large changes in the operating state lead to corresponding variations in the parameters of the linearized models of the plant about these operating states, it is well known that it is not possible therefore to design a controller to operate satisfactorily at one operating state and expect it to perform equally well elsewhere without re-tuning it. In closed loop ,system performance is degraded since the controller cannot track the changes in the operating states. The newly developed BF-PSO optimized Fuzzy Gain-Scheduling Controller assures smooth transitions in the control law, yet maintains essentially invariant closed system characteristics.
The design methodology is presented for the analysis and synthesis of different optimization techniques for the controller tuning. The aim of this thesis is to tune the controller optimally by using different soft computing techniques with the help of MATLAB software and to make a comparative study. Also it has been confirmed that the optimally tuned controller has a better response compared to the conventional systems.
To find the nonlinear mathematical model of CSTR process.
To develop different tuning algorithms for conventional controller
Apply the different optimization techniques.
To develop nonlinear controllers and perform the tuning.
To make the performance analysis of the controllers with different algorithms
based on the robustness and performance indices.
Numerous research works have been carried out to develop various algorithms in tuning PID controller for the CSTR process. This section summarizes the literature survey that was conducted as a part of the research reported in this thesis. It covers pertinent established concepts and techniques related to control system design for different types of process. Simulation and real time implementation results for CSTR process using conventional and intelligent control techniques employed in various literatures are analyzed and discussed. Literature survey was conducted on various optimization techniques involved in optimizing the gains of different controllers with various objectives. A detailed survey was conducted on various bio inspired optimization techniques and the performances of the algorithms in optimizing the controller structure in both simple feedback and complex modes of operation are discussed.
1.4.1 Overview of the System
The main purpose of this section is to provide essential background information about the Industrial Continuous Stirred Tank Reactor which is taken as the plant or process in this research .There are many excellent reference papers and books in the field of development of controllers for chemical process . This section describes, briefly, the general properties of reactor from a chemical perspective and continues with some explanations of the reaction process.
The model problem of a single, irreversible, exothermic reaction taking place in a perfectly mixed continuously stirred tank reactor, and its extensions, has been extensively studied by many authors. The early literature was reviewed by Ray (1977) , more recent review articles are those of Razon and Schmitz (1987), detailing the multiplicity and dynamic behaviour of chemically reacting systems, and Bequette (1991) , dealing with the nonlinear control of chemical processes. The classic problem has, at most, seven static steady-state diagrams as control parameters are varied Razon & Schmitz (1987), including the standard S-shaped combustion curve. We outline some recent extensions of the standard CSTR model which retain a single, one-step, exothermic reaction.
One of the motivations for studying the dynamic behaviour of a CSTR is to understand the mechanisms by which thermal runaway occurs, so as to eliminate this phenomenon in industry. The risk of thermal runaway may be reduced by using a control that changes one of the inflow parameters, e.g. the inflow temperature, inflow velocity, or inflow concentration, as a function of the outflow variables, usually the outflow temperature. Possio & Pellegrini (1996)  have conducted a detailed bifurcation analysis of a non-ideal CSTR, in which flow bypass and dead volume are taken into account, with a Proportional-Integral (PI) controller acting on the inlet temperature. They showed that the introduction of the control can cause unexpected patterns of dynamic behavior via the promotion of degenerate, type 1, Hopf bifurcations. Furthermore the PI controlled CSTR can destabilize a stable equilibrium point to a limit cycle for any value of the control constants. Sexton et al (1997)  considered the effect of control on limit cycle behaviour where the feed flow rate is continuously changed in response to the outflow temperature. They showed that oscillatory behaviour can be eliminated by linear proportional control, although the choice of the gain parameter is not simple. Furthermore they presented several nonlinear proportional control functions which ensure that oscillatory behaviour is eliminated.
Recently an ingenious way of controlling the standard reaction by placing the CSTR within a second CSTR in which an endothermic reaction occurs has been investigated Gray and Ball (1999) ; Ball and Gray (1999) . The basic idea is to carefully choose the endothermic reaction so that it `kicks in' when the exothermic reaction is on the point of thermal runaway, cooling it to the desired operating temperature. Although simple in principle, this concept translates into a differential dynamical system of formidable complexity.
The standard CSTR model implicitly assumes that the cooling jacket dynamics are negligible and that the coolant temperature, often referred to as `ambient temperature', is the experimentally manipulated variable. Russo & Bequette (1995), have included the cooling jacket energy balance, so that the coolant temperature becomes a third state variable. The principal control parameter then becomes the coolant flow rate.
Although research into CSTR models goes back over forty years, no universally accepted non-dimensionalisation scheme has emerged. This lack of consistency has resulted in many different dimensionless forms of CSTR models. The non-dimensionalisation chosen is unimportant in the cataloguing of the distinct types of steady-state diagrams exhibited by a system. However, it is not possible to relate the order in which these steady-state diagrams occurs in the non-dimensionalised model to the order in which they would be encountered experimentally unless the distinguished bifurcation parameters, which depend upon the non-dimensionalisation, correspond to experimentally distinct quantities.
This point has not been appreciated in most of the CSTR studies to-date and it is only recently that consistent use has been made of non-dimensionalised forms in which physically distinct quantities are retained as independent non-dimensionalised parameters (Ball 1999) .
Reactor control is an interesting and challenging research subject which has led to a large number of motivating and interesting published papers. Qing-Guo Wang, Tong-Heng Lee, Ho-Wang Fung, Qiang Bi and Yu Zhang (1999)  designed a simple PID controller that achieves high performance for a wide range of linear self-regulating processes. Satisfactory responses could be expected for processes with various dynamics, including those with low and high-order, small and large dead time, monotonic and oscillatory responses. It was developed based on a second-order plus dead time modeling technique and a closed-loop pole allocation strategy through the use of root-locus.
Detailed explanations were given by Majhi S. and Atherton D.P (2000)  in which they tested a single-relay feedback that can be used to identify the exact model parameters of an unstable First-Order Plus Dead Time (FOPDT) transfer function when a limit cycle exists. That was only true if the ratio of time delay to unstable time constant is less than 0.693. It has also been shown how adding an internal Proportional (P) feedback could increase this value. Jietae Lee and Ho-Cheol Park (2000)  successfully applied a two parameter model of a single fictitious weak acid with unknown dissociation constant that has been used to design a neutralization system for many multi-component acid streams. Hence, for better control of wide class of multi-component acid streams, a three parameter model of a strong acid and a weak acid with unknown dissociation constant was proposed. Therefore the model approximates effectively three types of largest gain variation nonlinearities. Based on this concept, a nonlinear pH control system was designed. Parameters could be easily estimated in that method since their combinations appear linearly in the model equations and nonlinear adaptive control system may also be constructed just as with the two parameter model.
Abdul Aziz Ishak, Mohamed Azlan Hussain and Elamin Elkanzi (2001)  implemented the dynamics and control of a semi batch wastewater neutralization process in modeling and simulation. The wastewater neutralization process has been modeled based on the reaction between a strong basic solution and a strong acidic solution in a semi batch reactor. It was assumed that the wastewater is acidic in nature. The semi batch system chosen there consists of two input streams and a mixing tank which contains an initial amount of acid solution. The initial simulation involves studying the dynamics and control of the model formulated. The objective of that control was to regulate the base flow, while keeping the acid flow rate constant, until the pH in the mixing tank was stabilized.
Later, Paraskevas N. Paraskevopoulos, George D. Pasgianos and Kostas G. Arvanitis (2004)  used Pseudo Derivative Feedback (PDF) in the control and identification of UFOPDT processes. Several new methods for tuning the PDF controller were also presented. In contrast to known tuning rules for conventional Proportional Integral Derivative (PID) controllers, which result in excessive overshoot in the closed-loop response, their proposed control structure and tuning methods ensure a smooth response to set-point changes, fast attenuation of step-load disturbances and satisfactory robustness against parametric uncertainty. Again Jose L. Rodriguez and Kenneth A. Loparo (2004)  presented a complete and detailed manner of the modeling of pH processes. The proposed model was strictly based on the physical balance equations of mass and charge. The same model exactly represents the balance equation. It proved that polyprotic substances could be represented uniquely by a combination of monoprotic substances equivalent to the real acids, bases and salts that compose the solution.
Another important paper on the investigation of the fundamental properties of an adaptive algorithm of Universal Learning Network (ULN) and its application to identify pure time delay of a plant model has been presented by Min Han et al. (2006) . Universal learning network could be used in model predictive control for stabilizing a class of nonlinear systems with long time delay. Depending on ULN model with single neuron controller, the control architectures were introduced and applied to pH neutralization process. In that paper the general architecture and adaptive learning algorithm gave ULN more representing abilities to model and control the nonlinear black box systems with long time delay.
The research group of Nick.J.Killingsworth and Miroslav Krstic (2006)  widely used Proportional-Integral-Derivative controllers in the process industry. Much effort has been invested by them in developing systematic tuning methods. Many of these methods require special experiments to identify a suitable plant model. Seok-Beom Lee (2006)  investigated a closed-loop estimation method of permanent magnet synchronous motor parameters by PI controller gain tuning. The idea of that method was to tune the controller to cancel the pole of the motor transfer function with a controller zero and estimate motor parameters from the tuned controller gains. The induced EMF is cancelled by a feed forward compensation term. A systematic tuning procedure was illustrated based on sound theoretical derivations. Their proposed method does not require complex data analysis or test configuration. Only simple calculation and tuning were sufficient to estimate motor parameters while meeting the desired closed-loop performance. Hence detailed idea about PID controller has been obtained from this paper.
Paraskevas N. Paraskevopoulos (2006)  controlled an Unstable First-Order plus Dead-Time (UFOPDT) processes using Proportional-Integral and Proportional-Integral-Differential type controllers. New tuning rules based on the exact satisfaction of gain and phase margin specifications were proposed in that paper. Their tuning rules were given in the form of iterative algorithms, as well as in the form of accurate, analytical approximations. Moreover, several specific functions, related to the crossover frequencies of the Nyquist plot and to the feasible design specifications for a given process, were derived. These functions, which were particularly useful for the general design of PI- and PID-type controllers for UFOPDT processes has been accurately approximated, in order to simplify the tuning procedure. With the proposed approximations, the tuning rules reported require relatively small computational effort and particularly useful for online applications. Again Ivan Zelinka et al. (2006)  studied the basic simulation on common used devices in chemical industry - Continuous Stirred Tank Reactor (CSTR). A mathematical model has been developed from material balances. Numerical techniques were used for steady-state and dynamic analysis. Also Lou Haichuan and Dai Wenzhan (2008)  proposed a novel predictive controller based on Minimal Resource Allocation Network (MRAN) for non-linear system. The controller combines the advantages of minimal resource allocation network algorithms and neural predictor. The algorithm was applied in a high non-linear Continuous Stirred Tank Reactor (CSTR) pH process model.
Rajani K. Mudi, Chanchal Dey and Tsu-Tian Lee (2008)  proposed Ziegler-Nichols tuned PI and PID controllers. It has been usually found to provide poor performances for high-order and nonlinear systems. There an improved auto-tuning scheme was presented for Ziegler-Nichols tuned PI controllers. With a view to improve the transient response, the proportional and integral gains of the proposed controller were continuously modified, based on the process trend. Robustness of the proposed scheme was established by varying the controller parameters as well as the dead-time of the process under control. Farhad Aslam and Mohd. Zeeshan Haider (2011)  proposed three level control models. Here the control is done with conventional control PID model. The auto tuning technique of PID controller has been adopted for more reliable and precise control action which incorporates the uncertain factors also. Comparison of the conventional PID and auto tuning is clarified.
Conventional control approaches are not convenient to solve the complexities. Fuzzy logic and neural networks control have emerged over the years and become one of the most active and fruitful areas of the research in the intelligent control applications. Fuzzy logic, neural networks and ANFIS are three popular artificial intelligence techniques that are widely used in many applications. Due to their distinct properties and advantages, they are currently being investigated and integrated to form new models or strategies in the areas of system control.
Seng et al (1998)  have proposed a neuro-fuzzy controller to a coupled-tank liquid-level laboratory process based on the radial basis function neural network tuned automatically using genetic algorithms (GA). They have used a linear mapping method to encode the GA chromosome, which consists of the width and centre of the membership functions, and also the weights of the controller. Dynamic crossover and mutation probabilistic rates are also applied for faster convergence of the GA evolution. Compared to a manually tuned conventional fuzzy logic controller and a Proportional-Integral-Derivative (PID) controller which are applied to the same process, the proposed controller shows considerable robustness and advantages. Human expert intelligence in framing the rule base of fuzzy logic controller is a major limitation.
Naman et al (2000)  have presented an adaptive model reference fuzzy controller (AMRFC) for controlling the water level in a water tank. Performance is compared with conventional methods of proportional-integral (PI) control and Model Reference Adaptive Control (MRAC). Unlike most of the literatures reviewed which use the error and error change as inputs to the fuzzy system, this method uses the theoretical background developed for MRAC in choosing these inputs. Although the controller uses many inference rules (441 rules), it is shown that the required mathematical calculations are not much, making implementation on a low-end microcontroller feasible. The control algorithms are implemented in simulation and real-time on an 8-bit microcontroller. It was found that the MRAC proved to be better compared to the PI controller. Limitation is the similarity in performance due to the linearity of the plant.
Han (2006)  have proposed an adaptive neural network control strategy based on fuzzy self-tuning to control the drum water level of coal-fired power plant. Fuzzy Inference Engine (FIE) is used to train neural network online. The control strategy possesses feed forward compensation ability for steam flow disturbance by introducing the steam flow signal to neural network controller. Robust controller is constructed to guarantee good regulating performance while dynamic behaviour of the controlled plant changes or external steam flow disturbances exists. In contrast to conventional cascade PID control, simulation results show the efficiency and superiority of the proposed strategy.
Yazdizadeh et al (2009)  have proposed two novel adaptive PID- like controllers such as Neural network PID and Neural network PID with internal dynamic feedbacks for controlling multivariable, nonlinear Multiple-Input Multiple-Output (MIMO) systems. Comparative analyses of the novel adaptive controllers are tested with conventional methods and results confirm that the algorithm exhibits excellent performance. It has been applied to control the water level of tanks in water refinement process, which is highly non linear and good performance and stability was achieved. Limitation is the learning rate and the system disturbances are not taken into consideration for the design.
Hasan et al (2011)  have investigated and found a solution by designing the intelligent controller such as neural network for controlling the water level system. The controller also can be specifically run under the circumstance of system disturbances. To achieve these objectives, a prototype of water level control system has been built and implementations of both PID and neural network control algorithms are performed. In PID control, Ziegler Nichols tuning method is used to control the system. In neural network control, the approach of Model Reference Adaptive Neural Network (MRANN) control based on the back propagation algorithm is applied on training the system. Both control algorithms are developed to embed into a standalone DSP-based micro-controller and their performances are compared. Limitation is that the non linear process characteristics are not preserved.
Allaoua et al (2001)  have proposed an Adaptive Neuro-Fuzzy Inference System (ANFIS) for the speed control of DC servo motor optimized with swarm collective intelligence. Initially, controller is designed based on fuzzy rules and then an adaptive neuro-fuzzy mechanism is adopted and ANFIS is optimized by Swarm Intelligence. ANFIS has the advantage of expert knowledge of the Fuzzy inference system and the learning capability of neural networks. Simulation results demonstrate that the deigned ANFIS-Swarm speed controller realize a good dynamic behaviour of the DC motor, a perfect speed tracking with no overshoot, gives better performance and high robustness than those obtained by the ANFIS alone. Adequate knowledge of experts required to design fuzzy logic rules for servo motor control is a major drawback.
Kang and Kim (2001)  have designed a neuro-fuzzy controller to improve some problems that occur when the non linear system isÂ controlledÂ by a fuzzy logic controller. Their model obtains fast time response, maximized learning effect and shortened settling time. To prove the capability of the designed neuro-fuzzy controller, the neuro-fuzzy model is applied to aÂ DCÂ servomotor. As a result, this controller does not produce overshoot, which occurs in the PID controller, and also does not produce the steady state error of FLC. Also, it shortens the settling time by about 10%. The model has only about 60% of the value of current peak of the PID controller. Limitation is that the training algorithm and the learning rate of neural networks have to be properly selected.
126.96.36.199 Fuzzy Logic Control (FLC)
Research on fuzzy logic controllers (FLCs) has increased recently, as FLCs allow a simple and human approach to controller design and do not require precise mathematical modeling knowledge. As the controlled systems become more complex, deriving their mathematical models becomes more difficult. Therefore, fuzzy controllers provide reasonable and effective alternatives to conventional controllers and many researchers have attempted to combine conventional PID controllers with fuzzy logic. Despite the significant improvement these fuzzy PID controllers represent over their classical counterparts, it should be noted that they still have disadvantages.
Extensive work on fuzzy logic has been done by Zadeh (1965 - 1985) [74-89]. In 1965, Lofti A. Zadeh published an interesting and ground-breaking paper on fuzzy sets. This paper describes the mathematics of fuzzy set theory which then led to the development of the fundamental ideas of fuzzy logic. As described in the paper, a fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership function which assigns to each object a grade of membership ranging between zero and one. The three main categories of papers are formal foundations, approximate reasoning and meaning representation.
Mamdani and Assilian (1975)  published a paper entitled "An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller". This paper described the first application of fuzzy set theory in a practical control systems context. The paper also presented the steps taken to control a steam engine and boiler combination by synthesizing a set of linguistic control rules obtained from experienced human operators. The inputs for the fuzzy logic control in this case were "error" and "change of error" and this was in many ways similar to the inputs used in conventional PI controllers. Other papers presented subsequently by Mamdani and his co-authors described the application of this concept of linguistic synthesis to a number of control applications [91-95]. This approach remains one of the most popular and commonly used methods in the development of fuzzy logic controllers. In that research the Mamdani type of approach has been used to develop a fuzzy logic controller for the pH neutralization pilot plant. Takagi T. and Sugeno M. (1983, 1985) [96-97] proposed fuzzy control rules for control applications and Yager, Ovhinnikov, Tong and Nguyen (1987)  published an edited volume entitled Fuzzy Set and Applications. This book is a compilation of selected papers by Zadeh on fuzzy logic. Also C.C. Lee [98-101] published fuzzy logic in control systems.
Qin S. J. and Borders G. (1994)  have found numerous practical applications of fuzzy logic control in control, prediction and inference. The fuzzy logic approach has been chosen in Tang et al. (2001)  in which an optimal fuzzy PID controller was developed. The fuzzy PID controller is a discrete-time version of the conventional PID controller. It preserves the same linear structure of the proportional, integral and derivative parts and has constant self-tuned control gains. The resulting controller does not need to execute any fuzzy rule base. It was actually a conventional PID controller with analytic formulas. Later Jing-Chung Shen (2001)  proposed the Fuzzy Neural Network (FNN) for tuning PID controller for plants with underdamped step responses. The underdamped systems were modeled by second-order plus dead-time transfer functions. For deriving the FNN, the dominant pole assignment method was applied to design the PID controllers for a batch of test plant models that represent the plants with underdamped responses.
A novel framework for fuzzy modeling and model-based control design was described by Janos Abony (2001) . The fuzzy model is of the Takagi-Sugeno (TS) type with constant consequents. The number and position of these points were determined by an iterative insertion algorithm. Constrained optimization has been used to estimate the consequent parameters, where the constraints were based on the modeled process. Feng Zheng et al. (2004)  studied the issue of designing robust adaptive stabilizing controllers for nonlinear systems in Takagi-Sugeno fuzzy model with both parameter uncertainties and external disturbances. It was assumed in that paper that the parameter uncertainties were norm-bounded and may be of some structure properties that the external disturbances satisfy matching conditions. Shinn-Jang Ho (2006)  solved an optimization problem of establishing a Fuzzy Neural Network Model (FNNM) for efficiently tuning PID controllers of various test plants with underdamped responses using a large number of training plants such that the mean tracking error of the obtained control systems was minimized. Their innovation was to directly and simultaneously optimize the four FNNs by using a novel Orthogonal Simulated Annealing algorithm (OSA). Feng Wan (2006)  formulated pH process control as a challenging problem due to the strong nonlinearity and extreme sensitivity to disturbances of the process. This paper also presented an application of one-step ahead adaptive fuzzy control scheme for a strong acid-strong base neutralization process.
Bharathi N. (2006)  investigated the control of chemical process using neural and fuzzy controller. Initially a conventional linear controller was tried to control the CSTR at different linear regions. Based on the process characteristics, the nonlinear operating region was divided into three linear regions like low, middle, and high. Within each region, a local linear model was used to represent the process, and a controller was designed in their work. Valarmathi et al. (2007)  successfully applied fuzzy logic to many applications in control with uncertainties. They have designed fuzzy rules and the membership functions. The rules and the membership functions were formed from the experience of the human experts. Lughofer E. and Guardiola C. (2008)  proposed On-line fault detection with data-driven evolving fuzzy models. An adaptive control scheme based on fuzzy logic systems, for pH control has been addressed by Shahin Salehi, Mohammad Shahrokhi and Ali Nejati (2009) . Stability of the closed-loop system was established in that paper and it was shown that the solution of the closed-loop system was uniformly bounded and under a certain condition, asymptotical stability was achieved.
Fuzzy logic is capable of managing complex applications efficiently, even with uncertainties or vague information about the system to be controlled. The fuzzy logic concept has also been shown to be capable of mimicking human decision making processes for applications where manual control is known to produce acceptable control performance. Thus the successful application of fuzzy control concepts in other fields has encouraged this research activity to investigate the benefits and limitations of fuzzy control in the pH neutralization process.
188.8.131.52 Adaptive Network based Fuzzy Inference System (ANFIS)
In general terms, these techniques are now recognized as the most successful technologies for developing and implementing control systems for a wide range of industrial applications. These research activities also reflect interest in improving the operation and control of systems involving highly nonlinear process plant using these algorithms.
Initially Jyh-Shing Roger Jang (1993)  presented the architecture and learning procedure underlying ANFIS, in which a fuzzy inference system was implemented in the framework of adaptive networks. By using a hybrid learning procedure, the proposed ANFIS can construct an input-output mapping based on both human knowledge (in the form of fuzzy if-then rules) and stipulated input-output data pairs. They have employed the ANFIS architecture to model nonlinear functions, identify on line nonlinear components in a control system and predict a chaotic time series. The node functions in the same layer are of the same function family as described in Luis E. Zarate, Peterson Resende and Benjamin M., (1991)  and Luis E. Zarate, Peterson Resende and Benjamin M. (2001) .
1.4.3 Bio Inspired Algorithms
Optimization is a powerful tool that has been used in the design of PID controllers. An optimization algorithm that takes care of the local minima, requires fewer computations and which settles at minimum time is said to be the best algorithm. A common feature of all population-based algorithms is that the population consisting of possible solutions to the problem is modiï¬ed by applying some operators on the solutions depending on the information of their ï¬tness. Hence, the population is moved towards better solution areas of the search space. This section involves the literatures on various research works in development and implementation of advanced control approaches involving Genetic Algorithm (GA) ,Simulated Annealing (SA),Particle Swam Optimization(PSO) and Ant Colony Optimization(ACO) techniques
184.108.40.206 Genetic Algorithm (GA)
Goldberg D.E (1989)  published a book on Genetic Algorithms in Search, Optimization, and Machine Learning that will enable both students and practitioners to apply genetic algorithms to problems in many fields. Zibo and Naghdt (1995)  applied genetic algorithms to identify the parameters of the Multi Input and Multi Output (MIMO) system that is assumed to have an Auto Regressive with Moving Average Exogenous (ARMAX) structure. Dangprasert P. and Avatchanakorn V. (1996)  presented a new intelligent regulator with self tuning scheme using the robust search feature of GAs incorporated with the basic idea of self-tuning regulators. The proposed regulator utilized GAs to search for the changes of system parameters and to calculate the corresponding control law. The optimum parameters and control law has been chosen by means of the selection mechanism of GAs which employed the squares of difference between the actual and the estimated outputs as the fitness function. The regulator has an online parameter identification function and requires neither prior knowledge nor training data for learning. The proposed GA based system was applied in their work to load frequency control of a power system to investigate its effectiveness.
Antonio et al. (2000)  worked on the investigation of the fundamental properties that simultaneously tune multiple power system damping controllers using Genetic Algorithms. The tuning method has taken robustness into consideration as it guarantees system stabilization over a pre-specified set of operating conditions. Modified GA operators were used in the simultaneous optimization of both phase compensations and gain settings for the stabilizers. Also Meng Joo Er and Ya Lei Sun (2001)  presented a new approach towards the optimal design of a hybrid PID controller applicable for controlling linear as well as nonlinear systems using genetic algorithms. The proposed hybrid PID controller has been derived by replacing the conventional PI controller by a two-input normalized linear fuzzy logic controller and executing the conventional controller in an incremental form.
Sanjay Kumar Sharma and George W. Irwin (2003)  presented a new chromosome encoding method, named fuzzy coding, which has been proposed for representing real number parameters in a genetic algorithm. In this investigation, fuzzy coding provides the value of a parameter on the basis of the optimum number of selected fuzzy sets and their effectiveness in terms of degree of membership. The knowledge associated with each parameter was an indirect method of encoding compared with alternatives, where the parameters were directly represented in the encoding. Mwembeshi and Kent (2004)  proposed a GA-based Internal Model Control (IMC) strategy for a chemical process. The research group of Yeong-Koo Yeo and Tae-In Kwon (2004)  has published a work that investigated a PID control strategy based on the genetic algorithm coupled with cubic spline interpolation method for the control of CSTR processes. The control scheme proposed consists of closed-loop identification based on the genetic algorithm and cubic spline method. The PID controller parameters were computed using relay feedback. Then approximate linear models corresponding to each process variable range were obtained by the closed-loop identification based on closed-loop operation data. The optimal parameters of the PID controller at each pH region were then computed by using the genetic algorithm
Shi Hongrui et al. (2004)  proposed several control strategies for controlling pH processes using Strong Acid Equivalent (SAE) scheme. Multiple model switching method was proposed in that work. For modeling, both fixed and adaptive models were considered. Also Andrey Popov, Adel Farag and Herbert Werner (2005)  characterized control engineering problems by several objectives, which have to be satisfied simultaneously. Two widely used methods for finding the optimal solution to such problems has been investigated by them using Pareto-optimal solutions. Also in that paper, Genetic Algorithm approach has been used to find a fixed-gain, discrete-time PID controller for a chemical neutralization plant. Valarmathi K., Devaraj D., and Radhakrishnan T.K. (2008)  proposed optimal values of Proportional Integral Derivative control gains, which is an important task in the design of PID controller. They also presented the application of sugeno fuzzy model for on-line tuning of PID controllers in pH process. The optimal PID controller parameters required to develop the Sugeno fuzzy model were estimated by genetic algorithm. Fuzzy controller which could give the PID parameters for different operating conditions was also developed.
220.127.116.11 Simulated Annealing (SA)
Giriraj Kumar et al. (2010)  discussed in detail about the simulated annealing algorithm, a computational intelligence technique, and its implementation in PID tuning. Here, a closed loop performance was achieved with few transducers and controllers, which could support a stable and efficient process. Salwani Abdullah, Laleh Golafshan and Mohd Zakree Ahmad Nazri (2011)  attributed reduction deals in finding all possible information of the problem in hand. An alternative approach to find the minimal attribute from a large set of attributes was done. Towards this goal, a Reheat Simulated Annealing (Reheat-SA) was proposed in that paper to solve an attribute reduction problem in rough set theory. It was a meta-heuristic approach that has a mechanism to escape from local optima. The concept of re-heat was introduced to help the algorithm to explore the search space in order to find a better solution.
18.104.22.168 Particle Swarm Optimization
Kennedy J. and Eberhart R. C. (1995, 2001) [129, 130] suggested the basics of Particle Swarm Optimization technique (PSO). Karray et al. (2002) investigated soft computing-based results pertaining to the hierarchical tuning process of PID controllers, located within the control loop of a class of nonlinear systems. Their work was motivated by the increasing need in the industry to design highly reliable and efficient controllers for dealing with regulation and tracking capabilities of complex processes characterized by nonlinearities and possibly time varying parameters. The soft computing-based controllers proposed by them were hybrid in nature in that they integrate within a well-defined hierarchical structure and the benefits of hard algorithmic controllers with those having supervisory capabilities. The proposed controllers also have the distinct features of learning and auto-tuning without the need for tedious and computationally extensive online systems identification schemes. Ciuprina G., Loan D. and Munteanu I. (2002)  described a new stochastic heuristic algorithm for global optimization. The proposed optimization algorithm, called Intelligent-Particle Swarm Optimization (IPSO), offers more intelligence to particles by using concepts such as: group experiences, unpleasant memories (tabu to be avoided), local landscape models based on virtual neighbors, and memetic replication of successful behavior parameters. The new individual complexity has been amplified at the group level and consequently generates a more efficient optimization procedure.
Clerc M. and Kennedy J. (2002)  analyzed a particle's trajectory as it moves in discrete time (the algebraic view), then progresses to the view of it in continuous time (the analytical view). A five-dimensional depiction has been developed, which describes the system completely. These analyses lead to a generalized model of the algorithm, containing a set of coefficients to control the system's convergence tendencies. Some results of the particle swarm optimizer, implementing modifications derived from the analysis, suggest methods for altering the original algorithm in ways that eliminate problems and increase the ability of the particle swarm to find optima of some well-studied test functions. Zwen-Lee Gaing (2004)  presented a novel design method for determining the optimal Proportional-Integral-Derivative controller parameters of an AVR system using the particle swarm optimization algorithm. It is demonstrated in detail how to employ the PSO method to search efficiently the optimal PID controller parameters of an AVR system.
Ho et al. (2006)  proposed new selection strategies to find the best solutions of the particle as well as for its neighbors to design a novel formula for velocity updating and the incorporation of an intensification search phase. Jose M. Simoes Moita (2006)  presented a finite element formulation based on the classical laminated plate theory for laminated structures with integrated piezoelectric layers or patches, acting as sensors and actuators. The finite element model was a single layer triangular nonconforming plate/shell element with 18 degrees of freedom for the generalized displacements, and one additional electrical potential degree of freedom for each surface bonded piezoelectric element layer or patch. The control has been initialized through an optimization of the core of the laminated structure, in order to minimize the vibration amplitude and maximize the natural frequency. Also the optimization of the patches position has been performed to maximize the piezoelectric actuators efficiency. The simulated annealing algorithm was used for those purposes.
Mehdi Nasri McAvoy, Hossein Nezamabadi-pour and Malihe Maghfoori (2007)  proposed a Particle Swarm Optimization method for determining the optimal PID controller parameters for speed control of a linear brushless DC motor. Also the same proposed method was more efficient in improving the step response characteristics such as reducing the steady-states error, rise time, settling time and maximum overshoot in speed control of a linear brushless DC motor. Amitava Chatterjee and Patrick Siarry (2007)  developed an adaptive neuro-fuzzy classifier which employs two relatively less explored and comparatively new problem solving domains in fuzzy systems. This relatively less explored field was the domain of the fuzzy linguistic hedges which has been employed there to define the flexible shapes of the fuzzy membership functions. To achieve finer and finer adaptation, and hence control, over the fuzzy MFs, each MF was composed of several piecewise MF sections and the shape of each such MF section was varied by applying a fuzzy linguistic operator on it. Therefore the system employed a Takagi-Sugeno based neuro-fuzzy system, where the rule consequences were described by zero order elements. Hence the proposed Linguistic Hedge Based Neuro-Fuzzy Classifier (LHBNFC) employed a relatively new field in the area of combinatorial metaheuristics, called particle swarm optimization, for its efficient learning. PSO has been employed in that scheme to simultaneously tune the shape of the fuzzy MFs as well as the rule consequences for the entire fuzzy rule base.
Tushar Jain and Nigam M. J. (2008)  deal a comparative study between Evolutionary Algorithms namely GAs and Swarm Intelligence, which has been carried out on the basis of performance indices: ITAE (Integral Time Absolute Error), ISE (Integral Square Error), IAE (Integral Absolute Error) and MSE (Mean Square Error). The idea of model generation and optimization was explored for PD -PI controller. Most commonly known, the highly nonlinear Inverted Pendulum system has been used as a test system for that approach.
In the recent years, Yinggan Tang, Leijie Qiao and Xinping Guan (2010)  developed a new approach to identify Wiener model. Firstly, a sequence of step signals with various amplitudes was supplied to the system. The structure of the static nonlinear function was obtained from the input step signals and their corresponding steady-state responses. Once the structure of the nonlinearity was determined, the parameters describing the nonlinearity were estimated through minimizing an objective function. Secondly, the parameters of the linear dynamic subsystem were estimated using random input from the view point of optimization. Particle swarm optimization has been used to solve the two optimization problems involved in parameter estimation of static nonlinear function and linear dynamic subsystem. The proposed method makes the identification problem of nonlinearity separate from that of linear part and simplifies the identification procedure significantly. Also, it does not require any structure information about the static nonlinear function.
Later Chien-Hung Liu and Yuan-Yih Hsu (2010)  proposed a self-tuning proportional-integral controller in which the controller gains were adapted using the particle swarm optimization technique for a Static Synchronous Compensator (STATCOM). An efficient formula for the estimation of system load impedance using real-time measurements was derived in that paper. It was suggested that based on the estimated system load, a PSO algorithm, which takes the best particle gains, the best global gains and previous change of gains into account, was also employed to reach the desired controller gains.
22.214.171.124 Bacterial Foraging Algorithm
A popular swarm-intelligence-based algorithm is the Bacterial Foraging Optimization Algorithm (BFOA) which was introduced by Passino in 2002 , . The BFOA is inspired by the chemotaxis behaviour of bacteria that will perceive chemical gradients in the environment (such as nutrients) and move towards or away from specific signals. The BFOA takes the optimal foraging decision making capabilities of the E.Coli bacteria . The coordinates of a bacterium represent an individual solution of the optimization problem. Such a set of trial solutions converges towards the optimal solution following the foraging group dynamics of the bacteria population .
126.96.36.199 Ant Colony Optimization
The first ACS algorithm was proposed by Dorigo in the early 1990s. The ACS belongs to biologically inspired heuristic algorithms. It was developed mainly based on the observation of the foraging behavior of a real ant. ACO algorithms were originally inspired by the ability of real ants to find the shortest path between their nest and a food source. The key to this ability lies in the fact that ants leave a pheromone trail behind while walking. Other ants can smell this pheromone, and follow it. When a colony of ants is resented with two possible paths, each ant initially chooses one randomly, resulting in 50% going over each path. It is clear, however, that the ants using the shortest path will be back faster. So, immediately after their return there will be more pheromone on the shortest path influencing other ants to follow this path. After some time, this results in the whole colony following the shortest path M Levine, F Ducatelle . Ant colony search algorithms have recently been introduced as powerful tools to solve a diverse set of optimization problems, such as the travelling salesman problem (TSP) Dorigo M, Gambardella LM (1997), the quadratic assignment problem Maniezzo V, Colorni (1999), and Shi L et.al (1999) proposed ACO for optimization problems in power systems, such as the generation scheduling problem, unit commitment problem (economic dispatch problem) Hou Y-H et.al(2002). Marco Dorigoa and Christian Blumb (2006)  focused a new metaheuristic for optimization initially on proof-of-concept applications. It is only after experimental work which has shown the practical interest of the method that researchers try to deepen their understanding of the method's functioning not only through more and more sophisticated experiments but also by means of an effort to build a theory. Marco Dorigo et al. (2006)  inspired swarm intelligence as a relatively new approach to problem solving from the social behaviors of insects and of other animals. ACO takes inspiration from the foraging behavior of some ant species. It is suggested that these ants deposit pheromone on the ground in order to mark some favorable decision that should be followed by other members of the colony. The particular optimization algorithm exploits a similar mechanism for solving optimization problems.
Ayla AltÄ±nten (2007)  applied Generalized Predictive Control (GPC) to a pH neutralization process. A continuous flow tubular reactor that employed a water solution of acetic acid and the titrating stream of sodium hydroxide is implemented. The pH value at a given set point value is kept when the process has to be subjected to variations in feed flow rate. Liqing Di, Zhihua Xiong and Xianhui Yang (2008)  introduced a MKPLS (Multiway Kernel Partial Least Squares) method used to model the batch processes from process operational data. To improve the optimization performance, a batch-to-batch optimization strategy has been proposed based on the idea of the similarity between the iterations during numerical optimization and successive batch runs. SQP (Sequential Quadratic Programming) coupling with MKPLS model was used to solve the optimization problem, and the plant data, instead of the MKPLS model predictions, that could be used in gradient calculation. The proposed strategy was illustrated on a simulated bulk polymerization of styrene. Later, Cus F. Balic J. and Zuperl U. (2009)  presented a new hybrid multi-objective optimization technique, based on Ant Colony Optimization algorithm, to optimize the machining parameters in turning processes. The proposed approach used in the same work was Adaptive Neuro-Fuzzy Inference System (ANFIS) to represent the manufacturer objective function and an ACO to obtain the optimal objective value. Masaya Yoshikawa (2010)  applied the route search problem to various engineering fields. A new hybrid routing algorithm which combines Tabu search with Ant Colony Optimization is proposed. This hybrid technique enabled to find the shortest route including the blind alley. Reza Akbaria, Vahid Zeighamib and Koorush Ziaratia (2011)  proved that the Resource Constrained Project Scheduling Problem (RCPSP) has an important role in the context of project scheduling. Considering a single objective RCPSP, their goal was to find a schedule that minimizes the make span. Recently, various meta-heuristics such as ACO, PSO, GA, and SA have been applied on RCPSP.
1.4.4 Fuzzy Sliding Mode Control (FSMC)
The Sliding Mode Control (SMC) is a type of non-linear control. SMC offers several advantages, namely large signal stability, robustness, good dynamic response and simple implementation. The design of an SMC does not require an accurate model of the system. The ideal nature of the controller is to operate at an infinite variable switching frequency. This nature enables the controlled variable to track a certain reference path to achieve the desired dynamic response and better steady state operation .
Habibi R. and Richards R.J. (1992)  proposed sliding mode control of an electrically powered industrial robot which has given an idea about the application of sliding mode control. Chen C. L. and Chang M. H. (1998)  and Chang W. et al. (2002)  discussed about the designing of Fuzzy Sliding-Mode as well as it's applications. Li T.H. S. and Shieh M. Y (2000)  developed a theme for a switching-type fuzzy sliding mode controller for the cart-pole system. The proposed control strategy is to make a vertical pole straight up and regulate the position of cart simultaneously. Bong Keun Kim, Wan Kyun Chung and Kohtaro Ohba (2009)  effectively determined the tuning method of controllers that could be used for the overall performance of positioning systems. In that method, high-speed and high-accuracy positioning systems were designed. Here, a sliding-mode controller that uses one of the well-known approaches