This paper analyzes the fragility issue of fractional-order proportional-integral-derivative controllers applied to integer first-order plus-dead-time processes. In particular, the effects of the variations of the controller parameters on the achieved control system robustness and performance are investigated. Results show that this kind of controllers is more fragile with respect to the standard proportional-integral-derivative controllers and therefore a significant attention should be paid by the user in their tuning.
Tuning of PID controllers for integrating systems using direct synthesis methodISA Interchange
A PID controller is designed for various forms of integrating systems with time delay using direct synthesis method. The method is based on comparing the characteristic equation of the integrating system and PID controller with a filter with the desired characteristic equation. The desired characteristic equation comprises of multiple poles which are placed at the same desired location. The tuning parameter is adjusted so as to achieve the desired robustness. Tuning rules in terms of process parameters are given for various forms of integrating systems. The tuning parameter can be selected for the desired robustness by specifying Ms value. The proposed controller design method is applied to various transfer function models and to the nonlinear model equations of jacketed CSTR to show its effectiveness and applicability.
Modeling and Control of MIMO Headbox System Using Fuzzy LogicIJERA Editor
The Headbox plays an important role in pulp supply system with sheet forming in paper making process. The air cushion headbox is a nonlinear & strong coupling system. In the air cushion headbox system there were two important parameters which include total head and the stock level for improving pulp product quality. These two parameters make this system MIMO output system so for this a decoupling controls strategy was required for interaction between these two control loops. In this paper fuzzy tuned PID control scheme is proposed for controlling the nonlinear control problem in air cushion headbox after the system being decoupled. An attempt has been made for comparison between classical (PID) and fuzzy tuned PID controller. It concludes that the fuzzy tuned PID controller is found most suitable for MIMO system in terms of obtaining steady state properties. The effects of disturbances are studied through computer simulation using Matlab/Simulink toolbox.
Adaptive Control Machining systems,Adaptive Control,Where to use adaptive control? Application:Sources of variability in machining,Types of Adaptive controls,Operation of ACC system,Relationship of AC software to APT program,Benefits of AC
Tuning of PID controllers for integrating systems using direct synthesis methodISA Interchange
A PID controller is designed for various forms of integrating systems with time delay using direct synthesis method. The method is based on comparing the characteristic equation of the integrating system and PID controller with a filter with the desired characteristic equation. The desired characteristic equation comprises of multiple poles which are placed at the same desired location. The tuning parameter is adjusted so as to achieve the desired robustness. Tuning rules in terms of process parameters are given for various forms of integrating systems. The tuning parameter can be selected for the desired robustness by specifying Ms value. The proposed controller design method is applied to various transfer function models and to the nonlinear model equations of jacketed CSTR to show its effectiveness and applicability.
Modeling and Control of MIMO Headbox System Using Fuzzy LogicIJERA Editor
The Headbox plays an important role in pulp supply system with sheet forming in paper making process. The air cushion headbox is a nonlinear & strong coupling system. In the air cushion headbox system there were two important parameters which include total head and the stock level for improving pulp product quality. These two parameters make this system MIMO output system so for this a decoupling controls strategy was required for interaction between these two control loops. In this paper fuzzy tuned PID control scheme is proposed for controlling the nonlinear control problem in air cushion headbox after the system being decoupled. An attempt has been made for comparison between classical (PID) and fuzzy tuned PID controller. It concludes that the fuzzy tuned PID controller is found most suitable for MIMO system in terms of obtaining steady state properties. The effects of disturbances are studied through computer simulation using Matlab/Simulink toolbox.
Adaptive Control Machining systems,Adaptive Control,Where to use adaptive control? Application:Sources of variability in machining,Types of Adaptive controls,Operation of ACC system,Relationship of AC software to APT program,Benefits of AC
Recovery from a process saturation condition benefits of using delta v pid_plusEmerson Exchange
The PIDPlus option of the PID function block in the DeltaV v11.3 allows improved recovery from a process saturation condition. In this workshop the technical basis for the change in the PID are presented. The impact this has on the time required to get to setpoint is examined and the improvement in response for the surge control will be demonstration using a synamic compressor simulation.
This paper presents an enhanced nonlinear PID (NPID) controller to follow a preselected speed profile of brushless DC motor drive system. This objective should be achieved regardless the parameter variations, and external disturbances. The performance of enhanced NPID controller will be investigated by comparing it with linear PID control and fractional order PID (FOPID) control. These controllers are tested for both speed regulation and speed tracking. The optimal parameters values of each control technique were obtained using Genetic Algorithm (GA) based on a certain cost function. Results shows that the proposed NPID controller has better performance among other techniques (PID and FOPID controller).
Study of PID Controllers to Load Frequency Control Systems with Various Turbi...IJERA Editor
This paper studies the load frequency control problem for various systems under various controller design
methods. Frequency should remain nearly constant for satisfactory operation of a power system because
frequency deviations can directly impact on a power system operation, system stability, reliability and
efficiency. A Load Frequency Control (LFC) scheme basically incorporates an appropriate control system for an
interconnected power system, which is having the capability to bring the frequencies of system to original set
point values or very nearer to set point values effectively after any load change. This can be achieved by the use
of conventional and modern controllers. In this proposed paper PID controller has been applied for LFC power
systems. The parameters of the PID controller are tuned by different methods names as Ziegler-Nichols (Z-N)
Method, and IMC method for better results. We use various tuning formulae in Z-N method and certain model
approximation methods and the responses of LFC with model approximation are studied. It is seen that the
results obtained are as good as the conventional controller.
Implementation of closed loop control technique for improving the performance...IJERA Editor
this review paper presents closed loop control techniques for controlling the inverter working under different load or KVA ratings. The control strategy of the inverter must guarantee its output waveforms to be sinusoidal with fundamental harmonic. For this purpose, close loop current control strategies such as H∞ repetitive controller, dual closed-loop feedback control, Adaptive Voltage Control, SRFPI controller, Optimal Neural Controller, etc. have been used to meet the power quality requirements imposed by IEEE Interconnection Standards. Based on present scenario regarding energy crises, immediate action is the use of different renewable energy sources (RESs) . Out of RESs, solar is gaining more attention. It is very important to design and developed a system which should be efficient enough to utilize the extracted energy for different types of load and feeding of energy into utility grid. Since experimentation and comparison of such inverter models on hardware being relatively expensive, the latest computing tool like MATLAB are considered to be a better alternative to simulate the outcomes of such expensive systems. The proposed closed loop control technique for the inverter working under linear and nonlinear system will be implemented in MATLAB/SIMULINK working platform and results will be analyzed to check its benefits.
Introduction, Feature of Control System, Requirement of Good Control System, Types of Control System, Open-loop control system, Closed-loop control system, Comparison of Closed-Loop and Open-Loop Control System, Signal flow graph, Conversion of Block Diagrams into Signal Flow Graphs, and Questions.
A coordinated mimo control design for a power plant using improved sliding mo...ISA Interchange
For the participation of the steam power plants in regulating the network frequency, boilers and turbines should be co-ordinately controlled in addition to the base load productions. Lack of coordinated control over boiler–turbine may lead to instability; oscillation in producing power and boiler parameters; reduction in the reliability of the unit; and inflicting thermodynamic tension on devices. This paper proposes a boiler–turbine coordinated multivariable control system based on improved sliding mode controller (ISMC). The system controls two main boiler–turbine parameters i.e., the turbine revolution and superheated steam pressure of the boiler output. For this purpose, a comprehensive model of the system including complete and exact description of the subsystems is extracted. The parameters of this model are determined according to our case study that is the 320 MW unit of Islam-Abad power plant in Isfahan/Iran. The ISMC method is simulated on the power plant and its performance is compared with the related real PI (proportional-integral) controllers which have been used in this unit. The simulation results show the capability of the proposed controller system in controlling local network frequency and superheated steam pressure in the presence of load variations and disturbances of boiler.
In this paper, the closed loop speed controller parameters are optimized for the permanent magnet synchronous motor (PMSM) drive on the basis of the indirect field-oriented control (IFOC) technique. In this derive system under study, the speed and current controllers are implemented using the fractional order proportional, integral, and derivative (FOPID) controlling technique. FOPID is considered as efficient techniques for ripple minimization. The hybrid grey wolf optimizer (HGWO) is applied to obtain the optimal controllers in case of implementing conventional PID as well as FOPID controllers in the derive system. The optimal controller parameters tend to enhance the drive response as ripple content in speed and current, either during steady state time or transient time. The drive system is modeled and tested under various operating condition of load torque and speed. Finally, the performance for PID and FOPID are evaluated and compared within MATLAB/Simulink environment. The results attain the efficacy of the operating performance with the FOPID controller. The result shows a fast response and reduction of ripples in the torque and the current.
FRACTIONAL ORDER PID CONTROLLER TUNING BASED ON IMC IJITCA Journal
In this work, a class of fractional order controller (FOPID) is tuned based on internal model control
(IMC). This tuning rule has been obtained without any approximation of time delay. Moreover to show
usefulness of fractional order controller in comparison with classical integer order controllers, an
industrial PID controller tuned in a similar way, is compared with FOPID and then robust stability of both
controllers is investigated. Robust stability analysis has been done to find maximum delayed time
uncertainty interval which results in a stable closed loop control system. For a typical system, robust
stability has been done to find maximum time constant uncertainty interval of system. Two clarify the
proposed control system design procedure, three examples have been given.
Simple robust autotuning rules for 2-DoF PI controllersISA Interchange
This paper addresses the problem of providing simple tuning rules for a Two-Degree-of-Freedom (2-DoF) PI controller (PI2) with robustness considerations. The introduction of robustness as a matter of primary concern is by now well established among the control community. Among the different ways of introducing a robustness constraint into the design stage, the purpose of this paper is to use the maximum sensitivity value as the design parameter. In order to deal with the well known performance/robustness tradeoff, an analysis is conducted first that allows the determination of the lowest closed-loop time constant that guarantees a desired robustness. From that point, an analytical design is conducted for the assignment of the load-disturbance dynamics followed by the tuning of the set-point weight factor in order to match, as much as possible, the set-point-to-output dynamics according to a first-order-plus-dead-time dynamics. Simple tuning rules are generated by considering specific values for the maximum sensitivity value. These tuning rules, provide all the controller parameters parameterized in terms of the open-loop normalized dead-time allowing the user to select a high/medium/low robust closed-loop control system. The proposed autotuning expressions are therefore compared with other well known tuning rules also conceived by using the same robustness measure, showing that the proposed approach is able to guarantee the same robustness level and improve the system time performance.
DESIGN OF PID CONTROLLERS INTEGRATOR SYSTEM WITH TIME DELAY AND DOUBLE INTEGR...ijics
In this paper first we investigate optimal PID control of a double integrating plus delay process and compare with the SIMC rules. What makes the double integrating process special is that derivative action is actually necessary for stabilization. In control, there is generally a trade-off between performance and
robustness, so there does not exist a single optimal controller. However, for a given robustness level (here defined in terms of the Ms-value) we can find the optimal controller which minimizes the performance J (here defined as the integrated absolute error (IAE)-value for disturbances). Interestingly, the SIMC PID controller is almost identical to the optimal pid controller. This can be seen by comparing the paretooptimal
curve for J as a function of Ms, with the curve found by varying the SIMC tuning parameter Tc.
Second, design of Proportional Integral and Derivative (PID) controllers based on internal model control (IMC) principles, direct synthesis method (DS), stability analysis (SA) method for pure integrating process with time delay is proposed. The performances of the proposed controllers are compared with the
controllers designed by recently reported methods. The robustness of the proposed controllers for the uncertainty in model parameters is evaluated considering one parameter at a time using Kharitonov’s theorem. The proposed controllers are applied to various transfer function models and to non linear model of isothermal continuous copolymerization of styrene-acrylonitrile in CSTR. An experimental set up of tank
with the outlet connected to a pump is considered for implementation of the PID controllers designed by
the three proposed methods to show the effectiveness of the methods.
DESIGN OF PID CONTROLLERS INTEGRATOR SYSTEM WITH TIME DELAY AND DOUBLE INTEGR...ijcisjournal
In this paper first we investigate optimal PID control of a double integrating plus delay process and compare with the SIMC rules. What makes the double integrating process special is that derivative action is actually necessary for stabilization. In control, there is generally a trade-off between performance and robustness, so there does not exist a single optimal controller. However, for a given robustness level (here defined in terms of the Ms-value) we can find the optimal controller which minimizes the performance J (here defined as the integrated absolute error (IAE)-value for disturbances). Interestingly, the SIMC PID controller is almost identical to the optimal pid controller. This can be seen by comparing the paretooptimal curve for J as a function of Ms, with the curve found by varying the SIMC tuning parameter Tc. Second, design of Proportional Integral and Derivative (PID) controllers based on internal model control (IMC) principles, direct synthesis method (DS), stability analysis (SA) method for pure integrating process with time delay is proposed. The performances of the proposed controllers are compared with the
controllers designed by recently reported methods. The robustness of the proposed controllers for the uncertainty in model parameters is evaluated considering one parameter at a time using Kharitonov’s theorem. The proposed controllers are applied to various transfer function models and to non linear model of isothermal continuous copolymerization of styrene-acrylonitrile in CSTR. An experimental set up of tank with the outlet connected to a pump is considered for implementation of the PID controllers designed by the three proposed methods to show the effectiveness of the methods.
Closed-loop step response for tuning PID fractional-order filter controllersISA Interchange
Analytical methods are usually applied for tuning fractional controllers. The present paper proposes an empirical method for tuning a new type of fractional controller known as PID-Fractional-Order-Filter (FOF-PID). Indeed, the setpoint overshoot method, initially introduced by Shamsuzzoha and Skogestad, has been adapted for tuning FOF-PID controller. Based on simulations for a range of first order with time delay processes, correlations have been derived to obtain PID-FOF controller parameters similar to those obtained by the Internal Model Control (IMC) tuning rule. The setpoint overshoot method requires only one closed-loop step response experiment using a proportional controller (P-controller). To highlight the potential of this method, simulation results have been compared with those obtained with the IMC method as well as other pertinent techniques. Various case studies have also been considered. The comparison has revealed that the proposed tuning method performs as good as the IMC. Moreover, it might offer a number of advantages over the IMC tuning rule. For instance, the parameters of the frac- tional controller are directly obtained from the setpoint closed-loop response data without the need of any model of the plant to be controlled.
Recovery from a process saturation condition benefits of using delta v pid_plusEmerson Exchange
The PIDPlus option of the PID function block in the DeltaV v11.3 allows improved recovery from a process saturation condition. In this workshop the technical basis for the change in the PID are presented. The impact this has on the time required to get to setpoint is examined and the improvement in response for the surge control will be demonstration using a synamic compressor simulation.
This paper presents an enhanced nonlinear PID (NPID) controller to follow a preselected speed profile of brushless DC motor drive system. This objective should be achieved regardless the parameter variations, and external disturbances. The performance of enhanced NPID controller will be investigated by comparing it with linear PID control and fractional order PID (FOPID) control. These controllers are tested for both speed regulation and speed tracking. The optimal parameters values of each control technique were obtained using Genetic Algorithm (GA) based on a certain cost function. Results shows that the proposed NPID controller has better performance among other techniques (PID and FOPID controller).
Study of PID Controllers to Load Frequency Control Systems with Various Turbi...IJERA Editor
This paper studies the load frequency control problem for various systems under various controller design
methods. Frequency should remain nearly constant for satisfactory operation of a power system because
frequency deviations can directly impact on a power system operation, system stability, reliability and
efficiency. A Load Frequency Control (LFC) scheme basically incorporates an appropriate control system for an
interconnected power system, which is having the capability to bring the frequencies of system to original set
point values or very nearer to set point values effectively after any load change. This can be achieved by the use
of conventional and modern controllers. In this proposed paper PID controller has been applied for LFC power
systems. The parameters of the PID controller are tuned by different methods names as Ziegler-Nichols (Z-N)
Method, and IMC method for better results. We use various tuning formulae in Z-N method and certain model
approximation methods and the responses of LFC with model approximation are studied. It is seen that the
results obtained are as good as the conventional controller.
Implementation of closed loop control technique for improving the performance...IJERA Editor
this review paper presents closed loop control techniques for controlling the inverter working under different load or KVA ratings. The control strategy of the inverter must guarantee its output waveforms to be sinusoidal with fundamental harmonic. For this purpose, close loop current control strategies such as H∞ repetitive controller, dual closed-loop feedback control, Adaptive Voltage Control, SRFPI controller, Optimal Neural Controller, etc. have been used to meet the power quality requirements imposed by IEEE Interconnection Standards. Based on present scenario regarding energy crises, immediate action is the use of different renewable energy sources (RESs) . Out of RESs, solar is gaining more attention. It is very important to design and developed a system which should be efficient enough to utilize the extracted energy for different types of load and feeding of energy into utility grid. Since experimentation and comparison of such inverter models on hardware being relatively expensive, the latest computing tool like MATLAB are considered to be a better alternative to simulate the outcomes of such expensive systems. The proposed closed loop control technique for the inverter working under linear and nonlinear system will be implemented in MATLAB/SIMULINK working platform and results will be analyzed to check its benefits.
Introduction, Feature of Control System, Requirement of Good Control System, Types of Control System, Open-loop control system, Closed-loop control system, Comparison of Closed-Loop and Open-Loop Control System, Signal flow graph, Conversion of Block Diagrams into Signal Flow Graphs, and Questions.
A coordinated mimo control design for a power plant using improved sliding mo...ISA Interchange
For the participation of the steam power plants in regulating the network frequency, boilers and turbines should be co-ordinately controlled in addition to the base load productions. Lack of coordinated control over boiler–turbine may lead to instability; oscillation in producing power and boiler parameters; reduction in the reliability of the unit; and inflicting thermodynamic tension on devices. This paper proposes a boiler–turbine coordinated multivariable control system based on improved sliding mode controller (ISMC). The system controls two main boiler–turbine parameters i.e., the turbine revolution and superheated steam pressure of the boiler output. For this purpose, a comprehensive model of the system including complete and exact description of the subsystems is extracted. The parameters of this model are determined according to our case study that is the 320 MW unit of Islam-Abad power plant in Isfahan/Iran. The ISMC method is simulated on the power plant and its performance is compared with the related real PI (proportional-integral) controllers which have been used in this unit. The simulation results show the capability of the proposed controller system in controlling local network frequency and superheated steam pressure in the presence of load variations and disturbances of boiler.
In this paper, the closed loop speed controller parameters are optimized for the permanent magnet synchronous motor (PMSM) drive on the basis of the indirect field-oriented control (IFOC) technique. In this derive system under study, the speed and current controllers are implemented using the fractional order proportional, integral, and derivative (FOPID) controlling technique. FOPID is considered as efficient techniques for ripple minimization. The hybrid grey wolf optimizer (HGWO) is applied to obtain the optimal controllers in case of implementing conventional PID as well as FOPID controllers in the derive system. The optimal controller parameters tend to enhance the drive response as ripple content in speed and current, either during steady state time or transient time. The drive system is modeled and tested under various operating condition of load torque and speed. Finally, the performance for PID and FOPID are evaluated and compared within MATLAB/Simulink environment. The results attain the efficacy of the operating performance with the FOPID controller. The result shows a fast response and reduction of ripples in the torque and the current.
FRACTIONAL ORDER PID CONTROLLER TUNING BASED ON IMC IJITCA Journal
In this work, a class of fractional order controller (FOPID) is tuned based on internal model control
(IMC). This tuning rule has been obtained without any approximation of time delay. Moreover to show
usefulness of fractional order controller in comparison with classical integer order controllers, an
industrial PID controller tuned in a similar way, is compared with FOPID and then robust stability of both
controllers is investigated. Robust stability analysis has been done to find maximum delayed time
uncertainty interval which results in a stable closed loop control system. For a typical system, robust
stability has been done to find maximum time constant uncertainty interval of system. Two clarify the
proposed control system design procedure, three examples have been given.
Simple robust autotuning rules for 2-DoF PI controllersISA Interchange
This paper addresses the problem of providing simple tuning rules for a Two-Degree-of-Freedom (2-DoF) PI controller (PI2) with robustness considerations. The introduction of robustness as a matter of primary concern is by now well established among the control community. Among the different ways of introducing a robustness constraint into the design stage, the purpose of this paper is to use the maximum sensitivity value as the design parameter. In order to deal with the well known performance/robustness tradeoff, an analysis is conducted first that allows the determination of the lowest closed-loop time constant that guarantees a desired robustness. From that point, an analytical design is conducted for the assignment of the load-disturbance dynamics followed by the tuning of the set-point weight factor in order to match, as much as possible, the set-point-to-output dynamics according to a first-order-plus-dead-time dynamics. Simple tuning rules are generated by considering specific values for the maximum sensitivity value. These tuning rules, provide all the controller parameters parameterized in terms of the open-loop normalized dead-time allowing the user to select a high/medium/low robust closed-loop control system. The proposed autotuning expressions are therefore compared with other well known tuning rules also conceived by using the same robustness measure, showing that the proposed approach is able to guarantee the same robustness level and improve the system time performance.
DESIGN OF PID CONTROLLERS INTEGRATOR SYSTEM WITH TIME DELAY AND DOUBLE INTEGR...ijics
In this paper first we investigate optimal PID control of a double integrating plus delay process and compare with the SIMC rules. What makes the double integrating process special is that derivative action is actually necessary for stabilization. In control, there is generally a trade-off between performance and
robustness, so there does not exist a single optimal controller. However, for a given robustness level (here defined in terms of the Ms-value) we can find the optimal controller which minimizes the performance J (here defined as the integrated absolute error (IAE)-value for disturbances). Interestingly, the SIMC PID controller is almost identical to the optimal pid controller. This can be seen by comparing the paretooptimal
curve for J as a function of Ms, with the curve found by varying the SIMC tuning parameter Tc.
Second, design of Proportional Integral and Derivative (PID) controllers based on internal model control (IMC) principles, direct synthesis method (DS), stability analysis (SA) method for pure integrating process with time delay is proposed. The performances of the proposed controllers are compared with the
controllers designed by recently reported methods. The robustness of the proposed controllers for the uncertainty in model parameters is evaluated considering one parameter at a time using Kharitonov’s theorem. The proposed controllers are applied to various transfer function models and to non linear model of isothermal continuous copolymerization of styrene-acrylonitrile in CSTR. An experimental set up of tank
with the outlet connected to a pump is considered for implementation of the PID controllers designed by
the three proposed methods to show the effectiveness of the methods.
DESIGN OF PID CONTROLLERS INTEGRATOR SYSTEM WITH TIME DELAY AND DOUBLE INTEGR...ijcisjournal
In this paper first we investigate optimal PID control of a double integrating plus delay process and compare with the SIMC rules. What makes the double integrating process special is that derivative action is actually necessary for stabilization. In control, there is generally a trade-off between performance and robustness, so there does not exist a single optimal controller. However, for a given robustness level (here defined in terms of the Ms-value) we can find the optimal controller which minimizes the performance J (here defined as the integrated absolute error (IAE)-value for disturbances). Interestingly, the SIMC PID controller is almost identical to the optimal pid controller. This can be seen by comparing the paretooptimal curve for J as a function of Ms, with the curve found by varying the SIMC tuning parameter Tc. Second, design of Proportional Integral and Derivative (PID) controllers based on internal model control (IMC) principles, direct synthesis method (DS), stability analysis (SA) method for pure integrating process with time delay is proposed. The performances of the proposed controllers are compared with the
controllers designed by recently reported methods. The robustness of the proposed controllers for the uncertainty in model parameters is evaluated considering one parameter at a time using Kharitonov’s theorem. The proposed controllers are applied to various transfer function models and to non linear model of isothermal continuous copolymerization of styrene-acrylonitrile in CSTR. An experimental set up of tank with the outlet connected to a pump is considered for implementation of the PID controllers designed by the three proposed methods to show the effectiveness of the methods.
Closed-loop step response for tuning PID fractional-order filter controllersISA Interchange
Analytical methods are usually applied for tuning fractional controllers. The present paper proposes an empirical method for tuning a new type of fractional controller known as PID-Fractional-Order-Filter (FOF-PID). Indeed, the setpoint overshoot method, initially introduced by Shamsuzzoha and Skogestad, has been adapted for tuning FOF-PID controller. Based on simulations for a range of first order with time delay processes, correlations have been derived to obtain PID-FOF controller parameters similar to those obtained by the Internal Model Control (IMC) tuning rule. The setpoint overshoot method requires only one closed-loop step response experiment using a proportional controller (P-controller). To highlight the potential of this method, simulation results have been compared with those obtained with the IMC method as well as other pertinent techniques. Various case studies have also been considered. The comparison has revealed that the proposed tuning method performs as good as the IMC. Moreover, it might offer a number of advantages over the IMC tuning rule. For instance, the parameters of the frac- tional controller are directly obtained from the setpoint closed-loop response data without the need of any model of the plant to be controlled.
Controller Tuning for Integrator Plus Delay Processes.theijes
A design method for PID controllers based on internal model control (IMC) principles, direct synthesis method (DS), stability analysis (SA) method for pure integrating process with time delay is proposed. Analytical expressions for PID controllers are derived for several common types of process models, including first order and second-order plus time delay models and an integrator plus time delay model. Here in this paper, a simple controller design rule and tuning procedure for unstable processes with delay time is discussed. Simulation examples are included to show the effectiveness of the proposed method
Tuning of IMC-based PID controllers for integrating systems with time delayISA Interchange
Design of Proportional Integral and Derivative (PID) controllers based on IMC principles for various types of integrating systems with time delay is proposed. PID parameters are given in terms of process model parameters and a tuning parameter. The tuning parameter is IMC filter time constant. In the present work, the IMC filter (Q) is chosen in such a manner that the order of the denominator of IMC controller is one less than the order of the numerator. The IMC filter time constant (λ) is tuned in such a way that a good compromise is made between performance and robustness for both servo and regulatory problems. To improve servo response of the controller a set point filter is designed such that the closed loop response is similar to that of first order plus time delay system. The proposed controller design method is applied to various transfer function models and to the non-linear model equations of jacketed CSTR to demonstrate its applicability and effectiveness. The performance of the proposed controller is compared with the recently reported methods in terms of IAE and ITAE. The smooth functioning of the controller is determined in terms of total variation and compared with recently reported methods. Simulation studies are carried out on various integrating systems with time delay to show the effectiveness and superiority of the proposed controllers.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Comparison of Tuning Methods of PID Controllers for Non-Linear Systempaperpublications3
Abstract: Modern days have seen vast developments in the field of controller’s .There are various controllers developed these days with various different specifications. But the only drawback is that, there is no fixed method for the tuning of these controllers, which is necessary for controlling of the system based on the variation of the input or for the changes in the system. In order to overcome this drawback, in this paper we have compared various tuning methods of PID controller for non-linear system. As a non-linear system we have taken the dc motor as a system. For the particular DC motor controller transfer function has been determined and control parameters such as Proportional Gain, Integral Time and Derivative time are identified. They are numerous methods of developing a Proportional Integral and Derivative (PID) Controller, amongst them some methods are adopted in this paper and Comparisons of Time Domain specifications of those controllers has been carried out.
Design of a model reference adaptive PID control algorithm for a tank system IJECEIAES
This paper describes the design of an adaptive controller based on model reference adaptive PID control (MRAPIDC) to stabilize a two-tank process when large variations of parameters and external disturbances affect the closed-loop system. To achieve that, an innovative structure of the adaptive PID controller is defined, an additional PI is designed to make sure that the reference model produces stable output signals and three adaptive gains are included to guarantee stability and robustness of the closed-loop system. Then, the performance of the model reference adaptive PID controller on the behaviour of the closed-loop system is compared to a PI controller designed on MATLAB when both closed-loop systems are under various conditions. The results demonstrate that the MRAPIDC performs significantly better than the conventional PI controller.
An optimal general type-2 fuzzy controller for Urban Traffic NetworkISA Interchange
Urban traffic network model is illustrated by state-charts and object-diagram. However, they have limitations to show the behavioral perspective of the traffic information flow. Consequently, a state space model is used to calculate the half-value waiting time of vehicles. In this study, a combination of the general type-2 fuzzy logic sets and the modified backtracking search algorithm (MBSA) techniques are used in order to control the traffic signal scheduling and phase succession so as to guarantee a smooth flow of traffic with the least wait times and average queue length. The parameters of input and output membership functions are optimized simultaneously by the novel heuristic algorithm MBSA. A comparison is made between the achieved results with those of optimal and conventional type-1 fuzzy logic controllers.
Embedded intelligent adaptive PI controller for an electromechanical systemISA Interchange
In this study, an intelligent adaptive controller approach using the interval type-2 fuzzy neural network (IT2FNN) is presented. The proposed controller consists of a lower level proportional - integral (PI) controller, which is the main controller and an upper level IT2FNN which tuning on-line the parameters of a PI controller. The proposed adaptive PI controller based on IT2FNN (API-IT2FNN) is implemented practically using the Arduino DUE kit for controlling the speed of a nonlinear DC motor-generator system. The parameters of the IT2FNN are tuned on-line using back-propagation algorithm. The Lyapunov theorem is used to derive the stability and convergence of the IT2FNN. The obtained experimental results, which are compared with other controllers, demonstrate that the proposed API-IT2FNN is able to improve the system response over a wide range of system uncertainties.
State of charge estimation of lithium-ion batteries using fractional order sl...ISA Interchange
This paper presents a state of charge (SOC) estimation method based on fractional order sliding mode observer (SMO) for lithium-ion batteries. A fractional order RC equivalent circuit model (FORCECM) is firstly constructed to describe the charging and discharging dynamic characteristics of the battery. Then, based on the differential equations of the FORCECM, fractional order SMOs for SOC, polarization voltage and terminal voltage estimation are designed. After that, convergence of the proposed observers is analyzed by Lyapunov’s stability theory method. The framework of the designed observer system is simple and easy to implement. The SMOs can overcome the uncertainties of parameters, modeling and measurement errors, and present good robustness. Simulation results show that the presented estima- tion method is effective, and the designed observers have good performance.
Fractional order PID for tracking control of a parallel robotic manipulator t...ISA Interchange
This paper presents the tracking control for a robotic manipulator type delta employing fractional order PID controllers with computed torque control strategy. It is contrasted with an integer order PID controller with computed torque control strategy. The mechanical structure, kinematics and dynamic models of the delta robot are descripted. A SOLIDWORKS/MSC-ADAMS/MATLAB co-simulation model of the delta robot is built and employed for the stages of identification, design, and validation of control strategies. Identification of the dynamic model of the robot is performed using the least squares algorithm. A linearized model of the robotic system is obtained employing the computed torque control strategy resulting in a decoupled double integrating system. From the linearized model of the delta robot, fractional order PID and integer order PID controllers are designed, analyzing the dynamical behavior for many evaluation trajectories. Controllers robustness is evaluated against external disturbances employing performance indexes for the joint and spatial error, applied torque in the joints and trajectory tracking. Results show that fractional order PID with the computed torque control strategy has a robust performance and active disturbance rejection when it is applied to parallel robotic manipulators on tracking tasks.
Fuzzy logic for plant-wide control of biological wastewater treatment process...ISA Interchange
The application of control strategies is increasingly used in wastewater treatment plants with the aim of improving effluent quality and reducing operating costs. Due to concerns about the progressive growth of greenhouse gas emissions (GHG), these are also currently being evaluated in wastewater treatment plants. The present article proposes a fuzzy controller for plant-wide control of the biological wastewater treatment process. Its design is based on 14 inputs and 6 outputs in order to reduce GHG emissions, nutrient concentration in the effluent and operational costs. The article explains and shows the effect of each one of the inputs and outputs of the fuzzy controller, as well as the relationship between them. Benchmark Simulation Model no 2 Gas is used for testing the proposed control strategy. The results of simulation results show that the fuzzy controller is able to reduce GHG emissions while improving, at the same time, the common criteria of effluent quality and operational costs.
Design and implementation of a control structure for quality products in a cr...ISA Interchange
In recent years, interest for petrochemical processes has been increasing, especially in refinement area. However, the high variability in the dynamic characteristics present in the atmospheric distillation column poses a challenge to obtain quality products. To improve distillates quality in spite of the changes in the input crude oil composition, this paper details a new design of a control strategy in a conventional crude oil distillation plant defined using formal interaction analysis tools. The process dynamic and its control are simulated on Aspen HYSYS dynamic environment under real operating conditions. The simulation results are compared against a typical control strategy commonly used in crude oil atmospheric distillation columns.
Model based PI power system stabilizer design for damping low frequency oscil...ISA Interchange
This paper explores a two-level control strategy by blending a local controller with a centralized controller for the low frequency oscillations in a power system. The proposed control scheme provides stabilization of local modes using a local controller and minimizes the effect of inter-connection of sub-systems performance through a centralized control. For designing the local controllers in the form of proportional-integral power system stabilizer (PI-PSS), a simple and straight forward frequency domain direct synthesis method is considered that works on use of a suitable reference model which is based on the desired requirements. Several examples both on one machine infinite bus and multi-machine systems taken from the literature are illustrated to show the efficacy of the proposed PI-PSS. The effective damping of the systems is found to be increased remarkably which is reflected in the time-responses; even unstable operation has been stabilized with improved damping after applying the proposed controller. The proposed controllers give remarkable improvement in damping the oscillations in all the illustrations considered here and as for example, the value of damping factor has been increased from 0.0217 to 0.666 in Example 1. The simulation results obtained by the proposed control strategy are favorably compared with some controllers prevalent in the literature.
A comparison of a novel robust decentralized control strategy and MPC for ind...ISA Interchange
Abstract: In this work we have developed a novel, robust practical control structure to regulate an industrial methanol distillation column. This proposed control scheme is based on a override control framework and can manage a non-key trace ethanol product impurity specification while maintaining high product recovery. For comparison purposes, an MPC with a discrete process model (based on step tests) was also developed and tested. The results from process disturbance testing shows that, both the MPC and the proposed controller were capable of maintaining both the trace level ethanol specification in the distillate (XD) and high product recovery (β). Closer analysis revealed that the MPC controller has a tighter XD control, while the proposed controller was tighter in β control. The tight XD control allowed the MPC to operate at a higher XD set point (closer to the 10 ppm AA grade methanol standard), allowing for savings in energy usage. Despite the energy savings of the MPC, the proposed control scheme has lower installation and running costs. An economic analysis revealed a multitude of other external economic and plant design factors, that should be considered when making a decision between the two controllers. In general, we found relatively high energy costs favor MPC.
Fault detection of feed water treatment process using PCA-WD with parameter o...ISA Interchange
Feed water treatment process (FWTP) is an essential part of utility boilers; and fault detection is expected for its reliability improvement. Classical principal component analysis (PCA) has been applied to FWTPs in our previous work; however, the noises of T2 and SPE statistics result in false detections and missed detections. In this paper, Wavelet denoise (WD) is combined with PCA to form a new algorithm, (PCA- WD), where WD is intentionally employed to deal with the noises. The parameter selection of PCA-WD is further formulated as an optimization problem; and PSO is employed for optimization solution. A FWTP, sustaining two 1000 MW generation units in a coal-fired power plant, is taken as a study case. Its operation data is collected for following verification study. The results show that the optimized WD is effective to restrain the noises of T2 and SPE statistics, so as to improve the performance of PCA-WD algorithm. And, the parameter optimization enables PCA-WD to get its optimal parameters in an auto- matic way rather than on individual experience. The optimized PCA-WD is further compared with classical PCA and sliding window PCA (SWPCA), in terms of four cases as bias fault, drift fault, broken line fault and normal condition, respectively. The advantages of the optimized PCA-WD, against classical PCA and SWPCA, is finally convinced with the results.
Model-based adaptive sliding mode control of the subcritical boiler-turbine s...ISA Interchange
As higher requirements are proposed for the load regulation and efficiency enhancement, the control performance of boiler-turbine systems has become much more important. In this paper, a novel robust control approach is proposed to improve the coordinated control performance for subcritical boiler-turbine units. To capture the key features of the boiler-turbine system, a nonlinear control-oriented model is established and validated with the history operation data of a 300 MW unit. To achieve system linearization and decoupling, an adaptive feedback linearization strategy is proposed, which could asymptotically eliminate the linearization error caused by the model uncertainties. Based on the linearized boiler-turbine system, a second-order sliding mode controller is designed with the super-twisting algorithm. Moreover, the closed-loop system is proved robustly stable with respect to uncertainties and disturbances. Simulation results are presented to illustrate the effectiveness of the proposed control scheme, which achieves excellent tracking performance, strong robustness and chattering reduction.
A Proportional Integral Estimator-Based Clock Synchronization Protocol for Wi...ISA Interchange
Clock synchronization is an issue of vital importance in applications of wireless sensor networks (WSNs). This paper proposes a proportional integral estimator-based protocol (EBP) to achieve clock synchronization for wireless sensor networks. As each local clock skew gradually drifts, synchronization accuracy will decline over time. Compared with existing consensus-based approaches, the proposed synchronization protocol improves synchronization accuracy under time-varying clock skews. Moreover, by restricting synchronization error of clock skew into a relative small quantity, it could reduce periodic re-synchronization frequencies. At last, a pseudo-synchronous implementation for skew compensation is introduced as synchronous protocol is unrealistic in practice. Numerical simulations are shown to illustrate the performance of the proposed protocol.
An artificial intelligence based improved classification of two-phase flow patte...ISA Interchange
Flow pattern recognition is necessary to select design equations for finding operating details of the process and to perform computational simulations. Visual image processing can be used to automate the interpretation of patterns in two-phase flow. In this paper, an attempt has been made to improve the classification accuracy of the flow pattern of gas/ liquid two- phase flow using fuzzy logic and Support Vector Machine (SVM) with Principal Component Analysis (PCA). The videos of six different types of flow patterns namely, annular flow, bubble flow, churn flow, plug flow, slug flow and stratified flow are re- corded for a period and converted to 2D images for processing. The textural and shape features extracted using image processing are applied as inputs to various classification schemes namely fuzzy logic, SVM and SVM with PCA in order to identify the type of flow pattern. The results obtained are compared and it is observed that SVM with features reduced using PCA gives the better classification accuracy and computationally less intensive than other two existing schemes. This study results cover industrial application needs including oil and gas and any other gas-liquid two-phase flows.
New Method for Tuning PID Controllers Using a Symmetric Send-On-Delta Samplin...ISA Interchange
In this paper we present a new method for tuning PI controllers with symmetric send-on-delta (SSOD) sampling strategy. First we analyze the conditions that produce oscillations in event based systems considering SSOD sampling strategy. The Describing Function is the tool used to address the problem. Once the conditions for oscillations are established, a new robustness to oscillation performance measure is introduced which entails with the concept of phase margin, one of the most traditional measures of relative stability in closed-loop control systems. Therefore, the application of the proposed robustness measure is easy and intuitive. The method is tested by both simulations and experiments. Additionally, a Java application has been developed to aid in the design according to the results presented in the paper.
Load estimator-based hybrid controller design for two-interleaved boost conve...ISA Interchange
This paper is devoted to the development of a hybrid controller for a two-interleaved boost converter dedicated to renewable energy and automotive applications. The control requirements, resumed in fast transient and low input current ripple, are formulated as a problem of fast stabilization of a predefined optimal limit cycle, and solved using hybrid automaton formalism. In addition, a real time estimation of the load is developed using an algebraic approach for online adjustment of the hybrid controller. Mathematical proofs are provided with simulations to illustrate the effectiveness and the robustness of the proposed controller despite different disturbances. Furthermore, a fuel cell system supplying a resistive load through a two-interleaved boost converter is also highlighted.
Effects of Wireless Packet Loss in Industrial Process Control SystemsISA Interchange
Timely and reliable sensing and actuation control are essential in networked control. This depends on not only the precision/quality of the sensors and actuators used but also on how well the communications links between the field instruments and the controller have been designed. Wireless networking offers simple deployment, reconfigurability, scalability, and reduced operational expenditure, and is easier to upgrade than wired solutions. However, the adoption of wireless networking has been slow in industrial process control due to the stochastic and less than 100% reliable nature of wireless communications and lack of a model to evaluate the effects of such communications imperfections on the overall control performance. In this paper, we study how control performance is affected by wireless link quality, which in turn is adversely affected by severe propagation loss in harsh industrial environments, co-channel interference, and unintended interference from other devices. We select the Tennessee Eastman Challenge Model (TE) for our study. A decentralized process control system, first proposed by N. Ricker, is adopted that employs 41 sensors and 12 actuators to manage the production process in the TE plant. We consider the scenario where wireless links are used to periodically transmit essential sensor measurement data, such as pressure, temperature and chemical composition to the controller as well as control commands to manipulate the actuators according to predetermined setpoints. We consider two models for packet loss in the wireless links, namely, an independent and identically distributed (IID) packet loss model and the two-state Gilbert-Elliot (GE) channel model. While the former is a random loss model, the latter can model bursty losses. With each channel model, the performance of the simulated decentralized controller using wireless links is compared with the one using wired links providing instant and 100% reliable communications. The sensitivity of the controller to the burstiness of packet loss is also characterized in different process stages. The performance results indicate that wireless links with redundant bandwidth reservation can meet the requirements of the TE process model under normal operational conditions. When disturbances are introduced in the TE plant model, wireless packet loss during transitions between process stages need further protection in severely impaired links. Techniques such as re-transmission scheduling, multi-path routing and enhanced physical layer design are discussed and the latest industrial wireless protocols are compared.
Fault Detection in the Distillation Column ProcessISA Interchange
Chemical plants are complex large-scale systems which need designing robust fault detection schemes to ensure high product quality, reliability and safety under different operating conditions. The present paper is concerned with a feasibility study of the application of the black-box modeling method and Kullback Leibler divergence (KLD) to the fault detection in a distillation column process. A Nonlinear Auto-Regressive Moving Average with eXogenous input (NARMAX) polynomial model is firstly developed to estimate the nonlinear behavior of the plant. Furthermore, the KLD is applied to detect abnormal modes. The proposed FD method is implemented and validated experimentally using realistic faults of a distillation plant of laboratory scale. The experimental results clearly demonstrate the fact that proposed method is effective and gives early alarm to operators.
Neural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank SystemISA Interchange
The paper is devoted to the problem of the robust actuator fault diagnosis of the dynamic non-linear systems. In the proposed method, it is assumed that the diagnosed system can be modelled by the recurrent neural network, which can be transformed into the linear parameter varying form. Such a system description allows developing the designing scheme of the robust unknown input observer within H1 framework for a class of non-linear systems. The proposed approach is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error, while guaranteeing the convergence of the observer. The application of the robust unknown input observer enables actuator fault estimation, which allows applying the developed approach to the fault tolerant control tasks.
A KPI-based process monitoring and fault detection framework for large-scale ...ISA Interchange
Large-scale processes, consisting of multiple interconnected sub-processes, are commonly encountered in industrial systems, whose performance needs to be determined. A common approach to this problem is to use a key performance indicator (KPI)-based approach. However, the different KPI-based approaches are not developed with a coherent and consistent framework. Thus, this paper proposes a framework for KPI-based process monitoring and fault detection (PM-FD) for large-scale industrial processes, which considers the static and dynamic relationships between process and KPI variables. For the static case, a least squares-based approach is developed that provides an explicit link with least-squares regression, which gives better performance than partial least squares. For the dynamic case, using the kernel re- presentation of each sub-process, an instrument variable is used to reduce the dynamic case to the static case. This framework is applied to the TE benchmark process and the hot strip mill rolling process. The results show that the proposed method can detect faults better than previous methods.
An adaptive PID like controller using mix locally recurrent neural network fo...ISA Interchange
Being complex, non-linear and coupled system, the robotic manipulator cannot be effectively controlled using classical proportional integral derivative (PID) controller. To enhance the effectiveness of the conventional PID controller for the nonlinear and uncertain systems, gains of the PID controller should be conservatively tuned and should adapt to the process parameter variations. In this work, a mix locally recurrent neural network (MLRNN) architecture is investigated to mimic a conventional PID controller which consists of at most three hidden nodes which act as proportional, integral and derivative node. The gains of the mix locally recurrent neural network based PID (MLRNNPID) controller scheme are initi- alized with a newly developed cuckoo search algorithm (CSA) based optimization method rather than assuming randomly. A sequential learning based least square algorithm is then investigated for the on- line adaptation of the gains of MLRNNPID controller. The performance of the proposed controller scheme is tested against the plant parameters uncertainties and external disturbances for both links of the two link robotic manipulator with variable payload (TL-RMWVP). The stability of the proposed controller is analyzed using Lyapunov stability criteria. A performance comparison is carried out among MLRNNPID controller, CSA optimized NNPID (OPTNNPID) controller and CSA optimized conventional PID (OPTPID) controller in order to establish the effectiveness of the MLRNNPID controller.
A method to remove chattering alarms using median filtersISA Interchange
Chattering alarms are the most found nuisance alarms that will probably reduce the usability and result in a confidence crisis of alarm systems for industrial plants. This paper addresses the chattering alarm reduction using median filters. Two rules are formulated to design the window size of median filters. If the alarm probability is estimated using process data, one rule is based on the probability of alarms to satisfy some requirements on the false alarm rate, or missed alarm rate. If there are only historical alarm data available, the other rule is based on percentage reduction of chattering alarms using alarm duration distribution. Experimental results for industrial cases testify that the proposed method is effective.
Buy Verified PayPal Account | Buy Google 5 Star Reviewsusawebmarket
Buy Verified PayPal Account
Looking to buy verified PayPal accounts? Discover 7 expert tips for safely purchasing a verified PayPal account in 2024. Ensure security and reliability for your transactions.
PayPal Services Features-
🟢 Email Access
🟢 Bank Added
🟢 Card Verified
🟢 Full SSN Provided
🟢 Phone Number Access
🟢 Driving License Copy
🟢 Fasted Delivery
Client Satisfaction is Our First priority. Our services is very appropriate to buy. We assume that the first-rate way to purchase our offerings is to order on the website. If you have any worry in our cooperation usually You can order us on Skype or Telegram.
24/7 Hours Reply/Please Contact
usawebmarketEmail: support@usawebmarket.com
Skype: usawebmarket
Telegram: @usawebmarket
WhatsApp: +1(218) 203-5951
USA WEB MARKET is the Best Verified PayPal, Payoneer, Cash App, Skrill, Neteller, Stripe Account and SEO, SMM Service provider.100%Satisfection granted.100% replacement Granted.
Business Valuation Principles for EntrepreneursBen Wann
This insightful presentation is designed to equip entrepreneurs with the essential knowledge and tools needed to accurately value their businesses. Understanding business valuation is crucial for making informed decisions, whether you're seeking investment, planning to sell, or simply want to gauge your company's worth.
Skye Residences | Extended Stay Residences Near Toronto Airportmarketingjdass
Experience unparalleled EXTENDED STAY and comfort at Skye Residences located just minutes from Toronto Airport. Discover sophisticated accommodations tailored for discerning travelers.
Website Link :
https://skyeresidences.com/
https://skyeresidences.com/about-us/
https://skyeresidences.com/gallery/
https://skyeresidences.com/rooms/
https://skyeresidences.com/near-by-attractions/
https://skyeresidences.com/commute/
https://skyeresidences.com/contact/
https://skyeresidences.com/queen-suite-with-sofa-bed/
https://skyeresidences.com/queen-suite-with-sofa-bed-and-balcony/
https://skyeresidences.com/queen-suite-with-sofa-bed-accessible/
https://skyeresidences.com/2-bedroom-deluxe-queen-suite-with-sofa-bed/
https://skyeresidences.com/2-bedroom-deluxe-king-queen-suite-with-sofa-bed/
https://skyeresidences.com/2-bedroom-deluxe-queen-suite-with-sofa-bed-accessible/
#Skye Residences Etobicoke, #Skye Residences Near Toronto Airport, #Skye Residences Toronto, #Skye Hotel Toronto, #Skye Hotel Near Toronto Airport, #Hotel Near Toronto Airport, #Near Toronto Airport Accommodation, #Suites Near Toronto Airport, #Etobicoke Suites Near Airport, #Hotel Near Toronto Pearson International Airport, #Toronto Airport Suite Rentals, #Pearson Airport Hotel Suites
As a business owner in Delaware, staying on top of your tax obligations is paramount, especially with the annual deadline for Delaware Franchise Tax looming on March 1. One such obligation is the annual Delaware Franchise Tax, which serves as a crucial requirement for maintaining your company’s legal standing within the state. While the prospect of handling tax matters may seem daunting, rest assured that the process can be straightforward with the right guidance. In this comprehensive guide, we’ll walk you through the steps of filing your Delaware Franchise Tax and provide insights to help you navigate the process effectively.
Personal Brand Statement:
As an Army veteran dedicated to lifelong learning, I bring a disciplined, strategic mindset to my pursuits. I am constantly expanding my knowledge to innovate and lead effectively. My journey is driven by a commitment to excellence, and to make a meaningful impact in the world.
Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
Putting the SPARK into Virtual Training.pptxCynthia Clay
This 60-minute webinar, sponsored by Adobe, was delivered for the Training Mag Network. It explored the five elements of SPARK: Storytelling, Purpose, Action, Relationships, and Kudos. Knowing how to tell a well-structured story is key to building long-term memory. Stating a clear purpose that doesn't take away from the discovery learning process is critical. Ensuring that people move from theory to practical application is imperative. Creating strong social learning is the key to commitment and engagement. Validating and affirming participants' comments is the way to create a positive learning environment.
Explore our most comprehensive guide on lookback analysis at SafePaaS, covering access governance and how it can transform modern ERP audits. Browse now!
3.0 Project 2_ Developing My Brand Identity Kit.pptxtanyjahb
A personal brand exploration presentation summarizes an individual's unique qualities and goals, covering strengths, values, passions, and target audience. It helps individuals understand what makes them stand out, their desired image, and how they aim to achieve it.
Remote sensing and monitoring are changing the mining industry for the better. These are providing innovative solutions to long-standing challenges. Those related to exploration, extraction, and overall environmental management by mining technology companies Odisha. These technologies make use of satellite imaging, aerial photography and sensors to collect data that might be inaccessible or from hazardous locations. With the use of this technology, mining operations are becoming increasingly efficient. Let us gain more insight into the key aspects associated with remote sensing and monitoring when it comes to mining.
Accpac to QuickBooks Conversion Navigating the Transition with Online Account...PaulBryant58
This article provides a comprehensive guide on how to
effectively manage the convert Accpac to QuickBooks , with a particular focus on utilizing online accounting services to streamline the process.
[Note: This is a partial preview. To download this presentation, visit:
https://www.oeconsulting.com.sg/training-presentations]
Sustainability has become an increasingly critical topic as the world recognizes the need to protect our planet and its resources for future generations. Sustainability means meeting our current needs without compromising the ability of future generations to meet theirs. It involves long-term planning and consideration of the consequences of our actions. The goal is to create strategies that ensure the long-term viability of People, Planet, and Profit.
Leading companies such as Nike, Toyota, and Siemens are prioritizing sustainable innovation in their business models, setting an example for others to follow. In this Sustainability training presentation, you will learn key concepts, principles, and practices of sustainability applicable across industries. This training aims to create awareness and educate employees, senior executives, consultants, and other key stakeholders, including investors, policymakers, and supply chain partners, on the importance and implementation of sustainability.
LEARNING OBJECTIVES
1. Develop a comprehensive understanding of the fundamental principles and concepts that form the foundation of sustainability within corporate environments.
2. Explore the sustainability implementation model, focusing on effective measures and reporting strategies to track and communicate sustainability efforts.
3. Identify and define best practices and critical success factors essential for achieving sustainability goals within organizations.
CONTENTS
1. Introduction and Key Concepts of Sustainability
2. Principles and Practices of Sustainability
3. Measures and Reporting in Sustainability
4. Sustainability Implementation & Best Practices
To download the complete presentation, visit: https://www.oeconsulting.com.sg/training-presentations
Sustainability: Balancing the Environment, Equity & Economy
On the fragility of fractional-order PID controllers for FOPDT processes
1. Research Article
On the fragility of fractional-order PID controllers for FOPDT processes
Fabrizio Padula a
, Antonio Visioli b,n
a
Dipartimento di Ingegneria dell'Informazione, University of Brescia - Italy, Italy
b
Dipartimento di Ingegneria Meccanica e Industriale, University of Brescia - Italy, Via Branze 38, I-25123 Brescia, Italy
a r t i c l e i n f o
Article history:
Received 2 December 2014
Received in revised form
26 August 2015
Accepted 9 November 2015
Available online 27 November 2015
This paper was recommended for publica-
tion by Dr. Y. Chen
Keywords:
Fractional-order controllers
PID control
Tuning
Fragility
a b s t r a c t
This paper analyzes the fragility issue of fractional-order proportional-integral-derivative controllers
applied to integer first-order plus-dead-time processes. In particular, the effects of the variations of the
controller parameters on the achieved control system robustness and performance are investigated. Results
show that this kind of controllers is more fragile with respect to the standard proportional-integral-
derivative controllers and therefore a significant attention should be paid by the user in their tuning.
& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
It is well known that a properly designed control system must
provide an effective trade-off between performance and robust-
ness. However, it has also been recognized that another important
issue to be addressed is the fragility of the control system to the
variation of the controller parameters, that is, the sensitivity of the
robustness and/or performance of the control system to changes in
the controller parameters.
This issue has been raised in the literature in some papers (see,
for example, [1]) and, in particular, in [2] where it has been
stressed that design techniques based on the minimization of the
H2, H1 and l1 norms can yield to high-order robust, optimal but
also extremely fragile controllers, namely, a very small variation of
the controller coefficients can result in an unstable system. How-
ever, in [3,4] it has been pointed out that this problem can be
solved by using a suitable controller parametrization.
As integer-order proportional-integral-derivative (IOPID)
controllers are the most used controllers in industry, the fragility
of such a kind for controllers has been specifically addressed in
[5,6]. Therein, authors suggest to tune the IOPID controller in
order to maximize the l2 norm of the controller parameter vector
in the stabilizing region for a given plant. However, the typical
industrial performance measures (related to the set-point
following and/or to the load disturbance rejection task) are not
taken into account. Further, it has been shown in [7] that this
kind of approach applied to first-order-plus-dead-time (FOPDT)
and integrator-plus-dead-time (IPDT) processes yields a tuning
similar to that obtained by using the Ziegler–Nichols step
response method [8] which is known to be improvable under
many points of view [9].
Thus, it has been recognized in the literature that one of the
main reasons to investigate the fragility of IOPID controllers is to
give to the user an idea of how a fine tuning of the controller can
be done [10–12]. In other words, as the IOPID parameters have a
clear physical meaning, the operator can modify them in order to
change the control system performance. In this context, it is useful
to evaluate the sensitivity of the robustness/performance behavior
with respect to (small) changes of the parameters. For this pur-
pose, a graphic tool called fragility rings providing a visual aid for
evaluation of the controller robustness/fragility has been proposed
in [13].
In the recent years, there has also been a significant interest
from the academic and industrial communities for fractional-
order-proportional-integral-derivative (FOPID) controllers because
they are capable to provide (as there are five parameters to tune)
more flexibility in the control system design (see, for example,
[14–17]). Many different tuning rules have been proposed in the
literature to facilitate their use (see, for example, [18–23]). In this
context, while the problem of stabilizing a (possibly fractional)
dynamic system using FOPID controllers has been already
addressed in the literature (see, for example, [24–26]), for such a
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2015.11.010
0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author. Tel.: þ39 030 3715460; fax: þ39 030 380014.
E-mail addresses: fabrizio.padula@unibs.it (F. Padula),
antonio.visioli@unibs.it (A. Visioli).
ISA Transactions 60 (2016) 228–243
2. kind of controllers, a fragility analysis has been only partially
exploited until now. In particular, in [27,28], the tuning of the
FOPID controllers is performed by considering the centroids of the
admissible regions in the parameter space so that a non-fragile
controller results. However, one of the main purposes for evalu-
ating the fragility of the controller is in evaluating the sensitivity of
the robustness/performance indexes to the (possibly fine) tuning
of the parameters.
Indeed, in order to foster a widespread use of FOPID controllers
in industrial plants, in addition to well-established tuning rules,
clear guidelines on how to modify the controller parameters
should be given to the operator in order for him/her to be con-
fident with them. Thus, the aim of this paper is to provide a fra-
gility analysis for FOPID controllers and to make a comparison
with IOPID controllers in order to understand the differences that
should be taken into account in the adjustment of the parameters
starting from a given tuning. For this purpose, the tuning rules
proposed in [23,29], which aim at minimizing the integrated
absolute error subject to constraints on the maximum sensitivity,
are used, both for FOPID and IOPID controllers. Both the tuning
rules for the set-point following and the load disturbance rejection
tasks are considered. They also have the significant feature of
providing a control action that is invariant when the time unit is
changed. These tuning rules are therefore suitable to perform a
fragility analysis with respect to both robustness and performance.
It is worth stressing that the calculated fragility depends on the
nominal parameters of the control system and for this reason, in
order to obtain a fair comparison, we select tuning rules that solve
the same optimization problem, so that the possible additional
complexity of adjusting the parameters of a FOPID controller, with
respect to a IOPID one, starting from a given tuning is clearly
addressed.
The fragility is evaluated by changing all the parameters at the
same time or just one of them by keeping the other ones fixed. The
latter case is performed in order to investigate which parameter
has more influence on the controller fragility.
The paper is organized as follows. The basic definitions
employed for the fragility evaluation are reviewed in Section 2, in
addition to the description of the tuning rules used for both integer-
order and fractional-order PID controllers. The fragility analysis
related to the robustness is presented in Section 3 while that related
to the performance is presented in Section 4. A discussion is made
in Section 5, while conclusions are drawn in Section 6.
2. Fragility indices
The fragility indices proposed in [10–12] are briefly reviewed in
this section for the sake of clarity and in order to introduce the
notation used in presenting the results.
Consider a unity feedback control system (see Fig. 1) where the
process (which is assumed to be self-regulating) is denoted as P
and the controller as C. In this paper, the controller is a FOPID
controller, which can be expressed either in series form, i.e.,
CðsÞ ¼ Kp
Tisλ þ1
Tisλ
Tdsμ þ1
Tf sþ1
ð1Þ
or in parallel (ideal) form, i.e.,
CðsÞ ¼ Kp 1þ
1
Tisλ
þTdsμ
1
Tf sþ1
: ð2Þ
In both expression, Kp is the proportional gain, Ti is the integral
time constant, Td is the derivative time constant and λ and μ are
the noninteger orders of the integral and derivative terms
respectively.
Note that it is important to consider both forms (1) and (2)
because it is not possible to transform (2) into an equivalent form
(1) and vice versa unless Ti Z4Td and λ ¼ μ [29]. In order to
implement the fractional-order controller, the well-known Ous-
taloup continuous integer-order approximation [30] has been
employed to approximate the fractional differintegrator. In this
paper 16 poles and zeros have been used in order to approximate
the fractional differintegrator in a frequency range ½ωl; ωhŠ, where
ωl and ωh have been selected as 0:0001ωc and 10000ωc
respectively, with ωc being the gain crossover frequency. It is
worth noting that the used number of poles and zeros leads to a
computationally demanding controller and, actually, the frac-
tional controller could be approximated with a lower order
integer one. Nevertheless, considering that the purpose of this
paper is the fragility analysis of the fractional controller, a higher
computational cost is accepted in order to achieve an improved
approximation. The approximated and the ideal open loop
transfer function in this way are virtually indistinguishable at
those frequencies that have an appreciable impact on the closed-
C
r ye
P
d
Fig. 1. The considered control scheme.
Table 1
The controller parameters for the considered example with L=T ¼ 0:5 and for the different control tasks, set-point (SP) following and load disturbance (LD) rejection with a
maximum sensitivity of 1.4 and 2.0 respectively.
Controller Kp Ti Td λ μ
FOPID series
SP 1.4 1.1060 0.9839 0.1554 1 1.2
SP 2.0 1.6698 1.0281 0.1975 1 1.1
LD 1.4 0.7818 0.4683 0.2617 1 1.1
LD 2.0 1.1182 0.4236 0.3105 1 1.1
FOPID parallel
SP 1.4 1.3307 1.1765 0.1384 1 1.1578
SP 2.0 2.0850 1.2507 0.1757 1 1.1351
LD 1.4 1.2786 0.8824 0.1686 1 1.1351
LD 2.0 2.3611 0.9079 0.1440 1 1.1525
IOPID series
SP 1.4 0.8676 0.8127 0.2074 – –
SP 2.0 1.4708 0.9568 0.2347 – –
LD 1.4 0.6369 1.0081 0.3031 – –
LD 2.0 1.007 0.4106 0.3304 – –
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 229
3. loop system dynamics. It can be also noted that an additional
first-order filter has been employed in both (1) and (2) in order to
make the controller proper. The selection of the time constant Tf
is done in such a way that the high-frequency noise is filtered
without influencing the dynamics of the controller significantly
[29,31,32]. Eventually, considering the Oustaloup approximation
and that only the fractional part μ (μÀ1 if μ41) of the derivative
action is approximated, an integer filter is enough to guarantee
the properness of the controller.
Then, it has also to be noted that by selecting λ ¼ μ ¼ 1, an
IOPID controller is obtained. In this paper, just for the sake of
comparison (as the analysis will be focused on FOPID controllers),
we consider the IOPID controller in series form, i.e.,
CðsÞ ¼ Kp
Tisþ1
Tis
Tdsþ1
Tf sþ1
ð3Þ
(note that with the employed tuning rules described below it
results in Ti 44Td and therefore an equivalent IOPID controller in
ideal form can always be considered, i.e., the optimal IOPID
controller is unique). The typical control specification requires
that a predefined performance is obtained in the set-point fol-
lowing and load disturbance rejection task. In both cases, a
typical performance index related to the step responses is the
integrated absolute error [33], which yields, in general, a small
overshoot and a small settling time at the same time and is
defined as
Je ¼
Z 1
0
jeðtÞj dt ¼
Z 1
0
jrðtÞÀyðtÞj dt; ð4Þ
where r is the set-point signal and y is the process variable.
From another point of view, it is often essential that the
control system is also robust to changes in the process dynam-
ics. A commonly employed measure of the robustness of the
system is the maximum sensitivity, which represents the
inverse of the minimum distance of the loop transfer function
from the critical point (À1, 0) in the Nyquist plot and it is
defined as
Ms ¼ max
ωA½0;þ1Þ
1
1þCðsÞPðsÞ
: ð5Þ
For this reason, specific tuning rules have been devised for
each controller (1)–(3) in order to minimize the Je value subject
to constraints on the maximum sensitivity [23,29]. In parti-
cular, both the set-point following and the load disturbance
Fig. 2. Resulting values of RFIΔε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243230
4. rejection tasks have been considered separately and, for each
task, the values of Ms ¼1.4 and Ms ¼2.0 have been selected
(note that tuning rules related to integral and unstable pro-
cesses have been proposed in [34]). In general, it has been
shown that the FOPID controller provides a better performance
than the IOPID controller and the improvement is achieved by
using an integer-order integrator and a fractional derivative
order μ41 [23].
The fragility of the controller can be evaluated with respect to
either the robustness or the performance. By denoting as θ
0
c the
vector of the controller parameters (that is, θ
0
c ¼ ½Kp; Ti; Td; λ; μŠ for
the FOPID controller and θ
0
c ¼ ½Kp; Ti; TdŠ for the IOPID controller),
the loss of robustness of the control system when the controller
parameters are perturbed can be expressed by the so-called Delta-
Epsilon-Robustness-Fragility Index which is defined as
RFIΔε ¼
Mm
sΔε
M0
s
À1 ¼
maxfMsðð17δεÞθ
0
c Þg
Msðθ
0
c Þ
À1; ð6Þ
where Mm
sΔε is the extreme maximum sensitivity, that is, the
highest loss of robustness of the control system that occurs
when all the parameters of the controller can vary of the
same δε quantity with respect to their nominal values θ
0
c ,
considering all the possible combinations of the perturbed
parameters. On the contrary, Msðθ
0
c Þ is the nominal sensitivity,
that is, the sensitivity obtained with the nominal controller. It
appears that, an index RFIΔε ¼ 0 implies that the controller is
absolutely robustness-non-fragile. It is however recognized
that a reasonable variation of the parameters is up to 20%. For
this reason, a controller is considered to be robustness resilient
if its delta 20 robustness fragility index is less than 0.10 (that is,
RFIΔ20 o0:10), robustness non-fragile if RFIΔ20 r0:50 and
robustness fragile if RFIΔ20 40:50.
It is also important to evaluate the relative influence of a single
parameter on the robustness fragility. In order to do that, the
Parametric-Delta-Epsilon-Robustness-Fragility Index has been defined
as
RFI
pi
δε ¼
M
pi
sδε
M0
s
À1 ¼
maxfMsðð17δεÞpi; θ
0
c Þg
Msðθ
0
c Þ
À1: ð7Þ
Similar to the previous case, the loss of performance of the
control system when the controller parameters are per-
turbed (note again that the tuning rules minimize the int-
Fig. 3. Resulting values of RFI
Kp
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 231
5. egrated absolute error) can be expressed by the so-called
Delta-Epsilon-Performance-Fragility Index which is defined as
PFIΔε ¼
Jm
eΔε
J0
e
À1 ¼
maxfJeðð17δεÞθ
0
c Þg
Jeðθ
0
c Þ
À1; ð8Þ
where Jm
eΔε is the extreme performance and Je
0
is the nominal
performance. The relative influence of a δε variation of a single
controller parameter pi on the performance fragility of the control
system can be expressed by the following Parametric-Delta-Epsi-
lon-Performance-Fragility Index:
PFI
pi
δε ¼
J
pi
eδε
J0
e
À1 ¼
maxfJeðð17δεÞpi; θ
0
c Þg
Jeðθ
0
c Þ
À1: ð9Þ
Similarly again to the robustness case, by assuming a reasonable
threshold of 20%, a controller is considered to be performance
resilient if its delta 20 performance fragility index is less than 0.10
(that is, PFIΔ20 o0:10), performance non-fragile if RFIΔ20 r0:50
and performance fragile if RFIΔ20 40:50.
Remark 1. It is worth stressing that the fragility indices
obviously depend on the tuning of the IOPID or FOPID
parameters. For this reason, in order to provide meaningful
results, it is important to compare the FOPID and IOPID con-
trollers with parameters selected in order to optimize the same
performance index.
3. Robustness fragility
The robustness fragility have been evaluated for the FOPID
controllers in both series and parallel form and the results are
compared with the IOPID controller. In particular, FOPDT pro-
cesses have been considered. They are described by the
transfer function
PðsÞ ¼
K
Tsþ1
eÀ Ls
: ð10Þ
Then, for the normalized gain K¼1 and for different values
of the normalized dead time L=T in the interval ½0:1; 1Š, the
tuning rules for the minimization of integrated absolute error
of the set-point step response and of the load disturbance step
response have been applied. In both cases, the values of the
target maximum sensitivity are set to either Ms ¼1.4 or Ms ¼2.0.
Fig. 4. Resulting values of RFITi
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243232
6. Thus, for each process and for each of the three controllers,
four cases have been considered and the RFIΔε index has been
calculated for different values of δε, by iteratively considering
all the possible variations of the parameters.
For the sake of brevity, only the results related to the process
with normalized dead time L=T ¼ 0:5 are shown. In this case,
assuming K¼1, T¼1 and L¼0.5, we obtain, for the different con-
trol specifications, the controller parameters shown in Table 1.
Actually, the results related to the other processes are very similar
to them. Results are shown in Fig. 2 where it has to be stressed
that when the data is missing it means that the overall control
system is unstable. Thus, it can be easily noted that the FOPID
controller (both in series and parallel form) is much more
robustness fragile (thus, the fine tuning is more critical) than the
IOPID controller.
In order to evaluate better the influence of the single con-
troller parameters, the Parametric-Delta-Epsilon-Robustness-
Fragility Index RFI
pi
δε has also been computed for the different
controller parameters. Results are shown in Figs. 3–7 (note that
the IOPID controller does not include the λ and μ parameters).
It appears that the robustness fragility of the FOPID controllers
is less critical if one parameter at a time is fine tuned and, in
any case, the fractional order of the derivative term is the most
dangerous parameter, especially when the employed tuning
rule aims at achieving a more aggressive controller (namely,
the target Ms ¼2.0 has been selected) and when a FOPID con-
troller in parallel form is used. A discussion about this issue
will be done in Section 5.
4. Performance fragility
The same analysis done for the robustness fragility has been
performed also for the performance fragility, that is, for each
FOPDT process and for each of the three controllers with the four
different considered tuning rules, the PFIΔε index has been cal-
culated for different values of δε. Results related again to the
process with a normalized dead time L=T ¼ 0:5 are shown in Fig. 8.
The same conclusions as for the robustness fragility can be made
for the performance fragility. The FOPID controllers (especially
that in parallel form) are more sensitive than the IOPID controller
with respect to changes in the parameters so that their fine tuning
can be more critical. By evaluating the Parametric-Delta-Epsilon-
Performance-Fragility Index PFI
pi
δε for each single parameter (see
Fig. 5. Resulting values of RFITd
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 233
7. Figs. 9–13), it can be deduced that the FOPID controllers are more
sensitive for changes in the fractional order of the integral and,
most of all, of derivative terms than in the other parameters.
5. Discussion
In the previous sections it has been pointed out that the fra-
gility of the FOPID controllers is mainly motivated by the presence
of the fractional-order derivative term. The reasons for this are
analyzed in this section by means of an illustrative example.
Consider the process
PðsÞ ¼
1
sþ1
eÀ0:5s
ð11Þ
and the tuning rules applied devised for the load disturbance
rejection with a target maximum sensitivity of Ms ¼2.0. For the
parallel FOPID controller (the series form is omitted for the
sake of brevity, but results are very similar to the parallel case)
these yield Kp ¼2.361, Ti ¼0.908, Td ¼0.144, λ ¼ 1, μ ¼ 1:153,
while for the IOPID controller we obtain Kp ¼1.008, Ti ¼0.411,
and Td ¼0.330. From the analysis of the Bode plots obtained in
the nominal case, shown in Fig. 14, it is evident that the FOPID
controller allows an increment of the bandwidth with respect
to the IOPID controller (the gain crossover frequency is ωgc ¼
2:01 for the FOPID controller and ωgc ¼ 1:73 for the IOPID
controller), with the same level of robustness. This is achieved
by exploiting the phase advance introduced by the fractional
derivative of order greater than one. While this implies a better
performance in the step response, it is also evident that the
frequency response function monotonicity is no longer guar-
anteed (see Fig. 14) because of the increased high frequency
roll-up that the fractional differentiator may exhibit and this
implies an incremented fragility of the controller. This can be
better analyzed by considering the frequency derivative of the
magnitude of the loop transfer function, which results in
d CðjωÞPðjωÞ
35. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N2
r ðωÞþN2
i ðωÞ
D2
r ðωÞþD2
i ðωÞ
s
ð13Þ
Fig. 6. Resulting values of RFIλ
δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right:
tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243234
44. dω
¼
À Kj jT2
ω
1þT2
ω2
3
2
ð15Þ
and
where the controller CðjωÞ has been expressed as
CðjωÞ ¼
NrðωÞþjNiðωÞ
DrðωÞþjDiðωÞ
ð17Þ
and
NrðωÞ ¼ 1þTiωλ cos
π
2
λ
þTiTdωλþ μ cos
π
2
ðλþμÞ
NiðωÞ ¼ Tiωλ sin
π
2
λ
þTiTdωλþ μ sin
π
2
ðλþμÞ
DrðωÞ ¼ Tiωλ cos
π
2
λ
þTiTf ωλþ 1
cos
π
2
ðλþ1Þ
DiðωÞ ¼ Tiωλ sin
π
2
λ
þTiTf ωλþ1
sin
π
2
ðλþ1Þ
ð18Þ
and
dNrðωÞ
dω
¼ TiλωλÀ1
cos
π
2
λ
þTiTdðλþμÞωλþ μÀ 1
cos
π
2
ðλþμÞ
dNiðωÞ
dω
¼ TiλωλÀ1
sin
π
2
λ
þTiTdðλþμÞωλþμÀ 1
sin
π
2
ðλþμÞ
Fig. 7. Resulting values of RFIμ
δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right:
tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
d CðjωÞ
53. dDrðωÞ
dω
¼ TiλωλÀ1
cos
π
2
λ
þTiTf ðλþ1Þωλ cos
π
2
ðλþ1Þ
dDiðωÞ
dω
¼ TiλωλÀ1
sin
π
2
λ
þTiTf ðλþ1Þωλ sin
π
2
ðλþ1Þ
: ð19Þ
Actually, a deeper analysis can be performed by evaluating the
frequency response function when one parameter at a time changes in
the FOPID controller. Results are shown in Figs. 15–19. As expected
from the results shown in Figs. 7 and 13, it appears that the (fractional)
derivative action is the most critical one. In particular, the increment of
the fractional order μ leads to high-frequency peaks in the sensitivity
function, to a loop gain with multiple gain crossover frequencies and,
eventually, to instability. Indeed, μ is the only parameter that is able to
destabilize the loop in spite of variations smaller than 30%. This hap-
pens because increasing μ also means an increased non-monotonic
behavior of the frequency response function (see Fig. 19) as a con-
sequence of the increased high frequency roll-up. Another critical
parameter is the derivative time constant Td. Indeed the optimal FOPID
controller has a derivative action with a derivative order μ greater than
1. This means that the optimal frequency response function is already
non-monotonic and an increased derivative time constant pushes up
the frequency response close to the 0 dB axes (see Fig. 17) creating
again high frequency peaks in the sensitivity function with a con-
sequent loss of robustness and performance. Again, this behavior is
expected from the results shown in Fig. 5 where robustness fragility is
considered. It can be appreciated that the IOPID controller always
results in less fragile compared to the FOPID one. This happens
because of its monotonic behavior.
On the contrary, variations in the integrator order do not generate
dramatic changes in the frequency response function (see Fig. 16).
Indeed, the monotonicity of the frequency response function is inde-
pendent from the selected value of λ, unless Ti⪢Td and a series FOPID
controller is considered, but, evidently, this is not a meaningful tuning
of the controller. This is inde66pc4.68 pendent from the fact that the
optimal FOPID controller is obtained with λ ¼ 1.
Summarizing, the relevant difference between FOPID and IOPID
controllers is that the former ones are capable (indeed because of
the fractional-order derivative action) of providing a reduction of
the integrated absolute error but their fragility should be carefully
considered when the tuning is performed and, in particular, when
the fractional-order derivative term is changed, as a small varia-
tion can modify the performance significantly.
Fig. 8. Resulting values of PFIΔε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243236
54. Fig. 9. Resulting values of PFI
Kp
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 237
55. Fig. 10. Resulting values of PFITi
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243238
56. Fig. 11. Resulting values of PFITd
δε for FOPID controller in series form (‘○’), for FOPID controller in parallel form (‘□’) and for IOPID controller (‘▵’). Top left: tuning for set-point
with Ms ¼1.4. Top right: tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 239
57. Fig. 12. Resulting values of PFIλ
δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right:
tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243240
58. Fig. 13. Resulting values of PFIμ
δε for FOPID controller in series form (‘○’) and for FOPID controller in parallel form (‘□’). Top left: tuning for set-point with Ms ¼1.4. Top right:
tuning for set-point with Ms ¼2.0. Bottom left: tuning for load disturbance with Ms ¼1.4. Bottom right: tuning for load disturbance with Ms ¼2.0.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 241
59. Fig. 14. Magnitude Bode plots in the nominal case for the illustrative example.
Solid line: FOPID controller. Dashed line: IOPID controller.
Fig. 15. Magnitude Bode plots for the illustrative example (FOPID controller) when
the proportional gain Kp changes in the range 730%.
Fig. 16. Magnitude Bode plots for the illustrative example (FOPID controller) when
the integral time constant Ti changes in the range 730%.
Fig. 17. Magnitude Bode plots for the illustrative example (FOPID controller) when
the derivative time constant Td changes in the range 730%.
Fig. 18. Magnitude Bode plots for the illustrative example (FOPID controller) when
the fractional integral order λ changes in the range 730%.
Fig. 19. Magnitude Bode plots for the illustrative example (FOPID controller) when
the fractional derivative order μ changes in the range 730%.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243242
60. 6. Conclusions
As the parameters of IOPID controllers have a clear physical
meaning and the relative easiness of their manual tuning has
determined (among other factors) their success, it is believed that
the same feature should be provided for FOPID controllers in order
to allow a more widespread use of them in industry. A key role in
this context is played by the robustness and performance fragility
of this kind of controllers which have been analyzed in this paper
in order to evaluate the criticalness of the fine tuning of the
parameters. It has been highlighted that FOPID controllers are
more fragile than IOPID controllers and a special attention should
be paid especially in changing the fractional order of the derivative
term. Thus, research effort should be provided in the future in
order to devise tuning rules for FOPID controllers that are able to
guarantee a resilient robustness and performance.
References
[1] Qiu L, Davison EJ. Feedback stability under simultaneous gap metric uncer-
tainties in plant and controller. Syst Control Lett 1992;18(1):9–22.
[2] Keel LH, Bhattacharyya SP. Robust, fragile, or optimal? IEEE Trans Autom
Control 1997;42(8):1098–105.
[3] Makila PM. Comments on robust, fragile, or optimal? IEEE Trans Autom Con-
trol 1998;43(9):1265–7.
[4] Paattilammi J, Makila PM. Fragility and robustness: a case study on paper
machine and headbox control. IEEE Control Syst Mag 2000;30(1):13–22.
[5] Ho M-T. Non-fragile PID controller design. In: Proceedings of IEEE interna-
tional conference on decision and control, Sydney, Australia; 2000. p. 4903–8.
[6] Silva GL, Datta A, Bhattacharyya SP. PID controllers for time-delay systems.
Boston, MA: Birkhauser; 2005.
[7] Bahavarnia M, Tavazoei MS. A new view to Ziegler–Nichols step response
tuning method: analytic non-fragility justification. J Process Control
2013;23:23–33.
[8] Ziegler JG, Nichols NB. Optimum settings for automatic controllers. Trans
ASME 1942;64:759–68.
[9] Åström KJ, Hägglund T. Advanced PID control. Research Triangle Park, NC: ISA
Press; 2006.
[10] Alfaro VM. PID controllers' fragility. ISA Trans 2007;46:555–9.
[11] Alfaro VM, Vilanova R, Arrieta O. Fragility analysis of PID controllers. In: Pro-
ceedings IEEE international conference on control applications, St. Petersburg,
Russia; 2009. p. 725–30.
[12] Alfaro VM, Vilanova R. Fragility evaluation of PI and PID controllers tuning
rules. In: Vilanova R, Visioli A, editors. PID control in the third millennium—
lessons learned and new approaches. London, UK: Springer; 2012. p. 349–80.
[13] Alfaro VM, Vilanova R. Fragility-rings—a graphic tool for PI/PID controllers
robustness-fragility analysis. In: Proceedings of IFAC conference on advances
in PID control, Brescia (I); 2012. p. 187–92.
[14] Podlubny I. Fractional-order systems and PIλ
Dμ
controllers. IEEE Trans Autom
Control 1999;44:208–14.
[15] Vinagre BM, Monje CA, Calderon AJ, Suarez JI. Fractional PID controllers for
industry application. A brief introduction. J Vib Control 2007;13:1419–29.
[16] Biswas A, Das S, Abraham A, Dasgupta S. Design of fractional-order PIλ
Dμ
controllers with an improved differential evolution. Eng Appl Artif Intell
2009;22:343–50.
[17] Zamani M, Karimi-Ghartemani M, Sadati N, Parniani M. Design of fractional
order PID controller for an AVR using particle swarm optimization. Control Eng
Pract 2009;17:1380–7.
[18] Monje CA, Vinagre BM, Calderon AJ, Feliu V, Chen YQ. On fractional PIλ
con-
trollers: some tuning rules for robustness to plant uncertainties. Nonlinear
Dyn 2004;38:369–81.
[19] Barbosa RS, Tenreiro Machado JA, Ferreira IM. Tuning of PID controllers based
on Bode's ideal transfer function. Nonlinear Dyn 2004;38:305–21.
[20] Valerio D, Sa da Costa J. Tuning of fractional PID controllers with Ziegler–
Nichols-type rules. Signal Process 2006;86:2771–84.
[21] Monje CA, Vinagre BM, Feliu V, Chen YQ. Tuning and auto-tuning of fractional
order controllers for industry applications. Control Eng Pract 2008;16:
798–812.
[22] Chen YQ, Bhaskaran T, Xue D. Practical tuning rule development for fractional
order proportional and integral controllers. ASME J Comput Nonlinear Dyn
2008;3:0214031–7.
[23] Padula F, Visioli A. Tuning rules for optimal PID and fractional-order PID
controllers. J Process Control 2011;21:69–81.
[24] Hamamci SE. Stabilization using fractional-order PI and PID controllers.
Nonlinear Dyn 2008;51:329–43.
[25] Lee YK, Watkins JM. Determination of All stabilizing fractional-order PID
controllers. In: Proceedings American control conference, San Francisco, CA,
2011. p. 5007–12.
[26] Gao Z, Yan M, Wei J. Robust stabilizing regions of fractional-order PDμ
con-
trollers of time-delay fractional-order systems. J Process Control 2014;24:37–47.
[27] Bahavarnia MS, Tavazoei MS, Mesbahi A. Non-fragile tuning of fractional-order
PD controllers for IPD-modelled processes. In: Proceedings of IFAC workshop
on fractional differentiation and its applications, Grenoble, France; 2013. p.
361–6.
[28] Mesbahi A, Haeri M. Robust non-fragile fractional order PID controller for linear
time invariant fractional delay systems. J Process Control 2014;24:1489–94.
[29] Padula F, Visioli A. Advances in robust fractional control. London, UK:
Springer; 2014.
[30] Oustaloup A, Levron F, Mathieu B, Nanot FM. Frequency-band complex non-
integer differentiator: characterization and synthesis. IEEE Trans Circuits Syst
I: Fundam Theory Appl 2000;47:25–39.
[31] Visioli A. Practical PID control. London, UK: Springer; 2006.
[32] Romero V, Hägglund T, Åström KJ. Measurement noise filtering for common
PID tuning rules. Control Eng Pract 2014;32:43–63.
[33] Shinskey FG. Feedback controllers for the process industries. New York, NY:
McGraw-Hill; 1994.
[34] Padula F, Visioli A. Optimal tuning rules for proportional-integral-derivative
and fractional-order proportional-integral-derivative controllers for integral
and unstable processes. IET Control Theory Appl 2012;6:776–86.
F. Padula, A. Visioli / ISA Transactions 60 (2016) 228–243 243