This document proposes a new method for tuning non-PID controllers to achieve high performance control of complex processes. The method involves 3 stages:
1) Determining an optimal closed-loop transfer function based on the process characteristics and limitations.
2) Deriving an ideal controller from the optimal transfer function, which is usually complex.
3) Applying model reduction to the ideal controller to obtain a simpler, realizable controller form.
Simulation examples are provided to demonstrate the effectiveness of the proposed non-PID controller tuning method for complex processes where PID control is insufficient.
Design PID controllers for desired time domain or frequency domain responseISA Interchange
The document discusses the design of PID controllers to achieve desired time-domain or frequency-domain responses for a nominal stable process with time delay. It presents an Hρ PID controller derived analytically based on optimal control theory. The controller parameters are derived such that they minimize the worst error resulting from system inputs. The properties of the Hρ PID controller are then investigated and compared to an H2 PID controller and a Maclaurin PID controller. All three controllers are shown to be able to provide quantitative time-domain and frequency-domain responses.
This document summarizes a research paper that proposes using differential evolution optimization to reduce the order of high-order discrete systems and design PID controllers for the reduced-order models. The paper presents a method for:
1) Reducing a high-order discrete system to a lower-order model by minimizing the error between step responses using differential evolution.
2) Designing a PID controller for the reduced-order model by optimizing controller parameters with differential evolution to minimize error.
3) Testing the PID controller on both the reduced and original high-order models. Differential evolution is an evolutionary algorithm that minimizes an objective function by mutating candidate solutions and selecting the best ones to propagate to the next generation.
The document describes a project to improve a PID controller for a DC motor system using fuzzy logic. The group designed fuzzy precompensator, PI, and PD controllers to improve the PID controller's ability to handle disturbances. Simulation results showed that the fuzzy logic controllers had better performance than the PID controller alone in terms of settling time, overshoot, and steady-state error during step responses and when disturbances were added.
A real-time system must respond to external stimuli within a finite time period. The correctness of real-time computations depends on both logical results and timeliness. Real-time systems require substantial design effort to ensure task deadlines are met. There are two types of real-time systems: hard where missing deadlines causes damage, and soft where missing deadlines is undesirable. Scheduling algorithms like earliest deadline first (EDF) and rate monotonic analysis (RMA) are used to ensure tasks meet deadlines in real-time systems.
Network analysis is a project planning technique that involves breaking down a project into individual tasks or activities and arranging them in a logical sequence. A network diagram is constructed to show the relationships between all project activities. Key concepts in network analysis include activities, events, predecessor and successor events, and the network diagram itself. Network analysis techniques like PERT and CPM help plan projects economically by analyzing activity sequences and resource requirements.
Design and Implementation of Discrete Augmented Ziegler-Nichols PID ControllerIDES Editor
Although designing and tuning a proportionalintegral-
derivative (PID) controller appears to be conceptually
intuitive, but it can be hard in practice, if multiple (and often
conflicting) objectives such as short transient and high
stability are to be achieved. Traditionally Ziegler Nichols is
widely accepted PID tuning method but it’s performance is
not accepted for systems where precise control is required. To
overcome this problem, the online gain updating method
Augmented Ziegler-Nichols PID (AZNPID) was proposed, with
the amelioration of Ziegler-Nichols PID’s (ZNPID’s) tuning
rule. This study is further extension of [1] for making the
scheme more generalized. With the help of fourth order
Runge-Kutta method, differential equations involved in PID
are solved which significantly improves transient performance
of AZNPID compared to ZNPID. The proposed augmented
ZNPID (AZNPID) is tested on various types of linear processes
and shows improved performance over ZNPID. The results of
the proposed scheme is validated by simulation and also
verified experimentally by implementing on Quanser’s real
time servo-based position control system SRV-02.
The document discusses fractional order PID tuning and control. It introduces fractional order systems and controllers (FOPID), and describes some of their advantages over traditional PID controllers, including better modeling of dynamic systems and more robust control design. It then discusses several FOPID tuning methods, focusing on the Taylor series expansion method which matches terms between the actual and desired closed-loop transfer functions to increase tracking accuracy.
IMPLEMENTATION OF FRACTIONAL ORDER TRANSFER FUNCTION USING LOW COST DSPIAEME Publication
In this paper, different fractional order transfer functions are taken first and discretized them using available methods and filters (i.e. Oustaloup or modified Oustaloup). Coefficients of discretized transfer function are calculated and scaled using Q15 number system to get the coefficients in the range between -1 to 1, and converted into equivalent hexadecimal number. These coefficients are entered into the Micro C code that is generated using filter design tool of Micro C for dsPIC microcontroller. Also the simulation results are validated using EasydsPIC4 development board.
Design PID controllers for desired time domain or frequency domain responseISA Interchange
The document discusses the design of PID controllers to achieve desired time-domain or frequency-domain responses for a nominal stable process with time delay. It presents an Hρ PID controller derived analytically based on optimal control theory. The controller parameters are derived such that they minimize the worst error resulting from system inputs. The properties of the Hρ PID controller are then investigated and compared to an H2 PID controller and a Maclaurin PID controller. All three controllers are shown to be able to provide quantitative time-domain and frequency-domain responses.
This document summarizes a research paper that proposes using differential evolution optimization to reduce the order of high-order discrete systems and design PID controllers for the reduced-order models. The paper presents a method for:
1) Reducing a high-order discrete system to a lower-order model by minimizing the error between step responses using differential evolution.
2) Designing a PID controller for the reduced-order model by optimizing controller parameters with differential evolution to minimize error.
3) Testing the PID controller on both the reduced and original high-order models. Differential evolution is an evolutionary algorithm that minimizes an objective function by mutating candidate solutions and selecting the best ones to propagate to the next generation.
The document describes a project to improve a PID controller for a DC motor system using fuzzy logic. The group designed fuzzy precompensator, PI, and PD controllers to improve the PID controller's ability to handle disturbances. Simulation results showed that the fuzzy logic controllers had better performance than the PID controller alone in terms of settling time, overshoot, and steady-state error during step responses and when disturbances were added.
A real-time system must respond to external stimuli within a finite time period. The correctness of real-time computations depends on both logical results and timeliness. Real-time systems require substantial design effort to ensure task deadlines are met. There are two types of real-time systems: hard where missing deadlines causes damage, and soft where missing deadlines is undesirable. Scheduling algorithms like earliest deadline first (EDF) and rate monotonic analysis (RMA) are used to ensure tasks meet deadlines in real-time systems.
Network analysis is a project planning technique that involves breaking down a project into individual tasks or activities and arranging them in a logical sequence. A network diagram is constructed to show the relationships between all project activities. Key concepts in network analysis include activities, events, predecessor and successor events, and the network diagram itself. Network analysis techniques like PERT and CPM help plan projects economically by analyzing activity sequences and resource requirements.
Design and Implementation of Discrete Augmented Ziegler-Nichols PID ControllerIDES Editor
Although designing and tuning a proportionalintegral-
derivative (PID) controller appears to be conceptually
intuitive, but it can be hard in practice, if multiple (and often
conflicting) objectives such as short transient and high
stability are to be achieved. Traditionally Ziegler Nichols is
widely accepted PID tuning method but it’s performance is
not accepted for systems where precise control is required. To
overcome this problem, the online gain updating method
Augmented Ziegler-Nichols PID (AZNPID) was proposed, with
the amelioration of Ziegler-Nichols PID’s (ZNPID’s) tuning
rule. This study is further extension of [1] for making the
scheme more generalized. With the help of fourth order
Runge-Kutta method, differential equations involved in PID
are solved which significantly improves transient performance
of AZNPID compared to ZNPID. The proposed augmented
ZNPID (AZNPID) is tested on various types of linear processes
and shows improved performance over ZNPID. The results of
the proposed scheme is validated by simulation and also
verified experimentally by implementing on Quanser’s real
time servo-based position control system SRV-02.
The document discusses fractional order PID tuning and control. It introduces fractional order systems and controllers (FOPID), and describes some of their advantages over traditional PID controllers, including better modeling of dynamic systems and more robust control design. It then discusses several FOPID tuning methods, focusing on the Taylor series expansion method which matches terms between the actual and desired closed-loop transfer functions to increase tracking accuracy.
IMPLEMENTATION OF FRACTIONAL ORDER TRANSFER FUNCTION USING LOW COST DSPIAEME Publication
In this paper, different fractional order transfer functions are taken first and discretized them using available methods and filters (i.e. Oustaloup or modified Oustaloup). Coefficients of discretized transfer function are calculated and scaled using Q15 number system to get the coefficients in the range between -1 to 1, and converted into equivalent hexadecimal number. These coefficients are entered into the Micro C code that is generated using filter design tool of Micro C for dsPIC microcontroller. Also the simulation results are validated using EasydsPIC4 development board.
Slide Solution oferece serviços de criação de apresentações profissionais em PowerPoint para empresas e profissionais, com foco em design atraente e mensagem clara, liderado por Rafael Borges, que pode ser contatado por telefone ou e-mail.
This document discusses the emergency response planning of Pakistan International Airlines. It outlines PIA's focus on passenger safety and efforts to achieve high safety standards. It also acknowledges that no airline is immune to accidents. The document then details PIA's emergency response planning efforts, which help manage major accidents, ensure humanitarian assistance for victims and families, and maintain the airline's image. It provides information on PIA's emergency response center, field teams, training programs, and procedures for responding to different levels of accidents. The document emphasizes the importance of effective communication and humanitarian support for victims' families in times of crisis.
Vincent Cusano was a member of the band KISS and left in 1984. He sold his rights in the band's songs, excluding performance royalties, to Horipro Entertainment Group. Cusano later sued Horipro claiming he did not intend to sell the mechanical royalties. The document analyzes the requirements for a valid offer and concludes that Cusano had serious intent as the offeror to sell the mechanical royalties based on the language of the agreement. Therefore, the court should grant Horipro's motion.
Nguyen Tan Huy has objectives to become a politician in the Nha Trang City Government within 5 years after completing a high training for politicians and an MBA. Long term, he aims to work in the Planning and Investing Department of the Vietnamese Government. He has a Bachelor's degree in Entrepreneurship and a Designer Diploma, as well as experiences selling sneakers online, interning with an architecture university, and teaching English to children.
Piecewise Controller Design for Affine Fuzzy SystemsISA Interchange
This document presents a new method for designing piecewise state feedback controllers for affine fuzzy systems. The method uses dilated linear matrix inequalities (LMIs) to characterize the system in a way that separates the system matrix from the Lyapunov matrix. This allows the controller parameters to be independent of the Lyapunov matrix. The results provide less conservative LMI characterizations than existing methods and can be applied to more general systems. The method is also extended to H-infinity state feedback synthesis. Numerical examples demonstrate the effectiveness of the new approach.
Generations In The Community A New MarketplacePresentMark
This document discusses generational cohorts and the nonprofit sector. It notes that Baby Boomers, Generation X, and Millennials each have defining characteristics and events that shaped their perspectives. It also discusses that the nonprofit sector business models are often outdated and should focus on social entrepreneurism, impact investing, and engaging various generations as advocates and investors. The key to success is seen as social capital and innovation to create new value propositions for clients, donors, and communities.
The document discusses the benefits of exercise for mental health. It notes that regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise has also been shown to enhance self-esteem and serve as a healthy distraction from daily stressors.
The document discusses synergy and positivity in achieving common goals, stating that working together through positive synergy helps achieve desired results, while business and political leaders should avoid negative synergy or passivity. It also mentions an e-paper in Pakistan that recommends daily positivity for 10 minutes.
This document presents a new method for designing predictive controllers called predictive feedback control. It uses a single prediction of the process output J time intervals ahead to compute the future error. The predictive feedback controller then weights the last w predicted errors, combining predictive capacity with feedback information.
The key aspects of the method are:
1) It uses a single-step prediction of the process output J intervals ahead, rather than the full predicted trajectory, to compute the future error.
2) The predictive feedback controller weights the last w predicted errors, incorporating feedback information to improve disturbance rejection.
3) The prediction time J is a tuning parameter that relates to the closed-loop response settling time. Weighting past errors with
The document discusses how the person's notebook laptop broke when they fell asleep on it, cracking the monitor. They checked online and saw replacement Acer laptops costing 5,900 pesos. For their birthday, their brother bought them a high-performance Dell laptop that they find very comfortable.
The century's greatest contributions to control practiceISA Interchange
This survey asked 100 professionals in industry and academia to identify the most significant contributions to control practice in the 20th century. The top contributions were grouped into 5 categories: concepts and theory, techniques, process instrumentation, control/decision/communication instrumentation, and organizations. Feedback principles and frequency analysis were most important in concepts/theory. Classical advanced control techniques and multivariable model predictive control topped techniques. Pneumatic valves and differential pressure cells were leaders in process instrumentation. The survey provides an overview of control history and can guide educational and technical development efforts.
101 Tips for a Successful Automation Career Appendix CISA Interchange
This document contains four checklists related to automation components and measurements:
1. A checklist for control valve performance with 12 items to ensure proper valve sizing, installation, and performance.
2. A checklist for inline flowmeters with 26 items to ensure proper meter selection, installation, and performance.
3. A checklist for analyzing process control loops using trend charts, with 14 items to ensure the historian setup allows for effective loop analysis.
4. A checklist for pH measurement with 22 items to ensure proper sensor material selection and installation for accuracy under operating conditions.
Tuning PI controllers for stable processes with specifications on gain and ph...ISA Interchange
In industrial practice, controller designs are performed based on an approximate model of the actual process. It is essential to design a control system which will exhibit a robust performance because the physical systems can vary with operating conditions and time. Gain and phase margins are well known parameters for evaluating the robustness of a control system. This paper presents a tuning algorithm to design and tune PI controllers for stable processes with a small dead time while meeting specified gain and phase margins. Simulation examples are given to demonstrate that the proposed design method can result, in a closed-loop system, in better performances than existing design methods which are also based on user-specified gain and phase margins.
This document describes a novel auto-tuning method for cascade control systems. The method uses a simple relay feedback test to simultaneously identify both the inner and outer loop process model parameters. This allows established PID tuning rules to then be applied to tune both loops. The method is simpler than existing approaches and can be directly integrated into commercial auto-tuning systems. It is illustrated through examples to be effective and robust.
Slide Solution oferece serviços de criação de apresentações profissionais em PowerPoint para empresas e profissionais, com foco em design atraente e mensagem clara, liderado por Rafael Borges, que pode ser contatado por telefone ou e-mail.
This document discusses the emergency response planning of Pakistan International Airlines. It outlines PIA's focus on passenger safety and efforts to achieve high safety standards. It also acknowledges that no airline is immune to accidents. The document then details PIA's emergency response planning efforts, which help manage major accidents, ensure humanitarian assistance for victims and families, and maintain the airline's image. It provides information on PIA's emergency response center, field teams, training programs, and procedures for responding to different levels of accidents. The document emphasizes the importance of effective communication and humanitarian support for victims' families in times of crisis.
Vincent Cusano was a member of the band KISS and left in 1984. He sold his rights in the band's songs, excluding performance royalties, to Horipro Entertainment Group. Cusano later sued Horipro claiming he did not intend to sell the mechanical royalties. The document analyzes the requirements for a valid offer and concludes that Cusano had serious intent as the offeror to sell the mechanical royalties based on the language of the agreement. Therefore, the court should grant Horipro's motion.
Nguyen Tan Huy has objectives to become a politician in the Nha Trang City Government within 5 years after completing a high training for politicians and an MBA. Long term, he aims to work in the Planning and Investing Department of the Vietnamese Government. He has a Bachelor's degree in Entrepreneurship and a Designer Diploma, as well as experiences selling sneakers online, interning with an architecture university, and teaching English to children.
Piecewise Controller Design for Affine Fuzzy SystemsISA Interchange
This document presents a new method for designing piecewise state feedback controllers for affine fuzzy systems. The method uses dilated linear matrix inequalities (LMIs) to characterize the system in a way that separates the system matrix from the Lyapunov matrix. This allows the controller parameters to be independent of the Lyapunov matrix. The results provide less conservative LMI characterizations than existing methods and can be applied to more general systems. The method is also extended to H-infinity state feedback synthesis. Numerical examples demonstrate the effectiveness of the new approach.
Generations In The Community A New MarketplacePresentMark
This document discusses generational cohorts and the nonprofit sector. It notes that Baby Boomers, Generation X, and Millennials each have defining characteristics and events that shaped their perspectives. It also discusses that the nonprofit sector business models are often outdated and should focus on social entrepreneurism, impact investing, and engaging various generations as advocates and investors. The key to success is seen as social capital and innovation to create new value propositions for clients, donors, and communities.
The document discusses the benefits of exercise for mental health. It notes that regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise has also been shown to enhance self-esteem and serve as a healthy distraction from daily stressors.
The document discusses synergy and positivity in achieving common goals, stating that working together through positive synergy helps achieve desired results, while business and political leaders should avoid negative synergy or passivity. It also mentions an e-paper in Pakistan that recommends daily positivity for 10 minutes.
This document presents a new method for designing predictive controllers called predictive feedback control. It uses a single prediction of the process output J time intervals ahead to compute the future error. The predictive feedback controller then weights the last w predicted errors, combining predictive capacity with feedback information.
The key aspects of the method are:
1) It uses a single-step prediction of the process output J intervals ahead, rather than the full predicted trajectory, to compute the future error.
2) The predictive feedback controller weights the last w predicted errors, incorporating feedback information to improve disturbance rejection.
3) The prediction time J is a tuning parameter that relates to the closed-loop response settling time. Weighting past errors with
The document discusses how the person's notebook laptop broke when they fell asleep on it, cracking the monitor. They checked online and saw replacement Acer laptops costing 5,900 pesos. For their birthday, their brother bought them a high-performance Dell laptop that they find very comfortable.
The century's greatest contributions to control practiceISA Interchange
This survey asked 100 professionals in industry and academia to identify the most significant contributions to control practice in the 20th century. The top contributions were grouped into 5 categories: concepts and theory, techniques, process instrumentation, control/decision/communication instrumentation, and organizations. Feedback principles and frequency analysis were most important in concepts/theory. Classical advanced control techniques and multivariable model predictive control topped techniques. Pneumatic valves and differential pressure cells were leaders in process instrumentation. The survey provides an overview of control history and can guide educational and technical development efforts.
101 Tips for a Successful Automation Career Appendix CISA Interchange
This document contains four checklists related to automation components and measurements:
1. A checklist for control valve performance with 12 items to ensure proper valve sizing, installation, and performance.
2. A checklist for inline flowmeters with 26 items to ensure proper meter selection, installation, and performance.
3. A checklist for analyzing process control loops using trend charts, with 14 items to ensure the historian setup allows for effective loop analysis.
4. A checklist for pH measurement with 22 items to ensure proper sensor material selection and installation for accuracy under operating conditions.
Tuning PI controllers for stable processes with specifications on gain and ph...ISA Interchange
In industrial practice, controller designs are performed based on an approximate model of the actual process. It is essential to design a control system which will exhibit a robust performance because the physical systems can vary with operating conditions and time. Gain and phase margins are well known parameters for evaluating the robustness of a control system. This paper presents a tuning algorithm to design and tune PI controllers for stable processes with a small dead time while meeting specified gain and phase margins. Simulation examples are given to demonstrate that the proposed design method can result, in a closed-loop system, in better performances than existing design methods which are also based on user-specified gain and phase margins.
This document describes a novel auto-tuning method for cascade control systems. The method uses a simple relay feedback test to simultaneously identify both the inner and outer loop process model parameters. This allows established PID tuning rules to then be applied to tune both loops. The method is simpler than existing approaches and can be directly integrated into commercial auto-tuning systems. It is illustrated through examples to be effective and robust.
A fuzzy model based adaptive pid controller design for nonlinear and uncertai...ISA Interchange
We develop a novel adaptive tuning method for classical proportional–integral–derivative (PID)
controller to control nonlinear processes to adjust PID gains, a problem which is very difficult to
overcome in the classical PID controllers. By incorporating classical PID control, which is well-known in
industry, to the control of nonlinear processes, we introduce a method which can readily be used by the
industry. In this method, controller design does not require a first principal model of the process which is
usually very difficult to obtain. Instead, it depends on a fuzzy process model which is constructed from
the measured input–output data of the process. A soft limiter is used to impose industrial limits on the
control input. The performance of the system is successfully tested on the bioreactor, a highly nonlinear
process involving instabilities. Several tests showed the method's success in tracking, robustness to noise,
and adaptation properties. We as well compared our system's performance to those of a plant with
altered parameters with measurement noise, and obtained less ringing and better tracking. To conclude,
we present a novel adaptive control method that is built upon the well-known PID architecture that
successfully controls highly nonlinear industrial processes, even under conditions such as strong
parameter variations, noise, and instabilities
Pid controller tuning using fuzzy logicRoni Roshni
This document provides an overview of tuning a PID controller with fuzzy logic. It introduces fuzzy logic and discusses how it can be applied to PID tuning. Specifically, it discusses using fuzzy set-point weighting to tune the PID controller by determining the proportional weighting factor b(t) using a fuzzy inference system based on the error e(t) and change in error. It also discusses traditional Ziegler-Nichols tuning and compares the performance of fixed versus fuzzy set-point weighting tuning. The conclusion is that fuzzy logic provides benefits like balancing rise time and overshoot to obtain better performance than traditional methods.
Design of PI controllers for achieving time and frequency domain specificatio...ISA Interchange
This document presents a new method for designing PI controllers that achieve both desired frequency domain and time domain specifications simultaneously. The method involves two main steps:
1. Computing the global and local stability regions of PI controller parameters that stabilize the system using a boundary locus approach.
2. Using the coefficient diagram method to design PI controllers within the local stability region such that the closed-loop system meets specifications for time domain measures like overshoot and settling time.
This provides a graphical tool called a frequency and time domain performances map that shows which PI controller parameters satisfy both the frequency and time domain performance criteria. Examples are given to demonstrate the benefits of the proposed design method.
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.
Design of controller using variable transformations for a nonlinear process w...ISA Interchange
This document presents a method for designing a globally linearized controller (GLC) for a first-order nonlinear system with dead time. Two methods are proposed - one based on Smith prediction in the transformed domain, and the other based on Newton's extrapolation method. The GLC design approach transforms the nonlinear system into a linear system through variable transformations. This allows a PI controller to be designed for the linearized system. The performance of the GLC is evaluated through simulations and experiments on a conical tank level process and is found to outperform conventional PI and Smith PI controllers.
Improving Structural Limitations of Pid Controller For Unstable ProcessesIJERA Editor
PID controllers have structural limitations which make it impossible for a good closed-loop performance to be achieved. A step response with high overshoot and oscillations always results. In controlling processes with resonances, integrators and unstable transfer functions, the PI-PD controller provides a satisfactory closed-loop performance. In this paper, a simple approach to extracting parameters of a PI-PD controller from parameters of the conventional PID controller is presented so that a good closed-loop system performance is achieved. Simulated results from this formation are carried out to show the efficacy of the technique proposed.
Performance based Comparison between Various Z-N Tuninng PID and Fuzzy Logic ...ijsc
The objective of this paper is to compare the time specification performance between conventional controller and Fuzzy Logic controller in position control system of a DC motor. The scope of this research is to apply direct control technique in position control system. Two types of controller namely PID and fuzzy logic PID controller will be used to control the output response. This paper was written to reflect on the work done on the implementation of a fuzzy logic PID controller. The fuzzy controller was used to control the position of a motor which can be considered for a general basis in any project design containing logic control. Motor parameters were taken from a datasheet with respect to a real motor and a simulated model was developed using Matlab Simulink Toolbox. The fuzzy control was also designed using the Fuzzy Control Toolbox provided within Matlab, with each rule consisting of fuzzy sets conditioned to provide appropriate response times with regards to the limitations of our chosen motor. The Fuzzy Inference Engine chosen for our control was the Mamdani Minimum Inference engine. The results of the control provided response times suitable for our application.
Performance Based Comparison Between Various Z-N Tuninng PID And Fuzzy Logic ...ijsc
The objective of this paper is to compare the time specification performance between conventional
controller and Fuzzy Logic controller in position control system of a DC motor. The scope of this research
is to apply direct control technique in position control system. Two types of controller namely PID and
fuzzy logic PID controller will be used to control the output response. This paper was written to reflect on
the work done on the implementation of a fuzzy logic PID controller. The fuzzy controller was used to
control the position of a motor which can be considered for a general basis in any project design
containing logic control. Motor parameters were taken from a datasheet with respect to a real motor and a
simulated model was developed using Matlab Simulink Toolbox. The fuzzy control was also designed using
the Fuzzy Control Toolbox provided within Matlab, with each rule consisting of fuzzy sets conditioned to
provide appropriate response times with regards to the limitations of our chosen motor. The Fuzzy
Inference Engine chosen for our control was the Mamdani Minimum Inference engine. The results of the
control provided response times suitable for our application.
This document discusses fuzzy logic control for liquid flow systems. It begins by introducing the basics of liquid flow measurement and control, noting that factors like viscosity, density, and pipe friction affect flow rate. It then discusses the main components of a fuzzy logic controller, including fuzzification, a rule base and inference engine, and defuzzification. The document formulates the problem of measuring and controlling liquid flow rates given various disturbances. It proposes using a fuzzy logic controller instead of a conventional PID controller due to the former's robustness with imprecise models. Finally, it mentions that MATLAB will be used to implement and test the fuzzy logic controller.
Autotuning of a new PI-PD Smith predictor based on time domain specificationsISA Interchange
This document presents a new PI-PD Smith predictor controller configuration for processes with long time delays. The controller aims to improve set point tracking and disturbance rejection performance over existing Smith predictor designs. It does this by adding additional controllers - a PI controller (Gc1(s)) for set point tracking, a PD controller (Gc2(s)) to stabilize unstable processes, and a disturbance rejection controller (Gd(s)). Autotuning formulas are developed to identify plant models from relay feedback tests and design the PI-PD controller parameters based on specifications for damping ratio, natural frequency, and settling time. Examples are given to show the effectiveness of the proposed method.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
This document presents a comparison of PID and fuzzy PID controllers for position control of a DC motor. It first describes the modeling of a DC motor transfer function. It then provides details on designing a PID controller using Ziegler-Nichols tuning methods. A fuzzy PID controller is also designed using triangular membership functions for error and change in error inputs. Simulation results in MATLAB/Simulink show that the fuzzy PID controller provides better tracking of setpoint changes with less overshoot compared to the ZN-tuned PID controller. The fuzzy PID controller therefore demonstrates better performance for position control of DC motors.
Adaptive predictive functional control of a class of nonlinear systemsISA Interchange
This document summarizes an adaptive predictive functional control algorithm for nonlinear systems.
1) It uses a concept called pseudo-partial derivative (PPD) to dynamically linearize the nonlinear system model based on input-output data.
2) It predicts future outputs using an internal model based on the estimated PPD.
3) Predictive functional control (PFC) is used to design the control algorithm, where only two coincidence points are used to calculate the manipulated variable.
4) The proposed algorithm provides bounded inputs/outputs and setpoint tracking without steady-state error using a simple structure that is easy to tune for real-time control of nonlinear systems.
Fuzzy controlled mine drainage system based on embedded systemIRJET Journal
This document proposes a fuzzy logic controlled mine drainage system based on an embedded system. Mines require proper drainage to improve stability, safety, and prevent equipment corrosion, but the variables involved like water levels and flow rates are unpredictable and non-linear, making an accurate empirical model difficult to design. The proposed system combines fuzzy logic control with an embedded system. Fuzzy logic handles the uncertainties while the embedded system provides better control, flexibility, compactness and user-friendliness. Sensors monitor water levels, flow rates, temperature, humidity and pressure, sending data to an operator. A fuzzy logic controller uses the sensor data and fuzzy rules to determine the number of pumps to operate, providing improved drainage control over traditional methods.
Optimization of Fuzzy Logic controller for Luo Converter using Genetic Algor...IRJET Journal
This document summarizes research on optimizing a fuzzy logic controller for a Luo converter using a genetic algorithm. A fuzzy logic controller was designed for the Luo converter but its parameters were determined through trial and error. The document proposes using a genetic algorithm to optimize the fuzzy logic controller's rules, membership functions, and scaling gains in order to improve the controller's performance for the Luo converter. Simulation results showed that the genetic algorithm-optimized fuzzy logic controller provided faster response, better transient performance, and more robustness to variations compared to the original fuzzy logic controller.
An intelligent hybrid control for paper machine systemeSAT Journals
Abstract
The aim of this paper is to present an intelligent hybrid controller for a paper machine system with ash content and dry weight as
outputs and filler valve and thick stock valve as inputs. Simulation studies have been carried out to check the robust performance
(10% increase in each process gain, 10% increase in each time delay, and 10% decrease in each time constant) of the system. The
improvement in the performance of proposed hybrid controller is compared with the Adaptive-neuro- fuzzy controller and Dahlin
controller and evaluated in terms of ISE.
Index Terms: Dahlin, Neuro-fuzzy, Hybrid and Robust
Model-based Approach of Controller Design for a FOPTD System and its Real Tim...IOSR Journals
The document summarizes a study on model-based controller design for a first-order plus time delay (FOPTD) system. The study identifies the process model of a level control system using process reaction curve methods. Various tuning rules for internal model control-proportional integral derivative (IMC-PID) controllers from literature are applied to the system, including rules from Rivera, Chien, Lee, Skogestad, and Panda. The performance of each controller is evaluated based on rise time, settling time, percentage overshoot, integral absolute error, and integral of time multiplied absolute error. The study finds that the Panda tuning rule has the smallest percentage overshoot and integral absolute error, while the Chien rule has
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
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2. 38 Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49
In this section, we will discuss the first stage of
our non-PID controller design. It is well known
that the best achievable control performance for a
process is limited by its nonminimum phase nature
͓4͔. To figure out the achievable performance, the
Fig. 1. Unity feedback control system. ˆ
process model G is factorized into
half plane zeros and time delays. The ideal con-
troller derived from the objective closed-loop
G ͑ s ͒ ϭG mp͑ s ͒
ˆ ˆ ͫ͟ m
iϭ1
ͬ
͑ z i Ϫs ͒ e Ϫ s , ͑2͒
transfer function turns out to be very complex.
ˆ
where G mp represents the stable and minimum
Model reduction with a recursive least squares al-
gorithm is thus applied to fit it into a much simpler ˆ
phase part of G . The nonminimum phase part of
form. The order of the controller is determined as G consists of nonminimum phase zeros ͟ iϭ1 ( z i
ˆ m
the lowest so as to make the controller of the least Ϫs ) and the time delay e Ϫ s . As nonminimum
complexity and yet achieve a specified approxima- phase zeros and time delays are inherent charac-
tion accuracy. Simulation results are provided to teristics of a process and cannot be altered by any
show that the proposed design is applicable to a feedback control, the achievable closed-loop trans-
wide range of complex processes with high perfor- fer function H is formulated as
ͫ͟ ͩ ͪͬ
mance where PID controllers are inadequate or
m
limited in performance. z i Ϫs
H͑ s ͒ϭ e Ϫs f ͑ s ͒, ͑3͒
The paper is organized as follows. Specification iϭ1 z i ϩs
of our desired objective loop performance is de-
tailed in Section 2. In Section 3, the model reduc- ˆ
that is, the nonminimum phase part of G has to
tion method employed in our proposed design is remain in H without any change. The f in Eq. ͑3͒
presented. Then the overall tuning procedure is usually has the format
summarized in Section 4 and simulation examples
are provided in Section 5. In Section 6, stability of f ͑ s ͒ ϭ f 1͑ s ͒ f 2͑ s ͒ , ͑4͒
the proposed method is analyzed. Finally, some
where
concluding remarks are made in Section 7.
1
ͩ ͪ
2. Determination of the objective loop f 1͑ s ͒ ϭ ␦ ͑5͒
1
performance sϩ1
Nn
Consider the conventional unity feedback con- is a filter to provide necessary high frequency gain
ˆ
trol system in Fig. 1, where G is a model of the reduction ͑or roll-off͒ rate and
given stable plant G to be controlled and K the
controller. Our proposed method for non-PID con- 2
n
f 2͑ s ͒ ϭ ͑6͒
troller design consists of the following three s 2 ϩ2 n sϩ 2
n
stages. First, the best achievable objective transfer
function H for the closed-loop system is derived is a standard second-order rational function that
from the process dynamic characteristics. Next, reflects typical performance requirements such as
the ideal controller K, which might be complex overshoot and settling time.
and not realizable, is obtained from The filter order ␦ in Eq. ͑5͒ determines the roll-
off rate of the system at high frequencies. Let
ˆ
GK ( G ) be the relative degree of G . ␦ is taken as
ˆ ˆ
Hϭ . ͑1͒
ˆ
1ϩG K ␦ ϭmax͕ 0, ͑ G ͒ Ϫ2 ͖
ˆ ͑7͒
ˆ
At the last stage, its approximation K , a simple to ensure that the resultant K is proper and physi-
rational function, is found through model reduc- cally realizable. N in Eq. ͑5͒ is a parameter that
tion. measures how much faster the roll-off rate of f 1 is
3. Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49 39
relative to that of f 2 . To provide necessary high- b
frequency gain reduction without changing the nϷ ͑11͒
working frequency range characteristics, N is cho- ͓ ͱ͑ 2 2 Ϫ1 ͒ 2 ϩ1Ϫ ͑ 2 2 Ϫ1 ͔͒ 1/2
sen in the range of 10–20. ignoring the effect of the f 1 .
The damping ratio is usually set according to
the overshoot specification on the closed loop step
response. It is generally acceptable to have the de- 3. Model reduction
fault value of as 0.707. n serves as a measure
of the closed-loop response speed, and is related to Once a suitable H is determined by the proce-
the closed-loop bandwidth, which is essentially dure described in the preceding section, the ideal
limited by the nonminimum phase part of the pro- controller can then be calculated from Eq. ͑1͒ as
cess. One knows that the response tends to be
H
heavily oscillatory if n is too large while a slug- Kϭ . ͑12͒
gish response results if n is too small. A suitable ͑ 1ϪH ͒ G
ˆ
n is found as follows.
It is shown by Astrom ͓4͔ that the gain crossover This controller is usually of a highly compli-
frequency g has to meet cated form and difficult to implement. Hence
model reduction is employed to find a controller in
ͩ ͪ
m
the least complex form and yet achieve the speci-
z iϪ j g fied approximation accuracy. A number of meth-
͚ arctan
iϭ1 z iϩ j g
Ϫ g уϪ ϩ m Ϫn g /2, ods for rational approximation are surveyed by
͑8͒ Pintelon et al. ͓6͔. A recursive least squares ͑RLS͒
algorithm is suitable for our application and is
briefly described as follows. The problem at hand
where the m is the required phase margin and n g
is to find a nth-order rational function approxima-
is the slope of the open loop gain at the crossover
tion:
frequency. It is generally accepted that a desired
shape for the open-loop gain ͉ L ( j ) ͉ b n s n ϩb nϪ1 s nϪ1 ϩ¯ϩb 1 sϩb 0
͉ G ( j ) K ( j ) ͉ should typically have a slope of
ˆ Kϭ
ˆ ͑13͒
s n ϩa nϪ1 s nϪ1 ϩ¯ϩa 1 s
about Ϫ1 ͑i.e., n g ϭϪ1 ͒ around the crossover fre-
quency, with preferably steeper slopes before and with an integrator such that the approximation er-
after the crossover. ror
Equation ͑8͒ thus reduces to M
J ͚ ͉ W ͑ j i ͒ „K ͑ j i ͒ ϪK ͑ j i ͒ …͉ 2
ˆ ͑14͒
ͩ ͪ
m iϭ1
z iϪ j g
͚
iϭ1
arctan
z iϩ j g
Ϫ g уϪ /2ϩ m . is minimized, where the original function K as
͑9͒ well as the weighting W are given and M is the
number of frequency points to be used in the al-
gorithm. The cost function J is rewritten as
The selection of m reflects the control system
ͯ
M
robustness to the process uncertainty ͓4͔. Typical W͑ j i͒
values for m could be /6, /4, and /3, respec- J͑k͒ ͚
iϭ1 ͑ j i ͒ n ϩa ͑ kϪ1 ͒ ͑ j i ͒ nϪ1 ϩ¯ϩa ͑ kϪ1 ͒ ͑ j i ͒
nϪ1 1
tively. Our study shows that m ϭ /6 is good
enough to determine the g . The bandwidth b of ϫ ͕ ͓ b ͑ k ͒ ͑ j i ͒ n ϩb ͑ k ͒ ͑ j i ͒ nϪ1 ϩ¯ϩb ͑ k ͒ ͑ j i ͒
n nϪ1 1
the closed-loop can be estimated ͓5͔ from g by
ϩb ͑ k ͒ ͔ ϪK ͑ j i ͓͒͑ j i ͒ n ϩa ͑ k ͒ ͑ j i ͒ nϪ1 ϩ¯
0 nϪ1
bϭ  g ,
ͯ
͑10͒ 2
ϩa ͑ k ͒ ͑ j i ͔͒ ͖ ,
1 ͑15͒
where  ͓ 1,2͔ . In our design, the  is selected as
2 to achieve a larger n and thus a quicker re- where k denotes the index for the kth recursion in
sponse. It follows from the definition of b for H the iterative weighted linear least squares method
in Eq. ͑3͒ that and
4. 40 Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49
W͑ j i͒ Table 1
W͑ j ͒ϭ
¯ Summary of simulation results.
͑ ji͒ n
ϩa nϪ1 ͑ j i ͒ nϪ1 ϩ¯ϩa ͑ kϪ1 ͒ ͑ j i ͒
͑ kϪ1 ͒
1
Plant Method M p ͑%͒ ts
operates as a weighting function derived from the
parameters generated in the last recursion. It is eϪ5s Proposed 4.2 20.5
Gϭ Wang-PID 4.3 36.1
͑ s2ϩsϩ1͒͑sϩ2͒
recommended ͓6͔ that the W is chosen as 1 and
¯
Zhuang-PID 20.1 40.3
standard LS is applied in each iteration. On con-
1.2e Ϫ10s Proposed 5.0 42.2
vergence, the resultant parameters will form one Gϭ Wang-PID 11.0 67.3
͑ 5sϩ1 ͒͑ 2.5sϩ1 ͒
solution that minimizes the cost function in Eq. Zhuang-PID 20.1 62.9
͑14͒. To derive the relevant recursive equations, ͑ Ϫ2sϩ1 ͒ e Ϫ4s Proposed 0.5 45.9
rearrange Eq. ͑15͒ to yield the matrix equation Gϭ
͑ 5sϩ1 ͒͑ 3sϩ1 ͒ Wang-PID 4.1 47.0
y kϭ k k , Zhuang-PID 20.1 47.1
where
y k ϭϪK ͑ s ͒͑ j i ͒ n , fied phase margin m and approximation accuracy
⑀.
k ϭ ͓ a ͑ k ͒ ¯a ͑ k ͒ b ͑ k ͒ b ͑ k ͒ ¯b ͑ k ͒ ͔ T ,
nϪ1 1 n nϪ1 0 • Step 1. Find out all the nonminimum phase zeros
z i , iϭ1,2,...,m, and the time delay of the plant
k ϭ ͓ K ͑ s ͒͑ j i ͒ nϪ1 ¯K ͑ s ͒͑ j i ͒ Ϫ ͑ j i ͒ n ˆ
model G .
Ϫ ͑ j i ͒ nϪ1 ¯Ϫ ͑ j i ͒ Ϫ1 ͔ . • Step 2. Obtain ␦ from Eq. ͑7͒ and choose N in
10–20.
The fitting range used in RLS is chosen as ⍀ • Step 3. Determine g from Eq. ͑9͒ and obtain n
( 0.01 g , g ) with M usually taken as 50–100 from Eqs. ͑10͒ and ͑11͒.
and i logarithmically equally spaced. This range • Step 4. Form H from Eq. ͑3͒, then evaluate K
is the most important for closed-loop stability and from Eq. ͑1͒.
robustness. • Step 5. Obtain a controller in Eq. ͑13͒ with the
Once a solution to Eq. ͑13͒ is found, the follow- RLS method in Section 3.
ing criterion is used to validate the solution:
Eϭmax
ͯ K ͑ j ͒ ϪK ͑ j ͒
ˆ
K͑ j ͒
ͯ
р⑀, ⍀,
5. Examples
In this section, simulation results for several
͑16͒ typical examples are provided to demonstrate the
where ⑀ is the user-specified fitting error threshold. effectiveness of our non-PID controller tuning al-
In our design, simulation results show that ⑀ gorithm. The simulation is done under the perfect
ϭ5% is good enough for control performance. To model matching condition, i.e., G ϭG ͑model
ˆ
find a model of the lowest order and yet meet Eq. mismatch will be considered in Section 6͒. Com-
͑16͒, we start with a model order nϭ2 until the parisons are made with two PID tuning methods.
smallest integer n is reached such that Eq. ͑16͒ is One is the technique proposed by Wang et al. ͓7͔
satisfied. and it is based on fitting the process frequency
response to a SOPDT structure. The other is the
modified Ziegler-Nichols method using the opti-
4. Tuning procedure mum ISTE criterion ͓8͔. For ease of presentation,
these two PID tuning methods will be referred to
In this section, the overall tuning procedure is as Wang-PID method and Zhuang-PID method, re-
summarized as follows. spectively, in this paper. The dynamic perfor-
mance indices of the closed-loop step responses in
4.1. Overall tuning procedure terms of overshoot in percentage ( M p ) and set-
tling time ͑to 1%͒ in seconds ( t s ) are shown in
ˆ
Given the model G ( s ) of a stable process G ( s ) , Table 1. In Figs. 2– 4, the system responses for the
seek a non-PID controller K ( s ) to meet the speci- proposed method, Wang-PID method, and
5. Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49 41
Fig. 2. Control performance for e Ϫ5s /(s 2 ϩsϩ1)(sϩ2) ͑ , step input; , proposed controller; ¯ , Wang-PID; -•-•-,
Zhuang-PID͒.
Zhuang-PID method are exhibited by solid, dot- 1.8454s 5 ϩ5.1757s 4 ϩ8.5743s 3
ted, and dashed-dotted lines, respectively. ϩ8.1018s 2 ϩ4.6103sϩ1.2594
Kϭ
ˆ , ͑17͒
s 5 ϩ7.0064s 4 ϩ12.8928s 3
5.1. Example 1
ϩ7.0838s 2 ϩ5.2692s
Consider the following oscillatory and high-
order plant: with the fitting error Eϭ0.22% less than ⑀
ϭ5%. Our design is completed. The PID control-
e Ϫ5s ler obtained by the Wang-PID method is
Gϭ .
͑ s 2 ϩsϩ1 ͒͑ sϩ2 ͒
0.1851
The process is of minimum phase and no z i ex- K PIDϭ0.2031ϩ ϩ0.1860s,
ists. The dead time is 5. The relative degree is 3, s
yielding ␦ ϭ1 from Eq. ͑7͒. N is taken as 20. With
default ϭ0.707, it follows that g ϭ0.2094 from while the Zhuang-PID method generates
Eq. ͑9͒ and n ϭ0.4189 from Eqs. ͑10͒ and ͑11͒.
H ( s ) is thus given by 0.2943
K PIDϭ0.9322ϩ ϩ1.7504s.
0.1755e Ϫ5s s
H͑ s ͒ϭ .
͑ s 2 ϩ0.5924sϩ0.1755͒͑ 0.1194sϩ1 ͒
We observe from the step responses plotted in
The ideal controller K is obtained from Eq. ͑1͒ Fig. 2 that the Zhuang-PID method does not pro-
and model reduction is invoked to give its ap- vide good tuning because the step response is
ˆ
proximation K as rather oscillatory and the settling time is relatively
6. 42 Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49
Fig. 3. Control performance for 1.2e Ϫ10s /(5sϩ1)(2.5sϩ1) ͑ , step input; , proposed controller; ¯ , Wang-PID;
-•-•-, Zhuang-PID͒.
long. The Wang-PID method results in a sluggish The Zhuang-PID method generates a PID control-
response. Our proposed design method achieves ler
the best performance.
0.0614
5.2. Example 2 K PIDϭ0.6482ϩ ϩ2.7242s.
s
Consider a plant with a long dead time used in The closed-loop step responses are given in Fig.
Zhuang and Atherton ͓8͔: 3. The controller from our method improves the
1.2e Ϫ10s closed-loop system performance significantly in
Gϭ . terms of overshoot reduction and response speed.
͑ 5sϩ1 ͒͑ 2.5sϩ1 ͒
It follows from our design procedure that the 5.3. Example 3
lowest order controller is
Consider a typical nonminimum phase plant
0.7916s 3 ϩ0.5488s 2 ϩ0.1351sϩ0.0116 from Chien ͓9͔:
Kϭ
ˆ ,
s 3 ϩ0.3999s 2 ϩ0.2135s
͑18͒ ͑ Ϫ2sϩ1 ͒ e Ϫ4s
Gϭ .
with the fitting error Eϭ0.02% less than ⑀ ͑ 5sϩ1 ͒͑ 3sϩ1 ͒
ϭ5%. The Wang-PID method gives a PID con-
troller Note that this plant has a right half plane zero at
0.5. Since our proposed design method has already
0.0451 taken into account this inherent characteristic of a
K PIDϭ0.3020ϩ ϩ0.4531s.
s plant, it succeeds in producing a controller
7. Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49 43
Fig. 4. Control performance for (Ϫ2sϩ1)e Ϫ4s /(5sϩ1)(3sϩ1) ͑ , step input; , proposed controller; ¯ , Wang-
PID; -•-•-, Zhuang-PID͒.
Kϭ
ˆ
0.5748s 2 ϩ0.1426sϩ0.0098
s 2 ϩ0.1338s
,
with the fitting error Eϭ0.18% less than ⑀
ϭ5%. The Wang-PID method gives a PID con-
troller
Gϭ
ͫ 12.8e Ϫs
16.7sϩ1
6.60e Ϫ7s
10.9sϩ1
Ϫ18.9e Ϫ3s
21.0sϩ1
Ϫ19.4e Ϫ3s
14.4sϩ1
ͬ .
The BLT method ͓11͔ gives a multiloop PI con-
0.0630 troller:
K PIDϭ0.3979ϩ ϩ0.0701s.
ͫ ͬ
s
k 1͑ s ͒ 0
The Zhuang-PID method produces a PID control- K͑ s ͒ϭ
0 k 2͑ s ͒
ͫ ͬ
ler
K PIDϭ0.9485ϩ
0.1075
ϩ2.8263s.
0.375 1ϩ ͩ 1
8.29s ͪ 0
ͩ ͪ
s ϭ .
1
0 Ϫ0.075 1ϩ
The step responses are given in Fig. 4. For this 23.6s
example, our proposed method gives almost no
overshoot and achieves the best performance. Now let k 1 remain as given by the BLT method,
but apply our method to design a new k 2 for the
5.4. Example 4 second loop. The equivalent plant for the second
loop with the first loop closed is obtained ͓5͔ as
Complex dynamics often come from multivari-
g 21g 12
able interactions. Consider the well-known Wood/ g 2 ϭg 22Ϫ Ϫ1
Berry binary distillation column plant ͓10͔: k 1 ϩg 11
8. 44 Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49
Fig. 5. Nyquist curve of g 2 in Example 4.
whose Nyquist curve is shown in Fig. 5. It is ob- with the fitting error Eϭ2.62% less than ⑀
vious that g 2 is not in the FOPDT or SOPDT ϭ5%. Hence, the corresponding multiloop con-
ͫ ͬ
form. Hence it is very difficult to control g 2 by a troller is formed as
ͩ ͪ
PID controller, as can be seen from Fig. 6. To
figure out what the equivalent nonminimum phase 1
0.375 1ϩ 0
zeros and dead time are in this g 2 , we first apply 8.29s
model reduction to get its rational plus dead time 0.0559s3ϩ0.0192s2
K͑ s ͒ϭ .
approximation g 2 as
ˆ ϩ0.0029sϩ0.0002
0 Ϫ
s 3 ϩ0.1512s 2
ϩ0.0144s
͑ 2.0197s 4 ϩ1.0793s 3 ϩ1.1030s 2
ϩ0.0732sϩ0.0054)e Ϫ3.1s The step response of the resultant feedback sys-
g 2 ϭϪ
ˆ , tem is shown in Fig. 6 with solid lines. The step
s 5 ϩ1.2517s 4 ϩ0.6339s 3
ϩ0.1570s 2 ϩ0.0107sϩ0.0006 response using the BLT method is given in
dashed-dotted lines. It is observed that the pro-
posed method achieves much better loop perfor-
from which we can determine the objective mance for this second loop. Since the second loop
closed-loop transfer function H ( s ) as usual. The is slower than the first one in the BLT design and
proposed controller design procedure then gives the slow loop dominates the system performance,
its improvement is more desirable and beneficial.
Our simulation shows that within the approxi-
0.0559s 3 ϩ0.0192s 2 ϩ0.0029sϩ0.0002 mation error bound ⑀ ϭ5%, the resultant closed-
K ϭϪ
ˆ
s 3 ϩ0.1512s 2 ϩ0.0144s ˆ
loop response achieved by K is so close to our
9. Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49 45
Fig. 6. Control performance for Example 4 ͑ , step input; , proposed controller; -•-•-, BLT͒.
specified objective closed-loop step response that 6. Stability analysis
one can hardly distinguish them from the graph.
Hence, no curves for the ideal loop are shown in In this section, we will give a detailed stability
Figs. 2– 4 and 6. The proposed design method re- analysis of our single-loop control system. Both
ally achieves better closed-loop responses with ˆ
nominal stability ( GϭG ) and robust stability ( G
much smaller overshoot and shorter settling times, ˆ
G ) will be discussed.
thus shows its superiority over PID control for
complex processes. The proposed method is a First, consider the case without model uncer-
simple and effective way to design high perfor- ˆ
tainty, i.e., GϭG . As our proposed method makes
mance controllers. ˆ
an approximation K to the ideal controller K, the
10. 46 Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49
Fig. 7. Block diagram of nominal system.
nominal single-loop system is actually as shown in Fig. 8. Block diagram of system with process uncertainty.
Fig. 7͑a͒, where K ϭK ( 1ϩ⌬ K ) . The system in
ˆ
Fig. 7͑a͒ can be redrawn ͓12͔ into Fig. 7͑b͒, where
Recall that in the proposed algorithm, the ap-
ˆ
GK
QϭϪHϭϪ . proximation accuracy is required to meet Eq. ͑16͒:
ͯ ͯ
ˆ
1ϩG K
K ͑ j ͒ ϪK ͑ j ͒
ˆ
Eϭmax р⑀, ͓ 0, g ͔ ,
It can be easily seen that Q is stable since H is K͑ j ͒
ˆ
stable. With the standard assumption that K has
the same number of unstable poles as K, the nomi- where ⑀ is usually specified as 5%. The resultant
nal single-loop feedback system is stable ͓12͔ if controller K then satisfies Eq. ͑21͒ with big mar-
ˆ
and only if gin and the nominal stability of the designed
single-loop system is thus expected.
ʈ H ͑ j ͒ ⌬ K ͑ j ͒ ʈ ϱ Ͻ1. ͑19͒
Under the situation where the model does not
From Eq. ͑3͒, it is easy to note that ʈ H ʈ ϭ ʈ f ʈ represent the plant exactly, nominal stability is not
ϭ ʈ f 1 f 2 ʈ . Hence Eq. ͑19͒ is equivalent to sufficient and robust stability of the closed-loop
system has to be considered. The single-loop sys-
ʈ f 1 ͑ j ͒ f 2 ͑ j ͒ ⌬ K ͑ j ͒ ʈ ϱ Ͻ1. ͑20͒ tem with model uncertainty is shown in Fig. 8͑a͒,
The term f 1 provides high frequency roll-off where ͉ ⌬ K ͉ р ␦ K ( ) and ͉ ⌬ G ͉ р ␦ G ( ) . It can be
rate and ʈ f 1 ( j ) ʈ ϱ р1 for all . Also note redrawn into the standard form in Fig. 8͑b͒, where
ʈ f 2 ( j ) ʈ ϱ р1 for all with chosen as 0.707 ⌬ is the normalized uncertainty ⌬ ϭdiag͕⌬K ,⌬G͖
˜ ˜ ˜ ˜
or below. Therefore, ͉ f ( j ) ͉ decays fast for with ͉ ⌬ K ͉ р1 and ͉ ⌬ G ͉ р1. The transfer function
˜ ˜
у g and we can assume that Eq. ͑20͒ is true matrix between z and x has no uncertainty and is
for у g . It follows that we now need to given by
ͬͫ ͬ
check Eq. ͑20͒ only for the working frequency
range ͓ 0, g ͔ . Note that ʈ f 1 ( j ) f 2 ( j ) ʈ ϱ
р ʈ f 1 ( j ) ʈ ϱ ʈ f 2 ( j ) ʈ ϱ р1 for all , hence the
Qϭ ͫ ␦K
0
0
␦G
ϪG K
ˆ
ˆ
G
ϪK
ϪG K
ˆ
͑ 1ϩG K ͒ Ϫ1
ˆ
ͫ ͬ
nominal closed loop is stable if
ͯ ͯ ͫ ͬ
H
K ͑ j ͒ ϪK ͑ j ͒
ˆ ␦K 0 ϪH Ϫ
͉ ⌬ K͉ ϭ р1, ͓ 0, g ͔ . ϭ ˆ
G ,
K͑ j ͒ 0 ␦G
͑21͒ G ͑ 1ϪH ͒
ˆ ϪH
11. Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49 47
Fig. 9. Performance robustness for e Ϫ5s /(s 2 ϩsϩ1)(sϩ2) ͑ , step input; , nominal performance; -•-•-, perfor-
mance after 20% gain change͒.
whose stability is guaranteed by our selection of The robust stability condition ͑22͒ becomes
H. It follows from the stability robustness theorem
͓13͔ that the uncertain feedback system remains ␦ K ͉ H ͉ 2 ϩ ␦ G ͉ H ͉ 2 ϩ2 ␦ K ␦ G ͉ ͑ 1ϪH ͒ H ͉
2 2
stable for all ⌬ ϭdiag͕⌬K ,⌬G͖ if and only if
˜ ˜ ˜
ʈ Q ʈ Ͻ1, ͑22͒ ϩ ͱ␦ ͓ K ͉ H ͉ ϩ ␦ G ͉ H ͉ ϩ2␦ K ␦ G ͉ ͑ 1ϪH ͒ H ͉ ͔
2 2 2 2
Ϫ4 ␦ K ␦ G ͉ H ͉ 2
2 2
2
where ʈ Q ʈ ϭsup „Q ( j ) … and ͑•͒ is the
р2, ᭙. ͑24͒
structured singular value with respect to ⌬ . In our
˜
case, the structured singular value „Q ( j ) … can Since 4 ␦ K ␦ G ͉ H ͉ 2 у0, ᭙, and ͉ 1ϪH ͉ р1
2 2
be calculated by ϩ ͉ H ͉ р2, Eq. ͑24͒ is satisfied if
„Q ͑ j ͒ …ϭ ͑ DQD Ϫ1 ͒ ϭinf ¯ ͑ DQD Ϫ1 ͒ ,
D ␦ K ͉ H ͉ 2 ϩ ␦ G ͉ H ͉ 2 ϩ4 ␦ K ␦ G ͉ H ͉ р1,
2 2
᭙,
͑25͒
where Dϭdiag͕d1 ,d2͖, d 1 ,d 2 Ͼ0, and ¯ ( • ) repre-
sents the largest singular value. By some calcula- i.e.,
tions, we can get
„Q ͑ j ͒ …
␦ K͑ ͉͒ f ͑ j ͉͒ 2ϩ ␦ G͑ ͉͒ f ͑ j ͉͒ 2
2 2
ͩ ͯ ͯ
ͪ
1ϪH 1/2 ϩ4 ␦ K ͑ ͒ ␦ G ͑ ͒ ͉ f ͑ j ͒ ͉ р1, ᭙.
␦ K ϩ ␦ G ϩ2 ␦ K ␦ G
2 2
H
ϩ ͱͩ 2 2
ͯ ͯͪ
␦ K ϩ ␦ G ϩ2 ␦ K ␦ G
1ϪH 2
H
ͯͯ
Ϫ4 ␦ K ␦ G
2 2
1
H
2
As ͉ f ( j ) ͉ decays fast for ϭ0.707 or below
for у g , Eq. ͑25͒ is likely to hold for high
ϭ͉H͉ .
2
frequencies. Thus, assume that Eq. ͑25͒ is true for
͑23͒ у g . We now need to check Eq. ͑25͒ only for
12. 48 Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49
Fig. 10. Performance robustness for 1.2e Ϫ10s /(5sϩ1)(2.5sϩ1) ͑ , step input; , nominal performance; -•-•-,
performance after 20% dominant time constant change͒.
the working frequency range ͓ 0, g ͔ . Due to indeed nominally stable. To see robustness, intro-
ʈ f ( j ) ʈ ϱ р1 for all , the closed loop is robustly duce a 20% perturbation in gain K, giving K
stable if ϭ1.2. Due to ␦ K ϭ0.22%р5% and ␦ G ϭ20%
р90.37%, the resultant system is expected to re-
␦ K ͑ ͒ ϩ ␦ G ͑ ͒ ϩ4 ␦ K ͑ ͒ ␦ G ͑ ͒ р1,
2 2
main stable, which is indeed the case, as exhibited
in Fig. 9 with the dashed-dotted line.
͓ 0, g ͔ .
In the proposed method, ͉ ⌬ K ͉ is made small, 6.2. Example 6
i.e., ␦ K ( ) р5%. Let ␦ K ϭ5%, then the closed
loop is robustly stable if Reconsider Example 2
␦ G ͑ ͒ р90.37%, ͓ 0, g ͔ . ͑26͒ 1.2e Ϫ10s
Gϭ ,
͑ 5sϩ1 ͒͑ 2.5sϩ1 ͒
6.1. Example 5
with the dominant time constant TϭT 0 ϭ5. Our
Reconsider Example 1
proposed method produces a non-PID controller in
Ke Ϫ5s Eq. ͑18͒ and the nominal performance is shown in
Gϭ , Fig. 10 with the solid line. It can be seen that the
͑ s 2 ϩsϩ1 ͒͑ sϩ2 ͒
system is nominally stable. To see robustness, in-
with the nominal KϭK 0 ϭ1. Our proposed troduce a 20% perturbation in T, giving Tϭ6. It
method generates a non-PID controller in Eq. ͑17͒ follows from Example 2 that ␦ K ϭ0.02%р5%.
and the nominal performance is shown in Fig. 9 Additionally, it can be found that ␦ G ( ) р5.75%
with the solid line. It can be seen that the system is р90.37% for ͓ 0, g ͔ . Based on our analysis
13. Qing-Guo Wang, He Ru, and Xiao-Gang Huang / ISA Transactions 41 (2002) 37–49 49
before, one thus concludes that the resultant sys- ͓4͔ Astrom, K. J., Limitations on control system perfor-
tem will remain stable, as confirmed in Fig. 9 with mance. European Journal of Control, submitted for
publication.
the dashed-dotted line. ͓5͔ Maciejowski, J. M., Multivariable Feedback Design.
Addison-Wesley, Workingham, UK, 1989.
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and Vanhamme, H., Parametric identification of trans-
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͓7͔ Wang, Q. G., Lee, T. H., and Ho, W. F., PID tuning for
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IFAC Symp. Adaptive Control Chem. Proc., 1988, p.
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