This paper details development of a Model Predictive Control (MPC) algorithm for a boiler-turbine unit, which is a nonlinear multiple-input multiple-output process. The control objective is to follow set-point changes imposed on two state (output) variables and to satisfy constraints imposed on three inputs and one output. In order to obtain a computationally efficient control scheme, the state-space model is successively linearized on-line for the current operating point and used for prediction. In consequence, the future control policy is easily calculated from a quadratic optimization problem. For state estimation the extended Kalman filter is used. It is demonstrated that the MPC strategy based on constant linear models does not work satisfactorily for the boiler-turbine unit whereas the discussed algorithm with on-line successive model linearisation gives practically the same trajectories as the truly nonlinear MPC controller with nonlinear optimization repeated at each sampling instant.
Turbofan Engine Modelling and Control Design using Linear Quadratic Regulator...theijes
There are many applications in which gas turbine engine is used today, including aircraft propulsion for both commercial and military purposes, and power generation and in all these Control systems technology has played a fundamental role in enhancing performance. Modelling plays a significant role in the development of the entire engine system performance. This paper investigated Linear Quadratic Regulator (LQR) model-based control method to obtain estimates of performance parameters. The main control variable selected is the fuel flow to control the rotational speed of high-pressure spool speed of the turbofan engine. Firstly a suitable mathematical model of the engine is developed in MATLAB Simulink environment with both the intercomponent volume and the constant mass flow methods used. Equations of the mass flow rate and the torque balance are incorporated in the steady state and dynamic state of the thermodynamic engine model. This represents the engine model by a set of first-order differential and algebraic equations and a linearized model is extracted for the analysis and design of a controller by LQR. It is demonstrated that LQR based controllers can perform better than conventional PID controllers. The settling time, rise time and maximum overshoot for LQR based controller are all less than those for PID based controller. The input also changes more accurately for LQR than the PID controller compared.
Synthesis of the Decentralized Control System for Robot-Manipulator with Inpu...ITIIIndustries
It is discussed the problem of synthesis of the decentralized adaptive-periodic control system for two degrees of freedom robotic manipulator which have an input limitations. The solution of the problem is based on the use of hyperstability criterion, L-dissipativity conditions and dynamic filter-corrector.
This paper presents the further developments and working principle of the speed-variable switched differential pump (SvSDP) concept proposed, designed and produced in [1]. The SvSDP system is designed to remove the throttling losses associated with typical valve driven control (VDC) systems. The hydraulic and mechanical system is modelled and linearised. The linearisation point is studied to provide an usable basis for controller design. It is proposed, in this paper, to model the converter and motor using a black box approach, where designed and informative input sequences are used to estimate the mathematical behaviour of the electrical drive based on the equivalent output data. The complete non linear model is verified against available trajectory data from the physical system, obtained from [1]. The linear model is analysed through a relative gain array (RGA) analysis to map the input output couplings present in the system. The results show that the system includes heavy cross-couplings. Results presented in [1] indicate, that it is possible to utilise a input output compensated decoupling to redefine the MIMO system into multiple SISO systems. The SvSDP concept is over-determined in relation to the amount of control inputs compared to possible outputs.
It is proposed in [1] to introduce two new input states and two new output states. The decoupling approach has been investigated in this paper. The decoupling results provided a basis of using decentralised control. The linear control strategies are designed independently based on the notion of decoupling. The first controller is related to the level flow, designed to maintain a desired minimum pressure level. The second load flow controller is related to the cylinder motion. The controller results indicate, that it is possible to achieve a good dynamic tracking performance with an error of maximum 0.5 mm for a given position trajectory.
This paper is also considering the energy consumption issues stated in [1], where two conceptual solutions are proposed, to solve the power loss associated with holding a load at a constant cylinder position. This paper is written as the product of an appendix report describing the whole project.
Optimal tuning linear quadratic regulator for gas turbine by genetic algorith...IJECEIAES
For multiple input-multiple output (MIMO) systems, the most common control strategy is the linear quadratic regulator (LQR) which relies on state vector feedback. Despite this strategy gives very good result, it still has trial and error procedure to select the values of its weight matrices which plays a important role in reaching to the desiered system performance. In order to overcome this problem, the Genetic algorithm is used. The design of genetic algorithm based linear quadratic regulator (GA-LQR) utilized Integral time absolute error (ITAE) as a cost function for optimization. The propsed procedure is implemented on a linear model of gas turbine to control the generator spool’s speed and the output power.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Turbofan Engine Modelling and Control Design using Linear Quadratic Regulator...theijes
There are many applications in which gas turbine engine is used today, including aircraft propulsion for both commercial and military purposes, and power generation and in all these Control systems technology has played a fundamental role in enhancing performance. Modelling plays a significant role in the development of the entire engine system performance. This paper investigated Linear Quadratic Regulator (LQR) model-based control method to obtain estimates of performance parameters. The main control variable selected is the fuel flow to control the rotational speed of high-pressure spool speed of the turbofan engine. Firstly a suitable mathematical model of the engine is developed in MATLAB Simulink environment with both the intercomponent volume and the constant mass flow methods used. Equations of the mass flow rate and the torque balance are incorporated in the steady state and dynamic state of the thermodynamic engine model. This represents the engine model by a set of first-order differential and algebraic equations and a linearized model is extracted for the analysis and design of a controller by LQR. It is demonstrated that LQR based controllers can perform better than conventional PID controllers. The settling time, rise time and maximum overshoot for LQR based controller are all less than those for PID based controller. The input also changes more accurately for LQR than the PID controller compared.
Synthesis of the Decentralized Control System for Robot-Manipulator with Inpu...ITIIIndustries
It is discussed the problem of synthesis of the decentralized adaptive-periodic control system for two degrees of freedom robotic manipulator which have an input limitations. The solution of the problem is based on the use of hyperstability criterion, L-dissipativity conditions and dynamic filter-corrector.
This paper presents the further developments and working principle of the speed-variable switched differential pump (SvSDP) concept proposed, designed and produced in [1]. The SvSDP system is designed to remove the throttling losses associated with typical valve driven control (VDC) systems. The hydraulic and mechanical system is modelled and linearised. The linearisation point is studied to provide an usable basis for controller design. It is proposed, in this paper, to model the converter and motor using a black box approach, where designed and informative input sequences are used to estimate the mathematical behaviour of the electrical drive based on the equivalent output data. The complete non linear model is verified against available trajectory data from the physical system, obtained from [1]. The linear model is analysed through a relative gain array (RGA) analysis to map the input output couplings present in the system. The results show that the system includes heavy cross-couplings. Results presented in [1] indicate, that it is possible to utilise a input output compensated decoupling to redefine the MIMO system into multiple SISO systems. The SvSDP concept is over-determined in relation to the amount of control inputs compared to possible outputs.
It is proposed in [1] to introduce two new input states and two new output states. The decoupling approach has been investigated in this paper. The decoupling results provided a basis of using decentralised control. The linear control strategies are designed independently based on the notion of decoupling. The first controller is related to the level flow, designed to maintain a desired minimum pressure level. The second load flow controller is related to the cylinder motion. The controller results indicate, that it is possible to achieve a good dynamic tracking performance with an error of maximum 0.5 mm for a given position trajectory.
This paper is also considering the energy consumption issues stated in [1], where two conceptual solutions are proposed, to solve the power loss associated with holding a load at a constant cylinder position. This paper is written as the product of an appendix report describing the whole project.
Optimal tuning linear quadratic regulator for gas turbine by genetic algorith...IJECEIAES
For multiple input-multiple output (MIMO) systems, the most common control strategy is the linear quadratic regulator (LQR) which relies on state vector feedback. Despite this strategy gives very good result, it still has trial and error procedure to select the values of its weight matrices which plays a important role in reaching to the desiered system performance. In order to overcome this problem, the Genetic algorithm is used. The design of genetic algorithm based linear quadratic regulator (GA-LQR) utilized Integral time absolute error (ITAE) as a cost function for optimization. The propsed procedure is implemented on a linear model of gas turbine to control the generator spool’s speed and the output power.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Analytical Evaluation of Generalized Predictive Control Algorithms Using a Fu...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Design and Control of a Hydraulic Servo System and Simulation AnalysisIJMREMJournal
This paper describes the system analysis, modeling and simulation of a Hydraulic Servo System (HSS) for
hydraulic mini press machine. Comparisons among linear output feed back PID control, Fuzzy control and
Hybrid of PID and Fuzzy control are presented. Application of hybrid controller to a nonlinear is investigated
by both position and velocity of the hydraulic servo system. The experiment is based on an 8 bit PIC
16F877 microcontroller, and the simulation is based on MATLAB Simulink. Simulation and hardware
experimental results show that the hybrid controller gave the best performance as it has the smallest overshoot,
oscillation, and setting time.
Dynamic Matrix Control (DMC) on jacket tank heater - Rishikesh BagweRishikesh Bagwe
The Dynamic Matrix Control (DMC) method of Model Predictive Control was simulated in MATLAB on Jacketed Tank Heater. The characteristics of the liquid being controlled are height and temperature
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.
Study and Development of an Energy Saving Mechanical SystemIDES Editor
A new energy-saving mechanical system with
automatically controlled air valves has been proposed by
investigator and the preliminary model setup has been tested.
The testing results indicated the proper function of this
energy-saving mechanical system. This mechanical system
model has been simulated and analyzed by the computational
aided engineering solution. The major advantages of this
mechanical system include: simple and compact in design,
higher efficiency in mechanical functioning, quiet in
manufacturing operation, less energy losses due to less
frictional forces in this free piston-cylinder setup, selfadjustable
in operational parameter to improve the system
performance, and etc.
It describe how to build a modelling solution with the Simple Water Budget component of JGrass-NewAGE. Se the companion documentation regarding the component itself
Nonlinear Control of Static Synchronous Compensator STATCOM for Transmission ...ijtsrd
The Static Synchronous Compensator STATCOM is a shunt controller, which is a member of FACTS devices. In this paper, an effective and robust controller for STATCOM device on transmission lines, a Single Machine Infinite Bus SMIB system is modeled. A state space mathematical model is constructed which considers both electromechanical oscillations and reactive current of the STATCOM at the installation site. Based the obtained third order model, state feedback linearization and linear quadratic regulation LQR approach are applied to obtain a nonlinear control law. The controllers are simulated and tested under different operating conditions comparing with the conventional PI controller. Yadana Aung | Phyu Phyu Win | Moe Phyu Thel "Nonlinear Control of Static Synchronous Compensator (STATCOM) for Transmission System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd28002.pdfPaper URL: https://www.ijtsrd.com/engineering/electrical-engineering/28002/nonlinear-control-of-static-synchronous-compensator-statcom-for-transmission-system/yadana-aung
EHR ATTRIBUTE-BASED ACCESS CONTROL (ABAC) FOR FOG COMPUTING ENVIRONMENTcsandit
Liquid level tanks are employed in many industrial and chemical areas. Their level must be keep
a defined point or between maximum-minimum points depending on changing of inlet and outlet
liquid quantities. In order to overcome the problem, many level control methods have been
developed. In the paper, it was aimed that obtain a mathematical model of an installed liquid
level tank system. Then, the mathematical model was derived from the installed system
depending on the sizes of the liquid level tank. According to some proportional-integralderivative
(PID) parameters, the model was simulated by using MATLAB/Simulink program.
After that, data of the liquid level tank were taken into a computer by employing data
acquisition cards (DAQs). Lastly, the computer-controlled liquid level control was successfully
practiced through a written computer program embedded into a PID algorithm used the PID
parameters obtained from the simulations into Advantech VisiDAQ software
Torque estimator using MPPT method for wind turbines IJECEIAES
In this work, we presents a control scheme of the interface of a grid connected Variable Speed Wind Energy Generation System based on Doubly Fed Induction Generator (DFIG). The vectorial strategy for oriented stator flux GADA has been developed To extract the maximum power MPPT from the wind turbine. It uses a second order sliding mode controller and Kalman observer, using the super twisting algorithm. The simulation describes the effectiveness of the control strategy adopted.For a step and random profiles of the wind speed, reveals better tracking and perfect convergence of electromagnetic torque and concellation of reactive power to the stator. This control limits the mechanical stress on the tansmission shaft, improves the quality of the currents generated on the grid and optimizes the efficiency of the conversion chain.
Multivariable Parametric Modeling of a Greenhouse by Minimizing the Quadratic...TELKOMNIKA JOURNAL
This paper concerns the identification of a greenhouse described in a multivariable linear system
with two inputs and two outputs (TITO). The method proposed is based on the least squares identification
method, without being less efficient, presents an iterative calculation algorithm with a reduced
computational cost. Moreover, its recursive character allows it to overcome, with a good initialization, slight
variations of parameters, inevitable in a real multivariable process. A comparison with other method s
recently proposed in the literature demonstrates the advantage of this method. Simulations obtained will be
exposed to showthe effectiveness and application of the method on multivariable systems.
Short-Term Load Forecasting Using ARIMA Model For Karnataka State Electrical ...IJERDJOURNAL
ABSTRACT: Short-term load forecasting is a key issue for reliable and economic operation of power systems. This paper aims to develop short-term electric load forecasting ARIMA Model for Karnataka Electrical Load pattern based on Stochastic Time Series Analysis. The logical and organised procedures for model development using Autocorrelation Function and Partial Autocorrelation Function make ARIMA Model particularly attractive. The methodology involves Initial Model Development Phase, Parameter Estimation Phase and Forecasting Phase. To confirm the effectiveness, the proposed model is developed and tested using the historical data of Karnataka Electrical Load pattern (2016). The forecasting error of ARIMA Model is computed and results have shown favourable forecasting accuracy.
Metal cutting tool position control using static output feedback and full sta...Mustefa Jibril
In this paper, a metal cutting machine position control have been designed and simulated using
Matlab/Simulink Toolbox successfully. The open loop response of the system analysis shows that the system needs
performance improvement. Static output feedback and full state feedback H 2 controllers have been used to increase
the performance of the system. Comparison of the metal cutting machine position using static output feedback and
full state feedback H 2 controllers have been done to track a set point position using step and sine wave input signals
and a promising results have been analyzed.
Performance prediction of a turboshaft engine by using of one dimensional ana...ijmech
Performance estimation of the axial flow gas turbines under variety of operating conditions like different speeds and pressure ratios has been hampered by lack of reliable experimental data and experiments cost.Simulation of gas turbine is a simple way to reduce testing costs and complexity.One-dimensional (1D).Simulation is a simple, fast and accurate method for performance prediction of turbine with different geometries. In this approach, inlet flow conditions and turbine geometry are known and by considering loss model, the turbine performance characteristics are predicted. In following work, that is based on one dimensional modelling method, after the presentation of solution algorithm by trial and error method and introduction of different loss models for modelling, this method were examined for a turbo shaft engine and
compared with experimental results. Comparison of the results with experimental data shows so good adaptation. Also according to these results, Kacker and Okapuu’s developed model gave the closest results to the reference data.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This slide show contains a detailed explanation of the following topics from Control System:
1. Open loop & Closed loop
2. Mathematical modeling
3. f-v and f-i analogy
4. Block diagram reduction technique
5. Signal flow graph
Multi objective control of nonlinear boiler-turbine dynamics with actuator ma...ISA Interchange
This paper investigates multi-objective controller design approaches for nonlinear boiler-turbine dynamics subject to actuator magnitude and rate constraints. System nonlinearity is handled by a suitable linear parameter varying system representation with drum pressure as the system varying parameter. Variation of the drum pressure is represented by suitable norm-bounded uncertainty and affine dependence on system matrices. Based on linear matrix inequality algorithms, the magnitude and rate constraints on the actuator and the deviations of fluid density and water level are formulated while the tracking abilities on the drum pressure and power output are optimized. Variation ranges of drum pressure and magnitude tracking commands are used as controller design parameters, determined according to the boiler-turbine's operation range.
Analytical Evaluation of Generalized Predictive Control Algorithms Using a Fu...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Design and Control of a Hydraulic Servo System and Simulation AnalysisIJMREMJournal
This paper describes the system analysis, modeling and simulation of a Hydraulic Servo System (HSS) for
hydraulic mini press machine. Comparisons among linear output feed back PID control, Fuzzy control and
Hybrid of PID and Fuzzy control are presented. Application of hybrid controller to a nonlinear is investigated
by both position and velocity of the hydraulic servo system. The experiment is based on an 8 bit PIC
16F877 microcontroller, and the simulation is based on MATLAB Simulink. Simulation and hardware
experimental results show that the hybrid controller gave the best performance as it has the smallest overshoot,
oscillation, and setting time.
Dynamic Matrix Control (DMC) on jacket tank heater - Rishikesh BagweRishikesh Bagwe
The Dynamic Matrix Control (DMC) method of Model Predictive Control was simulated in MATLAB on Jacketed Tank Heater. The characteristics of the liquid being controlled are height and temperature
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.
Study and Development of an Energy Saving Mechanical SystemIDES Editor
A new energy-saving mechanical system with
automatically controlled air valves has been proposed by
investigator and the preliminary model setup has been tested.
The testing results indicated the proper function of this
energy-saving mechanical system. This mechanical system
model has been simulated and analyzed by the computational
aided engineering solution. The major advantages of this
mechanical system include: simple and compact in design,
higher efficiency in mechanical functioning, quiet in
manufacturing operation, less energy losses due to less
frictional forces in this free piston-cylinder setup, selfadjustable
in operational parameter to improve the system
performance, and etc.
It describe how to build a modelling solution with the Simple Water Budget component of JGrass-NewAGE. Se the companion documentation regarding the component itself
Nonlinear Control of Static Synchronous Compensator STATCOM for Transmission ...ijtsrd
The Static Synchronous Compensator STATCOM is a shunt controller, which is a member of FACTS devices. In this paper, an effective and robust controller for STATCOM device on transmission lines, a Single Machine Infinite Bus SMIB system is modeled. A state space mathematical model is constructed which considers both electromechanical oscillations and reactive current of the STATCOM at the installation site. Based the obtained third order model, state feedback linearization and linear quadratic regulation LQR approach are applied to obtain a nonlinear control law. The controllers are simulated and tested under different operating conditions comparing with the conventional PI controller. Yadana Aung | Phyu Phyu Win | Moe Phyu Thel "Nonlinear Control of Static Synchronous Compensator (STATCOM) for Transmission System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd28002.pdfPaper URL: https://www.ijtsrd.com/engineering/electrical-engineering/28002/nonlinear-control-of-static-synchronous-compensator-statcom-for-transmission-system/yadana-aung
EHR ATTRIBUTE-BASED ACCESS CONTROL (ABAC) FOR FOG COMPUTING ENVIRONMENTcsandit
Liquid level tanks are employed in many industrial and chemical areas. Their level must be keep
a defined point or between maximum-minimum points depending on changing of inlet and outlet
liquid quantities. In order to overcome the problem, many level control methods have been
developed. In the paper, it was aimed that obtain a mathematical model of an installed liquid
level tank system. Then, the mathematical model was derived from the installed system
depending on the sizes of the liquid level tank. According to some proportional-integralderivative
(PID) parameters, the model was simulated by using MATLAB/Simulink program.
After that, data of the liquid level tank were taken into a computer by employing data
acquisition cards (DAQs). Lastly, the computer-controlled liquid level control was successfully
practiced through a written computer program embedded into a PID algorithm used the PID
parameters obtained from the simulations into Advantech VisiDAQ software
Torque estimator using MPPT method for wind turbines IJECEIAES
In this work, we presents a control scheme of the interface of a grid connected Variable Speed Wind Energy Generation System based on Doubly Fed Induction Generator (DFIG). The vectorial strategy for oriented stator flux GADA has been developed To extract the maximum power MPPT from the wind turbine. It uses a second order sliding mode controller and Kalman observer, using the super twisting algorithm. The simulation describes the effectiveness of the control strategy adopted.For a step and random profiles of the wind speed, reveals better tracking and perfect convergence of electromagnetic torque and concellation of reactive power to the stator. This control limits the mechanical stress on the tansmission shaft, improves the quality of the currents generated on the grid and optimizes the efficiency of the conversion chain.
Multivariable Parametric Modeling of a Greenhouse by Minimizing the Quadratic...TELKOMNIKA JOURNAL
This paper concerns the identification of a greenhouse described in a multivariable linear system
with two inputs and two outputs (TITO). The method proposed is based on the least squares identification
method, without being less efficient, presents an iterative calculation algorithm with a reduced
computational cost. Moreover, its recursive character allows it to overcome, with a good initialization, slight
variations of parameters, inevitable in a real multivariable process. A comparison with other method s
recently proposed in the literature demonstrates the advantage of this method. Simulations obtained will be
exposed to showthe effectiveness and application of the method on multivariable systems.
Short-Term Load Forecasting Using ARIMA Model For Karnataka State Electrical ...IJERDJOURNAL
ABSTRACT: Short-term load forecasting is a key issue for reliable and economic operation of power systems. This paper aims to develop short-term electric load forecasting ARIMA Model for Karnataka Electrical Load pattern based on Stochastic Time Series Analysis. The logical and organised procedures for model development using Autocorrelation Function and Partial Autocorrelation Function make ARIMA Model particularly attractive. The methodology involves Initial Model Development Phase, Parameter Estimation Phase and Forecasting Phase. To confirm the effectiveness, the proposed model is developed and tested using the historical data of Karnataka Electrical Load pattern (2016). The forecasting error of ARIMA Model is computed and results have shown favourable forecasting accuracy.
Metal cutting tool position control using static output feedback and full sta...Mustefa Jibril
In this paper, a metal cutting machine position control have been designed and simulated using
Matlab/Simulink Toolbox successfully. The open loop response of the system analysis shows that the system needs
performance improvement. Static output feedback and full state feedback H 2 controllers have been used to increase
the performance of the system. Comparison of the metal cutting machine position using static output feedback and
full state feedback H 2 controllers have been done to track a set point position using step and sine wave input signals
and a promising results have been analyzed.
Performance prediction of a turboshaft engine by using of one dimensional ana...ijmech
Performance estimation of the axial flow gas turbines under variety of operating conditions like different speeds and pressure ratios has been hampered by lack of reliable experimental data and experiments cost.Simulation of gas turbine is a simple way to reduce testing costs and complexity.One-dimensional (1D).Simulation is a simple, fast and accurate method for performance prediction of turbine with different geometries. In this approach, inlet flow conditions and turbine geometry are known and by considering loss model, the turbine performance characteristics are predicted. In following work, that is based on one dimensional modelling method, after the presentation of solution algorithm by trial and error method and introduction of different loss models for modelling, this method were examined for a turbo shaft engine and
compared with experimental results. Comparison of the results with experimental data shows so good adaptation. Also according to these results, Kacker and Okapuu’s developed model gave the closest results to the reference data.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This slide show contains a detailed explanation of the following topics from Control System:
1. Open loop & Closed loop
2. Mathematical modeling
3. f-v and f-i analogy
4. Block diagram reduction technique
5. Signal flow graph
Multi objective control of nonlinear boiler-turbine dynamics with actuator ma...ISA Interchange
This paper investigates multi-objective controller design approaches for nonlinear boiler-turbine dynamics subject to actuator magnitude and rate constraints. System nonlinearity is handled by a suitable linear parameter varying system representation with drum pressure as the system varying parameter. Variation of the drum pressure is represented by suitable norm-bounded uncertainty and affine dependence on system matrices. Based on linear matrix inequality algorithms, the magnitude and rate constraints on the actuator and the deviations of fluid density and water level are formulated while the tracking abilities on the drum pressure and power output are optimized. Variation ranges of drum pressure and magnitude tracking commands are used as controller design parameters, determined according to the boiler-turbine's operation range.
Output feedback trajectory stabilization of the uncertainty DC servomechanism...ISA Interchange
This work proposes a solution for the output feedback trajectory-tracking problem in the case of an uncertain DC servomechanism system. The system consists of a pendulum actuated by a DC motor and subject to a time-varying bounded disturbance. The control law consists of a Proportional Derivative controller and an uncertain estimator that allows compensating the effects of the unknown bounded perturbation. Because the motor velocity state is not available from measurements, a second-order sliding-mode observer permits the estimation of this variable in finite time. This last feature allows applying the Separation Principle. The convergence analysis is carried out by means of the Lyapunov method. Results obtained from numerical simulations and experiments in a laboratory prototype show the performance of the closed loop system.
Performance analysis of a liquid column in a chemical plant by using mpceSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Data-driven adaptive predictive control for an activated sludge processjournalBEEI
Data-driven control requires no information of the mathematical model of the controlled process. This paper proposes the direct identification of controller parameters of activated sludge process. This class of data-driven control calculates the predictive controller parameters directly using subspace identification technique. By updating input-output data using receding window mechanism, the adaptive strategy can be achieved. The robustness test and stability analysis of direct adaptive model predictive control are discussed to realize the effectiveness of this adaptive control scheme. The applicability of the controller algorithm to adapt into varying kinetic parameters and operating conditions is evaluated. Simulation results show that by a proper and effective excitation of direct identification of controller parameters, the convergence and stability of the implicit predictive model can be achieved.
Finite State Predictive Current and Common Mode Voltage Control of a Seven-ph...IAES-IJPEDS
The paper illustrates finite set predictive current control (FSPC) along with
common mode voltage control of a seven-phase voltage source inverter
(VSI). The current and common mode voltage (CMV) controls are done
considering a finite set of control actions. The space vector model of a sevenphase
voltage source inverter produces 27 = 128 space voltage vectors, with
126 active and two zero vectors. Out of 126 space vectors 112 are distinct
and 14 are redundant vectors. To control the current and the common mode
voltage, specific set of space vectors are chosen that minimizes the
magnitude of the CMV and makes it a dc signal and simultaneously track the
reference current. Hence no common mode current can flow. Three sets of
space vectors are used for switching actuation, in one case only 15 vectors
are used (14 active and one zero), in second case 14 vectors are used,
followed by use of 8 space vectors (7 large and one zero) and finally 7 large
vectors are employed. Optimal algorithm is employed to find the vector
which minimizes the chosen cost function. The effect of selecting the cost
function, the number of space vectors on current tracking and common mode
voltage is investigated and reported. The developed technique is tested for
RL load using simulation and experimental approaches.
Controlling a DC Motor through Lypaunov-like Functions and SAB TechniqueIJECEIAES
In this paper, state adaptive backstepping and Lyapunov-like function methods are used to design a robust adaptive controller for a DC motor. The output to be controlled is the motor speed. It is assumed that the load torque and inertia moment exhibit unknown but bounded time-varying behavior, and that the measurement of the motor speed and motor current are corrupted by noise. The controller is implemented in a Rapid Control Prototyping system based on Digital Signal Processing for dSPACE platform and experimental results agree with theory.
Model Predictive Current Control of a Seven-phase Voltage Source Inverteridescitation
The paper elaborate finite set model based predictive
current control of a seven-phase voltage source inverter. The
current control is carried out considering a finite set of control
actions. The space vector model of a seven-phase voltage source
inverter (VSI) yields 27 = 128 space voltage vectors, with 126
active and two zero vectors. The control method described in
this paper discard some switching states from the whole set
and employs reduced number of switching states to track the
commanded current. Three sets of space vectors are used for
switching actuation, in one case only 15 vectors are used (14
active and one zero), in second case 29 vectors are used (28
active and one zero) and finally 43 vectors (42 active and one
zero) are employed. Optimal algorithm is employed to find
the vector which minimizes the chosen cost function. The
effect of selecting the cost function, the number of space
vectors and the sampling time is investigated and reported.
The developed technique is tested for RL load using simulation
and experimental approaches.
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Metamodel-based Optimization of a PID Controller Parameters for a Coupled-tan...TELKOMNIKA JOURNAL
Liquid flow and level control are essential requirements in various industries, such as paper
manufacturing, petrochemical industries, waste management, and others. Controlling the liquids flow and
levels in such industries is challenging due to the existence of nonlinearity and modeling uncertainties of
the plants. This paper presents a method to control the liquid level in a second tank of a coupled-tank plant
through variable manipulation of a water pump in the first tank. The optimum controller parameters of this
plant are calculated using radial basis function neural network metamodel. A time-varying nonlinear
dynamic model is developed and the corresponding linearized perturbation models are derived from the
nonlinear model. The performance of the developed optimized controller using metamodeling is compared
with the original large space design. In addition, linearized perturbation models are derived from the
nonlinear dynamic model with time-varying parameters.
Comparison of backstepping, sliding mode and PID regulators for a voltage inv...IJECEIAES
In the present paper, an efficient and performant nonlinear regulator is designed for the control of the pulse width modulation (PWM) voltage inverter that can be used in a standalone photovoltaic microgrid. The main objective of our control is to produce a sinusoidal voltage output signal with amplitude and frequency that are fixed by the reference signal for different loads including linear or nonlinear types. A comparative performance study of controllers based on linear and non-linear techniques such as backstepping, sliding mode, and proportional integral derivative (PID) is developed to ensure the best choice among these three types of controllers. The performance of the system is investigated and compared under various operating conditions by simulations in the MATLAB/Simulink environment to demonstrate the effectiveness of the control methods. Our investigation shows that the backstepping controller can give better performance than the sliding mode and PID controllers. The accuracy and efficiency of the proposed backstepping controller are verified experimentally in terms of tracking objectives.
An efficient application of particle swarm optimization in model predictive ...IJECEIAES
Despite all the model predictive control (MPC) based solution advantages such as a guarantee of stability, the main disadvantage such as an exponential growth of the number of the polyhedral region by increasing the prediction horizon exists. This causes the increment in computation complexity of control law. In this paper, we present the efficiency of particle swarm optimization (PSO) in optimal control of a two-tank system modeled as piece- wise affine. The solution of the constrained final time-optimal control problem (CFTOC) is derived, and then the PSO algorithm is used to reduce the computational complexity of the control law and set the physical parameters of the system to improve performance simultaneously. On the other hand, a new combined algorithm based on PSO is going to be used to reduce the complexity of explicit MPC-based solution CFTOC of the two-tank system; consequently, that the number of polyhedral is minimized, and system performance is more desirable simultaneously. The proposed algorithm is applied in simulation and our desired subjects are reached. The number of control law polyhedral reduces from 42 to 10 and the liquid height in both tanks reaches the desired certain value in 189 seconds. Search time and apply control law in 25 seconds.
Current predictive controller for high frequency resonant inverter in inducti...IJECEIAES
In the context of this article, we are particularly interested in the modeling and control of an induction heating system powered by high frequency resonance inverter. The proposed control scheme comprises a current loop and a PLL circuit. This latter is an electronic assembly for slaving the instantaneous phase of output on the instantaneous input phase, and is used to follow the rapid variations of the frequency.To further improve the transient dynamics of the studied system and in order to reduce the impact of measurement noise on the control signal, a generalized predictive control has been proposed to control the current of the inductor. We discussed the main steps of this command, whose it uses a minimization algorithm to obtain an optimal control signals, its advantages are: its design is simple, less complexity and direct manipulation of the control signal. The results have shown the effectiveness of the proposed method, especially in the parameters variation and/or the change of the reference current.
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.
TORQUE RIPPLE MINIMIZATION OF MATRIX CONVERTER-FED PMSM DRIVES USING ADVANCED...ijscmcj
An advanced direct torque control (DTC) technique using Model predictive control (MPC) is proposed for matrix converter (MC)-based permanent-magnet synchronous motor (PMSM) drive system, which reduces the torque ripples, does not need the duty cycle calculation, and ensures the fixed switching frequency. Analytical expressions of change rates of torque and flux of PMSM as a function of MC - dqo components are derived. The predictive model of PMSM and MC is realized by means of State model. Then, the advanced MC-fed DTC algorithm is implemented based on Cost function evaluation. The simulation results exhibit remarkable torque ripple reduction with the help of MPC. As a result, the proposed strategy is proved to be effective in minimizing the torque ripples for MC-based PMSM drives.
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.
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.
Design of a new PID controller using predictive functional control optimizati...ISA Interchange
An improved proportional integral derivative (PID) controller based on predictive functional control (PFC) is proposed and tested on the chamber pressure in an industrial coke furnace. The proposed design is motivated by the fact that PID controllers for industrial processes with time delay may not achieve the desired control performance because of the unavoidable model/plant mismatches, while model predictive control (MPC) is suitable for such situations. In this paper, PID control and PFC algorithm are combined to form a new PID controller that has the basic characteristic of PFC algorithm and at the same time, the simple structure of traditional PID controller. The proposed controller was tested in terms of set-point tracking and disturbance rejection, where the obtained results showed that the proposed controller had the better ensemble performance compared with traditional PID controllers.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Pile Foundation by Venkatesh Taduvai (Sub Geotechnical Engineering II)-conver...
Nonlinear predictive control of a boiler turbine unit
1. Research article
Nonlinear predictive control of a boiler-turbine unit: A state-space approach
with successive on-line model linearisation and quadratic optimisation
Maciej Ławryńczuk
Institute of Control and Computation Engineering, Warsaw University of Technology, ul. Nowowiejska 15/19, 00-665 Warsaw, Poland
a r t i c l e i n f o
Article history:
Received 7 July 2015
Received in revised form
27 December 2016
Accepted 16 January 2017
Available online 30 January 2017
Keywords:
Process control
Model predictive control
Boiler-turbine unit
On-line successive linearisation
a b s t r a c t
This paper details development of a Model Predictive Control (MPC) algorithm for a boiler-turbine unit,
which is a nonlinear multiple-input multiple-output process. The control objective is to follow set-point
changes imposed on two state (output) variables and to satisfy constraints imposed on three inputs and
one output. In order to obtain a computationally efficient control scheme, the state-space model is
successively linearised on-line for the current operating point and used for prediction. In consequence,
the future control policy is easily calculated from a quadratic optimisation problem. For state estimation
the extended Kalman filter is used. It is demonstrated that the MPC strategy based on constant linear
models does not work satisfactorily for the boiler-turbine unit whereas the discussed algorithm with on-
line successive model linearisation gives practically the same trajectories as the truly nonlinear MPC
controller with nonlinear optimisation repeated at each sampling instant.
& 2017 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
A boiler-turbine unit is a crucial system in coal-fired power
plants. It is used to convert fuel's chemical energy into mechanical
one and then into electrical energy. The boiler generates high-
pressure and high-temperature steam to drive the turbine which
generates electricity. The boiler-turbine model most frequently
considered in literature was developed by R.D. Bell and K.J. Åström
[1], alternative models are described e.g. in [4,24]. The boiler-tur-
bine unit is a nonlinear multiple-input multiple-output process
with strong cross-coupling. The control objective is to follow the
set-points and to satisfy some constraints imposed on process
variables. Therefore, it usually cannot be efficiently controlled ef-
ficiently enough by simple, single-loop PID controllers [37], in
particular when the operating point must be changed fast and in
wide range. That is why more advanced control strategies should
be applied for the boiler-turbine unit.
A number of multivariable control strategies have been applied
to the boiler-turbine system, both linear and nonlinear, the state-
space model or its input-output approximations may be used for
controller synthesis. A multivariable PI controller and a linear
quadratic regulator (LQR) are discussed in [6], the controllers’
parameters are adjusted off-line by a genetic algorithm. The ∞H
controller and its transformation to a multivariable PI controller
are considered in [37]. A technique which allows to respect con-
straints in linear control schemes such as multivariable PI and ∞H
is considered in [36]. The gain-scheduled optimal controller which
takes into account the existing constraints is presented in [3], a
linear parameter varying state-space model is used. A multi-ob-
jective control of the process is discussed in [2], the constraints are
taken into account by a linear matrix inequality algorithm while
tracking capabilities are optimised in terms of ∞H performance. A
robust adaptive sliding mode controller is presented in [11], input-
output linearisation is performed to eliminate process non-
linearity. A two-level hierarchical control scheme is presented in
[10] in which a supervisory fuzzy reference governor generates the
set-points while a feedforward-feedback controller is used. Three
single loop PID controllers are used together with a nonlinear
fuzzy feedforward correctors.
As far as nonlinear control strategies are considered, a fuzzy
∞H tracking state feedback controller based on a fuzzy state-space
models may be used as described in [43]. An alternative is to use
a simplified fuzzy multiple-model controller which consists of
three single-loop fuzzy controllers [27]. An optimal multiple-
model state-space controller is described in [13]. An approach to
enforce constraints in a nonlinear state feedback controller is
presented in [19].
Model Predictive Control (MPC) strategy [21,30,39] is fre-
quently used to efficiently control numerous processes, e.g. ma-
nipulators [7], plastic injection moulding [8], distillation columns
[9], chemical reactors [32], cruise control [33], air conditioning
[35], and electric arc furnaces [40]. Due to its inherent ability to
take into account constraints and deal with multiple-input mul-
tiple-output processes with couplings, MPC is a straightforward
option for the boiler-turbine system. In the simplest case pure
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2017.01.016
0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.
E-mail address: M.Lawrynczuk@ia.pw.edu.pl
ISA Transactions 67 (2017) 476–495
2. linear models may be used. An application of Generalized Pre-
dictive Control (GPC) algorithm based on a linearised model using
an input-output feedback linearisation method and an MPC algo-
rithm based on a fuzzy model are presented in [18]. Multivariable
linear step-response input-output models obtained by two
methods and corresponding Dynamic Matrix Control (DMC) al-
gorithms are described in [26]. Unfortunately, the linear models of
the boiler-turbine system are inaccurate and the controller per-
formance is not satisfactory. Quite frequently fuzzy models or
multi-model structures are used in MPC due to their computa-
tional simplicity. An adaptive DMC algorithm with fuzzy step-re-
sponse models, which are recalculated on-line for the current
operating point of the process, is presented in [25]. A fuzzy input-
output model and a resulting MPC controller is discussed in [17]. A
conceptually similar control scheme based on a bank of local state-
space linear models used in MPC is studied in [28]. An MPC al-
gorithm based on a fuzzy state-space model is considered in [41].
A piecewise affine state-space model of the process is discussed in
[14] together with an explicit MPC algorithm which calculates on-
line the control law as an affine function of system states, the
control laws for a number of regions are precalculated off-line.
More advanced MPC schemes use the full nonlinear model of
the process or its transformation. An application of the extended
DMC technique based on a constant input-output step-response
model with an additional time-varying disturbances which ac-
counts for nonlinearity of the process is described in [12]. Con-
strained nonlinear MPC schemes with nonlinear optimisation re-
peated at each sampling instant are reported in [19], state-space
models are used for prediction. An application of a genetic algo-
rithm for optimisation in nonlinear MPC based on a fuzzy state-
space model is described in [16,19]. A predictive control scheme
based on a state-space hybrid model is discussed in [31], the re-
sulting optimisation problem is of a mixed integer linear or
quadratic type. Hierarchical set-point optimisation cooperating
with an MPC algorithm based on state-space a fuzzy model is
considered in [42].
Motivation of this work is to develop a computationally effi-
cient nonlinear MPC algorithm for the boiler-turbine unit. In
contrast to the MPC approaches with nonlinear on-line numerical
optimisation discussed in [16,19], in the presented MPC scheme
the original nonlinear model is successively linearised on-line for
the current operating point of the process and used for prediction.
Nomenclature
Fundamental model
f, f1, f2, f3 model state functions
g, g1, g2, g3 model output functions
qe evaporation rate (kg/s)
u1 valve positions for fuel flow (0–1)
u2 valve positions for steam control (0–1)
u3 valve positions for feedwater flow (0–1)
=x y1 1 drum pressure (kg/cm2)
=x y2 2 electric power output (MW)
x3 fluid density (kg/cm3)
y3 drum water level deviations (m)
αcs steam quality (–)
Model predictive control
×0m n zeros matrix of dimensionality m  n
A, B, C, D matrices of constant linear model for MPC
( )A k , ( )B k , ( )C k , ( )D k matrices of time-varying linearised
model for MPC
( )͠A k , ( )
∼
B k , ( )
∼
C k , ( )͠D k , ( )P k , ( )V k auxiliary matrices based on
parameters of linearised model
d(k) estimation of output disturbance
×Im n identity matrix of dimensionality m  n
J, JN auxiliary constant matrices
k discrete time (the current sampling instant)
N, Nu prediction and control horizons
SSEx sum of squared errors of controlled state variables
SSEy sum of squared errors of deviations of constrained
output variable
¯ ( − )u k 1 values of manipulated variables applied to process
umin, umax, umin, umax definition of range constraints imposed on
manipulated variables
¯ ( )x k , ¯ ( − )x k 1 measurements (estimation) of state variables
( + | )x k p k1
sp
, ( + | )x k p k2
sp
, ( + | )x k p k3
sp
set-point values of state
variables
( + | )x k p ksp , ( )x ksp state set-point trajectories
^ ( + | )x k p k1 , ^ ( + | )x k p k2 , ^ ( + | )x k p k3 predicted values of state
variables
^( + | )x k p k , ^( )x k predicted state trajectories
^ ( + | )y k p k3 predicted value of output
y3
min
, y3
max
, y3
min
, y3
max
definition of range constraints imposed
on controlled variable y3
^ ( )y k predicted output trajectory
μ p1, , μ p2, , μ p3, , Mp, M weights in MPC related to predicted errors
of controlled variables
λ p1, , λ p2, , λ Λ Λ, ,p p3, weights in MPC related to increments of
manipulated variables
▵ ( + | )u k p k1 , ▵ ( + | )u k p k2 , ▵ ( + | )u k p k3 future increments of
manipulated variables
▵ ( + | )u k p k , ▵ ( )u k trajectory of future increments of manipu-
lated variables
▵umax, ▵umax definition of rate constraints imposed on ma-
nipulated variables
ε ( )kmin , ε ( )kmax slack variables in MPC optimisation when the
output constraints are treated in a suboptimal way
ε ( + )k pmin , ε ( + )k pmax , ε ( )kmin , ε ( )kmax slack variables in MPC
optimisation when the output constraints are treated
in the optimal way
ρmin, ρmax weighting coefficients (penalties)
ν ( )k estimation of state disturbance
State estimation
( − )F k 1 , ( )H k matrices of linearised model for state estimation
( − )F k 1a , ( )H ka matrices of augmented linearised model for
state estimation
fa, ga state and output function of augmented model
( )K k filter gain matrix
( | − )P k k 1 , ( − | − )P k k1 1 covariance matrices of estimated
vectors ^( | − )x k k 1 , ^( − | − )x k k1 1
( − )Q k 1 , ( )R k noise covariance matrices
( )S k residual covariance matrix
( )u ka augmented input vector
v(k), w(k) process and observation noises
( )x ka augmented state vector
˜x3 estimated value of fluid density (kg/cm3)
z measured variable
˜z measurement residual
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495 477
3. Linearisation makes it possible to easily calculate on-line the fu-
ture control policy from a quadratic optimisation problem. In order
to show advantages of the discussed MPC algorithm, it is com-
pared with the fully-fledged nonlinear MPC in terms of control
accuracy and computational burden. In both approaches the con-
trol objective is to follow set-point changes imposed on two state
(output) variables and to satisfy constraints imposed on three in-
puts and one output.
This paper is organised as follows. First, Section 2 describes
the state-space model of the process and control objectives.
Next, Section 3 shortly reminds the general idea of MPC and
gives a mathematical formulation of the MPC optimisation
problem resulting from the control objectives. The main part of
the paper, given in Section 4, details derivation of the com-
putationally efficient MPC algorithm with on-line model line-
arisation for the boiler-turbine unit and discusses the state
estimation problem. Section 5 presents simulation results. In
particular, the discussed MPC strategy is compared with a
simple MPC algorithm based on constant linear models and
with a computationally demanding MPC algorithm with full
nonlinear optimisation repeated at each sampling instant. Fi-
nally, Section 6 concludes the paper.
2. Boiler-turbine unit
2.1. Continuous-time process description
The model considered in this paper is based on the basic con-
servation laws and their parameters have been estimated from the
data measured from the Synvendska Kraft AB Plant in Malmö,
Sweden. The process is described by the following continuous-
time state equations [1]
( )
= − ( ) ( ) + ( ) − ( )
( )
= ( ( ) − ) ( ) − ( )
( )
=
( ) − ( ( ) − ) ( )
( )
x t
t
u t x t u t u t
x t
t
u t x t x t
x t
t
u t u t x t
d
d
0.0018 0.9 0.15
d
d
0.073 0.016 0.1
d
d
141 1.1 0.19
85 1
1
2 1
9/8
1 3
2
2 1
9/8
2
3 3 2 1
where the state variables x1, x2, and x3 denote drum pressure
( kg/cm2), electric power output (MW), and fluid density
( kg/cm3), respectively. The manipulated inputs, u1, u2, and u3
are the valve positions for fuel flow, steam control, and feed-
water flow, respectively. The outputs y1, y2, and y3 of the sys-
tem are: drum pressure ( kg/cm2), electric output (MW) and
drum water level deviations (m). The continuous-time output
equations are
⎛
⎝
⎜
⎞
⎠
⎟α
( ) = ( )
( ) = ( )
( ) = ( ) + ( ) +
( )
−
( )
y t x t
y t x t
y t x t t
q t
0.05 0.13073 100
9
67.975
2
1 1
2 2
3 3 cs
e
where αcs and qe are steam quality (dimensionless) and eva-
poration rate (kg/s), respectively, and are given by
α ( ) =
( − ( ))( ( ) − )
( )( − ( )) ( )
t
x t x t
x t x t
1 0.001538 0.8 25.6
1.0394 0.0012304 3a
cs
3 1
3 1
( ) = ( ( ) − ) ( ) + ( ) − ( )
− ( )
q t u t x t u t u t0.854 0.147 45.59 2.514
2.096 3b
e 2 1 1 3
It is assumed that all the output signals are measured, which
means that the first two state variables x1 and x2 are also
measured, whereas the third state variable x3 is not available
for measurement. Therefore, for state estimation an observer
must be used. The estimated state variable is denoted by ˜x3.
2.2. Control objectives
The boiler-turbine controller must calculate the manipulated
variables u1, u2 and u3 in such a way that the discrepancy between
drum pressure ( =y x1 1), electric power output ( =y x2 2) and their
set-points ( =y x1
sp
1
sp
, )=y x2
sp
2
sp
are minimised whereas water level
deviations should be in the following range [3]
− ≤ ( ) ≤ ( )y t0.1 0.1 43
Due to actuators' limitations, the process inputs are subject to the
following magnitude constraints [3]
≤ ( ) ≤
≤ ( ) ≤
≤ ( ) ≤ ( )
u t
u t
u t
0 1
0 1
0 1 5
1
2
3
Additionally, rate constraints [3]
− ≤ ̇ ( ) ≤
− ≤ ̇ ( ) ≤
− ≤ ̇ ( ) ≤ ( )
u t
u t
u t
0.007 0.007
2 0.02
0.05 0.05 6
1
2
3
may be taken into account.
In this paper two notation methods are used: vectors and
scalars. For compactness of presentation the vectors
⎡⎣ ⎤⎦=u u u u1 2 3
T
, ⎡⎣ ⎤⎦=x x x x1 2 3
T
, ⎡⎣ ⎤⎦=y y y y1 2 3
T
are used. When it is
necessary or convenient, the elements of these vectors are also
used, i.e. the scalars un, xn and yn, where =n 1, 2, 3.
2.3. Model discretisation
The continuous-time state equations of the boiler-turbine
model (Eqs. (1)) are discreticised using the Euler's method. The
discrete-time state equations are
⎛
⎝
⎜
⎞
⎠
⎟
( )
( )
( + ) = ( ) + − ( ) ( ) + ( ) − ( )
( + ) = ( ) + ( ( ) − ) ( ) − ( )
( + ) = ( ) +
( ) − ( ( ) − ) ( )
( )
x k x k T u k x k u k u k
x k x k T u k x k x k
x k x k T
u k u k x k
1 0.0018 0.9 0.15
1 0.073 0.016 0.1
1
141 1.1 0.19
85 7
1 1 s 2 1
9/8
1 3
2 2 s 2 1
9/8
2
3 3 s
3 2 1
where =T 1 ss denotes the sampling time. From Eq. (2), the dis-
crtete-time output equations are
⎛
⎝
⎜
⎞
⎠
⎟α
( ) = ( )
( ) = ( )
( ) = ( ) + ( ) +
( )
−
( )
y k x k
y k x k
y k x k k
q k
0.05 0.13073 100
9
67.975
8
1 1
2 2
3 3 cs
e
where, from Eqs. (3a) and (3b), one has
α ( ) =
( − ( ))( ( ) − )
( )( − ( )) ( )
k
x k x k
x k x k
1 0.001538 0.8 25.6
1.0394 0.0012304 9a
cs
3 1
3 1
( ) = ( ( ) − ) ( ) + ( ) − ( )
− ( )
q k u k x k u k u k0.854 0.147 45.59 2.514
2.096 9b
e 2 1 1 3
The discrete-time state-space model may be described by the
general equations
( + ) = ( ( ) ( )) ( )x k f x k u k1 , 10a
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495478
4. ( ) = ( ( ) ( )) ( )y k g x k u k, 10b
where →f: 6 3 and →g: 6 3. The state equations may be
rewritten for the sampling instant k, which gives
( ) = ( ( − ) ( − )) ( )x k f x k u k1 , 1 11a
( ) = ( ( ) ( )) ( )y k g x k u k, 11b
Considering the actual relations between variables, one has
( ) = ( ( − ) ( − ) ( − ) ( − ))
( ) = ( ( − ) ( − ) ( − ))
( ) = ( ( − ) ( − ) ( − ) ( − )) ( )
x k f x k u k u k u k
x k f x k x k u k
x k f x k x k u k u k
1 , 1 , 1 , 1
1 , 1 , 1
1 , 1 , 1 , 1 12
1 1 1 1 2 3
2 2 1 2 2
3 3 1 3 2 3
and
( ) = ( ( ))
( ) = ( ( ))
( ) = ( ( ) ( ) ( ) ( ) ( )) ( )
y k g x k
y k g x k
y k g x k x k u k u k u k, , , , 13
1 1 1
2 2 2
3 3 1 3 1 2 3
3. Model predictive control problem formulation for the boi-
ler-turbine unit
MPC is a computer control technique which calculates re-
peatedly on-line not only the current values of control signals
(i.e. for the current sampling instant k), but the whole future
control policy, defined on some control horizon Nu [21,30,39].
Usually, for simplicity of implementation, control increments,
not their values, are calculated. The vector of decision variables
is
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
▵ ( ) =
▵ ( | )
⋮
▵ ( + − | )
u k
u k k
u k N k1u
Because the boiler-turbine system has 3 inputs, there are N3 u
decision variables. The control increments are defined as
⎧
⎨
⎩
▵ ( + | ) =
( | ) − ( − ) =
( + | ) − ( + − | ) ≥
u k p k
u k k u k p
u k p k u k p k p
1 if 0
1 if 1
n
n n
n n
where =n 1, 2, 3. It is assumed that ▵ ( + | ) =u k p k 0n for ≥p Nu.
Considering the control objectives detailed in Section 2, the
objective of the MPC algorithm is to minimise discrepancies
between predicted values of drum pressure x1 and its set-point
trajectory x1
sp
as well as discrepancies between predicted values
of electric power output x2 and its set-point trajectory x2
sp
.
These discrepancies are considered on the prediction horizon
≥N Nu. Thus, the following quadratic cost function is mini-
mised on-line
( )∑ ∑
∑ ∑
μ
λ
( ) = ( + | ) − ^ ( + | )
+ (▵ ( + | ))
( )
= =
= =
−
J k x k p k x k p k
u k p k
14
n p
N
n p n n
n p
N
n p n
1
2
1
,
sp 2
1
3
0
1
,
2
u
where μ ≥ 0n p, and λ > 0n p, are weighting coefficients. The
predictions ^ ( + | )x k p kn are calculated from a dynamic model of
the process. Additionally, the second part of the MPC cost-
function minimises excessive control increments. Having cal-
culated the decision variables at the sampling instant k, the
manipulated variables are updated using the first 3 elements of
the solution vector ▵ ( )u k , i.e.
( ) = ▵ ( | ) + ( − ) ( ) = ▵ ( | ) + ( − )
( ) = ▵ ( | ) + ( − )
u k u k k u k u k u k k u k
u k u k k u k
1 , 1 ,
1
1 1 1 2 2 2
3 3 3
In the next sampling instant, the measurements of the outputs
and available state variables are updated, and the whole opti-
misation procedure is repeated.
In the discrete-time domain and in the MPC framework, the
water level deviations constraints (4) may be rewritten as the
constraints imposed on the third output variable over the pre-
diction horizon
− ≤ ^ ( + | ) ≤ = … ( )y k p k p N0.1 0.1, 1, , 153
The predictions of the output variable, ^ ( + | )y k p k3 , are calculated
from the model. From Eq. (5), the constraints imposed on mag-
nitude of the inputs are used in MPC over the control horizon
≤ ( + | ) ≤ = … −
≤ ( + | ) ≤ = … −
≤ ( + | ) ≤ = … − ( )
u k p k p N
u k p k p N
u k p k p N
0 1, 0, , 1
0 1, 0, , 1
0 1, 0, , 1 16
1 u
2 u
3 u
From Eq. (6), taking into account that the sampling time is =T 1 ss
and the magnitude constraints imposed in MPC on the second
manipulated variable are ≤ ( + | ) ≤u k p k0 12 for = … −p N0, , 1u ,
the rate constraints used in MPC are
− ≤ ▵ ( + | ) ≤ = … −
− ≤ ▵ ( + | ) ≤ = … −
− ≤ ▵ ( + | ) ≤ = … − ( )
u k p k p N
u k p k p N
u k p k p N
0.007 0.007, 0, , 1
1 0.02, 0, , 1
0.05 0.05, 0, , 1 17
1 u
2 u
3 u
Taking into account Eqs. (14), (15), (16) and (17), the constrained
MPC optimisation problem which must be solved at each sampling
instant on-line may be formulated in the following way
⎪
⎪
⎪
⎪
⎧
⎨
⎩
⎫
⎬
⎭
∑ ∑
ρ ε ρ ε
ε ε
ε ε
( + | ) − ^( + | ) + ▵ ( + | )
+ (( ( )) + (( ( ))
≤ ( + | ) ≤ = … −
▵ ≤ ▵ ( + | ) ≤ ▵ = … −
− ( ) ≤ ^ ( + | ) ≤ + ( ) = …
( ) ≥ ( ) ≥ ( )
Λ▵ ( )
= =
−
ε
ε
( )
( )
x k p k x k p k u k p k
k k
u u k p k u p N
u u k p k u p N
y k y k p k y k p N
k k
min
subject to
, 0, , 1
, 0, , 1
, 1, ,
0, 0 18
u Mk
p
N
p
N
1
sp
2
0
1
2
min min 2 max max 2
min max
u
min max
u
3
min min
3 3
max max
min max
k
k
p p
min
max
u
The input constraints are defined by the vectors:
⎡⎣ ⎤⎦=u 0 0 0min T
, ⎡⎣ ⎤⎦=u 1 1 1max T
, ⎡⎣ ⎤⎦▵ = − − −u 0.007 2 0.05min T
,
⎡⎣ ⎤⎦▵ =u 0.007 0.02 0.05max T
, the output constraints by the scalars
= −y 0.13
min
, =y 0.13
max
. The matrices μ μ μ= ( ) ≥M diag , , 0p p p p1, 2, 3,
(with μ = 0p3, ), λ λ λ= ( ) >M diag , , 0p p p p1, 2, 3, are of dimensionality
3 Â 3. In order to guarantee that the feasible set of the MPC
optimisation problem is not empty, not the hard output con-
straints (15), but their soft versions are used. The hard con-
straints may be temporarily violated, but it enforces the ex-
istence of the feasible set. The degree of constraint violation is
minimised by the last two parts of the cost-function. The soft
approach to output variables increases the number of decision
variables of the MPC optimisation problem to +N3 2u , the
number of constraints is +N12 4u , ε ( )kmin and ε ( )kmax are ad-
ditional decision variables, ρ ρ >, 0min max are penalty
coefficients.
The MPC optimisation problem (18) may be further re-
formulated using the vector-matrix notation. The set-point
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495 479
5. trajectory and the predicted state trajectory vectors
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
( ) =
( + | )
⋮
( + | )
^( ) =
^( + | )
⋮
^( + | )
x xk
x k k
x k N k
k
x k k
x k N k
1
,
1
sp
sp
sp
are of length N3 , the weighting matrices = ( … )M M Mdiag , , N1 and
Λ Λ Λ= ( … )−diag , , N0 1u are of dimensionality ×N N3 3 and
×N N3 3u u, respectively. The input constraint vectors
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
= ⋮ = ⋮ ▵ =
▵
⋮
▵
▵ =
▵
⋮
▵
u u u u
u
u
u
u
u
u
u
u
, , ,min
min
min
max
max
max
min
min
min
max
max
max
are of length N3 u, the output constraint vectors
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= ⋮ = ⋮y y
y
y
y
y
,3
min
3
min
3
min
3
max
3
max
3
max
are of length N. The general MPC optimisation problem defined by
Eq. (18) can be expressed as
{
}
ρ ε
ρ ε
ε ε
ε ε
( ) − ^( ) + ▵ ( ) + ( ( ))
+ ( ( ))
≤ ( ) ≤
▵ ≤ ▵ ( ) ≤ ▵
− ( ) ≤ ^ ( ) ≤ + ( )
( ) ≥ ( ) ≥ ( )
Λ
▵ ( )
× ×
ε
ε
( )
( )
x x u
u u u
u u u
y I y y I
k k k k
k
k
k
k k k
k k
min
subject to
0, 0 19
u Mk
N N
sp
2
2 min min 2
max max 2
min max
min max
3
min min
1 3 3
max max
1
min max
k
k
min
max
One may note that in the MPC optimisation problems (18) and
(19) the constraints imposed on the predicted output variable y3
are used in a simple, suboptimal version, because the same coef-
ficients ε ( )kmin and ε ( )kmax are used over the whole prediction
horizon to soften the original hard constraints (15). In the general
optimal approach the vectors of additional variables are not con-
stant over the prediction horizon, but the degree of hard con-
straints violation is adjusted independently for consecutive sam-
pling instants over the prediction horizon. In place of the MPC
optimisation problem (18) one has
⎪
⎪
⎪
⎪
⎧
⎨
⎩
⎫
⎬
⎭
∑ ∑
∑ ∑ρ ε ρ ε
ε ε
ε ε
( + | ) − ^( + | ) + ▵ ( + | )
+ ( + ) + ( + )
≤ ( + | ) ≤ = … −
▵ ≤ ▵ ( + | ) ≤ ▵ = … −
− ( + ) ≤ ^ ( + | ) ≤ + ( + ) = …
( + ) ≥ ( + ) ≥ = … ( )
Λ▵ ( )
= =
−
= =
ε
ε
( + )
( + )
x k p k x k p k u k p k
k p k p
u u k p k u p N
u u k p k u p N
y k p y k p k y k p p N
k p k p p N
min
subject to
, 0, , 1
, 0, , 1
, 1, ,
0, 0, 1, , 20
u Mk
p
N
p
N
p
N
p
N
1
sp
2
0
1
2
min
1
min 2 max
1
max 2
min max
u
min max
u
3
min min
3 3
max max
min max
k p
k p
p p
min
max
u
The number of decision variables is +N N3 2u , the number of
constraints is +N N12 4u . Using the vectors of additional variables
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
ε ε
ε
ε
ε
ε
( ) =
( + )
⋮
( + )
( ) =
( + )
⋮
( + )
k
k
k N
k
k
k N
1
,
1
min
min
min
max
max
max
of length N, the MPC optimisation problem (20) may be also re-
written using the vector-matrix notation in the following way
⎧
⎨
⎩
⎫
⎬
⎭
( )
ε ε
ε ε
ε ε
ρ ρ( ) − ^ ( ) + ▵ ( ) + ( ) + ( )
≤ ( ) ≤
▵ ≤ ▵ ( ) ≤ ▵
− ( ) ≤ ^ ( ) ≤ + ( )
( ) ≥ ( ) ≥
ε
ε
Λ▵ ( )
( )
( )
× × 21
x x u
u u u
u u u
y y y
0 0
k k k k k
k
k
k k k
k k
min
subject to
,
u Mk
k
k
N N
min
max
sp
2 2 min min 2 max max 2
min max
min max
3
min min
3 3
max max
min
1
max
1
4. Nonlinear MPC algorithm with successive on-line model
linearisation (MPC-SL) for the boiler-turbine unit
If the nonlinear state-space model defined by Eqs. (7), (8), (9a)
and (9b) is used for prediction calculation in MPC, the predictions
of the state variables x1, x2 and of the output variable y3 are
nonlinear functions of the calculated on-line future control moves
▵ ( )u k . In consequence, the MPC optimisation problems (19) or (21)
are nonlinear tasks which must be solved on-line in real time.
In order to obtain a computationally efficient MPC algorithm,
the MPC scheme with Successive Linearisation (MPC-SL) is de-
veloped for the boiler-turbine system. A linear approximation of
the model is repeatedly, at each sampling instant, calculated on-
line for the current operating point of the process and used for
prediction, which makes it possible to calculate the decision
variables from an easy to solve quadratic optimisation problem (by
the active set method or the interior point method [23]). The
general formulation of the MPC-SL algorithm based on input-
output models is given in [39], algorithm implementation for
neural models is given in [20], the description for state-space
models is presented in [15], but in this work a simple approach to
prediction which guarantees offset-free control [38] is used.
4.1. On-line model linearisation
Using the Taylor series expansion method, the local linear
approximation of the nonlinear state-space model described by
Eqs. (11a) and (11b) is
( ) = (¯ ( − ) ¯ ( − )) + ( )( ( − ) − ¯ ( − ))
+ ( )( ( − ) − ¯ ( − )) ( )
A
B
x k f x k u k k x k x k
k u k u k
1 , 1 1 1
1 1 22a
( ) = (¯ ( ) ¯ ( − )) + ( )( ( ) − ¯ ( ))
+ ( )( ( ) − ¯ ( − )) ( )
C
D
y k g x k u k k x k x k
k u k u k
, 1
1 22b
where the vectors
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
¯ ( − ) =
¯ ( − )
¯ ( − )
˜ ( − )
¯ ( ) =
¯ ( )
¯ ( )
˜ ( )
¯ ( − ) =
¯ ( − )
¯ ( − )
¯ ( − )
x k
x k
x k
x k
x k
x k
x k
x k
u k
u k
u k
u k
1
1
1
1
, , 1
1
1
1
1
2
3
1
2
3
1
2
3
define the current operating point of the process. It is described by
real measurements of the first two state variables ( ¯x1, ¯x2) and by
the estimated state of the third state variable ( ˜x3) as well as by the
values of the manipulated variables applied to the process at the
previous sampling instant. The vectors ( − )x k 1 , x(k) and u(k) are
arguments (independent variables) of the linearised model. Be-
cause linearisation precedes optimisation of the future control
increments at each sampling instant, the current operating point is
defined in the output Eq. (22b) not by unavailable signals ¯ ( )u k1 ,
¯ ( )u k2 , ¯ ( )u k3 , but by the most recent values from the previous
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495480
6. sampling instant. The linearised model (22a)–(22b) may be ex-
pressed in the following form
δ( ) = ( ) ( − ) + ( ) ( − ) + ( ) ( )A Bx k k x k k u k k1 1 23ax
δ( ) = ( ) ( ) + ( ) ( ) + ( ) ( )C Dy k k x k k u k k 23by
where
δ
δ
( ) = (¯ ( − ) ¯ ( − )) − ( ) ¯ ( − ) − ( ) ¯ ( − )
( ) = (¯ ( ) ¯ ( − )) − ( ) ¯ ( ) − ( ) ¯ ( − ) ( )
A B
C D
k f x k u k k x k k u k
k g x k u k k x k k u k
1 , 1 1 1
, 1 1 24
x
y
The time-varying matrices of the linearised model, all of di-
mensionality 3 Â 3, are calculated analytically on-line by differ-
entiating the nonlinear model (11a)–(11b)
( ) =
( ( − ) ( − ))
( − )
( ) =
( ( − ) ( − ))
( − )
( ) =
( ( ) ( ))
( )
( ) =
( ( ) ( ))
( )
( )
( − )= ¯( − )
( − )= ¯( − )
( − )= ¯( − )
( − )= ¯( − )
( )= ¯( )
( )= ¯( − )
( )= ¯( )
( )= ¯( − )
A
B
C
D
k
f x k u k
x k
k
f x k u k
u k
k
h x k u k
x k
k
h x k u k
u k
d 1 , 1
d 1
d 1 , 1
d 1
d ,
d
d ,
d
25
x k x k
u k u k
x k x k
u k u k
x k x k
u k u k
x k x k
u k u k
1 1
1 1
1 1
1 1
1
1
Taking into account the linearisation rules (25) and the model
state Eq. (7) of the boiler-turbine system, one obtains
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
( )
(
)
( ) =
+ − ¯ ( − ) ¯ ( − )
¯ ( − ) ¯ ( − )
− ¯ ( − )
−
− ¯ ( − ) +
( )
A k
u k x k T
u k x k
x k T
T
u k
T
1 0.002025 1 1 0 0
0.082125 1 1
0.018 1
1 0.1 0
1.1 1 0.19
85
0 1
26
2 1
1/8
s
2 1
1/8
1
1/8
s
s
2
s
and
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
( )
( )( ) =
− ¯ ( − ) −
¯ ( − )
− ¯ ( − )
( )
B k
T T x k T
x k T
x k
T T
0.9 0.0018 1 0.15
0 0.073 1 0
0
1.1 1
85
141
85 27
s s 1
9/8
s
1
9/8
s
1
s s
Taking into account Eqs. (25) and the model output Eq. (8) of the
process, one has
where, from Eq. (9a)
α α∂ ( )
∂ ( )
=
( )
( )
∂ ( )
∂ ( )
=
( )
( )
( )
( )= ¯ ( )
( )= ¯ ( )
( )= ¯ ( )
( )= ˜ ( )
k
x k
N k
D k
k
x k
N k
D k
,
29
x k x k
x k x k
x k x k
x k x k
cs
1
1 cs
3
3
1 1
3 3
1 1
3 3
and
( )
( ) = ( − ˜ ( ))( ˜ ( ) − ¯ ( ) ˜ ( )) − ( ¯ ( )
− − ¯ ( ) ˜ ( ) + ˜ ( ))( − ˜ ( ))
( ) = (− ¯ ( ) + )( ˜ ( ) − ¯ ( ) ˜ ( )) − ( ¯ ( )
− − ¯ ( ) ˜ ( ) + ˜ ( ))( − ¯ ( ))
( ) = ( ˜ ( ) − ¯ ( ) ˜ ( )) 30
N k x k x k x k x k x k
x k x k x k x k
N k x k x k x k x k x k
x k x k x k x k
D k x k x k x k
0.8 0.0012304 1.0394 0.0012304 0.8
25.6 0.0012304 0.0393728 0.0012304
0.0012304 0.0393728 1.0394 0.0012304 0.8
25.6 0.0012304 0.0393728 1.0394 0.0012304
1.0394 0.0012304
1 3 3 1 3 1
1 3 3 3
3 1 3 1 3 1
1 3 3 1
3 1 3
2
From Eqs. (25), (8) and (9b), it follows that
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
( ) =
¯ ( − ) −
( )
D k
x k
0 0 0
0 0 0
2.2795
9
0.0427
9
1
0.1257
9 31
1
4.2. State and output prediction
In this work the state disturbance model discussed in [38,39] is
used. The predicted state vector for the sampling instant +k p is
calculated at the current instant k from
ν^( + | ) = ( + | ) + ( ) ( )x k p k x k p k k 32
where ( + | )x k p k denotes the state vector obtained from the model
and it is assumed that the same state disturbance ν ( )k acts on the
process over the whole prediction horizon. The unknown dis-
turbance vector may be assessed as the difference between the
state variables ( )x k1 , ( )x k2 (measured), ˜ ( )x k3 (estimated) and the
state variables calculated from the linearised state Eq. (23a) for the
sampling instant k
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
ν
ν
ν
ν
δ( ) =
( )
( )
( )
= ( ) − ( ) ( − ) − ( ) ( − ) − ( )
( )
A Bk
k
k
k
x k k x k k u k k1 1
33
1
2
3
x
The first two state variables are measured whereas the third one is
observed, which means that ⎡⎣ ⎤⎦( ) = ( ) ( ) ˜ ( )x k x k x k x k1 2 3
T
,
⎡⎣ ⎤⎦( − ) = ( − ) ( − ) ˜ ( − )x k x k x k x k1 1 1 11 2 3
T
. Similary to Eq. (32), the
predicted output for the sampling instant +k p is calculated at the
current instant k from
^ ( + | ) = ( + | ) + ( ) ( )y k p k y k p k d k 343 3
where ( + | )y k p k3 denotes the output obtained from the linearised
model and it is assumed that the same output disturbance d(k)
acts on the process over the whole prediction horizon. Taking into
account Eq. (23b), the output disturbance is assessed from
δ( ) = ( ) − ( ) ( ) − ( ) ( − ) − ( ) ( )C Dd k y k k x k k u k k1 353 y
Because the signals u(k) are not known before optimisation, the
vector ( − )u k 1 from the previous sampling instant is used. The
state and output predictions are calculated in a relatively
straightforward way from Eqs. (32) and (34), the state and output
disturbances are estimated from Eqs. (33) and (35). The prediction
equations are very similar to those used in MPC in which input-
output models are used for prediction. The discussed approach,
although simple, guarantees offset-free control, even in the case of
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
α α( ) = ∂ ( )
∂ ( )
+
¯ ( − ) −
+
∂ ( )
∂ ( )
( )
( )= ¯ ( )
( )= ¯ ( )
( )= ¯ ( )
( )= ˜ ( )
C k k
x k
u k k
x k
1 0 0
0 1 0
5
0.0427 1 0.00735
9
0 0.0065365 5
28
x k x k
x k x k
x k x k
x k x k
cs
1
2 cs
31 1
3 3
1 1
3 3
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495 481
7. deterministic constant-type disturbances acting on the process
and modelling errors, as proved and demonstrated in [38]. The
alternative offset-free formulations of MPC based on state-space
models require augmenting the process state by the states of de-
terministic disturbances [21,22,29] or using the velocity form
state-space model, in which the extended state consists of state
increments and the output signals [21].
From the state prediction Eq. (32) and using the linearised state
model (23a), state predictions for the consecutive sampling in-
stants are
δ ν
δ ν
δ ν
^( + | ) = ( ) ( ) + ( ) ( | ) + ( ) + ( )
^( + | ) = ( )^( + | ) + ( ) ( + | ) + ( ) + ( )
^( + | ) = ( )^( + | ) + ( ) ( + | ) + ( ) + ( )
⋮
A B
A B
A B
x k k k x k k u k k k k
x k k k x k k k u k k k k
x k k k x k k k u k k k k
1
2 1 1
3 2 2
x
x
x
The state predictions may be expressed as functions of the incre-
ments of the future control increments (i.e. the decision variables
of MPC)
δ ν
ν
δ
ν δ
^( + | ) = ( ) ( ) + ( )▵ ( | ) + ( ) ( − ) + ( ) + ( )
^( + | ) = ( ) ( ) + ( ( ) + ) ( )▵ ( | ) + ( )▵ ( + | )
+ ( ( ) + ) ( ) ( − ) + ( ( ) + ) ( )
+ ( ( ) + ) ( )
^( + | ) = ( ) ( ) + ( ( ) + ( ) + ) ( )▵ ( | )
+ ( ( ) + ) ( )▵ ( + | ) + ( )▵ ( + | )
+ ( ( ) + ( ) + ) ( ) ( − )
+ ( ( ) + ( ) + ) ( ) + ( ( ) + ( ) + ) ( )
⋮
×
× ×
×
×
×
×
× ×
A B B
A A I B B
A I B A I
A I
A A A I B
A I B B
A A I B
A A I A A I
x k k k x k k u k k k u k k k
x k k k x k k k u k k k u k k
k k u k k k
k k
x k k k x k k k k u k k
k k u k k k u k k
k k k u k
k k k k k k
1 1
2 1
1
3
1 2
1
x
2
3 3
3 3 3 3
3 3 x
3 2
3 3
3 3
2
3 3
2
3 3
2
3 3 x
where ×I3 3 stands for an identity matrix of dimensionality 3Â 3. The
predicted state trajectory over the prediction horizon (the vector of
length N3 ) may be expressed in the following compact manner
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
ν δ
^( ) =
^( + | )
⋮
^( + | )
= ( ) ( ) + ( ) ( − )
+ ( )( ( ) + ( )) + ( )▵ ( ) ( )
͠ ∼
x A B
V P u
k
x k k
x k N k
k x k k u k
k k k k k
1
1
36x
The matrices ( )͠A k , ( )
∼
B k and ( )V k , of dimensionality ×N3 3, are
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥∑
( ) =
( )
⋮
( )
( ) =
( ) +
⋮
( ) +
( ) = ( ) ( )
( )
͠ ∼
×
×
=
−
×
A
A
A
V
I
A I
A I
B V Bk
k
k
k
k
k
k k k, ,
37
N
i
N
i
3 3
3 3
1
1
3 3
whereas the matrix ( )P k , of dimensionality ×N N3 3 u, is
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
∑
∑
∑ ∑
( ) =
( ) …
( ( ) + ) ( ) …
⋮ ⋱ ⋮
( ) + ( ) … ( )
( ) + ( ) … ( ( ) + ) ( )
⋮ ⋱ ⋮
( ) + ( ) … ( ) + ( )
( )
×
× ×
=
−
×
=
× ×
=
−
×
=
−
×
P
B 0
A I B 0
A I B B
A I B A I B
A I B A I B
k
k
k k
k k k
k k k k
k k k k
38
i
N
i
i
N
i
i
N
i
i
N N
i
3 3
3 3 3 3
1
1
3 3
1
3 3 3 3
1
1
3 3
1
3 3
u
u
u
where ×03 3 stands for a zeros matrix of dimensionality 3Â 3.
From the output prediction Eq. (34), using the linearised output
model (23b) and the predicted state trajectory (36), predictions of
the third output variable for the consecutive sampling instants
over the prediction horizon are
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
δ
^ ( ) =
^ ( + | )
⋮
^ ( + | )
= ( ) ^( ) + ( ) ( )
+ ( ( ) + ( )) ( )
͠∼
×
y C x D u
I
k
y k k
y k N k
k k k k
k d k
1
39N
3
3
3
N
y 1
where the block-diagonal matrices are
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
( ) =
( ) …
⋮ ⋱ ⋮
… ( )
( ) =
( ) …
⋮ ⋱ ⋮
… ( )
͠∼
×
×
×
×
C
C 0
0 C
D
D 0
0 D
k
k
k
k
k
k
,
3 1 3
1 3 3
3 1 3
1 3 3
and the submatices ( )C k3 and ( )D k3 contains the third row of the
matrices ( )C k and ( )D k , respectively. The vector ( )u kN , of length
N3 , consists of all future control values over the prediction hor-
izon. It may be easily found from the future control increments
defined over the control horizon
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
( ) =
( + | )
⋮
( + | )
= ▵ ( ) + ( − )u J u uk
u k k
u k N k
k k
1
1N N N
The entries of the matrix JN, of dimensionality ×N N3 3 u, are
⎧
⎨
⎩
=
< +
≥ +
J
j i
j i
1 if 2
0 if 2i j,
N
and the vector ( − )u k 1N , of length N3 , is comprised of the values
of the manipulated variables from the previous sampling instant
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
( − ) =
( − )
⋮
( − )
u k
u k
u k
1
1
1
N
Using the state prediction Eq. (36), the output predictions given by
Eq. (39) may be rewritten in the form
⎡
⎣
⎤
⎦
( )
ν δ
δ
^ ( ) = ( ) ( ) ( ) + ( ) ( − ) + ( )( ( ) + ( )
+ ( ( ) ( ) + ( ) )▵ ( ) + ( ) ( − ) + ( ( ) + ( ))͠ ͠
∼
∼
∼∼
× 40
y C A B V
C P D J u D u I
k k k x k k u k k k k
k k k k k k k d k
1
1 N
3 x
N N
y 1
4.3. Quadratic programming MPC-SL optimisation problem
It is interesting to notice that as a result of model linearisation,
predictions of state and output variables (Eqs. (36) and (40)) are
linear functions of the future control increments ▵ ( )u k , i.e. the
decision variables of MPC. In consequence, it is possible to obtain
MPC quadratic optimisation problems, in which the minimised
cost-function is quadratic and the constraints are linear with re-
spect to the decision variables. The first version of the rudimentary
MPC optimisation problem (19) can be transformed to the fol-
lowing quadratic optimisation task
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495482
8. ⎧
⎨
⎩
⎡
⎣
⎤
⎦
}
( )
ε
ν δ
ρ ε ρ
ε ν δ
δ ε
ε ε
( ) − ( ) ( ) − ( ) ( − ) − ( )( ( ) + ( )) − ( )▵ ( )
+ ▵ ( ) + ( ( )) + ( ( ))
≤ ▵ ( ) + ( − ) ≤
▵ ≤ ▵ ( ) ≤ ▵
− ( ) ≤ ( ) ( ) ( ) + ( ) ( − ) + ( )( ( ) + ( )
+ ( ( ) ( ) + ( ) )▵ ( ) + ( ) ( − )
+ ( ( ) + ( )) ≤ + ( )
( ) ≥ ( ) ≥
͠
͠
͠ ͠
∼
∼
∼
∼
ε
ε
Λ
▵ ( )
( )
( )
×
× ×
41
x A B V P u
u
u J u u u
u u u
y I C A B V
C P D J u D u
I y I
k k x k k u k k k k k k
k k k
k k
k
k k k x k k u k k k k
k k k k k k
k d k k
k k
min 1
subject to
1
1
1
0, 0
u Mk
k
k
N
N N
min
max
sp
x
2
2 min min 2 max max 2
min max
min max
3
min min
1 x
N N
y 1 3
max max
1
min max
where the vector
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
( − ) =
( − )
⋮
( − ) ( )
u k
u k
u k
1
1
1 42
is of length N3 u and the matrix
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
…
…
⋮ ⋮ ⋱ ⋮
… ( )
× × ×
× × ×
× × ×
J
I 0 0
I I 0
I I I 43
3 3 3 3 3 3
3 3 3 3 3 3
3 3 3 3 3 3
is of dimensionality ×N N3 3u u. When different violations of the
predicted output constraints may be adjusted for consecutive
sampling instants of the prediction horizon, the general MPC op-
timisation problem (21) becomes
⎧
⎨
⎩
⎡
⎣
⎤
⎦
}
( )
ε ε
ε
ε
ε ε
ν δ
ρ ρ
ν δ
δ
( ) − ( ) ( ) − ( ) ( − ) − ( )( ( ) + ( )) − ( )▵ ( )
+ ▵ ( ) + ( ) + ( )
≤ ▵ ( ) + ( − ) ≤
▵ ≤ ▵ ( ) ≤ ▵
− ( ) ≤ ( ) ( ) ( ) + ( ) ( − ) + ( )( ( ) + ( )
+ ( ( ) ( ) + ( ) )▵ ( ) + ( ) ( − )
+ ( ( ) + ( )) ≤ + ( )
( ) ≥ ( ) ≥
͠
͠
͠ ͠
∼
∼
∼
∼
ε
ε
Λ
▵ ( )
( )
( )
×
× × 44
x A B V P u
u
u J u u u
u u u
y C A B V
C P D J u D u
I y
0 0
k k x k k u k k k k k k
k k k
k k
k
k k k x k k u k k k k
k k k k k k
k d k k
k k
min 1
subject to
1
1
1
,
u Mk
k
k
N
N N
min
max
sp
x
2
2 min min 2 max max 2
min max
min max
3
min min
x
N N
y 1 3
max max
min
1
max
1
4.4. MPC-SL algorithm summary
The following steps are repeated at each sampling instant k of
the MPC-SL algorithm:
1. The output values ( ) = ( )y k x k1 1 , ( ) = ( )y k x k2 2 , ( − )y k 13 are
measured, the state variable ˜ ( )x k3 is estimated using a state
observer.
2. A local linear approximation (Eqs. (22a), (22b)) of the nonlinear
model (Eqs. (7), (8), (9a), (9b)) of the process is calculated for
the current operating point, i.e. the matrices ( )A k , ( )B k , ( )C k ,
( )D k and are found from Eqs. (26), (27), (28), (29), (30), (31),
whereas the quantities δ ( )kx , δ ( )ky are found from Eq. (24).
3. The matrices ( )͠A k , ( )V k and ( )
∼
B k are calculated from Eqs. (37),
the matrix ( )P k is calculated from Eq. (38).
4. The state disturbance vector ν ( )k is estimated from Eq. (33), the
output disturbance vector d(k) is calculated from Eq. (35).
5. The quadratic optimisation problem (41) or (44) is solved to
calculate the decision variables of the algorithm.
6. The first 3 elements of the calculated sequence ▵ ( )u k are ap-
plied to the process, i.e. ( ) = ▵ ( | ) + ( − )u k u k k u k 1 ,1 1 1 ( )=u k2
▵ ( | ) + ( − )u k k u k 12 2 , ( ) = ▵ ( | ) + ( − )u k u k k u k 13 3 3 .
7. At the next sampling instant ( = +k k: 1) the algorithm goes to
step 1.
4.5. State estimation
When state-space models are used in MPC of the boiler-turbine
process, state estimation is necessary. In this work the extended
Kalman filter [34] is used for state estimation because it is char-
acterised by excellent estimation quality and low computational
complexity. Because the same state-space model is used for pro-
cess simulation and in MPC, at the sampling instant k the current
value of the output signal ( )y k3 is not available for measurement,
the most recent available signal comes from the previous sampling
instant, −k 1. Considering the model (12), it is because the output
( )y k3 depends on the inputs ( )u k1 , ( )u k2 , ( )u k3 , which are calculated
at the sampling instant k after state estimation. That is why the
measurement vector is
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
( ) =
( )
( )
( )
=
( )
( )
( − )
=
( )
( )
( ( − ) ( − ) ( − ) ( − ) ( − ))
z k
z k
z k
z k
y k
y k
y k
x k
x k
g x k x k u k u k u k
1
1 , 1 , 1 , 1 , 1
1
2
3
1
2
3
1
2
3 1 3 1 2 3
In order to deal with delayed measurement, the state augmenta-
tion technique discussed in [5] is used. For state estimation, the
following augmented model is used
( + ) = ( ( ) ( )) + ( ) ( )x k f x k u k w k1 , 45aa a a a
( ) = ( ( ) ( )) + ( ) ( )z k g x k u k v k, 45ba a a a
where w(k) and v(k) are the process and observation (measure-
ment) noises, respectively. They are assumed to be zero mean,
Gaussian and uncorelated noises with covariance matrices
( ) = ([ ( ) ( )])Q k E w k w kT and ( ) = ([ ( ) ( )])R k E v k v kT , respectively. The
augmented state and input vectors are
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
( ) =
( )
( )
( )
( − )
( − )
( − )
( ) =
( )
( )
( )
( − )
( − )
( − )
x k
x k
x k
x k
x k
x k
x k
u k
u k
u k
u k
u k
u k
u k
1
1
1
,
1
1
1
a
1
2
3
1
2
3
a
1
2
3
1
2
3
Taking into account Eqs. (12), the augmented state Eq. (45a) be-
comes
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
( + ) = ( ( ) ( ))
=
( ( − ) ( − ) ( − ) ( − ))
( ( − ) ( − ) ( − ))
( ( − ) ( − ) ( − ) ( − ))
( )
( )
( )
x k f x k u k
f x k u k u k u k
f x k x k u k
f x k x k u k u k
x k
x k
x k
1 ,
1 , 1 , 1 , 1
1 , 1 , 1
1 , 1 , 1 , 1
a a a a
1 1 1 2 3
2 1 2 2
3 1 3 2 3
1
2
3
State estimation requires successive on-line model linearisation
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495 483
9. ( − ) =
∂ ( ( − ) ( − ))
∂ ( − )
( ) =
∂ ( ( ) ( ))
∂ ( )
( − )= ˜( − | − )
( )= ˜( | − )
F
H
k
f x k u k
x k
k
g x k u k
x k
1
1 , 1
1
,
x k x k k
x k x k k
1 1 1
1
One may notice that linearisation for the MPC-SL algorithm and
for state estimation is performed in a very similar way. The matrix
( − ) = ( )F Ak k1 is calculated from Eq. (26). The matrix
( − ) = ( )H Ck k1 is calculated from Eqs. (28), (29) and (30), but for
calculation of the matrix ( − )H k 1 the operating point is defined
by the quantities ( − )x k 11 , ( − )x k 12 and ˜ ( − )x k 13 whereas for
model linearisation in the MPC-SL algorithm by the quantities
( )x k1 , ( )x k2 and ˜ ( )x k3 . When constant matrices F and H are used,
one obtains the Kalman filter for linear systems. The matrices of
the linearised model with the augmented state are easily calcu-
lated from the matrices which describe a linear approximation of
the rudimentary model used in the MPC-SL algorithm:
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
( ) =
( )
( )
=
( − ) ( − ) ( − )
×
× ×
F
A 0
0 0
H
H H H
k
k
k
k k k
,
1 0 0 0 0 0
0 1 0 0 0 0
0 0 0 1 1 1
a 3 3
3 3 3 3
a
3,1 3,1 3,1
The following steps are repeated for state estimation at each
sampling instant k (before execution of the MPC-SL algorithm)
1. The predicted state estimate is found from ˜( | − )=x k k 1
(˜( − | − ) ( − ))f x k k u k1 1 , 1 .
2. The predicted covariance estimate is calculated from
( | − ) = ( − ) ( − | − )( ( − )) + ( − )P F P F Qk k k k k k k1 1 1 1 1 1a a T .
3. The measurement residual is determined from ˜( ) = ( )−z k z k
(˜( | − ) ( − ))g x k k u k1 , 1 .
4. The residual covariance is found from ( ) = ( ) ( | −S H Pk k k ka
)( ( )) + ( )H Rk k1 a T .
5. The filter gain is calculated from ( ) =K k ( | − )P k k 1
( ( )) ( )−H Sk ka T 1 .
6. The state estimate is updated from ˜( | ) =x k k ˜( | − ) + ( ) ˜( )Kx k k k z k1 .
7. The covariance estimate is updated from ( | ) = ( − ( )P I K Hk k k a
( )) ( | − )Pk k k 1 .
5. Simulation results
The discrete-time dynamic model described by Eqs. (7), (8), (9a)
and (9b) is used for simulation of the process (i.e. as the simulated
process) and in MPC (for on-line linearisation and prediction). All
simulations are carried out in Matlab. Parameters of all algorithms
are the same: μ = 30p1, , μ = 1p2, , μ = 0p3, for = …p N1, , , λ = 1n p, for
=n 1, 2, 3 and = … −p N0, , 1u , ρ ρ= = 10min max 6. The extended
Kalman filter has the following parameters: ( | ) = ×P I1 0 6 6,
= ×Q I100 6 6, = ×R I0.0001 3 3.
Different operating points of the process are detailed in [1]. For
further simulations the operating points #1, #3 and #7 are chosen.
Considering control objectives described in Section 2, one may
easily check that the output constraints (4) are not satisfied at the
operating points described in [1]. That is why three modified op-
erating points are used in this study, the values of state, input and
output variables of these points are given in Table 1. The values of
the first two state variables are the same as in [1], but different
values of the third state variable and different values of process
inputs are necessary to guarantee that the output constraints are
satisfied. The chosen operating conditions are named operating
points A, B and C, respectively.
5.1. Model simulations
At first it is interesting to compare effectiveness of suc-
cessive model linearisation. Fig. 1 compares state and output
trajectories caused by step changes of the manipulated
variables u1, u2, u3 by δ = ± ± ±0.1, 0.2, 0.3 occurring at
k ¼100 of two models: the full nonlinear model and the suc-
cessively linearised one are considered (both in the open-loop
mode, i.e. with no controller). The initial operating point is B.
From the presented simulations two conclusions may be draw.
Firstly, the successively linearised model gives practically the
same trajectories as the full nonlinear model, which means
that local linearisation may be efficiently used for model ap-
proximation. Secondly, it is necessary to notice that the re-
sponses for positive and negative changes in the manipulated
variables have different shapes and final values, which means
that both dynamic and steady-state properties of the process
are nonlinear. It motivates development of nonlinear MPC
algorithms.
5.2. Selection of horizons
Selection of proper prediction and control horizons in MPC is
an important task. On the one hand, the horizons must be long
enough to give good control quality, on the other hand, they must
be short to minimise computational complexity. It is an interesting
issue to demonstrate that for the considered process it is sufficient
to use quite short prediction and control horizons in MPC. In this
work the horizons are determined experimentally, i.e. simulations
are carried out for different horizons and quality of control is as-
sessed. For this purpose two Sum of Squared Errors (SSE) perfor-
mance indices are used. They are calculated after completing si-
mulations of MPC in order to compare the influence of the tuning
parameters. The first performance index measures control errors
of the first two state variables
∑ ∑= ( ( ) − ( ))
( )= =
x k x kSSE
46n p
N
n nx
1
2
1
sp 2
sim
where ( )x kn
sp
denotes the state set-point whereas xn(k) is the ac-
tual value of the corresponding state variable calculated during
simulations, Nsim is the simulation horizon. It is necessary to point
out that the SSEx index is actually not minimised in MPC.
The second index measures the degree of hard output constraints
(Eq. (15)) violation
∑= ( − ( ( ))) ( ( ( ) − ))
( )=
y k ySSE 0.1 abs sgn max abs 3 0.1, 0
47p
N
y
1
3
2
sim
Table 1
Operating points of the boiler-turbine.
Variable Operating point A Operating point B Operating point C
=x y1 1 ×7.56 101 ×9.72 101 ×1.4 102
=x y2 2 ×1.53 101 ×5.05 101 ×1.28 102
x3 ×5.0897 102 ×4.6951 102 ×3.2368 102
u1 × −1.1926 10 1 × −2.7049 10 1 × −5.9589 10 1
u2 × −3.8063 10 1 × −6.2082 10 1 × −8.9447 10 1
u3 × −1.2262 10 1 × −3.3979 10 1 × −7.8829 10 1
y3 × −9.8414 10 2 × −9.7038 10 2 × −9.7970 10 2
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495484
10. Firstly, the influence of the prediction horizon on control
quality and constraint satisfaction is assessed. Table 2 shows ac-
curacy of the MPC-SL algorithm in terms of the performance in-
dices SSEx and SSEy for different prediction horizons for set-point
change from the operating point B to C (for different set-point
changes the conclusions are the same). A sufficiently long control
horizon =N 5u is used (its choice is explained next), real state
measurements are used (i.e. no observer is necessary), no con-
straints imposed on the rate of change of the manipulated vari-
ables are taken into account, the output constraints are softened
by means of two additional variables ε ( )kmin and ε ( )kmax , as in the
MPC-SL optimisation problem (19). Fig. 2 depicts selected simu-
lation results, for prediction horizons N¼5, N¼10 and N¼50.
The best control quality is obtained for the shortest horizon N¼5,
but for the steady-state, starting from the sampling instant k¼200,
Fig. 1. State and output trajectories caused by step changes of the manipulated variables u1, u2, u3 by δ = ± ± ±0.1, 0.2, 0.3 at k¼100: the nonlinear model (solid line with
dots) vs. the successively linearised model (dashed line with circles).
Table 2
Accuracy of the MPC-SL algorithm in terms of the performance indices SSEx and
SSEy for different prediction horizons N for set-point change from the operating
point B to C; =N 5u (real state measurements are used, no constraints imposed on
the rate of change of the manipulated variables are taken into account).
N SSEx SSEy
5 ×1.0858 105 7.7740
10 ×1.0918 105 7.6435
20 ×1.0957 105 7.7306
30 ×1.0959 105 7.7328
40 ×1.3034 105 7.7352
50 ×1.5222 105 8.0571
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495 485
11. the algorithm is unable to satisfy the constraints imposed on
the output variable y3. The best constraint satisfaction is possible
for N¼10, moderate lengthening the horizon results in practically
the same performance, but for very long horizons, e.g. N¼50, the
quality deteriorates. It is because for long-range prediction the
same linear approximation of the model is used and for long
horizons a significant discrepancy between the prediction calcu-
lated by a locally linearised model and the real process trajectory is
possible. Because for the prediction horizons < <N10 45 there is
no important difference in quality of control and constraint sa-
tisfaction, the quite short horizon N¼10 is finally chosen. Although
the very short horizons, e.g. N¼5, are also possible, they are not
recommended as the MPC algorithm with a too short horizon may
only work in the ideal situation with no disturbances, modelling
errors and noise.
Secondly, the influence of the control horizon on control
quality and constraint satisfaction is assessed. Table 3 shows
accuracy of the MPC-SL algorithm in terms of the performance
indices SSEx and SSEy for different control horizons for set-
point change from the operating point B to C. The prediction
horizon is fixed, N¼10, real state measurements are used, no
constraints imposed on the rate of change of the manipulated
variables are taken into account. Fig. 3 depicts selected simu-
lation results, for control horizons =N 1u , =N 2u and =N 5u . For
the shortest horizon =N 1u the trajectories are completely
different from those obtained for ≥N 2u . It is interesting to
notice that increasing the length of the control horizon (i.e. for
>N 2u ) does not lead to any improvement of control perfor-
mance and control satisfaction ability. That is why, in order to
reduce computational complexity of MPC, the control horizon
is fixed to =N 2u . Taking into account the simulations discussed
so far, in the following part of the paper the horizons are:
N¼10, =N 2u .
5.3. Classical MPC based on linear models
Poor performance of classical linear MPC algorithms applied
to the considered boiler-turbine process is demonstrated else-
where, e.g. simulation results for the DMC algorithm are given
in [26]. Nevertheless, it is interesting to evaluate the linear
Fig. 2. Simulation results: the nonlinear MPC-SL algorithm for set-point change from the operating point B to C for different prediction horizons: N¼5 (solid line), N¼10
(dashed line), N¼50 (dotted line); =N 5u (real state measurements are used, no constraints imposed on the rate of change of the manipulated variables are taken into
account).
Table 3
Accuracy of the MPC-SL algorithm in terms of the performance indicies SSEx and
SSEy for different control horizons Nu for set-point change from the operating point
B to C; N¼10 (real state measurements are used, no constraints imposed on the
rate of change of the manipulated variables are taken into account).
Nu SSEx SSEy
1 ×7.566243 105 0.5865
2 ×1.089781 105 7.6243
3 ×1.090982 105 7.6390
4 ×1.091857 105 7.6457
5 ×1.091822 105 7.6435
10 ×1.091564 105 7.6556
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495486
12. state-space MPC algorithm with the output constraints. Sa-
tisfaction of constraints imposed on the values and on the rate of
change of the manipulated variables is quite simple as such
constraints do not lead to an empty feasible set. Because a linear
model is only a very rough approximation of the process, when it
is used for prediction of the output y3, serious problems are
likely. The MPC algorithm based on linear models described in
[38] is used, it uses the same disturbance models as the MPC-SL
algorithm described in this work, but it is entirely based on
parameter-constant linear models. For the chosen operating
points (Table 1), using Eqs. (26), (27), (28), (29), (30), (31), the
matrices of the linear model are easily calculated off-line. For the
operating point A one obtains the matrices
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
×
×
− ×
=
− × −
− ×
=
× ×
=
× × − ×
−
−
−
−
−
− −
− − −
A
B
C
D
9.9868 10 0 0
2.2768 10 0.9 0
2.6904 10 0 1
,
0.9 2.3367 10 0.15
0 9.4768 0
0 9.7836 10 1.6588
,
1 0 0
0 1 0
2.8951 10 0 5.8251 10
,
0 0 0
0 0 0
2.5328 10 3.5868 10 1.3967 10
1
2
3
1
1
3 3
1 1 2
for the point B
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
×
×
− ×
=
− × −
×
−
=
× ×
=
× × − ×
−
−
−
−
− −
− − −
A
B
C
D
9.9777 10 0 0
5.8449 10 0.9 0
5.7989 10 0 1
,
0.9 3.1002 10 0.15
0 1.2573 10 0
0 1.2579 1.658823
,
1 0 0
0 1 0
4.927322 10 0 5.250236 10
,
0 0 0
0 0 0
2.53278 10 4.6116 10 1.3967 10
1
2
3
1
1
3 3
1 1 2
and for the point C
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
×
×
− ×
=
− × −
×
− ×
=
× − ×
=
× × − ×
−
−
−
−
−
− −
− − −
A
B
C
D
9.9664 10 0 0
1.0286 10 0.9 0
9.3402 10 0 1
,
0.9 4.6738 10 0.15
0 1.8955 10 0
0 1.8118 10 1.6588
,
1 0 0
0 1 0
1.800770 10 0 2.584602 10
,
0 0 0
0 0 0
2.5328 10 6.6422 10 1.3967 10
1
1
3
1
1
1
2 3
1 1 2
Fig. 3. Simulation results: the nonlinear MPC-SL algorithm for set-point change from the operating point B to C for different control horizons: =N 1u (solid line), =N 2u
(dashed line), =N 5u (dotted line); N¼10 (real state measurements are used, no constraints imposed on the rate of change of the manipulated variables are taken into
account).
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495 487
13. Fig. 4 depicts simulation results of the linear MPC algorithm for
set-point change from the operating point B to C when for pre-
diction the linear model for the operating point B is used. For
simplicity real state measurements are used and no constraints
imposed on the rate of change of the manipulated variables are
taken into account. Unfortunately, due to modelling errors the
algorithm is unable to guarantee that the state variables x1 and x2
follow precisely their set-points x1
sp
and x2
sp
, respectively. Fur-
thermore, the output variable y3 does not satisfy its constraints.
Unwanted oscillations may be eliminated by increasing the
penalty coefficients λn p, from their default value 1 to 100, but it
does not lead to any better set-point tracking and satisfaction of
the output constraints. Fig. 5 depicts simulation results of the
linear MPC algorithm when the linear model for the operating
point C is used with the same simplifications. When the output
constraints imposed on the variable y3 are taken into account, the
algorithm is unable to provide good state set-point tracking (in
particular of the variable x2) and the output constrains are not
satisfied. Better state set-point tracking may be achieved by re-
moving the output constraints from the MPC optimisation pro-
blem, but they are crucial from the technological point of view.
The same problems are encountered when the set-point is
changed from the operating point A to C. For example, Fig. 6
shows simulation results of the linear MPC algorithm based on
the linear model for the operating point C. When compared to
Fig. 5, due to big set-point change, input, state and output vari-
ables are characterised by bigger damped oscillations. All things
considered, the MPC algorithm based on linear models are unable
to provide offset-free control and satisfaction of the output
constraints.
5.4. Nonlinear MPC based on nonlinear model
The next simulations are concerned with two nonlinear MPC
strategies:
a) the MPC scheme with Nonlinear Optimisation (MPC-NO) re-
peated at each sampling instant, in which the full nonlinear
model of the process is used (for optimisation the Sequential
Quadratic Programming (SQP) method is used [23]),
b) the discussed MPC-SL scheme with on-line model linearisa-
tion, which means that a quadratic optimisation problem is
solved at each sampling instant (for optimisation the active set
method is used [23]).
Both algorithms use the same model of the process, but in a
different way. The figures presented next show the trajectories of
MPC algorithms in which the output constraints are softened by
means of two additional variables ε ( )kmin and ε ( )kmax , as in the
MPC-SL optimisation problem (19). Fig. 7 depicts simulation
results of MPC-NO and MPC-SL algorithms for set-point change
from the operating point B to C. For simplicity the same assump-
tions as in the case of the linear MPC algorithm are used, i.e. real
state measurements are used and no constraints imposed on the
rate of change of the manipulated variables are taken into account.
Unlike the results of the linear MPC algorithm (Figs. 4 and 5), the
state set-points are tracked precisely, with no steady-state error,
there are no oscillations. Furthermore, the constraints of the
output variable y3 are satisfied in some 2/3 period of the simula-
tion. In the first 1/3 period of simulation, when the state operating
point changes quickly, the hard output constraints are not satis-
fied, but it is necessary to enforce existence of a feasible set of the
MPC optimisation task (i.e. the additional variables ε ( ) >k 0min
Fig. 4. Simulation results: the linear MPC algorithm for set-point change from the operating point B to C (the linear model for the operating point B is used, real state
measurements are used, no constraints imposed on the rate of change of the manipulated variables are taken into account).
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495488
14. and ε ( ) >k 0min ). It is necessary to point out the fact that the MPC-
SL algorithm with on-line model linearisation and nonlinear
optimisation gives practically the same trajectories as the compu-
tationally demanding “ideal” MPC-NO algorithm with nonlinear
optimisation. Fig. 8 compares trajectories of both nonlinear MPC
control schemes for set-point change from the operating point A
to C. Unlike the linear algorithm (Fig. 6), they give good set-point
tracking and satisfaction of the output constraints. Moreover, the
trajectories of the MPC-SL algorithm are the same as in the MPC-
NO one. It is interesting to notice that for some portion of the
simulations the optimal values of the manipulated variables reach
their magnitude constraints which is necessary for fast transition
from the initial to the final operating point.
Figs. 9 and 10 compare MPC-NO and MPC-SL algorithms for set-
point changes from the operating point B to C and from the point A
to C, respectively, but the constraints imposed on the rate of
change of the manipulated variables are additionally taken into
account. For simplicity real state measurements are still used. In
comparison with Figs. 7 and 8 the state trajectories are sig-
nificantly slower, but it is caused by quite slow changes of the
input trajectories, which is enforced by the input rate constraints.
Also in the case of the additional rate constraints the MPC-SL al-
gorithm gives trajectories very similar to those possible in the
MPC-NO one. Comparing the situation when no rate constraints of
the manipulated variables are taken into account, i.e. Figs. 7 and 8,
introduction of such constraints reduces the time when the ma-
nipulated variables reach their magnitude constraints.
In all the simulation presented so far real state (undisturbed)
measurements are used. In reality state estimation of the variable
x3 is necessary. For this purpose the Extended Kalman Filter de-
scribed in Section 4.5 is used. Figs. 11 and 12 compare MPC-NO and
MPC-SL algorithms for set-point changes from the operating point
B to C and from the point A to C, respectively. All the constraints
imposed on the inputs, on the rate of change of the inputs and on
the output y3 are taken into account. It is assumed that the initial
condition of the process is not known, the initial state of the filter
is ⎡⎣ ⎤⎦˜ =x 50 25 300
T
. Additionally, the process is affected by un-
measured step disturbances which act on its inputs: at the sam-
pling instant k¼250 the step applied to the first input changes
from 0 to 0.3, at the instant k¼350 the step applied to the second
input changes from 0 to 0.2. Moreover, the process outputs y1, y2
and y3 are affected by noise with normal distribution with zero
mean and standard deviation 0.01, 0.01 and 0.001, respectively.
The simulation horizon is lengthen. It is evident that the nonlinear
MPC algorithms are able co compensate for disturbances and
noise, the wrong initial condition of the state observer does not
lead to any significant deterioration of set-point tracking and
constraints satisfaction. Also in this case the MPC-SL algorithm
with quadratic optimisation gives the trajectories almost the same
as the MPC-NO one.
Fig. 13 compares real and estimated state trajectories in the
nonlinear MPC-SL algorithm for two considered set-point changes
and state estimation error. Because the initial condition of the
process is not known precisely, the observer needs a few (6-7)
time steps to find the real state with quite a small error. The error
increases when the process is affected by the unmeasured input
step disturbances (in particular the second one, which starts for k
¼ 350).
Finally, the two nonlinear MPC algorithms are compared
in terms of the performance indicies SSEx (Eq. (46)) and SSEy
(Eq. (47)). Their values are given in Tables 4 and 5 for different set-
point changes and configurations: with or without the constraints
Fig. 5. Simulation results: the linear MPC algorithm (solid line) and the linear MPC algorithm when the predicted output variable y3 is not constrained (dashed line) for set-
point change from the operating point B to C (the linear model for the operating point C is used, real state measurements are used, no constraints imposed on the rate of
change of the manipulated variables are taken into account).
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495 489
15. Fig. 6. Simulation results: the linear MPC algorithm (solid line) and the linear MPC algorithm when the predicted output variable y3 is not constrained (dashed line) for set-
point change from the operating point A to C (the linear model for the operating point C is used, real state measurements are used, no constraints imposed on the rate of
change of the manipulated variables are taken into account).
Fig. 7. Simulation results: the nonlinear MPC-NO algorithm (solid line with dots) and the nonlinear MPC-SL algorithm (dashed line with circles) for set-point change from the
operating point B to C (real state measurements are used, no constraints imposed on the rate of change of the manipulated variables are taken into account).
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495490
16. Fig. 8. Simulation results: the nonlinear MPC-NO algorithm (solid line with dots) and the nonlinear MPC-SL algorithm (dashed line with circles) for set-point change from the
operating point A to C (real state measurements are used, no constraints imposed on the rate of change of the manipulated variables are taken into account).
Fig. 9. Simulation results: the nonlinear MPC-NO algorithm (solid line with dots) and the nonlinear MPC-SL algorithm (dashed line with circles) for set-point change from the
operating point B to C (real state measurements are used, the constraints imposed on the rate of change of the manipulated variables are taken into account).
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495 491
17. Fig. 10. Simulation results: the nonlinear MPC-NO algorithm (solid line with dots) and the nonlinear MPC-SL algorithm (dashed line with circles) for set-point change from the
operating point A to C (real state measurements are used, the constraints imposed on the rate of change of the manipulated variables are taken into account).
Fig. 11. Simulation results: the nonlinear MPC-NO algorithm (solid line with dots) and the nonlinear MPC-SL algorithm (dashed line with circles) for set-point change from the
operating point B to C (the state observer is used with a wrong initial condition, the constraints imposed on the rate of change of the manipulated variables are taken into
account, the inputs of the process are affected by step disturbances, the outputs are affected by measurement noise).
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495492
18. imposed on the rate of change of the manipulated variables, with
or without the state observer. When the observer is used, it is also
assumed that its initial condition is wrong, the process output
measurements are characterised by noise and two input step
disturbances act on the process. Two approaches to soft output
constraints are also compared, i.e. the suboptimal method in
which the same coefficients ε ( )kmin and ε ( )kmax are used over the
whole prediction horizon (as in the MPC-SL optimisation problem
(41) with +N3 2u decision variables) and the optimal method in
which different violations ε ε( + ) … ( + )k k N1 , ,min min and
ε ε( + ) … ( + )k k N1 , ,max max may be adjusted independently for
consecutive sampling instants (as in the MPC-SL optimisation
Fig. 12. Simulation results: the nonlinear MPC-NO algorithm (solid line with dots) and the nonlinear MPC-SL algorithm (dashed line with circles) for set-point change from the
operating point A to C (the state observer is used with a wrong initial condition, the constraints imposed on the rate of change of the manipulated variables are taken into
account, the inputs of the process are affected by step disturbances, the outputs are affected by measurement noise).
Fig. 13. Top panels: real state trajectories (solid line with dots) and the estimated state trajectories (dashed line) in the nonlinear MPC-SL algorithm (dashed line with circles),
bottom panels: state estimation errors, for set-point change from the operating point B to C (left panels) and from A to C (right panels), (the state observer is used with a
wrong initial condition, the constraints imposed on the rate of change of the manipulated variables are taken into account, the inputs of the process are affected by step
disturbances, the outputs are affected by measurement noise).
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495 493
19. problem (44) with +N N3 2u decision variables). For the simula-
tions carried out, two conclusions may be drawn: performance
(set-point accuracy) of the MPC-SL algorithm is very similar to that
of the MPC-NO one and the suboptimal approach to soft con-
straints does not lead to any significant worsening of constraint
satisfaction.
It is interesting to compare computational burden of the
discussed MPC-SL and MPC-NO algorithms. For this purpose the
cputime command in Matlab is used. Tables 6 and 7 give com-
putational effort of both algorithms for set-point changes from the
operating point B to C and from A to C, corresponding to the tra-
jectories depicted in Figs. 11 and 12. For comparison different
control horizons are taken into account ( =N 1, 2, 3, 4, 5, 10u ) and
two methods of treating the soft output constraints. First of all, it
may be easily noticed that the MPC-SL algorithm is many times
less computationally demanding than the MPC-NO one. For ex-
ample, for the first considered trajectory and the suboptimal soft
output constraints, the complexity reduction factor is 14.68 when
=N 2u , 18.50 when =N 5u and 25.50 when =N 10u . One may also
notice that as the control horizon is lengthened computational
burden of the MPC-SL algorithm grows, but the growth is quite
moderate, whereas in case of the MPC-NO algorithm lengthening
of the control horizon leads to a dramatic growth of the compu-
tational burden. Secondly, the suboptimal soft output constraints,
when compared with the optimal ones, are very computationally
efficient, in particular in the case of MPC-SL algorithm.
6. Conclusions
When the full nonlinear state-space model of the boiler-turbine
system is used in MPC, it is necessary to solve on-line a nonlinear
optimisation problem with constraints at each sampling instant
[16,19]. Nonlinear on-line optimisation in real time may be im-
possible in practice because of its significant computational bur-
den. This paper details derivation of the suboptimal MPC-SL al-
gorithm for the process and discusses simulation results. In the
MPC-SL algorithm the state-space model is successively linearised
on-line for the current operating point of the process and the
obtained linear approximation is used for prediction of state and
output variables. The MPC-SL algorithm has the following
advantages:
Table 4
Accuracy of compared nonlinear MPC algorithms in terms of the performance index SSEx.
Set-point change State observer Constraints ▵umin, ▵umax Algorithm
Suboptimal output soft constraints Optimal output soft constraints
MPC-SL MPC-NO MPC-SL MPC-NO
→B C No No ×1.0898 105 ×1.0894 105 ×1.0864 105 ×1.0895 105
→B C No Yes ×3.1675 105 ×3.1772 105 ×3.3168 105 ×3.1664 105
→B C Yes Yes ×3.3328 105 ×3.4133 105 ×3.3065 105 ×3.4102 105
→A C No No ×3.3110 105 ×3.3106 105 ×3.3077 105 ×3.3106 105
→A C No Yes ×9.3900 105 ×9.4059 105 ×9.8272 105 ×9.3635 105
→A C Yes Yes ×9.9041 105 ×1.0263 106 ×1.0374 106 ×1.0233 106
Table 5
Accuracy of compared nonlinear MPC algorithms in terms of the performance index SSEy.
Set-point change State observer Constraints ▵umin, ▵umax Algorithm
Suboptimal output soft constraints Optimal output soft constraints
MPC-SL MPC-NO MPC-SL MPC-NO
→B C No No 7.6243 7.6154 7.5697 7.6157
→B C No Yes 1.5203 1.5335 1.6464 1.5224
→B C Yes Yes × −9.2890 10 1 × −8.2773 10 1 × −9.4550 10 1 × −8.2173 10 1
→A C No No ×1.5753 101 ×1.5743 101 ×1.5696 101 ×1.5743 101
→A C No Yes 3.6320 3.6576 4.2028 3.6008
→A C Yes Yes 2.5186 2.4817 3.1905 2.4484
Table 6
Computational burden of the MPC-SL and MPC-NO algorithms; for set-point
change from the operating point B to C (the state observer is used with a wrong
initial condition, the constraints imposed on the rate of change of the manipulated
variables are taken into account, the inputs of the process are affected by step
disturbances, the outputs are affected by measurement noise).
Algorithm Output soft
constraints
=N 1u =N 2u =N 3u =N 4u =N 5u =N 10u
MPC-SL Suboptimal 5.67 5.69 6.11 6.55 7.06 10.06
MPC-SL Optimal 5.66 6.39 7.11 8.08 20.41 37.31
MPC-NO Suboptimal 38.66 83.56 91.78 113.61 130.61 256.51
MPC-NO Optimal 241.22 453.16 590.44 687.55 1052.60 4733.30
Table 7
Computational burden of the MPC-SL and MPC-NO algorithms; for set-point
change from the operating point A to C (the state observer is used with a wrong
initial condition, the constraints imposed on the rate of change of the manipulated
variables are taken into account, the inputs of the process are affected by step
disturbances, the outputs are affected by measurement noise).
Algorithm Output soft
constraints
=N 1u =N 2u =N 3u =N 4u =N 5u =N 10u
MPC-SL Suboptimal 6.56 6.81 7.23 7.44 8.33 10.61
MPC-SL Optimal 5.80 6.36 6.89 7.84 20.55 34.75
MPC-NO Suboptimal 47.76 51.78 61.95 73.51 126.30 325.34
MPC-NO Optimal 184.50 523.59 604.30 666.50 948.23 4742.80
M. Ławryńczuk / ISA Transactions 67 (2017) 476–495494
20. 1. Linearisation makes it possible to obtain a computationally not
demanding MPC quadratic optimisation problem. Its computa-
tional burden is many times lower than that of the MPC with
nonlinear optimisation.
2. Unlike the MPC algorithm based on linear models, the MPC-SL
technique gives good control accuracy and leads to satisfaction
of the output constraints.
3. For different set-point changes, different constraint configura-
tions and unmeasured disturbances, the suboptimal MPC-SL
algorithm gives process trajectories very similar to those ob-
tained in the MPC scheme with nonlinear optimisation.
4. The MPC-SL algorithm uses quite simple prediction method
discussed in [38], which guarantees offset-free control. It means
that the MPC-SL algorithm is able to compensate for unmea-
sured disturbances, i.e. the controlled variables of the process
follow their set-point trajectories even when the process is
affected by some unmeasured disturbances and noise.
Moreover, two approaches to treating the soft output constraints
are described. The first one is suboptimal, but it introduces only
two additional decision variables and four constraints whereas in
the second one (optimal) the number of decision variables and the
number of constraints are linearly dependent on the prediction
horizon. Taking into account the presented simulation results, it is
found that for the boiler-turbine process the suboptimal approach
to soft output constraints gives practically the same results as the
optimal one.
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