OPTIMIZATION OF PID CONTROLLER FOR A NONLINEAR MIMO SYSTEM AND COMPARATIVE PERFORMANCE ANALYSIS

Transcription

1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 8, August 2017, pp , Article ID: IJMET_08_08_085 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed OPTIMIZATION OF PID CONTROLLER FOR A NONLINEAR MIMO SYSTEM AND COMPARATIVE PERFORMANCE ANALYSIS S Anitha Janet Mary Research Scholar, Noorul Islam Centre for Higher Education, Kumaracoil, Tamilnadu, India L Padma Suresh Principal, Baselios Mathews II College of Engineering, Sasthamcotta, Kollam, Kerala, India ABSTRACT PID controllers are designed and optimized in this paper for wood-berry distillation column using simulated annealing algorithm and particle swarm optimization method PID controllers are the common instruments used in control applications Construction of PID controllers is smooth and easy It furnishes more stability and malleable while governing the processes Thousands of PID loops strive to keep processes safe, stable, and profitable Tuning of PID controller is the process of finding the values of constants named proportional constant K P, integral constant K i and derivative constant K d It is easy to tune PID controller gains of a continuous system But there are many nonlinear plants in industry Higher order, time delays, harmonics, poor damping, instability, time-varying dynamics, set points are not followed, valves swing around and creating oscillations etc are the problems related to the nonlinear system Thus the design of controller for a nonlinear process is a risky work This paper illustrates a comparison between the optimal tuning algorithms such as particle swarm optimization method and simulated annealing algorithm used to tune PID controller for a nonlinear system The comparison between the superior performance of SA tuned PID and PSO tuned PID controller are given Keywords: Wood-Berry Distillation Column, SA Tuned PID Controller, PSO Tuned PID Controller, MIMO System, Optimization, Performance Evolution Cite this Article: S Anitha Janet Mary and L Padma Suresh, Optimization of PID Controller for a Nonlinear MIMO System and Comparative Performance Analysis, International Journal of Mechanical Engineering and Technology 8(8), 2017, pp editor@iaemecom

2 Optimization of PID Controller for a Nonlinear MIMO System and Comparative Performance Analysis 1 INTRODUCTION Most of industrial control systems use PID controllers [1] It is salient that the PID parameters are correctly tuned to reach the high energy conservation made possible by industrial processes Now a day s not much time is spent for fine tuning the controller parameters when a system is deployed Usually safe parameters will be set which ensure that the system is stable These parameters mostly result in slow and inefficient control Calculating efficient parameters manually using conventional tuning methods is consuming long time In practical, 30% of loops have tuning that makes no sense whatsoever, and 85% of loops have sub-optimal tuning We all know why this happens most plants do not have a standard method, practices, training, or tools for loop tuning Many plants still tune by feel This leads to inconsistent and often terrible tuning results It is not surprising that you may find as many as 30% of control loops running in manual PID controller incorporates with three primary controller gains being proportional gain (K p ), integral gain (K i ) and derivative gain (K d ) [3], [4], [5] An output proportional to the present error value is produced by the proportional term Now the controller output is modified by multiplying the present error and a constant K p known as proportional gain constant An output proportional to the magnitude of the present error and the time period of the present error is produced by the integral term Then the controller output is rearranged by the product of accumulated error and the integral gain constant Ki An output proportional to the rate of change of error is generated by the derivative term This is changing the controller output by multiplying the rate of change of error and the derivative gain constant K d The raised gains are adjusted to get excellent output of the process This excellent control will make the deployment process run smoother and give industrial customers better results The conventional PID controllers are used in most of the industrial applications because of its simple structure and robustness When it comes to the control of multivariable processes, the controller parameters should be adjusted continuously There are many tuning methods for fine tuning the parameters of PID controller The PID parameters are obtained through the tuning approaches must be retuned before being implemented in the process under control in real time environment It can t be used in a continuous operation Therefore most researchers move towards optimization techniques Some of the optimization methods are genetic algorithm, ant colony algorithm, bat algorithm, tabu search algorithm, firefly algorithm, chaotic evolutionary programming [11], [15] These methods are used to tune the PID controller parameters They are giving good response for SISO systems Multivariable MIMO systems are extensively used in industrial applications Some optimization techniques are becoming inefficient due to more settling time, premature convergence and difficult to convert the off spring To overcome these disadvantages two optimization techniques are embraced in this paper They are simulated annealing (SA) algorithm and particle swarm optimization (PSO) algorithm Simulated Annealing is a global search method that is based on the analogy with the physical annealing process of solids [6], [7], [8], [9] SA requires little knowledge of the problem and need not require that the search space is either differentiable or continuous Therefore, they can solve nonlinear multi-objective optimization problems that difficult using other techniques SA has been successfully used in a wide variety of areas such as function optimization, design optimization, schedule optimization, travelling salesman problem, neural networks, image processing etc In the automatic control area, a few applications with SA have been accomplished Kokate employed SA to determine the compensator segmentation for rule-based control of slowly varying systems Chiang used SA for the optimal controller placements in large-scale linear systems editor@iaemecom

3 S Anitha Janet Mary and L Padma Suresh Particle swarm optimization (PSO) algorithm is a new population-based evolutionary optimization technique It is first introduced by Kennedy and Eberhart in 1995 The basic of PSO is based on the hypothesis that social sharing of information among conspecifics offers an evolutionary advantage It is inspired by animal social behaviour flocking of birds PSO is a population-based stochastic optimization paradigm, in which each agent, named particle, of the population, named swarm, is thought of as a collision-proof bird and used to represent a potential solution [10], [12] In this paper, a SA-based approach and a PSO-based approach are implemented to optimize the parameters of the PID controllers for distillation column process [16] Such effective and efficient approaches, with global optimal abilities and good robustness, are expected to overcome some shortcomings of conventional optimization approaches in this situation, and are more feasible for industrial applications 2 PROCESS MODEL Distillation is a process in which a liquid or vapour mixture of two or more substances is separated into its component fractions of desired purity by application and removal of heat [18] It is well known that pure liquids exhibit different volatilities ie vapour pressure at a given temperature, and thus if heat is applied to a liquid mixture of these substances, the vapour so generated will be richer in the more volatile substances those having higher vapour pressures If this vapour is condensed, it should be clear that a certain amount of purification will be achieved This is the basic principle underlying a distillation operation [17] A distillation process may be classified in one of two ways: Binary distillation refers to the separation of two substances and Multi component distillation involves more than two substances Distillation process is usually adopted in chemical, food, pharmaceutical and petroleum refining industries because of their reliability, simplicity and low capital cost Today, there are 40,000 distillation towers in operation Distillation causes about 95% of all present industrial separation processes And moreover distillation systems have relatively high energy consumption Therefore significant effort has been made to reduce the energy consumption and to improve the efficiency in distillation systems Wood and Berry distillation column is an example for a binary distillation column It is designed and modelled by an experiment It separates methanol and water from a feed mixture [10], [11] & [12] First the feed mixture is given into the distillation column Then the distillation column separates the volatile contents by vaporizing It is considered as top product named vapour rate It is coming out at the top of the distillation column At the bottom of the distillation column the remaining portions are collected as bottom product Due to the continuous flow of liquid feed mixture, there may be a chance of coupling and interaction between two loops With the help of decoupler it may avoid in between two controllers But the distillation column is efficiently operating along with the control of controller Then only the distillation column can give a specified product as required by the consumers For proving the ability of the SA algorithm and PSO algorithm, Wood-Berry (WB) distillation column is considered [13] It contains 8 trays thus it has eight differential equations Therefore it is an eighth order system But it is reduced into first order with time delay by wood berry using an experiment Now it has a 2*2 transfer function The WB distillation column has two input and two outputs so it is called as MIMO system It is a pure nonlinear process since the inputs and outputs are strongly interacting The top product composition and bottom product compositions are the controlled variables It is measured by the weight percentage of methanol The input variables are reflux flow rate and reboiler steam flow rate They are measured by lb/min Its transfer function is given in equation (1) editor@iaemecom

4 Optimization of PID Controller for a Nonlinear MIMO System and Comparative Performance Analysis! " (1) Figure1 Schematic of Distillation Column The two input signals reflux flow rate and steam flow rate are specified as u1 and u2 in the equation (1) And the two output signals y1 and y2 are given in the above equation They are the composition of top product and composition of bottom product in mole fraction The feed mixture flowing inside the distillation column is called feed flow rate which is specified as D in the transfer function equation D is the disturbance of the system The considered linear model is valid only for the given set point y1 = 096 and y2 = 002 Sampling time is one minute The schematic diagram of selected process wood berry distillation column is expressed in Figure1 3 DESIGN OF CONTROLLERS The basic conventional PID controller is first designed After that the PID controller parameters are tuned using simulated annealing algorithm and particle swarm optimization algorithm The basic form of PID controller diagram is shown in Figure 2 Kp e( t) Desired state + - e(t) t t0 Ki e( t) Control signal M d Kd e( t) dt Feedback signal Figure 1 Basic Diagram of PID Controller The above diagram shows three terms One is proportional gain term, next is gain of integral term and the last is gain of derivative term They are represented by Kp, Ki, and Kd The desire output will be reached by getting the optimal values of the controller parameters That will be achieved because of the error free and minimized system overshoots When the output is desired then the steady state error are removed and the system will become stable The basic transfer function of PID controller can be narrated by the equation (2) editor@iaemecom

5 S Anitha Janet Mary and L Padma Suresh # $ % & ' ( ) & * (2) The PID controller is operating as, when the input value changed the error detector calculating the error ie the change of value between the input and the current process output The error signal E(S) is used to generate the control action by the controllers The controller output called manipulated input variables U(S) is given to the process M It will rearrange the process and control the plant Thus the plant will be in controlled manner And the desired output is achieved [14] The PID controller is designed using basic empirical rule suggested by Ziegler / Nichols [2] The designed controller gain constant values are tabulated in the Table 1 Table 1 Conventional Controller Parameters Parameters Conventional PID K p, K i, K d, K p, K i, K d, Now the calculated controller parameter values are substituted in the simulated controller to activate the control action of the distillation plant Because of the nonlinearities of the system the output responses are not in satisfactory Thus the optimization is necessary 4 OPTIMIZATION METHODS Optimization means the perfect or accurate values of the controller parameters The layout of the multivariable control system design is shown in Figure 3 Input device, error detector, controller, process, optimization tool and output device are the parts of the layout Input device supplies the input signal Error signal is generated by the error detector after comparing the input and output signals Controller produces the control action with the help of error signal and the controller gain parameters Controller output is fed into the process The process makes it perfect and gives the desired output in an efficient way to satisfy the consumers Figure 3 Design Layout of Multivariable System Now the optimizing tools are used to produce the correct controller parameter to get the perfect manipulated input variable or the control action The optimizing tools are SA algorithm and PSO algorithm editor@iaemecom

6 Optimization of PID Controller for a Nonlinear MIMO System and Comparative Performance Analysis Simulated Annealing Algorithm The simulated annealing algorithm normally consists of two repeating steps One is the generation mechanism and the other is acceptance criterion For an optimization problem, SA starting with an initial random state, a sequence of iterations is created, in each iteration a perturbation mechanism is applied which transforms the present state into a next state by choosing the neighbourhood of the present state If the neighbouring state has a lower cost, the neighbouring state is declared as the present state If this neighbouring state has a higher cost, the neighbouring state is accepted with a certain probability determined by the acceptance criterion Figure 4 Flow Chart for SA Optimizing Algorithm The most important feature of the simulated annealing is that, besides accepting improvements in cost, it also, to a limited extent, accepts deteriorations in cost This feature ensures the simulated annealing to be a global search algorithm while it still has the favourable features of local search algorithms, ie simplicity and general applicability The methodology followed for optimizing the PID controller parameters by SA algorithm is explained by the flow chart in Figure 4The flow chart is explained by the following steps Step 1: Initialised the assumptions ie Initial values of controller parameters K p, K i and K d, termination criterion, higher temperature, and no of iteration in each temperature are assumed Step 2: The integral absolute error is the objective function That is evaluated in this step Step 3: The second step is repeated after selecting the neighbouring values of the controller parameters in random with in a particular boundary editor@iaemecom

7 S Anitha Janet Mary and L Padma Suresh Step 4: If the objective function value is lesser than the calculated value in step 2 for the initial controller parameters means save the selected neighbouring values of controller parameter otherwise decrease the temperature and go back to step 2 Step 5: If this continuous process reached it s stopping criterion means stop the iteration process and save the optimal solution of controller parameter Particle Swam Optimization The proposed method particle swam optimization is introduced by Kennedy & Eberhart in the year 1995 It is based on the inspiration of social behaviour and movement dynamics of insects, birds and fish All the particles are arranged at random position in PSO algorithm and are alleged to move randomly in the search space with a defined direction Each particle s direction is then changed gradually to hold to move along the direction of its best previous positions to detect even a new better position with respect to some fitness measures Both the initial velocity and initial position of the particle are chosen randomly and updated using the following equation, V = wv + c 1 R 1 (p b X) + c 2 R 2 (g b X) (3) X = X + V (4) Where, V-Velocity of the particle X-Position of the Particle R 1, R 2 -Random Numbers in Range (0, 1) W - Inertia Weight c 1 - Cognitive Parameter c 2 - Social Parameter This PSO technique has been applied in many real time engineering applications, it is extensively held that PSO algorithm gets captured in global optima Now the captured optimal values of controller parameters are tabulated in the Table 2 Table 2 Optimal Controller Parameters Parameters SA - PID PSO PID K p, K i, K d, K p, K i, K d, SIMULATION RESULTS The designed conventional PID controller, optimized SA-PID controller and the optimized PSO-PID controller are simulated in the selected wood berry distillation plant using MATLAB Simulink environment The performances of the controllers are evaluated Then the performances are compared The manipulated input for top product y1 is known as reflux flow rate u1 And the manipulated input for bottom product y2 is given by stream flow rate u2 Now the manipulated input produced by all the three controllers are given in Figure 4 and Figure editor@iaemecom

8 Optimization of PID Controller for a Nonlinear MIMO System and Comparative Performance Analysis PID SA-PID PSO-PID reflux flow rate, u Time (s) Figure 4: Manipulated Input for Top Product PID SA-PID PSO-PID 025 steam flow rate, u Time (s) Figure 5: Manipulated Input for Bottom Product The procured step response for the three controllers is compared and shown in Figure 6 for the top product In the same figure, the regulatory response of top product is shown by giving the disturbance at the time 120 second top product composition in mole fraction, y1(%) PID SA-PID PSO-PID Time (s) Figure 6 Output Response of Top Product editor@iaemecom

9 S Anitha Janet Mary and L Padma Suresh Then the step and regulatory response of the three controllers for the bottom product of wood berry distillation column process is given in Figure 7 Here also at time 120 second the disturbance is given bottom product composition in mole fraction, y2(%) PID SA-PID PSO-PID Time (s) Figure 7 Output Response of Bottom Product The performance measures of the top product of the selected system are examined for the three controllers and ordered in Table 3 Similarly the performance measures of the bottom product of the selected system are examined for the three controllers and organised in Table 4 Table 3 Top Product Performance Measures Performance Criteria(sec) Con-PID SA - PID PSO - PID Rise Time Settling Time Peak Overshoot Undershoot Peak Time Table 4 Bottom Product Performance Measures Performance Criteria (sec) Con-PID SA- PID PSO - PID Rise Time Settling Time Peak Overshoot Undershoot e e e+003 Peak Time The following observations are taken from the above two tables The rise time is less for the optimized PSO-PID controller compared to the other controllers in top product The settling time is less for the optimized PSO-PID controller in top and bottom products The peak time is also less for optimized PSO-PID controller than the other two controllers in top and bottom product Integral absolute error, integral square error and integral time absolute error are some of the integral errors They are calculated for the three different controllers for the top and bottom product of the process They are represented by IAE, ISE and ITAE The calculated absolute values of integral errors are tabulated in Table 5 and Table editor@iaemecom

10 Optimization of PID Controller for a Nonlinear MIMO System and Comparative Performance Analysis Table 5 Integral Errors for top Product Error PID SA - PID PSO - PID IAE ISE ITAE Table 6 Integral Errors for Bottom Product Error PID SA - PID PSO - PID IAE ISE ITAE CONCLUSION The conventional PID controller, optimized SA-PID controller and optimized PSO-PID controller are designed and simulated for the wood-berry distillation column process The methodologies are studied The performances and errors are analysed The conventional PID controller responses are not satisfied since the errors are little more than the optimized controllers for the top products The optimized controllers are tracking the system to their set point within the less time period And oscillations are less PSO- PID controller output responses are best when it is compared with other two controllers REFERENCES [1] Astrom, K J, & Hagglund, T (2001) The Future of PID Control Control Engineering Practice, Vol 9, No (11), pp [2] JG Ziegler and NB Nichols, Optimum Setting for Automatic Controllers, Trans ASME, Vol 64, Nov8, ppv , 1942 [3] KJ Åström, T Hägglund, PID Controllers: Theory, Design and Tuning, Second Ed, Instrument Society of America, Research Triangle Park, North Carolina, USA, 1995 [4] A O Dwyer, Handbook of PI and PID Controller Tuning Rules, Imperial College Press, London, England, 2003 [5] W Tan, J Liu, T Chen, HJ Marquez, Comparison of Some Well-Known PID Tuning formulas, Computers & Chemical Engineering, Vol 30, No (9), (2006), pp [6] L R Varela, R A Ribeiro and F M Pires, Simulated Annealing and Fuzzy Optimization, Proceedings of the 10 th Mediterranean Conference on Control and Automation - MED2002, Portugal, July 9-12, 2002 [7] Li - Sun Shu, Shinn - Ying Ho and Shinn - Jang Ho, A Novel Orthogonal Simulated Annealing Algorithm for Optimization of electromagnetic Problems, IEEE Transactions on Magnetics, Vol40, No 4, July 2004 [8] Hsien - Yu Tseng and Chang - Ching Lin, A Simulated Annealing Approach for Curve Fitting in Automated Manufacturing Systems, Journal of Manufacturing Technology Management, Vol 18, No2, pp , 2007 [9] SM Giriraj Kumar, Bodla Rakesh and N Anantharaman, Design of Controller Using Simulated Annealing for a Real Time Process, International Journal of Computer Applications ( ) Vol 6, No2, September 2010, pp [10] Willjuice I M, & Baskar S (2009) Evolutionary algorithm based design of multivariable PID controller Expert Systems with Applications, 36, editor@iaemecom

11 S Anitha Janet Mary and L Padma Suresh [11] Chang W D (2007) A Multi-Crossover Genetic Approaches to Multivariable PID Controllers tuning Expert Systems with Applications, Vol 33, pp [12] A Angelah Queen and D Jeraldin Auxillia, Simplified Discrete Binary PSO Tuned Multivariable PID Controller for Binary Distillation Column Plant, 2013 International Conference on Circuit, Power and Computing technologies IEEE, pp [13] Wood, R K, & Berry, M W (1973) Terminal composition control of a binary distillation column Chemical Engineering Science, 28(9), [14] S Anitha Janet Mary, L Padma Suresh, Rini Valsa Mathew and N Albert Singh, Comparative Study of PI and PID Controller for Non Linear MIMO System, Springer India 2015,Power Electronics and Renewable Energy Systems, Lecture Notes in Electrical Engineering, pp [15] Leandro dos Santos Coelho and Viviana Cocco Mariani, Firefly Algorithm Approach Based on Chaotic Tinkerbell Map Applied to Multivariable PID Controller Tuning Computers and Mathematics with Applications, Vol 64, (2012), pp [16] Anita Mary, L Padma Suresh and S H Krishna Veni, Comparative Performance Analysis of Different Controllers for a Nonlinear Multivariable System IEEE, International Conference on Emerging Technological Trends (ICETT), 2017 [17] Pradeep B Deshpande, Charles A Plank, Distillation Dynamics and Control, Instrument Society of America, 1985 New Jersey, 1989 [18] MT Tham, Introduction to Distillation Copy Right [19] D Lalitha Kumari and Prof M N Giri Prasad, A Review Paper on Performance Analysis of MIMO Based OFDMA System Under Fading Channel, International Journal of Electronics and Communication Engineering and Technology, 8(1), 2017, pp [20] Vaishak Dayanandan, Sudha T, A Novel Antenna for Uwb-Mimo Applications using a T- Shaped Stub, International Journal of Advanced Research in Engineering and Technology (IJARET), Volume 5, Issue 5, May (2014), pp [21] Prof S S Khade, Dr S L Badjate, A Pattern Diversity Compact MIMO Antenna Array Design for WLAN Application International Journal of Electronics and Communication Engineering & Technology (IJECET), Volume 4, Issue 7 (2013), pp editor@iaemecom