Linear and Nonlinear Model Predictive Control of Quadruple Tank Process

Transcription

1 Linear and Nonlinear Model Predictive Control of Qadrple Tank Process P.Srinivasarao Research scholar Dr.M.G.R.University Chennai, India P.Sbbaiah, PhD. Prof of Dhanalaxmi college of Engineering Thambaram Chennai, India ABSTRACT In a mlti inpt and otpt (MIMO) process mathematical modeling of the physical systems has gained importance de to the complexity of interactions within the system. All the parameters sed in a model cannot be determined accrately. The major problem in a mltivariable process is that loop interaction can arise and case difficlty in feedback control design. This problem can be solved sing centralized or decentralized controllers. One of the centralized techniqes is model predictive control, which can be measred crrent to predicted ftre vales of otpts. In this paper, model predictive control (MPC) technology for both linear and nonlinear model of qadrple tank process is proposed. It consists of for inter connected water tanks and two pmps as shown in figre. A general MPC control is presented, and approaches taken for the different aspects of the calclation are described. It is shown that MPC control is more stable, responsive and robst. Keywords MIMO, MPC, Qadrple Tank Process, Robst.. INTRODUCTION The major problem is to solve the control parameters for static as well as dynamic systems sing model predictive control (MPC). Here, MPC is a more advanced method to control the predicted otpt along with tning the parameters sch as prediction and control horizons and control weights. It can handle mltivariable processes, difficlt mltivariable control problems that inclde ineqality constraints []. Linear MPC refers to a family of MPC schemes in which linear models are sed to predict the system dynamics, even thogh the dynamics of the closed-loop system is nonlinear de to the presence of constraints []. Linear [5] and Nonlinear [6] MPC approaches have fond sccessfl applications, especially in the process indstries. The process is called the qadrple-tank process and consists of for interconnected water tanks and two pmps. The system is shown in Figre. The inpts are the voltages to the two pmps and the otpts are the water levels in the lower two tanks. The qadrpletank process can easily be bilt by sing two doble-tank processes. The linearized model of the qadrple-tank process has a mltivariable zero, which can be located in either the left or the right half-plane by simply changing a valve. Qadrple tank contains transmission zeros, which can vary from left half plane (minimm phase) to right half plane (nonminimm phase) depending on the ratio of the flow to pper and lower tanks. The step response of the qadrple tank system sing MPC with different control vales is obtained and compared to the step response of the controlled system sing Decopled control strategy [] and Comparative stdy []. This paper is organized as follows: section gives description of for tank process. The controller design for for tanks and simlation analysis for stability is explained in 3 &. Finally the conclsion is given in 5. DESCRITON OF FOUR TANK PROCESS Qadrple-tank process consists of for interconnected water tanks and two pmps as shown in Figre. The target is to control the level in the lower two tanks with two pmps [3]. The process inpts are v and v (inpt voltages to the pmps) y k h y k h (voltages and the otpts are c and c from level measrement devices). Mass balances and Bernolli s law yield the following model: dh a a k gh 3 gh3 v dt A A A dh a a k dt A A A dh gh gh v a 3 3 gh3 v 3 3 gh v dt A A dh a k dt A A where, i is the flow distribtion to lower and diagonal pper tank, A i is the cross-section area, a i is the otlet hole cross section and h i is the water level, in tank i respectively. The voltage applied to pmp i is v i and the corresponding flow is kv. i i The parameters,, are determined from how the valves are set prior to an experiment. The flow to tank is ykv and the flow to tank is y kv and similarly for tank and 3. The acceleration of gravity is denoted g. This typical system has two finite zeros for,, one always lies in the left half-plane, bt, the other can be placed either in the left or the right half-plane depending on the valve setting of, as explained in table,[]. k () 8

2 International Jornal of Compter Applications ( ) Fig.. Diagram for Qadrple Tank Process. Table. For Valve Setting. Valve Settings Process Zero Location If Minimm Zero is in left Phase plane Non If Zero is in left Minimm plane Phase If Zero at Origin Linearised the model has two sets of operating points with state space eqation at operating points xi hi hi and i vi vi. A k T AT A A k dx T AT A x dt k T A k T A () kc y x kc (3) Here the time constant Ai Ti a i i h g The dynamics for the process transfer fnction matrix is G s c c st st3 st c c st st st () (5) 3. DESCRIPTION OF MPC In MPC applications, the otpt variables are also referred to as controlled variables or CV s, while the inpt variables are called as maniplated variables or MV s. The predictions are made in two types of MPC calclations that are performed at each sampling instant: set-point calclations [5] and control calclations. Ineqality constraints as pper and lower limits can be inclded in either types of calclation [6]. In MPC the set points are typically calclated each time for MIMO process with inpt variables and y otpt variables The crrent vales of and y as (k) and y(k). The objective is to calclate the optimm set point y sp for the next control calclation (at k+) and also to determine the corresponding steady-state vale of, sp. This vale is sed as the set point for for the next control calclation. A general, linear steady-state process model can be written as [] y K Where K the steady-state is gain matrix and, y denotes steady-state changes in and y it is convenient define and y as sp y y yol k (7) Here yol k is steady-state vale of y sp k (8) To incorporate otpt feedback, the steady-state model Eqn (6) is modified as y K y k y k 3. The principle of linear and nonlinear model predictive control In general, the model predictive control problem is formlated for solving a finite horizon open-loop optimal control problem sbject to system dynamics and constraints involving states and controls. Fig. shows the basic principle of model predictive control. The measrements are obtained from plant at reglar intervals at time k,. The controller predicts the ftre dynamic behavior of the system over a predicted horizon P and control horizon M determines (over a control horizon P>M) the inpt sch that a predetermined open-loop as well as closed-loop performance objective fnctional is optimized. If there were no distrbances and no model-plant mismatch, and if the optimization problem cold be solved for infinite horizons, then one cold apply the inpt fnction fond at time k = to the system for all times k. However, this is not possible in general. De to distrbances and model-plant mismatch, the tre system behavior is different from the predicted behavior [6]. Here we have ignored distrbance and match plant model. (6) (9) 9

3 Fig. : Basic principle of Model Predictive Control. The main point of this optimization problem is to compte a new control inpt vector, k to be fed to the system, and at the same time take the process constraints into consideration. An MPC algorithm consists of: Cost Fnction, Constraint and A Model of the Process. 3. Cost Fnction The main idea with MPC is that the MPC controller calclates a seqence of ftre control actions sch that the cost fnction is minimized. The cost fnction often sed in MPC is like this (a linear qadratic fnction)[]: Np Np T T ˆ ˆ () J y r Q y r R Where: N p y k k Prediction horizon r Set point Past Ftre ŷ Predicted process otpt Predicted change in control vale, k k k Q Otpt error weight matrix R Control weight matrix ŷ Control horizon, M Set point (target) Past otpt Prediction horizon, P Predicted ftre otpt k - k k + k + k + M - k + P This works for MIMO systems (Mltiple Inpt and Mltiple Otpts) so we are dealing with vectors and matrices. 3.3 Constrains All physical systems have constraints. Generally, physical constraints like actator and valve limits, etc and performance constraints like overshoot, settling time, etc. In MPC one normally defines these constraints [] to minimize ineqalities. Constraints in the otpts: y y y min max Constraints in the inpts: min min max max Note: k k k () () (3) 3. Model The main drawback with MPC is that a model for the process, i.e., a model which describes the inpt to otpt behavior of the process, is needed. Mechanistic models derived from conservation laws can be sed. Usally, however in practice simply data-driven linear models are sed. In MPC it is assmed that the model represents a state-space model of the form [8]: xk Ax B, yk Cx D. () We consider the stabilization problem for a class of systems described by the following nonlinear of differential eqations [8]: x t A t f x t, t, t, x t x, c m c m x X R, U R, A. : R R (5) With the known smooth nonlinear map f xt, t the nknown parameter matrix At and. SIMULATION ANALYSIS In this paper, simlation reslts are compared with [] and [] on time based domain sing tning predictive control response, to a step response inpt. Here, response is plotted for the lower tanks for minimm and non minimm phase at two operating points is stdied at p - and p + of minimm and non minimm phase [9]. These operating points are at Table.. For minimm phase response of linear system with P=, M=, no of control interval= as shown in Fig. 3 when tning the parameters the response is shown in Fig., here P=, M=3, no of control interval=.5. Operating Points, h h 3, h h, v v, Table. Operating Points. Units p - p + [cm] [cm] [V] (.,.7) (.6, 3.) (.8,.) (.8,.9) (,) (, ) k k [cm 3 /Vs] (3.33, 3.35) (3., 3.9) (.7,.6) (.3,.3), 3

4 For non minimm phase response of linear system with P=, M=, no of control interval= as shown in Fig. 5. When tne parameters the response is shown in Fig.6, here P=, M=3, no of control interval=.5, along delay of second. Fig.3 Otpt response with specified inpt Fig.5 Otpt response with specified inpt Fig. Otpt response with specified inpt Fig.6 Otpt response with specified inpt 3

5 For minimm phase response of linear system withot applied inpt is shown at Fig. 7, similarly response obtained withot inpt is shown at Fig.8. Similarly for non minimm phase response of linear system withot applied inpt as shown in Fig. 9, similarly response obtained withot inpt as shown in Fig.. Fig.7 Otpt response with specified inpt Fig.9 Otpt response with specified inpt Fig. otpt response with specified inpt Fig.8 Otpt response with specified inpt For minimm phase response of non linear system P=, N=, no of control intervals= is shown in Fig., with slide adjsted the response is shown in Fig., here P=, M=3, no of control interval=.5 3

6 For non minimm phase response of non linear system with P=, M=, no of control interval=, along delay of second is shown in Fig. 3, when tne parameters the response is show in Fig., here P=, M=3, no of control interval =.5. Fig. otpt response with specified inpt Fig. 3 Otpt response with specified inpt Fig.Otpt response with specified inpt Fig. Otpt response with specified inpt 33

7 5. CONCLUSION The design procedre for MPC controller for Qadrple tank process has been proposed in this paper. The step response for process is compared with the reslts obtained in references listed as paper and paper for both minimm and non minimm phase of linear and nonlinear system. It is observed that linear and nonlinear process with different controller parameters offers stable response for step inpt. Nonlinear system exhibits stable response withot overshoots for minimm and non minimm phase. Similarly linear systems also exhibit stable response withot overshoot for certain tned controller parameters. Response of linear system for the lower two tanks, when absence of one inpt, one of them exhibits stable response for set point of respective step inpt. Remaining tank exhibits slit stable response not p to set point. Non minimm phase system has a transmission zero on right plane still exhibits stable response withot any compensation for linear and non linear system. MPC is a more advanced techniqe to handle mltivariable parameters. Finally, all transient and steady state responses have been obtained in all cases. 6. REFERENCES [] P. Srinivasarao and Dr. P. Sbbaiah, Decopled control strategy for qadrple tank process, CIIT International Jornal of Programmable Circit and System, Vol., No. 6, pp , May. [] P. Srinivasarao and Dr. P. Sbbaiah, Comparative stdy for Qadrple Tank Process with Coefficient Diagram Method, IFRSA International Jornal of Electronics Circits and System, Vol., Isse, pp. 9-99, Jly. [3] K. H. Johansson, The Qadrple-Tank Process: A Mltivariable Laboratory Process with an Adjstable Zero, IEEE Transactions on Control Systems Technology, pp ,. [] Qamar Saeed, Vali Uddin and Reza Katebi, Mltivariable Predictive PID Control for Qadrple Tank, World Academy of Science, Engineering and Technology,. [5] M. Alamir, G. Bornard, " Stability of a trncated infinite constrained receding horizon scheme: the general discrete nonlinear case, Elsevier Science Ltd, Vol. 3, No. 9, pp , Sep 995. [6] Zoltan K.Nagy Richard D.Braatz, Robost Nonlinear Model Predictive Control of Batch Processses, AIChe Jornal, Vol. 9 No.7, Jly 3. [7] M. J. Lengare, R. H. Chile, L. M. Waghmare and Bhavesh Parmar, Ato Tning of PID Controller for MIMO Processes, World Academy of Science, Engineering and Technology, 8. [8] Sergey Edward Lyshevski, Control system theory with Engineering Applications, Jaico Pblishers, 3. [9] Ashish Tewari, Modern Conrol Design with MATLAB and SIMULINK, John Willy & Sons, Ltd, 9. [] Dale E. Seborg,Thomas F. Edgar and Dncan A.Mellichamp, Process Dynamics and Control, John Willy & Sons, Ltd, 6. [] B.Wayane Beqette, Process Control Modeling, Design and Simlation, Prentice Hall,3. AUTHOR S PROFILE P. Srinivasarao working as Associate Professor in VITS Grop of Instittions, Sontyam, Visakhapatnam. Completed M.Tech from COE Pne, Ph.D Scholar, Dr. M.G..R. University. Dr.P.Sbbaiah is Professor in Dhanlaxmi college of Engineering, Thambaram, Chennai, Tamilnad, India. 3