Design of Centralised Pi Controller for Multivariable System

In this paper various methods for the designing of centralized PI controller for desalination system and distillation columns given in the literatures are discussed. Various tuning methods given by Davison and Maciejowski are used. The performance of the closed loop system servo responses are compared of terms of ISE. The better controller tuning method is chosen by means of least ISE value. INTRODUCTION Most process industries are dealing with multivariable process. Multivariable processes usually pose difficulties to design controllers, due to the interactions that occur between the input/output variables. Their control characteristics and tuning requirements change as operating conditions change. In addition, many operations experience some type of control loop interaction. When the process variable of one loop severely interacts with or disturbs the process variables of other loops, centralized control can eliminate the interaction. Two different processes are considered. One is desalination of sea water is by Multi Stage Flash Desalination [11]. Another process is distillation column which separates methanol and water [12]. In this centralized controller, if the transfer function of the process is n x n matrix, then it requires n x n PID controllers. This method reduces the interaction of the process [1-10]. However, the design methods of such Centralized controllers require first pairing of input-output variables, and tuning of controllers requires trial and error steps. Only well experienced operators can tune such control loops. For strongly interacting systems, decentralized controllers will not give satisfactory responses. There are some simpler methods of tuning centralized controllers. The centralized control system requires nxn controllers for controlling n output variables using n manipulated variables, which is an advantage if we use standard PI controllers. The tuning methods are evaluated based on the simple performance criteria which include the rise time, peak time, settling time, maximum peak overshoot and controller performance metrics which are the Integral of absolute Error (IAE) and the Integral square error (ISE), Integral of Time multiply by Absolute Error (ITAE). International Journal of Advances in Engineering Research http://www.ijaer.com (IJAER) 2014, Vol. No. 8, Issue No. IV, Oct ISSN: 2231-5152 13 INTERNATIONAL JOURNAL OF ADVANCES IN ENGINEERING RESEARCH DESIGN OF CONTROLLER The transfer function matrix of the system with s-domain is denoted by Gp(s) given by Y(s) = Gp(s). U(s) (1) Where Y is the vector of output variable and U is the vector of manipulated variables. The centralized controller are assumed to be in the form Gc(s) = Kc + (KI/s) (2) Where Kc and KI are matrices of size nxn   ij Kc Kc ,  , and          ij I ij Kc KI , ,  , i=1 to n and j=1 to n. (3) 2.1 Controller design for Square Process 2.1.1 Davison Method Davison has proposed a multivariable PI controller where the matrices Kc and KI are given by          0) (s G 1 δ c K (4)   0) (s G 1 ε KI    (5) Here   ) 0 ( 1   s G is inverse of steady state gain matrix and  and  are the fine tuning parameters. The fine tuning parameters range is from 0 to 1. The recommended values are 0.1-0.3. 2.1.2 Maciejowski Method In this method the matrix c K is given by   ) ( 1 b i s G c K      (6) Should be where b  the desired bandwidth of the system is. I K is the same as given in the Davison method. The matrix will be a complex matrix, and a real approximation of the matrix is required for the controller design and implementation. International Journal of Advances in Engineering Research http://www.ijaer.com (IJAER) 2014, Vol. No. 8, Issue No. IV, Oct ISSN: 2231-5152 14 INTERNATIONAL JOURNAL OF ADVANCES IN ENGINEERING RESEARCH 2.2 Controller design for Non-Square Process 2.2.1 Davison Method In the present work, this method is extended to non-square system. As inverse does not exist for non-square system, Moore–Penrose pseudo inverse is used. For matrix A, Moore–Penrose pseudoinverse is 1 1 ) * (    T T A A A A (7) T A is the transports matrix of A. So for non-square system, PID controller gains are: 1 )] 0 ( [    S G Kc  1 )] 0 ( [    S G KI  (8) This method is extended to non-square system. As inverse does not exist for non-square system, Moore–Penrose pseudo inverse is used. 2.2.2 Maciejowski Method In the present work, this method is extended to non-square system. As inverse does not exist for non-square system, Moore–Penrose pseudo inverse is used. For matrix A, Moore–Penrose pseudoinverse is same for Davison method For matrix Gp, Moore–Penrose pseudo-inverse is 1 T P P T P 1 P ) G * G ( G G    (9) In this method the matrix c K and I K are given by 1 )] ( [    b c i s G mag K   (10) 1 )] ( [    b I i s G K   (11) PROCESS DESCRIPTION 3.1 Multi-stage Flash Distillation (MSF) Process Transfer Function of MSF Desalination Process is given by