Simulation Of Multivariable Continuous-Time Decoupling Control

The paper is focused on an implementation of a decoupling multivariable controller in the Matlab/Simulink environment. The control algorithm is based on polynomial theory and pole – placement. A decoupling compensator is used to suppress interactions between control loops. The controller was realized both with fixed parameters and with recursive identification of a model of the controlled system. The internal structure of the controller enables its easy modification and implementation of further similar control algorithms. INTRODUCTION Typical technological processes require the simultaneous control of several variables related to one system. Each input may influence all system outputs. The design of a controller for such a system must be quite sophisticated if the system is to be controlled adequately. There are many different methods of controlling MIMO (multi input – multi output) systems. Several of these use decentralized PID controllers (Cui and Jacobsen, 2002) others apply single input-singleoutput (SISO) methods extended to cover multiple inputs (Chien et all, 1987). The classical approach to the control of multi-input–multi-output (MIMO) systems is based on the design of a matrix controller to control all system outputs at one time. The basic advantage of this approach is its ability to achieve optimal control performance because the controller can use all the available information about the controlled system. Controllers are based on various approaches and various mathematical models of controlled processes. A standard technique for MIMO control systems uses polynomial methods (Kučera, 1980, Kučera 1991, Vidyasagar 1985) and is also used in this paper. Controller synthesis is reduced to the solution of linear Diophantine equations (Kučera, 1993) . One controller, which enables decoupling control of TITO (two input-two output) systems, is presented. The proposed control algorithm applies a decoupling compensator (Krishnawamy et all, 1991, Peng, 1990, Tade et all, 1986) to suppress undesired interactions between control loops. The controller was realized both with fixed parameters and with recursive identification of a model of the controlled system. For purposes of simulation, the controller was realized in the Matlab/Simulink environment as a mask of subsystem. It can be easily inserted into Simulink schemes of the closed loop. No initialization is needed before the simulation start. Blocks for computation of the control law and for recursive identification were realized as S-functions. The internal structure of the controller enables its easy modification and implementation of further similar control algorithms. A simulation experiment is also introduced. MODEL OF THE CONTROLLED SYSTEM A general transfer matrix of a two-input–two-output system with significant cross-coupling between the control loops is expressed as ( ) ( ) ( ) ( ) ( )⎦ ⎤ ⎢ ⎣ ⎡ = s G s G s G s G s 22 21 12 11 G (1) ( ) ( ) ( ) s s s U G Y = (2) where ( ) s U and ( ) s Y are vectors of the manipulated variables and the controlled variables. ( ) ( ) ( ) [ ] s y s y s 2 1 , = Y ( ) ( ) ( ) [ ] s u s u s 2 1 , = U (3) It may be assumed that the transfer matrix can be transcribed to the following form of the matrix fraction: ( ) ( ) ( ) ( ) ( ) s s s s s 1 1 1 1 − − = = A B B A G (4) where the polynomial matrices [ ] [ ] s R s R 22 22 , ∈ ∈ B A represent the left coprime factorization of matrix ( ) s G and the matrices [ ] [ ] s R s R 22 1 22 1 , ∈ ∈ B A represent the right coprime factorization of ( ) s G .The further described algorithm is based on a model with polynomials of second order. This model proved to be effective for control of several TITO laboratory processes (Kubalčík and Bobál, 2006), where controllers based on a model with polynomials of the first order failed. In case of decoupling control using a compensator it is useful to consider matrix A(s) as diagonal. The reason is explained in the following section. Proceedings 28th European Conference on Modelling and Simulation ©ECMS Flaminio Squazzoni, Fabio Baronio, Claudia Archetti, Marco Castellani (Editors) ISBN: 978-0-9564944-8-1 / ISBN: 978-0-9564944-9-8 (CD) ( ) ⎢ ⎣ ⎡ + = 2 s s A ( ) ⎢ ⎣ ⎡ + + = 5 1 s b s b s B Differential e the system ar 1 1 1 y a y / // + + 2 3 2 y a y / // + + DESIGN OF One of possi systems is th the system (K et all, 1986) the controlle undesirable variables so variable. The shown in Fig and C is a de Figure 1: Clo The resulting omitted from simplification A GC H = = The decoupl is diagonal. A supposed to simplification matrix A is included int diagonal ma consequently The compens ( ) B C adj = The matrix H