State-Space Predictive Control Of Two Liquid Tanks System With Constraints Of Process Variables

This paper presents a method called the predictive control used to control a nonlinear process about a selected operating point. The system of the two funnel liquid tanks in series is chosen as an exemplar process. The parameters of the tanks simulate a large industry tanks used in a chemical industry. The state-space CARIMA mathematical model is used for the output values prediction. This paper describes the linearization process of the nonlinear system at the operating point and a possibility of constraints of the process variables. The designed controller is verified on the process without and with a time-delay. INTRODUCTION Many processes in the real world are nonlinear, complex and they often include a time-delay. These processes are very difficult to control. The situation is more complicated when some process variables require some sort of a limitation. The basic control methods do not handle with this situation so we need a more advanced method. The predictive control is a great example of the modern control method capable of solving the complex control problem (Bobál 2008). This method is based on the prediction of the output values on the chosen time horizon. This time horizon should be long enough to cover the step response of the controlled system and the prediction of the output values is based on the mathematical model of the controlled system. The predictive control in this paper uses state-space CARIMA model for multi-input multioutput (MIMO) system (Bars et al. 2011; Wang 2009). The control signal is obtain by minimization of a cost function. This cost function has usually a quadratic form and it minimize the differences between the reference value and the output value and the control signal increments. We can also take into account the constraints of the process variables in the cost function minimization process. This is done by using a quadratic programming method (Camacho and Bordons 2004; Maciejowski 2002; Rossiter 2003). However, the state-space CARIMA model is a linear mathematical model. So we have to do one more step to control the nonlinear system like the chosen system of the two funnel liquid tanks in series. This step is linearization of the nonlinear mathematical model in a selected operating point. The divergence linear model is result of the linearization. It means, that the selected operating point is a new origin state for the controller and the input and the output values are divergence from the equilibrium values (Albertos Peréz and Sala 20014; Hangos et al. 2004). MATHEMATICAL MODEL OF THE CONTROLLED SYSTEM The mathematical model of the chosen experimental system of the two liquid tanks system is taken from (Krhovják et al. 2015). This model represents a nonlinear system with two input variables and two output variables. Figure 1 shows a schematic diagram of the controlled process consists of two funnel liquid tanks in series. The first tank is filled by input stream q1f, the second tank is filled by input stream q2f and output stream from the first tank q1. Figure 1 : Schematic diagram of the process The mathematical model of this process can be obtained from balancing equations of the input and output mass streams. The equation (1) stand for the balancing equation of the input, output and accumulation of the first tank and the equation (2) stand for the balancing equation of the input, output and accumulation of the second tank (Richardson 1989). 2 2 1 1 1 1 2 4 f dh D h q q dt H π + = (1) 2 2 2 2 1 2 2 2 4 f dh D h q q q dt H π − + = (2) Proceedings 30th European Conference on Modelling and Simulation ©ECMS Thorsten Claus, Frank Herrmann, Michael Manitz, Oliver Rose (Editors) ISBN: 978-0-9932440-2-5 / ISBN: 978-0-9932440-3-2 (CD) In these equations, D is the maximum diameter, H is the total height and h1 and h2 are the liquid levels from the bottom of the tanks. The output liquid streams q1 and q2 depend on the valve constants k1, k2 and the liquid levels as well. 1 1 1 2 q k h h = − (3) 2 2 2 q k h = (4) However, the chosen predictive control method works only with linear mathematical models, so the model of the described process needs to be linearized about an operating point. First of all, the equations (1) and (2) have to be expressed as nonlinear state-space model by selecting the input variables as 1 1 f u q = and 2 2 f u q = and the output and state variables as 1 1 1 y x h = = and 2 2 2 y x h = = . This state-space model has form ( ) ( ) x = f x,u y = g x ɺ (5) where equations of single states and outputs are ( ) ( ) 2 1 1 1 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2