Digital Linear Quadratic Smith Predictor

Time-delays (dead times) are found in many processes in industry. Time-delays are mainly caused by the time required to transport mass, energy or information, but they can also be caused by processing time or accumulation. The contribution is focused on a design of universal digital algorithm for control of great deal of processes with time-delay. This requirement is successfully satisfied with digital Smith Predictor based on Linear Quadratic (LQ) method. A minimization of the quadratic criterion is realized using spectral factorization. The designed algorithm is suitable for control of stable, unstable and non-minimum phase processes. The algorithms for control of individual processes influenced by external disturbance were verified. The program system MATLAB/SIMULINK was used for simulation of designed algorithms. INTRODUCTION Time-delays appear not only in industrial processes, such as thermal, chemical, metallurgical or processes of plastic and rubber materials etc., but also in other fields, including economical and biological systems. They are caused by some of the following phenomena (NormyRico and Camacho 2007): • the time needed to transport mass, energy or information, • the accumulation of time lags in a great numbers of low order systems connected in series, • the required processing time for sensors, such as analyzers; controllers that need some time to implement a complicated control algorithms or process. The problem of controlling time-delay processes can be solved by several control methods (e. g. using PID controllers, time-delay compensators, model predictive control techniques). Time-delay in a process increases the difficulty of controlling it. However, the approximation of higherorder process by lower-order model with time-delay provides simplification of the control algorithms. When high performance of the control process is desired or the relative time-delay is very large, the predictive control strategy can be successfully applied. The predictive control method includes a model of the process in the structure of the controller. The first time-delay compensation algorithm was proposed by (Smith 1957). This control algorithm known as the Smith Predictor (SP) contained a dynamic model of the time-delay process and it can be considered as the first model predictive algorithm. First versions of Smith Predictors were designed in the continuous-time modifications, see e.g. (Normey-Rico and Camacho 2007). Although time-delay compensators appeared in the mid 1950s, their implementation with analog technique was very difficult and these were not used in industry. Because most of modern controllers are implemented on digital platforms, the discrete versions of the time-delay controllers are more suitable for time-delay compensation in industrial practice Since 1980s digital time-delay compensators can be implemented. The digital time-delay compensators are presented e.g. in (Vogel and Edgar 1980, Palmor and Halevi 1990, Normey-Rico and Camacho 1998). Some Self-tuning Controller (STC) modifications of the digital Smith Predictors (STCSP) are designed in (Hang et al. 1989; Hang et al. 1993; Bobál et al. 2011). Two versions of the STCSP were implemented into MATLAB/SIMULINK Toolbox (Bobál et al. 2012a; Bobál et al. 2012b). It is well known that classical analog Smith Predictor is not suitable for control of unstable processes. The designed digital LQ Smith Predictor eliminates this drawback. The paper is organized in the following way. The problem of a control of the time-delay systems is described in Section 1. The general principle of the Smith Predictor is described in Section 2. The discretization of analogue version, principle of digital Smith Predictor and polynomial two degrees of freedom (2DOF) controller is introduced in Section 3. Primary Linear Quadratic controller of the digital Smith Predictor is proposed in Section 4. Results of simulation experiments are summed in Section 5. Section 6 concludes the paper. 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) DIGITAL SMITH PREDICTOR The discrete versions of the SP and their modifications are suitable for time-delay compensation in industrial practice. Figure 1: Block diagram of a digital Smith Predictor The block diagram of a digital SP (see Hang, Lim, and Chong 1989; Hang, Tong, and Weng 1993) is shown in Fig. 1. The function of the digital version is similar to the classical analog version. The block ( ) 1 m G z represents process dynamics without the time-delay and is used to compute an open-loop prediction. The difference between the output of the process y and the model including time delay ŷ is the predicted error êp as shown in Fig. 1, whereas e and v are the error and the measured disturbance, w is the reference signal. If there are no modelling errors or disturbances, the error between the current process output y and the model output ŷ will be null. Then the predictor output signal ŷp will be the time-delay-free output of the process. Under these conditions, the controller ( ) 1 c G z can be tuned, at least in the nominal case, as if the process had no time-delay. The primary (main) controller ( ) 1 c G z can be designed by different approaches (for example digital PID control or methods based on algebraic approach). The outward feedback-loop through the block ( ) 1 d G z in Fig. 1 is used to compensate for load disturbances and modelling errors. Number of higher order industrial processes can be approximated by a reduced order model with a pure time-delay. In this paper the following second-order linear model with a time-delay is considered ( ) ( ) ( ) 1 1 2 1 1 2 1 2 1 1 2 1 d d B z b z b z G z z z a z a z A z − − − − − − − − − + = = + + (1) The term z represents the pure discrete time-delay. The time-delay is equal to 0 dT where 0 T is the sampling period. Design of Polynomial 2DOF Controller Previous simulation experiments demonstrated that polynomial theory is suitable method for design of the digital Smith Predictor. Polynomial control theory is based on the apparatus and methods of linear algebra (see e.g. Kučera 1993). The polynomial Smith Predictor based on the digital pole assignment was designed in (Bobál et al. 2011). The design of the controller algorithm is based on the general block scheme of a closed-loop with two degrees of freedom (2DOF) according to Fig. 2. Figure 2: Block diagram of a closed loop 2DOF control system The controlled process is given by the transfer function in the form 1 1 1 ( ) ( ) ( ) ( ) ( ) p Y z B z G z U z A z − − − = = (2) where A and B are the second order polynomials. The controller contains the feedback part Gq and the feedforward part Gr. Then the digital controllers can be expressed in the form of discrete transfer functions ( ) ( ) ( ) ( )( ) 1 1 0 1 1 1 1 1 1 r R z r G z P z p z z − − − − − = = + − (3) ( ) ( ) ( ) ( )( ) 1 1 2 1 0 1 2 1 1 1 1 1 1 q Q z q q z q z G z P z p z z − − − − − − − + + = = + − (4) According to the scheme presented in Fig. 2 and equations (1) – (4) it is possible to derive the characteristic polynomial 1 1 1 1 1 4 ( ) ( ) ( ) ( ) ( ) A z P z B z Q z D z − − − − − + = (5)