Digital Smith Predictors - Design And Simulation Study

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 algorithms for digital control for processes with time-delay. The algorithms are based on the some modifications of the Smith Predictor (SP). One modification of the SP based on the digital PID controller was applied and it was compared with two designed modifications based on polynomial approach. The program system MATLAB/SIMULINK was used for simulation verification of these algorithms. INTRODUCTION Time-delays appear in many processes in industry and other fields, including economical and biological systems (see Normey-Rico and Camacho 2007). They are caused by some of the following phenomena: 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. Consider a continuous time dynamical linear SISO (single input ( ) u t – single output ( ) y t ) system with time-delay d T . The transfer function of a pure transportation lag is d T s e where s is complex variable. Overall transfer function with time-delay is in the form ( ) ( ) d T s d G s G s e = (1) where ( ) G s is the transfer function without time-delay. Processes with significant time-delay are difficult to control using standard feedback controllers. When a high performance of the control process is desired or the relative time-delay is very large, a predictive control strategy must be used. The predictive control strategy 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. Historically first modifications of time-delay algorithms were proposed for continuous-time (analogue) controllers. On the score of implementation problems, only the discrete versions are used in practice in this time. Some modifications of the digital Smith Predictors are designed and verified by simulation in this paper. DIGITAL SMITH PREDICTORS Although time-delay compensators appeared in the mid 1950s, their implementation with analogue technique was very difficult and these were not used in industry. Since 1980s digital time-delay compensators can be implemented. In spite of the fact that all these algorithms are implemented on digital platforms, most works analyze only the continuous case. The digital time-delay compensators are presented e.g. in (Palmor and Halevi 1990, Normey-Rico and Camacho 1998). The discrete versions of the SP and its modifications are suitable for time-delay compensation in industrial practice. Structure of Digital SP The block diagram of a digital SP (see Vogel and Edgar 1980, Hang et al. 1986, Hang et al. 1989, Hang et al. 1993) is shown in Fig. 1. The function of the digital version is similar to the classical analogue 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 p ê as shown is in Fig. 1 where u , w and e are the control signal, the reference signal and the error. If there are no Proceedings 25th European Conference on Modelling and Simulation ©ECMS Tadeusz Burczynski, Joanna Kolodziej Aleksander Byrski, Marco Carvalho (Editors) ISBN: 978-0-9564944-2-9 / ISBN: 978-0-9564944-3-6 (CD) modelling errors or disturbances, the error between the current process output and the model output will be null and the predictor output signal p ŷ will be the timedelay-free output of the process. Under these conditions, the controller ( ) c G s can be tuned, at least in the nominal case, as if the process had no timedelay. The primary (main) controller ( ) 1 c G z can be designed by the 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. The dash arrows indicate the tuned parts of the Smith Predictor. Figure 1: Block Diagram of a Digital Smith Predictor Most industrial processes can be approximated by a reduced order model with some pure time-delay. Consider the following second order linear model with a time-delay ( ) ( ) ( ) 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 − − − − − − − − − + = = + + (2) for demonstration of some approaches to the design of the adaptive Smith Predictor. The term z represents the pure discrete time-delay. The time-delay is equal to 0 dT where 0 T is the sampling period. Identification of Time-delay In this paper, the time-delay is assumed approximately known or possible to be obtained separately from an off-line identification using the least squares method ( ) 1 ˆ − = T T Θ F F F y (3) where the matrix F has dimension (N-n-d, 2n), the vector y (N-n-d) and the vector of parameter model estimates Θ̂ (2n). N is the number of samples of measured input and output data, n is the model order. Equation (3) serves for a one-off calculation of the vector of parameter estimates Θ̂ using N samples of measured data. The individual vectors and matrix in equation (3) have the form ( ) ( ) ( ) ( ) ( ) ( )