Sensitivity of dynamic matrix control systems to model error
Authors:   , Hsu Mark Kwang-Jen 
Published: December 2024

A successful process controller design depends on matching the characteristics of controller and controlled process. One of the major problems in application of process control theory involves the relationship between the real process and the theoretical model. The mathematical model of the real process to be controlled is the basis on which the control strcture is built but in many cases that basis may be weak. Any mathematical model will have unavoidable inaccuracies in structure or in the values for process parameters. The mathematical model is always an incomplete picture of the physical reality. Although most of chemical processes are nonlinear, most theory is restricted to linear models. Nonlinearities are often ignored to simplify controller design, or because they cannot be fully characterized. While the assumption of linearity is often acceptable over a moderate range, the limits of applicability need to be considered. It is not too hard to devise examples in which the range of linearity is very small. With a nonlinear process, the model error will not be constant. Another source of model error may be due to limitations in measurement accuracy. Measurement error can be biased or unbiased, but will always introduce model bias in the form of inaccurate model parameters. Statistical methods may be used to reduce this type of model error. Such statisical methods may not be helpful, however, in the case where the process which changes with time. An accurate model parameter today may not be accurate tomorrow. An example would be scaling in a heat exchanger or a shift in catalyst activity. Such a model is actually deficient in structure, either linear or nonlinear, to represent the true global behavior of the process. Adaptative control methods may reduce this problem, but are not appropriate for many situations. Another circumstance which may be considered as a special case of model error is the presence of unmeasured (and unmodeled) loads or disturbances. These also, may cause output response that can not be predicted by the model. There are two courses to follow regarding these model limitations. First, one can try to determine a more exact model by multiple tests, statistical treatment of data, and other techniques. This may be done either off-line, or adoptively on-line. Nonetheless, some error will always remain although for some systems the error may be very small. The other approach for dealing with model error is to design the control system to be robust, i.e., to be as insensitive as possible to error in the model. or to error which may come into the model with time. Such robustnesss will be purchased at the expense of performance for the error-free system. It should be the objective of the designer to obtain robustness at minimal, or even negligible, cost in terms of optimal control performance. In the work described here, the performance and sensitivity of Dynamic Matrix Control will be studied for various values of system parameters and parameter errors. Dynamic Matrix Control ( DMC ) is a model reference method for digital process control. It was developed and has been used successfully in Shell Oil Company since 1973. As implied by its name, DMC algorithm uses a set of numerical coefficients, arranged in matrix form which is called the Dynamic Matrix, to represent the system dynamic behavior. This numerical approach makes it easier to solve complex control problems in digital computer control. As a model reference control method, DMC is particularly vulnerable to the effects of model error. The algorithm is composed of both feedforward and feedback controllers. Model error can have an adverse effect on both parts of the controller. The feedforward algorithm applies control effort to a process on the basis of a dynamic model and measured inputs. If the model contains error, the applied control will be, to some extent, faulty because of incorrectly calculated compensation. All feedback control systems have a potential problem with instability. Generally for feedback control processes, the more near-perfect the control, the more there exists a possibility for instability. This is especially true of samped data process which act in many ways like dead time processes. Any process dead time makes the situation worse. Model error in lightly damped process can increase the response error and cause instability. The formulation of DMC contains a suppression factor which constrains the amplitude of control effort. This dampens both control input and the response. It can also has the effect of reducing the sensitivity of the system to model error. In this study, the effect of the suppression factor and the effect that error in model parameters have on the controlled response will be evaluated. A study of the sensitivity of DMC control system to model error is given in this work; a DMC computer simulation model is studied for several different types of model error, i.e., error in the control gain, in dead time, and in the time constant.