Research On Ill-conditioned Problem In Grey Predictive Control Model

The gray system theory deals with the uncertain problem with no experience and little data. With twenty years' developments, it forms a self-contained theory system, and also becomes a new important member of system science theories. Meanwhile, the application of the gray theory becomes broader and broader, and the theory was applied in dozen fields, including image, life science, geology, agriculture, environment-protection, electric power and control system. The key technique of Grey predictive controller (GPC) is the GM(1,1) gray prediction control model, but there exists ill-conditioned problem in the equation group of GM(1,1), sometimes the condition is serious, it will affect the stability and efficiency of prediction control system. Professor Deng pointed out the ill-conditioned problem of gray model firstly, but how to research have not appear. This paper mainly research on the ill-conditioned problem of gray model.In order to show new developments of the gray theory and the gray technology, first, the paper introduces the theory's basis and its typical examples of development. Then introduces the new developments of the research on ill-conditioned problem in gray theory.Grey modeling theory is the key matter of gray theory. In modeling process, because using accumulate operation and least square method, we get better effect. Based on condition number and eigenvalue analytical theorem, this paper diagnose on the ill-conditioned problem, point out the advantage and disadvantage of this two methods. Through applying Gershgorin theorem, analyze and get that accumulate operation and least square method is the direction reason of the ill-conditioned in gray model. This paper introduces cumulative GM Method, a new method to estimate their parameter. Though practice, we get the condition number of cumulative GM Method is litter commonly, these models belong to well-conditioned model.GM(1,1) gray prediction control mode is the key technique of Grey Model, has been used widely in prediction control model . Through studying affined transformation on the original sequence and accumulated generating operation (AGO) sequence , this paper presents that selecting proper enter parallel movingtransformation to AGO sequence in GM (1, 1) modeling process is an effective method for solving ill-conditioned GM(1,1) model in Grey Prediction Controller.Due to GM(2,1) model has two eigenvalue, can reflect system case such as monotony, non- monotony or sway (shake ),so it has important station in gray model. Though studying the affect of multiply transformation on GM (2,1) model, we get multiply transformation on original sequence does not effect on the development parameter and prediction precision, can denote control parameter and prediction value linearity. If we choose proper multiply transformation, we can reduce the condition number.Preliminary researches are studied on ill-conditioned problem gray modeling with several innovations in this paper. However many problems are still existed, such as how to downright solve the ill-conditioned problem in GM (2, 1) model, gray Verhulst model ,and the theory of well-conditioned of Cumulative Grey Model .We shall continue research at these fields in the future.