Measuring Model Parameter Setting Errors’ Effects in the Control of an Order 4 Underactuated System

In the control-based approach of medical treatment of various illnesses such as diabetes mellitus, certain angiogenic cancers, or in anesthesia, the starting point used to be some “patient model” on the basis of which the appropriate administration of the drugs can be designed. The identification of the “patient model’s parameters” is always a hard and sometimes unsolvable mathematical task. Furthermore, these parameters have wide variability between patients. In principle, either robust or adaptive techniques can be used to tackle the problem of modeling imprecisions. In this paper, the potential application of a variant of Fixed Point Iteration-Based Adaptive Controllers was investigated in model-based control. The main point was the introduction of a “parameter estimation error significance metric” through the use of which the individual model parameter estimation can be avoided, and even the consequences of the deficiencies of the approximate model as a whole can be estimated. The adaptive controller forces the system to track the prescribed nominal trajectory; therefore, it brings about the “actual control situation” in which the consequences of the estimation errors are of interest. One component of the adaptive control is a “rotational block” that creates a multidimensional orthogonal (rotation) matrix that rotates arrays of identical Frobenius norms into each other. Since in a recent publication under review it was proved that the angle of the necessary rotation satisfies the mathematical criteria of metrics in a metric space, even in quite complicated nonlinear and multidimensional cases, this simple value can serve as a metric for this purpose. To exemplify the method, an under-actuated nonlinear system of 2 degree of freedom and relative order 4 was controlled by a special adaptive backstepping controller that was designed on a purely kinematic basis. From this point of view, it has a strong relationship with the PID controllers. This simple model was rich enough to exemplify parameters that require precise identification because their error produces quite significant consequences, and other parameters that do not require very precise identification. It was found that the method provided the dynamic models with reliable parameter sensitivity estimation metrics.