Control structures recursion

The point of departure for the constructive method was the consideration of a passive control structure in the light of a suitable definition of finite-time motion stability. Then, the related dynamical inverse yielded the controller and a recursive algorithm to design the nominal batch motion. The underlying solvability conditions were identifying. The combination of the controller with an observer with a compatible structure yielded the design of the output feedback tracking controller. 

Software to predict the properties of formulated products is made more powerful by a recursive procedure which can use formulas stored in files as raw materials. Particular care must be taken with program flow control and data structures for the recursion to be effective. This paper illustrates these issues using an example derived from a working formulation system for coatings development. 

Recursive procedures demand special attention to flow control and data structures. For instance, the flow control within the procedure must correctly handle an error, say missing file information or inconsistent data, that is discovered several iterations deep. Should the procedure break and return to the previous level only Should it force return to the level of the original invocation of the procedure Should it allow an interactive user a choice of supplying missing data, and if so, on any level of iteration If files are opened within the procedures, should they be closed when a recursive call is needed, or are new channel numbers to be requested, using up system resources ... 

The procedure illustrated here, besides containing only trivial technical calculations, lacks important features that are required in production programs. Extensive error checking and recovery must be performed. The procedure must detect the occurrence of a self-referential system of formulas, which would result in attempting endless recursive calls. Access to multiple raw material and formula databases adds power to the program, but must be implemented by complex code to allow flexible control of that access. The structural and input/output statements to support these features may greatly exceed the number of statements that perform modelling calculations. 

Recursive estimation methods are routinely used in many applications where process measurements become available continuously and we wish to re-estimate or better update on-line the various process or controller parameters as the data become available. Let us consider the linear discrete-time model having the general structure ... 

Based on current knowledge of the process and its disturbance characteristics, one may know or choose a reasonable difference equation structure for the controller algorithm. Starting with some assumed initial parameter values in the controller equation, the controller can be implemented on the process as shown. The control algorithm is coupled with an on-line recursive estimation algorithm which receives information on the inputs and outputs at each sampling interval and uses this to recursively estimate the optimal controller parameters on-line and to update the controller accordingly. The idea is to use the data collected from the on-line control manipulations to tune the controller directly. 

This chapter describes two new methods for obtaining frequency response and step response models from processes operating under relay feedback control. Both methods are based on the frequency sampling filter model structure and a recursive least squares estimator. 

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