In the field of noninear control two typical methodologies can be chosen.
A typical possibility is assuming “ideal controllers and sensors” of extremely fast response. In this case the equations of motion of the controlled system can mathematically be approximated by a set of differential equations. A considerable segment of the control literature using Lyapunov’s direct method (e.g. [R11]) proceeds along this rut. However, it must be emphasised that the great majority of the practical problems results in differential equations that do not have solutions in closed analytical form. If we wish to see numerical details on the operation of the controllers the stability of which has been mathematically proved we have to develop numerical simulations.
To achieve more realistic results it is expedient to take into account the limitations of our digital controllers and sensors of finite time-resolution. In this case the system originally described by differential equations must be completed by the insertion of event clocks and sample holders that represent the “cyclic” nature of the controllers. In this manner the “cycle time of the controller” can be distinguished from the time-resolution of the numerical simulations.
It must be emphasized that besides the discrete time-resolution applied various numerical simulators may apply different numerical integration methods and also allow setting certain numerical parameters that evidently concern the “results”
of the numerical simulations. In the lights of the “believabilty considerations”
expounded in the sequel I applied the following methods.
As the simplest and fastest approach, by the use of INRIA’s SCILAB programming environment I developed numerical programs applying simple Euler integration with fixed time resolution. It was found that for stable control rough approximate results can be obtained for making the assumed cycle time of the controller (1 ms) identical with the time-resolution of the numerical integration. For checking consistency this time step was halved and if the results did not show significant modification they have been accepted for illustrating the operation of the proposed controller.
A further step towards more reliable results the fixed time-resolution was distinguished from the controller’s cycle time and a control cycle was divided into 10 segments for numerical simulations. For such calculations I used the same simple SCILAB program language.
To make more professional simulations I applied the SCILAB’s numerical co-simulator, SCICOS, that gave a convenient graphical interface for calling more professional numerical integrators. For simulating the discrete nature of digital controllers sample holders and event clocks were built in these simulations.
Another aspect concerning the methodology of research is the fact that the question of “believability of the numerical results” arises in each of the above mentioned numerical solutions. Following the pioneering work by Lorenz who made numerical computations on simple meteorological model of Earth using the computer technology of the sixties it became evident that there are “stable” and “unstable”
systems in which the consequences of the initial errors remain finite or grow exponentially with time, respectively [R24]. Though for certain special systems of differential equations there are theoretical results for the proper application of finite element methods in general this problem cannot be tackled. In certain cases they can be tackled or understood by using the concepts of Riemannian Geometry if the solution of the equations corresponds to some geodesic line of a given geometry.
Using the concept of “parallel translation of vectors and tensors” two geodesic lines starting from neighboring points with identical initial velocity can be considered as it was discussed by Arnold [R25].
In my investigations I assumed that
• The successful adaptive control corresponds to a stable system;
• The actual numerical results obtained naturally depend on the time resolution applied but only in a slight extent;
• For a finite duration of motion the stable numerical results were declared to be believable if halving the finite time-step in the simulation did not lead to observable differences in them. This attitude is right since the convergence, or at least the possibility of the convergence within a region of attraction were theoretically proved before running the simulations that only illustrated but proved the stability and usability of the proposed methods.
• In certain cases I also used the ODE Solver of INRIA’s SCILAB and SCICOS software that generally applies various, quite sophisticated numerical integration methods, depending on the stiffness of the problem considered. (Its use is especially convenient when graphical programming can be applied to build up the appropriate environment in which the ODE Solver can be called.) It also modifies the density of the discrete time-resolution automatically to meet the prescribed precision requirements. By carefully prescribing the allowable maximal time step and the relative and absolute tolerance consistent results were obtained for the stable systems to illustrate the operation of the stable controller
• If the results were divergent their details were not “believed”. Such runs only illustrated the possibility of leaving the range of convergence of the applied method.
Another relevant point is the “believability or realistic nature of the models”
applied in the simulations. While in general it can be accepted that no any given model can fully and completely describe the reality, a good model can be regarded at least as a “cubist picture” that contains significant features of the reality, therefore it can be used as a “paradigm” i.e. as characteristic representative of a whole set or class of problems. In this sense the simulation results obtained cannot be regarded completely worthless or improper means of illustration, though it has to be admitted that any particular practical application of the proposed method needs further detailed investigations.
To technically realize the proposed novel approaches the observation of the behavior of the controlled system was necessary. For this purpose the “Expected – Realized Response Scheme” was introduced. According to that scheme a considerable part of the control tasks could be formulated by using the concepts of the appropriate “excitation” Q of the controlled system to which it is expected to respond by some prescribed or “desired response” rd. (The physical meaning of the appropriate excitation and response depend on the phenomenology of the system under consideration. In the case of Classical Mechanical Systems the excitation physically can be force and/or torque, while the response can be linear or angular acceleration, etc.) The appropriate excitation can be computed by the use of some available approximate “inverse dynamic model” as Q=φ(rd). Since normally this inverse model is neither complete nor exact, the actual response determined by the system's dynamics, ψ, results in a “realized response” rr that differs from the desired
one: rr=ψ(φ(rd))≠rd. It is worth noting that the functions φ() and ψ() may contain various hidden parameters that partly correspond to the dynamic model of the system, and partly pertain to unknown external dynamic forces acting on it. Due to phenomenological reasons the controller can manipulate or “deform” the input value from rd to some r*d so that rd=ψ(φ(r*d)). Other possibility is the manipulation of the output of the rough model.
The above structure evidently indicated that using the pairs of the “desired”
response known and set by the controller and comparing it to the observed “realized”
response mathematically can be formulated as seeking the solution of a Fixed Point Problem. From this point on the main direction of the research was seeking various deformations or fixed point transformations that were able to generate appropriate sequences of responses that can converge to the fixed point. In this approach in each control cycle one iterative step can be done with the actually available updated
“desired response”, and in the next cycle the deformation applied can be updated on the basis of the “observed response”. If the dynamics of the adaptive iteration is considerably faster than that of the control task such solution may result in practically acceptable tracking. (This idea is in strict analyogy with the use of Cellular Neural Networks in picture processing based on the concept of Complete Stability [R19].) Similar “dynamic approaches” were also applied in the literature as e.g. dynamic inversion of nonlinear maps by Getz, Getz and Marsden [R20], [R21], but these considerations extensively used the technique of the Lyapunov Functions.
In contrast to Lyapunov’s 2nd Method [R22], [R23] that normally can generate quadratic expressions with absolute minima in wide environments that can act as basins of attraction of convergent solutions, in the novel approach convergence can be achieved by applying contractive maps in Banach Spaces. In this manner iterative sequences converging to the fixed point of the appropriate map can be obtained. This latter solution can be more “fragile”, but in the same time far simpler than the application of some Lyapunov function. Furthermore, its realization may need far less complicated computations.