Nonlinear adaptive tracking‐control synthesis for functionally uncertain systems

Abstract

The paper is concerned with the problem of adaptive tracking system control synthesis. It is assumed that a nonlinear, feedback linearizable object dynamics (model structure) is (partially) unknown and some of its nonlinear characteristics can be approximated by a sort of functional approximators. It has been proven that proportional state feedback plus parameter adaptation are able to assure its asymptotic stability. This form of controller permits online compensation of unknown model nonlinearities and exogenous disturbances, which results in satisfactory tracking performance. An interesting feature of the system is that the whole process control is performed without requisite asymptotic convergence of approximator parameters to the postulated 'true' values. It has been noticed that the parameters play rather a role of slack variables on which potential errors (that otherwise would affect the state variables) cumulate. The system's performance has been tested via Matlab/Simulink simulations via an example of ship path-following problem.

On the ship path-following control system design by using robust feedback linearization

“…As we want to select γ to be minimal (provided the solution RE exists) we have to apply this procedure repeatedly. This way we get the proper control strategy (see (15)), which will be used later in the process simulation.…”

“…This approach, as an alternative to adaptive control techniques [15], is less susceptible to an unacceptable state transition process. However, one should be aware that the combination of robust and adaptive control techniques in one design may lead to a better effect.…”

On generalization of integral control to a class of nonlinear uncertain systems

“…polynomials). However, from practical point of view, it seems reasonable to use approximator-like structures where known model basis functions contain some portion of the plant specific knowledge [11]. In other words, the unknown functions α α α and β β β of the system (10a) are assumed to be the linear combinations of some known model related functions α α α i and β β β i and which represent elementary knowledge of the model.…”

“…The dynamic nonlinear controller (38), (or (41)) should be conceived not as a nonlinear controller but rather as a parameter adaptation scheme. This approach, from the adaptive control standpoint, can be summarized in the following theorem [11]. Note that the integrators (31) are interpreted here as the parameters adaptation laws (42).…”

“…In the adaptive control context this feature i.e. the fact that the variables θ θ θ do not tend to zero, has been considered as a drawback [11]; in the presented here 'integral action standpoint' it is a desired value. …”