Contradiction Resolution in the Adaptive Control of Underactuated Mechanical Systems Evading the Framework of Optimal Controllers

Abstract

In the practice, precise and efficient control is needed for certain state variables of multiple variable physical systems in which the number of the independent control variables is less than that of the independent state variables. In such cases, either the propagation of certain state variables is completely abandoned or the concept of the Model Predictive Control (MPC) is applied in which the model of the controlled system is embedded into the mathematical framework of the Optimal Controllers. This approach uses a cost function that summarizes the contributions of the frequently contradictory requirements. By minimizing this cost a kind of "compromise" is achieved. Whenever approximate and/or incomplete system models are available, the use of this controller is justified only for short time-intervals. The only way to reduce the accumulation of the effects of the modeling errors is the frequent re-design of the time horizon from the actual state as initial state that is done by the Receding Horizon Controllers. The more sophisticated Adaptive Controllers are designed by the use of Lyapunov's "Direct Method" that has a complicated mathematical framework that cannot easily be combined with that of the optimal controllers. As a potential competitor of the Lyapunov function-based adaptive controllers a Fixed Point Transformation-based approach was invented that in the first step transforms the the problem of computing the control signal into the task of finding an appropriate fixed point of a contractive map. The fixed point can be found by iteration in which the iterative sequence is generated by this contracting map. This method can be used for contradiction resolution without the minimization of any cost function by tracking the observable state components with time-sharing on a rotary basis.