Abstract
We design adaptive algorithms for both cancellation and estimation of unknown periodic disturbance, by feedback of state--derivatives ( i.e.,} without position information for mechanical systems) for the plants which are modeled as a linear time invariant system. We consider a series of unmatched unknown sinusoidal signals as the disturbance.The first step of the design consists of the parametrization of the disturbance model and the development of observer filters.The result obtained in this step allows us to use adaptive control techniques for the solution of the problem.In order to handle the unmatched condition, a backstepping technique is employed. Since the partial measurement of the virtual inputs is not available, we design a state observer and the estimates of these signals are used in the backstepping design.Finally, the stability of the equilibrium of the adaptive closed loop system with the convergence of states is proven.As a numerical example, a two-degree of freedom system is considered and the effectiveness of the algorithms are shown.
Highlights
We design adaptive algorithms for both cancellation and estimation of unknown periodic disturbance, by feedback of state–derivatives for the plants which are modeled as a linear time invariant system
The backstepping procedure consist of the following steps; (1) designing a controller bTx p(t) that both cancels the disturbance and stabilize the system, (2) defining the error between the desired and actual value bTx p(t), this step can be considered as a state transformation, (3) by taking the time derivative of the defined error term, we find the dynamics of error
By employing the developed algorithms, we present the results for estimation and active disturbance cancellation
Summary
We consider a general representation of a linear time invariant system which is given as follows x (t) =Anx(t) + Bn bTx p(t) + aTx x + ν(t) , (1) p(t) =Amp(t) + Bm bTp p(t) + aTp x + u(t) , (2). The disturbance signal and the control input are separated by m integrators. This situation makes the input u(t) and ν(t) unmatched. Because bTx p(t) is not the main control input, and it has its own dynamics given by (2). The main aim is to design a control law for u(t) such that the signal bTx p(t) cancels the effect of the disturbance while maintaining the boundedness of all signals. Since state p(t) and disturbance ν(t) are not measured, observers are designed to estimate these signals. The given assumptions are sufficient for observer based control design and show the boundedness of signals. This note provides a solution for more general systems
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