This paper extends the concept of the discrete describing function proposed previously for the analysis of sector type nonlinear sampled data control systems. The previously defined functions were only applicable to the signals which have sub-harmonics of sampling frequencies. We remove the restriction on frequencies and make it possible to apply to an arbitrary frequency. All frequencies less than a half of the sampling frequency are divided into many groups. For frequencies that belong to the same group, the evaluation of the discrete describing function provides the same region on the complex-plane. Practically, it is enough to consider a few shapes of these regions. Drawing the inverse Nyquist diagram of linear part of the system on the same complex-plane, limit cycles are predicted by the well-known. graphical method. To clarify the description, a few examples are given. The simulation results show that our proposed method is useful for predicting limit cycles which have a period not equal to the sampling period multiplied by an integer.
A method for designing a servomechanism using sliding mode is considered. This servomechanism of which the input-output relation is given by a transfer function is insensitive with respect to disturbances, parameter variations, and nonlinearity. The structure of this servomechanism satisfies the internal model principle. A servo-mechanism is also given which is obtained by quasi-sliding mode based on a continuous control law to avoid chattering. The properties of errors due to the application of the continuous control law are further investigated. It is shown that the design of the servomechanisms can easily extended to that of multivariable servomechanisms. An example is given in order to illustrate the efficiency of the proposed design method.
This paper presents an adaptive repetitive control scheme for discrete-time linear systems. The boundedness of input and output signals, and the convergence of the control error are proved. A discrete-time repetitive control structure with a specified convergence property for known parameters plants was considered by Shinnaka. However, when plant parameters are not known, the desired convergence speed is not accomplished. Since the algorithm with adaptive mechanism proposed here, is able to achieve the specified convergence rate of the control error for unknown parameters systems, the convergence speed is improved. Simulation examples are shown to confirm the effectiveness of the design scheme presented in this paper.
We consider the identification problem for a linear system when the variances of the system and/or the observation noises take on two unknown values. It is assumed that the system and observation noises can be represented in terms of Bernoulli sequences and two random variables with specified but unknown variances. We propose an algorithm to simultaneously estimate the unknown variances and the switching parameters by applying the EM (expectation-maximization) algorithm, avoiding non-linear optimizations that arise in the Maximum Likelihood (ML) and the Maximum A Posteriori (MAP) estimations. Several numerical examples are given to illustrate the effectiveness of the present algorithm.