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Observability and observable degree, which are derived from modern control theory and relative to control and estimation performances, are both important concepts for state estimation. For linear time-invariant systems, judgement of observability and evaluation of observable degree are irrelevant to filtering process and parameters. Thereby, it has limited application ability. Different from linear systems, observability and observable degree becomes more complex and are seldom studied deeply and integrally for nonlinear systems. Due to the complexity of nonlinear filtering, the observable degree depends strongly on filtering process, filter class and associated parameters. Thereby, we study the observable degree analysis of nonlinear estimation systems by using unscented information filter (UIF). A way to evaluate observable degree using singular value decomposition (SVD) of observability matrix is presented based on the UIF and the influencing factors are deeply analyzed. Moreover, differences of observable degree analysis are distinctly discussed between linear and nonlinear systems. The results reveal that evaluation of the observable degree for nonlinear systems can be affected by filtering estimate at last time, initial estimate and filter's parameters such as covariances of process and measurement noises. Some simulations are demonstrated to validate the results.

Rectangular hexahedron sonotrode has remarkable advantage as a platform to realize auxiliary ultrasonic vibration of workpiece with its abundant resonant frequency. However, the three-dimensional coupled vibration characteristics of rectangular hexahedron sonotrode is hard to meet the requirement of ultrasonic assisted micro forming process, which should be made structural optimization for improving its vibration characteristics. In this paper, the basic structure of 20 kHz rectangle hexahedron sonotrode was designed according to the apparent elastic method and R-L square transformer design theory. The influence of number, size and distribution of inner holes and structure transition arc radius was simulated analysed for optimizing the vibration properties of the sonotrode. The results show that the optimized rectangular hexahedron sonotrode can realize the conversion of ultrasonic vibration direction and get uniform vertical vibration amplitude on the whole radiant surface, so as to meet the requirement of ultrasonic assisted micro forming processing.

In order to avoid the complex adjustment of scaling factor and quantization factor in the design of fuzzy controller and the shortcomings that the fuzzy control rules can not be changed once identified, a fuzzy controller with self-optimizing rules is adopted . Particle swarm optimization (PSO) algorithm is used to optimize the parameters of fuzzy controller. A new evaluation function including the system adjusting time, rise time, over-shoot, and steady-state error is defined. A group of parameters of the fuzzy controller that minimize the evaluation function is calculated rapidly by searching in the given controller parameters area. Numerical simulations show that the controller can easily be applied to the first-order and second-order system with time delay, no overshoot, and its performance is also better than the classic PID optimized by PSO.