T (p) 2 iZ. With these properties and based on the direct method of the Lyapunov stability theory, a new sliding vector and two significant Lyapunov functions are introduced in the controller design and system stability analysis. Besides, the convergent rate of the error signal can be determined by suitably choosing the sliding vector. As for the chattering problem, the saturation functions have been suggested to replace the sign functions in the control laws. Finally, the example of spacecraft driven by pairs of opposing thrusters verifies the success and robustness of the sliding-mode controller.
The first observations of anomalous changes in the dielectric constants (ε) of the uniaxial antiferromagnet MnF2 near its TN≂67.3 K are reported. Measurements of ε∥(E∥c) and ε⊥(E⊥c) were carried out from 4.2 to 300 K at rf (10 kHz) using a three-terminal capacitive method. Beginning with 300 K where ε∥=7.645 and ε⊥=8.910, both ε∥ and ε⊥ decrease with decreasing T. The lattice contribution is determined from the fit to the equation ε(T)=ε(0)+A/[exp(T*/T)−1], with the constants ε∥(0)=7.3253, A(∥)=1.0487, T*(∥)=420 K and ε⊥(0)=8.4954, A(⊥)=0.7694, and T*(⊥)=314.7 K in the 100–300 K range. For T<100 K, magnetic contributions to ε viz. εm are observed, εm∥ being positive and εm⊥ negative. At 4.2 K εm⊥=−0.13 and εm∥=0.025. Below TN, ‖εm∥‖ and εm⊥ are found to vary as σ2 (σ=sublattice magnetization) whereas above TN, ‖∂εm⊥/∂T‖ yields specific-heat type singularity. Exchange-striction mechanism is invoked to yield εm∝ spin pair-correlation function, which in the mean-field approximation can explain the observed temperature variations. However the opposite signs of εm∥ and εm⊥ and their different magnitudes are not yet understood.
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