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In this paper we introduce a new kind of nonnegative matrices which is called (sp) matrices. We show that the zero solutions of a class of linear discrete dynamical systems are asymptotically stable if and only if the coefficient matrices are (sp) matrices. To determine that a matrix is (sp) matrix or not is very simple, we need only to verify that some elements of the coefficient matrices are zero or not. According to the result above, we obtain the conditions for the stability of several classes of discrete dynamical systems.

In this study, a new structural identification method is proposed to simultaneously evaluate the parameter and partial external excitation. The external excitation is decomposed as the linear combination of orthogonal bases, by which the problem of time-variant excitation identification is transformed into the identification of constant decomposition coefficients. A new constrained unscented Kalman filter (UKF) is proposed to identify the structural parameters and coefficients of excitation decomposition based only on measurement of the acceleration response. The proposed filter can retain the physical meaning of the structural parameters identified. A three-storey hysteretic nonlinear shear building is investigated numerically. The structural parameter and external force can be accurately identified with the proposed filter. The results of the simulation studies using the constrained UKF are compared with those from the conventional UKF. It is shown that some parameters identified by the conventional UKF may lose physical meaning, while the proposed constrained UKF can retain physical meaning. In the presence of measurement noise, the structural parameters and dynamic load can still be accurately identified using the proposed constrained UKF, which indicates the stability of the identification process.

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