In this study, the stability problem of descriptor second-order systems is considered. Lyapunov equations for stability of second-order systems are established by using Lyapunov method. The existence of solutions for Lyapunov equations are discussed and linear matrix inequality condition for stability of second-order systems are given. Then, based on the feasible solutions of the linear matrix inequality, all parametric solutions of Lyapunov equations are derived. Furthermore, the results of Lyapunov equations and linear matrix inequality condition for stability of second-order systems are extended to high-order systems. Finally, illustrating examples are provided to show the effectiveness of the proposed method.

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The partial eigenstructure assignment problem is to replace a small number of eigenvalues and eigenvectors of system while the remaining eigenvalues and corresponding eigenvectors are kept unchanged. The paper studies partial eigenstructure assignment problem for vibration system via accelerationvelocity feedback control. Two new orthogonality relations are established to construct a feedback control law which can solve the proposed partial eigenstructure assignment problem. The solvability condition and parametric expressions of feedback gains are derived for solving this problem. The minimum norm partial eigenstructure assignment problem is further considered by using gradient-based optimization method. An optimization algorithm is proposed for solving the minimum norm partial eigenstructure assignment problem. Numerical examples are provided to show the effectiveness of the algorithm.

In the paper, the partial eigenstructure assignment problems are investigated using acceleration–velocity–displacement active control in a singular vibrating structure. The problems are transformed into solving matrix equations using the receptance matrix method. Iterative sequences are constructed, and the iterative feasibility is presented for solving the matrix equations. The partial eigenvectors of the closed-loop system are reassigned by imposing modal constraints. An algorithm is proposed to get numerical solutions of the derived matrix equations. The initial value condition is discussed to obtain the minimum norm solution of the partial eigenstructure assignment problems. The designed acceleration–velocity–displacement active control can solve the partial eigenstructure assignment problems depending only on original vibrating structure information. The proposed numerical algorithm can obtain the minimum norms of controller gain, which implies minimum energy consumption. Numerical examples are given to illustrate the effectiveness of the proposed methods.

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