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In this paper, a parametric feed-forward compensator and a parametric state-feedback stabilization controller are proposed for the model reference control to a class of quasi-linear systems. Quasi-linear systems are a special type of nonlinear systems whose coefficient matrices contain the state variables and also a time-varying parameter vector. The parametric state-feedback stabilization controller guarantees the stability of the closed-loop system and the parametric feed-forward compensator compensates the effect of the reference model state to the tracking error. The complete parametrization of the parametric feed-forward compensator is established based on a complete parametric solution to a class of generalized Sylvester matrix equations and solution of a coefficient matrix such that two matrix equations are satisfied. The established parametric state-feedback stabilization controller only needs a complete parametric solution to the same generalized Sylvester matrix equations but with different sets of freely designed parameters that represent the degrees of design freedom and may be further utilized to improve the system performance. A linear closed-loop form with the desired eigenstructure can be derived with the proposed parametric feed-forward compensator and parametric state-feedback stabilization controller, and a constant linear can even be obtained in certain cases. A numerical example and the application in spacecraft rendezvous are provided to illustrate the effectiveness of the proposed approach.

This paper addresses the distributed finite-time tracking problem for multiple uncertain mechanical systems with dead-zone input and external disturbances. An observer-based adaptive finite-time consensus protocol is designed, which consists of two steps. Firstly, distributed observers are developed such that all the mechanical systems can obtain the leader’s state in finite settling time. Then, based on backstepping method and adding a power integrator technique, the finite-time consensus protocol and appropriate adaptive laws are designed to track the estimated leader’s state. Rigorous proofs show that the tracking errors between each mechanical system and the leader can converge to a small neighborhood of origin in finite time despite the presence of dead-zone nonlinearity and external disturbances. Finally, simulation example is provided to demonstrate the effectiveness of the proposed scheme.

In this paper, the rotating consensus tracking problem for second-order multi-agent systems with external disturbances in three-dimensional space is considered. It is assumed that only a subset of the agents is given direct access to the desired trajectory information. Based on a backstepping control technique, a distributed adaptive rotating consensus protocol, driving all followers to track the leader, is developed for the case when both the relative position and relative velocity measurements are available for feedback. Moreover, the proposed control protocol design method is also extended to the case that the relative velocities are not measured. Through Lyapunov stability analysis, it is shown that, under the proposed control protocols, all followers can track the leader rotating around a common point with and without relative velocity measurements. Simulation results are included to demonstrate the effectiveness of our schemes.

A robust model following control for a class of second-order dynamical systems subject to parameter uncertainties is considered in this article. The problem is decomposed into two subproblems: a robust state feedback stabilization problem for second-order dynamical systems subject to parameter uncertainties and a robust compensation problem. The latter concerns solution of three coefficient matrices such that five matrix equations are met and, simultaneously the effect of the uncertainties to the tracking error is minimized. Based on a complete parametric solution to a class of the second-order generalized Sylvester matrix equations, the robust compensation problem is turned into a minimization problem with linear constraints. The set of linear equations is derived that determines the solution to the minimization problem. A numerical example is given to demonstrate the validity of the results.

The present paper considers eigenvalue assignment with minimum sensitivity in matrix descriptor second-order linear (MDSOL) systems via proportional-derivative plus partial second-derivative state feedback. Based on a result in the perturbation theory of the eigenvalue problem of MDSOL systems, a closed-loop eigenvalue sensitivity measure in terms of the closed-loop normalized right and left eigenvectors is established directly in the matrix second-order framework. Also a general parametric eigenstructure assignment for MDSOL systems via proportional-derivative plus partial second-derivative state feedback is proposed. Based on this, the robust pole assignment problem is converted into an independent minimization problem. The closed-loop eigenvalues may be easily taken as part of the design parameters and optimized within certain desired fields on the complex plane to improve robustness. Using a simple sequential-order algorithm, the optimality of the obtained solution to the robust pole assignment problem is totally dependent on the solution to the independent minimization problem. An example of mass-spring system demonstrates the effect of the proposed approach.