Guang-Ren Duan

Solutions to generalized Sylvester matrix equation by Schur decomposition

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation. The primary feature of this approach is that the matrix F is firstly transformed into triangular form by Schur decomposition and then unimodular transformation or singular value decomposition are employed. The results can be easily extended to second order case and high order case and can provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.

On solutions of the matrix equations XF − AX = C and XF-AX¯=C

With the help of the concept of Kronecker map, an explicit solution for the matrix equation XF − AX = C is established. This solution is neatly expressed by a symmetric operator matrix, a controllability matrix and an observability matrix. In addition, the matrix equation XF-AX¯=C is also studied. An explicit solution for this equation is also proposed by means of the real representation of a complex matrix. This solution is neatly expressed by a symmetric operator matrix, two controllability matrices and two observability matrices.

Robust Pole Assignment in Descriptor Linear Systems via State Feedback

The problem of eigenvalue assignment with minimum sensitivity in multivariable descriptor linear systems via state feedback is considered. Based on the perturbation theory of generalized eigenvalues of matrix pairs, the sensitivity measures of the closed-loop finite eigenvalues are established in terms of the closed-loop normalized right and left eigenvectors. By combining these measures with a recently proposed general parametric eigenstructure assignment result for descriptor linear systems via state feedback, the robust pole assignment problem is converted into an independent minimization problem. The optimality of the obtained solution to the robust pole assignment problem is totally dependent on the solution to the independent minimization problem. The closed-loop eigenvalues are also taken as a part of the design parameters and are optimized, together with the other degrees of freedom, within certain desired regions on the complex plane. The approach takes numerical stability into consideration and also gives good robustness for the closed-loop regularity.

Robust stabilization of descriptor linear systems via proportional-plus-derivative state feedback

A sufficient condition, in terms of quadratic stability of matrices, for the class of regular descriptor linear systems which possess no infinite relative eigenvalues is presented. Utilizing this result, the problem of stabilization of descriptor linear system via proportional-plus-derivative (PD) state feedback control is converted into two related quadratically stabilization problems, which can be easily solved using linear matrix inequality (LMI) techniques. Based on the same idea, robust stabilization via PD state feedback for uncertain descriptor linear systems, with coefficient matrices belonging to some compact sets in proper matrix spaces, has also been dealt with.

Group consensus of networked multi-agent systems with directed topology

Abstract Group consensus problems of multi-agent systems in directed networks with fixed and switching topologies are addressed in this note. Under mild assumptions, necessary and/or sufficient conditions of group consensus are derived, based on algebraic graph theories and matrix theories. For the fixed topology case, the group consensus can be guaranteed by the Hurwitz stability of coefficient matrices. For the switching topology case, the group consensus is proved to be equivalent to the asymptotical stability of a class of switched linear systems under arbitrary switching signal. A simulation result is provided to demonstrate the effectiveness of the theoretical results.

A novel nested non-linear feedback law for global stabilisation of linear systems with bounded controls

A novel nested non-linear feedback law for global stabilisation of a chain of integrators with bounded controls is proposed, which can be directly applied on the system without state transformation as soon as a so-called -stable polynomial is constructed. Then the so-called state-dependent saturation function is introduced into this novel nested non-linear feedback law to replace the standard saturation function, which can significantly improve the convergence performance of the closed-loop system. The results are then extended to a wider class of linear system that can be globally stabilised by bounded controls. Two numerical examples show the effectiveness of the proposed approach.

Consensus of networked multi-agent systems with communication delays based on the networked predictive control scheme

The consensus problem of discrete-time networked multi-agent systems (NMASs) with a communication delay is investigated in this article, where the dynamics of agents described by discrete-time linear time-invariant systems can be either uniform or non-uniform. For the NMASs with a directed topology and constant delay, a novel protocol based on the networked predictive control scheme is proposed to compensate for communication delay actively. Using algebraic graph theories and matrix theories, necessary and/or sufficient conditions of achieving consensus are obtained, which indicates that, under the proposed protocol, the consensus is independent of the network delay and only dominated by agents dynamics and communication topology. Meanwhile, the protocol design and consensus analysis are also presented in the case of no network delay. Simulation results are further presented to demonstrate the effectiveness of theoretical results.

Complete parametric approach for eigenstructure assignment in a class of second-order linear systems

Abstract This paper deals with eigenstructure assignment in the second-order linear systems using proportional plus derivative feedback controllers. Under the controllability condition of the matrix pair [A B], veiy simple, general, and complete parametric expressions in direct closed forms for both the closed-loop eigenvector matrix and the proportional and derivative feedback gains are proposed. The approach utilises directly the original system data A, B and C, and involves manipulations on only n-dimensional matrices. An example illustrates the effect of the proposed approach.

Robust fault detection in linear systems based on PI observers

A parametric approach for robust fault detection in linear systems with unknown disturbances is presented. The residual is generated using full-order proportional-integral (PI) observers. The approach is based on a result for PI observer design recently proposed. In terms of the design degrees of freedom provided by the parametric PI observer design and a group of introduced parameter vectors, a sufficient and necessary condition for PI observer design with disturbance decoupling is established. By properly constraining the design parameters according to this proposed condition, the effect of the disturbance to the residual signal is decoupled, and a simple algorithm is presented. The presented approach offers all the degrees of design freedom. A numerical example illustrates the effect of the proposed approach.

Parametric eigenstructure assignment via state feedback: a simple numerically reliable approach

An extremely simple method for eigenstructure assignment in multivariable linear systems via state feedback controllers is presented. It is shown that as soon as a series of singular value decompositions are completed, the whole set of admissible right closed-loop eigenvectors as well as the whole set of state feedback gain matrices can be simply and neatly parameterized in terms of a group of parameter vectors. This group of parameter vectors provide all the design degrees of freedom and can be utilized to achieve additional desired specifications. The method involves mainly a series of singular value decompositions and the solution to a linear matrix equation and is numerically reliable. The proposed approach does not require the closed loop eigenvalues to be distinct or to be different from the open loop ones. Moreover, it allows repeated closed-loop eigenvalues, but produces only nondefective eigenstructure for robustness consideration. Based on the presented approach, two algorithms for robust pole assignment are proposed. Several computational examples demonstrate the numerical reliability of the proposed approach.