G.R. Duan

Robust Model Reference Control for Multivariable Linear Systems: A Parametric Approach

Abstract The problem of robust model reference control for multi variable linear systems with structural parameter uncertainties is considered. It is shown that the problem can be decomposed into two sub-problems. One is a robust state feedback stabilization problem for multi variable linear systems subject to parameter uncertainties; the other is the so-called robust compensation problem which concerns solution of three coefficient matrices such that four matrix equations are met and the effect of the uncertainties to the tracking error is minimized simultaneously. Based on a complete parametric solution of a class of generalized Sylvester matrix equations, the robust compensation problem is turned into a minimization problem with a quadratic objective and a set of linear constraints.

Robust model reference control for multivariable linear systems subject to parameter uncertainties

Abstract Robust model reference control for multivariable linear systems with structural parameter uncertainties is considered. It is shown that the problem can be decomposed into two subproblems: a robust state feedback stabilization problem for multivariable linear systems subject to parameter uncertainties and a robust compensation problem. The latter concerns solution of three coefficient matrices such that four 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 generalized Sylvester matrix equations, the robust compensation problem is turned into a minimization problem with quadratic cost and linear constraints. A set of linear equations is derived that determines the optimal solution to the minimization. An example illustrates the application of the proposed approach.

Robust Fault Detection in Descriptor Linear Systems Via Generalised Unknown Input Observers

Abstract A new parametric approach for robust fault detection in descriptor linear multivariable systems with unknown disturbances is proposed. The residual is generated using a full-order generalised state observer, which is designed based on a recently proposed parametric eigenstructure assignment approach. By establishing a condition for robust fault detection in terms of the observer eigenvectors, a simple algorithm for robust fault detection in descriptor linear systems via unknown observers is presented. An example shows the effect of the proposed approach.

Parametric eigenstructure assignment via output feedback based on singular value decompositions

The paper investigates eigenstructure assignment in multivariable linear systems via output feedback. Three problems are proposed and are related to a type of generalized Sylvester matrix equations. By proposing a general complete parametric solution to this type of generalized Sylvester matrix equations based on singular value decompositions, a general complete parametric approach is then presented for the proposed eigenstructure assignment problems. General parametric expressions for both the closed-loop eigenvector matrices and the output feedback gain are established in terms of certain parameter vectors. These parameter vectors provide the design degrees of freedom and can be utilized to achieve some desired specifications. Based on the proposed results and the Matlab Optimization Toolbox, a Matlab file is created, which finds a solution that gives minimum closed-loop eigenvalue sensitivities for the problem of eigenstructure assignment via output feedback.

Dynamical order assignment in linear descriptor systems via state derivative feedback

In this paper, the dynamical order assignment problem for linear descriptor systems via state derivative feedback is studied. A new, direct and simple parametric approach is presented, which assigns via state derivative feedback any desired allowable dynamical order of the closed-loop system. A simple, general direct and complete parametric expression for all the feedback gains which assign the desired dynamical order of the closed-loop system is established. Based on this result, a general complete parameterization of all those feedback gains possessing the minimum Frobenius norm is further presented. The proposed approach is convenient to use and possesses good numerical reliability since it mainly involves singular value decompositions and inverses of positive definite diagonal matrices. The presented example shows these advantages and the simplicity of the proposed approach.

Disturbance decoupling in descriptor systems via output feedback-a parametric eigenstructure assignment approach

Based on a complete parametric approach for eigenstructure assignment in descriptor linear systems via output feedback, disturbance decoupling using output feedback in descriptor linear systems is investigated. Both the dynamical and static disturbance decoupling problems are tackled. Necessary and sufficient conditions for both problems are proposed in terms of the closed-loop eigenvalues and eigenvectors. By arranging these conditions into constraints on the design parameters provided by eigenstructure assignment, the disturbance decoupling problems are converted into eigenstructure assignment problems with extra parameter constraints. The approach guarantees closed-loop regularity, offers certain flexibility and can provide all the degrees of design freedom. An example is investigated to show the effect of the proposed approach.

Separation principle for robust pole assignment-an advantage of full-order state observers

A separation principle for robust pole assignment in full-order observer-based control system designs is proposed, which reveals the fact that pole assignment with minimum sensitivities in a n-dimensional linear multivariable system using a full-order observer-based state feedback controller can easily be realized by solving two separate n-dimensional state feedback robust pole assignment problems.

Right coprime factorizations using system upper Hessenberg forms-the multi-input system case

Based on a method for right coprime factorizations of linear systems using matrix elementary transformations, it is shown that a very simple iteration formula exists for right coprime factorizations of multi-input linear systems in system upper Hessenberg forms. This formula gives directly the coefficient matrices of the pair of solutions to the right coprime factorization of the system Hessenberg form, and involves only manipulations of inverses of a few triangular matrices and some matrix productions and summations. Based on this formula, a simple, efficient procedure for determining a right coprime factorization of a multi-input linear system is proposed, which first converts a given linear system into its system Hessenberg form using some orthogonal similarity transformations and then applies the iteration formula to the converted system Hessenberg form. An example demonstrates the usage of the approach.

Disturbance Attenuation in Luenberger Function Observer Designs — A Parametric Approach

Abstract This paper proposes a simple approach for disturbance attenuation in the designs of Luenberger function observers for multi variable linear systems. The basic idea is to minimise the H 2 norm of the transfer function from disturbance to estimation error using the design freedom provided by a parametric approach for Luenberger observer design. Besides the design parameters, the observer poles are also optimised within desired regions on the left-half complex plane. With the proposed approach, additional specifications can be easily achieved.

Stable observer-based controller design for robust state-feedback pole assignment

Abstract The design of stable observer-based controllers for robust pole assignment is addressed in this paper. The stability problem of these dynamical controllers is investigated, which is often ignored during the controller design. A design formulation of stable observer controllers is presented using state-feedback pole assignment techniques. Although the design formulation is principally aimed at the design of a stable controller, the mixed sensitive function in the frequency domain is also considered to improve the robustness of the closed-loop system. This ensures that the closed-loop system has good robustness and the controller is stable.