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The problem of eigenstructure assignment in multivariable linear systems by decentralized output feedback is considered. By using a complete parametric solution of a generalized Sylvester matrix equation, parametric representations of both the left and right closed-loop eigenvectors and generalized eigenvectors as well as the feedback gains with respect to the closed-loop eigenvalues and two series of partially free parameter vectors are established. The obtained result does not require any conditions on the closed-loop eigenvalues, and generalizes some previous results in this area. An example shows the effect of the proposed approach. >

This note generalizes the previous results of the closed-loop eigenstructure assignment via output feedback in linear multivariable systems. Necessary and sufficient conditions for the closed-loop eigenstructure assignment by output feedback are presented. Some known results on entire eigenstructure assignment are deduced from this result.

This paper presents an algorithm for computing a feedback law which guarantees that any initial state of the closed loop of a linear time-invariant discrete system will be driven to the origin in minimum finite time. The feedback law is constructed by solving a sequence of algebraic linear equations. The designer has considerable freedom in choosing the generalized eigenvectors which would allow the shaping of the transient response of the system. The algorithm offers, as a byproduct, a new and easy method for computing the so-called "generalized control form" of any controllable pair. Several examples are worked out to demonstrate the feasibility of the proposed method.

The first-order, necessary condition for optimality is derived from a variational argument that involves an ad hoc modification of the Bliss method, resulting in a Hamiltonian characterization in terms of Edx,dt, rather than dx/dt, the former being smoother than the latter. This approach sidesteps the regularity conditions of the Lagrange multiplier theory. Under some mild assumptions, the necessary condition for optimality is also sufficient and the optimal control exists. The numerically relevant result is a generalized eigenvector, inverse-free characterization of optimality. >

Given a controllable and observable triple ( A, B, C ) describing a linear time invariant multivariable system Σ, which gives rise to a full rank transfer function matrix T_{o}(s) , the structure of the maximal ( A, B )- invariant subspace contained in \ker C is investigated using a polynomial matrix approach. Thus, certain connections between the geometric and the polynomial matrix approaches to linear system theory are established.

This note presents two generalized matrix continued-fraction descriptions in the second Cauer form obtained from a nonsingular rational matrix and/or polynomial matrix fraction descriptions. Computational algorithms are developed for the expansion and inversion of the generalized matrix continued-fraction descriptions which are useful in developing applications to multivariable control systems.

The eigenvalue-assignment approach of Brogan [1], [2] is generalized and extended to the assignment of the entire closed-loop eigenstructure of linear multivariable systems. The set of assignable eigenvectors and generalized eigenvectors emerges naturally in the solution and is given, moreover, in an explicit parametric form. Two numerical examples are worked out to demonstrate the application of the procedure.

The subspaces, to which closed-loop generalized eigenvectors are restricted, are described in terms of the controllability indexes of the pair ( A, B ) and the polynomials of minimal degree that span ker[ A- \lambda I, B ]. This characterization of the eigenspaces is then used to calculate the deadbeat controller of minimum Frobenius norm.

An elementary, but not sufficiently well-known algorithm for finding a basis for the null-space of a matrix is given. Its use is illustrated for the computation of generalized eigenvectors and the matrix pseudoinverse.

In a recent paper [1], a characterization has been given for the class of all closed-loop eigenvector sets which can be obtained with a given set of distinct closed-loop eigenvalues. This note extends these results to characterize the class of generalized eigenvector chains which can be obtained with a given set of nondistinct eigenvalues. Included is an algorithm for computing a feedback matrix which gives the selected closed-loop eigenvalues and generalized eigenvector chains. Although there are limitations on the Jordan structure of the closed-loop system, this algorithm allows one to realize any "allowable" closed-loop Jordan configuration.

It is shown that the positions of the zeros in the modal matrix of a system, and its inverse, which result from zeros in the state matrix, can be predicted by computing the corresponding reachability matrix.

The properties of the multiple-input time-optimal regulator state-transition matrix for linear discrete systems are developed by using the matrix concepts of nilpotency, eigenvector analysis, and Jordan canonical form.