In this note, an approach to the design of optimal fault detection systems for linear discrete-time periodic systems is proposed, which leads to an optimized compromise between robustness to unknown disturbances and sensitivity to faults. The needed computation mainly consists in solving a difference periodic Riccati system. The proposed approach is finally illustrated by a numerical example.

Fault detection of linear discrete-time periodic systems

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Totally Positive Matrices

Periodic ordered sets of structural indices and a new and simpler characterization of the notions of invariant zero, transmission zero, pole and eigenvalue are introduced for a linear periodic discrete-time system. Also the notions of input and output decoupling zeros (together with their ordered sets of structural indices) are extended to this type of system and are shown to have a meaning and relations with the structural properties of the system, wholly similar to the time-invariant ones. The ordered sets of structural indices of non-zero poles, eigenvalues and zeros of any type are independent of time, while those (and even the existence) of the null invariant zero, transmission zero, input and output decoupling zero and pole can depend on time, as well as those of the null eigenvalue (although its algebraic multiplicity is independent of time). The ordered sets of structural indices of invariant zeros are not altered by a linear periodic state feedback (as well as those of input decoupling zeros) and...

Finite zero structure of linear periodic discrete-time systems

The output stabilization problem for discrete-time linear periodic systems is solved. Both the state-feedback control law and the state-predictor are based on a suitable time-invariant state-sampled reformulation associated with a periodic system. Preliminary concepts of periodic system theory are briefly recalled. In particular, the structural properties of a linear discrete-time periodic system are properly related to those of a time-invariant system associated with it. By resorting to such a time-invariant reformulation, the output stabilization problem via pole placement is solved. The stabilizing controller is constituted by a control law and an asymptotic state predictor, both of which are shown to be periodic. >

Output stabilization via pole placement of discrete-time linear periodic systems

The geometric approach for linear periodic discrete-time systems

This paper considers the pole-assignment problem for discrete-time linear periodic systems through the use of linear periodic state-variable feedback control. It is shown that if the N-periodic system with m inputs and n states is completely reachable then the problem can be reduced to the pole-assignment problem for a discrete-time linear invariant system with Nm inputs and n states.

Pole-assignment problem for discrete-time linear periodic systems

P-matrix completions under weak symmetry assumptions

An $n \times n$ real matrix $A$ is said to be (totally negative) totally nonpositive if every minor is (negative) nonpositive. In this paper, we study the properties of a totally nonpositive matrix and characterize the case of a nonsingular totally nonpositive matrix $A$, with $a_{11}< 0$ in terms of its $LDU$ factorization ($L(U)$) is a unit lower- (upper-) triangular matrix, respectively, and $D$ is a diagonal matrix). This characterization allows us to significantly reduce the number of minors to be checked in order to decide the total nonpositivity of a nonsingular matrix with a negative (1,1) entry.

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$LDU$ Factorization of Nonsingular Totally Nonpositive Matrices

Given a square matrix A, a Brauer’s theorem [Brauer A, Limits for the characteristic roots of a matrix IV Applications to stochastic matrices, Duke Math J, 1952, 19(1), 75–91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues Older and newer results can be considered in the framework of the above theorem In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable Other applications presented are related to the Jordan form of A and Wielandt’s and Hotelling’s deflations An extension of the aforementioned Brauer’s result, Rado’s theorem, shows how to modify r eigenvalues of A at the same time via a rank-r perturbation without changing any of the remaining eigenvalues The same results considered by blocks can be put into the block version framework of the above theorem

/pdf/a-brauer-s-theorem-and-related-results-nbetbfjorl.pdf

A Brauer’s theorem and related results

The pole-placement problem for the monodromy matrix of a discrete-time linear periodic system is considered. Given a partition of the complex plane into a ‘good’ part ℂ and a ‘bad’ part ℂb, the result obtained by Wonham (1979) is extended to the periodic case. This extension is based on the decomposition of a completely controllable periodic system into two subsystems, such that the first subsystem is completely reachable and the eigenvalues of the monodromy matrix of the second one are equal to zero.

Ana M. Urbano

Papers

Pole-assignment problem for discrete-time linear periodic systems

P-matrix completions under weak symmetry assumptions

$LDU$ Factorization of Nonsingular Totally Nonpositive Matrices

A Brauer’s theorem and related results

Pole-placement problem for discrete-time linear periodic systems