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An existence result on diagonal solutions of a linear matrix inequality is used to study diagonal Hurwitz and Schur stability and to derive the secant condition for systems with cyclic structure.

A solution X of a discrete-time algebraic Riccati equation is called unmixed if the corresponding closed-loop matrix Φ(X) has the property that the common roots of det(sI−Φ(X)) and det(I−sΦ(X)∗) (if any) are on the unit circle. A necessary and sufficient condition is given for existence and uniqueness of an unmixed solution such that the eigenvalues of Φ(X) lie in a prescribed subset of C.

The existence of a solution of the discrete-time algebraic Riccati equation is established assuming modulus controllability and positive semidefiniteness on the unit circle of the Popov function. As an application a nonstrictly bounded real lemma is obtained.

If two solutions Y⩽Z of the DARE are given then the set of solutions X with Y⩽X⩽Z can be parametrized by invariant subspaces of the closed loop matrix corresponding to Y. The paper extends the geometric theory of Willems from the continuous-time to the discrete-time ARE making the weakest possible assumptions.

Difference equations of the form X(t) = F ∗ (t)X(t − 1)F(t) − F ∗ (t)X(t − 1)G(t)[I + G ∗ (t)X(t − 1)G(t) −1 G ∗ (t)X(t − 1)F(t) + Q(t) and their associated Hermitian matrices H(t) = ( F Q F ∗ −GG ∗ )(t) are studied. Solution of different Riccati equations can be compared if the difference of their corresponding Hermitian matrices is semidefinite for all t. An application to the discrete-time LQ optimal control problem is given.

With a non-singular polynomial matrix LeF n×n [z] over a field F we associate two vector spaces V L and U L with respective shift operators S + and S − . A duality between V L and V L T and between U L and V L T is established. The spaces V L and U L and the maps S + and S − determine L in the following way. Let the columns of the polynomial matrices Ĉ and B^ T form a pair of dual bases of V L and V L T and let A be the matrix of S + with respect to the basis Ĉ,S + Ĉ = ĈA , furthermore let the columns of Ĥ and Ĝ T be dual bases of U L and V L T and let N be the matrix of S − with respect to Ĥ,S − Ĥ = ĤN , then L is completely determined by these matrices. L can be factored as L ( z ) = ( C ^ ( z ) H ^ ( z ) ) ( z 1 - A 0 0 1 + z N ) ( B ^ ( z ) G ^ ( z ) ) .

Methods of linear systems theory are used to study matrices of rational numbers. An analogue to state space realizations of matrices over F ( z ) is defined and the construction of a state space model is adapted to yield state group realizations of matrices over Q . The approach relies on the concept of straight bases of a finite abelian group.

The Hilbert space generalization of the circle criterion is used for the finite dimensional compensator design for unstable infinite dimensional systems. A design algorithm which generalizes the small gain based algorithm in [5] is presented.

Nonsingular hermitian polynomial matrices are described by two inner product spaces and selfadjoint operators. The associated hermitian linear pencils are used to characterize isometries. As an application a new proof of a factorization result on polynomial matrices with constant signature is given.

A definition of Anderson and Jury is extended to define a Bezoutian B for a quadruple ( F, G; U, D ) of polynomial matrices which have the property that the strictly proper rational parts of F −1 G and UD −1 are equal. The right kernel of B is described by means of block Hankel matrices associated to D and to a greatest common right divisor of D and U .

Conditions are given under which a solution of the DARE is positive semidefinite if and only if all the eigenvalues of its associated closed-loop matrix are in the closed unit disc.

A monotonicity result for the maximal solution of the equation XBB * X − A ∗ X − XA − Q = 0, Q = Q ∗ , (A, B ) stabilizable, is proved.