Gilbert J. Groenewald

Structured Noncommutative Multidimensional Linear Systems

We introduce a class of multidimensional linear systems with evolution along a free semigroup. The transfer function for such a system is a formal power series in noncommuting indeterminates. Standard system-theoretic properties (the operations of cascade/parallel connection and inversion, controllability, observability, Kalman decomposition, state-space similarity theorem, minimal state-space realizations, Hankel operators, realization theory) are developed for this class of systems. We also draw out the connections with the much earlier studied theory of rational and recognizable formal power series. Applications include linear-fractional models for classical discrete-time systems with structured, time-varying uncertainty, dimensionless formulas in robust control, multiscale systems and automata theory, and the theory of formal languages.

Conservative Structured Noncommutative Multidimensional Linear Systems

We introduce a class of conservative structured multidimensional linear systems with evolution along a free semigroup. The system matrix for such a system is unitary and the associated transfer function is a formal power series in noncommuting indeterminates. A formal power series T(z1, ⋯ , zd) in the noncommuting indeterminates z1,⋯, zd arising in this way satisfies a noncommutative von Neumann inequality, i.e., substitution of a d-tuple of noncommuting operators δ = (δ1,⋯, δd) on a fixed separable Hilbert space which is contractive in the appropriate sense yields a contraction operator T(δ) = T(δ1,⋯, δd). We also obtain the converse realization theorem: any formal power series satisfying such a von Neumann inequality can be realized as the transfer function of such a conservative structured multidimensional linear system.

Equivalence of robust stabilization and robust performance via feedback

One approach to robust control for linear plants with structured uncertainty as well as for linear parameter-varying plants (where the controller has on-line access to the varying plant parameters) is through linear-fractional-transformation models. Control issues to be addressed by controller design in this formalism include robust stability and robust performance. Here robust performance is defined as the achievement of a uniform specified L2-gain tolerance for a disturbance-to-error map combined with robust stability. By setting the disturbance and error channels equal to zero, it is clear that any criterion for robust performance also produces a criterion for robust stability. Counter-intuitively, as a consequence of the so-called Main Loop Theorem, application of a result on robust stability to a feedback configuration with an artificial full-block uncertainty operator added in feedback connection between the error and disturbance signals produces a result on robust performance. The main result here is that this performance-to-stabilization reduction principle must be handled with care for the case of dynamic feedback compensation: casual application of this principle leads to the solution of a physically uninteresting problem, where the controller is assumed to have access to the states in the artificially-added feedback loop. Application of the principle using a known more refined dynamic-control robust stability criterion, where the user is allowed to specify controller partial-state dimensions, leads to correct robust-performance results. These latter results involve rank conditions in addition to linear matrix inequality conditions.

Standard versus strict Bounded Real Lemma with infinite-dimensional state space II: The storage function approach

For discrete-time causal linear input/state/output systems, the Bounded Real Lemma explains (under suitable hypotheses) the contractivity of the values of the transfer function over the unit disk for such a system in terms of the existence of a positive-definite solution of a certain Linear Matrix Inequality (the Kalman–Yakubovich–Popov (KYP) inequality). Recent work has extended this result to the setting of infinite-dimensional state space and associated non-rationality of the transfer function, where at least in some cases unbounded solutions of the generalized KYP-inequality are required. This paper is the second installment in a series of papers on the Bounded Real Lemma and the KYP-inequality. We adapt Willems’ storage-function approach to the infinite-dimensional linear setting, and in this way reprove various results presented in the first installment, where they were obtained as applications of infinite-dimensional State-Space-Similarity theorems, rather than via explicit computation of storage functions.

Standard versus strict bounded real lemma with infinite-dimensional state space. I. The state-space-similarity approach

The Bounded Real Lemma, i.e., the state-space linear matrix inequality characterization (referred to as Kalman-Yakubovich-Popov or KYP inequality) of when an input/state/output linear system satisfies a dissipation inequality, has recently been studied for infinite-dimensional discrete-time systems in a number of different settings: with or without stability assumptions, with or without controllability/observability assumptions, with or without strict inequalities. In these various settings, sometimes unbounded solutions of the KYP inequality are required while in other instances bounded solutions suffice. In a series of reports we show how these diverse results can be reconciled and unified. This first instalment focusses on the state-space-similarity approach to the bounded real lemma. We shall show how these results can be seen as corollaries of a new State-Space-Similarity theorem for infinite-dimensional linear systems.

The Bézout Equation on the Right Half-plane in a Wiener Space Setting

This paper deals with the Bezout equation \({G}(s){X}(s) = {I}_{m}, \mathfrak{R}{s} \leq {0}\), in the Wiener space of analytic matrix-valued functions on the right halfplane. In particular, G is an m × p matrix-valued analytic Wiener function, where p ≥ m, and the solution X is required to be an analyticWiener function of size p × m. The set of all solutions is described explicitly in terms of a p × p matrix-valued analyticWiener function Y , which has an inverse in the analytic Wiener space, and an associated inner function Θ defined by Y and the value of G at infinity. Among the solutions, one is identified that minimizes the H 2- norm. A Wiener space version of Tolokonnikov’s lemma plays an important role in the proofs. The results presented are natural analogs of those obtained for the discrete case in [11].

The Bezout-Corona Problem Revisited: Wiener Space Setting

The matrix-valued Bezout-corona problem

Bounded real lemma and structured singular value versus diagonal scaling: the free noncommutative setting

G(z)X(z)=I_m