On Strictly Positive Real Improper Transfer Matrices

The purpose of this paper is to give four new applications of the Quillen-Suslin theorem to mathematical systems theory. Using a constructive version of the Quillen-Suslin theorem, also known as Serre's conjecture, we show how to effectively compute flat outputs and injective parametrizations of flat multidimensional linear systems. We prove that a flat multidimensional linear system is algebraically equivalent to the controllable 1-D dimensional linear systems obtained by setting all but one functional operator to zero in the polynomial matrix defining the system. In particular, we show that a flat ordinary differential time-delay linear system is algebraically equivalent to the corresponding ordinary differential system without delay, i.e., the controllable ordinary differential linear system obtained by setting all the delay amplitudes to zero. We also give a constructive proof of a generalization of Serre's conjecture known as Lin-Bose's conjecture. Moreover, we show how to constructively compute (weakly) left-/right-/doubly coprime factorizations of rational transfer matrices over a commutative polynomial ring. The Quillen-Suslin theorem also plays a central part in the so-called decomposition problem of linear functional systems studied in the literature of symbolic computation. In particular, we show how the basis computation of certain free modules, coming from projectors of the endomorphism ring of the module associated with the system, allows us to obtain unimodular matrices which transform the system matrix into an equivalent block-triangular or a block-diagonal form. Finally, we demonstrate the package QuillenSuslin which, to our knowledge, contains the first implementation of the Quillen-Suslin theorem in a computer algebra system as well as the different algorithms developed in the paper.

PROBLEMS OF DISCRETE-TIME PERIODIC REALIZATION OF A PERIODIC COLLECTION OF RATIONAL MATRICES

The well-known Richards' Theorem of the continuous-time filter theory is reformulated in the digital domain in a convenient manner, leading to a simple derivation of cascaded lattice digital filter structures, realizing lossless bounded transfer functions. The theorem is also extended to the matrix case, leading to a derivation of m -input p -output cascaded lattice filter structures with lossless building blocks, that realize an arbitrary p \times m digital Lossless Bounded Real (LBR) transfer matrix. Extensions to the synthesis of arbitrary, stable p \times m transfer matrices in the form of such cascaded lattices is also outlined. The derivation also places in evidence a means of testing the stability of an arbitrary p \times m transfer matrix of a discrete-time linear system.

By employing differential vector spaces, we obtain results for linear systems with constant or varying coefficients and with lumped or distributed parameters which were only known for constant linear systems described by rational transfer matrices. Questions such as the characterization of input and output, input-output equivalence, invertibility, feedback decoupling and realization are investigated.

Eight unique mode configurations may occur in a multilayer planar waveguide.1 A correspondence exists between these modes and zero entries in the composite transfer matrix of the structure.2 The modes of such a structure are obtainable by equating transfer matrix entries to zero and solving for the complex propagation constants of individual mode types. Based on this theory to find the bound mode of waveguides with any number of layers, a numerical technique was developed that minimizes the magnitude of the bound entry in the transfer matrix.3 This downhill method can find minima of an arbitrary function, but it only converges with first-order accuracy and requires four evaluations of the transfer matrix at each iteration. A new method that takes advantage of the mathematical properties of the transfer matrix to achieve second-order convergence with a reduced computational stencil is presented. This method is capable of finding the propagation constants and field distributions of structures containing materials with complex permittivities. This new method offers improved accuracy and efficiency, compared to the first-order downhill method. Results accurate to within 0.01 dB/cm are typically obtained with one-seventh the computation of the downhill method.

The linear mapping effected by a coupled linear waveguide system on its input electromagnetic field can be defined by the system’s transfer matrix and this paper addresses the systematic design of the system’s internal dimensions to achieve any desired transfer matrix within a broad class. Manufacturing constraints limit the set of all realisable couplers and their transfer matrices can be used to define a Lie group, whose Lie algebra can be used to study the limitations imposed by such constraints. After having defined the group, the paper presents a design procedure that is guaranteed to realise any transfer matrix in the group by a finite concatenation of basic building-block coupler sections. This guarantee is afforded by a simple manipulation of canonical coordinate systems for suitable neighbourhoods of the group’s identity. The design procedure is illustrated for the group $\mathfrak{G}\mathfrak{U}( 2 )$, where the homomorphism with $\mathfrak{G}\mathfrak{D}( 3 )$ yields geometrical insight in terms of rigid rotations of the Poincaré sphere. A further example uses the procedure to realise a fully planar implementation of a $3 \times 3$ coupler of importance in optical sensing and signal processing.

Let A 0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C 0 be a bounded operator from D(A1/2 0 ) (with the norm ||z||2 1/2 = 〈A 0 z, z〉) to another Hilbert space U. It is known that the system of equations z(t) + A 0 z(t) + 1/2 C∗ 0 C 0 Z(t) = C∗ 0 u(t), y(t) = − C 0 z(t) + u(t), determines a well-posed linear system Σ with input u and output y, input and output space U and state space X = D(A1/2 0 ) × H. Moreover, Σ is conservative, which means that a certain energy balance equation is satisfied both by the trajectories of Σ and by those of its dual system. In this paper we present various conditions which are equivalent to the exponential stability of such a systems. Among the equivalent conditions are exact controllability and exact observability. Denoting V(s) = (s2 I + s/2 C∗ 0 C 0 + A 0 )−1, we also obtain that the system is exponentially stable if and only if s → A1/2 0 V(s) is a bounded L(H)-valued function on the imaginary axis. This is also equivalent to the condition that s → sV(s) is a bounded L(H)-valued function on the imaginary axis (or equivalently, on the open right half-plane).

This paper studies stabilization problems for linear systems with multiple delays in the input. Two types of delays are considered. The first type of delays is constant delays, which can be arbitrarily large, while the second type is time-varying with an arbitrarily large bound. With the first type of delays, under the condition that the open loop system is absolutely controllable with all its eigenvalues on the imaginary axis, (globally) stabilizing state and output feedback laws are constructed based on the solution to a family of parametric Riccati equations, which can be obtained explicitly through the solution of a parametric linear matrix equation. With the second type of delays, under the condition that the open-loop system is absolutely controllable with all its eigenvalues on the imaginary axis being zero, (global) state and output feedback laws are explicitly constructed based on the solution to a similar family of parametric Riccati equations. When the input is also subject to magnitude saturation, it is shown that semiglobal stabilization, instead of global stabilization, can still be achieved. Numerical examples illustrate the effectiveness of the proposed approach.

: This paper is concerned with the development of a complete set of invariants and canonical forms under feedback for linear systems characterized by proper rational transfer matrices. The invariants are determined in the frequency domain and consist of the Kronecker set of controllability indices together with a canonical form for the numerator of the transfer matrix under the action of a stabilizer subgroup of the unidoular group of polynomial matrices. Techniques used are algebro-geometric in nature. (Author)

Dynamics of rotationally periodic large space structures

On the Transfer Matrix in the Hubbard Model

Many multi-beam energy harvesters appearing in the literature require custom analytical or finite-element models to compute their eigensolutions and piezoelectric coupling effects. This paper discusses the use of the transfer matrix method to derive analytical solutions to beam structures with point-wise discontinuities or with lumped inertias between members or at the tip. In this method, transfer matrices are developed for the beam’s states (deflection, slope, shear force, and bending moment) analogously to the state transition matrix of a linear system. Euler-Bernoulli beam theory is used to derive transfer matrices for the uniform beam segments, and point transfer matrices are derived to handle discontinuities in the structure. The transfer matrix method is shown to be advantageous for analyzing complex structures because the size of the transfer matrix does not grow with increasing number of components in the structure. Furthermore, the same formulation can be used for a wide range of geometries, including arbitrary combinations of beam segments — single- or multi-layered — and lumped inertias. The eigensolution of the transfer matrix is shown to produce the natural frequencies and mode shapes for these structures. Subsequently, the electromechanical coupling effects are incorporated and the base excitation problem is considered. The electromechanical equations of motion are decoupled by mode and shown to be a generalization of existing analytical models. Parametric case studies are provided for beam structures with varying piezoelectric layer coverage.

The transfer matrix of the standard and fractional linear discrete-time linear systems is investigated. Necessary and sufficient conditions for zeroing of the transfer matrix of the linear discrete-time systems are established. The considerations are illustrated by examples of the standard and fractional linear discrete-time systems.

In this paper, the synthesis of robust memory modes for linear quantum passive systems in the presence of unknown inputs has been studied, aimed at facilitating secure storage and communication of quantum information. In particular, we can switch on decoherence-free (DF) modes in the storage stage by placing the poles on the imaginary axis via a coherent feedback control scheme, and these memory modes can further be simultaneously made robust against perturbations to the system parameters by minimizing the condition number associated with imaginary poles. The DF modes can also be switched off by tuning the controller parameters to place the poles in the left half of the complex plane in the writing/reading stage. We develop explicit algebraic conditions guiding the design of such a coherent quantum controller, which involves employing an augmented system model to counter the influence of unknown inputs. Examples are provided to illustrate the procedure of synthesizing robust memory modes for linear optical quantum systems.

The question of achieving a variety of specific design goals (e.g., arbitrary pole placement, decoupling, and exact model matching) for linear multivariable systems subjected to unknown step disturbances on the entire internal state and output is constructively resolved via a synthesis procedure which simultaneously combines integral feedforward compensation with asymptotic state estimation. The technique relies on the ability to express the desired closed-loop transfer matrix of the compensated system as the product of the open-loop transfer matrix and any proper rational transfer matrix, assuming the given system has no zeros at the origin and a stable closed-loop design is sought. The method employs some recent results involving the differential operator approach to multivariable system analysis and synthesis, and the treatment is confined almost exclusively to the frequency domain.