System Pencil Research Articles

Canonical Structure Transitions of System Pencils

We investigate the changes of the canonical structure information under small perturbations for a system pencil associated with a (generalized) linear time-invariant state-space system. The equivalence class of the pencil is taken with respect to feedback-injection equivalence transformations. The results allow us to track possible changes of important linear system characteristics under small perturbations.

Outer transfer functions of differential-algebraic systems

We consider differential-algebraic systems (DAEs) whose transfer function is outer : i . e ., it has full row rank and all transmission zeros lie in the closed left half complex plane. We characterize outer, with the aid of the Kronecker structure of the system pencil and the Smith–McMillan structure of the transfer function, as the following property of a behavioural stabilizable and detectable realization: each consistent initial value can be asymptotically controlled to zero while the output can be made arbitrarily small in the ℒ 2 -norm. The zero dynamics of systems with outer transfer functions are analyzed. We further show that our characterizations of outer provide a simple and very structured analysis of the linear-quadratic optimal control problem.

ℋ∞‐Norm Computation for Large and Sparse Descriptor Systems

AbstractIn this paper we discuss the problem of computing the ℋ∞‐norms of transfer functions associated to large‐scale descriptor systems. We exploit the relationship between the ℋ∞‐norm and the structured complex stability radius of the corresponding system pencil. To compute the structured stability radius we consider so‐called structured pseudospectra. Namely, we have to find the pseudospectrum that touches the imaginary axis. Therefore, we set up an iteration over the real part of the rightmost pseudoeigenvalue. For that we use a new fast iterative scheme which is based on certain rank‐1 perturbations of the system pencil. Finally, we illustrate the performance of our algorithm by using a real‐world example. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Monte Carlo verification of gel dosimetry measurements for stereotactic radiotherapy**Experimental aspects of this work were originally presented at the Engineering and Physical Sciences in Medicine Conference (EPSM-ABEC), Melbourne, 2010.

The quality assurance of stereotactic radiotherapy and radiosurgery treatments requires the use of small-field dose measurements that can be experimentally challenging. This study used Monte Carlo simulations to establish that PAGAT dosimetry gel can be used to provide accurate, high-resolution, three-dimensional dose measurements of stereotactic radiotherapy fields. A small cylindrical container (4 cm height, 4.2 cm diameter) was filled with PAGAT gel, placed in the parietal region inside a CIRS head phantom and irradiated with a 12-field stereotactic radiotherapy plan. The resulting three-dimensional dose measurement was read out using an optical CT scanner and compared with the treatment planning prediction of the dose delivered to the gel during the treatment. A BEAMnrc/DOSXYZnrc simulation of this treatment was completed, to provide a standard against which the accuracy of the gel measurement could be gauged. The three-dimensional dose distributions obtained from Monte Carlo and from the gel measurement were found to be in better agreement with each other than with the dose distribution provided by the treatment planning system's pencil beam calculation. Both sets of data showed close agreement with the treatment planning system's dose distribution through the centre of the irradiated volume and substantial disagreement with the treatment planning system at the penumbrae. The Monte Carlo calculations and gel measurements both indicated that the treated volume was up to 3 mm narrower, with steeper penumbrae and more variable out-of-field dose, than predicted by the treatment planning system. The Monte Carlo simulations allowed the accuracy of the PAGAT gel dosimeter to be verified in this case, allowing PAGAT gel to be utilized in the measurement of dose from stereotactic and other radiotherapy treatments, with greater confidence in the future.

SU‐E‐T‐827: The Impact of Artefact Contouring on the Optimization of Head and Neck Cancers Using IMRT

Purpose: IMRT planning of head and neck (H&N) cancers routinely employs artefact contouring in the oral cavity due to dental restorations (DR). The differentiation between teeth and soft tissue in the region immediately surrounding these DR however is uncertain. A study was carried out on patients with varying number of DR/artefacts to determine the impact on optimization. Methods: Five H&N patients who had been planned using IMRT were selected with varying PTV locations, degrees of DR and artefacts. These patients were originally planned with the artefacts surrounding the DR contoured and the density of these converted to water/soft tissue (HU=0). The DR were given a maximum CT number assigned for our CT scanner (HU∼2700). These patients were then re‐optimized (using identical constraints as in the original plan) without any compensation for artefacts. Optimizations were carried out in both scenarios using the Eclipse™ planning system Pencil Beam (PBC)8.6.15 and then carried out using the AAA algorithm with/without artefacts contouring as well. Results: In the cases where the PTV didnˈt overlap the teeth/DR, DVH as well as axial slice comparisons showed insignificant differences between contouring the artefacts or not. In those cases where the PTV did overlap the DR, a difference was seen between the two optimizations; with and without artefacts converted to water/soft tissue. For large PTV volumes, this may not translate into a significant effect based on DVH comparisons since the volume of DR artefact is relatively small compared to the PTV volume. However, for smaller PTV volumes, the impact could be quite significant. This effect was seen using both algorithms: PBC and AAA. Conclusion: For H&N cancers with PTVs within the oral cavity, artefact contouring does have an impact on the optimization of IMRT plans and the proper contouring of DR artefacts can minimize the impact.

Stratification of Controllability and Observability Pairs—Theory and Use in Applications

Cover relations for orbits and bundles of controllability and observability pairs associated with linear time-invariant systems are derived. The cover relations are combinatorial rules acting on integer sequences, each representing a subset of the Jordan and singular Kronecker structures of the corresponding system pencil. By representing these integer sequences as coin piles, the derived stratification rules are expressed as minimal coin moves between and within these piles, which satisfy and preserve certain monotonicity properties. The stratification theory is illustrated with two examples from systems and control applications, a mechanical system consisting of a thin uniform platform supported at both ends by springs, and a linearized Boeing 747 model. For both examples, nearby uncontrollable systems are identified as subsets of the complete closure hierarchy for the associated system pencils.

A Monte Carlo tool for combined photon and proton treatment planning verification

Photons and protons are usually used independently to treat cancer. However, at MGH patients can be treated with both photons and protons since both modalities are available on site. A combined therapy can be advantageous in cancer therapy due to the skin sparing ability of photons and the sharp Bragg peak fall-off for protons beyond the tumor. In the present work, we demonstrate how to implement a combined 3D MC toolkit for photon and proton (ph-pr) therapy, which can be used for verification of the treatment plan. The commissioning of a MC system for combined ph-pr involves initially the development of a MC model of both the photon and proton treatment heads. The MC dose tool was evaluated on a head and neck patient treated with both combined photon and proton beams. The combined ph-pr dose agreed with measurements in solid water phantom to within 3%mm. Comparison with commercial planning system pencil beam prediction agrees within 3% (except for air cavities and bone regions).

A Numerical Method for a Generalized Algebraic Riccati Equation

In this paper we develop a numerical method for computing the semistabilizing solution of a generalized algebraic Riccati equation (GARE). The semistabilizing solution of such a GARE has been used to characterize the solvability of the $(J, J^\prime)$‐spectral factorization problem for general rational matrices which have poles and zeros on the extended imaginary axis. The main difficulty for solving such a GARE is that its associated skew‐Hamiltonian/Hamiltonian pencil has eigenvalues on the extended imaginary axis; consequently, it is not clear which eigenspace of the associated skew‐Hamiltonian/Hamiltonian pencil can characterize the desired semistabilizing solution; i.e., it is not clear which eigenvectors and principal vectors corresponding to the eigenvalues on the extended imaginary axis should be contained in the eigenspace that we wish to compute, and hence the well‐known generalized eigenvalue approach for the classical algebraic Riccati equations cannot be directly employed for it. Our proposed method consists of computations of the eigendecomposition of the system pencil corresponding to the eigenvalues on the extended imaginary axis and the stable eigenspace of an augmented matrix pencil; hence, it is a generalization of the generalized eigenvalue approach for the classical algebraic Riccati equations.

Computing the zeros of periodic descriptor systems

In this paper, we give a numerically reliable algorithm to compute the zeros of a periodic descriptor system. The algorithm is a variant of the staircase algorithm applied to the system pencil of an equivalent lifted time-invariant state-space system and extracts a low-order pencil which contains the zeros (both finite and infinite) as well as the Kronecker structure of the periodic descriptor system. The proposed algorithm is efficient in terms of complexity by exploiting the structure of the pencil and is exclusively based on orthogonal transformations, which ensures some form of numerical stability.

On computing least order fault detectors using rational nullspace bases

We propose a numerically reliable computational approach to design least order fault detectors using descriptor system techniques. This approach is based on a new numerically stable algorithm to compute least order rational null space bases of rational matrices. The main computation in this algorithm is the onhogonal reduction of the system pencil matrix to a Kronecker-like form. The proposed approach can be combined with coprime factorization techniques to determine stable rational bases. Least order fault detectors can be detennined by selecting an appropriate linear combination of basis vectors by eliminating non-essential dynamics. The proposed approach is applicable to both standard and descriptor system descriptions.

Feedback design for regularizing descriptor systems

This paper surveys numerical techniques for the regularization of descriptor (generalized state-space) systems by proportional and derivative feedback. We review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the Weierstraß canonical form. Three condensed forms display the controllability and observability properties of a descriptor system. The condensed forms are obtained through orthogonal equivalence transformations and rank decisions, so they may be computed by numerically stable algorithms. In addition, the condensed forms display whether a descriptor system is regularizable, i.e., when the system pencil can be made to be regular by derivative and/or proportional output feedback, and, if so, what index can be achieved. Also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback.

An improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects

In this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil, and implicitly its Kronecker structure. The algorithm compares favorably to existing ones in terms of elegance, versatility, and complexity. In particular, the algorithm without any modification yields the structural invariants associated with a generalized state-space system and its system pencil. Two related geometric aspects are also discussed: we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form; we extend the notion of deflating subspace to the singular pencil case.

Linear SISO systems with extremely sensitive zero structure

If a system with regular system pencil and relative degree greater than one is perturbed, the relative degree will typically decrease, and new finite zeros will appear. These new zeros are singularly perturbed. This paper applies a new canonical parameterization to systems with singular system pencils. Such systems have undefined relative degree. In singular systems, new zeros also appear under small perturbation, but they are not necessarily singularly perturbed. Rather, these zeros may appear at any frequency.

117 Unfolding of singular system pencils with application to regulation

117 Unfolding of singular system pencils with application to regulation

Unfolding of Singular System Pencils with Application to Regulation

The system pencil is a convenient way of representing a linear time invariant control system. It is particularly appropriate when system zeros are important — in regulator design, for example. This paper presents an unfolding of certain system pencils, possibly, but not necessarily, singular. That is, it demonstrates a minimal parameterization of the nominal pencil such that any nearby pencil, up to strict equivalence, corresponds to some value of the parameter vector. This unfolding has potential practical and theoretical applications to systems with parameter dependence.

On the stability radius of a generalized state-space system

The concept of “distance to instability” of a system matrix is generalized to system pencils which arise in descriptor (semistate) systems. Difficulties arise in the case of singular systems, because the pencil can be made unstable by an infinitesimal perturbation. It is necessary to measure the distance subject to restricted, or structured, perturbations. In this paper a suitable measure for the stability radius of a generalized state-space system is defined, and a computable expression for the distance to instability is derived for regular pencils of index less than or equal to one. For systems which are strongly controllable it is shown that this measure is related to the sensitivity of the poles of the system over all feedback matrices assigning the poles.

Regularization of Descriptor Systems by Derivative and Proportional State Feedback

For linear multivariable time-invariant continuous or discrete-time singular systems it is customary to use a proportional feedback control in order to achieve a desired closed loop behaviour. Derivative feedback is rarely considered. This paper examines how derivative feedback in descriptor systems can be used to alter the structure of the system pencil under various controllability conditions. It is shown that derivative and proportional feedback controls can be constructed such that the closed loop system has a given form and is also regular and has index at most 1. This property ensures the solvability of the resulting system of dynamic-algebraic equations. The construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way. The problem of pole placement with derivative feedback alone and in combination with proportional state feedback is also investigated. A computational algorithm for improving the “conditioning” of the regularized closed loop system is derived.