Optimal stationary control of a linear system with state-dependent noise.
Linear optimal stationary control system with state dependent white noise disturbances described by differential equation
STEADY STATE↗
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Results for “Linear Optimization”
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Large-Scale Optimization with Linear Equality Constraints Using Reduced Compact Representation
For optimization problems with linear equality constraints, we prove that the (1,1) block of the inverse KKT matrix remains unchanged when projected onto the nullspace of the constraint matrix. In this work, we develop reduced compact representations of the limited-memory inverse BFGS Hessian to compute search directions efficiently when the constraint Jacobian is sparse. Orthogonal projections are implemented by a sparse QR factorization or a preconditioned LSQR iteration. In numerical experiments two proposed trust-region algorithms improve in computation times, often significantly, compared to previous implementations of related algorithms and compared to IPOPT.
97 MATHEMATICS AND COMPUTING↗
MCTP system model based on linear programming optimization of apertures obtained from sequencing patient image data maps
Purpose: The authors present a hybrid direct multileaf collimator (MLC) aperture optimization model exclusively based on sequencing of patient imaging data to be implemented on a Monte Carlo treatment planning system (MC-TPS) to allow the explicit radiation transport simulation of advanced radiotherapy treatments with optimal results in efficient times for clinical practice. Methods: The planning system (called CARMEN) is a full MC-TPS, controlled through aMATLAB interface, which is based on the sequencing of a novel map, called “biophysical” map, which is generated from enhanced image data of patients to achieve a set of segments actually deliverable. In order to reduce the required computation time, the conventional fluence map has been replaced by the biophysical map which is sequenced to provide direct apertures that will later be weighted by means of an optimization algorithm based on linear programming. A ray-casting algorithm throughout the patient CT assembles information about the found structures, the mass thickness crossed, as well as PET values. Data are recorded to generate a biophysical map for each gantry angle. These maps are the input files for a home-made sequencer developed to take into account the interactions of photons and electrons with the MLC. For each linac (Axesse of Elekta and Primus of Siemens) and energy beam studied (6, 9, 12, 15 MeV and 6 MV), phase space files were simulated with the EGSnrc/BEAMnrc code. The dose calculation in patient was carried out with the BEAMDOSE code. This code is a modified version of EGSnrc/DOSXYZnrc able to calculate the beamlet dose in order to combine them with different weights during the optimization process. Results: Three complex radiotherapy treatments were selected to check the reliability of CARMEN in situations where the MC calculation can offer an added value: A head-and-neck case (Case I) with three targets delineated on PET/CT images and a demanding dose-escalation; a partial breast irradiation case (Case II) solved with photon and electron modulated beams (IMRT + MERT); and a prostatic bed case (Case III) with a pronounced concave-shaped PTV by using volumetric modulated arc therapy. In the three cases, the required target prescription doses and constraints on organs at risk were fulfilled in a short enough time to allow routine clinical implementation. The quality assurance protocol followed to check CARMEN system showed a high agreement with the experimental measurements. Conclusions: A Monte Carlo treatment planning model exclusively based on maps performed from patient imaging data has been presented. The sequencing of these maps allows obtaining deliverable apertures which are weighted for modulation under a linear programming formulation. The model is able to solve complex radiotherapy treatments with high accuracy in an efficient computation time.
60 APPLIED LIFE SCIENCES↗
The linear time optimal control problem from a calculus of variations point of view
Linear time optimal control problem and calculus of variations
Hanafy, L. M.↗
SU-G-BRA-08: Diaphragm Motion Tracking Based On KV CBCT Projections with a Constrained Linear Regression Optimization
Purpose: To develop a novel strategy to extract the respiratory motion of the thoracic diaphragm from kilovoltage cone beam computed tomography (CBCT) projections by a constrained linear regression optimization technique. Methods: A parabolic function was identified as the geometric model and was employed to fit the shape of the diaphragm on the CBCT projections. The search was initialized by five manually placed seeds on a pre-selected projection image. Temporal redundancies, the enabling phenomenology in video compression and encoding techniques, inherent in the dynamic properties of the diaphragm motion together with the geometrical shape of the diaphragm boundary and the associated algebraic constraint that significantly reduced the searching space of viable parabolic parameters was integrated, which can be effectively optimized by a constrained linear regression approach on the subsequent projections. The innovative algebraic constraints stipulating the kinetic range of the motion and the spatial constraint preventing any unphysical deviations was able to obtain the optimal contour of the diaphragm with minimal initialization. The algorithm was assessed by a fluoroscopic movie acquired at anteriorposterior fixed direction and kilovoltage CBCT projection image sets from four lung and two liver patients. The automatic tracing by the proposed algorithm and manual tracking by a human operator were compared in both space and frequency domains. Results: The error between the estimated and manual detections for the fluoroscopic movie was 0.54mm with standard deviation (SD) of 0.45mm, while the average error for the CBCT projections was 0.79mm with SD of 0.64mm for all enrolled patients. The submillimeter accuracy outcome exhibits the promise of the proposed constrained linear regression approach to track the diaphragm motion on rotational projection images. Conclusion: The new algorithm will provide a potential solution to rendering diaphragm motion and ultimately improving tumor motion management for radiation therapy of cancer patients.
60 APPLIED LIFE SCIENCES↗
Digital program for solving the linear stochastic optimal control and estimation problem
A computer program is described which solves the linear stochastic optimal control and estimation (LSOCE) problem by using a time-domain formulation. The LSOCE problem is defined as that of designing controls for a linear time-invariant system which is disturbed by white noise in such a way as to minimize a performance index which is quadratic in state and control variables. The LSOCE problem and solution are outlined; brief descriptions are given of the solution algorithms, and complete descriptions of each subroutine, including usage information and digital listings, are provided. A test case is included, as well as information on the IBM 7090-7094 DCS time and storage requirements.
Geyser, L. C.↗
Closed-loop structural stability for linear-quadratic optimal systems
An explicit parametrization of a subclass of linear constant gain feedback maps that never destabilize an originally open-loop stable system is presented. These results are used to obtain several new structural stability results for multiinput linear-quadratic feedback optimal designs.
Wong, P. K.↗
Structural reliability and robustness properties of optimal linear-quadratic multivariable regulators
Strong sufficient conditions are derived for the robustness of optimal linear-quadratic (LQ) regulators to large parameter perturbations. In particular, it is shown that under certain conditions LQ designs remain stable in the presence of actuator channel failures. The general results can be specialized to provide insight into the gain margin, gain reduction, and phase margin properties of optimal LQ regulators.
Wong, P.-K.↗
Linear filtering and piecewise linear correlation functions.
Optimal linear mean square filtering for piecewise linear correlation functions
Blake, I. F.↗
Fast Linearized Coronagraph Optimizer (FALCO) I: A Software Toolbox for Rapid Coronagraphic Design and Wavefront Correction
The Fast Linearized Coronagraph Optimizer (FALCO) is an open-source toolbox of routines for coronagraphic focal plane wavefront correction. The goal of FALCO is to provide a free, modular framework for the simulation or testbed operation of several common types of coronagraphs. FALCO includes routines for pair-wise probing estimation of the complex electric field and Electric Field Conjugation (EFC) control, and we ask the community to contribute other wavefront correction algorithms. FALCO utilizes and builds upon PROPER, an established optical propagation library. The key innovation in FALCO is the rapid computation of the linearized response matrix for each deformable mirror (DM), which facilitates re-linearization after each control step for faster DM-integrated coronagraph design and wavefront correction experiments. FALCO is freely available as source code in MATLAB at github.com/ajeldorado/falco-matlab and will be available later this year in Python 3 at github.com/ajeldorado/falco-python.
Shaklan, Stuart B.↗
Fast Linearized Coronagraph Optimizer (FALCO) II: Optical Model Validation and Time Savings over Other Methods
We have developed the Fast Linearized Coronagraph Optimizer (FALCO), a new software toolbox for high-contrast, coronagraphic wavefront sensing and control. FALCO rapidly calculates the linearized deformable mirror (DM) response matrices, also called control Jacobians, and can be used for the design, simulation, or testbed operation of several types of coronagraphs. In this paper, we demonstrate that the optical propagation used in FALCO is accurate and matches PROPER. In addition, we demonstrate the drastic reduction in runtime when using FALCO for DM Jacobian calculations instead of the conventional method used, for example with a model of the Wide-Field Infrared Survey Telescope (WFIRST) Coronagraph Instrument (CGI). We then compare the relative accuracy between optical models in FALCO and PROPER.
Coker, Carl↗
Linear quadratic optimal control for symmetric systems
Special symmetries are present in many control problems. This paper addresses the problem of determining linear-quadratic optimal control problems whose solutions preserve the symmetry of the initial linear control system.
Lewis, J. H.↗
Nonlinear optimization with linear constraints using a projection method
Nonlinear optimization problems that are encountered in science and industry are examined. A method of projecting the gradient vector onto a set of linear contraints is developed, and a program that uses this method is presented. The algorithm that generates this projection matrix is based on the Gram-Schmidt method and overcomes some of the objections to the Rosen projection method.
Fox, T.↗
Robust computer-aided synthesis and optimization of linear multivariable control systems with varying plant dynamics via AUTOCON
AUTOCON is an automated computer-aided design tool for the synthesis and optimization of linear multivariable control systems based upon user-defined control parameter optimization. Violations in stability and performance requirements are computed from constraints on Single Input/Single Output (SISO) open- and closed-loop transfer function frequency responses, and from constraints on the singular-value frequency responses of Multiple Input/Multiple Output (MIMO) transfer functions, for all critical plant variations. Optimum nonlinear programming algorithms are used in the search for local constrained solutions in which violations in stability and performance are caused either to vanish or be minimized for a proper selection of the control parameters. Classical control system stability and performance design can, in this way, be combined with modern multivariable robustness methods to offer general frequency response loop-shaping via a computer-aided design tool. Complete Nichols, Nyquist, Bode, singular-value Bode magnitude and transient response plots are produced, including user-defined boundary responses. AUTOCON is used to synthesize and optimize the lateral/directional flight control system for a typical high-performance aircraft.
Lefkowitz, C. P.↗
Closed-loop structural stability for linear-quadratic optimal systems
This paper contains an explicit parameterization of a subclass of linear constant gain feedback maps that never destabilize an originally open-loop stable system. These results can then be used to obtain several new structural stability results for multi-input linear-quadratic feedback optimal designs.
Wong, P. K.↗
Closed-loop structural stability for linear-quadratic optimal system
This paper contains an explicit parameterization of a subclass of linear constant gain feedback maps that will not destabilize an originally open-loop stable system. These results can then be used to obtain several new structural stability results for multi-input linear-quadratic feedback optimal designs.
Wong, P. K.↗
Closed-loop structural stability for linear-quadratic optimal systems
This paper contains an explicit parametrization of a subclass of linear constant gain feedback maps that will not destabilize an originally open-loop stable system. These results can then be used to obtain several new structural stability results for multiinput linear-quadratic feedback optimal designs.
Wong, P. K.↗
Closed-loop structural stability for linear-quadratic optimal systems
This paper contains an explicit parametrization of a subclass of linear constant gain feedback maps that will not destabilize an originally open-loop stable system. These results can then be used to obtain several new structural stability results for multi-input linear-quadratic feedback optimal designs.
Wong, P. K.↗