Triangular Decoupling Research Articles

Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations

We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into sparse triangular form, where the diagonal blocks consist of spin-s scalar Regge-Wheeler equations (for spins s=0,1,2). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges.

Explicit triangular decoupling of the separated vector wave equation on Schwarzschild into scalar Regge-Wheeler equations

We consider the vector wave equation on the Schwarzschild spacetime, which can be considered as coming from the harmonic (or Lorenz) gauge fixed Maxwell equations. After a separation of variables, the radial mode equations form a complicated system of coupled linear ODEs. We outline a precise abstract strategy to decouple this system into triangular form, where the diagonal blocks consist of spin-s scalar Regge-Wheeler equations, with s = 0 or 1. This strategy is then implemented to give an explicit transformation of the radial mode equations (with nonzero frequency and angular momentum) into this triangular form. Our decoupling goes a step further than previous results in the literature by making the triangular form explicit and reducing it as much as possible. Also, with the help of our abstractly formulated decoupling strategy, we have significantly streamlined both the presentation of the final results and the intermediate calculations. Finally, we note that the vector wave equation is a simple model for more complicated equations, like harmonic (or de Donder) gauge fixed linearized gravity, and backgrounds, like Kerr, where we expect the same abstract decoupling strategy to work as well.

Common I/O Triangular Decoupling for multi model descriptor systems

The problem of Common Input/Output Triangular Decoupling (TD) for multi model descriptor linear systems is solved, using a static state feedback controller. The necessary and sufficient conditions for the solvability of the problem are established. The set of all state feedback controllers solving the problem under the solvability conditions is derived. For the derivation of the above results, the problem of the inversion of a multiparameter matrix pencil in a triangular matrix form and under linear inequality constraints is solved.

A backstepping approach to the output regulation of boundary controlled parabolic PDEs

In this article the output regulation problem for boundary controlled parabolic systems with spatially varying coefficients is solved by applying the backstepping approach. Thereby, the outputs to be controlled are not required to be measurable and can be pointwise, distributed or boundary quantities, whereas the measurement is located at the boundary. By solving the state feedback regulator problem in the backstepping coordinates regulator equations with a simple structure result, so that their analysis and solution is facilitated. The output feedback regulator design is completed by determining a finite-dimensional reference observer and an infinite-dimensional disturbance observer. For the latter a backstepping approach is presented that consists of a triangular decoupling in the backstepping coordinates. This allows a systematic design and the explicit derivation of directly verifiable existence conditions for the disturbance observer. It is shown that for the resulting compensator the separation principle holds implying output regulation for the exponentially stable closed-loop system with a prescribed stability margin. The output regulation results of the article are illustrated by means of a parabolic system with an in-domain pointwise controlled output.

Adaptive regulation of MIMO linear systems against unknown sinusoidal exogenous inputs

AbstractThis paper deals with the adaptive regulation problem in linear multi‐input multi‐output systems subject to unknown sinusoidal exogenous inputs, where the frequencies, amplitudes, and phases of the sinusoids are unknown and where the number of sinusoids is assumed to be known. The design of an adaptive regulator for the system under consideration is performed within a set of Q‐parameterized stabilizing controllers. To facilitate the design of the adaptive regulator, triangular decoupling is introduced in part of the closed‐loop system dynamics. This is achieved through the proper selection of the controller state feedback gain and the structure of the Q parameter. Regulation conditions are then presented for the case where the sinusoidal exogenous input properties are known. For the case where the sinusoidal exogenous input properties are unknown, an adaptation algorithm is proposed to tune the Q parameter in the expression of the parameterized controller. The online tuning of the Q parameter allows the controller to converge to the desired regulator. Convergence results of the adaptation algorithm are presented. A simulation example involving a retinal imaging adaptive optics system is used to illustrate the performance of the proposed adaptive system. Copyright © 2008 John Wiley & Sons, Ltd.

Transfer Matrix Approach to the Triangular Decoupling of General Neutral Multi-Delay Systems

The following major aspects of the problem of Input–Output Triangular Decoupling (TD) for general neutral multi–delay systems, via proportional realizable state feedback, are resolved for the first time: The necessary and sufficient conditions for the problem to have a realizable solution and the general analytical expressions of the proportional realizable TD controller matrices. The conditions and the solution of the controller matrices are computed using a finite step pure algebraic approach.

Robust triangular decoupling with application to 4WS cars

The necessary and sufficient conditions for the problem of robust triangular decoupling are established for the case of a linear time-invariant system with a nonlinear uncertain structure. The general analytic expressions of the feedback matrices and the robust triangularly decoupled closed-loop system are derived. For the solution of the problem of robust triangular decoupling with simultaneous Hurwitz invariability, sufficient conditions are established. Finally, all of the above results are successfully applied to control four-wheel steering (4WS) cars.

Triangular decoupling in singular systems

In this paper, the input-output triangular decoupling problem is studied for the case of regularizable singular systems. The problem is handled in a matrix polynomial equation setting. The necessary and sufficient conditions are introduced for the existence of a static state feedback that decouples the system in triangular form.

Anti-Windup Design for MIMO Plants Having General Infinite and Unstable Zero Structure

In this paper an anti-windup scheme for multi-input, multi-output (MIMO) plants is presented. While the proposed anti-windup scheme has general applicability, attention in this paper is focused on issues concerned with the infinite and unstable (or non-minimum phase) plant zeros, as represented in the factorization of rational transfer function matrices provided by the generalized interactor matrix. This factorization can be used directly to obtain controllers providing triangular decoupling of the closed-loop system. It is argued that design for triangular decoupling of non-minimum phase plants is one way of designing linear controllers whose performance degrades gracefully in the presence of input saturations. In particular, by forgoing the requirement that the closed-loop system be diagonally decoupled, it is possible to obtain controllers capable of limiting peak control amplitudes without necessarily compromising controller bandwidth.

Input-output triangular decoupling and data sensitivity

The problem of input-output triangular decoupling and data sensitivity is studied for the case of linear time-invariant systems. Using a static feedback law, the necessary and sufficient conditions for the problem to have a solution are established. The conditions are derived in terms of the variations of the system matrices. The general solution of the controller matrices is also derived. The necessary and sufficient conditions for triangular decoupling and data sensitivity with simultaneous asymptotic stability are established. The structural property of the perturbation-disturbance equivalence is also established.

The Simultaneous Block Triangular Decoupling, Disturbance Rejection and Pole Assignment Problem for Linear Multivariable Systems

The Simultaneous Block Triangular Decoupling, Disturbance Rejection and Pole Assignment Problem for Linear Multivariable Systems

Triangular Decoupling and Stabilization for Linear Control Systems in Hilbert Spaces

Triangular Decoupling and Stabilization for Linear Control Systems in Hilbert Spaces

Triangular decoupling for a binary distillation column under pressure control

The possibility of attaining complete decoupling of a distillation column under pressure control is studied using the state space model. It is shown that under the situation where the distillation column is under pressure control by the manipulation of the condenser duty, the complete decoupling using constant state feedback may not be attained, while the triangular decoupling can be attained with eight measurement variables.

Transfer matrix approach to the triangular block decoupling problem

The triangular block decoupling problem for linear multivariable systems is studied via the transfer matrix approach. This approach clearly separates the problem of admissible control laws from the one of desired decoupled system specifications. Necessary and sufficient triangular decoupling conditions are given for various control laws. These conditions are expressed in a very simple way in terms of linear dependance among the transfer matrix rows. It turns out that when the problem is solvable, this can be done by static state feedback on a minimal realization of the system. Furthermore it is shown that whenever triangular block decoupling is possible, it is also attainable with stability.

Triangular decoupling and pole placement in linear multivariable systems: a direct algebraic approach

This paper is concerned with an algebraic formulation of the state feedback triangular decoupling problem. Necessary and sufficient conditions for the existence of decoupling matrices for an (A, B, C, D) quadruple are presented. A procedure is outlined for simultaneously realizing a triangular structure and assigning any desired spectrum to the closed loop system.

Relationship between triangular decoupling and invertibility of linear multivariate systems†

Abstract For linear multivariable systems, which have square transfer function matrices, the necessary and sufficient conditions for the existence of state feedback laws for triangular decoupling are shown to be equivalent to the conditions of invertibility. A procedure of finding a state feedback law for this purpose is also given.