Diagonal Stabilization Research Articles

State Control Design of Ostensible Metzler Linear Systems with Unsigned Input Parameters

This paper deals with the design of a complete state control for input unsigned, rank deficient matrix parameters of a linear system with system dynamics defined by ostensible structures of Metzler matrices. The proposed solution is based on the principle of diagonal stabilization of positive systems and uses a stabilizing additional component over the decomposition of the Metzler matrix in solving the incomplete internal positivity of such linear system structures. The novelty of the proposed approach is the unified representation of the parametric constraints of the Metzler matrix and the structurally constrained system inputs using linear matrix inequalities, which guarantees that the closed-loop system will be asymptotically stable. Despite the complexity of the constraint conditions on this class of linear continuous systems, the design conditions are formulated using sharp linear matrix inequalities only. A detailed design process is presented using a system-linearized mathematical model to verify the superiority and practicality of the proposed method.

Ostensible Metzler Linear Uncertain Systems: Goals, LMI Synthesis, Constraints and Quadratic Stability

This paper deals with the design problem for a class of linear continuous systems with dynamics prescribed by the system matrix of an ostensible Metzler structure. The novelty of the proposed solution lies in the diagonal stabilization of the system, which uses the idea of decomposition of the ostensible Metzler matrix, preserving the incomplete positivity of the system during the synthesis. The proposed approach creates a unified framework that covers compactness of interval system parameter representation, Metzler parametric constraints, and quadratic stability. Combining these extensions, all of the conditions and constraints are expressed as linear matrix inequalities. Implications of the results, both for design and for research directions that follow from the proposed method, are discussed at the end of the paper. The efficiency of the method is illustrated by a numerical example.

Control Design for Linear Strictly Metzler Time-delay Systems

The relationships among structural constraints and involvement of the design condition are studied to synthesize state control for one class of linear strictly Metzler time-delay systems. These characterizations reflect the specific dynamical and structural attributes of the system class and outline the associated structures of linear matrix inequalities. Adjusting diagonal forms of linear matrix variables it is indicated how the proposed method gives a computable technique for the Metzler time-delay system, guaranteeing stabilising effect through implicit diagonal stabilization. The aim of this research is to describe conditions tying together inequality formulations and concepts of control theory in structures of Metzler systems

A novel domain of attraction based synthesis of inverse optimal control

AbstractThis article proposes a novel and systematic methodology for the inverse optimal control (IOC) of a class of input affine nonlinear systems. First, the system is represented in the pseudo‐linear state‐dependent coefficient (SDC) form so as to facilitate the use of matrix algebraic concepts in the design, while preserving the nonlinear system dynamics. Second, the IOC problem which seeks the solution to the Hamilton–Jacobi–Bellman equation is translated to a diagonal stability (D‐stability) problem. Therefore, the general notion of D‐stability, which deals with the characterization of a matrix for the existence of a diagonal matrix such that , is redefined with respect to SDC factored matrices in this work. Consequently, the criteria for the existence (necessary condition) and determination of a diagonal solution (sufficient condition) are formulated in terms of the SDC matrices. It is shown in this work that these criteria can be readily met by the closed‐loop system matrix , for any arbitrary domain in state‐space, by manipulating only the diagonal elements using the diagonal matrix constituted by the IOC feedback. In short, the IOC feedback is designed such that the criteria for D‐stability by the closed‐loop system matrix is fulfilled. Hence this novel design methodology is meaningfully named as “Inverse optimal control via diagonal stabilization” (IOC‐D). The main advantages of IOC‐D include: (i) guaranteed estimate of the domain of attraction (DOA), (ii) Immense flexibility of design, and (iii) guaranteed closed form solution which makes the design procedure analytic and computationally efficient. Numerical simulations validate the theoretical developments and the overall efficiency of the proposed approach.

Quadratic Stabilization of Linear Uncertain Positive Discrete-Time Systems

The paper provides extended methods for control linear positive discrete-time systems that are subject to parameter uncertainties, reflecting structural system parameter constraints and positive system properties when solving the problem of system quadratic stability. By using an extension of the Lyapunov approach, system quadratic stability is presented to become apparent in pre-existing positivity constraints in the design of feedback control. The approach prefers constraints representation in the form of linear matrix inequalities, reflects the diagonal stabilization principle in order to apply to positive systems the idea of matrix parameter positivity, applies observer-based linear state control to assert closed-loop system quadratic stability and projects design conditions, allowing minimization of an undesirable impact on matching parameter uncertainties. The method is utilised in numerical examples to illustrate the technique when applying the above strategy.

Inverse Optimal Control Via Diagonal Stabilization Applied to Attitude Tracking of a Reusable Launch Vehicle

In this work, a novel strategy for the inverse optimal control of a class of affine nonlinear systems is proposed. The proposed strategy involves translation of the infinite horizon nonlinear optimal control problem into a Diagonal stability problem, followed by the formulation of a set of criteria for the synthesis of a stabilizing feedback control law. Hence, the proposed controller design methodology is refered to as inverse optimal control via diagonal stabilization. Besides providing a closed-form solution, the methodology also possesses the added advantage of inherent robustness, on account of adequate stability margins. Most importantly, the methodology ensures an estimate of the associated domain of attraction, which is a highly desirable feature especially in the case of crucial and stringent aerospace applications. The proposed methodology is applied for the re-entry control of a reusable launch vehicle which provides a full envelope optimality-based design philosophy. For this, the control-oriented attitude model based on the three-degree-of-freedom dynamic model is utilized. The resulting control law possess desirable features of guaranteed estimate of stability domain, assured design flexibility, and inherent robustness. Simulation results verify the validity of these theoretically proven facets in terms of the efficacy and robustness of the proposed controller.

Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media

In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, generalizing the classical Biot equations. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators. It is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow equations. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration shows the best performance among the splitting schemes.

Parabolic Radon transform, sampling and efficiency

A good choice of the sampling in the transform domain is essential for a successful application of the parabolic Radon transform. The parabolic Radon transform is computed for each temporal frequency and is essentially equivalent to the nonuniform Fourier transform. This leads to new and useful insights in the parabolic Radon transform. Using nonuniform Fourier theory, we derive a minimum sampling interval for the curvature parameter and a maximum curvature range for which stability is guaranteed for general (irregular) sampling. A significantly smaller sampling interval requires stabilization. If diagonal stabilization is used, no gain in resolution is obtained. In contrast to conventional implementations, the curvature sampling interval is proposed to be inversely proportional to the temporal frequency. This results in improved quality of the transform and yields significant savings in computation time.

Robust diagonal stabilization: an LMI approach

Addresses the problem of robust diagonal stabilization for discrete-time systems. Linear matrix inequality (LMI) conditions are derived for the search of a diagonal solution to the associated Lyapunov equation. This approach can be applied to guarantee stability of quantized systems and to find numerous illustrative applications, e.g., for the design of robust digital filters.

Reconstruction of 3-D seismic signals irregularly sampled along two spatial coordinates

Seismic signals are often irregularly sampled along spatial coordinates, leading to suboptimal processing and imaging results. Least‐squares estimation of Fourier components is used for the reconstruction of band‐limited seismic signals that are irregularly sampled along two spatial coordinates. A simple and efficient diagonal weighting scheme, based on the areas surrounding the spatial samples, takes the properties of the noise (signal outside the bandwidth) into account in an approximate sense. Diagonal stabilization based on the energies of the signal and the noise ensures robust estimation. Reconstruction by temporal frequency component allows the specification of varying bandwidth in two dimensions, depending on the minimum apparent velocity. This parameterization improves the reconstruction capability for lower temporal frequencies. The shape of the spatial aperture affects the method of sampling the Fourier domain. Taking into account this property, a larger bandwidth can be recovered. The properties of the least‐squares estimator allow a very efficient implementation which, when using a conjugate gradient algorithm, requires a modest number of 2-D fast Fourier transforms per temporal frequency. The method shows signicant improvement over the conventionally used binning and stacking method on both synthetic and real data. The method can be applied to any subset of seismic data with two varying spatial coordinates.

Reconstruction of band‐limited signals, irregularly sampled along one spatial direction

Seismic signals are often irregularly sampled along spatial coordinates, leading to suboptimal processing and imaging results. Least squares estimation of Fourier components is used for the reconstruction of band‐limited seismic signals that are irregularly sampled along one spatial coordinate. A simple and efficient diagonal weighting scheme, based on the distance between the samples, takes the properties of the noise (signal outside the bandwidth) into account in an approximate sense. Diagonal stabilization based on the energies of the signal and the noise ensures robust estimation. Reconstruction for each temporal frequency component allows the specification of a varying spatial bandwidth dependent on the minimum apparent velocity. This parameterization improves the reconstruction capability for the lower temporal frequencies. In practical circumstances, the maximum size of the gaps in which the signal can be reconstructed is three times the (temporal frequency dependent) Nyquist interval. Reconstruction in the wavenumber domain allows a very efficient implementation of the algorithm, and takes a total number of operations a few times that of a 2-D fast Fourier transform corresponding to the size of the output data set. Quality control indicators of the reconstruction procedure can be computed which may also serve as decision criteria on in‐fill shooting during acquisition. The method can be applied to any subset of seismic data with one varying spatial coordinate. Applied along the cross‐line direction, it can be used to compute a 3-D stack with improved anti‐alias protection and less distortion of the signal within the bandwidth.

Quantitative Feedback Design of Decentralized Control Systems

In the spirit of practical applicability, design transparency and quantitative specifications, this paper presents a new robust decentralized control design framework—model reference quantitative feedback design (MRQFD)—for multivariable control systems with large plant uncertainty and strong cross-coupling. The MRQFD method provides a connection between Rosenbrock’s Nyquist array and Horowitz’s quantitative feedback theory. There are two main stages in the MRQFD method. First, an internal model reference loop, based on non-negative matrix theory, is used to obtain robust generalized diagonal dominance and hence the reduction of uncertainty in the resultant compensated internal loop system. Then a single-loop quantitative feedback design scheme is applied to solve the resulting series of individual loops to guarantee the satisfaction of predefined MIMO quantitative specifications. The design procedure can be carried out either in the direct or in the inverse plant domains, as in Rosenbrock’s Nyquist array. The conditions of robust diagonal stabilization are also given for the MRQFD framework. The effectiveness of the MRQFD method and its ease of application are demonstrated by a 3 × 3 uncertain multivariable example.

Decentralized diagonal dynamic stabilization of linear multivariable systems

The diagonal stabilization problem (DSP) is defined over the ring of proper rational functions which have no poles inside a prescribed region of the finite complex plane. Solvability is intimately related to systems which exhibit the property of cyclicity. Necessary and sufficient conditions are established for the existence of solutions to the DSP. A complete parametrization of stabilizing controllers for the 2 × 2 case is given. Conditions of nonsolvability and hence nonstabilizability yield an explicit expression for the fixed modes of the system.

Diagonal stabilization of linear multivariable systems

A new procedure is presented for the synthesis of diagonal compensators for N × N linear multivariable systems that are free of fixed modes with respect to constant diagonal output feedbacks. The synthesis procedure employs simple polynomial algebra and it is in the form of an N-step algorithm. The geometric configurations 2N- and 2N-cells in N-space are shown to be especially suitable for visualizing diagonal feedback and aiding the application of the synthesis algorithm.

Computer-aided nucleic acid secondary structure modeling incorporating enzymatic digestion data.

We present a computer-aided method for determining nucleic acid secondary structure. The method utilizes a program which has the capability to filter matrix diagonal data on the basis of diagonal length, stabilization energy, and chemical and enzymatic data. The program also allows the user to assign selected regions of the structure as uniquely single-stranded or paired, and to filter out "trade-off" structures on the basis of such pairing. In order to demonstrate the utility of the program we present a preliminary secondary structure for the 3' end of alfalfa mosaic virus RNA 4 (AMV-4 RNA). This structure is based on an analysis which includes the use of in vitro partial enzymatic digestion of the RNA.