Causal approximate inversion for control of structurally flexible manipulators using nonlinear inner–outer factorization

A control scheme for flexible‐link manipulators is advanced which is based on the notion of nonlinear inner–outer factorization. It is well known that the inverse of the forward dynamics map from joint torques to manipulator tip motion is noncausal and cannot be implemented in conjunction with real‐time path planning. The methods used here determine causal approximations for the inverse dynamics using the inverse of the outer (stable and minimum phase) factor and a static approximation for the inverse of the inner (lossless but nonminimum phase) factor. The Hamilton–Jacobi equation that arises is approximated by a state‐dependent Riccati equation at each time step. The factorization procedure yields the corresponding joint trajectories which can serve as reference trajectories for closing joint‐based feedback loops. Experimental results from a planar three‐link manipulator with two flexible links demonstrate the efficacy of the procedure. © 2001 John Wiley & Sons, Inc.     

Reduced-order compensators via balancing and central control design for a structural control problem

Real-time control of a physical system necessitates controllers that are low order. In this article, we compare two balanced truncation methods as a means of designing low-order controllers for a nonlinear cable-mass system. The first is the standard technique of balanced truncation. The second, linear quadratic Gaussian (LQG) balanced truncation, can be thought of as balancing based on the controller, and states that are important from the perspective of control and filter design are retained. The control design applied to each reduced-order model is the central controller. We provide an overview of the central controller and devote attention to the design of this controller in the presence of balancing. Also described in this article is a method for reducing computational time in solving algebraic Riccati equations for the design of low-order LQG balanced controllers.     

Parametric polynomial spectral factorization using the sum of roots and its application to a control design problem

This paper presents an algebraic approach to polynomial spectral factorization, an important mathematical tool in signal processing and control. The approach exploits an intriguing relationship between the theory of Gröbner bases and polynomial spectral factorization which can be observed through the sum of roots, and allows us to perform polynomial spectral factorization in the presence of real parameters. It is discussed that parametric polynomial spectral factorization enables us to express quantities such as the optimal cost in terms of parameters and the sum of roots. Furthermore an optimization method over parameters is suggested that makes use of the results from parametric polynomial spectral factorization and also employs two quantifier elimination techniques. This proposed approach is demonstrated in a numerical example of a particular control problem.     

Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations

Consideration was given to the problems of variational calculus and optimal control admitting impulsive controls and, correspondingly, discontinuous solutions. The evolution of the notion of singular space-time transformation beginning from the problem of classical variational calculus and ending with the problems of optimal control at the phase of impact was shown.     

Optimization method for solving bang-bang and singular control problems

In this paper we study optimal control problems with the control variable appearing linearly.A novel method for optimization with respect to the switching times of controls containing both bang-bang and singular arcs is presented.This method transforms the control problem into a finite-dimensional optimization problem by reformulating the control problem as a multi-stage optimization problem.The optimal control problem is partitioned as several stages, with each stage corresponding to a particular control arc.A control vector parameterization approach is applied to convert the control problem to a static nonlinear programming(NLP) problem.The control profiles and stage lengths act as decision variables.Based on the Pontryagin maximal principle,a multi-stage adjoint system is constructed to calculate the gradients required by the NLP solvers.Two examples are studied to demonstrate the effectiveness of this strategy.     

Size–density spectra and their application to image classification

We develop a density opening operator that is shown to satisfy the properties of an algebraic opening. This density opening enables the development of a number of variants of pattern spectra, which quantify the size or density information of a blob arrangement within the image. In contrast to regular morphological pattern size spectra, the proposed pattern spectra are spatially sensitive and robust to noise distortions. A pattern size–density spectrum or 2D signature was introduced and used for solving image classification tasks. Its application to the classification of real world medical images is illustrated.     

Unified dwell time–based stability and stabilization criteria for switched linear stochastic systems and their application to intermittent control

This paper considers the stability and stabilization problems for the switched linear stochastic systems under dwell time constraints, where the considered systems can be composed of an arbitrary combination of stable and unstable subsystems. First, a time‐varying discretized Lyapunov function is constructed based on the projection of a linear Lagrange interpolant and a switching‐time‐dependent “weighted” function. The “weighted” function not only enforces the Lyapunov function to decrease at switching instants but also coordinates the dynamical behavior of the subsystems. As a result, some unified criteria for mean square stability and almost sure stability of the switched stochastic systems are established in terms of linear matrix inequalities. Based on the obtained stochastic stability criteria, 2 types of state feedback controllers for the systems are designed. Moreover, the novel results are applied to solve the intermittent control or the controller failure problems. Finally, conservatism analysis and numerical examples are provided to illustrate the effectiveness of the established theoretical results.     

Application of the singular control test to periodic control problems of a continuous distillation process

The singular control test is applied to periodic control problems of a continuous distillation process in order to determine if the optimal periodic control is proper. It is also shown that the singular control test is easier to apply than the w-teat for the problems of concern. Some numerical calculations are carried out for seeking the optimal periodic controls.     

Basis factorization method for multistage linear programming problems with an application to optimal energy plant operation

An algorithm based on the basis factorization method for solving multistage linear programming problems is presented. In applying linear programming for optimal dynamic-plant operation, the constraint appears in the form of a staircase structure. When pivotings occur within each stage, the inversion of the basis matrix is easily obtained using inversions of small square diagonal blocks. In general cases, pivotings across stages occur. However, this basis matrix can be easily transformed into the former square block triangular form using the correction factor. The correction factor means multiple pivotings among the bases and contains structural columns corresponding to different columns among these bases. When these structural columns are maintained in computer storage, the relative cost factor for general bases is easily obtained using a set of elementary column operations. A program code using the above idea is developed and applied to in-plant energy operating systems. A computational example shows that the computing time of the proposed algorithm is less than 10 percent of the general purpose LP code MPS-II.     

An approach to a singular optimal control problem for non-linear systems using differential form theory

By applying differential form theory, we consider the singular control problem for non-linear systems with control variables appearing linearly in both the system dynamics and the performance index. First, we derive necessary conditions of singular optimality for a single-input system, including the relation to the Euler-Poisson equation and to the generalized Legendre-Clebsch condition. Defining the degree of singularity, we develop necessary conditions satisfied by the singular trajectory embedded in a reduced space. For a time-invariant system, we clarify the relation between the dynamic and the related static optimality. Second, we derive necessary conditions for singular optimality for a multi-input system where the dimension of the control vector is equal to that of the state space. We show that the Shima-Sawaragi condition for the optimality of boundary controls and the generalized Legendre-Clebsch condition are obtained from these conditions. The results are also applied to the analysis of a time-invariant system.     

Nonlinear Laguerre–Volterra observer‐controller and its application to process control

By expanding each kernel using the orthonormal Laguerre series, a Volterra functional series is used to represent the input/output relation of a nonlinear dynamic system. With the feedback of the modeling error, we design a novel nonlinear state observer, based on which an output feedback controller is derived for both the stabilization and tracking problems. The stability of the closed‐loop system is analyzed theoretically. The algorithm is effectively applied on the continuous stirring tank reactor and chemical reactor temperature control system. Copyright © 2009 John Wiley & Sons, Ltd.     

Algorithms for solving large-scale 0–1 goal programming and its application to reliability optimization problem

Goal programming(GP) is one of the most widely used Operations Research/Management Science/Industrial Engineering techniques for solving multiple criteria decision making (MCD M) problems. In the realistic decision making problems, many GP problems are involved a large number of 0–1 decision variables and a special type of system structures.

Inthis paper, we develop a computational algorithm for solving 0–1 goal programming with a generalized upper bounding (GUB) structures. From the views of the computational experience and storage requirement, we implemented an efficient software package for UN IX workstations in which we called it micro 0–1 GP(GUB). In the micro 0–1 GP(GUB) developed here, the GUB structures would be effectively handled and we designed user-friendly GP data entry subsystem.

As a real-world 0–1 goal programming problem with the GUB structures, we demonstrate an optimization problem of system reliability for allocating redundant units by the micro 0–1 GP(GUB) software package on an UN IX system.     

Worst‐case control policies for (terminal) linear–quadratic control problems under disturbances

We consider linear control systems under uncertainties. For such systems we solve the problem of constructing worst‐case feedback control policies that are allowed to be corrected at m fixed intermediate time moments. We propose two types of the approximative control policies. All of them guarantee that for all admissible uncertainties the terminal system state lies in a prescribed neighborhood of a given state x_* at a given final moment, and the value of the cost function does not exceed a given estimate. It is shown that computation of the estimate for each policy is equivalent to solving a corresponding convex mathematical programming (MP) problem with m decision variables. Based on the solution of the MP problem, we derive simple explicit rules (which can be easily implemented on‐line) for constructing the corresponding control policy in the original control problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Two singular value inequalities and their implications in H∞approach to control system design

In this note we prove that ifAandBare both nonnegative definite Hermitian matrices andA - Bis also nonnegative definite, then the singular values of A and B satisfy the inequalitiessigma_{i}(A)geq sigma_{i}(B), wherebar{sigma}(cdot) = sigma_{1}(cdot) geq sigma_{2}(cdot) geq '" geq sigma_{m}(cdot) = underbar{sigma}(.)denote the singular values of a matrix. A consequence of this property is that, in a nonsquare H^{infty} optimization problem, ifsup_{omega} bar{sigma}[Z(jsigma)] {underline{underline Delta}} sup_{omega} bar{sigma}[x(jomega)^{T}/ Y(jomega)^{T}]^{T} = lambda, then the singular values ofXandYsatisfy the inequalitylambda^{2} geq max_{i} sup_{omega} [sigma_{i}^{2}(X) + sigma_{m-i-1}^{2}(Y)]wheremis the number of columns of the matrixZ.     

Error analysis of variable stepsize Runge–Kutta methods for a class of multiply-stiff singular perturbation problems

In this paper, we present some results on the error behavior of variable stepsize stiffly-accurate Runge–Kutta methods applied to a class of multiply-stiff initial value problems of ordinary differential equations in singular perturbation form, under some weak assumptions on the coefficients of the considered methods. It is shown that the obtained convergence results hold for stiffly-accurate Runge–Kutta methods which are not algebraically stable or diagonally stable. Some results on the existence and uniqueness of the solution of Runge–Kutta equations are also presented.     

Heuristic algorithm based on molecules optimizing their geometry in a crystal to solve the problem of integer factorization

Integer factorization is a vital number theoretic problem finding frequent use in public key cryptography, and other areas like Fourier transforms. There has been growing interest among researchers in innovative alternative approaches to solving the problem by modeling some optimizing natural or biological behaviour in a form of an algorithm that can be used to solve the problem provided the problem can be represented as an optimization task. In this paper, we present a new model based on computational chemistry behind how molecules interact among each other to minimize their surface energy potential in a typical crystal. While this phenomenon itself is a research problem, it interestingly provides a new way of solving other problems like integer factorization which can be represented in different forms of discrete optimization task. However, we must note that the present methods based on such models are not scalable to the real-world scenario, and we present a brief discussion on this issue.     

Conditions for stability of the extended Kalman filter and their application to the frequency tracking problem

The error dynamics of the extended Kalman filter (EKF), employed as an observer for a general nonlinear, stochastic discrete time system, are analyzed. Sufficient conditions for the boundedness of the errors of the EKF are determined. An expression for the bound on the errors is given in terms of the size of the nonlinearities of the system and the error covariance matrices used in the design of the EKF. The results are applied to the design of a stable EKF frequency tracker for a signal with time-varying frequency.This research was supported by the Co-operative Research Centre for Robust and Adaptive Systems ((CR)² ASys). The authors wish to acknowledge the funding of the activities of (CR)² ASys by the Australian Commonwealth Government under the Co-operative Research Centre Program.