On the general solution of the state deadbeat control problem

In this note, the state deadbeat control problem is considered. It is shown that, after appropriate change of basis of input and state spaces, the general solution of the state deadbeat control problem can be expressed completely by the rows of the powers of system matrix. This result yields a very simple procedure for the calculation of a state feedback deadbeat control gain. It also provides the number of free parameters which could be used for further design purposes. The results are illustrated by an example at the end of the note     

An adaptive state estimation solution to the maneuvering target problem

A new approach to tracking a maneuvering target is presented. Modeling the randomly varying target as a semi-Markov process and then incorporating the statistics into the design of an adaptive state estimator produced an estimation scheme that greatly reduced the large bias errors that usually appear when a target makes an unexpected, large-scale maneuver.     

On the classical solution to the linear-constrained minimum energy problem

Minimum energy problems involving linear systems with quadratic performance criteria are classical in optimal control theory. The case where controls are constrained is discussed in Athans and Falb (1966 Athans, M and Falb, PL. 1966. Optimal Control. An Introduction to the Theory and Its Applications, New York: McGraw-Hill Book Co.  [Google Scholar]) [Athans, M. and Falb, P.L. (1966), Optimal Control: An Introduction to the Theory and Its Applications, New York: McGraw-Hill Book Co.] who obtain a componentwise optimal control expression involving a saturation function expression. We show why the given expression is not generally optimal in the case where the dimension of the control is greater than one and provide a numerical counterexample.     

On the numerical solution of state equations

Two new methods for the solution of state equations of a linear time-invariant system are suggested. These methods are based on Romberg's algorithm and utilize the special form of the function to be integrated. The suggested methods are compared with existing ones.     

Integration control and failure detection with application to the robot payload variation problem

The problem of integrated control and failure detection of robot payload variations during the execution of a task is investigated by incorporating failure diagnostic methods based on the novel 4-parameter control configuration. The combined controller and failure detection system is a complex design involving nontrivial combination of parameters related to two independent designs: the doubly coprime factorization-based compensator design and failure detection design. The combined design provides a generic setting to study fundamental tradeoffs present in a simultaneous control and diagnostic system. The PUMPA-560 robot arm is used as a case study for the combined design.     

Nonlinear feedback solution to a bounded Brachistochrone problem in a reduced state space

The optimal nonlinear feedback law for both an unbounded and bounded Brachistochrone problem is given. These control laws describe the Brachistochrone problem in a very simple manner. These laws are derived by using a dimensionless set of state variables which spans a lower dimensional space than the original state space. Also using this reduced state space a two-dimensional instead of three-dimensional field of extremals is constructed for a space bounded Brachistochrone problem. This illustrates that the state space may be reduced so that the storage of a field of extremal trajectories will not exceed the storage capacity of a practical on-board computer. In this way, optimal feedback control may become a practical technique for a larger class of nonlinear systems.     

Numerical solution to the Steklov problem

The Steklov problem is considered in a planar domain with a smooth boundary. A numerical algorithm without saturation is constructed. The algorithm allows calculating 3000 eigenvalues with 9 decimal digits.     

On a general solution to the problem of equivalent transformations of program schemes

Schemes of programs without procedures are considered, and a rich family of scheme equivalences is studied. The structure of equivalent schemes is analyzed in order to build transformation systems that are complete with respect to equivalences from this family. The first result of the research is reported, which is that all equivalences from the family in question are solvable.     

Differential evolution solution to the SLAM problem

A new solution to the Simultaneous Localization and Modelling problem is presented in this paper. The algorithm is based on the stochastic search for solutions in the state space to the global localization problem by means of a differential evolution algorithm. This non linear evolutive filter, called Evolutive Localization Filter (ELF), searches stochastically along the state space for the best robot pose estimate. The set of pose solutions (the population) focuses on the most likely areas according to the perception and up to date motion information. The population evolves using the log-likelihood of each candidate pose according to the observation and the motion errors derived from the comparison between observed and predicted data obtained from the probabilistic perception and motion model.The proposed SLAM algorithm operates in two steps: in the first step the ELF filter is used at local level to re-localize the robot based on the robot odometry, the laser scan at a given position and a local map where only a low number of the last scans have been integrated. In the second step, the aligned laser measures and the corrected robot poses are used to detect whether the robot is revisiting a previously crossed area (i.e., a cycle in the robot trajectory exists). Once a cycle is detected, the Evolutive Localization Filter is used again to estimate the accumulated residual drift in the detected loop and then to re-estimate the robot poses in order to integrate the sensor measures in the global map of the environment.The algorithm has been tested in different environments to demonstrate the effectiveness, robustness and computational efficiency of the proposed approach.     

On the disturbed fault detection and isolation problem

In this paper we study the fault detection and isolation problem in presence of disturbances. In the case of observer-based residual generation, the problem amounts to finding two gain matrices such that two problems are simultaneously solved. These problems are insensitivity of the residuals to disturbances and the existence of some special structure for the transfer from faults to residuals. We prove in this paper that this joint problem can be solved if and only if the usual (undisturbed) fault detection and isolation problem can be solved for a system with a reduced number of outputs.     

A faster solution to the horizon problem

An algorithm is developed for calculating the horizons for each point in a digital terrain grid in order N iterations, whereas all previous methods seem to be of O(N²) time complexity. The new method makes horizon computations reasonable, and ought to improve the accuracy of surface climate models in rugged terrain.     

On the solution of the traveling salesman problem once again

The procedure is suggested for the construction of Hamiltonian cycles optimized along the length in weighted graphs by the method of the stagewise isolation and lengthening of the linear portions of paths of the minimized length. The procedure makes it possible to process both nonoriented and oriented graphs, i.e., to solve symmetric and nonsymmetric problems of the traveling salesman. At each stage of the procedure (starting from the first stage), the subgraphs of the initial (original) graph of the problem are processed, the complexity of which decreases in the transitions from stage to stage. The isolation and the lengthening of linear portions of the paths are the simplest operations for the detection of vertices of degree 2 with the possible removal of some edges (arcs) of a subgraph. These characteristics of the procedure (the stagewise decrease of the complexity of the processable subgraphs and the exceptional simplicity of operations for the isolation and lengthening of the linear portions of paths) permits us to hope for a high effectiveness of the use of the procedure for the solution of the traveling salesman problems of large dimensions.     

A hybrid approach to minimize state space explosion problem for the solution of two stage tandem queues

Two stage open queuing networks are used for modeling the subsystem-behaviour in computers and communication networks, mass storage devices, memory servers, and queuing analysis of wireless mobile cellular networks. The queuing analysis of wireless systems is essential in order to quantify the impact of different factors on quality of service (QoS); performance measures so that wireless protocols can be designed and/or tuned in an optimal manner. In that sense two stage open queuing systems are particularly important to model handoff phenomena, especially for the integration of two different systems such as cellular and wireless local area networks (WLANs). Analytical solutions for two-dimensional Markov processes suffer from the state space explosion problem. The numerical difficulties caused by large state spaces, make it difficult to handle multiple servers at the second stage of a tandem queuing system together with server failures and repairs. This study presents a new approach to analytical modeling of open networks offering improvements in alleviating this problem. The proposed solution is a hybrid version, which combines well known spectral expansion, and hierarchical Markov reward rate approaches. Using this approach, two-stage open networks with multiple servers, break-downs, and repairs at the second stage and feedback can be modeled as three-dimensional Markov processes and solved for performability measures. Comparative results show that the new algorithm used for solution, provides a high degree of accuracy, and it is computationally more efficient than the existing approaches. The proposed model is capable of solving other three-dimensional Markov processes.     

On the direct solution of Dirichlet's problem in two dimensions

Computing - General methods for the solution of the two-dimensionalDirichlet problem are presented. They are general in the sense that they are independent of the numerical values of the entries of...     

On one approach to the solution of the problem of guaranteeing estimation for Volterra equations

The problem of minimax filtration for systems described by discrete Volterra equations is considered. It is suggested that a simpler estimation algorithm instead of an optimal one be used. With the aid of the theory of duality, the quality estimate of such an approximation is obtained.     

A distributed solution to the network reconstruction problem

It has been recently shown in Ren et al. (2010) that by collecting noise-contaminated time series generated by a coupled-oscillator system at each node of a network, it is possible to robustly reconstruct its topology, i.e. determine the graph Laplacian. Restricting ourselves to linear consensus dynamics over undirected communication networks, in this paper we introduce a new dynamic average consensus least-squares algorithm to locally estimate these time series at each node, thus making the reconstruction process fully distributed and more easily applicable in the real world. We also propose a novel efficient method for separating the off-diagonal entries of the reconstructed Laplacian, and examine several concepts related to the trace of the dynamic correlation matrix of the coupled single integrators, which is a distinctive element of our network reconstruction method. The theory is illustrated with examples from computer, power and transportation systems.     

A practical solution to the minimal design problem

In the past few years several techniqnes for solving the minimal design problem have appeared in the literature. In this paper, a new algorithm is presented which has computational advantages over existing techniques of solving the minimal design problem. The procedure is illustrated with a physical multivariable example.     

An explicit solution to the multi-level programming problem

The multi-level programming problem is defined as an n-person nonzero-sum game with perfect information in which the players move sequentially. The bi-level linear case is addressed in detail. Solutions are obtained by recasting this problem as a standard mathematical probram and appealing to its implicitly separable structure. The reformulated optimization problem is linear save for a complementarity constraint of the form 〈u, g〉 = 0. This constraint is decomposed in a manner that permits us to achieve separability with very little cost in dimensionality. A general branch and bound algorithm is then applied to obtain solutions. Unlike the conventional mathematical program though, the multi-level program may fail to have a solution even when the decision variables are defined over a compact set. An auxiliary optimization problem is employed to detect such failure. Finally, the general max-min problem is discussed within the bi-level programming framework. Examples are given for a variety of related problems.