Interval Constraint Satisfaction Research Articles

Dynamic ICSP Graph Optimization Approach for Car-Like Robot Localization in Outdoor Environments

Localization has been regarded as one of the most fundamental problems to enable a mobile robot with autonomous capabilities. Probabilistic techniques such as Kalman or Particle filtering have long been used to solve robotic localization and mapping problem. Despite their good performance in practical applications, they could suffer inconsistency problems. This paper presents an Interval Constraint Satisfaction Problem (ICSP) graph based methodology for consistent car-like robot localization in outdoor environments. The localization problem is cast into a two-stage framework: visual teach and repeat. During a teaching phase, the interval map is built when a robot navigates around the environment with GPS-support. The map is then used for real-time ego-localization as the robot repeats the path autonomously. By dynamically solving the ICSP graph via Interval Constraint Propagation (ICP) techniques, a consistent and improved localization result is obtained. Both numerical simulation results and real data set experiments are presented, showing the soundness of the proposed method in achieving consistent localization.

A Low-Cost Consistent Vehicle Localization Based on Interval Constraint Propagation

Probabilistic techniques (such as Extended Kalman Filter and Particle Filter) have long been used to solve robotic localization and mapping problem. Despite their good performance in practical applications, they could suffer inconsistency problems. This paper proposes an interval analysis based method to estimate the vehicle pose (position and orientation) in a consistent way, by fusing low-cost sensors and map data. We cast the localization problem into an Interval Constraint Satisfaction Problem (ICSP), solved via Interval Constraint Propagation (ICP) techniques. An interval map is built when a vehicle embedding expensive sensors navigates around the environment. Then vehicles with low-cost sensors (dead reckoning and monocular camera) can use this map for ego-localization. Experimental results show the soundness of the proposed method in achieving consistent localization.

Localization of a Vehicle: A Dynamic Interval Constraint Satisfaction Problem-Based Approach

This paper introduces a new interval constraint propagation (ICP) approach dealing with the real-time vehicle localization problem. Bayesian methods like extended Kalman filter (EKF) are classically used to achieve vehicle localization. ICP is an alternative which provides guaranteed localization results rather than probabilities. Our approach assumes that all models and measurement errors are bounded within known limits without any other hypotheses on the probability distribution. The proposed algorithm uses a low-level consistency algorithm and has been validated with an outdoor vehicle equipped with a GPS receiver, a gyro, and odometers. Results have been compared to EKF and other ICP methods such as hull consistency (HC4) and 3-bound (3B) algorithms. Both consistencies of EKF and our algorithm have been experimentally studied.

Evidential box particle filter using belief function theory

A box particle filtering algorithm for nonlinear state estimation based on belief function theory and interval analysis is presented. The system under consideration is subject to bounded process noises and Gaussian multivariate measurement errors. The mean and the covariance matrix of Gaussian random variables are considered bounded due to modeling errors. The belief function theory is a means to represent this type of uncertainty using a mass function whose focal sets are intervals. The proposed algorithm applies interval analysis and constraint satisfaction techniques. Two nonlinear examples show the efficiency of the proposed approach compared to the original box particle filter.

Design of robust fractional-order controllers and prefilters for multivariable system using interval constraint satisfaction technique

Design of robust controller for multivariable systems is very challenging and their automation is a key concern in control system design. In this paper, a new, simple, and reliable automated fractional-order multivariable Quantitative Feedback Theory (QFT) controllers design methodology is proposed. A fixed structure fractional-order multivariable QFT controller has been synthesized by solving QFT quadratic inequalities of robust stability and tracking specifications. The quadratic inequalities (constraints) are posed as Interval Constraint Satisfaction Problem (ICSP). The constraints are solved by constraint solver-RealPaver. The synthesis problem consists of obtaining a controller that ensures stability and meets a given set of performance specifications, in spite of disturbance and model uncertainties. In addition to perform the above tasks, a MIMO controller also has to perform the difficult task of minimizing interaction between the various control loops. Unlike existing manual or convex optimization based QFT design approaches, the proposed method gives a controller which meets all performance requirements in QFT, without going through the conservative and sequential design stages for each of the multivariable sub-systems. In the design method, we pose the prefilter design problem as an ICSP and solve it using the well-established constraint solver. The designed controllers and prefilters are tested for a quadruple tank process and their efficacy is demonstrated.

Robust Fault Diagnosis of Nonlinear Systems Using Interval Constraint Satisfaction and Analytical Redundancy Relations

In this paper, a robust fault diagnosis problem for nonlinear systems considering both bounded parametric modeling errors and noise is addressed using parity-equation-based analytical redundancy relations (ARR) and interval constraint satisfaction techniques. Fault detection, isolation, and estimation tasks are considered. Moreover, the problem of quantifying the uncertainty in the ARR parameters is also addressed. To illustrate the usefulness of the proposed approach, a case study based on the well-known wind turbine benchmark is used.

QFT prefilter design for multivariable systems using interval constraint satisfaction technique

In this paper, a computationally efficient method is proposed for automated design of the prefilters for multivariable systems. In quantitative feedback theory (QFT) method, proposed by Horowitz, the prefilter is designed to achieve the desired tracking specifications. In the proposed approach, we pose the prefilter design problem as an interval constraint satisfaction problem and solve it using the well-established interval constraint satisfaction techniques. The proposed method finds optimal values of the parameters of fixed structure prefilter within the initial search domain. An approach based on prefilter synthesis for single-input single-output is already developed. The purpose of this paper is to extend this approach to QFT prefilter design for general multivariable systems. To validate the above design approach, we applied the method to a laboratory setup of magnetic levitation system.

Automated synthesis of multivariable QFT controller using interval constraint satisfaction technique

Robust controller synthesis of Multi-Input–Multi-Output (MIMO) systems is of great practical interest and their automation is a key concern in control system design. The synthesis problem consists of obtaining a controller that ensures stability and meets a given set of performance specifications, in spite of the disturbance and model uncertainties. In addition to perform the above tasks, a MIMO controller also has to perform the difficult task of minimizing the interaction between the various control loops.Unlike existing manual or convex optimization based Quantitative Feedback Theory (QFT) design approaches, the proposed method gives a controller which meets all performance requirements in QFT, without going through the conservative and sequential design stages for each of the multivariable sub-systems. In this paper, a new, simple, and reliable automated MIMO QFT controllers design methodology is proposed. A fixed structure MIMO QFT controller has been synthesized by solving QFT quadratic inequalities of robust stability and tracking specifications. The quadratic inequalities (constraints) are posed as Interval Constraint Satisfaction Problem (ICSP). The constraints are solved by constraint solver — RealPaver. The main feature of this method is that the algorithm finds all the solutions to within the user-specified accuracy. The designed MIMO QFT controllers are tested on the experimental setup designed by Educational Control Product (ECP) Magnetic Levitation Setup ECP 730. From the experimental results presented, it is observed that, the designed controller satisfies the desired performance specifications. It is also observed that, the interactions between the loops are within the specified limits. The robustness of the designed controllers are verified by putting extra weights on the magnets.

Automated design of fractional PI QFT controller using interval constraint satisfaction technique (ICST)

In this paper, a fractional-order proportional-integral (FOPI) controller is proposed and designed for a class of nonlinear integer-order (IO) systems. For fair comparison, the proposed FOPI and the traditional integer-order PID (IOPID) controllers are designed following the same set of imposed tuning constraints, which can guarantee the desired control performance and the robustness of the designed controllers to the loop gain variations. This proposed design scheme offers a practical and systematic way of controller design for the nonlinear IO plant. We also propose a new and efficient method for automated synthesis of a fixed structure Quantitative Feedback Theory (QFT) FO controller. This is achieved by solving QFT quadratic inequalities of robust stability and performance specifications. The controller synthesis problem is posed as interval constraint satisfying problem (ICSP) and solved with interval constraint solver. The main feature of this method is that it guarantees to find all feasible controllers of given structure in the specified search domain. The designed FOPI and the traditional IOPID controllers are tested on the experimental setup designed by Educational Control Product (ECP) Magnetic Levitation Setup ECP 730. From the experimental results presented, it is observed that the designed FO controllers work more efficiently, with improved performance compared with the designed stabilizing IOPID controller.

Implementation of fixed structure QFT prefilter synthesised using interval constraint satisfaction techniques

Prefilter synthesis is one of the important design steps of Horowitz’s Quantitative Feedback Theory (QFT) to robust feedback system synthesis. The prefilter is designed to achieve tracking specifications. A new, computationally efficient approach for automation of the prefilter design step is proposed. In the proposed approach, the prefilter design problem is posed as an Interval Constraint Satisfaction Problem (ICSP) and solved using established Interval Constraint Satisfaction Techniques (ICST). To validate the above design approach we apply the method to a benchmark problem as well as laboratory set-up of industrial plant emulator and obtain simple, low-order QFT prefilters.

Computation of frequency responses for uncertain fractional-order systems

This paper proposes an algorithm for computation of Bode magnitude and phase responses for a large class of linear uncertain fractional-order systems using interval constraint propagation technique. It is first shown that the problem of finding the magnitude and phase of the uncertain fractional-order system over a frequency range can be formulated as a interval constraint satisfaction problem and then solved using branch and prune algorithm. The algorithm guarantees that the magnitude and phase responses are computed to prescribed accuracy. The other advantage of the method is that the magnitude and phase can be computed without any kind of integer-order approximation of the given fractional-order system. The proposed algorithm is demonstrated on three examples including a practical application of a gas turbine plant.

Computation of Stability Margins for Uncertain Linear Fractional-Order Systems

The present paper proposes an algorithm for finding the stability margins and cross over frequencies for an uncertain fractional-order system using the interval constraint propagation technique. It is first shown that the problem of finding the stability margins and crossover frequencies can be formulated as an interval constraint satisfaction problem and then solved using the branch and prune algorithm. The algorithm guarantees that the stability margins and the crossover frequencies are computed to the prescribed accuracy. The proposed algorithm is demonstrated on a noninductive cable system and also on a practical application of a gas turbine plant.

Computation of limit cycles for uncertain nonlinear fractional-order systems

The present paper proposes an algorithm to compute limit cycle points for an uncertain nonlinear fractional-order system with separable nonlinearities using interval constraint propagation. It is first shown that the problem of finding limit cycle points can be formulated as an interval constraint satisfaction problem and then solved using the branch and prune algorithm. The algorithm guarantees that all the points on the limit cycle locus are computed to the prescribed accuracy. The proposed algorithm is demonstrated with a fractional commensurate order system with uncertain coefficients and also with a practical application of a gas turbine plant.

Computation of spectral sets for uncertain linear fractional-order systems

The present paper proposes an algorithm to compute the spectral set of a family of fractional-order pseudo-polynomials. The algorithm makes use of interval constraint propagation technique to find out all the structural roots of the given uncertain fractional-order systems in the given search domain. It is first shown that the problem of finding the spectral set can be formulated as an interval constraint satisfaction problem and then solved using branch and prune algorithm. The algorithm guarantees that all the points of the spectral set are computed to prescribed accuracy. The proposed algorithm is demonstrated on a plant with nonlinear parametric dependencies and also on a practical application of a gas turbine plant.

Design and implementation of an optimal, reliable and robust control system for an industrial plant emulator

In this paper, we present the design of an optimal, reliable and robust control system for an industrial plant emulator that is representative of many 'real-life' electromechanical systems found in industry. The design approach is based on Quantitative Feedback Theory (QFT). The QFT controller synthesis problem is posed as an Interval Constraint Satisfaction Problem (ICSP) and the solution methodology uses interval box-consistency techniques. The control system designed is then experimentally implemented (real-time) on the industrial plant emulator system in a non-collocated configuration. Experimental tests are conducted to check the reliability and robustness of the designed control system, by subjecting it to disturbances, uncertainties, backlash and their various combinations. The experimental test results show the control system design to be reliable and robust.

Computation of Stability Margins for Uncertain Linear Fractional-order Systems using Interval Constraint Propagation

The present paper proposes an algorithm for finding the stability margins and cross over frequencies for an uncertain fractional-order system using interval constraint propagation technique. It is first shown that the problem of finding the stability margins and crossover frequencies can be formulated as a interval constraint satisfaction problem and then solved using branch and prune algorithm. The algorithm guarantees that the stability margins and the crossover frequencies are computed to prescribed accuracy and that theses values are reliable in the face of all kinds of computational errors. The other advantage of the method is that the stability margins and crossover frequencies can be computed without the need of frequency response plots or any kind of approximations. Two examples of uncertain fractional-order systems are taken from the literature and their stability margins and crossover frequencies are computed using the proposed algorithm.

Automated Synthesis of Fixed Structure QFT Controller using Interval Constraint Satisfaction Techniques

Abstract Robust controller synthesis is of great practical interest and its automation is a key concern in control system design. Automatic controller synthesis is still a open problem. In this paper a new, efficient method has been proposed for automated synthesis of a fixed structure quantitative feedback theory (QFT) controller by solving QFT quadratic inequalities of robust stability and performance specifications. The controller synthesis problem is posed as interval constraint satisfying problem (ICSP) and solved with interval constraint solver. The method is guaranteed to find all feasible controllers of given structure in the search domain. The proposed method is tested on two benchmark problems, and simple, low order controllers are successfully obtained in quick time.

Box particle filtering for nonlinear state estimation using interval analysis

In recent years particle filters have been applied to a variety of state estimation problems. A particle filter is a sequential Monte Carlo Bayesian estimator of the posterior density of the state using weighted particles. The efficiency and accuracy of the filter depend mostly on the number of particles used in the estimation and on the propagation function used to re-allocate weights to these particles at each iteration. If the imprecision, i.e. bias and noise, in the available information is high, the number of particles needs to be very large in order to obtain good performances. This may give rise to complexity problems for a real-time implementation. This kind of imprecision can easily be represented by interval data if the maximum error is known. Handling interval data is a new approach successfully applied to different real applications. In this paper, we propose an extension of the particle filter algorithm able to handle interval data and using interval analysis and constraint satisfaction techniques. In standard particle filtering, particles are punctual states associated with weights whose likelihoods are defined by a statistical model of the observation error. In the box particle filter, particles are boxes associated with weights whose likelihood is defined by a bounded model of the observation error. Experiments using actual data for global localization of a vehicle show the usefulness and the efficiency of the proposed approach.

A GMDH neural network-based approach to passive robust fault detection using a constraint satisfaction backward test

This paper proposes a new passive robust fault detection scheme using non-linear models that include parameter uncertainty. The non-linear model considered here is described by a group method of data handling (GMDH) neural network. The problem of passive robust fault detection using models including parameter uncertainty has been mainly addressed by checking if the measured behaviour is inside the region of possible behaviours based on the so-called forward test since it bounds the direct image of an interval function. The main contribution of this paper is to propose a new backward test, based on the inverse image of an interval function, that allows checking if there exists a parameter in the uncertain parameter set that is consistent with the measured system behaviour. This test is implemented using interval constraint satisfaction algorithms which can perform efficiently in deciding if the measured system state is consistent with the GMDH model and its associated uncertainty. Finally, this approach is tested on the servoactuator being a FDI benchmark in the European Project DAMADICS.

Algorithm 852

RealPaver is an interval software for modeling and solving nonlinear systems. Reliable approximations of continuous or discrete solution sets are computed using Cartesian products of intervals. Systems are given by sets of equations or inequality constraints over integer and real variables. Moreover, they may have different natures, being square or nonsquare, sparse or dense, linear, polynomial, or involving transcendental functions.The modeling language permits stating constraint models and tuning parameters of solving algorithms which efficiently combine interval methods and constraint satisfaction techniques. Several consistency techniques (box, hull, and 3B) are implemented. The distribution includes C sources, executables for different machine architectures, documentation, and benchmarks. The portability is ensured by the GNU C compiler.