Improved solutions for ill-conditioned problems involved in set-membership estimation for fault detection and isolation

Set-membership (SM) estimation implies that the computed solution sets are guaranteed to contain all the feasible estimates consistent with the bounds specified in the model. Two issues often involved in the solution of SM estimation problems and their application to engineering case studies are considered in this paper. The first one is the estimation of derivatives from noisy signals, which in a bounded uncertainty framework means obtaining an enclosure by lower and upper bounds. In this paper, we improve existing methods for enclosing derivatives using Higher-Order Sliding Modes (HOSM) differentiators combining filtering. Our approach turns the use of high order derivatives more efficiently especially when the signal to differentiate has slow dynamics. The second issue of interest is solving linear interval equation systems, which is often an ill-conditioned problem. This problem is reformulated as a Constraint Satisfaction Problem and solved by the combination of the constraint propagation Forward Backward algorithm and the SIVIA algorithm. The two proposed methods are tested on illustrative examples. The two methods are then used in a fault detection and isolation algorithm based on SM parameter estimation that is applied to detect abnormal parameter values in a biological case study.     

Guaranteed robust nonlinear minimax estimation

Minimax parameter estimation aims at characterizing the set of all values of the parameter vector that minimize the largest absolute deviation between the experimental data and the corresponding model outputs. It is well known, however, to be extremely sensitive to outliers in the data resulting, e.g., of sensor failures. In this paper, a new method is proposed to robustify minimax estimation by allowing a prespecified number of absolute deviations to become arbitrarily large without modifying the estimates. By combining tools of interval analysis and constraint propagation, it becomes possible to compute the corresponding minimax estimates in an approximate but guaranteed way, even when the model output is nonlinear in its parameters. The method is illustrated on a problem where the parameters are not globally identifiable, which demonstrates its ability to deal with the case where the minimax solution is not unique.     

Ellipsoidal parameter or state estimation under model uncertainty

Ellipsoidal outer-bounding of the set of all feasible state vectors under model uncertainty is a natural extension of state estimation for deterministic models with unknown-but-bounded state perturbations and measurement noise. The technique described in this paper applies to linear discrete-time dynamic systems; it can also be applied to weakly non-linear systems if non-linearity is replaced by uncertainty. Many difficulties arise because of the non-convexity of feasible sets. Combined quadratic constraints on model uncertainty and additive disturbances are considered in order to simplify the analysis. Analytical optimal or suboptimal solutions of the basic problems involved in parameter or state estimation are presented, which are counterparts in this context of uncertain models to classical approximations of the sum and intersection of ellipsoids. The results obtained for combined quadratic constraints are extended to other types of model uncertainty.     

On decentralised control of non-linear interconnected systems

The asymptotic stabilisation problem of a class of large-scale interconnected systems is considered, where the non-linear interconnections between subsystems satisfy quadratic constraints that are functions of the whole system's state vector. A decentralised combined observer-controller compensator is proposed and analysed, where the subsystems’ state vectors are estimated using local sliding mode observers. The closed-loop system driven by the proposed decentralised compensator is guaranteed to be asymptotically stable subject to two conditions that are easily verifiable. Simulation results illustrate the effectiveness of the proposed decentralised combined observer-controller compensator.     

Stability-constrained model predictive control

This paper presents a new approach to guaranteeing stability of model predictive control (MPC). A stability constraint is computed and propagated forward at each stage as a constraint on the magnitude of the predicted state vector for a state-space controllable-form realization. It is shown that asymptotic stability is guaranteed for the case of constrained multi-input linear time-invariant systems under full-state measurement or with state estimation for any prediction horizon and any optimization objective function     

Set-membership estimation from poor quality data sets: Modelling ammonia volatilisation in flooded rice systems

A set-membership (bounded-error) estimation approach can handle small and poor quality data sets as it does not require testing of statistical assumptions which is possible only with large informative data sets. Thus, set-membership estimation can be a good tool in the modelling of agri-environmental systems, which typically suffers from limited and poor quality observational data sets. The objectives of the paper are (i) to demonstrate how six parameters in an agri-environmental model, developed to estimate NH₃ volatilisation in flooded rice systems, were estimated based on two data sets using a set-membership approach, and (ii) to compare the set-membership approach with conventional non-linear least-squares methods. Results showed that the set-membership approach is efficient in retrieving feasible parameter-vectors compared with non-linear least-squares methods. The set of feasible parameter-vectors allows the formation of a dispersion matrix of which the eigenvalue decomposition reflects the parameter sensitivity in a region.     

On the normal vector estimation for point cloud data from smooth surfaces

Reliable estimation of the normal vector at a discrete data point in a scanned cloud data set is essential to the correct implementation of modern CAD/CAM technologies when the continuous CAD model representation is not available. A new method based on fitted directional tangent vectors at the data point has been developed to determine its normal vector. A local Voronoi mesh, based on the 3D Voronoi diagram and the proposed mesh growing heuristic rules, is first created to identify the neighboring points that characterize the local geometry. These local Voronoi mesh neighbors are used to fit a group of quadric curves through which the directional tangent vectors are obtained. The normal vector is then determined by minimizing the variance of the dot products between a normal vector candidate and the associated directional tangent vectors. Implementation results from extensive simulated and practical point cloud data sets have demonstrated that the present method is robust and estimates normal vectors with reliable consistency in comparison with the existing plane fitting, quadric surface fitting, triangle-based area weighted average, and triangle-based angle weighted average methods.     

Global bi-directional motion compensation frame interpolation algorithm

A frame interpolation algorithm for the application of low-bit-rate video coding is proposed in this paper. The global motion vectors are first estimated by computing global sum of absolute differences between frames. Then the block motion estimation is carried out according to the modified weighted correlation index criteria. The estimation is further adjusted around the moving object based on the modified weighted correlation index criteria but with a smaller factor k so that the global motion estimation is less influential to the movement vector estimation for the moving objects. Finally the motion estimation outliers are determined by the Box-and-Whisker analysis, and removed by two methods. Method 1 redoes the bi-directional estimation with a larger factor k; Method 2 computes the movement vectors of the outliers using neighboring movement vectors. The experiments showed that the proposed method can improve the peak signal to noise ratio for four test video sequences. Visual inspection also shows better frame interpolation quality.     

State estimation for linear and non-linear equality-constrained systems

This article addresses the state-estimation problem for linear and non-linear systems for the case in which prior knowledge is available in the form of an equality constraint. The equality-constrained Kalman filter (KF) is derived as the maximum-a-posteriori solution to the equality-constrained state-estimation problem for linear and Gaussian systems and is compared to alternative algorithms. Then, four novel algorithms for non-linear equality-constrained state estimation based on the unscented KF are presented, namely, the equality-constrained unscented KF, the projected unscented KF, the measurement-augmentation unscented KF, and the constrained unscented KF. Finally, these methods are compared on linear and non-linear examples.     

An asymmetric Lyapunov function for linear systems with asymmetric actuator saturation

This paper proposes an asymmetric Lyapunov function approach to the estimation of the domain of attraction and the domain with a guaranteed regional gain for a linear system subject to asymmetric actuator saturation. Depending on the sign of each of the m inputs, the input space is divided into 2^m regions. In each region, the linear system with asymmetrically saturated inputs can be expressed as a linear system with symmetric dead zone. A quadratic function of the augmented state vector containing the system state and the symmetric dead‐zone function is constructed for each region. From these quadratic functions, an asymmetric Lyapunov function is composed. Furthermore, based on the special properties of the intersections between regions, 2 generalized asymmetric Lyapunov functions are proposed that lead to reduced conservativeness. A set of conditions are established under which the level sets of these asymmetric Lyapunov functions are contractively invariant and are thus estimates of the domain of attraction. Another set of conditions are derived under which the level sets are subsets of the domain with a guaranteed regional gain. Based on these conditions, LMI‐based optimization problems are formulated and solved to obtain the largest level sets as the estimates of the domain of attraction and of the domain with a guaranteed regional gain. Simulation results demonstrate the effectiveness of the proposed approach.     

Adaptive stabilization of first-order systems using estimates modification based on a Sylvester determinant test

This note presents an adaptive control scheme for first-order continuous-time systems subject to bounded noise and unmodelled dynamics. The estimated plant model is controllable and then the adaptive scheme is free from singularities. The estimation scheme involves the use of a relative dead zone which freezes the estimation process when the size of the prediction error is small compared to the contribution of the unmodelled dynamics. The singularities are avoided through the use of a modification of the estimated plant parameter vector so that its associated Sylvester matrix is guaranteed to be nonsingular. That property is achieved by ensuring that the absolute value of its determinant does not lie below a prefixed positive threshold. In addition, the use of a hysteresis switching function is not used for a modification of the estimates while the absence of chattering is guaranteed in the eventual case when the Sylvester determinant tends to zero. The global stability of the closed-loop system is also guaranteed.     

Rigid body attitude estimation based on the Lagrange–d’Alembert principle

Estimation of rigid body attitude and angular velocity without any knowledge of the attitude dynamics model is treated using the Lagrange–d’Alembert principle from variational mechanics. It is shown that Wahba’s cost function for attitude determination from two or more non-collinear vector measurements can be generalized and represented as a Morse function of the attitude estimation error on the Lie group of rigid body rotations. With body-fixed sensor measurements of direction vectors and angular velocity, a Lagrangian is obtained as the difference between a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential obtained from Wahba’s function. An additional dissipation term that depends on the angular velocity estimation error is introduced, and the Lagrange–d’Alembert principle is applied to the Lagrangian with this dissipation. A Lyapunov analysis shows that the state estimation scheme so obtained provides stable asymptotic convergence of state estimates to actual states in the absence of measurement noise, with an almost global domain of attraction. These estimation schemes are discretized for computer implementation using discrete variational mechanics. A first order Lie group variational integrator is obtained as a discrete-time implementation. In the presence of bounded measurement noise, numerical simulations show that the estimated states converge to a bounded neighborhood of the actual states.     

Exponential stability of constrained receding horizon control withterminal ellipsoid constraints

In this paper, state- and output-feedback receding horizon controllers are proposed for linear discrete time systems with input and state constraints. The proposed receding horizon controllers are obtained from the finite horizon optimization problem with the finite terminal weighting matrix and the artificial invariant ellipsoid constraint, which is less restrictive than the conventional terminal equality constraint. Both hard constraints and mixed constraints are considered in the state-feedback case, and mixed constraints are considered in the output-feedback case. It is shown that all proposed state- and output-feedback receding horizon controllers guarantee the exponential stability of closed-loop systems for all feasible initial sets using the Lyapunov approach     

Synthesis of optimal fuzzy estimation algorithms

Conclusion The article shows that the well-known ellipsoidal estimation formalism can be constructively extended to the case of estimating sets of a more general form. Mathematical models of these sets are defined not only by positive definite quadratic forms (as in the case with ellipsoids), but also by Lyapunov, Bellman, and other functions. We have considered the parametric properties of estimates represented by fuzzy estimating sets. A formalism approximating the set-theoretical intersection operation has been developed for these fuzzy estimates. An important feature of the proposed approach is that the intersection procedure is optimized not in each step or cycle, but over the entire sequence of steps in accordance with an additive criterion which is fairly natural for the relevant class of problems. We have thus established a relationship between optimal fuzzy set-theoretical estimation procedures and standard optimal algorithms. In the concluding part of the article we have considered state estimation of a static object from observations distorted by additive noise. The state estimation problem has been solved in the class of fuzzy ellipsoidal estimates with a so-called tolerant part. The linear “size“ or “radius“ of the tolerant part provides an efficient measure of the “degree of fuzziness“ of the estimate. On the whole, the estimation (filtering) algorithm proposed in this article, like other fuzzy estimation algorithms, is robust to prior errors. The study has been supported by US CRDF grant VE 2300. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 167–183, July–August, 1998.     

Adaptive LFT control of a civil aircraft with online frequency‐domain parameter estimation

This paper describes the application of an indirect linear fractional transformation (LFT)–based state‐space adaptive control scheme to a transport aircraft, within the context of the European project REconfiguration of CONtrol in Flight for Integral Global Upset REcovery. The principle of the scheme is to design and validate off‐line a gain‐scheduled controller, depending on the plant parameters to be estimated, and to combine it online with a model estimator, so as to minimize the onboard computational time and complexity. A modal approach, very classical for the design of a flight control law, is used to directly synthesize the static output feedback LFT controller, depending on the control and stability derivatives, ie, the parameters of the linearized aerodynamic state‐space model to be estimated. Since the gain‐scheduled LFT controller online depends on the parameter estimates instead of the true values, its robustness to transient and asymptotic estimation errors needs to be assessed using μ and integral quadratic constraint analysis techniques. A primary concern being an online implementation, a fully recursive frequency‐domain estimation technique is proposed, with a low online computational burden and the capability to track time‐varying parameters. Full nonlinear simulations along a trajectory validate the good performance properties of the combined estimator and gain‐scheduled flight controller. To some extent, minimal guaranteed stability and performance properties of the adaptive scheme can be ensured by switching to a robust controller when the parameter estimates are not reliable enough, thus bypassing the Certainty Equivalence Principle.     

Memory annihilation of structured maps in bidirectional associativememories

Structured sets comprise Boolean vectors with equal pair-wise Hamming distances, h. An external vector, if it exists at an equidistance of h/2 from each vector of the structured set, is called the centroid of the set. A structured map is a one-one mapping between structured sets. It is a set of associations between Boolean vectors, where both domain and range vectors are drawn from structured sets. Associations between centroids are called centroidal associations. We show that when structured maps are encoded into bidirectional associative memories using outer-product correlation encoding, the memory of these associations are annihilated under certain mild conditions. When annihilation occurs, the centroidal association emerges as a stable association, and we call it an alien attractor. For the special case of maps where h=2, self-annihilation can take place when either the domain or range dimensions are greater than five. In fact, we show that for dimensions greater than eight, as few as three associations suffice for self-annihilation. As an example shows, annihilation occurs even for the case of bipolar decoding which is well known for its improved error correction capability in such associative memory models.     

Globally robust explicit model predictive control of constrained systems exploiting SVM‐based approximation

This paper presents a systematic method to address the reduction of online computational complexity and infeasibility problem of explicit model predictive control for constrained systems under external disturbance. In feasible state space, in order to avoid the expensive database searching procedure, support vector machine‐based approximation is proposed to yield a novel unified explicit optimal control law rather than a piecewise affine one developed by explicit model predictive control. In infeasible state space, through constructing finite maximum control invariant sets around fictitious equilibrium points, a reachable controller is devised to steer the infeasible state asymptotically to the feasible state space without violating the hard constraint. Consequently, global robustness is guaranteed by introducing a minimum robust positively invariant set by means of the tube‐based technique, despite the coexistence of external disturbance and training error. Finally, the performance of the presently proposed control law is evaluated through three groups of numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Non-rigid metric reconstruction from perspective cameras

The metric reconstruction of a non-rigid object viewed by a generic camera poses new challenges since current approaches for Structure from Motion assume the rigidity constraint of a shape as an essential condition. In this work, we focus on the estimation of the 3-D Euclidean shape and motion of a non-rigid shape observed by a perspective camera. In such case deformation and perspective effects are difficult to decouple – the parametrization of the 3-D non-rigid body may mistakenly account for the perspective distortion. Our method relies on the fact that it is often a reasonable assumption that some of the points on the object’s surface are deforming throughout the sequence while others remain rigid. Thus, relying on the rigidity constraints of a subset of rigid points, we estimate the perspective to metric upgrade transformation. First, we use an automatic segmentation algorithm to identify the set of rigid points. These are then used to estimate the internal camera calibration parameters and the overall rigid motion. Finally, we formulate the problem of non-rigid shape and motion estimation as a non-linear optimization where the objective function to be minimized is the image reprojection error. The prior information that some of the points in the object are rigid can also be added as a constraint to the non-linear minimization scheme in order to avoid ambiguous configurations. We perform experiments on different synthetic and real data sets which show that even when using a minimal set of rigid points and when varying the intrinsic camera parameters it is possible to obtain reliable metric information.     

Estimating differential quantities from point cloud based on a linear fitting of normal vectors

Estimation of differential geometric properties on a discrete surface is a fundamental work in computer graphics and computer vision. In this paper, we present an accurate and robust method for estimating differential quantities from unorganized point cloud. The principal curvatures and principal directions at each point are computed with the help of partial derivatives of the unit normal vector at that point, where the normal derivatives are estimated by fitting a linear function to each component of the normal vectors in a neighborhood. This method takes into account the normal information of all neighboring points and computes curvatures directly from the variation of unit normal vectors, which improves the accuracy and robustness of curvature estimation on irregular sampled noisy data. The main advantage of our approach is that the estimation of curvatures at a point does not rely on the accuracy of the normal vector at that point, and the normal vectors can be refined in the process of curvature estimation. Compared with the state of the art methods for estimating curvatures and Darboux frames on both synthetic and real point clouds, the approach is shown to be more accurate and robust for noisy and unorganized point cloud data. Supported in part by the National Natural Science Foundation of China (Grant Nos. 60672148, 60872120), the National High-Tech Research & Development Program of China (Grant Nos. 2006AA01Z301, 2008AA01Z301), and Beijing Municipal Natural Science Foundation (Grant No. 4062033)