New results on general observers for nonlinear systems

In this paper, we establish that detectability is a necessary condition for the existence ofgeneral observers (asymptotic or exponential) for nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential), for nonlinear systems with real parametric uncertainty, if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant.     

New results on general observers for discrete-time nonlinear systems

In this paper, we establish that detectability is a necessary condition for the existence of general observers (asymptotic or exponential) for discrete-time nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential) for discrete-time nonlinear systems with real parametric uncertainty, if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory for maps, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant.     

Additional necessary conditions for optimal control with state-variable inequality constraints

It is shown that, when the set of necessary conditions for an optimal control problem with state-variable inequality constraints given by Bryson, Denham, and Dreyfus is appropriately augmented, it is equivalent to the (different) set of conditions given by Jacobson, Lele, and Speyer. Relationships among the various multipliers are given.This work was done at NASA Ames Research Center, Moffett Field, California, under a National Research Council Associateship.     

Nonlinear observer for autonomous switching systems with jumps

This work deals with nonlinear observer synthesis for a particular class of Hybrid Dynamic Systems (HDS): autonomous switching systems with jumps. The jumps can result from the system’s dynamics or from the diffeomorphism which makes it possible to lead the system to an observability canonical form. The contribution of this work relates to the design of a second order sliding mode based observer (“Super Twisting Algorithm”). It allows estimating both continuous and discrete states related to the system active dynamic. On the other hand these observers ensure a finite time convergence of the estimation error.     

Perturbation theory for nonlinear feedback control systems and spencer-goldschmidt integrability of linear partial differential equations

This paper aims to develop the differential-geometric and Lie-theoretic foundations of perturbation theory for control systems, extending the classical methods of Poincaré from the differential equation-dynamical system level where they are traditionally considered, to the situation where the element of control is added. It will be guided by general geometric principles of the theory of differential systems, seeking approximate solutions of the feedback linearization equations for nonlinear affine control systems. In this study, certain algebraic problems of compatibility of prolonged differential systems are encountered. The methods developed by D. C. Spencer and H. Goldschmidt for studying over-determined systems of partial differential equations are needed. Work in the direction of applying theio theory is presented.Supported by grants from the Ames Research Center of NASA and the Applied Mathematics and Systems Research Programs of the National Science Foundation     

Invariants for feedback equivalence and Cauchy characteristic multifoliations of nonlinear control systems

Linearization of a nonlinear feedback control system under nonlinear feedback is treated as a problem of equivalence-under the Lie pseudogroup of feedback transformations-of distributions on the product manifold of the state and control variables. The new feature of this paper is that it introduces the Cauchy characteristic sub-distributions of these distributions and their derived distributions. These Cauchy characteristic distributions are involutive and nested, hence define a Multifoliate Structure. A necessary condition for feedback equivalence of two nonlinear control systems is that these multifoliations be transformed under the feedback pseudogroup. For linear systems, this Cauchy characteristic multifoliate structuee is readily computed in terms of the (A, B)-matrix that defines the linear system. Assuming that the conditions for local feedback linearization are satisfied, the existence of a global feedback linearizing transformation is dependent on computing an element of the first cohomology group of the space with coefficients in the sheaf of groupoid of infinitesimal feedback automorphisms of the linear system. The theorem quoted above about the Cauchy characteristic multifoliations provides some information about this groupoid. It is computed explicitly and directly for control systems with one- or two-state dimensions. Finally, these Cauchy characteristic sub-distributions must inevitably play a role in the numerical or symbolic computational analysis of the Hunt-Su partial differential equations for the feedback-linearizing transformation.Senior Research Associate of the National Research Council at the Ames Research Center of NASA.     

Detection,isolation and identification of multiple actuator and sensor faults in nonlinear dynamic systems: Application to a waste water treatment process

The goal in many fault detection and isolation schemes is to increase the isolation and identification speed. This paper, presents a new approach of a nonlinear model based adaptive observer method, for detection, isolation and identification of actuator and sensor faults. Firstly, we will design a new method for the actuator fault problem where, after the fault detection and before the fault isolation, we will try to estimate the output of the instrument. The method is based on the formation of nonlinear observer banks where each bank isolates each actuator fault. Secondly, for the sensor problem we will reformulate the system by introducing a new state variable, so that an augmented system can be constructed to treat sensor faults as actuator faults. A method based on the design of an adaptive observers’ bank will be used for the fault treatment. These approaches use the system model and the outputs of the adaptive observers to generate residues. Residuals are defined in such way to isolate the faulty instrument after detecting the fault occurrence. The advantages of these methods are that we can treat not only single actuator and sensor faults but also multiple faults, more over the isolation time has been decreased. In this study, we consider that only abrupt faults in the system can occur. The validity of the methods will be tested firstly in simulation by using a nonlinear model of waste water treatment process with and without measurement noise and secondly with the same nonlinear model but by using this time real data.     

Nonlinear feedback control and systems of partial differential equations

Finding an equivalence between two feedback control systems is treated as a problem in the theory of partial differential equation systems. The mathematical aim is to embed the Jakubzyk-Respondek, Hunt-Meyer-Su work on feedback linearization in the general theory of differential systems due to Lie, Cartan, Vessiot, Spencer, and Goldschmidt. We do this by using the functor taking control systems into differential systems, and studying the equivalence invariants of such differential systems. After discussing the general case, attention is focussed on the special situation of most immediate practical importance, the theory of feedback linearization. In this case, the general system for feedback equivalence becomes a system of linear partial differential equations. Conditions are found that the general solution of this system may be described in terms of a Frobenius system and certain differential-algebraic operations.This work was supported by grant from the Ames Research Center of NASA and the Applied Mathematics Program of the National Science Foundation.     

Designing SISO observers with unknown input

In this paper we investigate the observer design for Single-InputSingle-Output (SISO) systems with unknown input. We find thatthe design conditions proposed by several authors in the pastlead to unity relative degree of the plant and zeros of theplant in the open left half of the complex plane, for SISO systems.In other words an observer for a SISO plant with unknown inputcan be designed if and only if the plant is of unity relativedegree and the numerator of the transfer function is Hurwitz.These conditions have an interesting consequence in the contextof model reference adaptive control (MRAC) for the case of relativedegree one, in the sense that strictly real positive (SPR) conditionfor the model reference is not needed.     

Global optimality of extremals: An example

The question of the existence and the location of Darboux points (beyond which global optimality is lost) is crucial for minimal sufficient conditions for global optimality and for computation of optimal trajectories. Here, we investigate numerically the Darboux points and their relationship with conjugate points for a problem of minimum fuel, constant velocity, horizontal aircraft turns to capture a line. This simple second-order optimal control problem shows that ignoring the possible existence of Darboux points may play havoc with the computation of optimal trajectories.The authors are indebted to G. Moyer for his constructive comments. This research was supported, for the first author, by a National Research Council Associateship at NASA Ames Research Center.on leave from the Technion, Israel Institute of Technology, Haifa, Israel.     

Linear-size nonobtuse triangulation of polygons

We give an algorithm for triangulatingn-vertex polygonal regions (with holes) so that no angle in the final triangulation measures more than π/2. The number of triangles in the triangulation is onlyO(n), improving a previous bound ofO(n ²), and the running time isO(n log² n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm. The research of S. Mitchell was supported by the Applied Mathematical Sciences program, U.S. Department of Energy Research and by the U.S. Department of Energy under Contract DE-AC04-76DP00789. J. Ruppert's work was performed while he was at the NASA Ames Research Center as an employee of Computer Sciences Corporation, under NASA Contrast NAS 2-12961.     

Indirect synthesis of multidegree-of-freedom transient systems

It is shown that an indirect synthesis method can be used in the efficient optimal design of multidegree-of-freedon, multidesignelement, nonlinear, transient systems. The technique begins with a limiting performance analysis which requires linear programming for a kinematically linear system, following which the system is selected using system identification methods such that the designed system responds as closely as possible to the limiting performance. The efficiency is a result of the method avoiding the repetitive systems analyses accompanying other numerical optimization methods.This investigation was supported by the NASA Langley Research Center, Grant No. NGR-47-005-145.     

Wavelet sparse approximate inverse preconditioners

We show how to use wavelet compression ideas to improve the performance of approximate inverse preconditioners. Our main idea is to first transform the inverse of the coefficient matrix into a wavelet basis, before applying standard approximate inverse techniques. In this process, smoothness in the entries ofA ⁻¹ are converted into small wavelet coefficients, thus allowing a more efficient approximate inverse approximation. We shall justify theoretically and numerically that our approach is effective for matrices with smooth inverses. Supported by grants from ONR: ONR-N00014-92-J-1890, and the Army Research Office: DAAL-03-91-C-0047 (Univ. of Tenn. subcontract ORA4466.04 Amendment 1 and 2). The first and the third author also acknowledge support from RIACS/NASA Ames NAS 2-96027 and the Alfred P. Sloan Foundation as Doctoral Dissertation Fellows, respectively. the work was supported by the Natural Sciences and Engineering Research Council of Canada, the Information Technology Research Centre (which is funded by the Province of Ontario), and RIACS/NASA Ames NAS 2-96027.     

On observing the states of uncertain dynamical systems

For the implementation of a stable asymptotic reduced-order Luenberger observer, which is not sensitive to unknown disturbances entering the system, it is shown that specific columns of the measurement matrixC must be different from zero.This research was supported by the Technion VPR Fund, L. Rogow Aeronautical Research Fund.The author wishes to thank E. P. Ryan, G. Leitmann, and S. Gutman for helpful discussions.     

Minimum-order functional observers

The article considers the construction of a minimum-order observer for a linear state functional of a linear dynamical system. For a one-output system we derive a necessary and sufficient condition that allows reconstruction of the functional by a k th order observer. This result is generalized to a system with vector output. An algorithm for the construction of a minimal observer is proposed. The problem is solved by the method of decomposition into scalar observers.     

A Lie-theoretic setting for the classical interpolation theories

The three classical interpolation theories — Newton-Lagrange, Thiele and Pick-Nevanlinna — are developed within a common Lie-theoretic framework. They essentially involve a recursive process, each step geometrically providing an analytic map from a Riemann surface to a Grassmann manifold. The operation which passes from the (n−1)st to the nth involves the action of what the physicists call a group of gauge transformations. There is also a first-order difference operator which maps the set of solutions of the nth order interpolation to the (n−1)st: This difference operator is, in each case, covariant with respect to the action of the Lie groups involved. For Newton-Lagrange interpolation, this Lie group is the group of affine transformations of the complex plane; for Thiele interpolation the group SL(2, C) of projective transformations; and for Pick-Nevanlinna interpolation the subgroup SU(1, 1) of SL(2, C) which leaves invariant the disk in the complex plane. National Research Council Senior Research Associate at the Ames Research Center (NASA)}.     

Diagnosis for uncertain,dynamic and hybrid domains using Bayesian networks and arithmetic circuits

System failures, for example in electrical power systems, can have catastrophic impact on human life and high-cost missions. Due to an electrical fire in Swissair flight 111 on September 2, 1998, all 229 passengers and crew on board sadly lost their lives. A battery failure most likely took place on the Mars Global Surveyor, which unfortunately last communicated with Earth and thus ended its mission on November 2, 2006. Fault diagnosis techniques that seek to hinder similar accidents in the future are being developed in this article. We present comprehensive fault diagnosis methods for dynamic and hybrid domains with uncertainty, and validate them using electrical power system data. Our approach relies on the use of Bayesian networks, which model the electrical power system, compiled to arithmetic circuits. We handle in an integrated way varying fault dynamics (both persistent and intermittent faults), fault progression (both abrupt and drift faults), and fault behavior cardinality (both discrete and continuous behaviors). Our work has resulted in a software system for fault diagnosis, ProDiagnose, that has been the top performer in three of the four international diagnostics competitions in which it participated. In this paper we comprehensively present our methods as well as novel and extensive experimental results on data from a NASA electrical power system.     

Design of observers for nonlinear systems with H ∞ performance analysis

This paper investigates the problem of observer design for nonlinear systems. By using differential mean value theorem, which allows transforming a nonlinear error dynamics into a linear parameter varying system, and based on Lyapunov stability theory, an approach of observer design for a class of nonlinear systems with time‐delay is proposed. The sufficient conditions, which guarantee the estimation error to asymptotically converge to zero, are given. Furthermore, an adaptive observer design for a class of nonlinear system with unknown parameter is considered. A method of H _∞ adaptive observer design is presented for this class of nonlinear systems; the sufficient conditions that guarantee the convergence of estimation error and the computing method for observer gain matrix are given. Finally, an example is given to show the effectiveness of our proposed approaches. Copyright © 2013 John Wiley & Sons, Ltd.     

Horizontal variable-speed interception game solved by forced singular perturbation technique

The time-optimal pursuit-evasion game in the horizontal plane between two airplanes is analyzed by applying the technique of forced singular perturbations (FSPT). Based on the assumption of multiple time scale separation, a zeroth-order closed-form solution is obtained, enabling one to use realistic aerodynamic and propulsion data. Control strategies are approximated by explicit feedback expressions of the state variables and the aircraft performance parameters. The zeroth-order feedback approximation is compared to the optimal openloop solution of the game. This comparison confirms the validity of the FSPT approximation for sufficiently large initial distances of separation.This work was completed during the first author's visit as a Senior NRC Associate at NASA Ames Research Center, Moffett Field, California. Its earlier phase was partially supported by AFSC Contract No. F49620-79-C-0135.     

Local observer design for nonlinear systems

This paper is a geometric study of the local observer design for nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable nonlinear systems. As an application of our local observer design, we consider a class of nonlinear systems with an input generator (exosystem) and show that for this class of nonlinear systems, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs.