Computation of Lie Derivatives of Tensor Fields Required for Nonlinear Controller and Observer Design Employing Automatic Differentiation

Differential-geometric concepts are widely used in nonlinear controller and observer design. The design algorithms often require certain types of Lie derivatives. In the past, these derivatives have been computed symbolically. The author suggests an alternative approach based on automatic differentiation. The computation of Lie derivatives of different tensor fields is discussed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of Controllability and Observability Matrices – Duality and Automatic Differentiation

During the last decades, many new methods for controller and observer design for nonlinear state‐space systems have been developed. Several of these design methods require Lie derivaties and Lie brackets. These derivatives are also needed to compute controllability and observability matrices of nonlinear systems. Up to now, these derivatives have been computed symbolically. An alternative approach based on automatic differentiation is presented.     

High-Gain Scheduling of the Proportional-Integral-Observer

Estimation of states and unknown inputs simultaneously can be done by Proportional-Integral-Observer (PIO). This observer is used in nonlinear and robust control. It has additional integral feedback loop in comparison with the usual Luenberger observer. Increasing the gain of this observer the performance would be increased, however due to this high gain the performance is influenced by measurement noise. Advanced PI-Observer (API) algorithm is a method which is used for scheduling the optimal gain of PI-Observer based on the cost function and by using a bank of PI-Observer. In this contribution the proposed method obtains relative optimal gain by using a given cost function. Afterwards the absolute optimal gain is computed by changing the design parameters at each step based on the desired performance criteria. The absolute optimal gain is found instead of relative optimal gain in comparison with the API-Observer. Simulation results estimating contact forces acting to a vibrating elastic beam based on indirect measurements illustrate the improvement of the proposed method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An LMI approach to discrete-time observer design with stochastic resilience

Much of the recent work on robust control or observer design has focused on preservation of stability of the controlled system or the convergence of the observer in the presence of parameter perturbations in the plant or the measurement model. The present work addresses the important problem of stochastic resilience or non-fragility of a discrete-time Luenberger observer which is the maintenance of convergence and/or performance when the observer is erroneously implemented possibly due to computational errors i.e. round off errors in digital implementation or sensor errors, etc. A common linear matrix inequality framework is presented to address the stochastic resilient design problem for various performance criteria in the implementation based on the knowledge of an upper bound on the variance of the random error in the observer gain. Present results are compared to earlier designs for stochastic robustness. Illustrative examples are given to complement the theoretical results.     

A procedure for designing linear functional observers

We present low-order functional observer existence conditions; the proposed existence conditions always lead to a design of functional observers of order less than or equal to that of a Luenberger observer. The proposed low-order observer always exists if a Luenberger observer exists.     

Fault detection based on fractional order models: Application to diagnosis of thermal systems

The aim of this paper is to propose diagnosis methods based on fractional order models and to validate their efficiency to detect faults occurring in thermal systems. Indeed, it is first shown that fractional operator allows to derive in a straightforward way fractional models for thermal phenomena. In order to apply classical diagnosis methods, such models could be approximated by integer order models, but at the expense of much higher involved parameters and reduced precision. Thus, two diagnosis methods initially developed for integer order models are here extended to handle fractional order models. The first one is the generalized dynamic parity space method and the second one is the Luenberger diagnosis observer. Proposed methods are then applied to a single-input multi-output thermal testing bench and demonstrate the methods efficiency for detecting faults affecting thermal systems.     

Computation of multiple Lie derivatives by algorithmic differentiation

Lie derivatives are often used in nonlinear control and system theory. In general, these Lie derivatives are computed symbolically using computer algebra software. Although this approach is well-suited for small and medium-size problems, it is difficult to apply this technique to very complicated systems. We suggest an alternative method to compute the values of iterated and mixed Lie derivatives by algorithmic differentiation.     

Minimum dimension of a functional observer with specifiable dynamical properties

The article considers the construction of a functional observer with a given rate of convergence for the most general case: a vector state functional of a linear dynamical system with a vector output. An upper bound is derived on the minimum dimension of such an observer, which holds for almost all functionals. An algorithm is proposed for constructing an observer that achieves this bound. The algorithm can be used to construct a functional observer for almost all specified spectra (i.e., with the exception of a set of measure zero). The scalar observer method previously developed by the authors and proposed in the present article is based on canonical form Luenberger observability for systems with vector output.     

Boundary output feedback stabilization for a class of coupled parabolic systems

In this paper, the boundary output feedback stabilization problem is addressed for a class of coupled nonlinear parabolic systems. An output feedback controller is presented by introducing a Luenberger‐type observer based on the measured outputs. To determine observer gains, a backstepping transform is introduced by choosing a suitable target system with nonlinearity. Furthermore, based on the state observer, a backstepping boundary control scheme is presented. With rigorous analysis, it is proved that the states of nonlinear closed‐loop system including state estimation and estimation error of plant system are locally exponentially stable in the L²norm. Finally, a numerical example is proposed to illustrate the effectiveness of the presented scheme.     

Synchronization of integer order and fractional order Chua’s systems using robust observer

In this paper we present a fractional order Chua’s circuit that behaves chaotically based on the use of a fractional order low pass filter. Next, an integer order robust observer will be designed to synchronize the fractional order Chua’s circuit as well as integer order Chua’s circuit with unknown nonlinearity. This method consists in designing a Luenberger like observer appended with an estimator of the unknown nonlinear function. The estimator assumes that the nonlinear function is slowly varying and that the observer converges quickly and uses the backward difference formula to approximate the state derivative. The efficiency of the proposed method is confirmed using numerical simulations.     

Some remarks on tensor differentiation

Summary The Jacobian of classical tensor analysis is generalized to a transformation operator containing second derivatives. This operator and the Jacobian may be used to formulate a second order tensor calculus. In this calculus, a simple contraction scheme includes as special cases Lie differentiation, the Lie bracket, covariant differentiation, and differentiation with respect to the tensor connections of Bompiani. To Enrico Bompiani on his scientific Jubilee     

Design of a nonlinear observer using automatic differentiation

Observer design for a class of observable nonlinear systems is considered. By a coordinate transformation and linearization around a desired trajectory the problem is reduced to the design for a linear time varying system. Calculation of the observer gain, depending on the desired trajectory, requires several derivatives. These are calculated using automatic differentiation by a combination of forward and reverse mode. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust output-feedback SOSM control subject to unmatched disturbances and its application: A fixed-time observer-based method

This paper investigates the output-feedback control for a general class of multi-input multi-output (MIMO) linear systems in the presence of unmatched disturbances. Firstly, a new observer composed of a Luenberger observer and a novel hierarchical high-order sliding mode (HOSM) observer is proposed to identify the system states and disturbances, simultaneously. As one of the most remarkable properties, the convergence time of the proposed observer is bounded by a positive constant which is free of the system initial error conditions. Secondly, based on the proposed observer, a new second-order sliding mode (SOSM) controller is constructed by using a novel sliding surface with unmatched disturbances compensation. The proposed control law is a simply continuous function of time and thus can certainly reduce numerical chattering. Finally, to show the effectiveness of the theoretical results, an application to inverted pendulum system is used to make simulation comparison.     

Reduced order observer design for nonlinear systems

This work is a geometric study of reduced order observer design for nonlinear systems. Our reduced order observer design is applicable for Lyapunov stable nonlinear systems with a linear output equation and is a generalization of Luenberger’s reduced order observer design for linear systems. We establish the error convergence for the reduced order estimator for nonlinear systems using the center manifold theory for flows. We illustrate our reduced order observer construction for nonlinear systems with a physical example, namely a nonlinear pendulum without friction.     

Computation of mixed Lie derivatives in nonlinear control

Several algorithms in nonlinear control require Lie derivatives. The symbolic computation using computer algebra software is straightforward. However, the applicability of symbolic computation in nonlinear control is limited due to a considerable increase of the sizes of higher order derivatives. This problem can be circumvented using an alternative computation technique for derivatives called algorithmic differentiation. We extend existing approaches to calculate mixed Lie derivaties. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reduced order observer design for discrete-time nonlinear systems

This work is a geometric study of reduced order observer design for discrete-time nonlinear systems. Our reduced order observer design is applicable for Lyapunov stable discrete-time nonlinear systems with a linear output equation and is a generalization of Luenberger’s reduced order observer design for linear systems. We establish the error convergence for the reduced order estimator for discrete-time nonlinear systems using the center manifold theory for maps. We illustrate our reduced order observer construction for discrete-time nonlinear systems with an example.     

Controllability and Controller-Observer Design for a Class of Linear Time-Varying Systems

In this paper a class of linear time-varying control systems is considered. The time variation consists of a scalar time-varying coefficient multiplying the state matrix of an otherwise time-invariant system. Under very weak assumptions of this coefficient, we show that the controllability can be assessed by an algebraic rank condition, Kalman canonical decomposition is possible, and we give a method for designing a linear state-feedback controller and Luenberger observer.     

Sampled-data nonlinear observer design for chaos synchronization: A Lyapunov-based approach

This paper considers sampled-data based chaos synchronization using observers in the presence of measurement noise for a large class of chaotic systems. We study discretized model of chaotic systems which are perturbed by white noise and employ Lyapunov-like theorems to come up with a simple yet effective observer design. For the choice of observer gain, a suboptimal criterion is obtained in terms of LMI. We present semiglobal as well as global results. The proposed scheme can also be extended for discrete-time chaotic systems. Numerical simulations have been carried out to verify the effectiveness of theoretical results.     

Frobenius-type theorems for Lipschitz distributions

We generalize the classical Frobenius Theorem to distributions that are spanned by locally Lipschitz vector fields. The various versions of the involutivity conditions are extended by means of set-valued Lie derivatives—in particular, set-valued Lie brackets—and set-valued exterior derivatives. A PDEs counterpart of these Frobenius-type results is investigated as well.     

An observer for a class of disturbance driven nonlinear systems

An observer design for a class of nonlinear systems driven by disturbances or uncertainties is presented. The design is based on a high gain strategy, and the gain of the proposed observer is explicitly given.