Further results on adaptive full-order and reduced-order observers for Lur’e differential inclusions

This paper is concerned with the adaptive observer design of Lur’e differential inclusions with unknown parameters. Under a relaxed assumption on nonlinear perturbation functions, a sufficient condition for the existence of an adaptive full-order observer is established. Comparing with results in the literature, the present conditions are complemented with a numerically reliable computational approach, which can be checked by means of linear matrix inequalities. Furthermore, it is shown that, under the sufficient condition, the existence of a reduced-order observer is guaranteed. Also, the reduced-order observer is designed. The effectiveness of the proposed design is illustrated via a simulation example.     

Adaptive full-order and reduced-order observers for the Lur’e differential inclusion system

This paper deals with the observer design for the Lur’e differential inclusion system with unknown parameters. The set-valued mapping in the differential inclusion is upper semi-continuous, closed, convex, bounded and monotone. First, under some assumptions an adaptive full-order observer is designed for the system. Then, under the same assumptions, a reduced-order observer is proved to exist. An example is provided to show the validation of the designed observers.     

Actuator fault detection and estimation for the Lur’e differential inclusion system

This paper deals with actuator fault detection and estimation for the Lur’e differential inclusion system. An adaptive full-order observer is used to detect the occurrence of the actuator fault. Then, based on a reduced-order observer, an approach to estimate the actuator fault is presented. A simulation of rotor system is given to illustrate the effectiveness of the proposed method.     

Some Remarks on High Gain Observer Design

High gain observer design can be carried out for nonlinear systems, whose dynamics can be decomposed into a linear and a Lipschitz continuous nonlinear part. The observer uses a linear output injection, which is specified by a constant gain matrix. Due to this simple structure, high gain observers are frequently used in practical applications. The observer gain is often chosen via eigenvalue placement. However, the formal existence conditions are more complicated. In many cases, there is a finite bound on the maximum feasible Lipschitz constant of the nonlinear part for which the error dynamics can be stabilized. The existing results can be improved significantly, if the structure of the linear part is taken into account. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

2—D奇异系统正则观测器的设计

本文讨论了一类２－Ｄ奇异系统正则观测器的设计问题。为此，首先将２－Ｄ奇异系统等价为降阶正则系统和一纯代数等式的组合，在此基础上，从理论上解决了２－Ｄ奇异系统的一类正则观测器的设计问题。     

Output feedback vibration control of a string driven by a nonlinear actuator

In this paper, we design an observer-based output feedback controller to exponentially stabilize a system of nonlinear ordinary differential equation-wave partial differential equation-ordinary differential equation. An observer is designed to estimate the full states of the system using available boundary values of the partial differential equation. The output feedback controller is built via the combination of the ordinary differential equation backstepping which is applied to deal with the nonlinear ordinary differential equation, and the partial differential equation backstepping which is used for the wave partial differential equation-ordinary differential equation. The controller can be applied into vibration suppression of a string-payload system driven by an actuator with nonlinear characteristics. The global exponential stability of all states in the closed-loop system is proved by Lyapunov analysis. The numerical simulation illustrates the states of the actuator, string, payload and the observer errors are fast convergent to zero under the proposed output feedback controller.     

Direct Method of Constructing H 2-Suboptimal Controllers: Continuous-Time Systems

An H ₂-suboptimal control problem is defined and analyzed. Then, an algorithm called H ₂-suboptimal state feedback gain sequence algorithm (Algorithm A1) is developed. Rather than utilizing a perturbation method, which is numerically stiff and computationally prohibitive, Algorithm A1 utilizes a direct eigenvalue assignment method to come up with a sequence of H ₂-suboptimal state feedback gains. Also, although the sequence of H ₂-suboptimal state feedback gains constructed by Algorithm A1 depends on a parameter , the construction procedure itself does not require explicitly the value of the parameter . Next, attention is focused on constructing a sequence of H ₂-suboptimal observer-based measurement feedback controllers. Both full-order as well as reduced-order observer-based controllers are developed. For a given H ₂-suboptimal state feedback gain, a sequence of observer gains for either a full-order or reduced-order observer can be constructed by merely dualizing Algorithm A1. The direct method of constructing H ₂-suboptimal controllers developed here has a number of advantages over the perturbation method, e.g., it has the ability to design both full-order and reduced-order observer-based controllers while still maintaining throughout the design the computational simplicity of it.     

A reduction technique for discrete generalized algebraic and difference Riccati equations

This paper proposes a reduction technique for the generalized Riccati difference equation arising in optimal control and optimal filtering. This technique relies on a study on the generalized discrete algebraic Riccati equation. In particular, an analysis on the eigenstructure of the corresponding extended symplectic pencil enables to identify a subspace in which all the solutions of the generalized discrete algebraic Riccati equation are coincident. This subspace is the key to derive a decomposition technique for the generalized Riccati difference equation. This decomposition isolates a “nilpotent” part, which converges to a steady-state solution in a finite number of steps, from another part that can be computed by iterating a reduced-order generalized Riccati difference equation.     

Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n₁+n₂ into the pure-slow regular matrix differential Riccati equation of dimension n₁ and the pure-fast stiff matrix differential Riccati equation of dimension n₂. A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.     

A new model-free sliding observer to synchronization problem

In this paper, a new observer is proposed for the synchronization problem, this new observer presents a simple structure that contains a sliding mode term which turns out to be robust against output noises as well as sustained disturbances, the slave system is a pure sliding-mode observer. As far as we know in the literature this class of observers have not been used in the synchronization problem. Comparisons with other two model-based observers, Thau observer and Bestle-Zeitz observer, are proposed. The performances of these observers are shown by using Lorenz system and Chua’s circuit.     

Global observer design for nonlinear systems

This paper is a geometric study of the global observer design for nonlinear systems. Using the theory of foliations, we derive necessary and sufficient conditions for global exponential observers for nonlinear systems under some assumptions. Our proof for these necessary and sufficient conditions for global exponential observers if via defining two equivalence relations known as horizontal and vertical equivalence relations, and constructing two foliations known as horizontal and vertical foliations from these equivalence relations. Finally, as a corollary of our global theorem, we derive necessary and sufficient conditions for local exponential observers of critically Lyapunov stable nonlinear systems.     

State estimation in batch processes using a nonlinear observer

This paper deals with the problem of nonlinear states estimation in batch chemical processes. It presents a reduced-order nonlinear observer approach to perform the estimation. The proposed method allows adjustment of the speed of convergence towards zero of the estimation error. The stability properties of the model-based observer are analytically treated in order to show the conditions under which exponential convergence can be achieved. In addition, the performance of the proposed observer is evaluated on batch processes.     

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.     

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.     

Observer design for discrete-time nonlinear systems

This paper is a geometric study of the observer design for discrete-time nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable discrete-time nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable discrete-time nonlinear systems. As an application of our local observer design, we consider a class of discrete-time 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.     

Robust stabilization of large-scale time-delay systems with estimated state feedback

This paper proposes a class of observers and uses the estimated state feedback to stabilize a perturbed large-scale time-delay system in which the state is unmeasurable. An inequality representing the relationship among the perturbation bounds, interconnection magnitudes, and gains of observers and controllers is derived to ensure that the system is stabilized and the state is estimated. Moreover, this inequality does not need the solution of a Lyapunov equation or Riccati equation and is independent of time delays.This work was supported in part by the National Science Council, Taiwan, ROC, under Grant NSC-83-0404-E-008-040.     

The contraction rate in Thompson's part metric of order-preserving flows on a cone – Application to generalized Riccati equations

We give a formula for the Lipschitz constant in Thompson's part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This shows that in the special case of order-preserving flows, a general characterization of the contraction rate in Thompson's part metric, given by Nussbaum, leads to an explicit formula. As an application, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other invariant Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtkowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation.     

Error bounds for numerical solutions of ordinary differential equations

Summary An a posteriori error bound, for an approximate solution of a system of ordinary differential equations, is derived as the solution of a Riccati equation. The coefficients of the Riccati equation depend on an eigenvalue of a matrix related to a Jacobian matrix, on a Lipschitz constant for the Jacobian matrix, and on the approximation defect. An upper bound is computable as the formal solution of a sequence of Riccati equations with constant coefficients. This upper bound may sometimes be used to control step length in a numerical method.     

Observer synthesis method for Lipschitz nonlinear discrete-time systems with time-delay: An LMI approach

In this paper, we address the problem of observer design for a class of Lipschitz nonlinear discrete-time systems with time-delay. The main contribution lies in the use of a new structure of the proposed observer with a novel Lyapunov-Krasovskii functional. Thanks to these designs, new nonrestrictive synthesis conditions, expressed in terms of linear matrix inequalities (LMIs), are obtained. Indeed, the obtained LMIs contain more degree of freedom than those established by the approaches available in the literature which consider a simple Luenberger observer with a simple Lyapunov function for the stability analysis. An extension of the presented result to H_∞ performance analysis is given in order to take into account the noise (if it exists) affecting the considered system.