Full-order Observer Design Research Articles

Reduced-order intermediate variable observer based fault estimation and fault-tolerant control for fuzzy stochastic systems with exogenous disturbance

This paper studies fault estimation and fault-tolerant control problems for fuzzy systems with exogenous disturbance and stochastic noise. The fuzzy reduced-order intermediate variable observers and the disturbance observers are proposed to reconstruct the sensor fault, the actuator fault and the disturbance. According to the estimated results, the observer based fault-tolerant controllers are devised. Compared with the full-order observer design method, the designed reduced-order observers are with lower dimensions and less cost. Different from the classic intermediate observer, the feedback term of the output estimation error is introduced into our observers, which increases the design freedom. And the obtained linear matrix inequality conditions can ensure that the whole closed-loop systems are mean-square exponentially stable. Finally, two examples are offered to verify the feasibility of the proposed method.

Full-Order Observer Design for a Class of Nonlinear Port-Hamiltonian Systems

In this paper, we present a simple method to design a full-order observer for a class of nonlinear port-Hamiltonian systems (PHSs). We provide a sufficient condition for the observer to be globally exponentially convergent. This condition exploits the natural damping of the system. The observer and its design are illustrated by means of an academic example system. Numerical simulations verify the convergence of the reconstructions towards the unknown system variables.

Characterizations of Global Transversal Exponential Stability

We study the relationship between the global exponential stability of an invariant manifold and the existence of a positive semidefinite Riemannian metric which is contracted by the flow. In particular, we investigate how the following properties are related to each other (in the global case): 1) A manifold is globally “transversally” exponentially stable; 2) the corresponding variational system admits the same property; 3) there exists a degenerate Riemannian metric which is contracted by the flow and can be used to construct a Lyapunov function. We show that the transverse contraction rate being larger than the expansion of the shadow on the manifold is a sufficient condition for the existence of such a Lyapunov function. An illustration of these tools is given in the context of global full-order observer design.

Structure Preserving Observer Design for Port-Hamiltonian Systems

In this paper, a full-order observer design method is proposed for port-Hamiltonian systems. The proposed method is based on the notion of contractive port-Hamiltonian systems. It is the first structure preserving observer design for a broad class of input-state-output port-Hamiltonian systems. The design procedure consists of solving a matching equation, similar to the Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) for controller design. The matching equation, as shown in the paper, has some closed-form solutions for general class of mechanical and electromechanical systems. As another feature of the proposed method, it is shown that the existence of solution of the corresponding matching equation for a linear port-Hamiltonian system is equivalent to the detectability property of that system. Upon these facts, the proposed method can be considered as a counterpart of IDA-PBC for observer design. Simulations for some benchmark examples, including ball and beam, magnetic levitation, and permanent magnetic synchronous motor, show the potency and applicability of the method.

Cloud-based shaft torque estimation for electric vehicle equipped with integrated motor-transmission system

In order to improve oscillation damping control performance as well as gear shift quality of electric vehicle equipped with integrated motor-transmission system, a cloud-based shaft torque estimation scheme is proposed in this paper by using measurable motor and wheel speed signals transmitted by wireless network. It can help reduce computational burden of onboard controllers and also relief network bandwidth requirement of individual vehicle. Considering possible delays during signal wireless transmission, delay-dependent full-order observer design is proposed to estimate the shaft torque in cloud server. With these random delays modeled by using homogenous Markov chain, robust H∞ performance is adopted to minimize the effect of wireless network-induced delays, signal measurement noise as well as system modeling uncertainties on shaft torque estimation error. Observer parameters are derived by solving linear matrix inequalities, and simulation results using acceleration test and tip-in, tip-out test demonstrate the effectiveness of proposed shaft torque observer design.

Unknown input observer design for one-sided Lipschitz discrete-time systems subject to time-delay

In this paper, we address the problem of unknown input observer design, which simultaneously estimates state and unknown input, of a class of nonlinear discrete-time systems with time-delay. A novel approach to the state estimation problem of nonlinear systems where the nonlinearities satisfy the one-sided Lipschitz and quadratically inner-bounded conditions is proposed. This approach also allows us to reconstruct the unknown inputs of the systems. The nonlinear system is first transformed to a new system which can be decomposed into unknown-input-free and unknown-input-dependent subsystems. The estimation problem is then reduced to designing observer for the unknown-input-free subsystem. Rather than full-order observer design, in this paper, we propose observer design of reduced-order which is more practical and cost effective. By utilizing several mathematical techniques, the time-delay issue as well as the bilinear terms, which often emerge when designing observers for nonlinear discrete-time systems, are handled and less conservative observer synthesis conditions are derived in the linear matrix inequalities form. Two numerical examples are given to show the efficiency and high performance of our results.

Gain Scheduling of a Full-Order Observer for Sensorless Induction Motor Drives

This paper deals with the design of a speed-adaptive full-order observer for sensorless induction motor (IM) drives. A general stabilizing observer gain matrix, having three free design parameters, is used as a design framework. A gain-scheduled selection of the free design parameters is proposed. Furthermore, the full-order observer is augmented with the stator-resistance adaptation, and the local stability of the augmented observer is analyzed. The performance of the proposed full-order observer design is experimentally compared with a reduced-order observer using a 2.2-kW IM drive.

An Observer-Based Robust Load Frequency Control

A full-order observer-based state feedback linear controller design approach for a single area load frequency problem is presented. The load frequency control model incorporates integral control for improved command tracking response and disturbance rejection. A full state feedback is implemented through full-order state observer design. State feedback with integral controllers design is carried out through a pole-placement technique using symmetrical root locus method. Parametric model uncertainty has been considered and stability robustness study has been carried out assigning a bound on the deviations in parameters of single area load frequency model. The proposed design methodology achieves performance satisfying the specified stability margins. The methodology provides a control over peak values of the frequency and control signal deviations which may be utilized to meet hardware constraints. A numerical example illustrates the effectiveness of the developed methodology.

Non-fragile Observer Design for Fractional-order One-sided Lipschitz Nonlinear Systems

This paper is concerned with the problem of the full-order observer design for a class of fractional-order Lipschitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using an indirect Lyapunov approach, the sufficient condition for asymptotic stability of the full-order observer error dynamic system is presented. The stability condition is obtained in terms of LMI, which is less conservative than the existing one. A numerical example demonstrates the validity of this approach.

Full-order observer design for linear systems with unknown inputs

In this article, a full-order observer without unknown inputs reconstruction is suggested in order to achieve finite-time reconstruction of the state vector for a class of linear systems with unknown inputs. The observer is a simple one, its derivation being direct and easy. It will be shown that the problem of full-order observers for linear systems with unknown inputs can be reduced in this case to a standard one (the unknown input vector will not interfere in the observer equations). The effectiveness of the suggested design algorithm is illustrated by a numerical example (aircraft longitudinal motion), and, for the same aircraft dynamics, we make a comparison between our new observer and other already existing observers from the existence conditions and dynamic characteristics’ point of view; the superiority of the new designed observer is demonstrated.

Full-order observer design for a class of port-Hamiltonian systems

We consider a special class of port-Hamiltonian systems for which we propose a design methodology for constructing globally exponentially stable full-order observers using a passivity based approach. The essential idea is to make the augmented system consisting of the plant and the observer dynamics to become strictly passive with respect to an invariant manifold defined on the extended state-space, on which the state estimation error is zero. We first introduce the concept of passivity of a system with respect to a manifold by defining a new input and output on the extended state-space and then perform a partial state feedback passivation which leads to the construction of the observer. We then illustrate this observer design procedure on two physical examples, the magnetic levitation system and the inverted pendulum on the cart system.

Design of Full Order Observer in Speed Sensorless Induction Motor Drive

Design of Full Order Observer in Speed Sensorless Induction Motor Drive

Complements to full order observer design for linear systems with unknown inputs

This note completes the results obtained in [M. Darouach, M. Zasadzinski, S.J. Xu, Full-order observers for linear systems with unknown inputs, IEEE Trans. Automat. Control., AC-39 (3) (1994) 606–609]. It presents a full order observer for linear systems with unknown inputs in the state and in the measurement equations. It gives a more general approach for the observer’s design than that presented in the above mentioned reference and it shows that all the obtained results are independent of the choice of the generalized inverses considered in the observer’s design. Continuous- and discrete-time systems are considered.

OBSERVER DESIGN FOR LIPSCHITZ NONLINEAR SYSTEMS

This paper presents some new results on observer design for Lipschitz nonlinear systems. First, full-order observer design is considered. Based on a well-known result on the stability of the estimated error dynamics, a design procedure is given using linear matrix inequalities. A different result on the stability, and corresponding design procedure, are also presented, with less conservatism. Then on the basis of the discussion on full-order observer design, reduced-order observer design is discussed. The existence of reduced-order observer shares the same conditions as a full-order one.

Map Invariance and the State Reconstruction Problem for Nonlinear Discrete-time Systems

The role of map invariance is examined within the context of the dynamic state reconstruction problem for nonlinear discrete-time systems. In particular, the key notion of invariant manifold for maps in nonlinear discrete-time dynamics is shown to be conceptually insightful and technically quite effective to address important issues related to the deterministic observerbased nonlinear state estimation problem in the discrete-time domain. As a necessary first methodological step, the problem of quantitatively characterizing the asymptotic long-term behavior of nonlinear discrete-time systems with a skew-product structure using the notion of map invariance is revisited. The formulation of this problem can be naturally realized through a system of invariance functional equations (FEs), for which a set of existence and uniqueness conditions of a solution is provided. Under a certain set of conditions, it is shown that the invariant manifold computed attracts all system trajectories/orbits, and therefore, the asymptotic long-term dynamic behavior of the system is determined through the restriction of the discrete-time system dynamics on the invariant manifold. Within the above analytical framework, the nonlinear full-order observer design problem in the discrete-time domain is considered, appropriately formulated and an interpretation of previous work on the problem is attempted through the notion of invariant manifolds for maps. Furthermore, this framework allows the development of a new approach to the nonlinear reduced-order observer design problem for multiple-output systems in the discrete-time domain, which is also presented in the present work. Finally, the performance of the proposed nonlinear reduced-order discrete-time observer is assessed in an illustrative bioreactor example through simulations.

Robust fault detection in linear systems based on full-order state observers

A parametric approach to robust fault detection in linear systems with unknown disturbances is presented. The residual is generated using full-order state observers (FSO). Based on an analytical solution to a type of Sylvester matrix equations, the parameterization of the observer gain matrix is given. In terms of the design degrees of freedom provided by the parametric observer design and a group of introduced parameter vectors, a sufficient and necessary condition for full-order state observer design with disturbance decoupling is then established. By properly constraining the design parameters according to this proposed condition, the effect of the disturbance on the residual signal is also decoupled, and a simple algorithm is developed. The presented approach offers all the degrees of design freedom. Finally, a numerical example illustrates the effect of the proposed approach.

Delayed Observers for Linear Systems With Unknown Inputs

We present a method for constructing reduced-order state observers for linear systems with unknown inputs. Our approach provides a characterization of observers with delay, which eases the established necessary conditions for existence of unknown input observers with zero-delay. We develop a parameterization of the observer gain that decouples the unknown inputs from the estimation error, and we use the remaining freedom to ensure stability of the error dynamics. Our procedure is quite general in that it encompasses the design of full-order observers via appropriate choices of design matrices

An efficient method for observer design with approximately linear error dynamics

Exact error linearization is a well-known full-order observer design method which yields linear time-invariant, stable error dynamics in normal form coordinates. The difficult procedure associated with exact error linearization has led to the simpler extended Luenberger observer design. However, extended Luenberger observers can only guarantee a first-order approximation to the exact error linearization design. This paper provides an explicit formula involving only derivative (Lie derivative and bracket) and matrix inversion operations for computing an Nth-order approximation of the exact error linearization observer. Hence, the proposed method is a computational simple tool for designing observers for systems admitting exact error linearization observers. Two examples demonstrate the application of the method.

Fixed-order dynamic compensation for a high-performance fighter aircraft

This paper presents a robust fixed-order dynamic compensator design for a high-performance fighter aircraft. The compensator design methodology is based on an approximate loop transfer recovery technique. Full-state gains are first designed for the dual system, which is the equivalent of designing a full-order observer. Then the loop characteristics at the dual system input are approximated with' a compensator in controller canonical form. Finally, the optimal gains are implemented in the dual compensator to insure robustness and tracking performance at the plant output. The fixed-order dynamic compensator developed here is compared with a previously published full-order observer design.

Theory of LTR for non-minimum phase systems, recoverable target loops, and recovery in a subspace Part 2. Design

This Part focuses on the design of full order observer based controllers for the recovery of target loop transfer function or sensitivity and complimentary sensitivity functions when the given system is not necessarily left invertible and not necessarily of minimum phase. Four design tasks are considered. The first task concerns with arbitrarily specified target loop transfer functions and develops an observer design which has the capability to shape the inevitable recovery error according to the designer's needs whenever they are feasible. The second task considers observer design for exactly recoverable target loop transfer functions. The third task is similar to the second one in that it makes use of the specific properties of the target loop transfer function, but it considers observer design for asymptotically recoverable target loop transfer functions. The fourth task can be thought of as a generalization of the second and third tasks, and it considers observer design so that the achieved and target sensitivity and complimentary sensitivity functions match each other either exactly or asymptotically over a given subspace of the control space whenever it is possible. For all these tasks, observer design constraints and the available design freedom are reviewed. In view of the available freedom, possible specifications on the time-scale and/or eigenstructure of the observer dynamic matrix are formulated. In the case of first task, the conventional approach of designing observer based controllers by Kalman filter formalism which requires solving algebraic Riccati equations, is shown to have several fundamental limitations. A method of design based on asymptotic time-scale and eigenstructure assignment (ATEA) developed here overcomes these limitations. For the other tasks, no design methods other than the ones developed here are available in the literature. All the developed design methods are implemented in a ‘Matlab’ software package. A bank of examples illustrate that the proposed methods of design are capable of directly exploiting all the available freedom so as to achieve the desired results.