Observer Design For Systems Research Articles

Interval observer design for nonlinear discrete‐time dynamic systems

SummaryThe problem of interval observer design for systems described by nonlinear models with uncertainties is considered. The problem is solved based on special mathematical technique (the algebra of functions) which allows obtaining solution for systems containing non‐smooth nonlinearities. The relations to design interval observer insensitive or having minimal sensitivity to the disturbance and estimating the prescribed nonlinear function of the state vector of the original system are derived. Theoretical results are illustrated by example.

Interval Observer Design for Fault Diagnosis

The problem of interval observer design for systems described by linear models under the external disturbances is considered. The problem is solved based on the reduced-order model of the original system of minimal dimension. The relations allowing designing interval observer insensitive or having minimal sensitivity to the disturbance are obtained. The designed interval observers are used to solve the problem of fault diagnosis. Theoretical results are illustrated by example.

Adaptive observer design for systems with incremental quadratic constraints and nonlinear outputs – application to chaos synchronization

Adaptive observer design for systems with incremental quadratic constraints and nonlinear outputs – application to chaos synchronization

Interval observers design for systems with ostensible Metzler system matrices

This paper attempts to resolve the problem concerning the interval observers design for linear systems with ostensible Metzler system matrices. Because system dynamics matrices are partially different from strictly Metzler structures, a solution is achieved by constructing a composed system matrix representation, which combines pre-compensated interval matrix structures fixed with a prescribed region of D-stability and the reconstructed strictly Metzler matrix structure, related to the original interval system matrix parameter definition. A novel design procedure is presented, which results in a strictly positive observer gain matrix and guarantees that the lower estimates of the positive state variables are non-negative when considering the given system structure and the non-negative system state initial values. The design is computationally simple since it is reduced to the feasibility of the set of linear matrix inequalities.

H∞‐optimal observer design for systems with multiple delays in states, inputs and outputs: A PIE approach

AbstractThis article investigates the ‐optimal estimation problem of a class of linear system with delays in states, disturbance input, and outputs. The estimator uses an extended Luenberger estimator format which estimates both the present and history states. The estimator is designed using an equivalent Partial Integral Equation (PIE) representation of the coupled nominal system. The advantage of the resulting PIE representation is compact and delay free—obviating the need for commonly used bounding technique such as integral inequalities which typically introduces conservatism into the resulting optimization problem. The ‐optimal estimator synthesis problem is then reformulated as a Linear Partial Inequality (LPI)—a form of convex optimization using operator variables and inequlities. Such LPI‐based optimization problems can be solved using semidefinite programming via the PIETOOLS toolbox in Matlab. Compared with previous work, the proposed method simplifies the analysis and computation process and resulting in observers which are non‐conservtism to 4 decimal places when compared with Pad‐based ODE observer design methodologies. Numerical examples and simulation results are given to illustrate the effectiveness and scalability of the proposed approach.

Continuous–discrete time observers for homogeneous nonlinear systems with sampled-data outputs

In this article, we propose a new observer design for systems whose output is sampled in an aperiodic and asynchronous way and for which a continuous-time homogeneous observer of negative degree is available. The proposed method consists in adapting the existing continuous-time observer in order to cope with the fact that only sampled measurements are available instead of continuous ones. The obtained observer error is shown to be globally uniformly ultimately bounded for any upper bound on the sampling periods. Furthermore, the ultimate bound decreases as the upper bound on the sampling periods decreases. The stability analysis is based on a Lyapunov approach and the performances of the proposed observer are illustrated with simulations.

Equivariant Filters are Equivariant

Observers for systems with Lie group symmetries are an active area of research that is seeing significant impact in a number of practical domains, including aerospace, robotics, and mechatronics. This paper builds on the theory of the recently proposed Equivariant Filter (EqF), which is a general observer design for systems on homogeneous spaces that takes advantage of symmetries to yield significant performance advantages. It is shown that the EqF error dynamics are invariant to transformation of the input signal and equivariant as a parametrised vector field. The main theorem shows that two EqF's with different choices of local coordinates and origins and with equivalent noise modelling yield identical performance. In other words, the EqF is intrinsic to the system equations and symmetry. This is verified in a simulation of a 2D robot localisation problem, which also shows how the ability to choose an origin for the EqF can yield practical performance advantages by mitigating floating point precision errors.

Adaptive observer design for uncertain nonlinear ODE‐PDE cascades

SummaryWe are revisiting the problem of adaptive observer design for systems that are constituted of an Ordinary Differential Equation (ODE), containing a globally Lipschitz function of the state, and a linear Partial Differential Equation (PDE) of a diffusion–reaction heat type. The ODE and PDE are connected in series and both are subject to parametric uncertainties. In addition to nonlinearity and uncertainty, the system complexity also lies in the fact that no sensor can be implemented at the junction point between the ODE and the PDE. In the absence of parameter uncertainty, nonadaptive state observers are available featuring exponential convergence. However, convergence is guaranteed only under the condition that either the Lipschitz coefficient is sufficiently small or the PDE domain length is sufficiently small. To get around this limitation, and also to account for parameter uncertainty, we develop a design that involves two concatenated adaptive observers, covering the two subintervals of the PDE domain. The proposed design employs one extra sensor, providing the measurement of the PDE state at an inner position close to the ODE‐PDE junction point. Both observers are shown to be exponentially convergent, under ad‐hoc persistent excitation (PE) conditions, with no limitation on the Lipschitz coefficient and domain length.

Observer Design for ODE-PDE Cascade with Lipschitz ODE Nonlinear Dynamics and Arbitrary Large PDE Domain

This paper deals with observer design for systems constituted of an Ordinary Differential Equation (ODE), containing a Lipschitz function of the state, and a linear Partial Differential Equation (PDE) of diffusion-reaction heat type. In addition to nonlinearity, system complexity also lies in the fact that no sensor can be implemented at the junction point between the ODE and the PDE. Boundary observers providing state estimates of the ODE and PDE do exist but their exponential convergence is only ensured under the condition that either the Lipschitz coefficient is sufficiently small or the PDE domain length is sufficiently small. Presently, we show that limitation is removed by using two connected observers and implementing one extra sensor providing the PDE state at an inner position close to the ODE-PDE junction point. The measurements provided by both sensors are used by the first observer which estimates, not only the PDE state, but also its spatial derivative along the PDE subdomain located between both sensor positions. Using these estimates, the second observer determines estimates of the ODE state and the remaining PDE states. All state estimates are shown to be exponentially convergent to their true values.

Functional Observers Design for Linear Systems with Non-commensurate Time Varying Delays

This paper deals with the problem of functional observer design for systems with non-commensurate time varying delays. The main procedure of three types of functional observers are presented in this article, more precisely, observers with delays dependent, observers without internal delays and delay-free functional observers. A necessary condition for the existence of these observers is given as rank equality. The design conditions for functional observers are presented in terms of linear matrix inequalities (LMIs). The solution of these LMIs is used to determine the observers parameters. Finally, some numerical examples are given at the end of the paper to show the feasibility of the proposed approach.

Observer design for linear models of multi-rate asynchronous aerospace systems

This paper proposes an optimization-based computational approach to design multi-rate observers for linear models of aerospace systems with asynchronous measurements. A case study on the estimation of the state of an autonomous quadrotor UAV around its hovering operating point will be used as an example of the applicability of the proposed method. There are two theoretical contributions of the proposed method. The first is the derivation of a set of Linear Matrix Inequalities (LMIs) for the design of linear observers yielding convergence of the state estimation error given the maximum allowable sampling period (MASP) for each sensor. The second contribution is to propose an optimization problem subject to LMIs for finding the MASPs that guarantee exponential stability of the estimation error. The two contributions enable the analysis and design of multi-rate observers for systems with linear models, including quadrotor UAVs around a hovering operating point. The examples show a very intuitive result: as the sampling frequency of one sensor decreases, another sensor must sample faster to guarantee convergence of the estimated state to the real state. Simulations of a quadrotor with 12 states and 9 measurements show the effectiveness of the approach.

Nonlinear Observer design for Systems with Sampled Measurements: An LPV Approach

The aim of this work is to propose a design methodology of observers for a class of Lipschitz nonlinear dynamical systems with sampled measurements by using the differential mean value theorem (DMVT) which allows us to transform the nonlinear part of the estimation error dynamics into a linear parameter varying (LPV) system. The designed observer must ensure the stability of the estimation error subject to a sampled measurements. An LMI-based minimization problem is provided to ensure the stability and the existence of the observer using Lyapunov theory. Thus, the measurements sampling period is included in the LMI as a decision parameter. Indeed, this allows to widen the sampling period as much as possible, which helps optimization of energy consumption while guaranteeing the convergence of the observer. Finally, to illustrate the performance of the proposed methodology, a numerical example is presented.

Asymptotic and tracking guarantees in interval observer design for systems with unmeasured polytopic nonlinearities

For a class of nonlinear systems subject to disturbances, an observer is proposed to estimate time-varying intervals in which their state variables are guaranteed to stay all the time. The objective is to effectively deal with nonlinearities depending on unmeasured variables which have been usually treated as uncertainty in observer design. Focusing on nonlinearities in a polytopic form, this paper shows how an interval observer can replace nonlinearities in unmeasured variables by nonlinearities in estimated intervals. Theoretical guarantees and simulation comparisons are presented to demonstrate that the use of interval-dependent nonlinearities gives better estimates than the use of an overbounding observer.

Feedback Linearization Kalman Observer Based Sliding Mode Control for Semi-Active Suspension Systems

In this paper, a new approach for an observer based controller for semi-active suspension systems is presented. The observer part is a feedback linearization Kalman filter which is based on differential geometry. The original nonlinear system is transferred to a linear system by certain coordinate transfer after the verification of observability and solvability of the system observer design problem under certain sensor placement configurations. Then, a linear Kalman filter algorithm can be applied to the linearized system. The state information can be obtained through an inverse coordinate transfer of the estimation results of the linear Kalman filter. The observer is verified by a simulation test under different road profiles, and a comparison between the designed observer and extended Kalman filter shows that the feedback linearization Kalman filter has better performance. A model reference sliding mode controller based on the estimation results of the observer is also proposed. A rig test system for the semi-active suspension system is implemented, and, both the designed observer and controller are verified through the rig test. Experimental results show that the proposed new approach for semi-active suspension control can significantly improve vehicle ride comfort with common and low-cost sensors.

Higher Order Nonlinear Complementary Filtering on Lie Groups

Nonlinear observer design for systems whose state-space evolves on Lie groups is considered. The proposed method is similar to previously developed nonlinear observers in that it involves propagating the state estimate using a process model and corrects the propagated state estimate using an innovation term on the tangent space of the Lie group. In the proposed method, the innovation term is constructed by passing the gradient of an invariant cost function, resolved in a basis of the tangent space, through a linear time-invariant system. The introduction of the linear system completes the extension of linear complementary filters to nonlinear Lie group observers by allowing higher order filtering. In practice, the proposed method allows for greater design freedom and, with the appropriate selection of the linear filter, the ability to filter bias and noise over specific bandwidths. A disturbance observer that accounts for constant and harmonic disturbances in group velocity measurements is also considered. Local asymptotic stability about the desired equilibrium point is demonstrated. A numerical example that demonstrates the desirable properties of the observer is presented in the context of pose estimation.

A New Observer Design for Systems in Presence of Time-varying Delayed Output Measurements

This paper presents a state observer for linear systems and Lipschitz nonlinear systems with delayed output measurements, which are affected by a known and bounded time-varying delay. The structure of the proposed observer is based on a proportional-integral term, which allows to compensate the time-varying delay. The observer gain depends on the maximum bounded delay. This gain is computed by a Linear Matrix Inequality (LMI) approach. The observer exhibits good performance for state estimation of the system despite the presence of significantly long delay. A Lyapunov-Krasovskii functional is used to prove the asymptotical convergence to zero of the observation error. This observer is applied to the case of systems with time-varying delay whose dynamic is described by a piecewise differentiable function. Examples and numerical simulations are provided in order to support the validity of the main results.

New Designs of Linear Observers and Observer-Based Controllers for Singularly Perturbed Linear Systems

This paper considers the design of observers for systems with slow and fast modes. The existing methods are able to design independent slow and fast observers with $O(\epsilon)$ accuracy only, where $\epsilon$ is a small positive singular perturbation parameter that indicates separation of system state-space variables into slow and fast. In this paper, the design of independent slow and fast reduced-order observers was performed with very high accuracy $O(\epsilon ^{i}), i=2,3,....$ . The results obtained are extended to the design of corresponding observer driven controllers. The design allows complete time-scale separation for both the observer and controller through the decomposition into slow and fast time scales. This method reduces both offline and online computations. The effectiveness of the new methods is demonstrated through both theoretical and simulation results.

Sliding Mode Observer Design for a Parabolic PDE in the Presence of Unknown Inputs

AbstractThis paper considers observer design for systems modeled by linear partial differential equations (PDEs) of parabolic type, which may be subject to unknown inputs. The system is assumed to have only one spatial dimension, over which it is discretised to obtain what is referred to as the lattice system, which is a set of linear time invariant (LTI) ordinary differential equations (ODEs) having a canonical Toeplitz‐like structure with a specific sparsity pattern. This lattice structure is shown to be particularly appropriate for step‐by‐step sliding mode observer design that can reconstruct the state estimates at the points of discretisation and estimate the unknown input. Simulation results for both stable and unstable PDEs show that accurate state estimates can be provided at the points of discretisation. An approach to reconstruct the unknown input is demonstrated.

Robust observer design under measurement noise

We prove new results on robust observer design for systems with noisy measurement and bounded trajectories. A state observer is designed by dominating the incrementally homogeneous nonlinearities of the observation error system with its linear approximation, while gain adaptation and incremental observability guarantee an asymptotic upper bound for the estimation error depending on the limsup of the norm of the measuremen noise. The gain adaptation is implemented as the output of a stable filter using the squared norm of the measured output estimation error and the mismatch between each estimate and its saturated value.

Robust observer design under measurement noise with gain adaptation and saturated estimates

We use incremental homogeneity, gain adaptation and incremental observability for proving new results on robust observer design for systems with noisy measurement and bounded trajectories. A state observer is designed by dominating the incrementally homogeneous nonlinearities of the observation error system with its linear approximation, while gain adaptation and incremental observability guarantee an asymptotic upper bound for the estimation error depending on the limsup of the norm of the measurement noise. A characteristic and innovative feature of this observer is the mixed low/high-gain structure in combination with saturated state estimates and dynamically tuned gains and saturation levels. The gain adaptation is implemented as the output of a stable filter using the squared norm of the measured output estimation error and the mismatch between each estimate and its saturated value.