Design of observers for nonlinear systems with H ∞ performance analysis

International audience

In this paper, we investigate the impulse effect in nonlinear singular system, since the existence of the nonlinear dynamics, we use the Differential Mean Value Theorem based method to transfer the nonlinear singular system into a linear parameter varying system. It should be noted that Differential Mean Value Theorem is an identical transmission, thus, for the nonlinear system, if we transfer the nonlinear system into a linear system via differential mean value theorem, the properties of the original system will not be changed. Based on this point, we try to analyze the transferred singular system by some existence method which applied to the linear singular system.

In this note, we investigate a new approach for observer synthesis of nonlinear time delay systems. This approach is based on the differential mean value theorem (DMVT). Using the Lyapunov-Krasovskii functionals and the DMVT, we give new sufficient synthesis conditions for the observer design problem. These conditions are formulated as matrix inequalities which become linear whenever an unknown scalar variable is chosen a priori

In this paper, we present an adaptive observer for nonlinear systems that include unknown constant parameters and are not necessarily observable. Sufficient conditions are given for a nonlinear system to be transformed by state-space change of coordinates into an adaptive observer canonical form. Once a nonlinear system is transformed into the proposed adaptive observer canonical form, an adaptive observer can be designed under the assumption that a certain system is strictly positive real. An illustrative example is included to show the effectiveness of the proposed method.

Adaptive observer for a nonlinear system with partially unknown state matrix and delayed measurements

This paper presents an adaptive observer design for nonlinear systems that have parametric uncertainties in the unmeasured state dynamics. Without persistency of excitation (PE) the convergence of the state estimation error to zero is proved. Under PE conditions, uniform global asymptotic stability of the origin of the state and parameter error is shown. Furthermore, if the regressor is sufficiently smooth, uniform local asymptotic stability of the origin is proved. Two simulation examples are shown to illustrate the results.

ABSTRACTThis paper presents an observer design technique for a newly developed non-intrusive position estimation system based on magnetic sensors. Typically, the magnetic field of an object as a function of position needs to be represented by a highly nonlinear measurement equation. Previous results on observer design for nonlinear systems have mostly assumed that the measurement equation is linear, even if the process dynamics are nonlinear. Hence, a new nonlinear observer design method for a Wiener system composed of a linear process model together with a nonlinear measurement equation is developed in this paper. First, the design of a two degree-of-freedom nonlinear observer is proposed that relies on a Lure system representation of the observer error dynamics. To improve the performance in the presence of parametric uncertainty in the measurement model, the nonlinear observer is augmented to estimate both the state and unknown parameters simultaneously. A rigorous nonlinear observability analysis is also presented to show that a dual sensor configuration is a sufficient and necessary condition for simultaneous state and parameter estimation. Finally, the developed observer design technique is applied to non-intrusive position estimation of the piston inside a pneumatic cylinder. Experimental results show that both position and unknown parameters can be reliably estimated in this application.

This paper investigates the problem of observer design for nonlinear systems when only a transformed output is available for measurement. More precisely, we first assume that there exists an observer design that requires the output of the system. We then suppose that this disturbed output is only available under a transformation and we propose a new observer design. We establish small-gain type conditions ensuring the asymptotic convergence of the proposed observer and we provide an illustration example.

Observer design for nonlinear systems

A unified [formula omitted] adaptive observer synthesis method for a class of systems with both Lipschitz and monotone nonlinearities

A relation between the output regulation and the observer design for nonlinear systems

We present a functional observer design method for a class of nonlinear systems with output nonlinearities and incremental quadratic constraints. The system under consideration includes a wider class of nonlinear systems, such as, Lipschitz, one-sided Lipschitz, nondecreasing nonlinearity system as special cases. Based on Moore–Penrose pseudo-inverse and linear matrix inequality theory, the existence conditions of gain matrices of a functional observer are obtained. Moreover, exponential convergence of the estimate errors is achieved. Through the Lorenz system, it is shown that the proposed approach provides a smaller order of observer than that of the conventional approaches, while maintaining satisfactory performances.

In this study, differential stability is introduced for non-linear systems, and this concept is further exploited in reduced-order observer design for non-linear systems with non-linearities of unmeasured state variables, a more general class of non-linear systems than the systems with linear observer errors. It has been shown that if the dynamics of unmeasured state variables under a state transformation is differentially stable, a reduced-order observer can be designed to produce asymptotically convergent estimates of the unmeasured state variables. A systematic design method is then introduced for a class of multi-output non-linear systems. For such a system, a non-linear term of the unmeasured state variables enter the system through a coupling matrix. It is found that a reduced-order observer can be designed if the linear part with the coupling matrix as the input matrix has no unstable invariant zeros. A further exploitation is presented for a class of single-output non-linear systems with non-linearity of unmeasured state variables. In this case, the coupling vector is allowed to be a vector field which depends on the system output.

The problem of adaptive robust state observer design is considered for a class of uncertain nonlinear systems with delayed state perturbations. It is supposed that the upper bound of the nonlinearity and uncertainty, including delayed states, is a linear function of some parameters which are still assumed to be unknown. Here, it is not required that the nonlinear terms including delayed states are linear norm-bounded in the states. An improved adaptation law with sigma-modification is employed to estimate the unknown parameters. Then, by making use of the updated values of these unknown parameters a class of memoryless adaptive robust state observers is proposed for uncertain nonlinear time-delay systems. It is also shown that by employing the proposed adaptive robust state observer, the observation error between the observer state estimate and the true state can converge asymptotically to zero in the presence of significant uncertainties and time delays. Finally, a numerical example is given to demonstrate the validity of the results.

ABSTRACTThis paper provides a procedure for designing a sliding mode observer for a nonlinear system in the presence of Gaussian input disturbances and sensor noises. The paper first proposes a novel candidate for Lyapunov stability, termed a Time-averaged Lyapunov (TAL) function. The TAL averages the Lyapunov analysis over a small finite time interval, allowing for intuitive analysis of noises and disturbance. The paper then provides the necessary and sufficient condition for the design of the sliding observer gainsusing the TAL in the form of Linear Matrix Inequality (LMI). The LMIs can then be explicitly solved offline using commercial LMI solvers. The paper compares the LMIs for the two observer designs to demonstrate the design of the sliding mode observer using TAL can greatly enhance the scope of observer design for nonlinear systems.