Observer design of descriptor nonlinear system with nonlinear outputs by using -optimality criterion
Introduction
Due to the absence of a general and systematic design procedure, observer design for nonlinear systems has attracted the interest of many researchers of automatic control field. Several methods have been established and various classes of nonlinear systems have been investigated in the literature [4], [15], [24], [29], [31], [34], [37], [42]. Each established method is applicable on a specific class of systems, where each class of systems is defined by some specific assumptions on the nonlinearity. We can mention the high-gain methodology developed for uniformly observable systems that can be transformed into a triangular form [15]; the sliding mode observers [1], [23] and LMI-based observers [44] used for systems with globally Lipschitz nonlinearities. There are also different techniques using normal forms based on nonlinear transformations bringing the original system into a linear Brunovsky canonical form [21]. This approach, called exact linearization error method, is very efficient, however, it requires some conservative assumptions and constrained by solving a set of partial differential equations. There are also state observers working on systems with general nonlinearities, like the famous extended Kalman filter [33] and the receding-horizon estimators [2]. However, these estimators are only locally convergent and their global stability is not proved and remains a challenge. Finally, a recent observer design techniques have been proposed in [6] for non Lipschitz triangular systems satisfying Hölder inequality.
If the observer design problem for nonlinear systems is not easy in general, it is much more complicated when we have to face nonlinear descriptor systems in the presence of nonlinear outputs and disturbances. Few results are established in the literature. Some design techniques are based directly on the analysis of the estimation error, which has a regular and standard form [34] and some others consist to introduce pseudo-estimation error with descriptor form [12]. The first technique is simple while the second one is more complicated and requires some rank conditions and needs to solve a set of Sylvester type algebraic equations.
Investigating state observers for descriptor systems has many objectives. Some of them can be summarized in the following list:
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Unknown Input Observers (UIO): In the UIO problem for systems with unknown inputs, the procedure consists generally in rewriting the system under a convenient descriptor form and then all the theory developed for descriptor systems can be applied straightforwardly [8], [14], [19], [35] and [39].
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Fault diagnosis: In the context of studying nonlinear systems in the presence of actuator and sensor faults, the fault detection and isolation problem is transformed to equivalently to estimation problem for a class of descriptor systems. For a sample of techniques, we refer the reader to [20], [22], [36], [38], and the references therein.
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Chaotic communication schemes: Estimation problem for descriptor nonlinear systems is often used in synchronization and encryption/decryption using chaotic communication systems. Indeed, using the observer-based synchronization, the confidential messages to recover are considered as unknown inputs and then the systems can be transformed into a suitable descriptor form [11], [25]. Particularly in this context, disturbances are always encountered in the used transmission channels.
In presence of disturbances, usually the estimation is done according to optimality criteria, such as performance, coupled criterion for robust and performant control, filter for fault detection, and criterion, which replace the in some situations, namely when the criterion is unachievable. On the other hand, in some application problems, the -optimality offers nicer properties compared to the -performance, as well explained in [3] and [7]. Indeed, the Sobolev space is interesting for several reasons: first, all nice properties of classical Lebesgue space are still satisfied by this space (Hilbertian structure, isometry with its dual); in particular, the gain of any linear time-invariant system remains the -norm of its transfer matrix. In addition to the nice properties reported in [3] and [7], the advantage of the -optimality criterion lies in the fact that not only the estimation error is upper bounded, but also its velocity. Indeed, the criterion does not provide any information on the velocity of the estimation error, while the norm contains the estimation error and its derivative. Hence, the observer design parameters computed using the synthesis method often lead to a better estimation, as is the case in numerical examples given in this paper.
In this paper, we propose a new LMI design method to handle the problem of observer design for nonlinear descriptor systems with nonlinear measurements by considering the presence of disturbances in both the dynamics and outputs. Due to the nice properties discussed above, we chose to investigate the -optimality criterion. On the other hand, studying this criterion will offer the opportunity for some users of real-world applicative systems to exploit it because it is rarely investigated in the literature. The only limitation of -optimality criterion is that it requires the disturbances to belong to the Sobolev space instead of the Lebesgue space in the case, where is a compact set representing the domain of definition of the disturbance. Compared to the literature, the proposed LMI conditions are less conservative and contain additional decision variables coming from the convenient application of the Young’s inequality. In addition, the investigated class of systems is more general than those investigated in the literature, for instance [40], for the same estimation problem.
To clarify the contributions of the proposed design methodology, we summarize the advantages of the paper in the following items:
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-optimality criterion provides an upper bound not only on the error, but on its velocity also. Indeed, the -norm of a function contains informations on the derivative of such a function, but the -norm do not contains any information on the derivative. Then, as can be seen in the numerical example, the estimation is better in the case.
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Considering the -optimality criterion instead of one, allows avoiding additional rank conditions as in [5], which may require additional constraints on the number of measurements.
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In the context of systems with unknown inputs, considering the -optimality criterion instead of one developed in [5], allows estimating simultaneously the state and the unknown inputs, contrarily to [5] where the unknown inputs are estimated by using the derivative of the estimated state.
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The proposed design method allows handling a more general class of systems than that studied in [40], by considering nonlinearities in the output measurements.
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The proposed LMI conditions are less conservative than those established in [18]. Indeed, due to a convenient use of Young’s inequality, the LMIs given in this paper contain additional decision variables, which render the LMIs in [18] as a particular solution. This leads to better performances because the proposed LMIs offer a larger set of solutions.
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Combining a simple assumption (Assumption 1) with the -based analysis, we obtain a simple way to avoid several Sylvester equations and computation of complicated pseudo-inverse matrices as in [12]. Indeed, these Sylvester equations often require a set of additional rank conditions.
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In case of systems with unknown inputs, the proposed design method provides a simultaneous estimation of the system states and the unknown inputs, contrarily to the technique proposed in [30], where a dynamic model inversion has been used to compute the unknown inputs from output measurements. Such a method will fail for systems with disturbances in the output measurements, while the proposed one works successfully.
The rest of the paper is organized as follows. Section 2 provides some preliminaries followed by a detailed formulation of the considered estimation problem and introduces some necessary assumptions and definitions. Section 3 is devoted to the main contribution of this paper, which consists in proposing a new LMI design methodology. Two numerical examples are given in Section 4 to show the effectiveness and superiority of the proposed design technique compared to the method. Finally, we end the paper by a conclusion in Section 5.
Section snippets
Preliminaries and problem formulation
This section is devoted to some preliminaries, and provide clearly a formulation of the investigated problem.
New LMI design conditions
This section is devoted to the main contributions of the paper. A novel LMI design technique will be proposed.
First the following assumption is required in order construct an observer of the form (5) for the system (3). Assumption 1 The following condition is fulfilled:
Condition (12) implies Remark 2 Usually, the standard assumption is the rank condition (4) guaranteeing (8), which is important for the synthesis. In fact, since we have not the derivative of the entire state ξ(t), then
Numerical examples
This section is devoted to the numerical evaluation of the proposed design methodology. We provide two numerical examples. Through the first one, we will provide comparisons with the LMI conditions established in [18]. The second one aims to show the performance of the -based estimation compared to the -based performance criterion in the presence of nonlinearity in the output measurements.
Conclusion
We proposed in this paper a new LMI-based observer design technique to achieve the -convergence criterion for a class of Lipschitz systems with nonlinearities in both the dynamics process and the output measurements. Due to some convenient assumptions and Young’s inequality, new LMI conditions have been proposed. The -convergence context has been motivated by the fact that in some practical situations, the derivative of the disturbances are unavoidable from the Lyapunov analysis. This
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