Research Article | Open Access
Mohamed Zerrougui, Latifa Boutat-Baddas, Mohamed Darouach, " Observers Design for a Class of Continuous Time Nonlinear Singular Systems", Journal of Control Science and Engineering, vol. 2011, Article ID 967571, 8 pages, 2011. https://doi.org/10.1155/2011/967571
Observers Design for a Class of Continuous Time Nonlinear Singular Systems
This paper considers the problem of observers design for a class of Lipschitz continuous nonlinear singular systems. The method is based on the parameterization of the solution of the generalized Sylvester equations obtained from the estimation errors. Sufficient conditions for the existence of the observers which guarantee stability and the worst case observers error energy over all bounded energy disturbances is minimized are given. The approach also unifies the full-order, the reduced-order, and the minimal-order observers design. The solutions are obtained through linear matrix inequalities (LMIs) formulation. A numerical example is given to illustrate our results.
Observers design for nonlinear systems has been a very active field during the last two decades. This is due to the fact that a state estimation is generally required for the control when all states of the system are not available. The observers are also used in the monitoring and fault diagnosis. For standard nonlinear systems, there exist several approaches for the observers design including one based on coordinate transformations which lead to a linear error dynamics [1–3] and one where the problem of the observers design can be treated without the need of these transformations . An important class of standard nonlinear systems, the global Lipschitz, was considered by the authors in [5, 6], where the existence conditions for the observers are presented and constructive design methods were given for full-order and reduced-order cases.
On the other hand, singular systems (known as generalized, descriptor, or differential algebraic (DA) systems) describe a large class of systems. They are encountered in chemical and mineral industries; for example, the dynamic balances of mass and energy are described by differential equations, while thermodynamic equilibrium relations constitute additional algebraic constraints. The problem of the state estimation for these practical applications arises in data reconciliation, for example, . Singular systems are also frequently encountered in electronic and economics . In recent years a great deal of work has been devoted to the analysis and design techniques for singular systems [9–11]. On the other hand, the problem of observer design for linear systems has been greatly treated for the standard and singular systems with or without unknown inputs (see [12–14] and references therein). In , extension to observers design for Lipschitz singular systems has been presented; however, the observer considered has a singular system form. Recently a new method for the observers design is presented for a class of singular systems, where the nonlinearity is assumed to be composed of a Lipschitz one and an arbitrary one; the latter can be considered as an unknown disturbance. The approach is based on the parameterization of the generalized Sylvester equations solutions and unifies the design of full-, reduced-, and minimal-order observers the observer presented is causal and has a standard system form . However, only the case where the model and the measurement are free from noises was considered.
The state estimation problem for linear singular systems in presence of noises has been the subject of several studies in the past decades. We can distinguish two approaches, the Kalman observering approach and approach. In the Kalman observering, the system and the measurement noises are assumed to be Gaussian with known statistics [17–19]. When the noises are arbitrary signals with bounded energy, the observering permits to guarantee a noise attenuation level . Recently, a number of papers have appeared that deal with the observering for singular systems; see, for example, [21–23] and references therein. In all these works only full- or reduced-order observers were presented for the square singular systems.
In this paper, we consider the observers design problem for a class of Lipschitz nonlinear singular systems. The approach extends the work  to the case where the model and the measurement are affected by noises. Sufficient conditions in terms of LMIs are given for this problem. The method is more general than the one considered in  since it assumes only the impulse observability of the linear part. It also unifies the design for full-order, reduced-order, and minimal-order observers. A numerical example is given to illustrate our results.
2. Problem Formulation
Consider the following nonlinear singular system: with the initial state , where is the semi state vector, is the known input, is the disturbance vector containing both system and measurement noises, and is the measurement output. Matrix , and, when , matrix is singular. Matrices , , , , and . The nonlinearity verifies the Lipschitz constraints: where is a known Lipschitz constant and matrix is real with appropriate dimension.
Let be the rank of the matrix , and let be a full row rank matrix such that , with . Then, from (1), we obtain In the sequel we assume the following.
Assumption 1. Consider .
Before presenting our main results, we can make the following remarks.
Remark 2. When , Assumption 1 is exactly the impulse observability of the linear singular system .
Remark 3. Condition is more general than condition considered in . In fact when the matrix is of full row rank, matrix and Assumption 1 becomes which is more restrictive than the impulse observability condition.
Now, let us consider the following reduced-order observer for system (1): with the initial condition . Vector represents the state vector of the observer, and is the estimate of . Matrices , , , , , , and are unknown matrices of appropriate dimensions, which must be determined such that,(1)for , the error asymptotically converges to 0,(2)for , we solve the .
Let the error between and be then we obtain the following dynamics of : where .
Now the problem of the observer design is reduced to find the matrices , and such that (8) is satisfied and the worst case observers error energy over all bounded energy disturbances is minimized.
Remark 4. The observer given in (5) is more general than that presented in ; this can be seen from the fact that, for , and , we obtain the observer given in . As in  this observer is general, and its design unifies the full-order (), the reduced-order (), and the minimal-order observers design.
3. Main Results
In this section we will present the observers design. The observer design problem can be formulated as follows: given the nonlinear singular system (1)-(2) and a prescribed level of noise , find a suitable observer of the form (5), such that,(1)for , estimation error (11) converges asymptotically to 0, that is, as ,(2)under the zero initial condition, the error satisfies for any , where is the prescribed constant.
3.1. Stability Analysis
Before giving the observers design method for system (1)-(2), let us deal with the stability analysis problem and derive a sufficient condition, in a strict LMI form, for system (12) to be asymptotically stable for . The following lemma gives this condition.
Lemma 5. For , system (12) is asymptotically stable, if there exists a matrix , such that the following LMIs are satisfied: with .
Proof. For stability analysis we construct the following Lyapunov candidate function:
with . The derivative of along the solution of (12), for , is given by
Let and be two vectors of appropriate dimensions, then for all scalar the following inequality holds:
By using (17), it is not difficult to check that
In this case we have . On the other hand, by using the Lipschitz conditions and the fact that , we have , where . Then we obtain the following inequality:
and , if .
Using the Schur complement we obtain (14), then Lemma 5 is proved.
From this lemma, one can see that the stability conditions (13) and (14) are nonstrict LMIs, which contain equality constraints; this may result in numerical problems when checking such conditions. Therefore, strict LMIs conditions are more desirable than nonstrict ones from the numerical point of view. The following lemma presents the stability conditions in a strict LMI formulation.
Lemma 6. For , system (12) is asymptotically stable if there exist a positive definite matrix and matrices , and and a scalar such that the following LMI is satisfied:
Proof. Let , where is the orthogonal matrix of satisfying and , and , and . It is easy to see that , since . In this case (13)-(14) reduce toOn the other hand by inserting the values of , and into (21) we obtain (20).
3.2. Observes Design
In this section we shall present the observer design. The following lemma gives the sufficient conditions for system (12) to be stable for and for .
Lemma 8. The error given by (11) is asymptotically stable for and for , if there exist a positive definite matrix and matrices and scalars and such that the following LMI is satisfied:
Proof. Consider the following Lyapunov candidate function , with . Then we have
From (17) we have the following inequality: Then becomes
On the other hand, we have , and we know that then , where .Then Let , then from (27) and (12) we obtain the following inequality: with where .
If , we obtain Integrating the two sides of this inequality gives or equivalently under zero initial conditions, we obtain which leads to Using Schur complement we obtain Let with we obtain And by substituting and by their values we obtain (23), which proves the lemma.
Remark 9. The results of Lemmas 6 and 8 are independent of the choice of matrix ; this can be seen from the fact that the general form of is , where is an arbitrary nonsingular matrix; in this case we have , where and , which shows that it suffices to choose .
Before giving the design method for the observer (5), let us consider (8) and let , where is an arbitrary matrix of appropriate dimension; they can be written as Equations (38) and (39) have a solution if and only if Now, from Assumption 1 and (40), we have Let be any full row rank matrix such that then there always exist matrices parameter and such that or equivalently Then, under Assumption 1, there exists a solution to (43) given by In this case matrices and are given by Also, under Assumption 1, the general solution to (38) is given by here and is an arbitrary matrix of appropriate dimension.
And also under Assumption 1 one solution to (39) is given by Now, define the following matrices: then we obtain where
Now, the observer design can be obtained from the following theorem.
Theorem 10. There exists an observer of the form (5) such that the error given by (10)-(11) is asymptotically stable for and for if there exist a symmetric positive definite matrix and matrices and scalars and such that the following LMIs are satisfied: where
4. Numerical Example
Let us consider the following continuous nonlinear singular system of the form (1) with and . The nonlinearity . For this system, the matrix . In this case it is easy to see that Assumption 1 is verified. We will design a reduced-order observer of dimension ; let , then . For , from Section 3 we obtain the following results: The observer is given by the following model:
In this paper a new method for the observers design for a class of Lipschitz nonlinear singular systems has been developed. The obtained results unify the observers design of full, reduced, and minimal orders. Sufficient conditions for the existence of these observers are given in terms of LMIs. The advantage of these LMIs conditions is that they can be performed by using convex optimization techniques available in Matlab LMI toolbox, for example. A numerical example has been presented to show the applicability of our approach. The extension of our work to more general nonlinear singular systems is under study.
- J. Krener and W. Respondek, “Nonlinear observer design with linearizable error dynamics,” SIAM Journal on Control and Optimization, vol. 23, no. 2, pp. 197–216, 1985.
- H. Keller, “Nonlinear observer design by transformation into generalized observer canonical form,” International Journal of Control, vol. 46, no. 6, pp. 1915–1930, 1987.
- X. H. Xia and W. Gao, “Nonlinear observer design by observer error linearization,” SIAM Journal on Control and Optimization, vol. 27, no. 1, pp. 199–216, 1989.
- M. Arcak and P. Kokotovi, “Nonlinear observers: a circle criterion design and robustness analysis,” Automatica, vol. 37, no. 12, pp. 1923–1930, 2001.
- R. Rajamani, “Observers for Lipschitz nonlinear systems,” IEEE Transactions on Automatic Control, vol. 43, no. 3, pp. 397–401, 1998.
- F. Zhu and Z. Han, “A note on observers for Lipschitz nonlinear systems,” IEEE Transactions on Automatic Control, vol. 47, no. 10, pp. 1751–1754, 2002.
- M. Darouach and M. Zasadzinski, “Data reconciliation in generalized linear dynamic systems,” AICHE Journal, vol. 37, no. 2, pp. 193–201, 1991.
- D. J. Luenberger, “Non-linear descriptor systems,” Journal of Economic Dynamics and Control, vol. 1, no. 3, pp. 219–242, 1979.
- S. L. Campbell, Singular Systems of Differential Equations, vol. I, Pitman, San Francisco, Calif, USA, 1980.
- S. L. Campbell, Singular Systems of Differential Equations, vol. II, Pitman, San Francisco, Calif, USA, 1982.
- L. Dai, Singular Control Systems, Springer, New York, NY, USA, 1989.
- M. Darouach, M. Zasadzinski, and S. J. Xu, “Full-order observers for linear systems with unknown inputs,” IEEE Transactions on Automatic Control, vol. 39, no. 3, pp. 606–609, 1994.
- M. Darouach and M. Boutayeb, “Design of observers for descriptor systems,” IEEE Transactions on Automatic Control, vol. 40, no. 7, pp. 1323–1327, 1995.
- M. Darouach, M. Zasadzinski, and M. Hayar, “Reduced-order observer design for descriptor systems with unknown inputs,” IEEE Transactions on Automatic Control, vol. 41, no. 7, pp. 1068–1072, 1996.
- G. Lu and D. W. C. Ho, “Full-order and reduced-order observers for Lipschitz descriptor systems: the unified LMI approach,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 53, no. 7, pp. 563–567, 2006.
- M. Darouach and L. Boutat-Baddas, “Observers for a class of nonlinear singular systems,” IEEE Transactions on Automatic Control, vol. 53, no. 11, pp. 2627–2633, 2008.
- M. Darouach, M. Zasadzinski, and D. Mehdi, “State estimation of stochastic singular linear systems,” International Journal of Systems Science, vol. 24, pp. 345–354, 1993.
- R. Nikoukhah, A. Willsky, and B. Levy, “Kalman filtering and Riccati equations for descriptor systems,” IEEE Transactions on Automatic Control, vol. 37, no. 9, pp. 1325–1342, 1992.
- R. Nikoukhah, S. Campbell, and F. Delebecque, “Kalman filtering for general discrete-time linear systems,” IEEE Transactions on Automatic Control, vol. 34, no. 10, pp. 1829–1839, 1999.
- J. Watson and K. Grigoriadis, “Optimal unbiased filtering via linear matrix inequalities,” Systems and Control Letters, vol. 35, no. 2, pp. 111–118, 1998.
- S. Xu, J. Lam, and Y. Zou, “H∞ filtering for singular systems,” IEEE Transactions on Automatic Control, vol. 48, no. 12, pp. 2217–2222, 2003.
- S. Xu and J. Lam, “Reduced-orde H∞ observering for singular systems,” Systems & Control Letters, vol. 56, pp. 48–57, 2007.
- M. Darouach, “H∞ unbiased observering for descriptor systems via LMI,” IEEE Transactions on Automatic Control, vol. 54, no. 8, pp. 1966–1972, 2009.
Copyright © 2011 Mohamed Zerrougui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.