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Journal of Control Science and Engineering
VolumeΒ 2011, Article IDΒ 967571, 8 pages
http://dx.doi.org/10.1155/2011/967571
Research Article

𝐻∞ Observers Design for a Class of Continuous Time Nonlinear Singular Systems

CRAN-CNRS, UHP NancyI, IUT de Longwy 186, rue de Lorraine, 54400 Cosnes-et-Romain, France

Received 29 April 2011; Revised 13 July 2011; Accepted 17 July 2011

Academic Editor: MohamedΒ Zribi

Copyright Β© 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.

Abstract

This paper considers the problem of 𝐻∞ 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.

1. Introduction

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 [4]. 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, [7]. Singular systems are also frequently encountered in electronic and economics [8]. 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 [15], 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 [16]. 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 𝐻∞ 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 𝐻∞ observering permits to guarantee a noise attenuation level [20]. Recently, a number of papers have appeared that deal with the 𝐻∞ 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 𝐻∞ observers design problem for a class of Lipschitz nonlinear singular systems. The approach extends the work [16] to the case where the model and the measurement are affected by noises. Sufficient conditions in terms of LMIs are given for this 𝐻∞ problem. The method is more general than the one considered in [16] 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: 𝐸̇π‘₯(𝑑)=𝐴π‘₯(𝑑)+𝐡𝑒(𝑑)+𝐷𝑓𝑑,𝐹𝐿π‘₯,𝑒+𝐷1𝑀(𝑑),(1)𝑦(𝑑)=𝐢π‘₯(𝑑)+𝐷2𝑀(𝑑),(2) with the initial state π‘₯(0)=π‘₯0, where π‘₯(𝑑)βˆˆβ„π‘› is the semi state vector, 𝑒(𝑑)βˆˆβ„π‘š is the known input, 𝑀(𝑑)βˆˆβ„π‘›π‘€ is the disturbance vector containing both system and measurement noises, and 𝑦(𝑑)βˆˆβ„π‘ is the measurement output. Matrix πΈβˆˆβ„π‘›1×𝑛, and, when 𝑛1=𝑛, matrix 𝐸 is singular. Matrices π΄βˆˆβ„π‘›1×𝑛, π΅βˆˆβ„π‘›1Γ—π‘š, πΆβˆˆβ„π‘Γ—π‘›, π·βˆˆβ„π‘›1×𝑛𝑓, 𝐷1βˆˆβ„π‘›1×𝑛𝑀 and 𝐷2βˆˆβ„π‘Γ—π‘›π‘€. The nonlinearity 𝑓(𝑑,𝐹𝐿π‘₯,𝑒) verifies the Lipschitz constraints: ‖‖𝑓𝑑,𝐹𝐿π‘₯1ξ€Έξ€·,π‘’βˆ’π‘“π‘‘,𝐹𝐿π‘₯2‖‖‖‖𝐹,π‘’β‰€πœ†πΏξ€·π‘₯1βˆ’π‘₯2ξ€Έβ€–β€–ξ€Έ,(3) where πœ† is a known Lipschitz constant and matrix 𝐹𝐿 is real with appropriate dimension.

Let π‘Ÿ be the rank of the matrix 𝐸, and let Ξ¦βˆˆβ„π‘Ÿ1×𝑛1 be a full row rank matrix such that Φ𝐸=0, with π‘Ÿ1=𝑛1βˆ’π‘Ÿ. Then, from (1), we obtain Φ𝐴π‘₯(𝑑)+Φ𝐷1𝑀(𝑑)+Φ𝐷𝑓𝑑,𝐹𝐿π‘₯,𝑒=βˆ’Ξ¦π΅π‘’(𝑑).(4) In the sequel we assume the following.

Assumption 1. Consider rank𝐸𝐢Φ𝐴=𝑛.

Before presenting our main results, we can make the following remarks.

Remark 2. When 𝑛1=𝑛, Assumption 1 is exactly the impulse observability of the linear singular system (𝐸,𝐴,𝐡,𝐢).

Remark 3. Condition Φ𝐸=0 is more general than condition Ξ¦[𝐸𝐷]=0 considered in [16]. In fact when the matrix [𝐸𝐷] is of full row rank, matrix Ξ¦=0 and Assumption 1 becomes ξ€Ίrank𝐸𝐢=𝑛 which is more restrictive than the impulse observability condition.

Now, let us consider the following reduced-order observer for system (1): Μ‡πœξ€·(𝑑)=π‘πœ(𝑑)+𝐽𝑦(𝑑)+𝐻𝑒(𝑑)+𝑇𝐷𝑓𝑑,𝐹𝐿,ξ€·Μ‚π‘₯,𝑒̂π‘₯(𝑑)=π‘ƒπœ(𝑑)βˆ’π‘„Ξ¦π΅π‘’(𝑑)+𝐹𝑦(𝑑)βˆ’π‘„Ξ¦π·π‘“π‘‘,𝐹𝐿,Μ‚π‘₯,𝑒(5) with the initial condition 𝜁(0)=𝜁0. Vector 𝜁(𝑑)βˆˆβ„π‘ž represents the state vector of the observer, and Μ‚π‘₯(𝑑)βˆˆβ„π‘› is the estimate of π‘₯(𝑑). Matrices 𝑁, 𝐽, 𝑇, 𝐻, 𝑃, 𝑄, and 𝐹 are unknown matrices of appropriate dimensions, which must be determined such that,(1)for 𝑀(𝑑)=0, the error 𝑒(𝑑)=Μ‚π‘₯(𝑑)βˆ’π‘₯(𝑑) asymptotically converges to 0,(2)for 𝑀(𝑑)β‰ 0, we solve the minsupπœ”βˆˆπΏ2βˆ’{0}(‖𝑒‖𝐿2/β€–πœ”β€–πΏ2).

Let the error between 𝜁(𝑑) and 𝑇𝐸π‘₯(𝑑) be πœ€(𝑑)=𝜁(𝑑)βˆ’π‘‡πΈπ‘₯(𝑑),(6) then we obtain the following dynamics of πœ€(𝑑):Μ‡ξ€·Μ‡πœ€(𝑑)=𝜁(𝑑)βˆ’π‘‡πΈΜ‡π‘₯(𝑑)=π‘πœ€+(π‘π‘‡πΈβˆ’π‘‡π΄+𝐽𝐢)π‘₯(𝑑)+(π»βˆ’π‘‡π΅)𝑒(𝑑)+𝑇𝐷Δ𝑓+𝐽𝐷2βˆ’π‘‡π·1ξ€Έξ‚ƒξ‚„βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ€·π‘€(𝑑),Μ‚π‘₯(𝑑)=π‘ƒπœ€(𝑑)+𝑃𝑄𝐹𝑇𝐸Φ𝐴π‘₯(𝑑)βˆ’π‘„Ξ¦π·Ξ”π‘“+𝑄Φ𝐷1+𝐹𝐷2𝑀(𝑑),(7) where Δ𝑓=𝑓(𝑑,𝐹𝐿̂π‘₯(𝑑),𝑒)βˆ’π‘“(𝑑,𝐹𝐿π‘₯(𝑑),𝑒).

Under Assumption 1, if there exists a matrix 𝑇 such that ,π‘π‘‡πΈβˆ’π‘‡π΄+𝐽𝐢=0(8)ξ‚ƒξ‚„βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦𝐻=𝑇𝐡,𝑃𝑄𝐹𝑇𝐸Φ𝐴=𝐼,(9) then (7) becomes Μ‡πœ€(𝑑)=π‘πœ€(𝑑)+𝑇𝐷Δ𝑓+πœ‘1𝑀(𝑑),(10)𝑒(𝑑)=π‘ƒπœ€(𝑑)βˆ’π‘„Ξ¦π·Ξ”π‘“+πœ‘2𝑀(𝑑),(11) where πœ‘1=𝐽𝐷2βˆ’π‘‡π·1 and πœ‘2=𝑄Φ𝐷1+𝐹𝐷2.

Now the problem of the 𝐻∞ observer design is reduced to find the matrices 𝑁,𝐽,𝐻,𝑃,𝑄,𝐹, and 𝑇 such that (8) is satisfied and the worst case observers error energy over all bounded energy disturbances 𝑀(𝑑) is minimized.

Remark 4. The observer given in (5) is more general than that presented in [16]; this can be seen from the fact that, for Φ𝐷=0, and 𝑀(𝑑)=0, we obtain the observer given in [16]. As in [16] this observer is general, and its design unifies the full-order (π‘ž=𝑛), the reduced-order (π‘ž=π‘›βˆ’π‘), and the minimal-order observers design.

3. Main Results

In this section we will present the 𝐻∞ observers design. The 𝐻∞ observer design problem can be formulated as follows: given the nonlinear singular system (1)-(2) and a prescribed level of noise 𝛾>0, find a suitable observer of the form (5), such that,(1)for 𝑀(𝑑)=0, estimation error (11) converges asymptotically to 0, that is, 𝑒(𝑑)β†’0 as π‘‘β†’βˆž,(2)under the zero initial condition, the error 𝑒(𝑑) satisfies ‖𝑒(𝑑)‖𝐿2<𝛾‖𝑀(𝑑)‖𝐿2 for any 𝑀(𝑑)∈𝐿2βˆ’{0}, where 𝛾>0 is the prescribed constant.

Now, the dynamics error (10)-(11) can be written in a singular form as π”ΌΜ‡πœ‰(𝑑)=π”Έπœ‰(𝑑)+𝔹Δ𝑓+𝔻𝑀(𝑑),(12) where [πœ‰(𝑑)=πœ–π‘’], 𝔼=𝐼000ξ€», 𝔸=𝑁0π‘ƒβˆ’πΌξ€», 𝔹=π‘‡π·βˆ’π‘„Ξ¦π·ξ€», and 𝐷=πœ‘1πœ‘2ξ€». Then, the error 𝑒(𝑑)β†’0 as π‘‘β†’βˆž, for 𝑀(𝑑)=0 if system (12) is asymptotically stable.

3.1. Stability Analysis

Before giving the 𝐻∞ 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 𝑀(𝑑)=0. The following lemma gives this condition.

Lemma 5. For 𝑀(𝑑)=0, system (12) is asymptotically stable, if there exists a matrix π‘Œ, such that the following LMIs are satisfied: π”Όπ‘‡π‘Œ=π‘Œπ‘‡,𝔼⩾0(13)βŽ‘βŽ’βŽ’βŽ£π”Έπ‘‡π‘Œ+π‘Œπ‘‡π”Έ+πœ‡πœŒπ‘Œπ‘‡π”Ήπ”Ήπ‘‡βŽ€βŽ₯βŽ₯βŽ¦π‘Œβˆ’πœ‡πΌ<0(14) with ξ‚ƒπœŒ=000πœ†21𝐹𝑇𝐿𝐹𝐿.

Proof. For stability analysis we construct the following Lyapunov candidate function: 𝑉(𝑑)=πœ‰π‘‡(𝑑)π”Όπ‘‡π‘Œπœ‰(𝑑)=πœ‰π‘‡(𝑑)π‘Œπ‘‡π”Όπœ‰(𝑑)(15) with π”Όπ‘‡π‘Œ=π‘Œπ‘‡π”Όβ©Ύ0. The derivative of 𝑉(𝑑) along the solution of (12), for 𝑀(𝑑)=0, is given by Μ‡π‘‰Μ‡πœ‰(𝑑)=𝑇(𝑑)π”Όπ‘‡π‘Œπœ‰(𝑑)+πœ‰π‘‡(𝑑)π‘Œπ‘‡π”ΌΜ‡πœ‰(𝑑)=(π”Έπœ‰(𝑑)+𝔹Δ𝑓)π‘‡π‘Œπœ‰(𝑑)+πœ‰π‘‡(𝑑)π‘Œπ‘‡(π”Έπœ‰(𝑑)+𝔹Δ𝑓)=πœ‰π‘‡ξ€·π”Έ(𝑑)π‘‡π‘Œ+π‘Œπ‘‡π”Έξ€Έπœ‰(𝑑)+Ξ”π‘“π‘‡π”Ήπ‘‡π‘Œπœ‰(𝑑)+πœ‰π‘‡(𝑑)π‘Œπ‘‡π”ΉΞ”π‘“.(16) Let 𝑒 and 𝑣 be two vectors of appropriate dimensions, then for all scalar πœ‡>0 the following inequality holds: 𝑒𝑇𝑣+π‘£π‘‡π‘’β©½πœ‡π‘’π‘‡1𝑒+πœ‡π‘£π‘‡π‘£.(17) By using (17), it is not difficult to check that Δ𝑓𝑇(𝔹)π‘‡π‘Œπœ€(𝑑)+πœ€π‘‡(𝑑)π‘Œπ‘‡(β©½1𝔹)Ξ”π‘“πœ‡πœ€π‘‡(𝑑)π‘Œπ‘‡π”Ήπ”Ήπ‘‡π‘Œπœ€(𝑑)+πœ‡Ξ”β„±π‘‡Ξ”π‘“.(18) In this case we have ̇𝑉(𝑑)β©½πœ‰π‘‡(𝑑)(π”Έπ‘‡π‘Œ+π‘Œπ‘‡π”Έ)πœ‰(𝑑)+πœ‡Ξ”π‘“π‘‡Ξ”π‘“+(1/πœ‡)πœ‰π‘‡(𝑑)π‘Œπ‘‡π”Ήπ”Ήπ‘‡π‘Œπœ‰(𝑑). On the other hand, by using the Lipschitz conditions and the fact that [𝑒(𝑑)=0𝐼]πœ‰(𝑑), we have πœ‡Ξ”π‘“π‘‡Ξ”π‘“β©½πœ‡πœ‰π‘‡(𝑑)πœŒπœ‰(𝑑), where ξ‚ƒπœŒ=000πœ†21𝐹𝑇𝐿𝐹𝐿. Then we obtain the following inequality: ̇𝑉(𝑑)β©½πœ‰π‘‡ξ€·π”Έ(𝑑)π‘‡π‘Œ+π‘Œπ‘‡π”Έξ€Έπœ‰(𝑑)+πœ‰π‘‡+1(𝑑)πœ‡πœŒπœ‰(𝑑)πœ‡πœ‰π‘‡(𝑑)π‘Œπ‘‡π”Ήπ”Ήπ‘‡π‘Œπœ‰(𝑑)=πœ‰π‘‡(𝔸𝑑)π‘‡π‘Œ+π‘Œπ‘‡1𝔸+πœ‡π‘Œπ‘‡π”Ήπ‘‡ξ‚Άπ”Ήπ‘Œ+πœ‡πœŒπœ‰(𝑑)(19) and ̇𝑉(𝑑)<0, if π”Έπ‘‡π‘Œ+π‘Œπ‘‡π”Έ+(1/πœ‡)π‘Œπ‘‡π”Ήπ”Ήπ‘‡π‘Œ+πœ‡πœŒ<0.
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 𝑀(𝑑)=0, system (12) is asymptotically stable if there exist a positive definite matrix 𝑋1 and matrices 𝑋2,𝑄1, and 𝑄2 and a scalar πœ‡ such that the following LMI is satisfied:βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘π‘‡π‘‹1+𝑋𝑇1𝑁+𝑋2𝑃+𝑃𝑇𝑋𝑇2+𝑃𝑇𝑄1+𝑄𝑇1π‘ƒβˆ’π‘‹2βˆ’π‘„π‘‡1+𝑃𝑇𝑄2𝑋𝑇1π‘‡π·βˆ’π‘„π‘‡1π‘„Ξ¦π·βˆ’π‘‹2π‘„Ξ¦π·βˆ—βˆ’π‘„2βˆ’π‘„π‘‡2+πœ‡πœ†2πΉπ‘‡πΏπΉπΏβˆ’π‘„π‘‡2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘„Ξ¦π·βˆ—βˆ—βˆ’πœ‡πΌ<0.(20)

Proof. Let π‘Œ=𝑋𝔼+π”ΌβŸ‚π‘‡π‘„, where π”ΌβŸ‚ is the orthogonal matrix of 𝔼 satisfying π”ΌβŸ‚π”Ό=0 and π”ΌβŸ‚π”ΌβŸ‚π‘‡>0, and 𝑋=𝑋1𝑋2𝑋𝑇2𝑋3ξ‚„, and [𝑄=𝑄1𝑄2]. It is easy to see that π”Όπ‘‡π‘Œ=π‘Œπ‘‡πΈβ©Ύ0, since 𝑋1>0. In this case (13)-(14) reduce toβŽ‘βŽ’βŽ’βŽ£π”Έπ‘‡π‘‹π”Ό+π”Έπ‘‡π”ΌβŸ‚π‘‡π‘„+𝔼𝑇𝑋𝑇𝔸+π‘„π‘‡π”ΌβŸ‚π”Έπ”Όπ‘‡π‘‹π‘‡π”Ή+π‘„π‘‡π”ΌβŸ‚π”Ήπ”Ήπ‘‡π‘‹π”Ό+π”Ήπ‘‡π”ΌβŸ‚π‘‡βŽ€βŽ₯βŽ₯βŽ¦π‘„βˆ’πœ‡πΌ<0.(21)On the other hand by inserting the values of 𝔼,𝔸,𝔹, and 𝜌 into (21) we obtain (20).

Remark 7. For Φ𝐷=0, inequality (20) becomesβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘π‘‡π‘‹1+𝑋𝑇1𝑁+𝑋2𝑃+𝑃𝑇𝑋𝑇2+𝑃𝑇𝑄1+𝑄𝑇1π‘ƒβˆ’π‘‹2βˆ’π‘„π‘‡1+𝑃𝑇𝑄2𝑋𝑇1π‘‡π·βˆ’π‘‹π‘‡2βˆ’π‘„1+𝑄𝑇2π‘ƒβˆ’π‘„2βˆ’π‘„π‘‡2+πœ‡πœ†21𝐹𝑇𝐿𝐹𝐿0(𝑇𝐷)𝑇𝑋1⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦0βˆ’πœ‡πΌ<0.(22)
By premultiplying (22) by 𝐼𝑃𝑇000𝐼 and postmultiplying it by 𝐼0𝑃00𝐼, we obtain 𝑁𝑇𝑋1+𝑋𝑇1𝑁+πœ‡πœ†1𝐹𝑇𝐿𝐹𝐿𝑋𝑇1𝑇𝐷(𝑇𝐷)𝑇𝑋1βˆ’πœ‡πΌξ‚„<0, which is exactly the inequality (15) of [16].

3.2. 𝐻∞ Observes Design

In this section we shall present the 𝐻∞ observer design. The following lemma gives the sufficient conditions for system (12) to be stable for 𝑀(𝑑)=0 and ‖𝑒(𝑑)‖𝐿2<𝛾‖𝑀(𝑑)‖𝐿2 for 𝑀(𝑑)β‰ 0.

Lemma 8. The error 𝑒(𝑑) given by (11) is asymptotically stable for 𝑀(𝑑)=0 and ‖𝑒(𝑑)‖𝐿2<𝛾‖𝑀(𝑑)‖𝐿2 for 𝑀(𝑑)β‰ 0, if there exist a positive definite matrix 𝑋1 and matrices 𝑋2,𝑄1,𝑄2 and scalars πœ‡ and 𝛾 such that the following LMI is satisfied:βŽ‘βŽ’βŽ’βŽ’βŽ£π‘π‘‡π‘‹1+𝑋𝑇1𝑁+𝑋2𝑃+𝑃𝑇𝑋𝑇2+𝑃𝑇𝑄1+𝑄𝑇1π‘ƒβˆ’π‘‹2βˆ’π‘„π‘‡1+𝑃𝑇𝑄2𝑋𝑇1π‘‡π·βˆ’π‘„π‘‡1π‘„Ξ¦π·βˆ’π‘‹2𝑋𝑄Φ𝐷𝑇1πœ‘1+𝑋2πœ‘2+𝑄𝑇1πœ‘2ξ€Έβˆ—βˆ’π‘„2βˆ’π‘„π‘‡2+𝐼+πœ‡πœ†2πΉπ‘‡πΏπΉπΏβˆ’π‘„π‘‡2𝑄Φ𝐷𝑄𝑇2πœ‘2βˆ—βˆ—βˆ’πœ‡πΌ0βˆ—βˆ—βˆ—βˆ’π›Ύ2𝐼⎀βŽ₯βŽ₯βŽ₯⎦<0.(23)

Proof. Consider the following Lyapunov candidate function 𝑉(𝑑)=πœ‰π‘‘π”Όπ‘‡π‘Œπœ‰=πœ‰π‘‡(𝑑)π‘Œπ‘‡π”Όπœ‰(𝑑), with π”Όπ‘‡π‘Œ=π‘Œπ‘‡π”Ό. Then we haveΜ‡π‘‰Μ‡πœ‰(𝑑)=𝑇(𝑑)π”Όπ‘‡π‘Œπœ‰(𝑑)+πœ‰π‘‡(𝑑)π‘Œπ‘‡π”ΌΜ‡πœ‰(𝑑)=(π”Έπœ‰(𝑑)+𝔹Δ𝑓+𝔻𝑀(𝑑))π‘‡π‘Œπœ‰(𝑑)+πœ‰(𝑑)π‘‡π‘Œπ‘‡(π”Έπœ‰(𝑑)+𝔹Δ𝑓+𝔻𝑀(𝑑))=πœ‰π‘‡ξ€·π”Έ(𝑑)π‘‡πœ‰+π‘Œπ‘‡π”Έξ€Έπœ‰(𝑑)+Ξ”π‘“π‘‡π”Ήπ‘‡π‘Œπœ‰(𝑑)+πœ‰(𝑑)π‘‡π‘Œπ‘‡π”ΉΞ”π‘“+𝑀𝑇(𝑑)π”»π‘‡π‘Œπœ‰(𝑑)+πœ‰(𝑑)π‘‡π‘Œπ‘‡π”»π‘€(𝑑).(24)
From (17) we have the following inequality: Δ𝑓𝑇(𝔹)π‘‡π‘Œπœ€(𝑑)+πœ€π‘‡(𝑑)π‘Œπ‘‡(β©½1𝔹)Ξ”π‘“πœ‡πœ€π‘‡(𝑑)π‘Œπ‘‡π”Ήπ”Ήπ‘‡π‘Œπœ€(𝑑)+πœ‡Ξ”π‘“π‘‡Ξ”π‘“.(25) Then ̇𝑉(𝑑) becomes ̇𝑉(𝑑)β©½πœ‰π‘‡ξ€·π”Έ(𝑑)π‘‡π‘Œ+π‘Œπ‘‡π”Έξ€Έπœ‰(𝑑)+πœ‡Ξ”π‘“π‘‡1Δ𝑓+πœ‡πœ‰π‘‡(𝑑)π‘Œπ‘‡π”Ήπ”Ήπ‘‡π‘Œπœ‰(𝑑)+𝑀𝑇(𝑑)π”»π‘‡π‘Œπœ‰(𝑑)+πœ‰π‘‡(𝑑)π‘Œπ‘‡π”»π‘€(𝑑).(26)
On the other hand, we have Ξ”π‘“π‘‡Ξ”π‘“β©½πœ†2𝑒𝑇(𝑑)𝐹𝑇𝐿𝐹𝐿𝑒(𝑑), and we know that [𝑒(𝑑)=0𝐼]πœ‰(𝑑) then Ξ”π‘“π‘‡Ξ”π‘“β©½πœ‰π‘‡(𝑑)πœŒπœ‰(𝑑), where ξ‚ƒπœŒ=000πœ†2𝐹𝑇𝐿𝐹𝐿.Then ̇𝑉(𝑑)β©½πœ‰π‘‡ξ€·π”Έ(𝑑)π‘‡π‘Œ+π‘Œπ‘‡π”Έξ€Έπœ‰(𝑑)+πœ‡πœ†2πœ‰π‘‡1(𝑑)πœŒπœ‰(𝑑)+πœ‡πœ‰π‘‡π‘£π‘Œπ‘‡π”Ήπ”Ήπ‘‡π‘Œπœ‰(𝑑)+π‘€π‘‡π”»π‘‡π‘Œπœ‰(𝑑)+πœ‰π‘‡(𝑑)π‘Œπ‘‡π”»π‘€(𝑑).(27) Let ξ€Ίπœ‚(𝑑)=πœ€(𝑑)𝑀(𝑑)ξ€», then from (27) and (12) we obtain the following inequality: ̇𝑉+𝑒𝑇(𝑑)𝑒(𝑑)βˆ’π›Ύ2𝑀𝑇(𝑑)𝑀(𝑑)β©½πœ‚π‘‡(𝑑)Ξ£πœ‚(𝑑)(28) with βŽ‘βŽ’βŽ’βŽ’βŽ£π”ΈΞ£=π‘‡π‘Œ+π‘Œπ‘‡1𝔸+πœ‡π‘Œπ‘‡π”Ήπ”Ήπ‘‡π‘Œ+πœŒπ‘Œπ‘‡π”»π”»π‘‡π‘Œβˆ’π›Ύ2𝐼⎀βŽ₯βŽ₯βŽ₯⎦,(29) where ξ‚ƒπœŒ=000𝐼+πœ‡πœ†2𝐹𝑇𝐿𝐹𝐿.
If Ξ£<0, we obtain ̇𝑉<𝛾2𝑀𝑇(𝑑)π‘€βˆ’π‘’π‘‡(𝑑)𝑒(𝑑).(30) Integrating the two sides of this inequality gives ξ€œβˆž0Μ‡ξ€œπ‘‰(𝜏)π‘‘πœ<∞0𝛾2π‘€π‘‡ξ€œ(𝜏)𝑀(𝜏)π‘‘πœβˆ’βˆž0𝑒𝑇(𝜏)𝑒(𝜏)π‘‘πœ(31) or equivalently 𝑉(∞)βˆ’π‘‰(0)<𝛾2‖‖𝑀(𝑑)22β€–βˆ’β€–π‘’(𝑑)22,(32) under zero initial conditions, we obtain 𝑉(∞)<𝛾2‖‖𝑀(𝑑)22β€–βˆ’β€–π‘’(𝑑)22(33) which leads to ‖‖𝑒(𝑑)22<𝛾2‖‖𝑀(𝑑)22.(34) Using Schur complement we obtain βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π”ΈΞ£=π‘‡π‘Œ+π‘Œπ‘‡π”Έ+πœŒπ‘Œπ‘‡π”Ήπ‘Œπ‘‡π”»π”Ήπ‘‡π”»π‘Œβˆ’πœ‡πΌ0π‘‡π‘Œ0βˆ’π›Ύ2𝐼⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦<0.(35) Let π‘Œ=𝑋𝐸+π”ΌβŸ‚π‘‡π‘„ with 𝑄𝑄=1𝑄2ξ‚„βŽ‘βŽ’βŽ’βŽ£π‘‹,𝑋=1𝑋2𝑋𝑇2𝑋3⎀βŽ₯βŽ₯⎦,(36) we obtain π”Όπ‘‡π‘Œ=π‘Œπ‘‡πΈβ©Ύ0.(37) And by substituting 𝔼,𝔸,𝔹 and 𝔻 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 π”ΌβŸ‚; this can be seen from the fact that the general form of π”ΌβŸ‚ is π”ΌβŸ‚=[0𝑀], where 𝑀 is an arbitrary nonsingular matrix; in this case we have π”ΌβŸ‚π‘‡ξ€Ίπ‘„=0𝐼[𝑄1𝑄2], where 𝑄1=𝑀𝑄1 and 𝑄2=𝑀, which shows that it suffices to choose π”ΌβŸ‚=[0𝐼].
Before giving the design method for the observer (5), let us consider (8) and let 𝑇=𝑇+ΨΦ, where Ξ¨ is an arbitrary matrix of appropriate dimension; they can be written as ξ‚ƒξ‚„βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£ξ‚πΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦=𝑁Ψ𝐽𝑇𝐸Φ𝐴𝑇𝐴,(38)ξ‚ƒξ‚„βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£ξ‚πΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘ƒπ‘„πΉπ‘‡πΈΞ¦π΄=𝐼𝑛.(39) Equations (38) and (39) have a solution if and only if βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£ξ‚πΆξ‚πΌrankπ‘‡πΈΞ¦π΄π‘‡π΄π‘›βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£ξ‚πΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦=rank𝑇𝐸Φ𝐴=𝑛.(40) Now, from Assumption 1 and (40), we have βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£ξ‚πΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦=⎑⎒⎒⎒⎒⎣𝐸𝐢⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦rank𝑇𝐸Φ𝐴Φ𝐴=𝑛.(41) Let 𝑅 be any full row rank matrix such that βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘…πΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣𝐸𝐢⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£ξ‚πΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦rankΦ𝐴=rankΦ𝐴=rank𝑇𝐸Φ𝐴=𝑛,(42) then there always exist matrices parameter 𝐾 and 𝑇 such that ξ‚βŽ‘βŽ’βŽ’βŽ£πΆβŽ€βŽ₯βŽ₯βŽ¦π‘‡πΈ=π‘…βˆ’πΎΞ¦π΄(43) or equivalently ξ‚ƒξ‚ξ‚„βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πΈπΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘‡πΎΞ¦π΄=𝑅.(44) Then, under Assumption 1, there exists a solution to (43) given by ξ‚ƒξ‚ξ‚„βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πΈπΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘‡πΎ=𝑅Φ𝐴†.(45) In this case matrices 𝑇 and 𝐾 are given by ξ‚βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πΈπΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘‡=π‘…Ξ¦π΄β€ βŽ‘βŽ’βŽ’βŽ£πΌ0⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎒⎒⎣𝐸𝐢⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦𝐾=π‘…Ξ¦π΄β€ βŽ‘βŽ’βŽ’βŽ£0𝐼⎀βŽ₯βŽ₯⎦.(46) Also, under Assumption 1, the general solution to (38) is given by =ξ‚π‘Ξ¨π½π‘‡π΄Ξ©β€ βˆ’π‘1ξ€·πΌβˆ’Ξ©Ξ©β€ ξ€Έ,(47) here 𝑇Ω=𝐸𝐢Φ𝐴 and 𝑍1 is an arbitrary matrix of appropriate dimension.
And also under Assumption 1 one solution to (39) is given by 𝑃𝑄𝐹=𝐼𝑛Ω†.(48) Now, define the following matrices: Λ𝑃=Ξ©β€ βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πΌ00⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,Λ𝑄=Ξ©β€ βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£0𝐼0⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,Λ𝑁=ξ‚βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πΌ00⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘‡π΄,ΛΨ=𝑇𝐴Λ𝑄,Λ𝐽=𝑇𝐴Λ𝐹,Δ𝑁=ξ€·πΌβˆ’Ξ©Ξ©β€ ξ€ΈβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πΌ00⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,ΔΨ=ξ€·πΌβˆ’Ξ©Ξ©β€ ξ€ΈβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£0𝐼0⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,Δ𝐽=ξ€·πΌβˆ’Ξ©Ξ©β€ ξ€ΈβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£00𝐼⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,Λ𝐹=Ξ©β€ βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£00𝐼⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,(49) then we obtain 𝑁=Ξ›π‘βˆ’π‘1Δ𝑁,Ξ¨=Ξ›Ξ¨βˆ’π‘1ΔΨ,𝐽=Ξ›π½βˆ’π‘1Δ𝐽,𝑃=Λ𝑃,𝑄=Λ𝑄,𝐹=Λ𝐹,πœ‘1=Ξ›πœ‘1βˆ’π‘1Ξ”πœ‘1,πœ‘2=Ξ›πœ‘2,𝑇𝐷=Ξ›π‘‡π·βˆ’π‘1Δ𝑇𝐷,(50) where Ξ›πœ‘1=Λ𝐽𝐷2βˆ’ξ‚π‘‡π·1+ΛΨΦ𝐷1,Ξ”πœ‘1=Δ𝐽𝐷2+ΔΨΦ𝐷1ξ€Έ,Δ𝑇𝐷=βˆ’Ξ”Ξ¨Ξ›Ξ¦π·,πœ‘2=Λ𝑄Φ𝐷1+Λ𝐹𝐷2ξ€Έ,Λ𝑇𝐷=ξ‚π‘‡π·βˆ’Ξ›Ξ¨Ξ¦π·.(51)
Now, the 𝐻∞ observer design can be obtained from the following theorem.

Theorem 10. There exists an observer of the form (5) such that the error 𝑒(𝑑) given by (10)-(11) is asymptotically stable for 𝑀(𝑑)=0 and ‖𝑒(𝑑)‖𝐿2<𝛾‖𝑀(𝑑)‖𝐿2 for 𝑀(𝑑)β‰ 0 if there exist a symmetric positive definite matrix 𝑋1 and matrices 𝑋2,Ω𝑋1 and scalars πœ‡ and 𝛾 such that the following LMIs are satisfied: ⎑⎒⎒⎒⎒⎒⎒⎣((1,1)(1,2)(1,3)(1,4)1,2)𝑇(2,2)(2,3)(2,4)(1,3)𝑇(2,3)π‘‡βˆ’πœ‡πΌ0(1,4)𝑇(2,4)𝑇0βˆ’π›Ύ2𝐼⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦<0,(52) where (1,1)=Λ𝑇𝑁𝑋1βˆ’Ξ”π‘‡π‘Ξ©π‘‹1+𝑋1Ξ›π‘βˆ’Ξ©π‘‡π‘‹1Δ𝑁+𝑋2Λ𝑃+Λ𝑇𝑃𝑋𝑇2+Λ𝑇𝑃𝑄1+𝑄𝑇1Λ𝑃,(1,2)=βˆ’π‘‹2βˆ’π‘„π‘‡1+Λ𝑇𝑃𝑄2,(1,3)=𝑋1Ξ›π‘‡π·βˆ’Ξ©π‘‡π‘‹1Ξ”π‘‡π·βˆ’π‘„π‘‡1Ξ›π‘„Ξ¦π·βˆ’π‘‹2Λ𝑄Φ𝐷,(1,4)=𝑋1Ξ›πœ‘1βˆ’Ξ©π‘‡π‘‹1Ξ”πœ‘1+𝑄𝑇1Ξ›πœ‘1+𝑋2Ξ›πœ‘2,(2,2)=βˆ’π‘„2βˆ’π‘„π‘‡2+𝐼+πœ‡πœ†2𝐹𝑇𝐿𝐹𝐿,(2,3)=βˆ’π‘„π‘‡2Λ𝑄Φ𝐷,(2,4)=𝑄𝑇2Ξ›πœ‘2.(53)

Proof. Let Ω𝑋1=𝑍𝑇1𝑋1, and by substituting 𝑁,𝑃,𝑄,πœ‘1, and πœ‘2 by their values in (23) we obtain (52).

Procedure for the Observers Design
(1) Under Assumption 1, compute matrices Λ𝑁, Δ𝑁, Λ𝐽, Δ𝐽, Λ𝑃, Λ𝑄,Λ𝐹, ΛΨ, ΔΨ. Ξ›πœ‘1, Ξ”πœ‘1, Ξ›πœ‘2, Λ𝑇𝐷, Δ𝑇𝐷.(2) Solve the LMI (52) to obtain the matrix parameter 𝑍1.(3) Compute the observers parameters 𝑁,𝐽,𝑇,𝐻,𝑃,𝑄, and 𝐹.

4. Numerical Example

Let us consider the following continuous nonlinear singular system of the form (1) with ⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣111⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐷𝐸=010001000,𝐴=100010101,𝐡=1=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣111⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ‚ƒξ‚„,111βˆ’111,𝐷=,𝐢=100(54)𝐷2=[11] and 𝑒(𝑑)=sin(2𝑑). The nonlinearity 𝑓(π‘₯,𝑒,𝑑)=sin(π‘₯2(𝑑)). For this system, the matrix [Ξ¦=001]. In this case it is easy to see that Assumption 1 is verified. We will design a reduced-order observer of dimension π‘ž=2; let 𝑅=010001ξ€», then rank𝑅𝐢Φ𝐴=3. For 𝛾2=9.762, from Section 3 we obtain the following results: Ω𝑋1=102⎑⎒⎒⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’4.1830.5βˆ’0.331βˆ’0.11βˆ’0.5670.276βˆ’0.0450.627,𝑋1=⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦.3.919βˆ’2.628βˆ’2.6285.139(55) The 𝐻∞ observer is given by the following model:Μ‡βŽ‘βŽ’βŽ’βŽ£βŽ€βŽ₯βŽ₯⎦⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦+⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦ξ€·πœ(𝑑)=0βˆ’1.9960.666βˆ’2.121𝜁(𝑑)+βˆ’0.331βˆ’1.414𝑦(𝑑)βˆ’0.331βˆ’0.747𝑒(𝑑)+βˆ’0.331βˆ’0.747sinΜ‚π‘₯2(ξ€Έ,⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣0⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦+⎑⎒⎒⎒⎒⎣0⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣0⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ€·π‘‘)Μ‚π‘₯(𝑑)=0βˆ’0.3521000.705𝜁(𝑑)βˆ’0.2350.529𝑒(𝑑)0.764βˆ’0.529𝑦(𝑑)βˆ’0.2350.529sinΜ‚π‘₯2ξ€Έ.(𝑑)(56)

Simulation results are presented in Figures 1–3. Figure 1 presents the noises 𝑀1(𝑑) and 𝑀2(𝑑). Figures 2 and 3 show the estimation of the states π‘₯1, π‘₯2, and π‘₯3. It can be seen that the observer performs as expected.

fig1
Figure 1: Perturbation 𝑀(𝑑).
fig2
Figure 2: π‘₯𝑖 (blue) and Μ‚π‘₯𝑖 (red).
fig3
Figure 3: Errors 𝑒𝑖=Μ‚π‘₯π‘–βˆ’π‘₯𝑖.

5. Conclusion

In this paper a new method for the 𝐻∞ 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.

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