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

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: 𝐸𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝐷𝑓𝑡,𝐹_𝐿𝑥,𝑢+𝐷₁𝑤(𝑡),(1)𝑦(𝑡)=𝐶𝑥(𝑡)+𝐷₂𝑤(𝑡),(2) with the initial state 𝑥(0)=𝑥₀, where 𝑥(𝑡)∈ℝ^𝑛 is the semi state vector, 𝑢(𝑡)∈ℝ^𝑚 is the known input, 𝑤(𝑡)∈ℝ^𝑛𝑤 is the disturbance vector containing both system and measurement noises, and 𝑦(𝑡)∈ℝ^𝑝 is the measurement output. Matrix 𝐸∈ℝ^𝑛1×𝑛, and, when 𝑛₁=𝑛, matrix 𝐸 is singular. Matrices 𝐴∈ℝ^𝑛1×𝑛, 𝐵∈ℝ^𝑛1×𝑚, 𝐶∈ℝ^𝑝×𝑛, 𝐷∈ℝ^{𝑛1×𝑛𝑓}, 𝐷₁∈ℝ^{𝑛1×𝑛𝑤} and 𝐷₂∈ℝ^{𝑝×𝑛𝑤}. The nonlinearity 𝑓(𝑡,𝐹_𝐿𝑥,𝑢) verifies the Lipschitz constraints: ‖‖𝑓𝑡,𝐹_𝐿𝑥₁,𝑢−𝑓𝑡,𝐹_𝐿𝑥₂,𝑢‖‖≤𝜆‖‖𝐹_𝐿𝑥₁−𝑥₂‖‖,(3) where 𝜆 is a known Lipschitz constant and matrix 𝐹_𝐿 is real with appropriate dimension.

Let 𝑟 be the rank of the matrix 𝐸, and let Φ∈ℝ^𝑟1×𝑛1 be a full row rank matrix such that Φ𝐸=0, with 𝑟₁=𝑛₁−𝑟. Then, from (1), we obtain Φ𝐴𝑥(𝑡)+Φ𝐷₁𝑤(𝑡)+Φ𝐷𝑓𝑡,𝐹_𝐿𝑥,𝑢=−Φ𝐵𝑢(𝑡).(4) In the sequel we assume the following.

Assumption 1. Consider rank^{𝐸𝐶Φ𝐴}=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,𝐵,𝐶).

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

Now, let us consider the following reduced-order observer for system (1): 𝑥̂(𝑡𝜁)̇(=𝑡)𝑃=𝜁𝑁(𝑡𝜁)−(𝑡)𝑄+Φ𝐽𝐵𝑦𝑢(𝑡()𝑡)++𝐻𝐹𝑢𝑦((𝑡𝑡))+−𝑇𝑄𝐷Φ𝑓𝐷𝑓𝑡,𝐹𝑡_𝐿,𝐹𝑥̂,_𝐿𝑢,𝑥̂,𝑢,(5) with the initial condition 𝜁(0)=𝜁₀. Vector 𝜁(𝑡)∈ℝ^𝑞 represents the state vector of the observer, and 𝑥̂(𝑡)∈ℝ^𝑛 is the estimate of 𝑥(𝑡). Matrices 𝑁, 𝐽, 𝑇, 𝐻, 𝑃, 𝑄, and 𝐹 are unknown matrices of appropriate dimensions, which must be determined such that,(1)for 𝑤(𝑡)=0, the error 𝑒(𝑡)=𝑥̂(𝑡)−𝑥(𝑡) asymptotically converges to 0,(2)for 𝑤(𝑡)≠0, we solve the minsup_{𝜔∈𝐿2−{0}}(‖𝑒‖_𝐿2/‖𝜔‖_𝐿2).

Let the error between 𝜁(𝑡) and 𝑇𝐸𝑥(𝑡) be 𝜀(𝑡)=𝜁(𝑡)−𝑇𝐸𝑥(𝑡),(6) then we obtain the following dynamics of 𝜀(𝑡):𝜀̇(𝑡)==+𝜁𝑁̇((𝜀𝐻𝑡)+−−(𝑁𝑇𝑇𝐸𝐵𝑇𝑥𝐸)̇𝑢(−𝑡()𝑡𝑇)𝐴++𝑇𝐷𝐽Δ𝐶𝑓)𝑥+(𝑡)𝐽𝐷₂𝑥̂(𝑡)=−𝑃𝜀𝑄(Φ𝑡)𝐷+Δ𝑓𝑃+𝑄𝑄𝐹Φ𝐷⎡⎢⎢⎢⎢⎣Φ𝑇₁𝐶𝐸𝐴⎤⎥⎥⎥⎥⎦𝑥(𝑡)−𝑇𝐷₁𝑤(𝑡),+𝐹𝐷₂𝑤(𝑡),(7) where Δ𝑓=𝑓(𝑡,𝐹_𝐿𝑥̂(𝑡),𝑢)−𝑓(𝑡,𝐹_𝐿𝑥(𝑡),𝑢).

Under Assumption 1, if there exists a matrix 𝑇 such that 𝑁𝑇𝐸−𝑇𝐴+𝐽𝐶=0,(8)𝑃𝑄𝐻𝐹=⎡⎢⎢⎢⎢⎣𝑇Φ𝑇𝐶𝐵𝐸𝐴,⎤⎥⎥⎥⎥⎦=𝐼,(9) then (7) becomes 𝜀̇(𝑡)=𝑁𝜀(𝑡)+𝑇𝐷Δ𝑓+𝜑₁𝑤(𝑡),(10)𝑒(𝑡)=𝑃𝜀(𝑡)−𝑄Φ𝐷Δ𝑓+𝜑₂𝑤(𝑡),(11) where 𝜑₁=𝐽𝐷₂−𝑇𝐷₁ and 𝜑₂=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 𝑤(𝑡) 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 𝑤(𝑡)=0, we obtain the observer given in [16]. As in [16] this observer is general, and its design unifies the full-order (𝑞=𝑛), the reduced-order (𝑞=𝑛−𝑝), 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 𝑤(𝑡)=0, estimation error (11) converges asymptotically to 0, that is, 𝑒(𝑡)→0 as 𝑡→∞,(2)under the zero initial condition, the error 𝑒(𝑡) satisfies ‖𝑒(𝑡)‖_𝐿²<𝛾‖𝑤(𝑡)‖_𝐿² for any 𝑤(𝑡)∈𝐿₂−{0}, where 𝛾>0 is the prescribed constant.

Now, the dynamics error (10)-(11) can be written in a singular form as 𝔼𝜉̇(𝑡)=𝔸𝜉(𝑡)+𝔹Δ𝑓+𝔻𝑤(𝑡),(12) where 𝜉(𝑡)=[^𝜖𝑒], 𝔼=^𝐼000, 𝔸=^{𝑁0𝑃−𝐼}, 𝔹=^{𝑇𝐷−𝑄Φ𝐷}, and 𝐷=^𝜑1_𝜑2. Then, the error 𝑒(𝑡)→0 as 𝑡→∞, for 𝑤(𝑡)=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 𝑤(𝑡)=0. The following lemma gives this condition.

Lemma 5. For 𝑤(𝑡)=0, system (12) is asymptotically stable, if there exists a matrix 𝑌, such that the following LMIs are satisfied: 𝔼^𝑇𝑌=𝑌^𝑇𝔼⩾0,(13)⎡⎢⎢⎣𝔸^𝑇𝑌+𝔹𝑌^𝑇_𝑇𝔸+𝜇𝜌𝑌^𝑇𝔹𝑌−𝜇𝐼⎤⎥⎥⎦<0(14) with 𝜌=^000𝜆21_𝐹^𝑇𝐿_𝐹𝐿.

Proof. For stability analysis we construct the following Lyapunov candidate function: 𝑉(𝑡)=𝜉^𝑇(𝑡)𝔼^𝑇𝑌𝜉(𝑡)=𝜉^𝑇(𝑡)𝑌^𝑇𝔼𝜉(𝑡)(15) with 𝔼^𝑇𝑌=𝑌^𝑇𝔼⩾0. The derivative of 𝑉(𝑡) along the solution of (12), for 𝑤(𝑡)=0, is given by 𝑉̇(𝑡)==(𝜉𝔸̇^𝑇𝜉(𝑡()𝑡𝔼)+^𝑇𝑌𝔹𝜉Δ(𝑡𝑓)+)_𝑇𝜉^𝑇(𝑡)𝑌^𝑇𝔼𝜉̇(𝑡)=𝜉_𝑇𝑌𝜉(𝑡)+𝜉^𝑇(𝑡)𝑌^𝑇(𝔸𝜉(𝑡)+𝔹Δ𝑓)+Δ(𝑓𝑡)_𝑇𝔸^𝑇𝑌+𝑌^𝑇𝔸𝜉(𝑡)𝔹^𝑇𝑌𝜉(𝑡)+𝜉^𝑇(𝑡)𝑌^𝑇𝔹Δ𝑓.(16) Let 𝑢 and 𝑣 be two vectors of appropriate dimensions, then for all scalar 𝜇>0 the following inequality holds: 𝑢^𝑇𝑣+𝑣^𝑇𝑢⩽𝜇𝑢^𝑇𝑢+1𝜇𝑣^𝑇𝑣.(17) By using (17), it is not difficult to check that Δ𝑓^𝑇⩽(𝔹)1^𝑇𝑌𝜀(𝑡)+𝜀^𝑇(𝑡)𝑌^𝑇(𝔹)Δ𝑓𝜇𝜀^𝑇(𝑡)𝑌^𝑇𝔹𝔹^𝑇𝑌𝜀(𝑡)+𝜇Δℱ^𝑇Δ𝑓.(18) In this case we have 𝑉̇(𝑡)⩽𝜉^𝑇(𝑡)(𝔸^𝑇𝑌+𝑌^𝑇𝔸)𝜉(𝑡)+𝜇Δ𝑓^𝑇Δ𝑓+(1/𝜇)𝜉^𝑇(𝑡)𝑌^𝑇𝔹𝔹^𝑇𝑌𝜉(𝑡). On the other hand, by using the Lipschitz conditions and the fact that 𝑒(𝑡)=[^0𝐼]𝜉(𝑡), we have 𝜇Δ𝑓^𝑇Δ𝑓⩽𝜇𝜉^𝑇(𝑡)𝜌𝜉(𝑡), where 𝜌=^000𝜆21_𝐹^𝑇𝐿_𝐹𝐿. Then we obtain the following inequality: 𝑉̇(𝑡)⩽+𝜉^𝑇1(𝑡)𝔸^𝑇𝑌+𝑌^𝑇𝔸𝜉(𝑡)+𝜉^𝑇(𝑡)𝜇𝜌𝜉(𝑡)𝜇𝜉^𝑇=𝜉_𝑇(𝑡)𝑌^𝑇𝔹𝔹^𝑇𝑌𝜉(𝑡)(𝑡)𝔸^𝑇𝑌+𝑌^𝑇𝔸+1𝜇𝑌^𝑇𝔹^𝑇𝔹𝑌+𝜇𝜌𝜉(𝑡)(19) and 𝑉̇(𝑡)<0, if 𝔸^𝑇𝑌+𝑌^𝑇𝔸+(1/𝜇)𝑌^𝑇𝔹𝔹^𝑇𝑌+𝜇𝜌<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 𝑤(𝑡)=0, system (12) is asymptotically stable if there exist a positive definite matrix 𝑋₁ and matrices 𝑋₂,𝑄₁, and 𝑄₂ and a scalar 𝜇 such that the following LMI is satisfied:⎡⎢⎢⎢⎢⎣𝑁^𝑇𝑋₁+𝑋^𝑇1𝑁+𝑋₂𝑃+∗𝑃^𝑇𝑋^𝑇2+𝑃^𝑇𝑄₁+𝑄^𝑇1𝑃−𝑄−𝑋₂₂−𝑄^𝑇1+𝑃^𝑇𝑄₂𝑋^𝑇1𝑇𝐷−𝑄^𝑇1𝑄Φ𝐷−𝑋₂𝑄Φ𝐷∗−𝑄^𝑇2∗+𝜇𝜆²𝐹^𝑇𝐿𝐹_𝐿−𝑄−^𝑇2𝜇𝑄𝐼Φ𝐷⎤⎥⎥⎥⎥⎦<0.(20)

Proof. Let 𝑌=𝑋𝔼+𝔼^⟂𝑇𝑄, where 𝔼^⟂ is the orthogonal matrix of 𝔼 satisfying 𝔼^⟂𝔼=0 and 𝔼^⟂𝔼^⟂𝑇>0, and 𝑋=^𝑋1𝑋2_{𝑋𝑇2𝑋3}, and 𝑄=[^𝑄₁^𝑄₂]. It is easy to see that 𝔼^𝑇𝑌=𝑌^𝑇𝐸⩾0, since 𝑋₁>0. In this case (13)-(14) reduce to⎡⎢⎢⎣𝔸^𝑇𝑋𝔼+𝔸𝔹^𝑇𝔼^⟂𝑇_𝑇𝑄+𝔼^𝑇𝑋^𝑇𝔸+𝑄^𝑇𝔼^⟂𝔸𝔼^𝑇𝑋^𝑇𝔹+𝑄^𝑇𝔼^⟂𝔹𝑋𝔼+𝔹^𝑇𝔼^⟂𝑇𝑄−𝜇𝐼⎤⎥⎥⎦<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) becomes⎡⎢⎢⎢⎢⎣𝑁^𝑇𝑋₁+𝑋^𝑇1𝑁−+𝑋𝑋_2𝑇2𝑃+𝑃^𝑇𝑋^𝑇2+𝑃^𝑇𝑄₁+𝑄^𝑇1𝑃−𝑋₂−𝑄^𝑇1+𝑃^𝑇𝑄₂𝑋^𝑇1𝑇𝐷(𝑇−𝐷𝑄)_1𝑇+𝑄^𝑇2𝑃−𝑄₂−𝑄^𝑇2+𝜇𝜆²¹𝐹^𝑇𝐿𝐹_𝐿0𝑋₁0−𝜇𝐼⎤⎥⎥⎥⎥⎦<0.(22)
By premultiplying (22) by ^{𝐼𝑃𝑇000𝐼} and postmultiplying it by ^{𝐼0𝑃00𝐼}, we obtain ^{𝑁𝑇𝑋1+𝑋𝑇1𝑁+𝜇𝜆1𝐹𝑇𝐿𝐹𝐿𝑋𝑇1𝑇𝐷(𝑇𝐷)𝑇}_{𝑋1−𝜇𝐼}<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 𝑤(𝑡)=0 and ‖𝑒(𝑡)‖_𝐿2<𝛾‖𝑤(𝑡)‖_𝐿2 for 𝑤(𝑡)≠0.

Lemma 8. The error 𝑒(𝑡) given by (11) is asymptotically stable for 𝑤(𝑡)=0 and ‖𝑒(𝑡)‖_𝐿2<𝛾‖𝑤(𝑡)‖_𝐿2 for 𝑤(𝑡)≠0, if there exist a positive definite matrix 𝑋₁ and matrices 𝑋₂,𝑄₁,𝑄₂ and scalars 𝜇 and 𝛾 such that the following LMI is satisfied:⎡⎢⎢⎢⎣^{𝑁𝑇𝑋1+𝑋𝑇1𝑁+𝑋2𝑃+𝑃𝑇𝑋𝑇2+𝑃𝑇𝑄1+𝑄𝑇1𝑃−𝑋2−𝑄𝑇1+𝑃𝑇𝑄2𝑋𝑇1𝑇𝐷−𝑄𝑇1𝑄Φ𝐷−𝑋2𝑋𝑄Φ𝐷𝑇1𝜑1+𝑋2𝜑2+𝑄𝑇1𝜑2∗−𝑄}_{2−𝑄𝑇2+𝐼+𝜇𝜆2𝐹𝑇𝐿𝐹𝐿−𝑄𝑇2𝑄Φ𝐷𝑄𝑇2𝜑2∗∗−𝜇𝐼0∗∗∗−𝛾2𝐼}⎤⎥⎥⎥⎦<0.(23)

Proof. Consider the following Lyapunov candidate function 𝑉(𝑡)=𝜉^𝑡𝔼^𝑇𝑌𝜉=𝜉^𝑇(𝑡)𝑌^𝑇𝔼𝜉(𝑡), with 𝔼^𝑇𝑌=𝑌^𝑇𝔼. Then we have𝑉̇(𝑡)==(𝜉𝔸̇^𝑇𝜉(𝑡()𝑡𝔼)+^𝑇𝑌𝔹𝜉Δ(𝑡𝑓)++𝜉𝔻^𝑇𝑤(𝑡()𝑡𝑌))^𝑇_𝑇𝔼𝜉̇(𝑡)+𝜉(𝑡)_𝑇𝑌𝜉(𝑡)=𝜉_𝑇𝑌^𝑇(𝔸𝜉(𝑡)+𝔹Δ𝑓+𝔻𝑤(𝑡))+𝜉((𝑡𝑡))_𝑇𝔸^𝑇𝜉+𝑌^𝑇𝔸𝜉(𝑡)+Δ𝑓^𝑇𝔹^𝑇𝑌𝜉(𝑡)𝑌^𝑇𝔹Δ𝑓+𝑤^𝑇(𝑡)𝔻^𝑇𝑌𝜉(𝑡)+𝜉(𝑡)^𝑇𝑌^𝑇𝔻𝑤(𝑡).(24)
From (17) we have the following inequality: Δ𝑓^𝑇⩽(𝔹)1^𝑇𝑌𝜀(𝑡)+𝜀^𝑇(𝑡)𝑌^𝑇(𝔹)Δ𝑓𝜇𝜀^𝑇(𝑡)𝑌^𝑇𝔹𝔹^𝑇𝑌𝜀(𝑡)+𝜇Δ𝑓^𝑇Δ𝑓.(25) Then 𝑉̇(𝑡) becomes 𝑉̇(𝑡)⩽+𝜉^𝑇𝜇(Δ𝑡)𝑓_𝑇𝔸^𝑇𝑌+𝑌^𝑇𝔸𝜉(𝑡)Δ𝑓+1𝜇𝜉^𝑇+𝑤_𝑇(𝑡)𝑌^𝑇𝔹𝔹^𝑇𝑌𝜉(𝑡)(𝑡)𝔻^𝑇𝑌𝜉(𝑡)+𝜉^𝑇(𝑡)𝑌^𝑇𝔻𝑤(𝑡).(26)
On the other hand, we have Δ𝑓^𝑇Δ𝑓⩽𝜆²𝑒^𝑇(𝑡)𝐹^𝑇𝐿𝐹_𝐿𝑒(𝑡), and we know that 𝑒(𝑡)=[^0𝐼]𝜉(𝑡) then Δ𝑓^𝑇Δ𝑓⩽𝜉^𝑇(𝑡)𝜌𝜉(𝑡), where 𝜌=^000𝜆2_𝐹^𝑇𝐿_𝐹𝐿.Then 𝑉̇(𝑡)⩽+𝜉^𝑇𝜇(𝜆𝑡)₂𝔸^𝑇𝑌+𝑌^𝑇𝔸𝜉(𝑡)𝜉^𝑇(𝑡)𝜌𝜉(𝑡)+1𝜇𝜉^𝑇+𝑤_𝑇𝑣𝑌^𝑇𝔹𝔹^𝑇𝑌𝜉(𝑡)𝔻^𝑇𝑌𝜉(𝑡)+𝜉^𝑇(𝑡)𝑌^𝑇𝔻𝑤(𝑡).(27) Let 𝜂(𝑡)=^{𝜀(𝑡)𝑤(𝑡)}, then from (27) and (12) we obtain the following inequality: 𝑉̇+𝑒^𝑇(𝑡)𝑒(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)⩽𝜂^𝑇(𝑡)Σ𝜂(𝑡)(28) with Σ=⎡⎢⎢⎢⎣𝔸^𝑇𝑌+𝑌^𝑇𝔸+1𝜇𝑌^𝑇𝔻_𝑇𝔹𝔹^𝑇𝑌+𝜌𝑌^𝑇𝔻𝑌−𝛾²𝐼⎤⎥⎥⎥⎦,(29) where 𝜌=^{000𝐼+𝜇𝜆2}_𝐹^𝑇𝐿_𝐹𝐿.
If Σ<0, we obtain 𝑉̇<𝛾²𝑤^𝑇(𝑡)𝑤−𝑒^𝑇(𝑡)𝑒(𝑡).(30) Integrating the two sides of this inequality gives ^∞0𝑉̇(𝜏)𝑑𝜏<^∞0𝛾²𝑤^𝑇(𝜏)𝑤(𝜏)𝑑𝜏−^∞0𝑒^𝑇(𝜏)𝑒(𝜏)𝑑𝜏(31) or equivalently 𝑉(∞)−𝑉(0)<𝛾²‖𝑤(𝑡)‖²²−‖𝑒(𝑡)‖²²,(32) under zero initial conditions, we obtain 𝑉(∞)<𝛾²‖𝑤(𝑡)‖²²−‖𝑒(𝑡)‖²²(33) which leads to ‖𝑒(𝑡)‖²²<𝛾²‖𝑤(𝑡)‖²².(34) Using Schur complement we obtain Σ=⎡⎢⎢⎢⎢⎣𝔸^𝑇𝑌+𝔹𝑌^𝑇_𝑇𝔸+𝜌𝑌^𝑇𝔹𝑌^𝑇𝔻𝔻𝑌−𝜇𝐼0_𝑇𝑌0−𝛾²𝐼⎤⎥⎥⎥⎥⎦<0.(35) Let 𝑌=𝑋𝐸+𝔼^⟂𝑇𝑄 with 𝑄=𝑄₁𝑄₂,𝑋=⎡⎢⎢⎣𝑋¹𝑋₂𝑋^𝑇2𝑋₃⎤⎥⎥⎦,(36) we obtain 𝔼^𝑇𝑌=𝑌^𝑇𝐸⩾0.(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 𝔼^⟂; this can be seen from the fact that the general form of 𝔼^⟂ is 𝔼^⟂=[^0𝑀], where 𝑀 is an arbitrary nonsingular matrix; in this case we have 𝔼^⟂𝑇𝑄=^0𝐼[^𝑄1𝑄2], where 𝑄₁=𝑀𝑄₁ and 𝑄₂=𝑀, which shows that it suffices to choose 𝔼^⟂=[^0𝐼].
Before giving the design method for the observer (5), let us consider (8) and let 𝑇=𝑇+ΨΦ, where Ψ is an arbitrary matrix of appropriate dimension; they can be written as 𝑁Ψ𝐽⎡⎢⎢⎢⎢⎣𝑇Φ𝐶𝐸𝐴⎤⎥⎥⎥⎥⎦=𝑇𝐴,(38)𝑃𝑄𝐹⎡⎢⎢⎢⎢⎣𝑇Φ𝐶𝐸𝐴⎤⎥⎥⎥⎥⎦=𝐼^𝑛.(39) Equations (38) and (39) have a solution if and only if rank⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑇Φ𝑇𝐼𝐶𝐸𝐴𝐴_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=rank⎡⎢⎢⎢⎢⎣𝑇Φ𝐶𝐸𝐴⎤⎥⎥⎥⎥⎦=𝑛.(40) Now, from Assumption 1 and (40), we have rank⎡⎢⎢⎢⎢⎣𝑇Φ𝐶𝐸𝐴⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣Φ𝐸𝐶𝐴⎤⎥⎥⎥⎥⎦=𝑛.(41) Let 𝑅 be any full row rank matrix such that rank⎡⎢⎢⎢⎢⎣Φ𝐶𝑅𝐴⎤⎥⎥⎥⎥⎦=rank⎡⎢⎢⎢⎢⎣Φ𝐸𝐶𝐴⎤⎥⎥⎥⎥⎦=rank⎡⎢⎢⎢⎢⎣𝑇Φ𝐶𝐸𝐴⎤⎥⎥⎥⎥⎦=𝑛,(42) then there always exist matrices parameter 𝐾 and 𝑇 such that 𝑇𝐸=𝑅−𝐾⎡⎢⎢⎣Φ𝐶𝐴⎤⎥⎥⎦(43) or equivalently 𝑇𝐾⎡⎢⎢⎢⎢⎣Φ𝐸𝐶𝐴⎤⎥⎥⎥⎥⎦=𝑅.(44) Then, under Assumption 1, there exists a solution to (43) given by 𝑇𝐾=𝑅⎡⎢⎢⎢⎢⎣Φ𝐸𝐶𝐴⎤⎥⎥⎥⎥⎦^†.(45) In this case matrices 𝑇 and 𝐾 are given by 𝑇=𝑅⎡⎢⎢⎢⎢⎣Φ𝐸𝐶𝐴⎤⎥⎥⎥⎥⎦^†𝐾=𝑅⎡⎢⎢⎢⎢⎣Φ𝐸𝐶𝐴⎤⎥⎥⎥⎥⎦_†⎡⎢⎢⎣𝐼0⎤⎥⎥⎦,⎡⎢⎢⎣𝐼0⎤⎥⎥⎦.(46) Also, under Assumption 1, the general solution to (38) is given by 𝑁Ψ𝐽=𝑇𝐴Ω^†−𝑍₁𝐼−ΩΩ^†,(47) here Ω=𝑇^{𝐸𝐶Φ𝐴} and 𝑍₁ is an arbitrary matrix of appropriate dimension.
And also under Assumption 1 one solution to (39) is given by 𝑃𝑄𝐹=𝐼^𝑛Ω^†.(48) Now, define the following matrices: Λ_𝑃=Ω^†⎡⎢⎢⎢⎢⎣𝐼00⎤⎥⎥⎥⎥⎦,Λ_𝑄=Ω^†Λ_𝑁⎡⎢⎢⎢⎢⎣𝐼00⎤⎥⎥⎥⎥⎦,Δ_𝑁=𝑇𝐴⎡⎢⎢⎢⎢⎣𝐼00⎤⎥⎥⎥⎥⎦,Λ_Ψ=𝑇𝐴Λ_𝑄,Λ_𝐽=𝑇𝐴Λ_𝐹,=Δ𝐼_𝐽−ΩΩ^†⎡⎢⎢⎢⎢⎣𝐼00⎤⎥⎥⎥⎥⎦,Δ_Ψ=𝐼−ΩΩ^†⎡⎢⎢⎢⎢⎣𝐼00⎤⎥⎥⎥⎥⎦,=𝐼−ΩΩ^†⎡⎢⎢⎢⎢⎣𝐼00⎤⎥⎥⎥⎥⎦,Λ_𝐹=Ω^†⎡⎢⎢⎢⎢⎣𝐼00⎤⎥⎥⎥⎥⎦,(49) then we obtain 𝑁=Λ𝐽_𝑁=−Λ𝑍_1𝐽Δ_𝑁,Ψ=Λ_Ψ−𝑍₁Δ_Ψ,𝑄=−Λ𝑍_1𝑄Δ_𝐽,𝑃=Λ_𝑃,𝜑₁,𝐹=Λ_𝐹,=Λ𝑇𝐷_𝜑1−=𝑍Λ₁Δ_{𝜑1𝑇𝐷},𝜑₂=Λ_𝜑2,−𝑍₁Δ_𝑇𝐷,(50) where Δ_𝜑1Λ_𝜑1=Λ_𝐽𝐷₂−𝑇𝐷₁+Λ_ΨΦ𝐷₁,Λ=Δ_𝐽𝐷₂+Δ_ΨΦ𝐷₁,Δ_𝑇𝐷=−Δ_ΨΦ𝐷,_𝜑2=Λ_𝑄Φ𝐷₁+Λ_𝐹𝐷₂,Λ_𝑇𝐷=𝑇𝐷−Λ_ΨΦ𝐷.(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(𝑡) given by (10)-(11) is asymptotically stable for 𝑤(𝑡)=0 and ‖𝑒(𝑡)‖_𝐿2<𝛾‖𝑤(𝑡)‖_𝐿2 for 𝑤(𝑡)≠0 if there exist a symmetric positive definite matrix 𝑋₁ and matrices 𝑋₂,Ω_𝑋1 and scalars 𝜇 and 𝛾 such that the following LMIs are satisfied: ⎡⎢⎢⎢⎢⎢⎢⎣((11,,21))_𝑇(1,2)(1,3)(1,4)(1,3)_𝑇(2,2)(2,3)(2,4)(1,4)_𝑇(2,3)^𝑇−𝜇𝐼0(2,4)^𝑇0−𝛾²𝐼⎤⎥⎥⎥⎥⎥⎥⎦<0,(52) where (1,1)=Λ^𝑇𝑁𝑋₁−Δ^𝑇𝑁Ω_𝑋1+𝑋₁Λ_𝑁−Ω^𝑇𝑋₁Δ_𝑁(1,2)=+−𝑋𝑋₂₂Λ_𝑃+Λ^𝑇𝑃𝑋^𝑇2+Λ^𝑇𝑃𝑄₁+𝑄^𝑇1Λ_𝑃,(1,3)=𝑋₁−𝑄^𝑇1+Λ^𝑇𝑃𝑄₂,(1,4)=𝑋₁Λ_𝑇𝐷−Ω^𝑇𝑋₁Δ_𝑇𝐷−𝑄^𝑇1Λ_𝑄Φ𝐷−𝑋₂Λ_𝑄Φ𝐷,(2,2)=−𝑄Λ_𝜑12−Ω^𝑇𝑋₁Δ_𝜑1+𝑄^𝑇1Λ_𝜑1+𝑋₂Λ_𝜑2,(2,3)=−𝑄_𝑇2−𝑄^𝑇2+𝐼+𝜇𝜆²𝐹^𝑇𝐿𝐹_𝐿,(2,4)=𝑄_𝑇2Λ_𝑄Φ𝐷,Λ_𝜑2.(53)

Proof. Let Ω_𝑋1=𝑍^𝑇1𝑋₁, and by substituting 𝑁,𝑃,𝑄,𝜑₁, and 𝜑₂ by their values in (23) we obtain (52).

Procedure for the Observers Design
(1) Under Assumption 1, compute matrices Λ_𝑁, Δ_𝑁, Λ_𝐽, Δ_𝐽, Λ_𝑃, Λ_𝑄,Λ_𝐹, Λ_Ψ, Δ_Ψ. Λ_𝜑1, Δ_𝜑1, Λ_𝜑2, Λ_𝑇𝐷, Δ_𝑇𝐷.(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 𝐷𝐸=⎡⎢⎢⎢⎢⎣000100010⎤⎥⎥⎥⎥⎦,𝐴=⎡⎢⎢⎢⎢⎣101010001⎤⎥⎥⎥⎥⎦,𝐵=₁⎡⎢⎢⎢⎢⎣111⎤⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎣111−111⎤⎥⎥⎥⎥⎦,𝐷=⎡⎢⎢⎢⎢⎣111⎤⎥⎥⎥⎥⎦,𝐶=100,(54)𝐷₂=[¹¹] and 𝑢(𝑡)=sin(2𝑡). The nonlinearity 𝑓(𝑥,𝑢,𝑡)=sin(𝑥₂(𝑡)). For this system, the matrix Φ=[⁰⁰¹]. In this case it is easy to see that Assumption 1 is verified. We will design a reduced-order observer of dimension 𝑞=2; let 𝑅=⁰¹⁰⁰⁰¹, then rank^{𝑅𝐶Φ𝐴}=3. For 𝛾²=9.762, from Section 3 we obtain the following results: Ω_𝑋1=10²⎡⎢⎢⎢⎢⎢⎢⎣−−−−4000....135083643175−000..026..5721671⎤⎥⎥⎥⎥⎥⎥⎦,𝑋₁=⎡⎢⎢⎣−32.9.61298−52.1.63298⎤⎥⎥⎦.(55) The 𝐻_∞ observer is given by the following model:𝜁̇(𝑡)=+⎡⎢⎢⎣0⎡⎢⎢⎣.−−606006..3734−−1712⎤⎥⎥⎦..91𝑢92(61𝑡)⎤⎥⎥⎦+𝜁(⎡⎢⎢⎣𝑡−−)00+..37⎡⎢⎢⎣34−−1701⎤⎥⎥⎦..34si31n14⎤⎥⎥⎦𝑥𝑦̂₂(𝑡)𝑥̂(𝑡)=+⎡⎢⎢⎢⎢⎣010⎡⎢⎢⎢⎢⎣−−000.07..075.0630245592⎤⎥⎥⎥⎥⎦⎤⎥⎥⎥⎥⎦𝑦𝜁((𝑡)𝑡)−−⎡⎢⎢⎢⎢⎣⎡⎢⎢⎢⎢⎣0000..25..025032325959⎤⎥⎥⎥⎥⎦⎤⎥⎥⎥⎥⎦s𝑢in(𝑡)𝑥̂₂(𝑡),(𝑡).(56)

Simulation results are presented in Figures 1–3. Figure 1 presents the noises 𝑤₁(𝑡) and 𝑤₂(𝑡). Figures 2 and 3 show the estimation of the states 𝑥₁, 𝑥₂, and 𝑥₃. It can be seen that the observer performs as expected.

(a)

(b)

Figure 1 

Perturbation 𝑤(𝑡).

(a)

(b)

(c)

Figure 2 

𝑥𝑖 (blue) and 𝑥̂𝑖 (red).

(a)

(b)

(c)

Figure 3 

Errors 𝑒𝑖=𝑥̂𝑖−𝑥𝑖.

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.