ℒ₂ Gain Performance for a Class of Lipschitz Uncertain Nonlinear Systems via Variable Gain Robust Output Feedback Controllers

Abstract

We consider a design problem of a variable gain robust output feedback controller with guaranteed ℒ₂ gain performance for a class of Lipschitz uncertain nonlinear systems. The proposed variable gain robust output feedback controller achieves not only robust stability but also a specified ℒ₂ gain performance. In this paper, we show that sufficient conditions for the existence of the proposed variable gain robust output feedback controller with guaranteed ℒ₂ gain performance are given in terms of linear matrix inequalities (LMIs). Finally, a simple numerical example is included.

1. Introduction

In general, there exists a gap between controlled systems and their mathematical models. Therefore controller design methods dealing with the model uncertainties explicitly have been required, and, for linear dynamical systems with unknown parameters, a large number of design methods of robust state feedback controllers have been presented (e.g., [1] and references therein). In particular, there are lots of existing results for state feedback robust control such as quadratic stabilizing control, ℋ^∞ control (see [2, 3] and references therein). Besides, some design methods of variable gain robust controllers for uncertain dynamical systems have also been suggested (e.g., [4, 5]). These controllers consist of a fixed gain controller and a variable gain one, and the variable gain controller is tuned by updating laws.

By the way, since not all the states are measurable in practical systems because of technical, physical, and/or economic reasons, the control scheme may be designed via observer-based robust controllers [6] or robust output feedback one [7], which is of interest in this paper, and these robust controllers have also been well studied. Geromel et al. [8] use LMI approach to design static output feedback controllers based on a set of Lyapunov inequalities coupled by the constraint that one Lyapunov matrix is the inverse of another. Additionally for a class of linear systems with uncertainties of which upper bounds are unknown, an adaptive robust output feedback stabilizing controller has been proposed [9].

On the other hand in recent years, much attention has been focusing upon global stabilization for nonlinear systems via output feedback control (e.g., [10, 11]). Mazenc et al. [10] have shown that, through counter examples, some extra growth conditions on the unmeasurable states of the plant are usually necessary for the global stabilization of nonlinear systems via output feedback. Additionally, some researchers have studied the control problem for a selective class of nonlinear systems by placing some structural constraints on the nonlinearities in order to derive output feedback control. In Tsinias [12], the problem of backstepping design for time-varying nonlinear systems with unknown parameters was considered. Pailla and Zhu [13] have proposed a nonlinear observer-based controller for a class of Lipschitz nonlinear systems. Choi and Lim [14] have presented a solution to the output feedback stabilization problem for a class of single-input single-output Lipschitz nonlinear systems and the nonlinearity characterization function (NCF) concept. Besides, a design method of a variable gain robust output feedback stabilizing controller for a class of Lipschitz uncertain nonlinear systems has also been suggested in [15].

In this paper on the basis of the works of [9, 15], we propose a design method of a variable gain robust output feedback controller with guaranteed ℒ₂ gain performance for a class of Lipschitz uncertain nonlinear systems. In this paper, we show that sufficient conditions for the existence of the variable gain robust output feedback controller which achieves not only internal stability but also a specified ℒ₂ gain performance are given in terms of LMIs. Therefore, one can easily see that there are crucial difference between our new one and the existing results [15] which merely deal with stabilization problem via output feedback controllers. This paper is organized as follows. In Section 2, we show the notation used in this paper. In Section 3, we define the class of Lipschitz uncertain nonlinear systems under consideration and introduce a variable gain robust output feedback controller. Section 4 contains our main results. Finally, numerical examples are presented.

2. Preliminaries

In this section, we show notations and useful and well-known lemmas which are used in this paper.

For a matrix 𝒜, the transpose of matrix 𝒜 and the inverse of one are denoted by 𝒜^𝑇 and 𝒜⁻¹, respectively. Also, 𝐻_𝑒{𝒜} means 𝒜+𝒜^𝑇, and 𝐼_𝑛 represents 𝑛-dimensional identity matrix. For real symmetric matrices 𝒜 and ℬ, 𝒜>ℬ(resp.,𝒜≥ℬ) means that 𝒜−ℬ is positive (resp., nonnegative) definite matrix. For a vector 𝛼∈ℝ^𝑛, ‖𝛼‖ denotes standard Euclidian norm, and, for a matrix 𝒜, ‖𝒜‖ represents its induced norm. The symbol “≜” means equality by definition. Besides, ℒ₂[0,∞) is ℒ₂-space (i.e., the collection of all square integrable functions) defined on [0,∞), and, for a signal 𝑓(𝑡)∈ℒ₂[0,∞), ‖𝑓(𝑡)‖_ℒ2 denotes its ℒ₂-norm.

Lemma 1. For arbitrary matrices 𝒢 and ℋ and vectors 𝜆 and 𝜉 which have appropriate dimensions, the following inequality holds: 𝐻_𝑒𝜆^𝑇𝒢Δ(𝑡)ℋ𝜉≤2𝒢^𝑇𝜆ℋ𝜉, (1) where Δ(𝑡)∈ℝ^𝑝×𝑞 is a time-varying unknown matrix satisfying ‖Δ(𝑡)‖≤1.

Proof. This relation is easily obtained by Schwartz’s inequality [16].

Lemma 2 (Schur complement formula). For a given constant real symmetric matrix Ξ, the following items are equivalent:(i)Ξ=^Ξ11Ξ12_Ξ𝑇12Ξ22>0, (ii)Ξ₁₁>0 and Ξ₂₂−Ξ^𝑇₁₂Ξ⁻¹₁₁Ξ₁₂>0,(iii)Ξ₂₂>0 and Ξ₁₁−Ξ₁₂Ξ⁻¹₂₂Ξ^𝑇₁₂>0.

Proof. See Boyd et al. [17].

3. Problem Formulations

Consider the following Lipschitz uncertain nonlinear systems: 𝑑𝑥(𝑡)𝑑𝑡=𝐴_Δ(𝑡)𝑥(𝑡)+𝐵𝑢(𝑡)+𝛿(𝑥,𝑡)+Γ₁𝜔(𝑡),𝑦(𝑡)(𝑡),𝑧(𝑡)(𝑡)=𝐶𝑥=𝐷𝑥+Γ₂𝜔(𝑡), (2) where 𝑥(𝑡)∈ℝ^𝑛,𝑢(𝑡)∈ℝ^𝑚,𝑦(𝑡)∈ℝ^𝑙,𝑧(𝑡)∈ℝ^𝑝, and 𝜔(𝑡)∈ℝ^𝑞 are the vectors of the state, the control input, the measured output, the controlled output, and the disturbance input, respectively, and the disturbance input is assumed to be square integrable; that is, 𝑤(𝑡)∈ℒ₂[0,∞). In (2), 𝐴_Δ(𝑡)∈ℝ^𝑛×𝑛is supposed to have the following structure,𝐴_Δ(𝑡)=𝐴+𝐵Δ(𝑡)𝐶. (3) In (2) and (3), the matrices 𝐴,𝐵,𝐶,and𝐷, and Γ_𝑘(𝑘=1,2) are known system parameters and the matrix Δ(𝑡)∈ℝ^𝑚×𝑙 denotes unknown time-varying parameters, which satisfy ‖Δ(𝑡)‖≤1. In this paper, we introduce the following assumption for the nonlinear term 𝛿(𝑥,𝑡) in (2): 𝛿(𝑥,𝑡)=𝐵𝜉(𝑥,𝑡). (4) In addition we assume that for the function 𝜉∶ℝ^𝑛×𝑅→ℝ^𝑚 in (4) there exists a constant scalar 𝜒^∗ such that for all 𝑥₁,𝑥₂∈ℝ^𝑛𝜉𝑥₁,𝑡−𝜉𝑥₂,𝑡≤𝜒^∗𝑥₁−𝑥₂. (5) Besides, we assume that there exists a constant matrix 𝒯∈ℝ^𝑚×𝑙 satisfying the following relation [15]: 𝐵^𝑇=𝒯𝐶. (6)

The nominal system, ignoring unknown parameters and nonlinearities in (2), is given by 𝑑𝑑𝑡𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑦(𝑡)=𝐶𝑥(𝑡) (7) and the nominal system of (7) is supposed to be stabilizable via static output feedback control. Namely, there exists a fixed gain matrix 𝐾∈ℝ^𝑚×𝑙 such that the matrix 𝐴_𝐾≜𝐴+𝐵𝐾𝐶 is asymptotically stable. In other words, there exists an output feedback control 𝑢(𝑡)=𝐾𝑦(𝑡) which stabilizes the nominal system of (7). Note that the static output feedback gain matrix 𝐾∈ℝ^𝑚×𝑙 is designed by using the existing results (e.g., [7, 18]). Besides, in this paper, we consider the following target model so as to generate the desirable trajectory for the Lipschitz uncertain nonlinear system of (2) 𝑑𝑥𝑑𝑡^𝑡(𝑡)=𝐴𝑥_𝑡(𝑡)+𝐵𝑢_𝑡(𝑡)+𝛿𝑥_𝑡,𝑦,𝑡_𝑡(𝑡)=𝐶𝑥_𝑡(𝑡),𝑧_𝑡(𝑡)=𝐷𝑥_𝑡(𝑡). (8) For the target model of (8), we select the control input 𝑢_𝑡(𝑡)=𝐾_ℒ𝒬𝑥_𝑡(𝑡)−𝜉(𝑥_𝑡,𝑡). Thus the target model can be written as 𝑑𝑥𝑑𝑡^𝑡(𝑡)=𝐴+𝐵𝐾_ℒ𝒬𝑥_𝑡(𝑡),𝑦_𝑡(𝑡)=𝐶𝑥_𝑡(𝑡),𝑧_𝑡(𝑡)=𝐷𝑥_𝑡(𝑡). (9) Note that the fixed state feedback gain matrix 𝐾_ℒ𝒬∈ℝ^𝑚×𝑛 can be determined by applying the standard LQ optimal control problem for the nominal system of (7).

Now on the basis of the work of   [9], we introduce the error vectors 𝑒(𝑡)≜𝑥(𝑡)−𝑥_𝑡(𝑡), 𝑒_𝑦(𝑡)≜𝑦(𝑡)−𝑦_𝑡(𝑡), and 𝑧_𝑒(𝑡)≜𝑧(𝑡)−𝑧_𝑡(𝑡). Beside, using the fixed gain matrices 𝐾∈ℝ^𝑚×𝑙 and 𝐾_ℒ𝒬∈ℝ^𝑚×𝑛, we consider the following control input for the Lipschitz uncertain nonlinear system of (2): 𝑢(𝑡)≜𝐾𝑒_𝑦(𝑡)+𝐾_ℒ𝒬𝑥_𝑡(𝑡)−𝜉𝑥_𝑡,𝑡+𝜓𝑒_𝑦,ℒ,𝑡, (10) where 𝜓(𝑒_𝑦,ℒ,𝑡)∈ℝ^𝑚 is a compensation input [5, 9] and has the following form 𝜓𝑒_𝑦,ℒ,𝑡≜ℒ𝑒_𝑦,𝑡𝑒_𝑦(𝑡), (11) where ℒ(𝑒_𝑦,𝑡)∈ℝ^𝑚×𝑙 is a variable gain matrix. Then one can see from (2) and (8)–(11) that the following uncertain error system with nonlinear terms can be derived: 𝑑𝑒(𝑡)𝑑𝑡=𝐴_𝐾𝑒(𝑡)+𝐵Δ(𝑡)𝐶𝑥(𝑡)+𝐵𝜉(𝑥,𝑡)−𝜉𝑥_𝑡𝑒,𝑡+𝐵ℒ_𝑦,𝑡𝑒_𝑦(𝑡)+Γ₁𝜔(𝑡),𝑒_𝑦(𝑡)(𝑡),𝑧=𝐶𝑒_𝑒(𝑡)=𝐷𝑒(𝑡)+Γ₂𝜔(𝑡), (12) where 𝐴_𝐾 is a stable matrix given by 𝐴_𝐾≜𝐴+𝐵𝐾𝐶.

Now we will give the definition of the variable gain robust output feedback control with guaranteed ℒ₂ gain performance 𝛾^∗>0 [19].

Definition 3. For the Lipschitz uncertain nonlinear system of (2), the control input of (10) is said to be a variable gain robust output feedback control with guaranteed ℒ₂ gain performance 𝛾^∗>0 if the uncertain nonlinear error system of (12) is internally stable, and ℋ^∞-norm of the uncertain nonlinear error system which transfers function from the disturbance input 𝑤(𝑡) to the controlled output 𝑧_𝑒(𝑡) is less than or equal to a positive constant 𝛾^∗.

By introducing a symmetric positive definite matrix 𝒫∈ℝ^𝑛×𝑛, we consider a quadratic function 𝒱(𝑒,𝑥_𝑡,𝑡)≜𝑒^𝑇(𝑡)𝒫𝑒(𝑡)+𝑥^𝑇_𝑡(𝑡)𝒮𝑥_𝑡(𝑡). Besides, we define the Halmitonian as ℋ𝑒,𝑥_𝑡≜𝑑,𝑡𝒱𝑑𝑡𝑒,𝑥_𝑡,𝑡+𝑧^𝑇_𝑒(𝑡)𝑧_𝑒(𝑡)−𝛾^∗2𝑤^𝑇(𝑡)𝑤(𝑡). (13) Then we have the following lemma for the variable gain robust output feedback control with guaranteed ℒ₂ gain performance 𝛾^∗.

Lemma 4. Consider the uncertain nonlinear error system of (12) and the control input of (10).
For the quadratic function 𝒱(𝑒,𝑥_𝑡,𝑡) and the signals 𝑧_𝑒(𝑡) and 𝜔(𝑡), if there exist symmetric positive definite matrices 𝒫∈ℝ^𝑛×𝑛 and 𝒮∈ℝ^𝑛×𝑛 and a positive scalar 𝛾^∗ satisfying the inequality ℋ𝑒,𝑥_𝑡,𝑡<0, (14) then control input of (10) is a variable gain robust output feedback control with guaranteed ℒ₂ gain performance 𝛾^∗.

Proof. By integrating both sides of the inequality of (14) from 0 to ∞ with 𝑒(0)=0 and 𝑥_𝑡(0)=0, we easily see from 𝒱(𝑒,𝑥_𝑡,0)=0 that the inequality ^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)𝑑𝑡−𝛾^∗2∞₀𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡+𝒱(𝑥,∞)<0 (15) holds. We see from the inequality of (15) that the uncertain nonlinear error system of (12) is robustly stable (internally stable). Note that if 𝜔(𝑡)≡0, then from the inequality of (14) the quadratic function 𝒱(𝑒,𝑥_𝑡,𝑡) becomes a Lyapunov function for the augmented system consisting of the target model of (9) and the uncertain nonlinear error system of (12). Namely, internal stability is guaranteed for the uncertain nonlinear error system of (12) and that the ℋ^∞-norm of the uncertain nonlinear error system which transfers function from the disturbance input 𝑤(𝑡)∈ℝ^𝑞 to the controlled output 𝑧_𝑒(𝑡)∈ℝ^𝑝 is less than a given positive constant 𝛾^∗, because the inequality of (15) means the following relation: ‖𝑧(𝑡)‖_ℒ2<𝛾^∗‖𝑤(𝑡)‖_ℒ2. (16) Thus the proof of Lemma 4 is completed.

Therefore, our control objective is to design the variable gain robust output feedback controller with guaranteed ℒ₂ gain performance 𝛾^∗ for the Lipschitz uncertain nonlinear system of (2). That is to derive the symmetric positive definite matrices 𝒫∈ℝ^𝑛×𝑛 and 𝒮∈ℝ^𝑛×𝑛, a positive scalar 𝛾^∗, and the variable gain matrix ℒ(𝑒_𝑦,𝑡)∈ℝ^𝑚×𝑙 which satisfies the inequality condition of (14) for all admissible nonlinear perturbations and the disturbance input 𝜔(𝑡)∈ℒ₂[0,∞).

4. Main Results

In this section, we show our main results.

The following theorem gives an LMI-based design synthesis of the variable gain robust output feedback control with guaranteed ℒ₂ gain performance 𝛾^∗.

Theorem 5. Consider the uncertain nonlinear error system of (12) with the variable gain matrix ℒ(𝑒_𝑦,𝑡)∈ℝ^𝑚×𝑙.
If there exist symmetric positive definite matrices 𝒫∈ℝ^𝑛×𝑛, 𝒮∈ℝ^𝑛×𝑛, Ξ∈ℝ^𝑙×𝑙, and Ψ∈ℝ^𝑙×𝑙 and positive scalars 𝜖, 𝜇 and 𝛾 satisfying the LMIs 𝐻_𝑒𝐴^𝑇_𝐾𝒫+𝐷^𝑇𝐷+𝜇𝐼_𝑛𝒫Γ₁+𝐷^𝑇Γ₂⋆Γ^𝑇₂Γ₂−𝛾𝐼_𝑞<0,−𝐶^𝑇Ψ𝐶𝒫𝐶^𝑇𝒯^𝑇⋆−𝐼_𝑚𝐶≤0,_𝑇Ξ𝐶−𝐻_𝑒𝒫𝐶^𝑇𝒯^𝑇𝐻𝒯𝐶≤0,_𝑒𝐴+𝐵𝐾_ℒ𝒬^𝑇𝒮+𝜖𝐶^𝑇𝐶<0, (17) then, by using the solution of the LMIs of (17) and the known scalar 𝜒^∗, the variable gain matrix ℒ(𝑒_𝑦,𝑡)∈ℝ^𝑚×𝑙 is determined as ℒ𝑒_𝑦≜−𝜒𝑒,𝑡^𝑦,𝑡Ξ^1/2𝑒_𝑦(𝑡)²𝒯𝑖𝑓𝑒_𝑦(𝑡)−𝜒𝑒≠0,_𝑦,𝑡⁻Ξ^1/2𝑒_𝑦(𝑡⁻)²𝒯𝑖𝑓𝑒_𝑦(𝑡)𝜒𝑒≡0,_𝑦,𝑡≜𝜗Ψ^1/2𝑒_𝑦(𝑡)²+𝜎𝑒_𝑦(𝑡)². (18) In (18), 𝑡⁻=lim_{𝜈>0,𝜈→0}(𝑡−𝜈) [5], and 𝜗 is a positive scalar defined as 1𝜗≜𝜎+1𝜖+𝜒^∗2𝜇, (19) where 𝜎∈ℝ¹>0 is a design parameter. Then the control input of (10) is a variable gain robust output feedback control with guaranteed ℒ₂ gain performance 𝛾^∗=√𝛾.

Proof. Consider the quadratic function 𝒱(𝑒,𝑥_𝑡,𝑡), the Hamiltonian ℋ(𝑒,𝑥_𝑡,𝑡) of (13), and the inequality of (14).
The time derivative of the quadratic function 𝒱(𝑒,𝑥_𝑡,𝑡) along the trajectory of the target model of (9) and the one of the uncertain nonlinear error systems of (12) is given by 𝑑𝒱𝑑𝑡𝑒,𝑥_𝑡,𝑡=𝑒^𝑇(𝑡)𝐻_𝑒𝐴^𝑇_𝐾𝒫𝑒(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫𝐵Δ(𝑡)𝐶𝑥(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫Γ₁𝜔(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫𝐵ℒ𝑒_𝑦,𝑡𝑒_𝑦(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫𝐵𝜉(𝑥,𝑡)−𝜉𝑥_𝑡,𝑡+𝑥^𝑇_𝑡(𝑡)𝐻_𝑒𝐴+𝐵𝐾_ℒ𝒬^𝑇𝒮𝑥_𝑡(𝑡). (20) Namely, the condition of (14) can be written as ℋ𝑒,𝑥_𝑡,𝑡=𝑒^𝑇(𝑡)𝐻_𝑒𝐴^𝑇_𝐾𝒫𝑒(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫𝐵Δ(𝑡)𝐶𝑥(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫Γ₁𝜔(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫𝐵ℒ𝑒_𝑦,𝑡𝑒_𝑦(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫𝐵𝜉(𝑥,𝑡)−𝜉𝑥_𝑡,𝑡+𝑥^𝑇_𝑡(𝑡)𝐻_𝑒𝐴+𝐵𝐾_ℒ𝒬^𝑇𝒮𝑥_𝑡(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝐷^𝑇Γ₂𝜔(𝑡)+𝜔^𝑇(𝑡)Γ^𝑇₂Γ₂−𝛾^∗2𝐼_𝑞𝜔(𝑡)+𝑒^𝑇(𝑡)𝐷^𝑇𝐷𝑒(𝑡). (21) Besides, one can see from the assumption of (5) for the nonlinear term and Lemma 1 that the following inequality holds: ℋ𝑒,𝑥_𝑡,𝑡≤𝑒^𝑇(𝑡)𝐻_𝑒𝐴^𝑇_𝐾𝒫+𝐷^𝑇𝐷𝑒(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫𝐵Δ(𝑡)𝐶𝑒(𝑡)+𝑥_𝑡(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫Γ₁𝜔(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫𝐵ℒ𝑒_𝑦,𝑡𝑒_𝑦(𝑡)+2𝜒^∗𝐵^𝑇𝒫𝑒(𝑡)𝑒(𝑡)+𝑥^𝑇_𝑡(𝑡)𝐻_𝑒𝐴+𝐵𝐾_ℒ𝒬^𝑇𝒮𝑥_𝑡(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝐷^𝑇Γ₂𝜔(𝑡)+𝜔^𝑇(𝑡)Γ^𝑇₂Γ₂−𝛾^∗2𝐼_𝑞𝜔(𝑡). (22) From the assumption of (6), we have the following inequality as a sufficient condition for the inequality of (22): ℋ𝑒,𝑥_𝑡,𝑡≤𝑒^𝑇(𝑡)𝐻_𝑒𝐴^𝑇_𝐾𝒫+𝐷^𝑇𝐷𝑒(𝑡)+1𝜎𝑒^𝑇(𝑡)𝒫𝐶^𝑇𝒯^𝑇𝒯𝐶𝒫𝑒(𝑡)+𝜎𝑒^𝑇_𝑦(𝑡)𝑒_𝑦(𝑡)+1𝜖𝑒^𝑇(𝑡)𝒫𝐶^𝑇𝒯^𝑇𝒯𝐶𝒫𝑒(𝑡)+𝜖𝑥^𝑇_𝑡(𝑡)𝐶^𝑇𝐶𝑥_𝑡(𝑡)+𝜒^∗2𝜇𝑒^𝑇(𝑡)𝒫𝐶^𝑇𝒯^𝑇𝒯𝐶𝒫𝑒(𝑡)+𝜇𝑒^𝑇(𝑡)𝑒(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫Γ₁𝜔(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫𝐵ℒ𝑒_𝑦,𝑡𝑒_𝑦(𝑡)+𝑥^𝑇_𝑡(𝑡)𝐻_𝑒𝐴+𝐵𝐾_ℒ𝒬^𝑇𝒮𝑥_𝑡(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝐷^𝑇Γ₂𝜔(𝑡)+𝜔^𝑇(𝑡)Γ^𝑇₂Γ₂−𝛾^∗2𝐼_𝑞𝜔(𝑡). (23) Here we have used the well-known following relation: 2𝑎^𝑇𝑏≤𝜇𝑎^𝑇1𝑎+𝜇𝑏^𝑇𝑏. (24) Additionally, introducing the variable 𝛾≜(𝛾^∗)² and some algebraic manipulations give the following inequality for Hamiltonian ℋ(𝑒,𝑥_𝑡,𝑡): ℋ𝑒,𝑥_𝑡≤𝑒(𝑡)𝜔(𝑡),𝑡^𝑇Ψ𝑒(𝑡)𝜔(𝑡)𝒫,𝜇,𝛾+𝐻_𝑒𝑒^𝑇(𝑡)𝒫𝐶^𝑇𝒯ℒ𝑒_𝑦,𝑡𝑒_𝑦(𝑡+1)𝜎+1𝜖+𝜒^∗2𝜇𝑒^𝑇(𝑡)𝒫𝐶^𝑇𝒯^𝑇𝒯𝐶𝒫𝑒(𝑡)+𝜎𝑒^𝑇_𝑦(𝑡)𝑒_𝑦(𝑡)+𝑥^𝑇_𝑡(𝑡)𝐻_𝑒𝐴+𝐵𝐾_ℒ𝒬^𝑇𝒮+𝜖𝐶^𝑇𝐶𝑥_𝑡(𝑡), (25) where Ψ(𝒫,𝜇,𝛾)∈ℝ^𝑛×𝑛 is given by Ψ≜𝐻𝒫,𝜇,𝛾^𝑒𝐴^𝑇_𝐾𝒫+𝐷^𝑇𝐷+𝜇𝐼_𝑛𝒫Γ₁+𝐷^𝑇Γ₂⋆Γ^𝑇₂Γ₂−𝛾𝐼_𝑞. (26)
Now we consider the case of 𝑒_𝑦(𝑡)≠0. Note that the second LMI of (17) is equivalent to the following inequality: −𝐶^𝑇Ψ𝐶+𝒫𝐶^𝑇𝒯^𝑇𝒯𝐶𝒫≤0. (27) Thus in this case, considering the variable gain matrix ℒ(𝑒_𝑦,𝑡)∈ℝ^𝑚×𝑙 given by (18) and the second LMI, the third one, and the fourth one of (17) and some trivial algebraic manipulations giveℋ𝑒,𝑥_𝑡≤𝑒(𝑡)𝜔(𝑡),𝑡_𝑇Ψ𝑒(𝑡)𝜔(𝑡)𝒫,𝜇,𝛾+𝜗𝑒^𝑇(𝑡)𝒫𝐶^𝑇𝒯^𝑇𝒯𝐶𝒫𝑒(𝑡)+𝜎𝑒^𝑇_𝑦(𝑡)𝑒_𝑦(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒫𝐶^𝑇𝒯−𝜒𝑒_𝑦,𝑡Ξ^1/2𝑒_𝑦(𝑡)²𝒯𝑒_𝑦(𝑡)=𝑒(𝑡)𝜔(𝑡)^𝑇Ψ𝑒(𝑡)𝜔(𝑡)𝒫,𝜇,𝛾. (28) In (28), 𝜗 is the positive constant given by (19). Besides, one can see from the first LMI of (17) and the relation of (28) that the inequality of (14) is satisfied.
Next, we consider the case of 𝑒_𝑦(𝑡)≡0. In this case, we see from the definition of the compensation input of (11) and the variable gain matrix of (18) that if the LMIs of (17) are satisfied, then the inequality of (14) also holds.
From the above discussion, if the LMIs of (17) are feasible, then the inequality condition of (14) for Hamiltonian ℋ(𝑒,𝑥_𝑡,𝑡) is always satisfied. Namely, the proposed control input of (10) is a variable gain robust output feedback control with guaranteed ℒ₂ gain performance. This completes the proof of Theorem 5.

The LMIs of (17) define a convex solution set of 𝒫>0, 𝒮>0, Ψ≥0, Ξ≥0, and 𝜇>0. Therefore various efficient convex optimization algorithms can be used to test whether the linear matrix inequality (LMI) is solvable and to generate particular solutions. Since our interest is in establishing ℒ₂ gain performance, we can also minimize the parameter 𝛾∈ℝ¹ (see Appendix A).

5. Illustrative Examples

In order to demonstrate the efficiency of the proposed control scheme, we have run a simple example. The control problem considered here is not necessarily practical. However, the simulation results stated below illustrate the distinct feature of the proposed adaptive robust controller.

Consider the uncertain nonlinear system described by 𝑑𝑥(𝑡)=𝑥(𝑡)+Δ(𝑡)𝑥(𝑡)+𝑢(𝑡)+𝜉()+𝜔(),𝑦(𝑡)=𝑥(𝑡),𝑧(𝑡)=𝑥(𝑡)(𝑡),𝑑𝑡−2.00.0−6.00.01.01.03.00.0−7.02.01.00.01.00.01.00.03.01.02.01.00.02.01.00.0𝑥,𝑡1.00.01.0𝑥,𝑡1.00.00.00.01.00.01.00.00.0+𝜔 (29) that is, 𝒯=(2.01.0). In this example we assume that the positive scalar 𝜒^∗∈ℝ¹ in (5) is given by 𝜒^∗=5.0.

Firstly, selecting the weighting matrices such as 𝒬_𝑡=𝐼₃ and ℛ_𝑡=1.0×10¹, we have the solution of algebraic Riccati equation for the standard LQ control problem and the optimal feedback gain matrix 𝐾_ℒ𝒬∈ℝ^1×3 such as 𝒳_𝑡=2.2366×10⁻¹1.04461.9538×10⁻²⋆1.6170×10¹1.2562⋆⋆2.2216×10⁻¹,𝐾_ℒ𝒬=−1.4919×10⁻¹−1.8259−1.2953×10⁻¹. (30)

Next, adopting the LMI-based algorithm based on the work of [18] (see Appendix B), we design an output feedback gain matrix 𝐾∈ℝ^1×2 for the nominal system. We select the design parameter 𝛼 such as 𝛼=3.0; then, by applying the LMI-based design algorithm, we obtain the following output feedback gain matrix 𝐾∈ℝ^1×2 for the nominal system: 𝐾=−8.0484×10⁻¹−1.0525×10¹. (31)

Finally, in order to derive the proposed controller, solve the LMIs of (17). By solving the LMIs of (17), we have the solution of (31) and 𝒫=,1.5983−2.0562−1.0664⋆4.87372.1874⋆⋆2.4921𝒮=2.2743×10⁻¹−6.4569×10⁻¹−8.0576×10⁻¹⋆3.85546.1920×10⁻¹,⋆⋆3.3330𝛾=1.0204,𝜇=3.5659×10⁻¹,,𝜖=1.7205,Θ=2.88021.38671.38671.2897Ψ=1.71543.1942×10⁻¹.⋆1.6381 (32) Therefore, guaranteed ℒ₂ gain performance 𝛾^∗=√𝛾 via the proposed variable gain controller is given by 𝛾^∗=1.0102. (33)

6. Conclusions

In this paper, we have proposed a variable gain robust output feedback controller with guaranteed ℒ₂ gain performance for a class of Lipschitz uncertain nonlinear systems. The proposed control scheme is adaptable when some assumptions are satisfied, and, in cases where only the output signal of the controlled system is available, the proposed method can be used widely.

The future research subject is the extension of proposed robust controller to such a broad class of systems as large-scale interconnected system, time-delay systems, and so on. Furthermore in future work, we will examine the assumption of (6).

Appendices

A. Optimal Guaranteed ℒ₂ Gain Performance

Since the LMIs of (17) define a convex solution set, we consider minimizing the parameter 𝛾∈ℝ¹, because our interest is in establishing ℒ₂ gain performance. Thus our design problem can be reduced to the following constrained convex optimization problem Minimize_{𝒫>0,𝒮>0,Ψ≥0,Ξ≥0,𝜇>0}𝛾subjectto(17). (A.1) If the optimal solution 𝒫>0, 𝒮>0, Ψ≥0, Ξ≥0, and 𝜇>0 of the constrained convex optimization problem of (28) is obtained, then the control input of (10) with the variable gain matrix ℒ(𝑒_𝑦,𝑡) in (18) is a variable gain robust output feedback control with guaranteed optimal ℒ₂ gain performance 𝛾^∗=√𝛾.

As a result, the following theorem is obtained.

Theorem A.1. Consider the Lipschitz uncertain nonlinear system of (2) and the control input of (10).
The control input of (10) is a variable gain robust output feedback control with guaranteed optimal ℒ₂ gain performance 𝛾^∗=√𝛾 provided that the constrained convex optimization problem of (A.1) is feasible.

B. LMI-Based Design Algorithm for an Output Feedback Gain

Consider the following linear dynamical system: 𝑑𝑥(𝑡)(𝑡)(𝑡),𝑦(𝑡)(𝑡),𝑑𝑡=𝐴𝑥+𝐵𝑢=𝐶𝑥 (B.1) where 𝑥(𝑡)∈ℝ^𝑛, 𝑢(𝑡)∈ℝ^𝑚 and 𝑦(𝑡)∈ℝ^𝑙 are the vectors of the state, the control input, and the measurement output, respectively, and the matrices 𝐴, 𝐵, and 𝐶 denote the nominal values of system parameters.

For the linear system (B.1), we consider the static output feedback control 𝑢(𝑡)=𝐾𝑦(𝑡). The following LMI-based algorithm to derive the static output feedback gain matrix 𝐾∈ℝ^𝑚×𝑙 has been developed by the existing result of [18].

A LMI-Based Algorithm

Step 1. Define 𝐴_𝛼=𝐴+𝛼𝐼_𝑛, where 𝛼∈ℝ¹ is the desired prescribed degree of stability, as described in Anderson and Moore [20].

Step 2. Solve the following LMI problem:Minimize_𝒳,𝒴[Trace{𝒳}]subjectto𝒳−𝐼_𝑛𝐻>0,_𝑒𝐴_𝛼𝒳+𝐵𝒴<0. (B.2)

Step 3. By using the matrices 𝒳∈ℝ^𝑛×𝑛 and 𝒴∈ℝ^𝑚×𝑛, set the state feedback gain matrix such as 𝐾_𝑠𝑓=𝒴𝒳⁻¹.

Step 4. Solve the following LMI feasibility problem: Find𝜎and𝒫suchthat𝒫>𝐼_𝑛,𝐻_𝑒𝒫𝐴_𝛼+𝐵𝐾_𝑠𝑓𝐻<0,_𝑒𝒫𝐴_𝛼−𝜎𝐶^𝑇𝐶<0,𝜎>0. (B.3)

Step 5. In order to derive an output feedback gain 𝐾∈ℝ^𝑚×𝑙, fix the matrix 𝒫∈ℝ^𝑛×𝑛 and solve the following LMI minimization problem: Minimize_𝜆,𝐾[𝜆]subjectto𝜆𝒱^𝑇_𝐾𝒱_𝐾𝐼_𝑚×𝑙𝐻>0,_𝑒𝒫𝐴_𝛼+𝐵𝐾𝐶<0, (B.4) where 𝒱_𝐾=vec(𝐾).

Acknowledgment

The authors would like to thank CAE Solutions Corporation for providing its support in conducting this study.