Synthesis of Adaptive Gain Robust Output Feedback Controllers for a Class of Lipschitz Nonlinear Systems with Unknown Upper Bound of Uncertainty

Abstract

We propose a new adaptive gain robust output feedback controller for a class of the Lipschitz nonlinear systems with unknown upper bound of uncertainty. The proposed adaptive gain robust output feedback controller is designed so as to reduce the effect of uncertainties and Lipschitz nonlinearities. In this paper, we show that sufficient conditions for the existence of the proposed adaptive gain robust output feedback controller are reduced to LMI conditions. Finally, the effectiveness of the proposed robust output feedback controller is demonstrated by numerical simulations.

1. Introduction

Robustness of control systems to uncertainties has always been the central issue in feedback control, and therefore the problems of stability analysis and stabilization for uncertain systems have received much attention for a long time (e.g., [1, 2] and references therein). In particular, there are lots of existing results for state feedback robust control such as quadratic stabilizing control and ℋ^∞ control (see [3, 4] and references therein). Besides, some design methods of variable gain robust state feedback controllers for uncertain systems have been suggested (e.g., [5–8]). Yamamoto and Yamauchi [5] proposed a design method of a robust controller with the ability to adjust control performances adaptively. In [6], an adaptive robust controller with adaptation mechanism has been presented and the adaptive robust controller is tuned on-line based on the information about parameter uncertainties. Besides, we have proposed robust controllers with adaptive compensation inputs [7, 8]. These controllers consist of a fixed gain controller and a variable gain one, and the variable gain controller is tuned by updating laws.

However, not all the states are measurable in practical systems because of technical, physical, and/or economic reasons. Therefore, the control strategies via observer-based robust controllers (e.g., [9, 10]) or robust output feedback one (e.g., [11–13]), which is of interest in this paper, have also been well studied. For robust output feedback controllers, Moheimani and Petersen [11] have presented a set of cross-coupled algebraic Riccati equations and algebraic Lyapunov equations. Geromel et al. [12] and Iwasaki et al. [13] adopted linear matrix inequality (LMI) approaches to design static output feedback controllers based on a set of the Lyapunov inequalities coupled by the constraint that one Lyapunov matrix is the inverse of another. Additionally the work of Matsuoka and Hagino [14] has presented an observer-based variable gain controller for a class of linear systems with uncertainties of which upper bounds are unknown, and we have also proposed an adaptive robust output feedback controller for a class of linear systems with uncertainties [15].

By the way, in recent years, there have been increasing attention for the problem of global stabilization of nonlinear systems via output feedback control (e.g., [16–18]). In the work of Mazenc et al. [16], it was presented through counter examples that some extra growth conditions on the unmeasurable states of the controlled system are usually necessary for the global stabilization of nonlinear systems via output feedback. Additionally, finite-time method and homogeneous domination approach for the output feedback control problem have also been proposed (e.g., [19–21]). Polendo and Qian [20] have suggested output feedback controllers for a class of uncertain nonlinear systems via homogeneous domination approach, and Li and Qian [21] adopted the concept of finite-time stabilization so as to design a dynamic output feedback controller for a class of continuous but nonsmooth nonlinear systems. Besides, some results have focused on considering a selective class of nonlinear systems by placing some structural constraints on the nonlinearities in order to derive output feedback control. The nonlinear systems whose nonlinearity is in a triangular form are considered in [22]. In the work of Choi and Lim [23], 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 has been presented. However, in the existing results, the design parameters are determined by trial and error, and uncertainties in the system dynamics have not been considered.

From these viewpoints on the basis of our results [15, 24], we propose a new adaptive gain robust output feedback controller for a class of uncertain Lipschitz nonlinear systems. The uncertainties and the nonlinearities under consideration supposed to satisfy the matching condition, and the perturbation region of uncertainty is bounded, but its upper bound is unknown. Besides, the proposed adaptive gain robust output feedback controller consists of a fixed gain controller, a variable gain one, and an adjustable parameter tuned by updating laws. The fixed gain controller is determined by using the nominal system, and the variable gain controller and the adjustable parameter are designed so as to reduce the effect of both uncertainties and nonlinearities. In this paper, we show that sufficient conditions for the existence of the proposed adaptive gain robust output feedback controller are reduced to LMI conditions.

This paper is organized as follows. In Section 2, we introduce the class of uncertain Lipschitz nonlinear systems under consideration. Section 3 contains the main results. The design method of the adaptive robust output feedback controller for the uncertain Lipschitz nonlinear systems is presented. Finally, numerical examples are included to illustrate the results developed in this paper.

2. Preliminaries

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

In the sequel, we use the following notation. For a matrix 𝒜, the transpose of matrix 𝒜 and the inverse of one are denoted by 𝒜^𝑇 and 𝒜⁻¹, respectively, and rank{𝒜} represents the rank of the matrix 𝒜. Also, 𝐻_𝑒{𝒜} and 𝐼_𝑛 represent 𝒜+𝒜^𝑇 and 𝑛-dimensional identity matrix, respectively, and vec(𝒜) denotes the column vector of the matrix 𝒜; that is, the operator “vec” vectorizes a matrix by stacking its columns. The notation diag(𝒜₁,…,𝒜_𝑁) denotes a block diagonal matrix composed of matrices 𝒜_𝑖 for 𝑖=1,…,𝑁. 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 symbols “≜” and “★” mean equality by definition and symmetric blocks in matrix inequalities, respectively. Besides, for a symmetric matrix 𝒫, 𝜆_min{𝒫} (resp., 𝜆_max{𝒫}) represents the minimal eigenvalue (resp. maximal eigenvalue). It is well-known that for any symmetric matrix 𝒫∈ℝ^𝑛×𝑛, eigenvalues of 𝒫∈ℝ^𝑛×𝑛 are real number [25].

Furthermore, the following well-known lemmas are used in this paper.

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

Proof. The above relation can be easily obtained by Schwartz’s inequality [25].

Lemma 2 (Schur complement). 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. [26].

Lemma 3 (𝒮-procedure). Let ℱ(𝑥) and 𝒢(𝑥) be two arbitrary quadratic forms over ℝ^𝑛. Then ℱ(𝑥)<0 for all 𝑥∈ℝ^𝑛 satisfying 𝒢(𝑥)≤0 if and only if there exists a nonnegative scalar 𝜏 such that ℱ(𝑥)−𝜏𝒢(𝑥)≤0for∀𝑥∈ℝ^𝑛. (2)

Proof. See Boyd et al. [26].

Lemma 4 (Barbalat’s lemma). Let 𝜙∶ℝ→ℝ be a uniformly continuous function on [0,∞). Suppose that lim_𝑡→∞∫^𝑡₀𝜙(𝜏)𝑑𝜏 exists and is finite. Then 𝜙(𝑡)→0as𝑡→∞. (3)

Proof. See Khalil [27].

3. Problem Formulation

Consider the uncertain Lipschitz nonlinear system described by the following state equation: 𝑑𝑥(𝑡)=((𝑡)𝐸)𝑥(𝑡)(𝑡)()𝑦(𝑡)(𝑡),𝑑𝑡𝐴+𝐵Δ+𝐵𝑢+𝛿𝑥,𝑡=𝐶𝑥 (4) 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 measured output, respectively. In (4), the matrices 𝐴,𝐵, and 𝐶 are the nominal values of system parameters, and the matrix Δ(𝑡)∈ℝ^𝑝×𝑞 denotes unknown time-varying parameters which satisfy ‖Δ(𝑡)‖≤𝜗^∗, (5) where the upper bound 𝜗^∗ is bounded, but it is unknown. Additionally in this paper, we assume that the nonlinear term 𝛿(𝑥,𝑡)∈ℝ^𝑛 in (4) is given by 𝛿(𝑥,𝑡)=𝐵𝜉(𝑥,𝑡), (6) and for the function 𝜉∶ℝ^𝑛×𝑅→ℝ^𝑚, there exists a known positive constant scalar 𝜒^∗ such that for all 𝑥₁,𝑥₂∈ℝ^𝑛𝜉𝑥₁,𝑡−𝜉𝑥₂,𝑡≤𝜒^∗𝑥₁−𝑥₂. (7) Note that since not all the states are measurable, the nonlinear term 𝛿(𝑥,𝑡) is unknown. Besides, we introduce the following assumption for the system parameters [15, 24]: 𝐵^𝑇=𝒯𝐶, (8) where 𝒯∈ℝ^𝑚×𝑙 is a known constant matrix.

The nominal system, ignoring unknown parameters and nonlinearities in (4), is given by 𝑑𝑑𝑡𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑦(𝑡)=𝐶𝑥(𝑡), (9) and it is supposed to be stabilizable via static output feedback control. Namely, there exists a static output feedback stabilizing control 𝑢(𝑡)=𝐾𝑦(𝑡) (i.e., a fixed gain matrix 𝐾∈ℝ^𝑚×𝑙). In other words since the nominal system of (9) is stabilizable via static output feedback control, the matrix 𝐴_𝐾≜𝐴+𝐵𝐾𝐶 is asymptotically stable. Note that the output feedback gain matrix 𝐾∈ℝ^𝑚×𝑙 is designed by using the existing results (e.g., [28, 29]). Besides, in this paper, we consider the following target model so as to generate the desirable trajectory: 𝑑𝑥𝑑𝑡^𝑡(𝑡)=𝐴𝑥_𝑡(𝑡)+𝐵𝑢_𝑡(𝑡)+𝛿𝑥_𝑡,𝑦,𝑡_𝑡(𝑡)=𝐶𝑥_𝑡(𝑡). (10) In order to generate the desirable trajectory for the uncertain system of (4), we select the control input for the target model such as 𝑢_𝑡(𝑡)=𝐾_ℒ𝒬𝑥_𝑡(𝑡)−𝜉(𝑥_𝑡,𝑡) where 𝐾_ℒ𝒬∈ℝ^𝑚×𝑛 is determined by adopting the standard LQ problem. Namely, by using the solution of the algebraic Riccati equation 𝐻_𝑒{𝐴^𝑇𝒳_𝑡}−𝒳_𝑡𝐵ℛ_𝑡⁻¹𝐵^𝑇𝒳_𝑡+𝒬_𝑡=0, the gain matrix 𝐾_ℒ𝒬 is determined as 𝐾_ℒ𝒬=−ℛ_𝑡⁻¹𝐵^𝑇𝒳_𝑡. Of course, some other design methods can also be utilized. Note that the target model with the control input 𝑢_𝑡(𝑡)=𝐾_ℒ𝒬𝑥_𝑡(𝑡)−𝜉(𝑥_𝑡,𝑡) can be written as the following form: 𝑑𝑥𝑑𝑡^𝑡(𝑡)=𝐴+𝐵𝐾_ℒ𝒬𝑥_𝑡(𝑡),𝑦_𝑡(𝑡)=𝐶𝑥_𝑡(𝑡). (11)

Now on the basis of the works of Oya and Hagino ([15, 24]), by introducing the error vectors 𝑒(𝑡)≜𝑥(𝑡)−𝑥_𝑡(𝑡) and 𝑒_𝑦(𝑡)≜𝑦(𝑡)−𝑦_𝑡(𝑡), we consider the following control input for the uncertain Lipschitz nonlinear system of (4): 𝑢(𝑡)≜𝐾𝑒_𝑦(𝑡)+𝐾_ℒ𝒬𝑥_𝑡(𝑡)−𝜉𝑥_𝑡,𝑡+𝜓𝑒_𝑦,𝑥_𝑡,𝜃,𝑡. (12) In (12), 𝜓(𝑒_𝑦,𝑥_𝑡,𝜃,𝑡)∈ℝ^𝑚 is an adaptive compensation input where 𝜃(𝑡)∈ℝ¹ is an adjustable parameter. Then one can see from (4), (6), and (10)–(12) that the following uncertain error system with nonlinear terms can be derived: 𝑑𝑒(𝑡)𝑑𝑡=𝐴_𝐾𝑒(𝑡)(𝑡)(𝑡)𝜉()𝑥+𝐵Δ𝐸𝑥+𝐵𝑥,𝑡−𝜉_𝑡,𝑡+𝐵𝜓𝑒_𝑦,𝑥_𝑡,,𝑒𝜃,𝑡_𝑦(𝑡)=𝐶𝑒(𝑡). (13)

From the above, our control objective is to design the adaptive gain robust output feedback controller which achieves not only robust stability for the uncertain Lipschitz nonlinear system of (4) but also satisfactory transient behavior as closely as possible to desired trajectory generated by the target model. That is to derive the adaptive compensation input 𝜓(𝑒_𝑦,𝑥_𝑡,𝜃,𝑡)∈ℝ^𝑚 which stabilizes the uncertain nonlinear error system of (13).

4. Main Results

In this section, we show an LMI-based design method of the adaptive gain robust output feedback controller for the uncertain Lipschitz nonlinear system of (4). The following theorem gives an LMI-based design method of an adaptive gain robust output feedback controller.

Theorem 5. Consider the uncertain nonlinear error system of (13) with the adaptive compensation input 𝜓(𝑒_𝑦,𝑥_𝑡,𝜃,𝑡)∈ℝ^𝑚.
If there exist symmetric positive definite matrices 𝒮∈ℝ^𝑛×𝑛, Ξ∈ℝ^𝑙×𝑙, and Ψ∈ℝ^𝑙×𝑙 and the positive scalars 𝛾₁,𝛾₂, and 𝜖 satisfying the LMIs: 𝐻_𝑒𝒮𝐴_𝐾+𝛾₁𝐸^𝑇𝐸+𝜖𝜒^∗2𝐼_𝑛𝐶≤−𝒬,_𝑇Ξ𝐶−𝐻_𝑒𝒮𝐶^𝑇𝒯^𝑇𝒯𝐶≤0,−𝐶^𝑇Ψ_𝛾𝐶𝒮𝐶^𝑇𝒯𝒮𝐶^𝑇𝒯★−𝛾₁𝐼_𝑚0★★−𝛾₂𝐼_𝑚≤0,−𝐶^𝑇Ψ_𝜖𝐶𝒮𝐶^𝑇𝒯^𝑇★−𝜖𝐼_𝑚≤0, (14) then by using the solution of the LMIs of (14), we consider the adaptive compensation input 𝜓𝑒_𝑦,𝑥_𝑡,1𝜃,𝑡≜−Ξ^1/2𝐶𝑒(𝑡)²𝜔𝑒_𝑦,𝑥_𝑡,𝜃,𝑡𝒯𝑒_𝑦(𝑡), (15) where 𝜔(𝑒_𝑦,𝑥_𝑡,𝜃,𝑡) is the positive scalar function given by 𝜔𝑒_𝑦,𝑥_𝑡,≜Ψ𝜃,𝑡^𝛾1/2𝑒_𝑦(𝑡)⁴𝜃²(𝑡)Ψ_𝛾^1/2𝑒_𝑦(𝑡)²𝜃(𝑡)(𝑡)+Ψ+𝜎_𝜖1/2𝑒_𝑦(𝑡)⁴Ψ_𝜖^1/2𝑒_𝑦(𝑡)²(𝑡)+𝛾+𝜎₂₂𝐸𝑥_𝑡(𝑡)⁴𝛾₂𝐸𝑥_𝑡(𝑡)²(𝑡).+𝜎 (16) Besides, we introduce the following updating law for the adjustable parameter 𝜃(𝑡)∈ℝ¹: 𝑑𝜃(𝑡)1𝑑𝑡≜−𝜎(𝑡)𝜃(𝑡)2𝜗−𝜗^∗−Ψ_𝛾^1/2𝑒_𝑦(𝑡)²+𝜎(𝑡)𝜗^∗1=−𝜎(𝑡)−Ψ2𝜗^𝛾1/2𝑒_𝑦(𝑡)². (17) Hereby asymptotical stability of the uncertain nonlinear error system of (13) is guaranteed. In (14), 𝒬∈ℝ^𝑛×𝑛 is a symmetric positive definite matrix selected by designers, and 𝜎(𝑡)∈ℝ¹ in (16) is any positive uniform continuous and bounded function which satisfies ^𝑡_𝑡0𝜎(𝜏)𝑑𝜏≤𝜎^∗<∞, (18) where 𝑡₀ and 𝜎^∗ are an initial time and any positive constant, respectively.

Proof of Theorem 5. Firstly, we introduce the quadratic function 𝒱𝑒,𝜃,𝑡≜𝑒^𝑇(𝑡)𝒮𝑒(𝑡)+𝜗𝜃(𝑡)−𝜗^∗2. (19) The time derivative of the quadratic function 𝒱(𝑒,𝜃,𝑡) can be written as 𝑑𝒱𝑑𝑡𝑒,𝜃,𝑡=𝑒^𝑇(𝑡)𝐻_𝑒𝒮𝐴_𝐾𝑒(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐵Δ(𝑡)𝐸𝑥(𝑡)+𝑒^𝑇(𝑡)𝒮𝐵𝜓𝑒_𝑦,𝑥_𝑡,𝜃,𝑡+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐵𝜉(𝑥,𝑡)−𝜉𝑥_𝑡𝜃(𝑡),𝑡+2𝜗−𝜗^∗𝑑𝜃(𝑡).𝑑𝑡 (20) Now, using Lemma 1 and the assumptions of (6) and (7), we can obtain the following relation for the time derivative of the quadratic function 𝒱(𝑒,𝜃,𝑡): 𝑑𝒱𝑑𝑡𝑒,𝜃,𝑡≤𝑒^𝑇(𝑡)𝐻_𝑒𝒮𝐴_𝐾𝑒(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐵Δ(𝑡)𝐸𝑥(𝑡)+𝑒^𝑇(𝑡)𝒮𝐵𝜓𝑒_𝑦,𝑥_𝑡,𝜃,𝑡+2𝜒^∗𝐵^𝑇𝒮𝑒(𝑡)𝜉(𝑥,𝑡)−𝜉𝑥_𝑡𝜃(𝑡),𝑡+2𝜗−𝜗^∗𝑑𝜃(𝑡)𝑑𝑡≤𝑒^𝑇(𝑡)𝐻_𝑒𝒮𝐴_𝐾𝑒(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐵Δ(𝑡)𝐸𝑒(𝑡)+𝑒^𝑇(𝑡)𝒮𝐵Δ(𝑡)𝐸𝑥_𝑡(𝑡)+𝑒^𝑇(𝑡)𝒮𝐵𝜓𝑒_𝑦,𝑥_𝑡,𝜃,𝑡+2𝜒^∗𝐵^𝑇(𝑡𝑒(𝑡𝜃(𝑡)𝒮𝑒)‖)‖+2𝜗−𝜗^∗𝑑𝜃(𝑡).𝑑𝑡 (21)
Notice the fact that, for any positive constant 𝜇 and any vectors 𝑎 and 𝑏 with appropriate dimensions, 2𝑎^𝑇𝑏≤𝜇𝑎^𝑇1𝑎+𝜇𝑏^𝑇𝑏. (22) Then some algebraic manipulations yield 𝑑𝒱𝑑𝑡𝑒,𝜃,𝑡≤𝑒^𝑇(𝑡)𝐻_𝑒𝒮𝐴_𝐾+𝛾₁𝐸^𝑇𝐸+𝜖𝜒^∗2𝐼_𝑛𝑒(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇𝜓𝑒_𝑦,𝑥_𝑡,𝜃,𝑡+𝛾₂𝑥^𝑇_𝑡(𝑡)𝐸^𝑇𝐸𝑥_𝑡(𝑡)+𝜗^∗1𝛾₁𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇(𝑡)+1𝒯𝐶𝒮𝑒𝛾₂𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇(𝑡)+1𝒯𝐶𝒮𝑒𝜖𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇(𝑡)𝜃(𝑡)𝒯𝐶𝒮𝑒+2𝜗−𝜗^∗𝑑𝜃(𝑡).𝑑𝑡 (23) Besides, we obtain the following inequality for the time derivative of the quadratic function 𝒱(𝑒,𝜃,𝑡), 𝑑𝒱𝑑𝑡𝑒,𝜃,𝑡≤𝑒^𝑇(𝑡)𝐻_𝑒𝒮𝐴_𝐾+𝛾₁𝐸^𝑇𝐸+𝜖𝜒^∗2𝐼_𝑛𝑒(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇𝜓𝑒_𝑦,𝑥_𝑡,𝜃,𝑡+𝜗^∗𝑒^𝑇(𝑡)𝐶^𝑇Ψ_𝛾𝐶𝑒(𝑡)+𝑒^𝑇(𝑡)𝐶^𝑇Ψ_𝜖𝐶𝑒(𝑡)+𝛾₂𝑥^𝑇_𝑡(𝑡)𝐸^𝑇𝐸𝑥_𝑡(𝑡)+2𝜗𝜃(𝑡)−𝜗^∗𝑑𝜃(𝑡),𝑑𝑡 (24) because, by using Lemma 2 (Schur complement), one can see that the third LMI of (14) can be written as −𝐶^𝑇Ψ_𝛾1𝐶+𝛾₁𝒮𝐶^𝑇𝒯^𝑇1𝒯𝐶𝒮+𝛾₂𝒮𝐶^𝑇𝒯^𝑇𝒯𝐶𝒮≤0, (25) and fourth LMI of (14) is equivalent to the following matrix inequality: −𝐶^𝑇Ψ_𝜖1𝐶+𝜖𝒮𝐶^𝑇𝒯^𝑇𝒯𝐶𝒮≤0. (26) Using the first LMI and the second one of LMIs of (14) and introducing the adaptive compensation input of (15) and (16) and the updating law of (17), we have 𝑑𝒱𝑑𝑡𝑒,𝜃,𝑡≤−𝑒^𝑇(𝑡)𝒬𝑒(𝑡)+𝜗^∗𝑒^𝑇(𝑡)𝐶^𝑇Ψ_𝛾𝐶𝑒(𝑡)+𝑒^𝑇(𝑡)𝐶^𝑇Ψ_𝜖𝐶𝑒(𝑡)+𝛾₂𝑥^𝑇_𝑡(𝑡)𝐸^𝑇𝐸𝑥_𝑡(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇×−1Ξ^1/2𝐶𝑒(𝑡)²×𝜔𝑒_𝑦,𝑥_𝑡,𝜃,𝑡𝒯𝑒_𝑦(𝑡)+2𝜗𝜃(𝑡)−𝜗^∗𝑑𝜃(𝑡)𝑑𝑡=−𝑒^𝑇(𝑡)𝒬𝑒(𝑡)+Ψ_𝛾^1/2𝑒_𝑦(𝑡)²𝜃(𝑡)−𝜎(𝑡)𝜃(𝑡)−𝜗^∗2−𝜎(𝑡)𝜗^∗𝜃(𝑡)−𝜗^∗−Ψ_𝛾^1/2𝑒_𝑦(𝑡)⁴𝜃²(𝑡)Ψ_𝛾^1/2𝑒_𝑦(𝑡)²𝜃(𝑡)(𝑡)+Ψ+𝜎_𝜖1/2𝑒_𝑦(𝑡)²Ψ_𝜖^1/2𝑒_𝑦(𝑡)²(𝑡)𝜎(𝑡)+𝛾+𝜎₂𝐸𝑥_𝑡(𝑡)²𝛾₂𝐸𝑥_𝑡(𝑡)²(𝑡)𝜎(𝑡).+𝜎 (27) In addition, utilizing the well-known inequality for any positive constants 𝛼 and 𝛽, 0≤𝛼𝛽𝛼+𝛽≤𝛼∀𝛼,𝛽>0, (28) and some trivial manipulations give the relation 𝑑𝒱𝑑𝑡𝑒,𝜃,𝑡≤−𝑒^𝑇(𝑡)𝒬𝑒(𝑡)+3𝜎(𝑡)−𝜎(𝑡)𝜃(𝑡)−𝜗^∗2−𝜎(𝑡)𝜗^∗𝜃(𝑡)−𝜗^∗. (29) Namely, we have the following inequality: 𝑑𝒱𝑑𝑡𝑒,𝜃,𝑡≤−𝑒^𝑇(𝑡)𝒬𝑒(𝑡)+𝜑𝜎(𝑡). (30) Here we have used the well-known inequality of (22), and 𝜑 in (30) is a positive constant given by 𝜑≜3+(𝜗^∗)². By letting 𝜁^∗≜min{𝜆_min{𝒬}}, one can see that the inequality of (30) can be rewritten as 𝑑𝒱𝑑𝑡𝑒,𝜃,𝑡≤−𝜁^∗‖𝑒(𝑡)‖²+𝜑𝜎(𝑡). (31)
On the other hand, letting 𝑒𝜃(𝑡)=(𝑒^𝑇(𝑡)𝜃(𝑡))^𝑇, we see from the definition of the quadratic function 𝒱(𝑒,𝜃,𝑡) that there always exist two positive constants 𝛿_min and 𝛿_max such that, for any 𝑡≥𝑡₀, 𝜉⁻𝑒𝜃(𝑡)≤𝒱𝑒,𝜃,𝑡≤𝜉⁺𝑒𝜃(𝑡), (32) where 𝜉⁻𝜃(‖𝑒(𝑡)‖) and 𝜉⁺𝜃(‖𝑒(𝑡)‖) are given by 𝜉⁻𝑒𝜃(𝑡)≜𝛿_min𝑒𝜃(𝑡)²,𝜉⁺𝑒𝜃(𝑡)≜𝛿_max𝑒𝜃(𝑡)². (33)
It is obvious that any solution 𝑒(𝑡;𝑡₀,𝑒(𝑡₀)) of the uncertain nonlinear error system of (13) is continuous. In addition, it follows from (31) and (32) that, for any 𝑡≥𝑡₀, the relations of 0≤𝜉⁻𝜃(‖𝑒(𝑡)‖)≤𝒱(𝑒,𝜃,𝑡) and the following inequality holds: 𝒱𝑒,𝜃,𝑡=𝒱𝑒,𝜃,𝑡₀+^𝑡_𝑡0𝑑𝒱𝑑𝑡𝑒,𝜃,𝜏𝑑𝜏≤𝜉⁺𝑒𝜃𝑡₀−^𝑡_𝑡0𝜉^∗(‖𝑒(𝜏)‖)𝑑𝜏+𝜑^𝑡_𝑡0𝜎(𝜏)𝑑𝜏. (34) In (34), 𝜉^∗(‖𝑒(𝑡)‖) is defined as 𝜉^∗(‖𝑒(𝑡)‖)≜𝜁^∗‖𝑒(𝑡)‖². (35)
Therefore, from (34) we can obtain the following two results. Firstly, taking the limit as 𝑡 approaches infinity on both sides of the inequality of (34), we have the inequality 0≤𝜉⁺𝑒𝜃𝑡₀−lim_𝑡→∞^𝑡_𝑡0𝜉^∗(‖𝑒(𝜏)‖)𝑑𝜏+𝜑lim_𝑡→∞^𝑡_𝑡0𝜎(𝜏)𝑑𝜏. (36) Thus one can see from (18) and (36) that lim_𝑡→∞^𝑡_𝑡0𝜉^∗(‖𝑒(𝜏)‖)𝑑𝜏≤𝜉⁺𝑒𝜃𝑡₀+𝜑𝜎^∗. (37)
On the other hand, from (34), we obtain 0≤𝜉⁻𝑒𝜃(𝑡)≤𝜉⁺𝑒𝜃𝑡₀+𝜑^𝑡_𝑡0𝜎(𝜏)𝑑𝜏. (38) It follows from (18) and (38) that 0≤𝜉⁻𝑒𝜃(𝑡)≤𝜉⁺𝑒𝜃𝑡₀+𝜑𝜎^∗. (39)
The relation of (39) implies that 𝑒(𝑡) is uniformly bounded. Since 𝑒(𝑡) has been shown to be continuous, it follows that 𝑒(𝑡) is uniformly continuous. Therefore, one can see that 𝜉^∗(‖𝑒(𝑡)‖) is also uniformly continuous. Thus applying Lemma 4 (Barbalat’s lemma) to (37) yields lim_𝑡→∞𝜉^∗(‖𝑒(𝑡)‖)=lim_𝑡→∞𝜁^∗‖𝑒(𝑡)‖=0. (40) Namely, asymptotical stability of the uncertain nonlinear error system of (13) is ensured. Thus the uncertain Lipschitz nonlinear system of (4) is also stable.
It follows that the result of the theorem is true. Thus the proof of Theorem 5 is completed.

Theorem 5 provides a sufficient condition for the existence of an adaptive gain robust output feedback controller for uncertain Lipschitz nonlinear system of (4). Next, we consider a special case. In this case, we deal with the uncertain Lipschitz nonlinear system described by 𝑑𝑥(𝑡)=((𝑡)𝐶)𝑥(𝑡)(𝑡)𝑑𝑡𝐴+𝐵Δ+𝐵𝑢+𝛿_𝐵(),𝑦(𝑡)(𝑡).𝑥,𝑡=𝐶𝑥 (41) In (41) the nonlinear term 𝛿_𝐵(𝑥,𝑡)∈ℝ^𝑛 satisfies 𝛿_𝐵(𝑥,𝑡)=𝒮⁻¹𝐶^𝑇𝜉(𝑥,𝑡), (42) where the matrix 𝒮∈ℝ^𝑛×𝑛 is symmetric positive definite and it is a solution of the following LMIs: 𝐻_𝑒𝒮𝐴_𝐾+𝜖𝜒^∗2𝐼_𝑛𝐶≤−𝒬,_𝑇Ξ𝐶−𝐻_𝑒𝒮𝐶^𝑇𝒯^𝑇𝒯𝐶≤0,−𝐶^𝑇Ψ_𝛾𝐶𝒮𝐶^𝑇𝒯★−𝛾𝐼_𝑚≤0. (43) In (43), 𝒬∈ℝ^𝑛×𝑛 is a symmetric positive definite matrix selected by designers. Thus one can see from (6), (10)–(12), (42), and (43) that we have 𝑑𝑒(𝑡)𝑑𝑡=𝐴_𝐾𝑒(𝑡)+𝐵Δ(𝑡)𝐶𝑥(𝑡)+𝒮⁻¹𝐶^𝑇𝜉(𝑥,𝑡)−𝜉𝑥_𝑡,𝑡+𝐵𝜓𝑒_𝑦,𝑥_𝑡,,𝑒𝜃,𝑡_𝑦(𝑡)=𝐶𝑒(𝑡). (44)

Next theorem gives an LMI-based design method of an adaptive gain robust output feedback controller for this case.

Theorem 6. Consider the uncertain nonlinear error system of (44) with the adaptive compensation input 𝜓(𝑒_𝑦,𝑥_𝑡,𝜃,𝑡)∈ℝ^𝑚.
If there exist symmetric positive definite matrices 𝒮∈ℝ^𝑛×𝑛, Ψ∈ℝ^𝑙×𝑙, and Ξ∈ℝ^𝑙×𝑙 and the positive scalars 𝛾 and 𝜖 satisfying the LMIs of (43), then by using the solution of the LMIs of (43), one considers the adaptive compensation input 𝜓(𝑒_𝑦,𝑥_𝑡,𝜃,𝑡)∈ℝ^𝑚 described as 𝜓𝑒_𝑦,𝑥_𝑡,1𝜃,𝑡≜−Ξ^1/2𝐶𝑒(𝑡)²𝜔_𝐵𝑒_𝑦,𝑥_𝑡,𝜃,𝑡𝒯𝑒_𝑦(𝑡),𝜔_𝐵𝑒_𝑦,𝑥_𝑡,≜Ψ𝜃,𝑡^𝛾1/2𝑒_𝑦(𝑡)⁴𝜃²(𝑡)Ψ_𝛾^1/2𝑒_𝑦(𝑡)²𝜃(𝑡)(𝑡)+𝛾+𝜎₂𝑦(𝑡)⁴𝛾𝑦(𝑡)²(𝑡)+𝜖+𝜎₋₁𝑒_𝑦(𝑡)⁴𝜖⁻¹𝑒_𝑦(𝑡)²(𝑡),+𝜎 (45)(46) and the updating law of (17) for the adjustable parameter 𝜃(𝑡)∈ℝ¹. Hereby asymptotical stability of the uncertain error system with nonlinear terms of (4) is guaranteed. In (46), 𝜎(𝑡)∈ℝ¹ is any positive uniform continuous and bounded function satisfying (18).

Proof of Theorem 6. By using the symmetric positive definite matrix 𝒮∈ℝ^𝑛×𝑛, we consider the quadratic function 𝒱(𝑒,𝜃,𝑡) of (19). Then using the assumptions of (6) and (42) we have 𝑑𝒱𝑑𝑡𝑒,𝜃,𝑡=𝑒^𝑇(𝑡)𝐻_𝑒𝒮𝐴_𝐾𝑒(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇Δ(𝑡)𝐶𝑥(𝑡)+𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇𝜓𝑒_𝑦,𝑥_𝑡,𝜃,𝑡+𝐻_𝑒𝑒^𝑇(𝑡)𝐶^𝑇𝜉(𝑥,𝑡)−𝜉𝑥_𝑡𝜃(𝑡),𝑡+2𝜗−𝜗^∗𝑑𝜃(𝑡)𝑑𝑡≤𝑒^𝑇(𝑡)𝐻_𝑒𝒮𝐴_𝐾𝑒(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇Δ(𝑡)𝐶𝑥(𝑡)+2𝜒^∗‖𝐶𝑒(𝑡)‖‖𝑒(𝑡)‖+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇𝜓𝑒_𝑦,𝑥_𝑡,𝜃(𝑡)𝜃,𝑡+2𝜗−𝜗^∗𝑑𝜃(𝑡).𝑑𝑡 (47) In addition, applying the well-known inequality of (22) to the second term and the third one on the right-hand side of (47), we obtain the following relation for the time derivative of the quadratic function 𝒱(𝑒,𝜃,𝑡): 𝑑𝒱𝑑𝑡𝑒,𝜃,𝑡≤𝑒^𝑇(𝑡)𝐻_𝑒𝒮𝐴_𝐾+𝜖𝜒^∗2𝐼_𝑛𝑒(𝑡)+𝜗^∗𝛾𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇𝒯𝐶𝒮𝑒(𝑡)+𝛾𝑦^𝑇(𝑡)𝑦(𝑡)+1𝜖𝑒^𝑇_𝑦(𝑡)𝑒_𝑦(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇𝜓𝑒_𝑦,𝑥_𝑡,𝜃(𝑡)𝜃,𝑡+2𝜗−𝜗^∗𝑑𝜃(𝑡).𝑑𝑡 (48)
Now, one can see from (17), (43), (45), and (46) that if the following inequality holds, then the condition of (48) is also satisfied 𝑑𝒱𝑑𝑡𝑒,𝜃,𝑡≤−𝑒^𝑇(𝑡)(𝑡)+𝜗𝒬𝑒^∗𝛾𝑒^𝑇(𝑡)𝐶^𝑇Ψ_𝛾𝐶𝑒(𝑡)+𝛾𝑦^𝑇(𝑡)𝑦(𝑡)+1𝜖𝑒^𝑇_𝑦(𝑡)𝑒_𝑦(𝑡)+𝐻_𝑒𝑒^𝑇(𝑡)𝒮𝐶^𝑇𝒯^𝑇𝜓𝑒_𝑦,𝑥_𝑡,𝜃(𝑡)𝜃,𝑡+2𝜗−𝜗^∗𝑑𝜃(𝑡)𝑑𝑡≤−𝑒^𝑇(𝑡)𝒬𝑒(𝑡)+Ψ_𝛾^1/2𝑒_𝑦(𝑡)²𝜃(𝑡)−𝜎(𝑡)𝜃(𝑡)−𝜗^∗2−𝜎(𝑡)𝜗𝜃(𝑡)−𝜗^∗−Ψ_𝛾^1/2𝑒_𝑦(𝑡)⁴𝜃²(𝑡)Ψ_𝛾^1/2𝑒_𝑦(𝑡)²𝜃(𝑡)(𝑡)+𝛾𝑦(𝑡+𝜎)²𝛾𝑦(𝑡)²(𝑡)𝜎(𝑡)+𝜖+𝜎₋₁𝑒_𝑦(𝑡)²𝜖⁻¹𝑒_𝑦(𝑡)²(𝑡)𝜎(𝑡).+𝜎 (49) Besides, from (49) we obtain the inequality of (30). Therefore, one can see from Proof of Theorem 5 that the rest of proof of Theorem 6 is straightforward.

Remark 7. The proposed control scheme is adaptable when some assumptions are satisfied, and in cases where only the output signal of the system is available, the proposed scheme can be used widely. In addition, the proposed controller is more effective for systems with large uncertainties and Lipschitz constants. Note that the adjustable parameter 𝜃(𝑡)∈ℝ¹ is not an estimate of the unknown bound 𝜗^∗.

5. Numerical 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 necessary practical. However, the simulation results stated below illustrate the distinct feature of the proposed output feedback controller.

Consider the uncertain Lipschitz nonlinear system described by the following state equation 𝑑𝑥(𝑡)=𝑥(𝑡)+Δ(𝑡)𝑥(𝑡)+𝑢(𝑡)+𝜉(),𝑦(𝑡)=𝑥(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,𝑡1.00.00.00.01.00.0 (50) that is, 𝒯=(2.01.0). In this example we assume that the function 𝜉(𝑥,𝑡) and the positive scalar 𝜒^∗∈ℝ¹ in (5) are given by 𝜉(𝑥,𝑡)=√3.0×sin(𝑥₂(𝑡)) and 𝜒^∗=√3.0, respectively.

Firstly by adopting the similar way to the standard linear quadratic control problem, we consider designing the fixed gain matrix 𝒦_ℒ𝒬∈ℝ^2×3. Thus selecting the design parameters 𝒬_𝑡∈ℝ^3×3 and ℛ_𝑡∈ℝ¹ such that 𝒬_𝑡=1.0×𝐼₃ and ℛ_𝑡=1.0×10¹, respectively, and solving the algebraic Riccati equation 𝐻_𝑒{𝐴^𝑇𝒳_𝑡}−𝒳_𝑡𝐵ℛ_𝑡⁻¹𝐵^𝑇𝒳_𝑡+𝒬_𝑡=0, we obtain 𝒳_𝑡=2.23662×10⁻¹1.044591.95383×10⁻²★1.6170145×10¹1.25624★★2.22160×10⁻¹𝐾_ℒ𝒬,=−1.49191×10⁻¹−1.82593−1.29532×10⁻¹. (51)(52)

Next we design an output feedback gain matrix 𝐾∈ℝ^1×2 by using the the nominal system of (9). We select the design parameter 𝛼∈ℝ¹ such as 𝛼=5.0, then by applying the LMI-based design algorithm (see [29] and the Appendix section in [15]), we obtain the following solutions for LMI problems in Appendix in [15] and the following gain matrices, respectively: 𝒳=1.26046×10³1.25379×10²3.19323×10²★8.97329×10¹−1.13357×10²★★5.45874×10²,𝒴=1.02905×10³−5.78548×10²−4.11077×10²,𝒫=1.34029×10¹−4.46413×10¹−2.31136★1.74644×10²4.12700,★★1.03904𝜌=4.38355×10⁴,𝐾_sf=2.18703−1.63634×10¹−5.43048,𝐾=3.17745×10⁻¹−1.20809×10¹. (53)(54)

Now, we use Theorem 5 to design the proposed adaptive gain robust output feedback controller; that is, we solve the LMIs of (14). By selecting the symmetric positive definite matrix 𝒬∈ℝ^3×3 such as 𝒬=0.1×𝐼₃, we have 𝒮=1.67330−1.49618−6.50900×10⁻³★4.177251.48476×10⁻¹,,Ψ★★6.58185Ξ=4.300032.32356★2.04206_𝛾=1.070158.119016×10⁻¹,Ψ★1.00293_𝜖=,𝛾9.561257.12983★9.13154₁=7.14302,𝛾₂=7.57592,𝜖=9.55061. (55)

In this example, we consider the following two cases for the unknown parameter Δ(𝑡)∈ℝ^1×2 and its unknown upper bound 𝜗^∗ in (5): Case (1): 𝜗^∗Δ(𝑡)=5.0,=𝜗^∗×8.113249−5.56040×10⁻¹. (56) Case (2): 𝜗^∗=5.0×10⁻¹,Δ(𝑡)=𝜗^∗×sin(10𝜋𝑡)cos(10𝜋𝑡). (57) Furthermore, initial values for the uncertain system of (50), its target model, and the adjustable parameter 𝜃(𝑡) are selected as 𝑥(0)=(1.52.0−4.5)^𝑇,𝑥_𝑡(0)=(2.02.0−5.0)^𝑇, and 𝜃(0)=1.0, respectively. Besides, we choose 𝜎(𝑡)∈ℝ⁺ in (16) and the design parameter 𝜗∈ℝ¹ in (17) such as 𝜎(𝑡)=5.0×10⁴×exp(−1.0×10⁻⁴𝑡) and 𝜗=2.5×10³, respectively.

The results of the simulation of this example are depicted in Figures 1, 2, 3, 4, and 5. In these figures, the time histories of the state variables 𝑥₁(𝑡), 𝑥₂(𝑡), and 𝑥₃(𝑡), the control input 𝑢(𝑡), and the adjustable parameter 𝜃(𝑡)∈ℝ¹ are shown. “Desired” in these figures represents the trajectories for the state variables and the control input generated by the target model; that is, “Desired” for state variables shows a desirable response for the uncertain nonlinear systems.

Figure 1 

Time histories of the state variable 𝑥1(𝑡).

Figure 2 

Time histories of the state variable 𝑥2(𝑡).

Figure 3 

Time histories of the state variable 𝑥3(𝑡).

Figure 4 

Time histories of the control input 𝑢(𝑡).

Figure 5 

Time histories of the adjustable parameter 𝜃(𝑡).

From Figures 1, 2, 3, and 4, we find that the proposed adaptive gain robust output feedback controller stabilizes the controlled system of (50) in spite of plant uncertainties and nonlinearities. Besides, one can see that the proposed adaptive gain robust output feedback controller achieves good transient performance and the proposed control input is tuned by the measurable signals and the adjustable parameter. In particular the time responses of the uncertain nonlinear system of (50) for Case (2) close to the trajectory of the target model and the control input also achieves satisfactory trajectory. This result shows that the proposed adaptive robust output feedback controller reflects the effect of uncertainties and nonlinearities as online information. Furthermore, we can see from Figure 5 that the parameter 𝜃(𝑡) is tuned by the updating law and is not an estimate of the upper bound 𝜗^∗.

6. Conclusions

In this paper, we have proposed an adaptive gain robust output feedback controller for a class of uncertain Lipschitz nonlinear system of which upper bounds are unknown. For the uncertain Lipschitz nonlinear system, we have shown that the proposed adaptive gain robust output feedback controller can be obtained by solving LMIs. Besides, by numerical simulations, the effectiveness of the proposed adaptive robust controller has been presented. One can see that the crucial difference between the existing results [24] and our new one is that information is not required on the upper bound of the unknown parameter Δ(𝑡) in the system matrix. The proposed controller design method is adaptable when some assumptions are satisfied and in cases where only the output signal of the system is available. Namely, if for uncertain Lipschitz nonlinear systems which satisfy the assumptions for the system parameters, the LMIs of (14) are feasible, then the proposed adaptive gain robust output feedback controller is applicable.

The future research subjects are extension of the proposed adaptive gain robust output feedback controller synthesis to such a broad class of systems as uncertain time-delay systems and uncertain large-scale interconnected systems. Furthermore in future work, we will examine the assumption of (7).

Acknowledgment

The authors would like to thank CAE Solutions Corp. for providing its support in conducting this study. Additionally, the authors would like to thank Professor Zhiyong Chen with the University of Newcastle and the anonymous reviewers for their valuable and helpful comments that greatly contributed to this paper.