L 2 Gain Performance for a Class of Lipschitz Uncertain Nonlinear Systems via Variable Gain Robust Output Feedback Controllers

We consider a design problem of a variable gain robust output feedback controller with guaranteed L 2 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 specifiedL 2 gain performance. In this paper, we show that sufficient conditions for the existence of the proposed variable gain robust output feedback controller with guaranteedL 2 gain performance are given in terms of linear matrix inequalities (LMIs). Finally, a simple numerical example is included.


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 observerbased 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 timevarying 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 L 2 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 L 2 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.

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  and A −1 , respectively.Also,   {A} means A + A  , and   represents -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  ∈ R  , ‖‖ denotes standard Euclidian norm, and, for a matrix A, ‖A‖ represents its induced norm.The symbol "≜" means equality by definition.Besides, L 2 [0, ∞) is L 2 -space (i.e., the collection of all square integrable functions) defined on [0, ∞), and, for a signal () ∈ L 2 [0, ∞), ‖()‖ L 2 denotes its L 2 -norm.Lemma 1.For arbitrary matrices G and H and vectors  and  which have appropriate dimensions, the following inequality holds: where Δ() ∈ R × is a time-varying unknown matrix satisfying ‖Δ()‖ ≤ 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: Proof.See Boyd et al. [17].

Problem Formulations
Consider the following Lipschitz uncertain nonlinear systems:    () =  Δ ()  () +  () +  (, ) + Γ 1  () , where () ∈ R  , () ∈ R  , () ∈ R  , () ∈ R  , and () ∈ R  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, () ∈ L 2 [0, ∞).In (2),  Δ () ∈ R × is supposed to have the following structure, In ( 2) and ( 3), the matrices , , , and , and Γ  ( = 1, 2) are known system parameters and the matrix Δ() ∈ R × denotes unknown time-varying parameters, which satisfy ‖Δ()‖ ≤ 1.In this paper, we introduce the following assumption for the nonlinear term (, ) in (2): In addition we assume that for the function  : R  ×  → R  in (4) there exists a constant scalar  * such that for all Besides, we assume that there exists a constant matrix T ∈ R × satisfying the following relation [15]: The nominal system, ignoring unknown parameters and nonlinearities in (2), is given by and the nominal system of ( 7) is supposed to be stabilizable via static output feedback control.Namely, there exists a fixed gain matrix  ∈ R × such that the matrix   ≜  +  is asymptotically stable.In other words, there exists an output feedback control () = () which stabilizes the nominal system of (7).Note that the static output feedback gain matrix  ∈ R × 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) For the target model of ( 8), we select the control input   () =  LQ   () − (  , ).Thus the target model can be written as Note that the fixed state feedback gain matrix  LQ ∈ R × 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 () ≜ () −   (),   () ≜ () −   (), and   () ≜ () −   ().Beside, using the fixed gain matrices  ∈ R × and  LQ ∈ R × , we consider the following control input for the Lipschitz uncertain nonlinear system of (2): (10) where (  , L, ) ∈ R  is a compensation input [5,9] and has the following form where L(  , ) ∈ R × 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: where   is a stable matrix given by   ≜  + .Now we will give the definition of the variable gain robust output feedback control with guaranteed L 2 gain performance  * > 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 L 2 gain performance  * > 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 () to the controlled output   () is less than or equal to a positive constant  * .By introducing a symmetric positive definite matrix P ∈ R × , we consider a quadratic function V(,   , ) ≜   ()P()+   ()S  ().Besides, we define the Halmitonian as Then we have the following lemma for the variable gain robust output feedback control with guaranteed L 2 gain performance  * .
Lemma 4. Consider the uncertain nonlinear error system of (12) and the control input of (10).
For the quadratic function V(,   , ) and the signals   () and (), if there exist symmetric positive definite matrices P ∈ R × and S ∈ R × and a positive scalar  * satisfying the inequality then control input of (10) is a variable gain robust output feedback control with guaranteed L 2 gain performance  * .
Proof.By integrating both sides of the inequality of ( 14) from 0 to ∞ with (0) = 0 and   (0) = 0, we easily see from V(,   , 0) = 0 that the inequality holds.We see from the inequality of ( 15) that the uncertain nonlinear error system of ( 12) is robustly stable (internally stable).Note that if () ≡ 0, then from the inequality of ( 14) the quadratic function V(,   , ) 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 () ∈ R  to the controlled output   () ∈ R  is less than a given positive constant  * , because the inequality of (15) means the following relation: Thus the proof of Lemma 4 is completed.
Therefore, our control objective is to design the variable gain robust output feedback controller with guaranteed L 2 gain performance  * for the Lipschitz uncertain nonlinear system of (2).That is to derive the symmetric positive definite matrices P ∈ R × and S ∈ R × , a positive scalar  * , and the variable gain matrix L(  , ) ∈ R × which satisfies the inequality condition of ( 14) for all admissible nonlinear perturbations and the disturbance input () ∈ L 2 [0, ∞).

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 L 2 gain performance  * .Theorem 5. Consider the uncertain nonlinear error system of (12) with the variable gain matrix L(  , ) ∈ R × .
If there exist symmetric positive definite matrices P ∈ R × , S ∈ R × , Ξ ∈ R × , and Ψ ∈ R × and positive scalars ,  and  satisfying the LMIs then, by using the solution of the LMIs of (17) and the known scalar  * , the variable gain matrix L(  , ) ∈ R × is determined as In (18),  − = lim ]>0,] → 0 ( − ]) [5], and  is a positive scalar defined as where  ∈ R 1 > 0 is a design parameter.Then the control input of (10) is a variable gain robust output feedback control with guaranteed L 2 gain performance  * = √ .
Proof.Consider the quadratic function V(,   , ), the Hamiltonian H(,   , ) of ( 13), and the inequality of ( 14).The time derivative of the quadratic function V(,   , ) along the trajectory of the target model of ( 9) and the one of the uncertain nonlinear error systems of ( 12) is given by Namely, the condition of ( 14) can be written as Besides, one can see from the assumption of ( 5) for the nonlinear term and Lemma 1 that the following inequality holds: From the assumption of ( 6), we have the following inequality as a sufficient condition for the inequality of ( 22): Here we have used the well-known following relation: Additionally, introducing the variable  ≜ ( * ) 2 and some algebraic manipulations give the following inequality for Hamiltonian H(,   , ): where Ψ(P, , ) ∈ R × is given by Now we consider the case of   () ̸ = 0. Note that the second LMI of ( 17) is equivalent to the following inequality: Thus in this case, considering the variable gain matrix L(  , ) ∈ R × given by ( 18) and the second LMI, the third one, and the fourth one of ( 17) and some trivial algebraic manipulations give In (28),  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   () ≡ 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(,   , ) is always satisfied.Namely, the proposed control input of ( 10) is a variable gain robust output feedback control with guaranteed L 2 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  > 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 L 2 gain performance, we can also minimize the parameter  ∈ R 1 (see Appendix A).

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 that is, T = (2.0 1.0).In this example we assume that the positive scalar  * ∈ R 1 in ( 5) is given by  * = 5.0.Firstly, selecting the weighting matrices such as Q  =  3 and R  = 1.0 × 10 1 , we have the solution of algebraic Riccati equation for the standard LQ control problem and the optimal feedback gain matrix  LQ ∈ R 1×3 such as Next, adopting the LMI-based algorithm based on the work of [18] (see Appendix B), we design an output feedback gain matrix  ∈ R 1×2 for the nominal system.We select the design parameter  such as  = 3.0; then, by applying the LMI-based design algorithm, we obtain the following output feedback gain matrix  ∈ R 1×2 for the nominal system:  = (−8.0484× 10 −1 −1.0525 × 10 1 ) . (31) Finally, in order to derive the proposed controller, solve the LMIs of (17).By solving the LMIs of ( 17

Conclusions
In this paper, we have proposed a variable gain robust output feedback controller with guaranteed L 2 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 largescale interconnected system, time-delay systems, and so on.Furthermore in future work, we will examine the assumption of (6).

A. Optimal Guaranteed L 2 Gain Performance
Since the LMIs of ( 17) define a convex solution set, we consider minimizing the parameter  ∈ R 1 , because our interest is in establishing L 2 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, >0 [] subject to (17) . (A.1) If the optimal solution P > 0, S > 0, Ψ ≥ 0, Ξ ≥ 0, and  > 0 of the constrained convex optimization problem of (28) is obtained, then the control input of (10) with the variable gain matrix L(  , ) in ( 18) is a variable gain robust output feedback control with guaranteed optimal L 2 gain performance  * = √ .
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 L 2 gain performance  * = √  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: where () ∈ R  , () ∈ R  and () ∈ R  are the vectors of the state, the control input, and the measurement output, respectively, and the matrices , , and  denote the nominal values of system parameters.For the linear system (B.1),we consider the static output feedback control () = ().The following LMI-based algorithm to derive the static output feedback gain matrix  ∈ R × has been developed by the existing result of [18].

A LMI-Based Algorithm
Step 1. Define   =  +   , where  ∈ R 1 is the desired prescribed degree of stability, as described in Anderson and Moore [20].Step 3. By using the matrices X ∈ R × and Y ∈ R × , set the state feedback gain matrix such as   = YX −1 .