Adaptive Regulation of a Class of Uncertain Nonlinear Systems with Nonstrict Feedback Form

This paper focuses on the semiglobal stabilization for a class of nonlinear systems with nonstrict feedback form. Based on a generalized scaling technique, an adaptive control algorithm with dynamic high gain is developed for a class of nonstrict feedback nonlinear systems. It can be proved that, under some appropriate design parameters, all signals of the resulting closed-loop system are bounded semiglobally, and the system statewill be convergent to origin exponentially. Finally, a numerical simulation is provided to confirm the effectiveness of the proposed method.

System (1) can be stabilized by numerous adaptive controllers by means of backstepping technique [1][2][3] if (1) is of a strict feedback structure; that is, in the th subsystem, the uncertain nonlinear function   (, ) is the function with respect to (w.r.t.) the first state variables  1 , . . .,   and time .Furthermore, if the function   (, ) depends on whole state variables but its bounding function is assumed to be a function of  1 , . . .,   and , thus system is called semistrict feedback systems, and the backstepping technique is also an alternative method to controller design [4][5][6][7].However, when the subsystem functions contain whole state variables and the above assumption of semistrictness is removed, that is, system (1) is of a nonstrict feedback structure, the aforementioned control methodology would be invalidated.
The control problem of nonstrict feedback nonlinear systems arises and attracts some researchers' attention.In [8,9], by utilizing the monotonically increasing property of the bounding functions, the authors developed the variables separation technique and by means of the approximation property of fuzzy logic systems and neural networks (NN) proposed adaptive fuzzy and NN control design methods, respectively.Subsequently, the design method was extended and applied to MIMO nonlinear systems [10], stochastic nonlinear systems [11,12], and uncertain switched nonlinear systems [13].Furthermore, [14] eliminated the assumption in [8][9][10][11][12][13] that the unknown nonlinear functions must satisfy the monotonically increasing property of the bounding functions.
Different from [8][9][10][11][12][13][14], a semiglobal output feedback controller with high gain for a class of nontriangular nonlinear systems was proposed in [15], in which the authors introduced dilation   () = ( 1 ,  2 , . . .,  −1   ) with constant 2 Mathematical Problems in Engineering  ≥ 1 and assumed that the nonlinear functions   (, )'s should satisfy the following inequalities: where  ∈ [0, 1) and continuous function   () ≥ 0 were known.In [15], the up-bounds of some terms containing   ()'s, such as  2 and  3 , should be known, and these up-bounds might be very large, so that a larger gain  would be required to guarantee the stability of the system.Consequently, the amplitude of the control input  would be very large.Besides, the large high gain would amplify the noise [16].On the other hand, in fact, the high gain sometimes need not be larger than the up-bounds but be just larger than the terms.To determine the gain is not easy; a trial-and-error method is often adopted [17].Different from the constant high gain used in controller design in [15], a dynamic high gain is employed in this paper.The dynamic high gain will increase until it is larger than the terms and the closed system is stable.It can be found that the dynamic high gain or the so-called adaptive high gain was employed in [7] for dealing with the unknown growth rate of nonlinear functions and was used to avoid amplifying the noise in Extended Kalman Filters in [16].
In this paper, we focus on the stabilization scheme via state feedback for uncertain nonstrict feedback nonlinear system (1).Inspired by the generalized scaling technique of [18], a dynamic high-gain-based adaptive controller is designed for system (1), which can guarantee that all signals of the resulting closed-loop system are bounded semiglobally and the system state is convergent to the origin exponentially.
The rest of this paper is organized as follows.Two assumptions imposed on system (1) and the problem statements are presented in Section 1.The main results and theoretical analysis are given in Section 2. Simulation examples are given in Section 3, followed by Section 4 that concludes the work.

Assumptions and Statements
Rewrite system (1) as the following form: where And the following two assumptions are imposed on system (3).
Assumption 2 is about the nonlinear functions   (, )'s.Compared with the corresponding assumption in [15] given by ( 2), the high gain  in ( 6) is a variable rather than a constant  in (2), and, in the th inequality, the power of high gain in the right-hand side of the inequality is   in (6) instead of  − 1 in (2).It can be seen that the assumption of this paper is more general, and ( 2) is a special case of (6) with   =  − 1,   = , and () =  = , where  is an appropriate positive constant.In addition, it should be pointed out that some terms containing functions   ()'s in [15] are assumed to be bounded with up-bounds  2 and  3 , so a large enough  is required to guarantee the stability of the system.Too large gain will result in too large amplitude of the control input and will amplify the unavoidable noise [16].Therefore, an appropriate high gain is required, and it is often achieved by using trial-and-error method [17].In this paper, a dynamic high gain instead of constant gain will be employed in controller design.
The main objective of this paper is to design a state feedback controller  =   () to stabilize the nonstrict feedback nonlinear system (3) under Assumptions 1 and 2.

Dynamic High-Gain-Based Control and Stability Analysis
In this section, the main result of this paper is given, an adaptive controller with dynamic high gain is presented for system (3), and the rigorous theoretical analysis is then provided.
Noting  =   , inequality (6) in Assumption 2 imposed on system (1) can be written as follows: The controller V is designed as where  = [ hold, and if we choose an appropriate gain , then controller (9) can stabilize the state of system ( 7) exponentially.
The real controller  can be directly obtained from (9) as the following form: However, controller ( 16) is invalid because (, ) is unknown.Under Assumption 1, the bounds of (, ) could be used instead of (, ); the controller is designed as And it should be pointed out that, in Proposition 3, the high gain  should be equal to or larger than   () to guarantee the convergence of .If a constant gain  is chosen, for example, in [15], the bound of ‖()‖ should be known, and a very large gain is required; meanwhile, with higher , more noise is amplified [16].And, from (17), it is easy to see that large  will bring about large control input  and may lead to the control input saturation due to physical limit in practice.To reduce this defect of large high gain, a dynamic high gain is employed in this paper, and the update law of the high gain  is given as follows: From (18), it is obvious that  is increasing until  ≥   () or  1 ≡ 0. Theorem 4.Under Assumptions 1 and 2, if there exist PSD matrices ,  1 , and  2 , such that (10) and (11) hold, controller (17) with gain update law (18) can guarantee that all the signals of the closed-loop system are uniformly ultimately bounded and the states are convergent to origin exponentially.
The next step is to illustrate the boundedness of the high gain .
In the case of  1 ≡ 0, it follows that ṙ = 0 regardless of whether  ≤   () or not, which also means  is bounded.
Then, we can obtain that the high gain  is bounded.
To sum up, we can conclude that all the signals of the closed-loop system of ( 6), (17), and ( 18) are bounded, and the system state  converges to origin exponentially.That completes the proof.

Numerical Simulation
To illustrate the effectiveness of the proposed method, a numerical simulation is presented in this section.
The simulation results are shown in Figures 1-3.It is clearly shown that, under the action of the proposed control (17) and adaptive law (18), all the signals are bounded and the system states are convergent to origin.
Furthermore, it is easy to see that when larger (0) is chosen, () is driven faster to steady state (see Figure 1); correspondingly, () and () achieve steady state more quickly (see Figures 2 and 3).
In practice, the control input is often limited due to some physical limits.Too large high gain could lead to the control input saturation, which would degrade the performance of the control system and even cause system instability.To illustrate the above problem caused by too large high gain, some more simulations are given as follows.
From Figure 3, the high gain  can be approximately regarded as a constant,  = 6.And if a constant high gain is employed in the controller design, to ensure the stability of the system,  = 6 or a larger one would be chosen using trialand-error strategy.So, the simulations are run with the same design parameters and the same initial condition of state and (0) = 6.And two cases of the control input saturation are considered; namely, || ≤ 65 and || ≤ 63.The simulation results are shown in Figures 4 and 5.
In comparison between the state responses with (0) = 6 in Figure 1 and the state responses in Figure 4, it can be found that the performance of the control system degrades when the control input is restricted by || ≤ 65 but is still stable, while the system would be unstable when the control input is limited by || ≤ 63.
Under the control input limit || ≤ 63, the controller with constant high gain  = 6 or a larger one will be invalid.But a small gain cannot ensure the stability of the system.To solve the above problem, this paper gives an alternative method.We can find that if we choose (0) = 1, the input saturation does not occur (see Figure 2(a)).Therefore, a smaller (0) will bring a smaller control input and will avoid the input saturation.

Conclusions
In this paper, an adaptive controller with dynamic high gain has been designed for a class of nonstrict feedback nonlinear systems.The main difference from the existing related works is that a dynamic high gain rather than a constant one is employed in the controller design.The advantage of the dynamic high gain is that it would not result in a large control input, and it is more likely to avoid the input saturation occurring which usually exists due to physical limits in practice.By Lyapunov stability theory method, a rigorous theoretical proof is provided to show that the proposed controller can guarantee the semiglobal stability of the system state.Finally, a numerical simulation is provided to illustrate the effectiveness of the proposed method.