Constrained Adaptive Neural Control of Nonlinear Strict-Feedback Systems with Input Dead-Zone

This paper focuses on a single neural network tracking control for a class of nonlinear strict-feedback systems with input dead-zone and time-varying output constraint via prescribed performance method. To release the limit condition on previous performance function that the initial tracking error needs to be known, a new modified performance function is first constructed. Further, to reduce the computational burden of traditional neural back-stepping control approaches which require all the virtual controllers to be necessarily carried out in each step, the nonlinear items are transmitted to the last step such that only one neural network is required in this design. By regarding the input-coefficients of the dead-zone slopes as a system uncertainty and introducing a new concise system transformation technique, a composite adaptive neural state-feedback control approach is developed.Themost prominent feature of this scheme is that it not only owes low-computational property but also releases the previous limitations on performance function and is capable of guaranteeing the output confined within the new form of prescribed bound. Moreover, the closed-loop stability is proved using Lyapunov function. Comparative simulation is induced to verify the effectiveness.


Introduction
In recent years, actuated by practical requirements and theoretical developments, numerous adaptive back-stepping control schemes have been proposed for uncertain nonlinear systems in lower-triangular form including strict-feedback and pure-feedback systems [1].Specially, fuzzy logic systems (FLS) or neural networks (NNs) based control schemes have attracted great concern due to their inherent approximation capabilities as well as relaxing linearly in parameters assumption [2,3].The benefit of applying FLS or NNs is that the problem of spending much effort on system modeling can be elegantly overcome.Although there has been significant progress in aforementioned literatures, the problem of complexity (POC) is the main drawback of traditional adaptive back-stepping schemes caused by two reasons [4,5]: the first one is the reduplicative derivations of virtual controllers, and the other one is that there exist numerous NNs/FLS.In the existing literature, several so-called dynamic surface control [6], sliding mode differentiator (or tracking differentiator) [7], and command filter techniques [8] have been incorporated into adaptive back-stepping control design to avoid the repeated derivations problem.Meanwhile, those techniques have been extensively employed to multifarious practical systems such as flexible-joint robot system [9], flight control [10], and static Var compensator [11].By using those techniques, the repeated differentiations of virtual control laws can be avoided.However, numerous NNs employed to construct the virtual and actual control laws are still necessary for high-order systems.To thoroughly refrain from this problem, single neural (SN) approximation-based adaptive back-stepping control has been proposed for strictfeedback nonlinear systems where the control gains in each subsystem are equal to one [4].After that, Pan and Sun have extended SN control structure to a general class of uncertain strict-feedback [5] and pure-feedback nonlinear systems [12], respectively.Meanwhile, some researchers have also utilized the single neural control technique (SNC) to discrete-time nonlinear systems [13,14].However, the works described in [4,5,[12][13][14] only focus on the stabilization of interested systems, and the output-constraint problems are rarely considered.Meanwhile, it must be pointed out that violation of the output constraint may lower the system performance or even lead to systems instability [15,16].

Mathematical Problems in Engineering
To manage the problem of output constraint, a number of attempts regarding model predictive control [17], barrier Lyapunov function (BLF) [18][19][20][21][22][23][24], and prescribed performance control (PPC) [25][26][27][28] techniques have been investigated from both academia and industries.Using the property of BLF, a constrained back-stepping control approach is proposed for nonlinear strict-feedback systems to ensure that the static constraint is not transgressed [20].More specifically, some constrained adaptive neural/fuzzy back-stepping schemes are investigated for nonlinear systems subject to external disturbance and unknown functions [19,22,29].Apart from the technique about BLF, Bechiloulis et al. have also proposed an alternative approach named PPC to conquer the problem of output constraint [25].
The dead-zone characteristic on behalf of the most important input nonlinearities widely exists in actuators and sensors among numerous industrial processes, which also severely deteriorate the system performance [30][31][32][33][34]. Thus, many researchers have devoted themselves to improving the performance of the control systems in the presence of deadzones.Some researches apply an inverse dead-zone model to minimize the influence of the dead-zone; see reference [30].On the other hand, by exploring the bounds of the deadzone slopes and regarding the coefficients of the dead-zone as a system uncertainty, some robust adaptive neural/fuzzy control schemes are proposed of nonlinear systems with nonsymmetric dead-zone inputs [31,[35][36][37].Na [38] and Chen [39] both proposed an adaptive neural prescribed performance control for nonlinear system with dead-zone input.Unfortunately, aforementioned PPC approaches still suffer from the POC and use numerous NNs, which result in a complex control structure.Moreover, there also exists restriction that the initial tracking errors must be known in advance.However, actually, the initial tracking error is hard to obtain in the presence of uncertainties and additional disturbances.Thus, how to concurrently tackle such restriction and aforementioned problems needs further investigations.
To simultaneously solve the aforementioned problems, a systematic design procedure is developed to derive a composite single neural network control scheme of nonlinear strict-feedback systems subject to dead-zone input, system uncertainties, and time-varying output constraint.In the controller design, to release the limit condition on previous performance function that the initial tracking error needs to be known, a modified performance function is first constructed.Meanwhile, depending on a new system transformation technique, the prescribed performance of the constrained system can be equivalent to ensure the stability of the transformed system.Regarding the coefficients of the dead-zone as a system uncertainty, an adaptive NN approach is developed via single NN control structure to stabilize the transformed system.Finally, stability analysis is conducted by using Lyapunov theory.The main contributions of this paper are summarized as follows: (1) A new modified prescribed performance function is constructed to stipulate the predefined region of the tracking error.Thus, the previous restriction on initial tracking error is removed.
(2) In construct to previous output-constraint neural/ fuzzy back-stepping control approach [15,25,38,40,41], the proposed control scheme does not need numerous NNs/FLS to construct virtual and practical control law in each step, only one neural network including one adaptive laws is required to approximate the lumped unknown function, thus deriving a low-computational control scheme.
(3) By regarding time-varying input-coefficients of input dead-zone as a system uncertainty and using a new system transformation technique in the control design, an integrated single NN adaptive controller, which is capable of arbitrarily prescribing the system performance and dealing with dead-zone input nonlinearity, simultaneously, is first presented for a class of nonlinear strict-feedback systems.

Problem Formulation and Preliminaries
2.1.Problem Formulation.Consider uncertain nonlinear strict-feedback systems in the following form: where   = [ 1 ,  2 , . . .,   ]  ∈   ,  = 2, . . ., ,  ∈ , and  ∈  are state variables, system input, and output, respectively.  (  ) are the unknown smooth functions;   (  ) denote unknown smooth virtual control gain functions.V is the output from the controller.According to [30],  is a nonsymmetric dead-zone input nonlinearity which is defined as follows: where   and   denote breakpoints of the input nonlinearity.
To facilitate control design, we need the following assumptions.
The dead-zone nonlinearity (2) can be rewritten as a slowly time-varying function: where Here, we can easily conclude that | V | ≤  * ,  * = max{    ,     }, and there exist positive constants  and  such that  ≤  ≤ .
Control Objective.The control goal is to design a controller such that (1) the output  tracks the desired trajectory   well; (2) meanwhile, the output  can be always confined within the predefined time-varying constraint.That is to say, the output  is needed to satisfy where () and () are the bounds of the output.
Assumption 4 (see [15]).The output bounds () and () are smooth functions and their derivatives from 1 to th are all available.
RBF network (8) has the ability to approximate any continuous function to arbitrary accuracy over the set where  * is ideal constant weights and  is the approximation error.
Lemma 7 (see [43]).The Gaussian function described in (9) with Ẑ =  −  being the input vector, where  is a bounded vector and  is a positive, is given as where Φ  is a bounded vector.
Remark 8. Assumption 1 imposes a controllability condition for system (1) and can be found in most existing adaptive back-stepping neural network control approaches.

System Transformation.
Inspired by [15], the tracking error is defined as By subtracting   from both sides of (5), we have For clarity, we define  1 () = () −   () and Then, (14) becomes Therefore, the output constraints given by ( 5) are now transformed into tracking error constraints shown as (15).To establish the relationship between  1 () and  1 (), Then, the following novel transformed error is established as where  1 () is an error variable.
The inverse transformation of ( 16) with respect to  1 () is shown as Therefore, if  1 () is bounded, we have Adding   to every side of (18), condition (5) can be easily got.Therefore, the goal is presently converted to designing a controller to ensure  1 to be bounded.The derivative of ( 16) is shown as Therefore, ( 19) can be rewritten as where Apparently, we have the transformed systems shown as Remark 9.It should be noted that  1 ,  −1 1 , φ 1 ,  1 , φ 1 , , and  are definitely known and can be used in the controller design.
Remark 10.From ( 20) and ( 18), we know that Define a new function the derivative of   1 with respect to  1 can be described as It is easy for us to obtain that function   1 will get minimum in the set  ), which also means that there exists a time instant   such that  ≥  min with  min = 2/( 1 (  ) −  1 (  )).
Remark 12.In contrast to previous prescribed performance functions with constant parameters  1 and  2 , where the designed  1 () and  2 () are time-varying functions.It implies that the proposed constrained function owes more degrees of freedom to adjust the prescribed performance bounds of the tracking error.This is quite beneficial for the controller design.(), respectively.Meanwhile, we define the virtual tracking errors   =   −   with  = 2, . . ., , where   are virtual control inputs which will be defined later.Let   = [   ,   (+1) ,  (+1) (),  (+1) ()]  with  = 1, 2, . . ., .   denotes the neural input vector which will be used in the following derivation.
Proof.Consider the following Lyapunov candidate: By using ( 28), ( 35), (41), and (53), the derivative of ( 56) is shown Since Ŵ1 is bounded as presented in Lemma 13, it follows that ‖ W1 ‖ ≤   , where   denote the upper bound of the approximation error.We note that ‖Φ(  )‖ ≤ Φ.Consider the following facts, we have we have By choosing   and  such that   > 0,  = 1, 2, . . .,  − 1,   >  1 / 0 , and  > 0 and defining . The derivative of  can be reformed as Thus, we can conclude that the transformed error  1 , tracking errors   , and W are uniformly ultimately boundedness.Then, the boundedness of  1 can ensure the time-varying output constraints.
3.5.Further Design.It must be noted that the control design described in (51) needs the high-order derivatives of the reference signal   .In practice, high-order derivatives are hard to obtain in the implantation of the controller.Motivated by [46,47], we can employ a high-gain observer (HGO) or high-order sliding mode observer (HOSM) to estimate  () ,  = 2, . . ., .As indicated in [46], the prominent features of those observers lie in the fact that they can achieve finitetime observer error convergence and thus can be utilized in almost any feedback with separation principle being trivially fulfilled.If we use HGO to estimate the high-order derivatives of   , the actual control law (51) and updated law (52) can be modified as where ŷ and X are the estimation of   and   , respectively.By using HGO to estimate the high-order derivative of   , the actual controller and updated law have been changed to (61).The stability of the closed-loop system can be demonstrated the same as Section 3.4.From Lemma 7, it has ‖Φ( X )‖ ≤ Φ with Φ = ‖Φ(  )‖+‖Φ  ‖.Employing the same procedure of Lemma 13, we can easily prove that there also exists a compact set where ‖Φ( X )‖ ≤ Φ, such that Ŵ() ∈ Ω  , ∀ ≥ 0 in case that Ŵ(0) ∈ Ω  .
Using the property of Gaussian radial basis function ( 12), (53) can be rewritten as where Substituting (63) into the derivative of Lyapunov function, we have where  =  − Φ  +  1 + WΦ( X ).Since Ŵ1 is bounded, it follows that ‖ W1 ‖ ≤   where   denote the upper bound of the approximation error.We note that ‖Φ( X )‖ ≤ Φ.Using Lemma 5, we can also get that there exist a constant  1 such that | 1 | ≤  1 .Until now, it can be easily concluded that there is a constant  with  = ‖‖ + ‖Φ  ‖ +  1 + ‖ WΦ( X )‖ such that || ≤ .
Consider the following fact: we have The same analysis with Theorem 14, the signals  1 , and   ( = 2, . . ., ) in the closed-loop are bounded.
Remark 15.Compared with the previously proposed outputconstraint [40] or prescribed performance control [41,44,45] adaptive back-stepping approach, the proposed structure is extremely simple and the computational burden is really low since there is only actual controller required to be implemented and there exists single neural network to approximate the lump uncertainty.It also be noted that the issue of explosion of the complexity inherent in the traditional backstepping approach is completely removed without employing DSC, command filter, or differentiator technique.
Remark 16.In contrast to the results [4,5,12] using cascade low-pass filter to approximate the high-order derivatives of the reference without considering the influence of estimated error in the closed-loop stability analysis, the HGO is applied to approach the derivatives of the reference and the estimated error effect is also taken into account in the control design and stability analysis.

Simulation Studies
In this section, the effectiveness of the proposed method is illustrated by one-link manipulator with a brush DC (BDC) motor.One-link manipulator actuated by a BDC motor can be expressed as [48]  q +  q +  sin () =   İ = − −   q + , where , q , and q represent the link angular position, velocity, and acceleration, respectively. denotes the motor current. represents the input control voltage.The parameter values with appropriate units are given by  = 1,  = 1,  = 0.05,  = 0.5,  = 10, and   = 10.
To verify the proposed controller, a comparative simulation study [49] has been conducted in this section.For fair comparison, the control gains and initial conditions of control scheme in [49] are the same with this approach.The comparative simulation results are depicted in Figures 1, 2, 3,  and 4, where   ,   ,   , and ‖ Ŵ ‖ denote the simulation results of comparative study [49].In Figures 1 and 2, we plot the controller tracking performance and tracking errors trajectory along with their bounds.Figures 1 and 2 imply that the proposed control method can ensure the output and its tracking error both confined within their bounds.However, the tracking performance and tracking error of comparative study are given an unsatisfactory effect.The control input and norm of NN weights are depicted in Figures 3 and 4, respectively.It is easy for us to find that the control input and norm of NN weights are bounded.Apparently, as opposed to the output tracking performance achieved by control scheme [49], the proposed control approach has a satisfactory result which can ensure the output and tracking error without overstepping their bounds.

Conclusion
In this paper, a new concise adaptive neural tracking control scheme has been developed for a class of strict-feedback system subject to input dead-zone and time-varying output constraint.Firstly, to release the limit condition on previous performance function that the initial tracking error needs to be known, a new modified performance function is constructed.By utilizing a system transformation technique, after we transform the original constrained nonlinear system into an unconstrained one, a composite adaptive neural state-feedback control approach is investigated.All unknown functions from 1 to −1 steps are integrated to the  step such that only one neural network is required to approximate the lump uncertainty, thus deriving a low-computational scheme.Meanwhile, the output constraint and input dead-zone are both considered in this scheme.Finally, the closed-loop stability is proved using Lyapunov technique.Comparative