Low-cost adaptive fuzzy neural prescribed performance control of strict-feedback systems considering full-state and input constraints

: A low-cost adaptive neural prescribed performance control (LAFN-PPC) scheme of strict-feedback systems considering asymmetric full-state and input constraints is developed in this paper. In the controller design procedure, one-to-one nonlinear transformation technique is employed to handle the full-state constraints and prescribed performance requirement. The Nussbaum gain technique is introduced for solving the unknown control direction and the input constraint nonlinearity simultaneously. Furthermore, a fuzzy wavelet neural network (FWNN) is utilized to approximate the unknown nonlinearities. Compared with traditional approximation-based backstepping schemes, the constructed controller can not only overcome the so-called “explosion of complexity” (EOC) problem through command filter, but also reduce filter errors by error compensation mechanism. Moreover, by constructing a virtual parameter, only one parameter is required to be updated online without considering the order of system and the dimension of system parameters, which significantly reduces the computational cost. Based on the Lyapunov stability theory, the presented controller can ensure that all the closed-loop signals are ultimate boundedness, and all state variables and tracking error are restricted in the prespecified regions. Finally, the simulation results of comparison study verify the effectiveness of the constructed controller.


Introduction
The control problems of nonlinear systems have received a great deal of attention, and a considerable amount of literatures have been published [1][2][3][4][5]. A two-layer neural networks (NNs) based robust control for nonlinear induction motors is proposed in [1]. In [2] the unknown nonlinear items in the dynamic model of robotic manipulators are identified by NNs. Dynamic properties and optimal stabilization issues of fractional-order (FO) self-sustained electromechanical seismograph system is investigated in [3]. A neural adaptive control scheme is raised for a class of uncertain multiinput/multi-output nonlinear systems [4] and only one learning parameter is updated in the parameter identification. And coupled FO chaotic electromechanical devices are studied in [5]. Specifically, an adaptive dynamic programming policy is proposed to address the zero-sum differential game issue in the optimal neural feedback controller. Obviously, approximation-based adaptive backstepping control has been widely utilized in designing controllers for various nonlinear systems [6][7][8]. However, traditional approximation-based adaptive backstepping technique faces two crucial issues that hinder its application. The first issue is so-called "EOC" arising from the derivations of virtual control input [7]. The second one is a large computational burden caused by high precision approximation requirements [8].
For the backstepping technique, the design of the control law relies on intermediate state variables as virtual control signals. The controller of each subsystem requires the virtual control signal and its derivative. The lower-order derivatives of the virtual control signals are likely simple in theory, but the higher-order derivatives in higher-order systems are quite complex, which is called the "EOC" problem. For handling the "EOC" problem, a first-order filter was used to calculate the virtual control signal derivatives in each recursive step [9]. This technique is the so-called dynamic surface control (DSC). The utilization of the first-order filter overcomes the EOC problem, but its own characteristics lead to the derivative errors of virtual control signal, which will certainly affect the tracking performance of the system. Based on this, a modified scheme of the DSC method named command filter based control (CFC) method was proposed in [10]. On the one hand, the EOC problem is avoided by substituting a first-order filter with a second-order one to obtain the derivatives of the virtual control signal. On the other hand, the filter errors are compensated by a constructed error compensation mechanism for obtaining better tracking performance of the system. Moreover, constructing an effective error compensation mechanism needs to be further studied.
For the approximation-based control, the NNs or fuzzy logic systems (FLSs) are utilized to approximate unknown functions and external disturbances for ensuring better tracking performance [11,12], which is also a kind of learning control [13]. In [11] a NNs-based approximator is utilized to solve the unmodeled dynamics of the system. FLSs are employed to identify unknown functions existing in systems [12]. Iterative learning control schemes are designed to suppress the vibrations in bending and twisting of the flexible micro aerial vehicle [13]. The approximation accuracy improves with the increase of the number of neural network nodes or fuzzy rules in general, but it also significantly increases the number of estimated parameters. Therefore, the burdens of computation required for online learning will become very heavy. For decreasing the computational burden of approximationbased control, a tuning-function is inserted in the controller of strict-feedback systems [14,15], in which the number of parameter to be updated is the same as the number of unknown parameters. Recently, a kind of one-parameter estimation approach is proposed in [16,17], which needs only one parameter to be updated online and can significantly reduce the computational burden. Nevertheless, the mentioned control schemes do not take issues of input constraints, state constraints and prescribed performance into account. Therefore, for applying to a broader range of control problems with security, reliability and performance consideration in reality, further research is needed.
In the real world, many physical constraints are generated with security and reliability consideration, such as the output of MEMS resonator [18], the state constraint of aircraft engine [19] and the input constraint of magnetic-field electromechanical transducer [20]. Obviously, severe security matters, performance degradation, and other troubles can be caused without considering these constraints. For the issue of input constraint, a considerable amount of literatures have been published about it [21][22][23]. A non-smooth and piecewise input constraint model is described in [21]. Furthermore, the model in [22] is a smooth but piecewise function. The input constraint nonlinearity is tackled by asymmetric smooth input constraint model in [23]. For the issues of output constraint and state constraint, Barrier Lyapunov Function (BLF) is seen as an effective tool, and a significant number of typical works have been published including symmetric BLF [24,25] and asymmetric BLF [26,27]. However, the aforementioned BLF-based controllers have the following three drawbacks: 1) Discontinuous actions exist when constructing asymmetric BLF deals with asymmetric constraints; 2) Output/state constraints are achieved indirectly through error constraint, which leads to a more conservative initial output and state; 3) It is not allowed to handle both constrained and unconstrained systems without changing the controller structure. Although the integral BLF (IBLF)-based approach is possible to tackle output/state constraints directly [28], it can only overcome the disadvantages 1) and 2). By constructing a novel state transformation nonlinear function in [29,30], all those shortcomings can be overcome simultaneously. However, in practical applications, ensuring system security and reliability is the foundation, and achieving high performance control of the system is surely the ultimate goal. To be the best of our knowledge, no relevant results have been reported which can overcome the all above drawbacks and ensure safety reliability and high performance of systems simultaneously.
In this paper, with consideration of security, reliability and high performance, a LAFN-PPC of strict-feedback systems considering asymmetric full-state and input constraints is raised. In the controller design procedure, the constrained system is transformed into an unconstrained system using one-to-one nonlinear state transformation technique. One-to-one nonlinear error transformation technique is used to guarantee the prescribed performance. Furthermore, the unknown control direction and the input constraint nonlinearity are resolved by Nussbaum gain technique simultaneously. By introducing command filter and an error compensation mechanism, the constructed scheme can not only overcomes the so-called "EOC" problem, but also reduces filter errors. Moreover, the maximum values of the norm of optimal weight vector in FWNN is constructed as a virtual parameter, and the only one virtual parameter is estimated instead of the optimal weight vectors (OWVs). Regardless of the order of the system and the dimension of the system parameters, only one parameter is required to be updated online, which significantly reduces the computational burdens. The major contributions comparing with the existing ones are listed as: 1) In order to ensure the controlled systems with higher security, faster response speed and lower tracking error simultaneously, we combine a simple state transformation function with an error transformation function. All states and tracking error are always in symmetric or asymmetric prescribed bounds. Compared with the BLF-based methods [24][25][26][27], the LAFN-PPC can overcome all the three drawbacks, because we utilize the state transformation function instead of BLF, by which the constrained system is converted to an unconstrained system. In contrast to state transformation based methods [29,30], the tracking error is always remained within the prescribed performance bound by using an error transformation function.
2) By using command filtering to get the virtual control signal derivatives, the "EOC" problem of traditional backstepping method is overcome, the filter error caused by command filter is compensated by the carefully constructed error compensation mechanism. Compared with [1][2][3][4][5], the method we take only requires the reference signal and its first derivative, which greatly reduces the amount of calculation and meets many practical engineering requirements.
3) To significantly improve the computational efficiency of FWNN-based approximator and to replace estimating the OWVs in each step of backstepping, we construct the maximum value of the norm of OWVs in the FWNN as a virtual parameter. Only one virtual parameter needs to be estimated in the FWNN-based approximator, with this one-parameter estimation-based approach, the number of parameters updated online is independent of the order of the system and the dimension of OWVs, and the computational burden is significantly reduced, while the computational efficiency is significantly improved.

System descriptions
The considered strict-feedback systems with input constraint nonlinearity are given as As is known that input saturation of actuator is a common problem. The input saturation nonlinearity can seriously affect the safety and performance of the system. How to cope with the saturation nonlinearity has become an urgent and challenging research issue. In this paper, we take the input constraint nonlinearity into account. Mostly, the input constraint nonlinearity [21] can be expressed as where + and − are the upper/lower bounds of ( ).
To simplify the design of the control, we can define ( ) = − , where is a positive 8267
constant. We can rewrite the input constraint model as: Based on the above strict-feedback systems, the control object of this paper is to design a LAFN-PPC for system (2.1) to realize the following purposes: (a) All signals of the system are in the sense of uniformly ultimate boundedness.
The input signal and full state variables can be strictly restricted in asymmetric upper and lower bounds.
(c) The output signal can track the reference signal very well. And the output tracking error can be strictly restricted in upper and lower bounds.

FWNN
The FWNN [20] has strong power in function approximation, which consists of a series of fuzzy IF-THEN rules as: The FWNN shown in Figure 1 consists of five lawyers, including an input layer, a fuzzification layer, a membership layer, a rule layer, and an output layer. The firing degrees of rulers are defined as  The firing degrees are defined as The FWNN can be described as is the input vector, n is the input dimension of neural network.
denotes the weight vector, and 1 N  is the node number of neuron. and indicates the basic function vector.  is the estimation error.
And there is a positive constant Mi Lemma 1 [20]. Continuous function (ℤ) is defined on a compact set ℧. And for represents the estimation of  .

State transformation
For resolving physical constraints generated with security and reliability consideration, the following state transformation function [29,30] is introduced to achieve asymmetric constraints, symmetric constraints and no constraints on states simultaneously in a unified form: where i  and i  are positive constants, the initial state If no constraints need to be handled, let i (2.7) The above state transformation function can solve the control problems with asymmetric, symmetric and no state constraints in a unified form. Furthermore, it can handle above three kinds of control problems without changing adaptive laws. Figure 2 shows that the state transformation function can constrain  The state transformation function can be rewritten as

Remark 2.
i  are also bounded in the sets Proof. See Appendix A. Taking the time derivative of the state transformation function (2.5): The constrained system is converted to an unconstrained system as

Error transformation
For achieving the prescribed performance and guaranteeing the transformed output tracking error to converge within the prescribed performance bounds. Firstly, we define transformed output tracking error, virtual control errors and the command filters [10] as: where r y is the reference signal.
To ensure the transformed output tracking error 1 z strictly converges in the prescribed performance region during the whole time, we define where the design parameters k and k are positive constants, and  , Proof. See Appendix B. The error transformation can be defined as: where ℤ 1 ( ) is the transformed error and ϒ(ℤ 1 ) is defined as: is strictly constrained in symmetric or asymmetric domain ( ) , kk − showed in Figure 3 (For the symmetric one: . For the asymmetric one: ). It means that the error transformation function can deal with the issues considering symmetric or asymmetric exponential constraint simultaneously. We finally define the transformed tracking error as

(3.22)
Choosing a Lyapunov function and taking its time derivative, one can obtain where the compensation error 2 Step n. Taking the time derivative of n e , it has: ̇=̇−̇= + −̇ . (3.45) The error compensation signal is constructed as   (3.58) Up to now, the whole construction of LAFN-PPC is completed.

Stability analysis
For any given positive constant p , consider a closed set  We denote F  as the set of all bounded functions. According to the above analysis, we can get Vp  . Therefore, we can obtain that 1 v , 2 v ,L ,  , h and   determine the initial error, the error convergence rate and the steady-state error of the transformed output tracking error bound, which can be selected according to the performance requirements. k and k are always within (  0,1 and can deal with the issues considering symmetric or asymmetric transformed output tracking error constraint.

Simulations
In order to prove the effectiveness and feasibility of our control scheme, this section provides comparison simulation cases. Meanwhile, the control schemes based on works in [29,30] are compared with our suggested control scheme. A rigid manipulator system [30] is given as  In the simulation, we set  manipulator system (5.3). The controller, adaptive laws and first-order filter of work in [29] are given as ̂̇= ‖ 2 ‖ 2 − , Detailed controller design and parameter meaning is found in [29]. The controller, adaptive laws and first-order filter of works in [30]. are given as and parameter meaning is found in [30]. Figures 4-7 reflect the main results of our control scheme. Figure 4 shows that output y of rigid manipulator system can track desired trajectory well without violating the output constraint. Figure 5 reveals the output tracking error r e is in the region boundary all the times. The actual controls u is showed in Figure 6. And the state 2 x is illustrated in Figure 7. From Figure 8, the proposed control scheme tracks the desired signal well with different prescribed performance, and the boundedness of prescribed performance is not violated. Figures 9-11 present the comparative results of tracking trajectories, output tracking errors and input signals. We can easily find that the results of LAFN-PPC is better than controllers in [29,30]. Hence, for strict-feedback systems with purpose of high-precision tracking performance, full-state constraints and input constraint, we can conclude that our suggested control scheme has a potential to control them.        [29,30]. Figure 11. Comparison of input signal with works in [29,30].

Conclusions
The LAFN-PPC is newly constructed for strict-feedback systems with prescribed output performance, full-state constraints and input constraint. The newly constructed command filter based adaptive control scheme with an error compensation mechanism can not only overcome the so-called "EOC" problem, but also reduce filter errors. By introducing a one-to-one nonlinear state transformation function, the full-state constraints are resolved. The prescribed performance can be guaranteed by using the one-to-one nonlinear error transformation function. The unknown control direction and the input constraint nonlinearity are resolved by introducing the Nussbaum function simultaneously. Moreover, the large computational cost is solved by introducing a virtual parameter of adaptive laws. Only one parameter needs to be updated online. Future works can focus on commandfilter based optimal control of nonstrict-feedback systems considering performance constraint and address how to minimize resources consumption while ensuring performance. This finishes the proof.