Finite-Time Sliding Mode Control Design for Unknown Nonaffine Pure-Feedback Systems

A class of unknown nonaffine pure-feedback nonlinear systems is investigated and a novel output feedback control scheme with low complexity is proposed, based on the sliding mode control theory. The scheme is capable of guaranteeing output tracking error with finite-time convergence and bounded closed loop signals. In this scheme, a novel transformation method is included, which can easily transform the state-feedback control of nonaffine systems into output feedback control of strict-feedback affine systems. Based on the transformed affine systems, a novel finite-time sliding mode control is designed, which is continuous and nonsingular. The control scheme proposed in this work is simple and easy to implement, in which the “explosion of complexity” caused by backstepping-like scheme is completely avoided. And the finite-time convergence is successfully achieved. In addition, the scheme is designed based on output feedback control. And the dynamics of the nonaffine nonlinear systems is unknown in the design process. Thus, the system knowledge needed is reduced.


Introduction
In the past decade, control design for complex nonlinear systems has attracted a lot of attention.And many breakthrough results have been obtained, such as feedback linearization technical [1] and adaptive backstepping technical [2].The aforementioned results assumed the input is affine in the systems considered.In practice, there are many nonlinear systems with nonaffine structure, such as aircraft dynamics [3] and mechanical systems [4].Explicit inverting control design is impossible for nonaffine systems, because the input does not appear linearly in the systems.Therefore, the control design for nonaffine systems is more complex and more challenging.
Many researches have been done for nonaffine purefeedback nonlinear systems.In the literatures [5][6][7][8][9][10], nonaffine pure-feedback systems are studied, in which the model is known.In practice, it is difficult to obtain an exact dynamic model of the system.And in the literature [11][12][13][14], the controller design for unknown nonaffine pure-feedback systems is studied.However, the proposed control methods above are all designed based on the backstepping technical.
A drawback with the back-stepping technique is the problem of "explosion of complexity," which is caused by the repeated differentiations of certain nonlinear functions such as virtual controls [15,16].To overcome the problem, dynamic surface control (DSC) was proposed in the controller design for nonaffine system [17], by employing first-order filtering of the synthetic virtual control input at each step of backstepping approach.However, this method will produce an algebraic loop, because the controller is employed in the approximation algorithm, whose output is simultaneously utilized in the controller.The problem of algebraic loop is solved in [18] by combining the DSC technique and input-to-state stability-(ISS-) modular design method [6].However, in [18] the corresponding neural network (NN) and filter are needed to be designed in each recursive step.Thus, the method is still complex.Recently, a low-complexity global approximation-free backstepping-like approach is proposed for pure-feedback nonaffine systems in [19], under the assumption that the full states are known.And the complexity explosion issue is avoided.However, it is still a recursive method.And it is usually impossible that all the states are available in the actual process.Hence, for nonaffine systems, it needs to 2 Mathematical Problems in Engineering further study the low-complexity controller design that is not based on the backstepping technique and needs less system knowledge.
It has been recognized that sliding mode control is efficient to design a robust controller.In most of the sliding mode controllers, the sliding mode surface is linear, which can only guarantee the infinite-time convergence of the states [20,21].To implement finite-time convergence, nonlinear sliding mode surfaces are employed.One of such sliding mode surfaces is the terminal sliding mode (TSM) [22].However, TSM controller may cause the singularity problem in the closed-loop system [23].To overcome the singularity problem, the nonsingular terminal sliding mode control (NTSM) is proposed in [24,25].However, NTSM can only be used in the second-order system.For high-order systems, a finite-time sliding mode controller is designed in [26].However, the sliding mode controllers mentioned above are all designed for affine systems and are difficult to directly be applied to nonaffine systems.According to the authors' best knowledge, all the sliding mode control designed for nonaffine systems [27][28][29][30][31][32] can only guarantee the infinitetime convergence of the states.The finite-time controllers have faster convergence rate than that of infinite-time controllers.Furthermore, they possess better robustness and better disturbance rejection performance, because fractional power items are contained in them [33,34].Thus, research on finite-time control of nonaffine systems has important practical and theoretical significance.
In this paper, a novel finite-time convergence control scheme is proposed for unknown nonaffine pure-feedback system, based on sliding mode control theory.In the design procedure, the input of the system is augmented with a lowpass filter firstly.Then the nonaffine system is transformed into an affine strict-feedback system through states transformation.Because the states in the transformed system are unavailable for controller design, a finite-time differentiator is introduced to obtain the unavailable states.Finally, a finitetime sliding mode controller is designed.Main contributions of the paper are the following: (1) Complexity of the control for nonaffine system is reduced dramatically.A novel transformation method is proposed, capable of completely avoiding "explosion of complexity" caused by backsteppinglike scheme.Moreover, in the proposed scheme, only one estimator is needed to obtain the states used in the controller.(2) The finite-time convergence of the closed system is achieved for the first time.For nonaffine purefeedback nonlinear systems, a nonsingular, finitetime sliding mode control is designed for the first time.And the controller is continuous; thus, the chattering of actuators is alleviated.(3) The system knowledge needed is reduced.In the proposed scheme, only the outputs of the system, other than all the states, are used.Furthermore, the control is designed based on the unknown nonaffine pure-feedback systems; thus, accurate model of the system is not required.
Assumption 1 is a commonly used sufficient condition for controllability of nonaffine system [19].
In this paper, we focus on the control of unknown nonaffine pure-feedback system.The objective is to obtain a low-complexity controller, such that the tracking error with respect to the desired trajectory can converge to the equilibrium point with finite time, and all the states are bounded.

Finite-Time Output Feedback Control
Based on Sliding Mode Control Theory

Design of a Novel System Transformation Method.
Recently, in [21], a low-pass filter used to attenuate the chattering is introduced in the input channel (i.e., u = +V), when designing the sliding mode control for affine systems.
Inspired by this, we proposed a novel transformation method, in which we will show that the low-pass filter can also be used to transform the nonaffine systems into affine systems.The transformation procedure is given as follows.Augment the nonaffine system (1) through introducing a low-pass filter in the input channel.And the augmented system can be denoted by the following: In ( 3),  > 0,  becomes a new state, and V denotes the new control input. 1),  = 2, . . .,  + 1.Based on (3), one can obtain Now, the initial nonaffine system denoted by ( 1) is transformed into an affine strict-feedback integral chain system denoted by the following: Base on Assumption 1, one can obtain 0 < ∏  =1  0 ≤  ≤ ∏  =1  1 .Moreover, if || ≤ , combing ( 5) and ( 6), the following differential inclusion can be concluded: in which

Design of a Finite-Time
Differentiator.In transformed system (5), only the states  1 =  1 are measurable, and the states   ,  = 2, . . .,  + 1, are not available.All the unmeasurable states can be obtained through solving the derivatives of  1 .In this section, a finite-time differentiator is given by where  is a Lipschitz constant of  (+1) 1 and   is the estimate of  (−1) 1 .Define the estimated errors and their derivatives as   =   −  (−1) 1 =   −   ,  = 1, . . .,  + 1.The following dynamic equations can be obtained: It follows from [35] that the estimated errors can converge to zero in finite time.
Because ẽ ,  = 1, . . ., , will converge to zero in finite time, system (12) will then reduce to Substituting ( 11) into (23) yields Noting that  ∈ [− 0 ,  1 ],  1 >  0 , 0 <  0 < 1,  1 >  0 > 0 and  > 0, and further substituting (11) into (24), it follows that Substituting  denoted by ( 11) into (25), and combing with  0  > , yields According to (26), we know that the sliding mode surfaces  and ṡ will converge to zero in finite time.And (9) implies that ẽ will converge to zero with finite time.So dynamic (13) will reduce to From ( 27), it can be concluded that   ( = 1, . . .,  + 1) and ς +1 will converge to zero in finite time [21].And at this time   =   ( = 1, . . .,  + 1) denote the tracking error and its derivatives, so the tracking error and its derivatives are bounded and can converge to zero in finite time.Because   ,  = 1, . . .,  + 1, are all bounded, the new input V in (11) is bounded.And according to the bounded input and bounded output properties of one-order linear system [36], Then it can be concluded that  2 is bounded according to the fact that  1 / 2 is bounded.Through using the procedure recursively, it can be concluded that all the other states   ,  = 2, . . ., , are bounded too.
Remark 4. Compared with the commonly used backstepping-like controller, the complexity is reduced dramatically in the proposed controller through using a novel system transformation.And the finite-time convergence of the tracking error can also be insured.The system knowledge needed is reduced, and only the output state is used in the proposed controller, while all the states are usually needed in the backstepping-like controller.
As a comparison, an improved backstepping-like controller designed in [18] is also simulated.In [18] the complexity issue is alleviated through using the dynamic surface technique, which can avoid calculating the derivative of the virtual control.The controller is designed as  = − 3  3 + θ 3  3 ( 1 ,  2 ,  3 , ż 3 ), in which  3 =  3 −  3 and  3 is tuned by one-order filter      According to the simulation results, the following can be concluded: (1) The proposed controller can insure the tracking error and its derivatives have a finite-time convergence property (Figures 2-5).This is because the proposed controller can insure a finite-time convergence of the sliding mode surface (Figure 6); moreover, the sliding mode surface in this paper is a nonsingular finitetime terminal sliding mode surface, so the states on the sliding mode surface can converge to zero in finite time.
(2) The proposed sliding mode controller is chatteringfree.Though the augmented input is desecrated (Figure 9), the actual input  is continuous and smooth, because  is produced by u =  + V.
Thus, the chattering of the conventional sliding mode control is avoided.
(3) The proposed controller is simple and easy to implement.And the complexity of the current controllers designed for nonaffine systems is greatly avoided.It is noted that three NN units, two filters, and all the three states are needed in the backstepping-like controller proposed in [18].And the number of NN units and filters will increase with the increase of the system order.However, only one output state, one differentiator, and one filter are needed in the controller proposed in this paper.In addition, the number of the needed states, differentiators, and filters will not increase with the increase of the system order.So the complexity of the controller for nonaffine pure-feedback systems is greatly avoided.
(4) Compared to the controller in [18], the needed knowledge of system is reduced.In the proposed controller, only the output and the order of the system, other than all the states and the accurate system model, are needed.
(5) According to , it is shown that the control performance of the proposed controller in this paper is as good as that of the improved backstepping-like controller proposed in [18].

Conclusion
In the paper, a novel control scheme with low complexity is proposed for unknown nonaffine pure-feedback systems.The scheme involves a novel transformed method, which can transform the nonaffine systems into affine strict-feedback system.And a nonsingular finite-time sliding mode controller is also implemented.Compared with the related article, the proposed scheme not only can achieve finite-time convergence, but also is much simpler and easy to implement and needs less system knowledge.And the simulation results demonstrate that the tracking performance is also very good.