Approximation-free Quantitative Prescribed Performance Control of Unknown Strict-feedback Systems

 Abstract —Prescribed performance control (PPC) has been proved to be a powerful tool which seeks transient performance for tracking errors. Unfortunately, the existing PPC schemes only can qualitatively design transient performance, while they cannot quantitatively set the convergence time and meanwhile minimize overshoot. In this article, we propose a new quantitative PPC strategy for unknown strict-feedback systems, capable of quantitatively designing convergence time and minimizing overshoot. Firstly, a new quantitative prescribed performance mechanism is proposed to impose boundary constraint on tracking errors. Then, back-stepping is used to exploit virtual controllers and actual controller based on Nussbaum Function, without requiring any prior knowledge of system unknown dynamics. Compared with the existing methodologies, the main contribution of this paper is that it can guarantee predetermined convergence time and zero overshoot for tracking errors, and meanwhile there is no need of any fuzzy/neural approximation. Finally, compared simulation results are given to validate the effectiveness and


I. INTRODUCTION
ECENTLY, prescribed performance control (PPC) has caused increasing interests and there have been many successful uses of PPC in a series of dynamic systems [1]- [5]. Compared with the existing other control strategies, the clear superiority of PPC is that it can impose boundary constraints on the convergence process of control errors, being expected to seek satisfactory transient performance and steady-state performance. The common technique of current PPC theories is to develop various types of performance functions which are further used to constrain control errors. Owing to the different considered focuses, lots of new performance functions [6]- [8] have been developed, getting rid of the dependence on initial error [6], ensuring the finite time convergence of error [7], and owing the ability of boundary readjustment [8].
Despite the excellent developments mentioned above, it is worthy to point out all of the existing PPC methodologies are qualitative ones, while the mechanism of quantitative constraint is still unclear. Based on PPC, the starting point is to exploit performance functions to constrain control errors. Then, the error transformation approach is adopted such that the "constrained" system is equivalently transformed into an unconstrained one which is convenient for controller design [1]. The newly defined transformed error, instead of the initial tracking error, is used to exploit feedback controllers. The boundedness of transformed error is equivalent to the guarantees of the spurred prescribed performance. The original intention of PPC is to achieve good transient performance which is determined by the formulations of performance functions. Performance functions contain several design parameters whose values directly affect the transient performance such as overshoot and convergence time of control errors. Unfortunately, the current PPC methods only qualitatively select appropriate design parameters for performance functions for the sake of obtaining satisfactory performance indices. However, what design parameters are appropriate? And, how to select appropriate design parameters for quantitative design of performance indices? None of this is clear. Actually, no schemes exist to quantitatively design overshoot and convergence time for tracking errors.
It is well-known that strict-feedback system is one of the most representational dynamic systems. Control system design for such systems has attracted extensive attention and achieved gratifying results [9]- [12]. Many practical systems such as robots, manipulators, servo mechanisms, aircraft, spacecraft and vessels can be represented as strict-feedback formulations. It is impossible to develop a model which can completely describe the actual system accurately. As a result, system uncertainties are inevitable. Considering a more rigorous condition, it is usually supposed that the system model is completely unknown. On this basis, fuzzy/neural approximation is a commonly used strategy [13]- [18]. The existing studies [9]- [10], [12], [14]- [15], [19]- [20] consider a class of partially unknown strict-feedback systems, that is, the system functions are unknown while the control gains are assumed to be known constants. Thanks to the universal approximation property, fuzzy systems and neural networks are applied to estimate the unknown system functions, and the Approximation-free quantitative prescribed performance control of unknown strict-feedback systems Xiangwei Bu*, Qiang Qi, Baoxu Jiang convergences of estimation errors are ensured by regulation laws exploited for the elements of fuzzy/neural weight vectors [9]- [10], [12], [14]- [15], [19]- [20]. Further, more general cases are investigated [17]- [18] and the authors suppose that both the system functions and control gains are unknown. Despite all this, a strict precondition that the bounds of unknown control gains must be known in advance, still is necessary for control design. The fuzzy/neural approximation approach is used to estimate the hybrid function consisting of system functions and control gains, avoiding repeated approximations of both of them. Such strict precondition [17]- [18] is released in another study [21], but the sign of unknown control gain still is a priori information.
In the last several years, lots of PPC-based control researches of strict-feedback systems have been reported [2]- [3], [22]- [23]. Unfortunately, all of them cannot quantitatively set prescribed performance (i.e., overshoot and convergence time) for control errors. Moreover, the uncertainty rejection ability is accomplished via fuzzy/neural approximation at the expense of reducing real-time performance because of high computational online learning schemes. Besides, very strict preconditions for control gains seriously damage the operability and application prospect. For this reason, this paper proposes a novel quantitative PPC strategy for a type of unknown strict-feedback systems, capable of minimizing overshoot and quantitatively setting the convergence time, while no strict precondition or neural/fuzzy approximation is required. The special contributions are summarized as follows.
1) Unlike the existing qualitative PPC [1]- [6], we develop a new type of performance functions which guarantee quantitative prescribed performance (i.e., minimize the overshoot and quantitatively set the convergence time), being able to achieve given time convergence of control errors without overshoot.
2) In the current studies [9]- [10], [12], [14]- [15], [17]- [21], there needs to be the prior knowledge of the signs and the bounds of control gains. However, the proposed method is addressed based on a weaker precondition that both the control gains and the system functions are completely unknown continuous functions.
3) Different from traditional fuzzy/neural approximation strategies [9]- [16] which suffer from high computational burden caused by multifarious online learning parameters required for fuzzy/neural weight vectors, this study exploits an approximation-free approach, utilizing the Nussbaum-type function, for unknown strict-feedback systems.

Remark 2.
There exists another common formulation of strict-feedback system, denoted by (2) The subsequently developed controller is also valid for this formula (2) since system (2) can be rewritten as system (1) via a concise model transformation approach [4], [24], [25].

B. Quantitative prescribed performance
To accomplish the spurred control objective, we propose a new prescribed performance approach, being different from all of the existing ones, namely quantitative prescribed performance which is capable of quantitatively designing the convergence time and minimizing the overshoot.
The addressed prescribed performance boundary is where the tracking error () et is constrained by the newly developed performance functions () The constraint performance by boundary (3) is clearly shown in Fig. 1. Fig. 1 reveals that if the tracking error e(t) is limited to the constraint boundary (3), then its transient performance and steady state performance can be quantitatively set as needed.
Moreover, e(t) can converge its steady-state value in a given time f T  , and there is no convergence overshoot. If we choose 1 are linearly convergent in the transient process (See Fig. 1 (b)). Otherwise, the convergences of ( )  We make the following transformation to facilitate control design.

  
From (7), we further have It is sorted out that The right side of (9) satisfies 01 .
The left side of (9) also satisfies From (11), we finally get It is concluded that the boundedness of T  is equivalent to (3). This completes the proof. ▇ Theorem 1 indicates that the boundedness of T  is a guarantee of prescribed performance (3). In what follows, the controller will be devised using the transformed error (6), and the design objective is to guarantee the boundedness of T  via Lyapunov approach.

A. Controller design
This subsection presents the design process of quantitative prescribed performance controller for unknown strict-feedback system (1) without using fuzzy/neural approximation based on back-stepping.
The following filters are introduced to handle the problem of "explosion of terms".
The existing study [20] shows that there exist non-negative continuous functions Substituting (20a), (22) and (26) 1, 2 Substituting (20b) into (34), we get , , ,  [8], the special novelty of the addressed method is that it can guarantee tracking errors to converge their steady-state values in a given time, and the convergence time can be arbitrary quantitatively designed. Furthermore, the overshoot of tracking error convergence also is minimal.
Remark 4. To reject system unknown dynamics, fuzzy/neural approximations [13]- [18] are common methods. Despite the global approximation ability of fuzzy systems and neural networks, the real-time performance may not be satisfied due to high computational load caused by lots of online learning parameters. While, in this paper, we propose an approximation-free PPC scheme, which eliminates fuzzy/neural approximation, for completely unknown strict-feedback systems.

IV. SIMULATION RESULTS
In this section, compared simulation results are presented to verify the superiority. The following strict-feedback system is considered. The reference command is chosen as 1, sin( ) . Based on the results in Section Ⅲ, we design the following controllers, regulation law and filter.
The proposed method is compared with a direct neural control (DNC) strategy [28]. Design parameters are designed as:

V. CONCLUSIONS
A new PPC strategy, being able to quantitatively design transient performance indices such as convergence time and overshoot, is investigated for a type of strict-feedback systems with unknown dynamics. A new kind of performance functions are designed to quantitatively constrain tracking errors with prescribed performance guarantees. Then, error transformed functions are introduced to facilitate control developments, from which transformed errors are derived for back-stepping controller design. Furthermore, from the benefit of Nussbaum Function, the addressed controller can reject unknown system dynamics without any fuzzy/neural approximation. The stability of closed-loop system is proved, and the superiority of the proposed method is verified via simulation results.

CONFLICTS OF INTEREST
The authors declare that they have no conflicts of interest.    Please see the Manuscript PDF le for the complete gure caption Figure 5 Please see the Manuscript PDF le for the complete gure caption Figure 6 The control input in Case 1.

Figure 7
Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption Figure 10 Please see the Manuscript PDF le for the complete gure caption Figure 11 The control input in Case 2.