Adaptive Dynamic Surface Control for a Class of Dead-Zone Nonlinear Systems via Output Feedback

The paper aims to address output feedback problem for a class of nonlinear systems subjected to unknown dead-zone. High-gain K-filters is firstly designed such that unavailable states of system can be reconstructed. Then, with the help of dynamic surface control (DSC) method, an adaptive output feedback controller design is established, which can alleviate undesired influence of the dead-zone and guarantee the semi-global stability of the system. Furthermore, the $\mathcal {L_{\infty }}$ performance of tracking error is achieved by means of initialization technique. Ultimately, numerical and practical examples clarify the efficiency of the proposed solution.


I. INTRODUCTION
Motivated by the widely application of the industrial motion control systems such as electric servo system and hydraulic servo system, dead-zone characteristic existed in a large number of actuators of the industrial control systems has been extensively discussed.
The pioneering work on control for nonlinear systems subjected to dead-zone could be dated back to [1], where adaptive dead-zone inverse compensation proposal was presented for a linear system with known bound of dead-zone parameters. The proposal ensured global stability and asymptotical tracking for a full state available system. Similarly, adaptive dead-zone inverse was constructed in [2], then linear model reference controller was presented. More recently, the research has been extended to strict-feedback nonlinear systems. In [3] smooth dead-zone inverse was delivered in order that control performance of adaptive backstepping design can be improved.
It is quite difficult to construct the exactly inverse of various dead-zone models from another point of view. Thus robust adaptive thought has been extensively applied to mitigate dead-zone. Instead of constructing dead-zone inverse, The associate editor coordinating the review of this manuscript and approving it for publication was Ahmed A. Zaki Diab .
separating thought is accepted based on the dead-zone characteristic: one is linear part and the other is ''disturbancelike'' part with unknown bound. In [4], a robust adaptive dynamic surface control (DSC) was provided in order to address the influence from unknown non-symmetric dead-zone in a MIMO nonlinear systems, and initialization technique was offered so as to obtain the L ∞ performance. When only input and output signals are available, [5] investigated an output feedback modified DSC based on tracking differentiator for a dead-zone nonlinear system. Both steady state and transient performance are achieved under assumption that all control functions are positive as well as bounded. Furthermore, with the aid of the fuzzy logic systems and neural networks techniques, lots of semi-global adaptive robust strategies were published for nonlinear systems disturbed by unknown dead-zone [6]- [9]. Other control schemes concerning the dead-zone can be refer to [6]- [9] and the references cited therein.
Based on the aforementioned results with respect to the dead-zone, we note that most existing control methods require the measurement of all the system states. Thus adaptive output feedback control is still challenging for nonlinear systems with dead-zone. Moreover, [10] proposed an adaptive output feedback control strategy for system subjected to dead-zone input, but the transient performance VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ could not be guaranteed, which may lead to some undesired behaviors, such as bursting. While in [11], although the transient performance is achieved, a complex dynamic surface design has to be needed due to tracking error transformation. Until now, the difficulties of output feedback control on system subjected to the dead-zone are both states observer construction and relatively simple control strategy in order that the undesirable catastrophic caused by dead-zone can be counteracted and closed-loop stability is obtained. Especially, steady state and transient performance could be guaranteed. This paper devotes attention to propose an adaptive DSC control via output feedback for uncertain nonlinear systems preceded by dead-zone input. The motivation for the scheme is the high-precision control for some mechanical plans subjected to dead-zone. The study makes these contributions: • Unlike the common K-filters employed in existing output feedback approach [11], unmeasurable states are reconstructed by high-gain K-filters, which pave the way for improvement of transient performance without tracking error transformation.
• An initialization technique is incorporated into the DSC for purpose of guaranteeing the L ∞ tracking performance. That is, just by adjusting the value of some design parameters, steady state error as well as maximum overshoot of tracking error could be made arbitrarily small. The structure of the paper: Section II: Problem statement and some preliminary knowledge are described. Section III: An adaptive output feedback DSC design procedure by means of high-gain K-filters is presented. Section IV: Stability and L ∞ tracking performance analysis are given. Section V: Two examples are provided to demonstrate effectiveness of this design.

II. PROBLEM STATEMENT AND PRELIMINARIES
In this paper, uncertain nonlinear SISO system is considered: where . . .
where x := [x 1 , x 2 , . . . , x n ] T ∈ R n and y ∈ R are state vector and output signal, respectively; θ i , i = 1, . . . , r and b are unknown parameters; the nonlinear smooth functions f i,j (y) , i = 1, . . . , r, j = 1, . . . , n, and g i (y), i = 0, . . . , m, (g m (y) = 0) are known; ρ (≥ 1) = n-m stands for the system relative order, and n, m are known constants; d ∈ R n is the external disturbance; u is unknown dead-zone's output formulated as [12], [13] with where m, h r , h l are unknown bounded positive constants, and h (t) satisfies Invoking (1) and (3) yieldṡ with b m = bm. The control objective: besides the boundedness of all signals of the closed-loop system, the tracking performance can be guaranteed under the proposed adaptive DSC control scheme.
Assumption 1: withd i (≥ 0) being unknown constants. Assumption 2: The sign of b is assumed to be positive for simplicity.
Assumption 3: The reference trajectory y d is smooth with y d (0) at designer's disposal, and there exists a known compact set such that [y d ,ẏ d ,ÿ d ] T belongs to it for all t ≥ 0.

Assumption 4:
The systeṁ are bounded-input to bounded-output (BIBO) stable, where y 0 , y, x 2 , . . . , x ρ are inputs and x ρ+1 is output. Remark 1: In contrast to the most related work [14]- [17], system (1) takes g i (y) into account and the term bG (y) h (t) caused by dead-zone cannot be directly seen as the bounded disturbance, which will increase the difficulty of our control design.

III. HIGH-GAIN OBSERVER BASED ADAPTIVE DSC DESIGN A. HIGH-GAIN K-FILTER OBSERVER
Firstly, design a proper vector k = [k 1 , k 2 , . . . , k n ] T to make become Hurwitz matrix. Now, (6) can be rewritten as: Owing to the only available output signal, the high-gain K-filter observer should be designed to compensate effect of dead-zone as well as to reconstruct unavailable states: where = diag 1, q, . . . , q n−1 and q(≥ 1) is observer gain chosen as a positive parameter. Then estimation of states is given asx To proceed, the estimation error is By means of (10)- (13), and (14), one can concludė where 1 is the first entry of . Lemma 1: Let where a symmetric matrix P = −1 TP −1 , andP = P T > 0, andP satisfies the equation with I being the unit matrix. Then the following inequality holdsV with nonnegative smooth function (y), and where λ max (•) denotes the largest eigenvalue of •, d max presents the maximum value of d(t) . Proof: We can get the relationship T P =¯ TP¯ by using the coordinate transformation¯ = −1 . And¯ could result in˙ Since A 0 is Hurwitz, (18) holds for positive definite matrixP. Define and time derivative of V¯ along (23) iṡ Meanwhile, there is a nonnegative smooth function (y) such that Together with Young's inequality and (5), one havė Hence we arrive at (19). This completes the proof.

B. ADAPTIVE DSC DESIGN
On the basis of above observer, the adaptive DSC design can be proceed smoothly.
The virtual controlζ i+1 is with positive design parameter c i . Letζ i+1 pass through filters with constant τ i+1 : Step ρ. The ρ-th surface error is Finally, the actual control v is with positive design parameter c ρ . Remark 2: As shown in (33), the estimated parameter ϑ is a scalar instead of a vector with n + r + 1 − ρ parameters, which not only can greatly reduce the computational burden, but also make the controller easy to implement. Moreover, the adaptive laws with dimension one are only used at first step.
Viewing (55)-(59), the time derivative of (46) satisfieṡ with Q defined by (51). Let r satisfy Then if V = C 2 , we haveV ≤ 0. Hence, given V (0) ≤ C 2 , one can obtain that V (t) ≤ C 2 , t ≥ 0. Further, according to (60), we obtain and from which we note steady state error could be made arbitrarily small under sufficiently large r . From the above analysis, V is proven to be bounded, and then S 1 , . . . , S ρ , y 2 , . . . , y ρ ,θ,p, and are bounded. Moreover, y d is bounded due to Assumption 3, which implies y is bounded. What is more, the nonlinear smooth functions f i,j (y), g i (y), (y) and¯ (y) are bounded, and then ω, ξ i and ψ are bounded. Note that and Assumption 1, which implies that ζ 1 is bounded. Viewing (31) and (35), we have z 2 is bounded. Then it can be checked from (36) that ζ 2 is bounded. Similarly, we obtain that z 3 , . . . , z ρ , ζ 3 , . . . , ζ ρ are bounded due to (36)-(40). From (65), we can know that x 2 , . . . , x ρ are bounded. With Assumption 4, x ρ+1 is bounded. Furthermore, ζ ρ+1 is bounded. Then, according to (42), we can get bounded v. Also, from (11), we can get bounded ζ . Hence, we have shown that uniformly ultimate boundedness is guaranteed for all signals in the system. Until now, we only discussed steady state tracking performance. Transient state performance also should be considered. What follows is a conclusion aims to solve the problem.
Remark 3: From (75), the high gain q of the high gain K-filters is crucial to guarantee the transient tracking performance. And the common K-filters only ensure the steady state tracking performance, cannot obtain the transient tracking performance.
Remark 4: In the proposed controller, design parameters q, c i , i = 1, . . . , ρ, τ i+1 , i = 1, . . . , ρ − 1, η ϑ , η p , and adaptive gains γ ϑ , γ p should be adjusted for a good transient tracking performance, then tracking error in (50) and (67) can be reduced. Since there are no explicit guideline to quantify the relationship of these parameters, some suggestions are given for obtaining good tracking performance.
• By (50), increasing the value of r can reduce the tracking error. Further, in view of (59), increasing the values of q, c i , i = 1, . . . , ρ, η ϑ , η p and adaptive gains γ ϑ , γ p and decreasing the values τ i+1 , i = 1, . . . , ρ − 1 is able to make r larger. However, the increase of η ϑ , η p will cause the increase of Q in (51). Based on the above observation, we can fix η ϑ , η p first and Q is independent of r . By this means, r can be arbitrarily increased without increasing Q and the maximum overshoot of tracking error is arbitrarily small.

VI. CONCLUSION
An adaptive DSC strategy with high-gain K-filters is proposed to cancel undesirable influence from input dead-zone in this paper. Also, it can be shown that design complexity of controller is greatly reduced due to the first order filters and simplified adaptive update laws. Moreover, the explicit function of L ∞ norm of tracking performance is given. However, our control design only can guarantee stability of closed-loop system in the sense of semi-global stability, and thus more attention should be paid on this problem in the further research.