Saturated adaptive barrier sliding mode control with state-dependent uncertainty limit

This paper proposed a saturated adaptive barrier sliding mode control (SABSMC) for disturbed systems with parameter uncertainty, external disturbance, and input saturation. One distinctive feature of the proposed method is that even under the inﬂuence of input saturation, it is able to drive the system trajectory into a prespeciﬁed neighbourhood of the sliding surface, independent of the upper limit of lumped disturbance. Moreover, attributing to the designed adaptive laws, the switching gain of the proposed method can be adaptively adjusted in a wise manner. Another attractive feature of the proposed method is that it removes any presumption of constant upper limit on the lumped disturbance. Instead, SABSMC utilises a more rational state-dependent upper limit assumption. The system dynamics of the closed-loop system is analysed via the Lyapunov method in the presence of parameter uncertainty, external disturbance, and input saturation. The effectiveness of the proposed method is conﬁrmed through numerical simulations.


INTRODUCTION
As an efficient control method, sliding mode control (SMC) has developed greatly owing to its strong robustness to parameter uncertainty and external disturbance [1][2][3], and has been broadly implemented in various actual situations [4,5]. In SMC construction, the switching gain is one of the key parameters that affect the controller performance. The choice of the switching gain, nevertheless, is a tough task. To guarantee system stability, the switching gain should be adjusted to be greater than the upper limit of the overall parameter uncertainty and disturbance (or we called the lumped disturbance) [6]. However, due to the complicacy of the lumped disturbance, such a limit is time-varying and, furthermore, commonly it is hard to know in numerous real scenarios [7]. Therefore, the chosen switching gain should be sufficiently large to deal with different types of lumped disturbances, which will eventually lead to parameter overestimation, undesired chattering, and control accuracy deterioration [8].
In order to improve the robustness and adaptability of controlled systems, the SMC strategy is hoped to be adaptive to various disturbances. The pursuit of this objective has led to , where the switching gain is automatically tuned for the establishment of a sliding mode (SM) regardless of the upper limit of the lumped disturbance [9,10]. Various approaches have been presented for constructing ASMC. These approaches roughly fall into the next three cases: (a) ASMC on the basis of the equivalent control [11][12][13]. This method is realised by adopting low-pass-filtered equivalent control to approximate the value of disturbance. Nevertheless, the use of a low-pass filter requires the information on the upper limit of disturbance for the purpose of appropriately selecting the filter parameter; (b) ASMC on the basis of monotonically increasing the gain [14][15][16]. The main idea of this method is to increase the gain until reaching the upper limit of disturbance and SM and then keeping the gain fixed. The main drawback of this method is that it does not prevent the gain from increasing even if the disturbance or control error is reduced, resulting in the overestimation problem; (c) ASMC on the basis of increasing and decreasing the gain [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. To tackle the disadvantages in method (b), method (c) has been presented, where the gain increases (resp. decreases) until SM is achieved (resp. lost). This method avoids the big overestimation of the gain in method (b).
It is worth noting that the aforementioned ASMC methods  may provide conservative results since the lumped disturbance is supposed to be upper limited by a constant a priori. As parameter uncertainty, which is part of the lumped disturbance and is commonly coupled with system states, consideration of this constant limit restricts the system states a priori (i.e. before performing closed-loop stability), which is a restrictive assumption. Recently presented ASMC strategies reported in [35,36] do not require the unreasonable constant upper limit assumption on the lumped disturbance. The main drawback of the strategies is that the ultimate bounds in [35,36] are greatly affected by the unknown lumped disturbance. Therefore the control performance is difficult to be pre-evaluated and prespecified, and the lumped disturbance makes a great impact on the control accuracy. Furthermore, the ASMC strategies in  have ignored a very important issue, that is, input saturation, which always results in performance deterioration and even system instability if it is ignored. Although ASMC with input saturation issue has been studied in [37][38][39][40], they employ monotonically increasing adaptive gains, which lead to the overestimation problem. In addition, the lumped disturbance in [37][38][39][40] is assumed to have a priori constant upper limit, which is a pretty restrictive assumption. To the author's knowledge, no work has been carried out on the design of an ASMC controller, which can solve the above-mentioned problems simultaneously.
To solve the above-mentioned problems, a new ASMC strategy termed saturated adaptive barrier SMC (SABSMC) is formulated for a class of disturbed systems with parameter uncertainty, external disturbance, and input saturation. One distinguishing characteristic of SABSMC is that it is capable of driving the system trajectory into a prespecified neighbourhood of the sliding surface and remaining inside it even under the influence of input saturation. This prespecified neighbourhood is not affected by the lumped disturbance, which allows predesigning and minishing the ultimate bound and tracking error ahead of time. Hence, the proposed method is promising for actual applications. The switching gain can be adaptively adjusted in a wise way by utilising the barrier function. Another distinctive feature of SABSMC is that it utilises a more proper state-dependent structure of the upper limit on the lumped disturbance and does not presume the lumped disturbance to be upper limited by a constant a priori. The effectiveness of the controlled system is proved in theory by considering parameter uncertainty, external disturbance, and input saturation. Numerical simulations are conducted to test the superiority of SABSMC.

PROBLEM DESCRIPTION
Consider a class of continuous-time systems with parameter uncertainty, external disturbance, and input saturatioṅ where x n ∈ ℝ n , v ∈ ℝ, and f ∈ ℝ n denotes the state vector, control input, and external disturbance, respectively. f is assumed to satisfy the matching condition. The system matrix Φ is supposed to be unknown. In actual applications, input saturation will inevitably occur as a result of the physical limitation of the control capacity. Input saturation is represented by sat(ν(t)) where ν M > 0 stands for the maximum value of the control input. In addition, the saturation function sat(ν(t)) is further expressed as sat(ν(t)) = Θ(ν(t))⋅ν(t) with Θ (ν(t)) ∈ (0, 1] can be regarded as an index for the saturation level of the control input [37]. There exists a constant δ 1 satisfying 0 < δ 1 ≤ min(Θ(ν(t))) ≤ 1 [37,38].
The switching function is defined as follows: where G is a constant gain vector. For the convenience of analysis, the parameter uncertainty Φ along with the external disturbance f(t) is treated as a lumped disturbance ξ(t)

SABSMC
In this part, a SABSMC scheme is first designed. Then the controlled system performance is theoretically analysed.

Design of SABSMC
The following assumption of the lumped disturbance is introduced. Assumption 1 [35]: where * 1 and * 2 are unknown positive constants. Remark 1: Previous ASMC methods assume that ξ(t) oṙ(t ) is upper limited by a constant a priori. It can be observed that Assumption 1 is more general as the constant upper limit is substituted by a state-dependent upper limit. ■ In view of the system (1) and Assumption 1, the new SABSMC is designed where the adaptive switching gain u s (t, η(t)) is defined as with and the update laws are chosen aṡ̂1 with the barrier function where τ, ε, α 1 , α 2 , α 3 > 0, μ > 1 are the control gains, and Recalling (15), it can be found that κ a (η(t)) : [25]. κ a (η(t)) allows the switching gain to increase and decrease based on the current value of η(t). If η(t) reaches the sliding surface, κ a (η(t)) and the switching gain will also become zero. Therefore, the switching gain of the reported method can be adaptively adjusted in an appropriate manner.

Stability analysis
Theorem 1: Consider the system given in (1) with parameter uncertainty, external disturbance, and input saturation, and the lumped disturbance ξ(t) satisfies Assumption 1, then the pro-posed SABSMC controller (7) with update laws (8)-(15) can drive the system trajectory into and remain always in a region |η(t)| ≤ η 1 < ε with Proof: The stability of the controlled system is studied for the two feasible cases, that is, cases (i) and (ii).
It can be concluded from the above analyses that the system trajectory of (1) can enter into and ultimately bounded within a certain region |η(t)| ≤ η 1 < ε of the sliding surface.
Theorem 1 is proven. Remark 2: Compared to the state-of-the-art ASMC methods like , the most significant merits of the proposed SAB-SMC lie in the following two aspects: First, it forces the system trajectory into a prespecified neighbourhood ε of the sliding surface even under the influence of input saturation, uncertainty, and disturbance. The prespecified neighbourhood ε is independent of the lumped disturbance ξ(t), which allows to design the ultimate bound of the system trajectory in advance and further improve the control accuracy. This is in contrast with previous methods whose upper limits depend on the lumped disturbance ξ(t) or the derivative of lumped disturbancė(t). Second, the reported method does not impose a priori constant upper limit on the lumped disturbance ξ(t). Instead, a more rational state-dependent assumption (6) is adopted in the proposed method. Moreover, the switching gain of SAB-SMC is adaptively regulated in a wise manner via the barrier function.
Remark 3: It should be pointed out that in previous methods [25][26][27], the convergence timet is designed as a control gain of FIGURE 1 Block diagram of the proposed method the controller, which is difficult to find. In addition, η(t) cannot converge to the ultimate bound again if an escape from the ultimate bound takes place at a certain time. In contrast,t is replaced by a more reasonable gain ε in the proposed method. Moreover, if an escape occurs at a certain time instant, the proposed method can drive η(t) into the ultimate bound again as verified in Theorem 1. ■ Based on the formulation in Section 3, the block diagram of the proposed method is illustrated in Figure 1.

Simulation results
In this section, the effectiveness and virtue of SABSMC are demonstrated through a series of numerical simulations. Throughout simulations, the saturated system (1) with the following parameters [35] is employed: ). An adaptive SM controller presented in [35] and in the form of (44) is tested for comparison.
The switching functions of the method in [35] and the proposed method are illustrated in Figures 2 and 3, respectively. It is seen that the method in [35] produces the width of the ultimate bound of 0.021. Concerning the proposed FIGURE 2 Switching function of the method in [35] FIGURE 3 Switching function of the proposed method method, the switching function can converge into and ultimately constrained by a prespecified bound ε = 0.001. As compared with the method in [35], the proposed method mitigates the width of the ultimate bound to 8.4 × 10 −4 , which substantially reduces the width by 96%. The control inputs of the two methods are illustrated in Figures 4 and 5. It is observed that the control inputs satisfy the input constraints and are limited within ±2.5 along the entire evolution. The behaviour of the adaptive switching gain u s (t, η(t)) of the proposed method is depicted in Figure 6, which illuminates that u s (t, η(t)) changes according to the variation of η(t). Hence, the adaptive switching gain of SAB-SMC is regulated in a wise manner. The simulation results verify the superiority of the proposed method over the method in [35] under the influence of parameter uncertainty, external disturbance, and input saturation.

CONCLUSION
A novel SABSMC method was developed in this paper dedicated to the robust control of saturated systems with respect to  Adaptive switching gain u s (t, η(t)) of the proposed method parameter uncertainty and external disturbance. In contrast to existing ASMC methods, the SABSMC drives the system trajectory into a prespecified, that is, independent of the lumped disturbance, region of the sliding surface in the presence of input saturation. Furthermore, SABSMC performs better control accuracy without imposing a priori constant upper limit assumption on the lumped disturbance. The property of SAB-SMC has been analysed in theory. Simulation results show that SABSMC outperforms the previous method under the influence of parameter uncertainty, external disturbance, and input saturation. In the future, the proposed method will be extended to the second-order SM case [41], and it will be applied to practical systems.