Command-Filter-Based Finite-Time Adaptive Control for Nonlinear Systems With Quantized Input

This article considers the finite-time adaptive control problem of nonlinear systems with quantized input signal. Compared with existing results, the quantized parameters are unknown and the bound of the disturbance is not required. By utilizing the command filter backstepping method, an adaptive switching-type controller is designed and a novel switching mechanism is also proposed. By regulating the controller parameters online, practical finite-time stability can be guaranteed for the closed-loop system. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.


I. INTRODUCTION
In recent years, nonlinear quantized systems have attracted great attention because of the application of digital control systems and networked systems [1], [2]. However, quantization may lead to undesired system performance or even instability. Due to this practical and theoretical significance, the control problem of nonlinear quantized systems has been a hot topic in the nonlinear control area and numerous results have been reported (see [3]- [7] and references therein). However, some challenging problems still exist.
On one hand, most results just ensured that the states of closed-loop systems were bounded or asymptotically converged to zero [3]- [7]. Specifically, backstepping-based adaptive stabilization was considered for strict-feedback nonlinear systems with the quantized input signal in [3]. By choosing suitable quantization and controller parameters, the ultimate boundedness was achieved. However, global Lipschitz conditions were required for the nonlinear functions. In [4], a simple adaptive backstepping method was proposed such that the Lipschitz conditions required in [3] were removed. Then, the output feedback control problems were solved for nonlinear systems and interconnected nonlinear systems in [6] and [7], respectively. Compared with asymptotical stability, finite-time stability has unique advantages, such as faster convergence, higher accuracy, and wonderful disturbance rejection [8]- [10]. Considering these factors, a finite-time control problem was solved for strict-feedback nonlinear systems with input quantization in [11]- [13]. By using the adding a power integrator technique, the adaptive finitetime controllers were designed in [11] and [12] to ensure the global stability of the quantized systems. By utilizing the neural networks and backstepping techniques, an adaptive neural network output feedback controller was also designed in [13] where semiglobal practical finitetime stability can be achieved. However, it should be mentioned that the aforementioned results [3]- [7], [11]- [13] were all obtained based on the assumption that the quantized parameters were known. When the quantized parameters are unknown, the control problem will become more difficult for the nonlinear systems. Recently, the adaptive control problem was investigated for nonlinear quantized systems in [14] where the quantized parameters were assumed to be unknown. However, only asymptotical stability could be ensured. Therefore, it is a challenging and difficult problem to design an adaptive finite-time controller for nonlinear quantized systems whose quantized parameters are totally unknown. To the author's knowledge, there is no result reported on this issue because of the challenging problem involved.
On the other hand, it should be pointed out that all aforementioned papers [3]- [7], [11]- [14] utilized the backstepping technique. As we all know, an inherent problem of backstepping is the "explosion of complexity," which was caused by the repeated differentiation of the virtual controllers. To handle this problem, considerable efforts have been devoted [15]- [18]. In [15] and [16], dynamic surface control was proposed where first-order filters were introduced at each step of the backstepping design procedures. However, both papers ignored the compensation errors caused by the filter-order filters, which may degrade the system performance. Then, a modified technique, command filter backstepping, was first introduced in [17] where command filters were designed to approximate the derivative of the virtual controllers. Based on the technique in [17], fruitful results have been reported (see [18]- [20]). Besides, command-filter-based adaptive finite-time control has also been investigated in [21]- [25]. In [21], a neural network finite-time adaptive controller was designed for multiagent systems by constructing novel error compensation signals. Then, modified error compensation systems were designed in [22] and [23] where sign functions were utilized to eliminate the filter error in finite time. It should be mentioned that in the aforementioned references [21]- [23], Levant differentiator was utilized to approximate the derivative of the virtual controller in finite time. However, a drawback of this method is that the nth order derivative of the virtual controller should exist and be bounded. Obviously, it is hard to check this condition in advance. Consequently, it is also a challenging problem to design a commandfilter-based finite-time controller without the restriction in [21]- [23].
Motivated by the aforementioned discussions, command-filter-based adaptive finite-time control is considered for a class of nonlinear systems with the quantized input signal, unknown control direction, and This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ disturbance. Based on the command filter and the modified error compensation system, a novel adaptive finite-time controller is designed. By tuning the controller parameters via the switching mechanism, the system output can converge to a desired set in finite time. Compared with the existing results, the contributions of this article can be listed as follows. 1) Different from [3]- [7], [11]- [13], the quantization parameters are not required to be known. The bounds of the control coefficient and the disturbance are also not required. 2) Unlike the results in [21]- [23], a modified error compensation system is constructed in this article. By regulating the system's parameters online, finite-time stability of the error compensation system can be achieved. 3) Compared with the results in [14] where asymptotic stability was achieved, finite-time stability can be obtained for closed-loop system such that a better performance can be obtained. The remainder of this article is organized as follows. In Section II, the problem formulation is shown and some useful lemmas are also provided. In Section III, command-filter-based controller is designed. In Section IV, we provide the stability analysis. In Section V, a numerical simulation example is given to illustrate the effectiveness of the proposed method.

II. PROBLEM FORMULATION AND PRELIMINARIES
Consider the following nonlinear system with input quantization: . , x n ] ∈ R n denotes the system state; y is the system output; u is the system input and q(u) represent the quantizer; g(x) is an unknown function satisfying g(x) = 0; f i (x i ) are assumed to be known wherex i = [x 1 , . . . , x i ] T ; and d(x, t) denotes the external disturbance.
In this article, the quantizer is described as [12] q(u) = . More details about the quantizer can be found in [12]. Quantizer (2) satisfies The control objective of this article is to design an adaptive controller such that all the states of the closed-loop system are bounded and the system output y can converge to a desired set in finite time. The following assumptions are made for the system (1).
Assumption 1: It is assumed that the sign of g(x) is known. There exist unknown positive constants g andḡ such that g ≤ |g(x)| ≤ḡ.
Assumption 3: The parameters of the quantizer are unknown, which means σ and u min are unknown.
Remark 1: In the existing results about finite-time control [11]- [13], [21], a priori knowledge of the control coefficient and disturbance is always required. As shown in Assumptions 1 and 2, these restrictions are removed in this article. Therefore, a more general result can be achieved. Without loss of generality, it is assumed that g(x) > 0.
Remark 2: In the existing works [3]- [7], [14], only global boundedness or asymptotical stability could be obtained for quantized nonlinear systems. Although finite-time stability was achieved in [11]- [13], the quantized parameters were required to be known. In this article, the finite-time control problem is studied based on the assumption that the quantized parameters are unknown. Therefore, the proposed result is quite different from the existing results.
Besides, the following lemma is also introduced. Lemma 1 (see [10]): Let V (x) : R n → R be a positive definite Lyapunov function. There exist positive constants c 1 > 0, c 2 > 0, and Lemma 2 (see [25]): Then, the trajectory of the system is practical finite-time stable, and the residual set of the solution of system is given by

III. CONTROLLER DESIGN
In this section, based on the command filter, an adaptive controller will be designed by using the backstepping scheme. First, introduce the following change of coordinates: whereᾱ i−1 is the output of the command filter with α i−1 as the input.
Here, the command filter is designed as follows: where β > 0 and 0 < γ ≤ 1.ᾱ i = ζ i1 is the output of the command filter. The initial condition is ζ i1 (0) = α i (0) and ζ i2 (0) = 0. The virtual controllers α i , i = 1, . . . , n − 1 are designed as where c i and k i > 0 are designed parameters, μ is a positive constant satisfying 0 < μ = μ 1 μ 2 < 1, where μ 1 and μ 2 are odd integers. z is the compensation error signal defined as and the error compensation signal ξ is defined aṡ where l := l(s) + √ 2 2 l 0 . l 0 is a positive constant and l(s) is a strictly increasing function with respect to s, which satisfies lim s→+∞ l(s) = +∞.
Remark 3: Different from the existing results [21]- [23], Levant differentiator is not utilized in this article to approximate the virtual controller. In a different way, a novel error compensation system with switching parameters is developed in this article. By tuning the parameter online, the finite-time stability of ξ can also be guaranteed.

A. Controller Design
In this part, we will give the design procedures of the controller.
Then, design the controller as where h(k) a strictly increasing function, which satisfies lim k→+∞ h(k) = +∞.

B. Switching Mechanism
In this section, we will propose a switching mechanism to regulate the parameter k. Define the Lyapunov function ..,n {2c i }, and c 0 is a positive constant. Then, design the switching mechanism as follows.
then k → k + 1, and reset t k = t. Otherwise, k and t k remain unchanged.

2) For t > t s , if
then s → s + 1, and reset t s = t. Otherwise, s and t s remain unchanged.

IV. STABILITY ANALYSIS
By the work in Section III, we can obtain the following results, as shown in Theorem 1.
Theorem 1: Consider the nonlinear system (1), which satisfies Assumptions 1-3. Under the adaptive control law (20) and the switching mechanism, all the signals of the closed-loop system are bounded and finite-time stability can be achieved.
Proof: As shown in [8] and [9], for any fixed k and s, the solution of the closed-loop system exists until the state escapes to the infinity or time t tends to infinity. Define the maximum internal of the solution as [0, t m ). In the following part, we will first prove that the closed-loop system states are bounded and t m = +∞. The details are as follows.
During every switch of k, According to (25), it is obvious that z is uniformly bounded on [t k , t k+1 ). Then, we assume that the last switching occurs at time t k * .
which also implies the boundedness of z. Based on these facts, we have that closed-loop system states are bounded on [0, t m ). Since no finite-time escape phenomenon occurs, we have t m = +∞. Next, we can prove that the switching times are finite. The reason is: define g 0 = inf t≥0 g 0 andd 0 = sup t≥0 |d 0 |. According to Assumptions 1 and 2, it is easy to prove that g 0 > 0 andd 0 is bounded. Since h(k) is a strictly increasing function with respect to k, there exists a switching numberk such that Substituting (27) into (21) yieldṡ where ω > 0 is a design parameter satisfying nω ≤ δ. Select c j = c 0 2 + l 2 4ω . Then, we havė Based on (30) and Lemma 2, it is easy to prove which also implies that switching condition (23) will also never be satisfied. Therefore, the switching times are finite. Besides, we also have k * ≤k. Finally, finite-time stability can be obtained for z. The reason is which results in a contradiction. Thus, the finite-time stability is proved. Since (32) hold for t ≥ t k * , based on the results in [18], [19], [22], and [25], we have Then, consider the Lyapunov function V ξ . According to (9)-(11), we haveV Now, we will prove the finite-time stability of ξ. First of all, we can prove that s can just be switched a finite number of times. Then, we can prove that ξ can converge to zero in finite time. Then, reasons are shown as follows. 1) It is claimed that the switching number of s is finite. The reason is: since l(s) is a strictly increasing function, there exists a switching number s 1 such that l(s 1 ) −ς ≥ 0, whereς = max j=1,...,n {ς j }.
Hence, for t ≥ t s 1 , we havė By using Lemma 1 and the results in [10], we have . Hence, the following inequality can be obtained: which means the switching condition (24) can never be satisfied for t ≥ t s 1 . Therefore, the switching times for s are finite. 2) Lets be the last switching number of s. Then, we haves ≤ s 1 . For t ≥ ts, we have From (37), we can conclude that ξ can converge to zero in finite time. The reason is: if ξ = 0, ∃t ≥ ts which results in a contradiction. Therefore, ξ = 0 for t ≥ ts + 1 c(1−γ ξ ) ln(1 + c l 0 V (ts) 1−γ ξ ). In conclusion, finite-time stability of z and ξ is ensured. According to the definition of z and η, we can easily prove the finite-time stability of x.
Remark 4: In [14], parameter updating laws were developed to handle the unknown quantized parameters where two estimates were introduced in the adaptive controller. However, once introducing the adaptive estimates, the finite-time stability cannot be achieved. Different from the result in [14], switching parameters are introduced in the adaptive controller and error compensation systems. By tuning the switching parameters in a finite number of times, the effect of the unknown quantized parameters can be eliminated such that the finite-time convergence can be achieved.
Remark 5: Our proposed method can also be utilized to solve the finite-time control problem of nonlinear systems with actuator fault where f (u) denotes the actuator fault, which can be modeled as [12] f where b(t) > 0 and u d (t) is a bounded signal. Obviously, system (39) has similar form with system (1). Therefore, our proposed method can be applied to system (39). Remark 6: The finite-time fault-tolerant control has been investigated in [25] where the error signal can converge to a residual set in finite time. Compared with Li [25], the advantages of all papers are: first, the unknown system parameters are compensated by regulating the controller parameters; and second, the bound knowledge of |ᾱ i − α i | is not required to be known in this article.

V. SIMULATION EXAMPLE
In this section, to show the effectiveness of the proposed approach, the pendulum system in [26] is considered The detailed physical descriptions about the parameters m, l, g, and k can be found in [26]. d is the unknown disturbance. Let x 1 = θ and x 2 =θ, then we havė According to the results in Section III, we can design the command filter and the error compensation system first. Then, the system controller can be designed as follows: u = −h(k)(c 2 z 2 + k 2 z μ 2 + sgn(z 2 f 0 )f 0 + sgn(z 2 )).
The design parameters are chosen as: c 0 = 2, k = 1, l 0 = √ 2, β = 1000, γ = 0.75, h(k) = 0.8 k, and l(s) = 0.3s + 1. In the simulation, the initial conditions are chosen as:    Fig. 1 provides the response of the system state x. The response of the compensation system signal ξ is shown in Fig. 2. Figs. 3 and 4 provide the response of switching parameters k, s and system input u. From Fig. 1, it can be seen that the finite-time stability can be obtained for the considered system, and we can also find that the compensation system ξ is finite-time stable from Fig. 2.

VI. CONCLUSION
In this article, the command-filter-based finite-time adaptive control problem has been investigated for a class of nonlinear systems with quantized input signal and disturbance. Unlike the existing results, the quantized parameters and the bound of the disturbance are not required in this article. By introducing the novel command filter and the error compensation system, the "explosion of complexity" problem can be solved. By tuning the control parameters online, the designed controller can ensure that the finite-time stability can be guaranteed for the closedloop system. Moreover, in the future, we will focus on the commandfilter-based finite-time control problem for nonlinear systems whose functions are unknown.