Adaptive tracking control for uncertain nonlinear systems subject to unknown control coefficients, state constraints, and input saturation


 This article is committed to studying the tracking control problem for a class of uncertain nonlinear system with unknown control coefficients. The system is subject to full state constraints, input saturation constraint, and external disturbances simultaneously. By introducing a hyperbolic tangent function to approximate the saturated input function, the sharp corner caused by the input saturation is avoided. In the meanwhile, an auxiliary system is constructed to compensate the resulting approximation error. By using the barrier Lyapunov function (BLF) based adaptive backsteping control, the Nussbaum-type adaptive controllers are constructed for the augmented system with unknown control direction. It not only ensures the system states are always within the constrained range, but also guarantees the tracking performance of the system, no matter whether the control direction of the system is known or not. Meanwhile, dynamic surface control (DSC) is used in the controller design, which avoids ”computation explosion” caused by the repeated derivation of virtual control law. Aiming at the nonparametric uncertainty of the system, a common adaptive law is designed by combining the unknown constant bounds of the external disturbance with the error term caused by input saturation estimation. It improves the tracking performance of the system and reduces the burden of the controller greatly. Finally, a simulation example is given to demonstrate the effectiveness of the proposed control scheme in three scenarios.


Introduction
In the past few decades, the tracking control problem for uncertain nonlinear system has attracted great public attention due to its wide applications [1,2,3]. Self-driving cars and ships, for example, follow the trajectory set by their navigation systems [4]. The tracking performance is directly related to the safety and efficiency of automatic driving. System uncertainties are important factors leading to system instability, poor tracking performance or other undesirable consequence [5,6]. In a broad sense, system uncertainty mainly can be classified into parametric uncertainty and nonparametric uncertainty. It is well known that the adaptive backstepping control is a classical control methods for parametric nonlinear system [1,7]. When there are nonparametric uncertainties in nonlinear system, the unknown dynamics can be approximated by neural networks or fuzzy systems [8,9,10]. Although great achievements have been made in the tracking control of uncertain nonlinear systems, it is worth noting that all the mentioned results are based on the fact that the control coefficients of the system are known, or at least the control directions (the signs of the control coefficients) are available.
The control coefficients or called control gains, which represent the control directions are usually unknown, especially in practical systems [11,12]. For instance, a priori knowledge of the control direction of a the introduced error, which solves the input saturation constraint problem of the system well. On the other hand, a BLF-based adaptive controller is constructed for the augmented system, which ensures the stability of the system and averts the violation of state constraints for all the system running time. Compared with our control scheme, the nonlinear uncertainty in [8,36] is approximated by neural network, which leading to a heavy computational burden as the nodes of NN increase.
(2) Compared with the existed results [8,36], the considered system of this paper is more general, where the signs of control gains are not required to be with a prior knowledge. To facilitate the controller design, Nussbaum gain functions are utilized to solve the difficulty caused by unknown control coefficient.
(3) The adaptive law is designed by combining the unknown constant boundary of the system external disturbance with the error term caused by the input saturation estimation. By minimizing the design parameters, the burden of the controller is reduced and the tracking performance of the system is improved. Additionally, dynamic surface control is adopted which solved the problem of computation explosion caused by repeated derivation of virtual control law.
The organization of the rest of this article is as follows. Section 2 describes the statement of the problem and some necessary definitions, assumptions, and lemmas are given. Section 3 shows the explicit controller design procedure and stability analysis. Meanwhile, the main results of this article are summarized in a theorem. Following, a simulation example is given in Section 4 to illustrate the effectiveness of the presented controller. Finally, Section 5 ends this paper with the conclusions.

Preliminaries
Consider the following semi-strict feedback nonlinear systems (j = 1, 2, · · · , n − 1) x n (t) = θ n ψ n (x n (t)) + g n (x n (t))u(v) + D n (x, t), wherex j (t) = (x 1 (t), · · · , x j (t)) T ∈ R j (j = 1, 2, · · · , n) and y(t) ∈ R are the system states and output, respectively; g j (·) : R j → R is unknown smooth control gain function, θ j is unknown constant parameter with known boundry; ψ j (·) : R j → R is known continuous function; D j (x, t) : R n × R + → R represents the unknown external disturbance; u(v(t)) is the input saturation function, it is obtained by passing the control input signal v(t) through a saturation constraintū, which described by whereū is a known input saturation constant bound. Obviously, there is a sharp corner between u(v(t)) and v(t) when |v(t)| =ū. By referring to [8,36], the following smooth function can be used to approximate the input saturation function: Then, the saturation function sat(v(t)) can be expressed as where ρ(v) is the approximation error. Obviously, it is bounded, which can be seen from the detailed expression below On the other hand, it is worth mentioning that the system states are measurable and subject to constraints: where k c j are positive constants. They are required to remain in the predefined constrained regions for all the system running time.
To handle the input saturation constraint, with the help of equations (3)-(5), we use the neural network approximation to augment the n-order system (1) to the following (n + 1)-order system, where v and µ are the auxiliary state and control signals, ζ is a positive constant,D n (x, t) = D n (x, t) + g n (x n (t))ρ(v).
Remark 1. By introducing the hyperbolic tangent function to approximate the saturated input function, the problem of the sharp corner caused by the input function is avoided. Further more, the auxiliary system (6) is constructed to compensate the resulting approximation error. In system (6), the termD n (x, t) is the combination of the unknown constant boundary of the system external disturbance and the upper bound of the input saturation approximation error. We combine these uncertainties together and design a common adaptive law, which greatly reduces the burden of the controller.
The control aim of this article is to design a state feedback control strategy for the system (1), which drives the system output y to track a reference trajectory y d (t) as close as possible. In the meanwhile, all state constraints should be guaranteed during the system running time. To achieve the control goal, the following assumptions, definition and lemmas are necessary and practical. Assumption 1. θ j (j = 1, · · · , n) in system (1) represent the unknown parametric uncertainties with known boundry. Suppose that there exist a positive constant θ M such that |θ j | ≤ θ M . On the other hand, D j (x, t)(j = 1, · · · , n) in system (1) denote the external disturbances, which is bounded. Suppose that there exist unknown positive constants d j such that |D j (x, t)| ≤ d j .
Assumption 2. The functions g j (·)(j = 1, · · · , n) are unknown but bounded. Suppose that there exist positive constantsḡ j such that 0 < |g j (·)| ≤ḡ j for ∀x j ∈ Ω x j . Assumption 3. The reference signal y d (t) is smooth and available, which satisfies where ω is a constant.
Remark 2. The above assumptions are standard and necessary. Assumptions 1 is the standard assumption for semi-strict feedback nonlinear system, and the constant bounds for θ j and D j (x, t) are necessary and reasonable in adaptive backstepping tracking control. Assumptions 2 indicates that the control direction of system (1) is unknown but boundedness, which is not only reasonable and necessary but also widely available in many important practical systems, such as high speed trains, ship steering system, and aerial vehicles, etc. Assumptions 3 is also the standard assumption in adaptive backstepping tracking control.
Remark 3. Compared with the system studied in [8,36], the considered system of this paper is more general. Assumptions 2 indicates that the control gains g i of system (1) are unknown, while the control gains of the system in articles [8,36] are known constants. So the systems discussed in [8,36] are our special case. Additionally, state constraints are not considered in [36].
Lemma 1 [28] . Given U ∈ R, there exist a hyperbolic function tanh(·) such that the following inequality holds for any ε > 0, where k = e −(k +1) , ie, k = 0.2785. Lemma 2 [14,17] . Let V (·) and ξ(·) be smooth functions defined on [0, t f ) and V (·) ≥ 0, N (ξ) is a smooth even Nussbaum-type function. For ∀ t ∈ [0, t f ), if there exist positive constants c 0 and c 1 , such that where g(τ ) = 0 is a bounded time-varying function, then V (t), ζ(t) and Remark 4. Lemma 2 is developed by basing on the definition of Nussbaum function, which is always used to deal with the issue of uncertain control gain. What's more, according to [17,18], if the solution of closed-loop system is bounded on [0, t f ) for any given t f > 0, then t f = ∞.

Robust Adaptive controller design and stability analysis
In this section, we design the robust adaptive tracking controller for the expanded system (6) based on the backstepping control. A first order dynamic surface is established to avoid the repeated derivation for the virtual controllers, and the BLF technique is utilized to avert the violation of state constraints. From this point onwards, for the sake of convenience in writing and structure clear, if there are no confusion arise, some arguments of the functions will be omitted.

Preliminaries
To achieve the predetermined control objective, we first do the following coordinate transformations: where S j (j = 1, · · · , n + 1) denote the actual/virtual tracking errors; α j (j = 2, · · · , n + 1) represent the virtual control laws; α jf (j = 2, · · · , n + 1) are intermediate variables which are produced by the following first-order filters, where τ j are time constants; Z j denote the filter surface errors.
Since θ j and d j (j = 1, · · · , n) are parametric uncertainties and unknown external disturbance bounds, respectively, we define variablesθ j andd j to estimate them. The estimation errors are denoted as follows For dealing with the full state constraints, the following tan-type BLFs are introduced [28, ?]: where k b j are positive constants. Computing the time derivative of V * j , it follows thaṫ where Remark 5. Obviously, the tan-type BLF (15) restrains S j within the region Ω s j : It is through the bounds of S j that we ensure the constraint satisfaction of x j . The specific proving process will be given in the proof of the theorem later. What's more, compared with the most commonly used log-type BLF for symmetric state constraints [25,27,37], the tan-type BLF has significant advantages. When there are no constraints on system states, that is k c j → ∞, hence k b j → ∞, j = 1, · · · , n. Using L'Hospital rule we have If some states in the system are constrained and some are free, we can design a unified tan-type BLF without the need for separate case discussion as in [37]. Hence, the tan-type BLF is advantageous to integrate the constraint analysis into a unified algorithm.

Robust adaptive controller design
In this subsection, we present the detailed adaptive backstepping design procedure step by step for system (6), which is essential to achieve our control goal.
Step 1. Starting with the x 1 -subsystem of system (6), differentiating S 1 with respect to time and invoking equations (6), (8) and (9), it yieldṡ To deal with the system state constraint, parametric uncertainty, and external disturbance simultaneously, the following Lyapunov candidate function is constructed where λ θ 1 and λ d 1 are positive constants. Base on the equations (11), (16) and (18), the time derivative of V * 1 can be rewritten asV * Utilizing Assumption 1 and Lemma 1, it follows that where ε is a positive parameter, k = 0.2785. In the meanwhile, involving the Young's inequality, it follows that where l 1 is a positive parameter,ḡ 1 is the upper bound of |g 1 |. Then, computing the time derivative of V 1 based on (13), (14) and (19)-(23), it yieldṡ By appealing to the Nussbaum-gain technique to deal with the unknown control gains, the virtual control law and adaptive laws can be designed as: where K 1 , σ d 1 and σ θ 1 are positive constants. Remark 6. Note that, by invoking L'Hospital rule, the first term of η 1 will tend to zero when S 1 → 0: Hence, the first term of η 1 will not be singular even if S 1 close to zero. In simulation, we should take it to be zero when |S 1 | < for some small , because " 0 0 " cannot be computed in a digital computer. Since α 1 is continuous, differentiable and bounded, so the time derivative of α 1 is bounded. By appealing to the equations (11) and (12), using the Young's inequality, the term of Z 2Ż2 in (24) can be dealt with as follows where m 1 is the supremum of |α 1 |. Substituting (25)- (30) into (24), after some deformation, we havė Using the Young's inequality, the terms − (31) can be handled as: Substituting (32) and (33) into (31), it follows thaṫ where (19), one haṡ Multiplying both sides of (35) by e β 1 t leads to Integrating the above inequality over [0, t), it yields By invoking the Lemma 2, it can be concluded that V 1 (t), ξ 1 (t) and t 0 g 1 N (ξ 1 )ξ 1 dτ are all bounded on [0, t) as long as the last term of (37) is bounded. As a result, V * 1 will be bounded and then |S 1 | < k b 1 holds. Therefore, the problem boils down to the boundedness of S 2 , which will be obtained in Step 2.
Step j(j = 2, · · · , n − 1). From Step 2 onwards, a similar design procedure can be involved recursively at each step. Considering the x j -subsystem of system (6), take the time derivative of S j based on equations (6), (9), (11) and (12), it yieldsṠ Analogous to the design approach in Step 1, the following Lyapunov function candidate is employed: where λ θ j and λ d j are positive constants. According to (16) and (38), the time derivative of V * j can be written asV * By invoking Assumption 1 and Lemma 1, it follows that Utilizing the Young's inequality, it leads to where l j is a positive constant. With the help of (39)-(43), differentiating V j with respect to time yieldṡ By employing the Nussbaum-gain technique to handle the unknown control gain, the virtual control law and adaptive laws are designed as: where K j , σ d j and σ θ j are positive parameters. Analogous to the discussion in Step 1, the boundedness oḟ α j also can be verified. According to the Young's inequality, with the help of the equations (11) and (12), Z j+1Żj+1 in (44) can be dealt with as follows where m j is a positive constant. Substituting (45)-(50) into (44), adding and subtractingξ j on the right side of (44), after some deformation, one haṡ Using the Young's inequality, the terms − σ d j λ d jd jdj and − σ θ j λ θ jθ jθj in (51) can be handled as: Substituting (52) and (53) into (51), it follows thaṫ where Multiplying both sides of inequality (55) by e β j t , and then integrating both sides over [0, t), it yields Invoking the Lemma 2, it can be seen from the above result, V j (t), ξ j (t) and t 0 g j N (ξ j )ξ j dτ are all bounded on [0, t) as long as the last term of (56) is bounded. As a result, V * j is bounded and then |S j | < k b j holds. Therefore, the problem boils down to the boundedness of S j+1 , which will be obtained in Step j + 1.
For the external disturbance termD n , invoking Assumption 1-2 and equation (5), one has whered n is a positive constant. Choose the Lyapunov function candidate V n as: where λ θn and λ dn are positive constants,d n =d n −d n withd n be an estimate ofd n . According to (16) and (57), taking the time derivative of V * n , it yieldṡ V * n = ϑ sn θ n ψ n + g n (S n+1 + Z n+1 + α n ) +D n + Z n τ n .
Utilizing the Young's inequality, we can obtain: where l n is a positive parameter. With the help of (59)-(63), taking the time derivative of V n , it follows thaṫ V n ≤ ϑ sn · θ n ψ n + g n α n + Z n τ n + l n ϑ 2 sn +ḡ 2 n 2l n Z 2 n+1 +d n k ε + ϑ sn tanh In the same way as before, the following Nussbaum type controller and adaptive laws are designed where K n , σ dn and σ θn are positive parameters. In the same way as the previous steps, the boundedness oḟ α n can be verified. By appealing to the equations (11) and (12), the term of Z n+1Żn+1 in (64) can be dealt with as follows by the Young's inequality, where m n is a positive constant. Substituting (65)-(70) into (64), adding and subtracting the termξ n on the right side of (64), we havė Using the Young's inequality, the terms − σ dn λ dnd ndn and − σ θn λ θnθ nθn in (71) can be handled as: Substituting (72) and (73) into (71), it follows thaṫ where c n = m 2 n +d n k ε + It can be seen from the above result, invoking the Lemma 2, as long as the last term of (75) is bounded, then V n (t), ξ n (t) and t 0 g n N (ξ n )ξ n dτ are all bounded on [0, t). As a result, V * n is bounded and then |S n | < k bn holds. Therefore, the problem boils down to the boundedness of S n+1 , which will be obtained in Step n + 1.
Step n + 1. Considering the auxiliary (n + 1)-subsystem of system (6). Since the virtual error S n+1 only needs to be bounded, rather than restricted to a predefined region, the BLF is no longer required in this step. Define the following quadratic Lyapunov function candidate: From equations (3), (10) and (12), computing the time derivative of V n+1 , it yieldṡ where Since the control coefficient is still unknown, the auxiliary control law can be designed as where K n+1 is a positive parameter. Substituting (78)-(80) into (77), adding and subtracting the termξ n+1 on the right side of (77), one haṡ where β n+1 is a positive constant and supposing β n+1 ≤ 2K n+1 . Multiplying both sides of (81) by e β n+1 t and integrating it over [0, t), it follows that By invoking the Lemma 2, because inequality (82) holds, then V n+1 (t), ξ n+1 (t) and t 0 G(t)N (ξ n+1 )ξ n dτ are all bounded on [0, t). As a result, S n+1 is bounded, and then it is concluded that V n (t), ξ n (t) and t 0 g n N (ξ n )ξ n dτ are all bounded on [0, t) from Step n. Back to the previous steps one by one, we can summarize V i (t), ξ i (t) and t 0 g i N (ξ i )ξ i dτ are all bounded on [0, t f ), and |S i | < k b i (i = 1, · · · , n) hold. Furthermore, based on Remark 2, t can be extended to infinity.

Theoretical results and proof
Based on the adaptive controller design of the system (1) given in subsection B, the following theorem is obtained, which summarizes the main results of this paper.
(i). All the signals of the closed-loop system are bounded; (ii). All the states are always within the predefined constrained regions; (iii). The actual/virtual tracking errors S j (t)(j = 1, · · · , n) converge to a small neighbourhood of zero.
Proof: (i). From the adaptive controller design process given in subsection B , we know that V j (j = 1, · · · , n) are bounded. Note that V j = are all bounded. The boundedness of θ j ,θ j , d j , andd j ensure the boundedness ofθ j andd j , and further make η j bounded. Since α j = N (ξ j )η j , the boundedness of ξ j and η j guarantee the boundedness of α j . As a result, α jf is bounded. In the same way, µ is bounded. Then, x j (j = 1, · · · , n) are bounded. Conclusively, all the signals of the closed-loop system are bounded.
(ii). The boundedness of y d , α jf and S j have been stressed in (i). In view of holds. Since α jf is bounded, we assume there exist a positive constantᾱ j0 which make |α jf | ≤ᾱ j0 . In view of S j = x j − α jf , |S j | ≤ k b j and |α jf | ≤ᾱ j0 , it follows that |x j | ≤ |S j | + |α jf | < k b j +ᾱ j0 . So, only given k b j = k c j −ᾱ j0 , then |x j | < k c j (j = 2, · · · , n) hold. Thus we conclude that all the states can remain within the predefined constrained regions by proper selection of k b j .

Simulation
In this section, in order to demonstrate the effectiveness of the proposed control scheme, the following states and input constrained nonlinear system is considered: where ψ 1 (x 1 ) = x 2 1 , ψ 2 (x 2 ) = 0.2x 1 x 2 + x 1 , θ 1 and θ 2 are parametric uncertainties, D 1 and D 2 are external disturbances.
The control objective is to design a saturation controller with constraintū = 0.5, which make the output y(t) to track the given reference signal y d (t) = 0.5 * cos(t/2) + 0.2 * sin(t) as closely as possible, and the states within the constrained regions : Ω x = {|x 1 | < k c 1 = 0.8, |x 2 | < k c 2 = 2} for all the system running time.
Case one: g 1 = −1 − sin(x 1 ), g 2 = −1.5 − sin(x 1 x 2 ); D 1 = D 2 = 0. This is the normal case without external disturbance, the direction of the gain function g 1 and g 2 are the same. Using the matlab routine we get the constraints of the virtual errors are k b 1 = k c 1 − sup{|y d |} = 0.4 and k b 2 = 1.73. In light of the design procedure proposed in section 2.2, the state feedback controller can be constructed and the simulation results are shown in Fig. 1-7. Fig. 1 shows the trajectory of the system output following the reference signal. On the one hand, it illustrates the accuracy of tracking performance, on the other hand, it also shows that the system state x 1 is always within the constrained region. The trajectory of the another state x 2 is shown in Fig. 2, it also within its predefined constrained region for all the time. Fig. 3 gives the trajectories of the uncertain parameters. Fig. 4 illustrates the trajectories of the Nussbaum functions and their arguments. The system input signal is given in Fig. 5 and Fig. 6, of which Fig. 6 is the input saturation signal. It can be seen that the input of the system is always between -0.5 and 0.5, so as to meet the requirements of saturation. The trajectories of the tracking errors are shown in Fig. 7. It indicates that the output tracking error converges to a very small neighbourhood of zero. Then, the state feedback controller can be constructed as where the even Nussbaum-type function is selected as N (ξ i ) = ξ 2 i cos(ξ).   Case two: g 1 = −1 − sin(x 1 ), g 2 = 1.5 + sin(x 1 x 2 ); D 1 = D 2 = 0. In this case we are talking about the case where there is no external disturbance, but the signs of the control gains g 1 and g 2 are opposite. Based on the same design parameters and initial values with case one,    Remark 8. In this case, the signs of the control gain functions g 1 and g 2 are opposite. The simulation results demonstrate the crucial role of the Nussbaum function in the controller design. Nussbaum function can effectively adjust the unknown control direction. No matter the control direction is positive or negative, the controller we designed has a very good performance. By comparing the tracking errors of the two cases from Fig. 7 and Fig. 14, it can be seen that Fig. 14 only has a significant error within the first 1 second of operation due to the change of control direction. Then the controller quickly adjusted to achieve the similar tracking performance as the first case. From Fig. 12 and 13, it can be seen that the oscillation of the control input is     more obvious because of the change of control direction. Nevertheless, all of our preset control requirements are met.
In this case we are talking about the case where there exist unknown external disturbance, and the signs of the control gains g 1 and g 2 are opposite. Based on the same design parameters and initial values with above two cases, the simulation results are shown in Fig. 15-22.
Remark 9. In this case, different control direction and unknown external disturbance are considered. The simulation results illustrate the effectiveness of the proposed control scheme even in the presence of external interference. It can be seen from Fig. 15 and Fig. 22 that our tracking performance is better than the previous two cases in the case of external disturbance. That's because in the first two cases, where there is no external disturbance, we have do nothing with the redundant terms. In the third case, because of the disturbance, we combine the unknown bounds of the disturbance with other redundancies and design the adaptive law for them. It is not only eliminates the influence of the external disturbance, but also addresses other redundancy issues. That is the main reason for the better control performance.

Conclusions
In this paper, a novel adaptive tracking control scheme is proposed for a class of nonlinear systems with full state constraints, input saturation, unknown control direction, and external disturbances. Using a hyperbolic      tangent function to approximate the saturated input function, which avoids the sharp corner caused by the input function. The unknown control directions of system are solved by the Nussbaum-gain technique. And the state constraints are maintained by the involving of BLF. Meanwhile, dynamic surface control (DSC) is used in the controller design, which solved the problem of computation explosion caused by the repeated derivation of virtual control law. The proposed control strategy not only solves the problem of unknown control direction, the states are always within the constrained regions, but also makes the output tracking error of the system converges to a small neighborhood of zero closely. In future work, the control methods proposed in this paper can be further extended, such as systems with asymmetric time-varying state constraints, nonlinear systems with hysteresis, event-triggered constrained nonlinear systems, etc.