Stabilization analysis of a class of nonlinear time delay systems with time-varying full-state constraints

In this paper, a novel tracking control strategy is proposed to address the problem of stabilization of a class of nonlinear time delay systems with time-varying full-state constraints. The effect of the nonlinear system resulting from the time delays is canceled out with the utilization of the novel iterative procedures optimized by dynamic surface control (DSC) and the appropriate time-varying asymmetric barrier Lyapunov functions (ABLFs) are employed to stem the violation of time-varying states constraints. Finally, it is proved that the proposed control method guarantees the uniformly ultimate boundedness of all the signals in the closed-loop system, meanwhile, the tracking errors converge to a small interval. The effectiveness of the presented control strategy is confirmed by a simulation example provided in this paper.


Introduction
Constraints are significant factors leading to instability and undesirable performance existing ineluctably in a wide variety of practical engineering systems. Among the numerous effective research results on constraint-handling approaches promoted by practical requirements, such as reference governors [1,2] model predictive control [3,4] and the set invariance notions [5], the barrier Lyapunov function (BLF) is the leading method to handle the constraint problems. An adaptive controller based on asymmetrical Barrier Lyapunov Functions was first proposed in [6] to handle parametric uncertainties meanwhile deter constraints from being breached. Subsequently, some different forms of novel adaptive controllers based on Integral Barrier Lyapunov Functionals (IBLF) and tangent BLFs (TBLFs) were proposed for nonlinear singleinput single-output (SISO) output-constrained systems in [7][8][9] respectively. A BLF-based backstepping was presented in [10] for strict-feedback systems with partial state constraints to deal with the control design problem. The full state constraint control problem of nonlinear systems in pure-feedback was addressed in [11,12]. Relying on the combination of neural networks (NNs) and BLF, adaptive controllers were designed in [13][14][15] for full state constrained systems. Meanwhile, time delay, which has an impact that can not be neglected on the performance of control systems and even leads to the deterioration of system stability, is also frequently encountered in many practical engineering systems, such as spacecraft attitude control systems [16], active suspension system, vehicle control system [17,18], microwave oscillators.
The issue of time-delay systems has received considerable critical attention. With regard to the robust stability proof for systems affected by time delay, two theorems are employed by a large proportion of effective methods, one is the Lyapunov-Krasovskii theorem [19][20][21][22][23], the other is the Lyapunov-Razumikhin theorem [24][25][26][27], and a great deal of research based on Lyapunov-Krasovskii functionals (LKFs) and Lyapunov-Razumikhin functions (LRFs) has been done on how to deal with time delay problems. In [28], LRF was combined with an adaptive stability control scheme designed by backstepping to process a class of nonlinear time-delay systems with a state feedback control problem. In [29], aiming at the input-to-state stability (ISS) of nonlinear time-delay systems, a class of LRFs with a more relaxed requirement on derivative was constructed for the first time. Based on this, the stability of a class of event-triggered stabilization of switched nonlinear time-varying systems was studied. An iterative robust controller for a class of SISO nonlinear time-delay systems had been designed with the novel use of Lyapunov-Krasovskii functionals see [30]. In [31,32], unknown functions containing time-delayed states could be resolved into a series of continuous functions with the utilization of the separation technique, which canceled out the constraints of assumption about the functions containing delayed states. The desired control performance of nonlinear time-delay systems was achieved in [31,33] due to the combination of LKFs and adaptive NNs backstepping. A class of strict-feedback nonlinear systems with time delays were presented in [32,34,35] and adaptive fuzzy control based on the backstepping technique as well as LKFs had been developed to handle the problem of output tracking. In practice, time delay and state constraints usually exist in practical engineering systems simultaneously, how to address the control problem of nonlinear systems with both the state constraints and time delay remains urgent.
It is noteworthy that the problem of symmetric constant state constraints in time delay systems can be solved by using the tangent barrier Lyapunov function (TBLF) in [9], or NNs based on BLF in [31,33,36]. However, symmetric and non-time-varying state constraints considered in [30,31,33,36] were just some special cases of our scheme and may not meet the requirements when applied to the actual engineering systems. And the method proposed in [30,31,32,33], which used a traditional backstepping algorithm, suffered from a large amount of calculation caused by repetitive differentiations. To overcome these problems, the dynamic surface control (DSC) technique, which obtained the information of the virtual control by introducing a firstorder low-pass filter at each step of the design procedure was presented in [37,38] and adopted in [39][40][41][42][43]. Therefore, through the study of previous papers, it can be concluded that there are few works combining DSC technology to solve the problem of time delay and timevarying full-state constraint at the same time, which motivates our research.
In this paper, the time-varying asymmetric Barrier Lyapunov Functionals (ABLF)-based control is employed to stabilize a class of nonlinear strictfeedback systems with the time-delays and the timevarying full state constraints. The main contributions of this paper are summarized as follows: (i) Based on the separation technique, a novel unified framework is proposed, in the meantime, time delay and the full state time-varying constraint problems are taken into account more comprehensively in this current work, which means that the limitation on initial conditions can be relaxed and is able to satisfy the constraint requirements of the state variables better in practical engineering systems. (ii) Compared with approaches using traditional iterative backstepping, repetitive differentiation of stabilizing functions, which will lead to tedious and complicated calculation, especially in higher order systems, can be averted by choosing the DSC strategy. (iii) With the utilization of the timevarying asymmetric BLFs, all the states are always within the prescribed time-varying scopes.
The rest of this paper is organized as follows: Section 2 formulates the problem, and some necessary preliminary knowledge is provided in this section as well. The design process of the controller and stability analysis are given in Section 3. The effectiveness of the proposed approach is illustrated by the example provided in Section 4. The conclusions are drawn in Section 5.

Problem formulation
Consider the following strict-feedback nonlinear timedelay systems with output constraints . . n represents system states, y ∈ R, u ∈ R are system output and control input respectively; g i (x T i ) and f i (x i ) are smooth functions; h i (·) are unknown smooth functions affected by time delays; τ i ,i = 1, 2, · · · n denote time-delay constants. The control objective of this paper is to design an adaptive controller u for system (1) to ensure that the system output y tracks the desired trajectory y d , while all signals in the close-loop system remain bounded and all the corresponding constraints k ci < x i <k ci , ∀t ≥ 0 are not violated.
The necessary Assumptions for establishing constraint satisfaction and performance bounds are as follows.

Assumption 2.1 ([31]):
For the uncertain non-linear time-delay functions, the following inequality holds where q ij (·) are known continuous functions,

Preliminaries
is continuous, and monotonic, i.e. for any |x| > c, where c is a positive constant, k(x) ≥ k(c).

Remark 1:
Observing the system studied in this paper, each state variable has a separate derivative expression, so we design a BLF function for each state variable based on the backstepping technique, and the function will involve the constraint boundaries of that state variable, the LKFs, and the design terms of the dynamic surfaces. The controller is then designed based on the derivatives of each BLF so that the final overall BLF derivative satisfies our requirements.

Controller design and stability analysis
In this section, a combination of backstepping DSC design and ABLF will be proposed to develop a controller for strict-feedback nonlinear time-delay systems. The specific design process is shown below.
Step 1: Suppose the tracking error as s 1 (t) = x 1 − y d , its time derivative according to the first equation of (1) is given bẏ then the virtual error of the next step is defined as s 2 (t) = x 2 − z 2 , simultaneously, letting a designed stabilizing function α 1 passing through a first-order filter that contains a time constant ϑ 2 in the following way and the corresponding time domain expression wherefore we could get the first-order filter error χ 2 = z 2 − α 1 , the derivative of z 2 can be defined aṡ then, referring to [6] we choose the time-varying asymmetric barrier function candidate as for ease of notation, we abbreviate q(•) by q i throughout this paper, unless otherwise stated. And the timevarying barriers in the above formula are given by and due to assumptions 2.3 and 2.4, we know that positive constants k b1 ,k b1 , k a1 ,k a1 exist, which leads to Remark 2: log(·) denotes the natural logarithm of ·, define a set s : It is clear that V 1 is positive definite and continuously differentiable, then the time derivative of V 1 is given bẏ similarly, for ease of notation, the following notation definitions will be used ⎧ ⎨ where the time-varying gain is given bȳ and denoteK 1 (t) as follows in (14), β 1 is a positive constant, which ensures that Design stabilizing function α 1 as where k 1 and ε 1 are positive design parameters, substituting (12), (13), (15), and (17) into (11), we could obtain thaṫ where μ 1 λ 1 = 1 and the function D 1 is designed as with respect to χ 2χ2 , according to first-order filter error χ 2 = z 2 − α 1 and (3), we have the time derivative ofχ 2 where B 2 (x 1 , y d · · · ) = −α 1 , due to assumptions 2.3 and 2.4, we note that |B 2 | ≤ M 2 , which means that B 2 is bounded and continuous with a maximum absolute value. Thus, we can rewrite χ 2χ2 with following Young's inequality we have combining (22) and the following Young's inequality we obtaiṅ In order to guarantee the closed-loop stability, the design of constant gain k 1 and time-delay con- > 0, and the term μ 1 g 1 s 1 s 2 will be canceled in the subsequent step.
Step i, 2 ≤ i ≤ n − 1 Define the tracking error as s i+1 (t) = x i+1 − z i+1 , we introduce a filtering virtual control z i+1 and let designed stabilizing function α i passing through a firstorder filter that contains a time constant τ i+1 , similar to step 1, Where α i is a stabilizing function to be designed. The first-order filter error in this step is choose the time-varying asymmetric barrier function candidate as noting that V i is positive definite and continuously differentiable, then the differentiating of V i yieldṡ for ease of notation, we have following notation definitions where the time-varying gaink i (t) andK i (t) are given by β i is a positive constant, which ensures thatK i (t) > 0. The time-varying barriers k ai (t), k bi (t) are given as As mentioned above, it is easy to know that s i (t) = due to tracking error and first-order filter error, x i+1 (t) can be represented as follows stabilizing function is designed as where k i , ε i are designed positive constant. We Substituting (31), (32), (34) and (37) into (30), it leads tȯ where μ i λ i = 1, and the function D i is designed as with respect to the term χ i+1χi+1 , according to firstorder filter error χ i+1 = z i+1 − α i and (19), we have the time derivative ofχ i+1 (39) we note that B i+1 is bounded and continuous with a maximum absolute value under assumptions 2.3 and 2.4, which can be expressed as |B i+1 | ≤ M i+1 . Thus we can rewrite χ i+1χi+1 with following Young's inequality we have combining (41) and the following Young's inequality we can simplify (36) aṡ In order to guarantee the closed-loop stability, the design of constant gain k i and time-delay constant τ i should make sure k i +K i (t) − g i max (x i ) > 0 and − 1 2o i+1 M 2 i+1 + 1 ϑ i+1 − 1 4 μ i g i max (x i ) > 0, the term μ i g i s i s i+1 will be canceled in the subsequent step.
Step n Define the tracking error as s n (t) = x n − z n , we can get a time derivative of s n (t) In step i, when i = n − 1, we can obtain choose the time-varying asymmetric barrier function candidate as for ease of notation, we have the following notation definitions where the time-varying gaink n (t) andK n (t) are given by ⎧ ⎪ ⎨ ⎪ ⎩k  (1) and (44), (45), we could obtaiṅ introduce the actual controller u(t) as where k n , ε n are designed positive constants and define where μ n λ n = 1 and define using following Young's inequality we could obtaiṅ whereK n (t) + k n > 0. Combining (7), (26), (46) and (24), (43), (58) we could obtain V n andV n respectively according to assumption 2.1, we obtain So far, the controller design process has been completed, and the schematic diagram of the proposed control scheme is illustrated in Figure 1.  Proof: In order to express more conveniently, we present gain parameters in the following way: due to the parameter selection requirements in the previous article, we can see E i > 0, i = 1 · · · , n, e i > 0, i = 1 · · · , n − 1,V n can be rewritten aṡ therefore, (63) further becomeṡ depending on the size of s j , concerning the analysis of stability, there are three cases that need to be considered.

Remark 3:
Based on Lemma 2.2, there are three cases of error variables in the control process, all in the predetermined set, all not in the predetermined set, and partially in the predetermined set. The proof is developed for these three cases.
Case 1: (s j ∈ c1 , ∀j = 1, 2 . . . , n): it is easy to ensure the boundedness of s j due to the positive constant v j and |s j | < 0.5549v j , and we can also have y i d (t) are bounded considering |y i d (t)| ≤ Y i in assumption 2.4. Because of |y d (t)| ≤ Y 0 and s 1 = x 1 − y d we get k c1 = −|k a1 | + Y 0 < x 1 < |k b1 | + Y 0 =k c1 , we can also conclude that α 1 in (20) is bounded, therefore, k c2 = −|k a2 | + |α 1 | < x 2 < |k b2 | + |α 1 | =k c2 . Following the same way, we can get k ci (t) ≤ x i ≤k ci (t) and the actual controller u is bounded, which means that all the closed-loop signals are bounded.
we define σ as follows based on (59), we obtaiṅ multiplying (61) by e σ t on its both sides yields to for s i > 0, q = 1,(70) can be simplified to multiplying both sides of (71) by k 2 bi (t) − s 2 i , which yields to Likewise, when s i ≤ 0, q = 0, we can obtain , thus the conclusion can be drawn that −D s i ≤ s i ≤D s i , where . From the previous derivation, we have k c1 (t) = −k a1 (t) + y d (t) and k c1 (t) = k b1 (t) + y d (t), due to s 1 (t) = x 1 − y d and s 1 (t) ∈ s , which has been proved, it can be learned that k c1 (t) ≤ x 1 ≤k c1 (t). Similarly, it can also be proved that k ci (t) ≤ x i ≤k ci (t) on account of Hence, the conclusion can be drawn that the full state constraints are not breached. It has been proven that the states x i (t) and the error signals s i (t) are all bounded, according to assumption 2.4, y d ,ẏ d are bounded. From (13) we know that α 1 is a function of x 1 ,ẏ d , so α 1 must be bounded, the upper boundᾱ 1 of α 1 exists, so we can infer that the signals α i , i = 2, . . . n − 1 are bounded, furthermore u(t) is bounded. Since z 1 and y d are bounded, y is also bounded.

Case 3: (s j /
∈ c1 , s i ∈ c1 ): Denote s j / ∈ c1 as J, s i ∈ c1 as I,define the Lyapunov function candidate as follow for J: eliminated in the backstepping design in both case 1 and case 2, but in the present case, these two terms have errors at different steps and are coupled terms, which cannot be canceled in the backstepping process when s i and s i−1 or s i and s i−1 do not happen to be in the same set, so a separate analysis is needed for this case, and a similar analysis can be found in the paper [31].
where μ j is positive and bounded, according to (76), (74) can be rewritten as With a similar reasoning in case 2, It is easy to conclude the boundedness of all the signals in the closed-loop system, the tracking errors are able to converge in a compact set and the time-varying full state constraints are never breached for j ∈ J. In regard to j ∈ I The proof of the stability of the closed-loop system is the same as in case 1. So far, we complete the proof of Theorem 3.1.

Simulation example
A simulation example containing terms affected by uncertain time delay is provided in this section in order to verify the effectiveness and feasibility of the proposed method, the proposed approach is used to control the following nonlinear systems with time-varying full state constraints: in (76), x 1 (t) and x 2 (t) are the state variables of systems, which need to be constrained, and denote initiate values x 1 (0) = 0.8 and x 2 (0) = 0, y and u(t) are the output and the input of systems, the terms affected by uncertain time delay are described by the relevant system parameters are chosen as τ 1 = 0.1, τ 2 = 0.2, q 11 = q 12 = 1. y d (t), the desired trajectory tracked by y is 0.5 cos(t). k c1 (t) < x 1 (t) < k c1 (t), k c2 (t) < x 2 (t) <k c2 (t) are the asymmetric timevarying full state constraints, where the following time-varying asymmetric barrier function are chosen according to the design process above mentioned DSC laws are designed as follows according to the DSC method demonstrated in the paper the design parameters are given as k 1 = 12, k 2 = 20, β 1 = β 2 = 0.1, ε 1 = 0.01, ε 2 = 0.01 and ϑ 2 = 0.5. The tracking performance is illustrated in Figure 2, it can be observed that the output trajectories are always within the scope of the asymmetric constraint k c1 (t) <   y(t) <k c1 (t), in the meantime, good tracking performance is achieved. The trajectory of state variables shown in Figure 3, it is obvious that time-varying state constraints are not violated. From Figures 4 and 5, it can be seen that tracking errors are within their constraints. The boundedness trajectory of the actual controller is illustrated in Figure 6. Therefore, according to the  figures of the simulation results above, it can be learned the constraints are not violated and all the signals in the closed-loop system are bounded.

Remark 5:
The common simulation example used in this paper is referenced from the following papers [30,31,33]. Compared with the papers [30,31,33,36] using non-time-varying BLF, this paper takes the timevarying boundary into account in the design process of the control algorithm, and the simulation results also show that the control algorithm can satisfy the tracking control of the state variables, and the state variables do not cross the time-varying boundary throughout the simulation process. The DSC technique used in this paper largely simplifies the design steps of the control algorithm as well as the computational effort, which can be compared with the papers [30,31,32,33] that also uses the backstepping control technique, but this cannot be presented in the final simulation results.

Conclusion
In this paper, an ABLF-based backstepping DSC strategy has been studied for a class of nonlinear systems with time delays and full state constraints. The influence of the time-delays terms was canceled by adopting appropriate assumptions. By employing the appropriate ABLFs, the full state constraints, which are asymmetric and time-varying, were ensured not to be breached. The result that the tracking errors and all the signals of the closed-loop system are asymptotic stable is guaranteed by the proposed approach and choosing appropriate parameters. In addition, compared with approaches using iterative backstepping, repetitive differentiation of stabilizing functions, which will lead to tedious and complicated calculations, especially in higher-order systems, can be averted by choosing the DSC strategy. Finally, the effectiveness of the DSC in conjunction with the ABLF in dealing with time delay problems can be verified by the simulation results and the desired performance can be achieved. Our future research work is to apply this control algorithm to the actual engineering system.